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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
×
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
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Suggested Citation:"Chapter 4: Findings." National Academies of Sciences, Engineering, and Medicine. 2018. A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers. Washington, DC: The National Academies Press. doi: 10.17226/25304.
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16 CHAPTER 4. FINDINGS Introduction This chapter on the findings of the research project first reviews the methods to determine the complex modulus and modulus gradient of asphalt field cores with different aging times. Due to the non-uniform aging of asphalt pavements, the testing of field cores needs to capture the material responses at different locations of the specimens. The analysis of test data also becomes much more complex to obtain a complex modulus gradient for asphalt mixtures. In addition, in order to construct a master curve for asphalt field cores, the time-temperature-aging-depth shift functions are needed. Compared to an ordinary asphalt mixture that only has the time- temperature shift factor, an asphalt field core requires a long-term aging shift function and depth shift function to consider the effects of field aging. Since aging plays an important role, a method to accurately and conveniently predict field aging of asphalt pavement is desirable. A kinetics- based aging modeling technique is proposed by making use of the field core testing data; moreover, the field deflection test data, i.e. FWD data. This aging model targets directly aging of asphalt mixtures and utilizes the aging activation energies to evaluate the aging speed of the material. The proposed aging prediction model takes into account the major factors that affect field aging speed of an asphalt pavement, such as the binder type, aggregate type, air void content, pavement depth, aging temperature, and aging time. The top-down crack initiation is defined as the stage for microcracks initiate then coalesce into a visible macro-crack at the pavement surface. A mechanistic-empirical model is developed to predict the top-down cracking initiation time using the available LTPP data. For the crack propagation phase, finite element models are used to determine the J-integral at the crack tip that drives the crack growth downward from the pavement surface. Different combinations of material properties, pavement structure, and crack depth are the inputs to the finite element models so as to develop a database for ANN modeling. The finite element analysis reveals the features of the J-integral at the crack tip of a top-down crack in the asphalt layer under different tire-pavement contact stresses and thermal stresses. In order to provide the Pavement ME Design software with the capability to calculate the J-integrals for top-down crack growth, ANN models are prepared using the data generated from the finite element modeling. The models can reproduce the results of thousands of runs with the finite element models, representing a wide variety of pavement structures and layer material properties. Finally, a cumulative damage model combines the computations of top-down crack growth under different traffic load spectra levels and thermal stresses to produce a net result of prediction of top-down crack severity as recorded in the LTPP data base. Determination of Complex Modulus Gradient of Field-Aged Asphalt Mixtures The dynamic modulus of asphalt mixtures is a material property and one of the most important inputs in the AASHTO Pavement ME Design (123). It is also used as an indicator for either the level of aging or damage of the asphalt mixtures. In general, the field-aged asphalt

17 mixtures become stiffer after a long-term aging period, which is similar to the laboratory-mixed- laboratory-compacted (LMLC) mixtures under long-term aging in the laboratory. In addition to the long-term aging, there is another unique aging feature for the field cores: non-uniform aging in the pavement depth. It is known that the surface of the asphalt layer suffers from the solar radiation and oxidative aging more than deeper layers, as the oxygen needs time to diffuse through the interconnected air voids into the pavement structure from the surface of the pavement. Thus less carbonyl area is formed in the deeper layers due to the lower temperature and lower volume of oxygen. As a result, the modulus at the surface is higher than the other layers due to this non-uniform aging effect, and finally a modulus gradient is developed. In order to take into account the field aging of asphalt mixtures in the Pavement ME Design, considerable research efforts have been made to either simulate or analyze the field aging in the laboratory, in both binder level and mixture level, or extract binders from the field cores using chemical solvent then determine the complex shear modulus and phase angle of the aged binders. However, there is one main problem with this method: some effects such as air void distribution, aggregate gradation, binder absorption, and aggregate-binder interaction on the modulus of the mixtures are not considered. As a result, it is preferred to obtain the material properties of the field cores directly. Under this circumstance, this project presents a new mechanistic method to determine the complex modulus and modulus gradient of field cores using the direct tension test. The direct tension test is adopted because of the three key advantages: 1) it is simpler to conduct and only takes less than 1 minute for a given temperature; 2) it causes no damage to the specimen if the strain limitation is carefully controlled; and 3) the tensile modulus is determined instead of compressive modulus. It has been shown that the tensile modulus and compressive modulus of asphalt mixtures are different in both the magnitudes and phase angles (124). However, most tests are conducted in the compression mode (123). The tensile modulus is necessary, especially for the characterization of various types of cracking in the asphalt pavements. The test protocol to measure the complex modulus and modulus gradient is presented in Appendix A. A comparison of the material responses between the field core specimen and LMLC specimen is shown in Figure 4.1. (a) Measured Strain at Different Depths of Field Core Specimen

18 (b) Measured Strain of Laboratory Fabricated Specimen Figure 4.1. Measured Vertical Strains at Top, Center, and Bottom of Tested Specimen An inverse approach is proposed to accurately determine the complex modulus and modulus gradient at different temperatures using the elastic theory, pseudo strain concept and elastic-viscoelastic correspondence principle. The modulus gradient of a field core specimen at a specific loading frequency and temperature is modeled by Equations 4.1 and 4.2: 0( ) ( ) n d d d zE z E E E d         (4.1) 0 d Ek E  (4.2) where E (z) is the dynamic modulus (MPa) in pavement depth z at a specific loading frequency and temperature; Ed and E0 are the dynamic moduli (MPa) at the top and bottom at the same loading condition, respectively; d is the thickness of the field core specimen (cm); n is the model parameter, which presents the shape of the stiffness gradient; and k is the ratio of the modulus at the top to the modulus at the baseline 1.5 inches (38 mm) below the surface. The reason why the pseudo strain needs to be determined and used other than the measured strain to solve for the modulus gradient is that the parameters n and k are viscoelastic properties of the asphalt mixture. In this study, due to the complexities of stress and strain in the field core specimen, an inverse analysis with an iteration process is used to determine the pseudo strain and the gradient parameters. More specifically, it contains the following steps: I. In the first iteration: 1) Use the measured tensile strain of an undamaged field core specimen as the seed value for the pseudo strain. In other words, temporarily, there is an elastic relationship between the measured stress and the measured strain; 2) Utilize the elastic theory along with the measured loads and strains to determine the modulus gradient parameters. The values of n and k are determined and checked for their dependence on loading time and frequency;

19 3) Convert the functions of the measured loads, strains and modulus gradient parameters using the Laplace transform to calculate the corresponding viscoelastic property: complex modulus; 4) Calculate the relaxation modulus and then the reference modulus using the calculated complex modulus; and 5) Calculate the pseudo strain using the reference modulus. II. In the second iteration: replace the measured strain with the calculated pseudo strain as the seed value after the first iteration and repeat steps 2 to 5. III. In the following iterations (normally 3 to 5): 1) Replace the pseudo strain in the previous iteration with the newest one and repeat steps 2 to 5; and 2) Stop the iteration when the pseudo strain is stable. Then the modulus gradient parameters converge, the complex modulus and pseudo strain will not change. The major steps are elaborated in Appendix B. Once the relationships of the pseudo strains and time are determined, the measured strains used in the first iteration are replaced by the pseudo strains to recalculate the values of n and k. Then the updated values of n and k are used to obtain the new dynamic modulus master curve and relaxation modulus again. This procedure is repeated until the convergence requirement of the values of n and k are met. In general, the values of n and k become stable within 5 iterations. For instance, in Figure 4.2, the change of the pseudo strain at 30°C is minimal after 3 iterations. Once the convergence is reached, the complex modulus and the modulus gradient parameters are the actual material properties. The three complex moduli are determined with the updated n and k. The modulus gradient is then extracted from the dynamic modulus curves at the three depths and three temperatures for 8 and 22 months aged field core specimens when the loading frequency is selected as 0.1 Hz, which is shown in Figure 4.3. Figure 4.2. Measured Strain at the Bottom of a Field Core Specimen and Associated Pseudo Strains at Different Iterations

20 Figure 4.3. Modulus Gradients of 8 and 22 Months Aged Field Specimens at Three Temperatures and 0.1 Hz Time-Temperature-Aging-Depth Shift Functions for Dynamic Modulus Master Curves A single dynamic modulus master curve cannot be constructed including the effects of temperature and aging for both of the LMLC and field-aged asphalt mixtures. This is because the laboratory mixture does not have a modulus gradient as does the field core. It is preferred to develop a new method to evaluate and predict the dynamic modulus master curves of aged asphalt mixtures, which takes into account the long-term aging and non-uniform aging. The desired result is, a single dynamic modulus master curve with the effects of temperature, loading time, long-term aging, and non-uniform aging needs to be constructed by applying time- temperature and time-aging shift functions. As mentioned above, a mechanistic-based analysis method is developed to determine the complex modulus gradient of asphalt field cores using the direct tension test. The same test protocol is used to obtain the dynamic modulus at different field aging times and pavement depths of field-aged mixtures. The construction of the field dynamic modulus master curve follows the same process as used in developing the dynamic modulus master curve for LMLC sample with some exceptions as will be discussed in a subsequent section. Both master curves have the following shift functions: (a) Horizontal time-temperature shift (b) Glassy modulus vertical aging shift (c) Rheological index aging shift (d) Horizontal aging time shift The field master curve has one additional shift function which provides for the change of master curve with depth below the surface. The field master curve is developed for the baseline depth of 1.5 inches (38 mm) below the surface. Below this depth, aging of the asphalt mixture is uniform.

21 The effect of aging shifts (a) and (b) is to raise and rotate the master curve, as will be illustrated in Figures 4.5 through 4.6. By developing aging master curves for lab samples in which aging is accelerated with elevated temperatures separately from field cores, it is possible to derive an acceleration factor, A, which gives the lab-to-field aging ratio. The asphalt field cores include one type of hot mix asphalt (HMA) and one type of warm mix asphalt (WMA) treated by a foaming process. They are fabricated with a PG 70-22 asphalt binder and Texas limestone aggregates. The field cores are taken from one WMA section and one HMA section near Austin, Texas. The thicknesses of the field cores range from 38 to 51 mm. A total of 16 field cores are collected at the center of two lanes of the HMA section and the WMA section at 1, 8, 14 and 22 months after construction. It is reasonable to assume that the collected cores are not damaged by traffic within the aging periods while they are in the field. Figure 4.4a shows the examples of the dynamic modulus at different depths of the field core specimens. At a given frequency, the dynamic modulus at the top of the specimen is largest and the one at the baseline is smallest. In addition, the dynamic modulus at the center is closer to the one at the baseline. This indicates that aging in the top half of the field cores is much more severe than in the bottom half. It also shows that as the frequency increases, the dynamic modulus becomes larger, which reflects the viscoelasticity of asphalt mixtures. Figure 4.4b shows the examples of the dynamic modulus of field specimens at different aging times and temperatures. (a) Different Depths (b) Different Aging Times Figure 4.4. Dynamic Modulus of Field Cores at Different Conditions Construction of Aging Dynamic Modulus Master Curve for LMLC Mixtures The modified Christensen-Anderson-Marasteanu (CAM) model (126) is used to construct the dynamic modulus master curve of the asphalt mixtures, as shown in Equation 4.3. log 2 log2 ( ) 1 E E g e e R R cE T E E E E a                  (4.3)

22 where gE and eE are the glassy modulus (i.e., modulus at infinite frequency) and equilibrium modulus (i.e., modulus at zero frequency), respectively; cE is crossover frequency; Ta is time- temperature shift factor; ER is rheological index of the asphalt mixtures. The time-temperature shift factor for the CAM model uses the Williams-Landel-Ferry (WLF) equation:  1 2 ( )10 R R C T T C T T Ta     (4.4) where C1 and C2 are the fitting parameters of the WLF equation; RT is the reference temperature in Kelvin (293°K); T is the test temperature in Kelvin. The CAM model is adopted since its parameters have specific physical significance. In the CAM model, the crossover frequency cE is the frequency at which the magnitude of the real part of the complex modulus equals the imaginary part of the mixtures, Ee and Eg are the low and high asymptotes of the master curve, respectively; and rheological index RE is the shape factor of the master curve. The parameters of Eg and cE are employed as two aging parameters for the dynamic modulus master curves at different aging periods. Using the method described above, the dynamic modulus master curves for the LMLC mixtures are determined. The parameters gE , ER and cE in the CAM model are calculated and shown in Table 4.1 for the tested LMLC mixtures. The master curves for the LMLC mixtures with air void contents are shown in Figure 4.5. Table 4.1. Calculated CAM Model Parameters Mixture Type Glassy Modulus (MPa) Crossover Frequency (rad/s) Rheological Index 4% 0 Month 31801 0.109718 1.16520 4% 3 Months 39402 0.010827 1.53283 4% 6 Months 41403 0.000935 1.63738 7% 0 Month 31827 0.084390 1.65467 7% 3 Months 38248 0.002436 1.87641 7% 6 Months 39751 0.000735 1.93958

23 Figure 4.5. Dynamic Modulus LMLC Master Curves Constructed by CAM Model Determination of Aging Vertical Shift and Rotation The aging shift function is defined at the aging level for an asphalt mixture at a specific aging time compared to the same asphalt mixture at an unaged or reference aging level. To efficiently characterize the aging effect on dynamic modulus master curves at different aging times, the aging shift function should be capable of shifting the dynamic modulus master curves at longer aging times to the reference aging time, and eventually form a single dynamic modulus master curve, which includes the aging effect. As can be seen from Table 4.1, the glassy modulus for dynamic modulus master curves at different aging periods is not identical, showing a larger value of the glassy modulus asymptote for the asphalt mixtures at longer aging periods. It indicates that the aging rheological property of asphalt mixtures is different from that of the asphalt binders, since the glassy modulus for the binders is normally assumed to be identical at different aging times. As a result, a vertical shift is needed to normalize the higher asymptotes at the longer aging times to that at the reference aging time (i.e., unaged state). The following equation is used to determine the vertical shift factor. 0 0 En gx x g E a E a             (4.5) where gxE , 0gE are glassy modulus (MPa) at x months and at the reference aging time, respectively; xa , 0a are the aging time of interest and reference aging time (month), respectively; En is the model parameter. As indicated earlier, the rheological index RE is the shape factor of the master curve. More specifically, the master curve at a longer aging period normally has a flatter shape, which means a more gradual transition. The rheological index RE can also be used to determine the relaxation spectrum. In order to form a single master curve, it is important to rotate the master curves at other aging times to be identical to that at the reference aging time before applying the 1 10 100 1000 10000 100000 0.001 0.1 10 1000 100000 10000000 Dy na m ic  M od ul us  (M Pa ) Reduced Angular Frequency (rad/s) 0 Month 3 Months 6 Months

24 horizontal aging shift factor. The power law equation is used to determine the change of the rheological index for the rotation: 0 0 Rn Ex x E R a R a             (4.6) where ExR and 0ER are the rheological index at x months and at the reference aging time, respectively; xa , 0a are the age of interest, and reference aging time, respectively. Rn is the model parameter. Examples are given using the constructed dynamic modulus master curves for the LMLC mixtures with 7% air void content. The ratios for both calculated and fitted glassy modulus and rheological index are shown in Figure 4.6a. Figure 4.6b shows the modified 6 months aged master curve after applying the vertical shift and rotation to the original 6 months aged modulus curve. As can be seen, the modified master curve and original master curve have different shapes and magnitudes. The vertical shifted and rotated master curves for the 3 and 6 months aged mixtures and the original master curve for the 0 month aged mixtures are shown in Figure 4.6c. (a) Fitted Aging Parameters 1 1.1 1.2 1.3 1.4 1.5 0 2 4 6 8 Ra tio Aging Time (Month) Glassy Modulus Rheological Index Glassy Modulus (Fitted) Rheological Index (Fitted) R = 0.98 R = 0.98

25 (B) Modified and Original 6 Months Master Curves (c) Original and Modified Master Curves Figure 4.6. Vertical Shift and Rotation for LMLC Master Curves Determination of Horizontal Aging Shift Factor The next step is to develop a horizontal aging shift factor to construct a single aging dynamic modulus master curve. With the horizontal aging shift factor, it is possible to shift the modified master curves at longer aging periods horizontally to that at the reference aging time. 100 1000 10000 100000 0.001 0.1 10 1000 100000 10000000 Dy na m ic  M od ul us  (M Pa ) Reduced Angular Frequency (rad/s) Original Vertical Shift and Rotation 10 100 1000 10000 100000 0.001 0.1 10 1000 100000 10000000 Dy na m ic  M od ul us  (M Pa ) Reduced Angular Frequency (rad/s) 0 Month (Original) 3 Months (Modified) 6 Months (Modified)

26 As shown before, the aging is mainly affected by two factors: aging time and aging temperature. Therefore, the horizontal aging shift factor has the following form which includes an Arrhenius relation: 0 0 1 1( ) ( , ) aEb a a A R T T Aa a T e             (4.7) where ( , )Aa a T is the horizontal aging shift factor at the aging time a and aging temperature T in Kelvin; 0a is the initial aging time or unaged time ( 0 0a  ), A is the acceleration factor to account for the laboratory accelerated aging process; aE is the activation energy, R is the universal gas constant ( 3 1 18.314 10 kJ mol K   ) , T is the average monthly temperature in a field location or the laboratory aging temperature for an aging period a , and 0T is the initial average monthly temperature in that field location or the reference aging temperature in the laboratory. In the construction of the dynamic modulus aging master curve, the aging reduced frequency 'f is determined as ' ( , ) ( , )a T Af a a T f (4.8) where ' ( , )a Tf is the reduced frequency and f is the previous frequency after the time- temperature shift. Once applying this horizontal aging shift factor, a final single master curve is formed by shifting simultaneously the other two modified master curves of the 3 and 6 months aged mixtures horizontally to the right to superimpose upon the master curve of the 0 month aged mixtures. The final LMLC aging master curve is formed and shown in Figure 4.7. Figure 4.7. Final Aging LMLC Dynamic Modulus Master Curve 100 1000 10000 100000 0.001 0.1 10 1000 100000 10000000 1E+09 Dy na m ic  M od ul us  (M Pa ) Reduced Angular Frequency (rad/s) 0 Month (Original) 3 Months (Shifted) 6 Months (Shifted)

27 Construction of Dynamic Modulus Master Curve of Field Cores The process for constructing a final LMLC dynamic modulus master curve illustrated above is also used for constructing a final dynamic modulus master curve for field-aged mixtures but with several modifications, as follows. (a) The test that is made on the aged cores is a monotonically loaded tension test which is conducted at different temperatures. The details of this test are in Appendix A. (b) There is an additional shift that accounts for the differential aging with depth below the surface that results in the mixture modulus gradient. The master curves that are developed at depths at or near the surface are shifted to superimpose on the master curve at the baseline depth of 1.5 inches (38 mm). Examples are given using the dynamic modulus master curves of HMA. The dynamic modulus master curves of the HMA mixtures at the standard baseline depth (38 mm) of the asphalt layer for the four field aging times are shown in Figure 4.8. Figure 4.8. Baseline Dynamic Modulus Master Curve for HMA Field Cores Application of Aging Shift Function In order to construct a final field core aging master curve, the first step is to determine the aging parameters for field mixtures for the vertical shift and rotation. The final master curve of the baseline of the aged asphalt layer is developed after the horizontal shifts of the 8, 14 and 22 months aged master curves, which are shown in Figure 4.9a. Furthermore, by the same approach, the aging master curves of the surface and center of the asphalt layer are obtained after the vertical shift, rotation and horizontal aging shift. The final three aging master curves for the surface, center and baseline are shown in Figure 4.9b. 10 100 1000 10000 100000 0.0001 0.01 1 100 10000 1000000 Dy na m ic  M od ul us  (M Pa ) Reduced Angular Frequency (rad/s) 1 Month 8 Months 14 Months 22 Months

28 (a) Baseline Aging Master Curve of Field Asphalt Mixtures (b) Aging Master Curves for Top, Center and Baseline. Figure 4.9. Development of Aging Master Curve for Field Mixtures Determination of Horizontal Depth Shift Function Unlike the LMLC aged mixtures, the dynamic modulus of an asphalt layer also changes with pavement depth for a long-term field aging time due to the non-uniform aging effect. In order to take into account this unique aging effect for field-aged mixtures, another shift function 1 10 100 1000 10000 100000 0.001 0.1 10 1000 100000 10000000D yn am ic  M od ul us  (M Pa ) Reduced Angular Frequency (rad/s) 1 Month (Original) 8 Months (Horizontal Shift) 14 Months (Horizontal Shift) 22 Months (Horizontal Shift) 1 10 100 1000 10000 100000 0.001 0.1 10 1000 100000D yn am ic  M od ul us  (M Pa ) Reduced Angular Frequency (rad/s) Bottom Center Topaseline

29 is needed to quantify this non-uniform aging. The bottom of the asphalt layer is selected as the reference pavement depth of 38 mm. Due to some similarities between the non-uniform aging and the long-term aging, it is possible to shift the aging master curves of the surface and center to superimpose on that of the baseline. Therefore, a vertical shift and rotation are needed for this purpose. Figure 4.10a shows the dynamic modulus master curves at the center and surface after the vertical shift and rotation. Also on the graph is the unshifted master curve at the baseline depth of 1.5 inches (38 mm). At this stage, the glassy modulus and rheological index for the master curves of top, center and baseline of the layer are identical. A depth shift function is developed to shift the surface and center modulus master curves horizontally to superimpose the baseline one. The horizontal shift has the form: 0( )db d d da e  (4.9) where bd is the fitting parameter for the depth shift function, d and d0 are the depth of interest and the thickness of the layer down to the baseline depth, respectively. The final master curve for the HMA is shown in Figure 4.10b. After determining the aging shift function and depth shift function, it is possible to predict the dynamic modulus master curve at any depth and any aging time for an asphalt pavement layer. At this point, a final master curve for a field-aged mixture is established after the time- temperature superposition, long-term aging superposition, and non-uniform aging superposition. (a) Vertical Shift and Rotation 1 10 100 1000 10000 100000 0.001 0.1 10 1000 100000 Dy na m ic  M od ul us  (M Pa ) Reduced Angular Frequency (rad/s) Bottom Center (Vertical Shift and Rotation) Top (Vertical Shift and Rotation)aseline

30 (b) Horizontal Depth Shift Figure 4.10. Application of Depth Shift Function to Top and Center Dynamic Modulus Master Curves Kinetics-Based Aging Prediction for Asphalt Pavements in the Field Aging refers to the process of change of chemical and physical properties of asphalt binder due to oxidation and/or the loss of volatile oils. The prevalent chemical property used to describe aging of the asphalt binder is the carbonyl content or carbonyl area (CA) since carbonyl compounds are the major products of oxidation (127-130). Aging of asphalt pavements has two distinct stages. The first stage is called short term aging, which occurs during plant mixing and laydown construction in a relatively short period of time. The mixing temperature, time, plant type, asphalt type, and rate of cooling after laydown are primary factors that affect short term aging (131). The second stage is called long term in-service aging, or in-situ field aging, which starts immediately after construction and depends on the asphalt type, mixture composition (especially air void content), and climatic effects (particularly temperature) (131-133). The short term aging is a combined effect of oxidation and volatilization, while the in-situ field aging is mainly due to oxidation. Prediction of Field Aging Gradient Aging prediction models of asphalt binders usually make use of statistical or theoretical equations to describe the change of physical or chemical properties of the binder due to aging. For example, the change of viscosity is predicted by several regression equations at three stages of aging of asphalt binders in a comprehensive work (132). The field-aged viscosity has a gradient with higher values near the pavement surface. In order to consider this important phenomenon, a viscosity-depth relationship is proposed to produce viscosity gradient in the field aging condition. In addition, an adjustment factor is used to take into account the notable influence of air voids on the field-aged viscosity. In addition to physical properties of asphalt binders as mentioned above, their chemical properties are frequently used to develop aging 1 10 100 1000 10000 100000 0.001 0.1 10 1000 100000 10000000D yn am ic  M od ul us  (M Pa ) Reduced Angular Frequency (rad/s) Bottom Center (Horizontal Shift) Top (Horizontal Shift)aseline

31 prediction models. For example, the reaction rate of CA in an asphalt binder is predicted using an Arrhenius equation for temperature variation and pressure dependence (127-128). Depending on the speed of the reaction rate, the oxidative aging of asphalt binders is divided into two stages in chronological sequence: fast-rate period and constant-rate period. The fast-rate period is an initial jump stage that is rapid and nonlinear. Then the reaction rate declines and a linear constant-rate period is reached. In this project, the research team develops a kinetics-based aging prediction model making use of popular physical properties (e.g. dynamic modulus) of field cores from existing asphalt pavements instead of laboratory measured chemical properties of asphalt binders. The developed aging prediction model predicts the dynamic modulus gradient with pavement depth at any aging time at specific field aging temperatures. Laboratory Measurement of Modulus Gradient in Aged Field Cores The testing and analysis of asphalt field cores to determine the modulus gradient are documented in Appendix A. The asphalt mixtures include one type of HMA, one type of WMA treated by a foaming process (FWMA), and one type of WMA with Evotherm additive (EWMA). The field asphalt mixtures were cored from HMA, FWMA, EWMA sections 1 month after of construction, 8 months, and 14 months. These field cores are then subjected to the direct tension test to measure the modulus gradient at different ages. Table 4.2 presents the information on the number of tested field core specimens, air void content, laboratory testing temperature and results, aging time, and field aging temperature. Considering that all the field cores are aged under similar climatic conditions, the field aging temperature is taken as the average of the average monthly temperature in Austin for the corresponding aging time. Figure 4.11 shows examples of measured dynamic modulus with depth of the tested field core specimen. Figure 4.11 Laboratory Measured and Field Modulus Gradients in Asphalt Field Cores

32 Table 4.2 Laboratory Testing Results and Calculation Results for Field Condition Ty pe  o f  Fi el d  Co re s  Sa m pl e  ID   AV*  (%)  Laboratory Testing   Field Condition  Tlab**  (˚C)  * b E  at  Tlab (MPa)  * s E  at  Tlab (MPa)  n at Tlab  Aging  Time  (month)  Tfield***  (˚C)  * b E  at  Tfield  (MPa)  * s E  at  Tfield  (MPa)  n at  Tfield  HM A  H1  9.99  10  2660  3836  4.93  1  14  2381  3103  4.86  20  2005  3073  4.89  H2  8.44  10  2584  3357  3.78  1  14  2239  3002  4.78  20  1633  2460  4.93  H3  9.28  10  2471  3349  4.98  1  14  2211  2998  4.88  20  2246  3101  4.74  H4  7.41  10  2824  4271  4.77  1  14  2513  3795  4.70  H5  6.64  10  3677  6235  3.93  8  25  2786  4884  4.31 20  2935  5202  4.15  H6  10.43  10  3475  5727  3.71  8  25  2546  4632  4.50  20  2756  4561  4.07  H7  11.09  10  3611  6198  3.51  8  25  2726  4951  4.54  H8  6.03  10  3572  7377  4.11  14  21  2959  6256  4.22 20  3206  4620  3.33  H9  10.13  10  3902  6798  4.16  14  21  3270  5867  4.17  FW M A  F1  5.01  10  2243  2898  5.12  1  14  1924  2452  4.92  20  1530  2115  4.93  F2  5.65  10  2282  2775  4.75  1  14  1866  2378  4.92  20  1419  1743  4.86  F3  6.89  10  2634  3287  4.93  1  14  2205  2816  4.81  F4  10.59  10  2463  2917  5.57  1  14  2052  2567  4.60  F5  7.00  10  3456  5580  3.96  8  25  2444  4021  4.53  20  2784  4885  4.07  F6  9.03  10  3292  6411  4.69  8  25  2311  4717  4.38  20  2460  4968  4.31  F7  10.82  10  3057  5348  4.15  8  25  2201  3984  4.20  20  2618  4313  4.10  F8  7.33  10  2828  5310  4.30  8  25  1974  3925  4.62  F9  7.49  10  3041  5661  4.10  14  21  2584  4581  4.35  20  2668  4833  3.88  F10  7.92  10  3378  5206  4.35  14  21  2836  4427  4.02  20  2813  4778  4.17  F11  9.84  10  3599  6260  4.10  14  21  3012  5399  4.14  EW M A  E1  7.16  10  2000  2730  4.86  1  14  1796  2472  4.63  20  1743  2219  4.76  E2  9.74  10  1708  2235  4.94  1  14  1535  1966  4.38  20  1309  1805  4.93  E3  10.10  10  1993  2440  4.86  1  14  1725  2090  4.44  E4  7.74  10  2966  5209  4.50  8  25  1945  3550  4.30  20  2129  3907  4.23  E5  8.33  10  2608  4788  4.09  8  25  1805  3304  4.30  20  1867  3576  4.17  E6  9.22  10  2182  4626  4.73  8  25  1757  3199  4.29  E7  7.51  10  2813  5095  4.42  14  21  2283  4393  4.23 20  2496  4771  3.27  E8  8.44  10  2944  4671  4.18  14  21  2447  4178  3.80  20  2427  3861  4.13  E9  9.22  10  3365  5109  3.89  14  21  2503  4521  3.84  *: AV = air void content; **: Tlab = testing temperature of field cores in the laboratory; ***: Tfield = field aging temperature

33 Establishment of Aging Prediction Model for Field Modulus Gradient Based on the measured modulus gradient of field cores, the modulus gradient in an asphalt pavement can be idealized as illustrated in Figure 4.12. As aging time increases, the modulus within the top 1.5-inch increases and changes nonuniformly with depth. Below the 1.5- inch depth, the measurements show that the modulus increases with age and changes uniformly with depth. The modulus at the 1.5-inch depth is called the baseline modulus (i.e. * b E ); the one at the surface is called the surface modulus (i.e. * s E ). The modulus gradient within the top 1.5- inch at any age is described by Equation 4.1; the modulus below the 1.5-inch depth is given by the baseline modulus. Figure 4.12. Idealization of Modulus Gradient in Asphalt Pavements As a result, in order to completely characterize the change of aged modulus gradient in an asphalt pavement, predictive equations are developed for the baseline modulus, surface modulus, and aging exponent, respectively. The three submodels are formulated as follows:  Baseline modulus aging submodel:   * * * *0 1 fbk t cbb bi b biE E E E e k t     (4.10) in which afb field E RT fb fbk A e   (4.11) acb field E RT cb cbk A e   (4.12)  Surface modulus aging submodel:   * * * *0 1 fsk t css si s siE E E E e k t     (4.13)

34 in which: afs field E RT fs fsk A e   (4.14) acs field E RT cs csk A e   (4.15)  Aging exponent submodel:    0 1 fnk ti i cnn n n n e k t     (4.16) in which: afn field E RT fn fnk A e   (4.17) acn field E RT cn cnk A e   (4.18) where * b E and * s E are the baseline modulus and surface modulus, respectively; * bi E and * si E are the initial baseline modulus and initial surface modulus, respectively; * 0b E and * 0s E are the intercept of the constant-rate line of baseline modulus and that of surface modulus, respectively; in is the initial aging exponent; 0n is the intercept of the constant-rate line of the aging exponent; fbk , fsk , fnk are the fast-rate reaction constants for the baseline modulus, surface modulus, and aging exponent, respectively; cbk , csk , cnk are the constant-rate reaction constants for the baseline modulus, surface modulus, and aging exponent, respectively; t = the aging time in days; fbA , fsA , fnA are the fast-rate pre-exponential factors for the baseline modulus, surface modulus, and aging exponent, respectively; afbE , afsE , afnE are the fast-rate aging activation energies for the baseline modulus, surface modulus, and aging exponent, respectively; cbA , csA, cnA are the constant-rate pre-exponential factors for the baseline modulus, surface modulus, and aging exponent, respectively; acbE , acsE , acnE are the constant- rate aging activation energies for the baseline modulus, surface modulus, and aging exponent, respectively; and fieldT are the field aging absolute temperature. Conversion of Laboratory Measured Modulus to Field Conditions It is known that the modulus of asphalt mixtures is temperature dependent and aging- dependent. More specifically, with the increase of the aging time, the field modulus gradient is a combined effect of temperature change and aging. Thus, the modulus gradient used to develop the aging model must be the one at the given age and at the corresponding temperature. However, the laboratory aging and testing temperatures are different from the field aging temperatures. Laboratory aged samples do not develop modulus gradients. Because of different

35 aging with depth below the surface, field cores always have modulus gradients. As the first step of developing the aging mixture modulus prediction model, the laboratory test condition must be converted to the field aging condition. The solution to this problem of making this conversion is to introduce the rheological activation energy to quantify the temperature dependence of asphalt field cores at different ages. The change of the modulus of asphalt mixtures from one temperature to another is caused by the change of the viscosity of asphalt binders, which can be regarded as a thermally activated rate process. The asphalt molecules must overcome an energy barrier to move and the rate of this movement depends on the temperature. This phenomenon can be described by an Arrhenius equation as follows: ar lab E RT rA e  (4.19) where  is the bitumen viscosity of field cores; rA is the rheological pre-exponential factor; arE is the rheological activation energy of an asphalt binder; and labT = the absolute temperature at which field cores are tested in the laboratory. Based on Equation 4.19, the rheological activation energy and pre-exponential factor can be determined for the field core specimens tested at 10°C and 20°C in Table 4.2. The values of  are calculated from the moduli in Table 4.2 using the dynamic modulus predictive equation in the Mechanistic-Empirical Pavement Design Guide (122, Appendix CC, Figure 26). Figure 4.13 presents two examples of Equation 4.19 using the calculated viscosity of field cores in Table 4.2. The data points of “FWMA_S-1 month” correspond to the surface moduli of samples F1 and F2; the data points of “EWMA_B-14 months” correspond to the baseline moduli of samples E7 and E8. Figure 4.13. Examples of Arrhenius Plot of Bitumen Viscosity for Field Cores Tested at 10 and 20˚C

36 With known rheological parameters, Equation 4.14 is then used to determine the modulus gradient of field cores in the field aging condition. The viscosity at the field aging temperature at the given age is calculated by substituting fieldT for labT and selecting the rheological parameters at the corresponding age. Then the determined viscosity is substituted into the same dynamic modulus predictive equation used above (Mechanistic-Empirical Pavement Design Guide (122, Appendix CC, Figure 26)) to calculate the modulus for each field core. In this way, the laboratory measured baseline modulus, surface modulus, and center modulus are converted to their counterparts in the field. The calculated results are given in the last three columns of Table 4.2. The difference between the modulus gradient measured in the laboratory (labeled “lab”) and that in the field (labeled “field”) is demonstrated in Figure 4.11. Determination of Aging Prediction Model Parameters After obtaining the modulus gradient of field cores at the field condition, the next step of developing the aging mixture modulus prediction model is to determine the parameters in Equations 4.10 to 4.18. First, the field cores in Table 4.2 are separated into two groups: ones with the air void content less than or approximately equal to 7% and the others larger than 7%. The purpose is to consider the significant impact of air voids on aging of asphalt mixtures. Second, the change of the moduli and aging exponents in Table 4.2 with age is separated into two phases: fast-rate period and constant-rate period. As examples, Figure 4.14 presents the surface moduli of HMA and the aging exponents of EWMA, denoted “HMA_S” and “EWMA_n”, respectively. A power function is fitted to the measured modulus as shown in Figure 4.14. Then the entire modulus curve is divided into three portions: 1) the beginning of field aging to the end of the fast-rate period; 2) the beginning of constant-rate period to 8 months with an field aging temperature 1fieldT , denoted “HMA_S-I” and “EWMA_n-I”; and 3) the end of 8 months to 14 months with 2fieldT , denoted “HMA_S-II” and “EWMA_n-II”. The key problem in separating these three portions is to determine the end of the fast-rate period, or the beginning of the constant-rate period. According to the characteristic of the constant-rate period, it should have a constant rate that is visualized as a straight line in the modulus versus age curve. Therefore, a linear function is fitted to the second and third portions, respectively, in Figure 4.14. The third portions are fixed with the R-squared values nearly equal to 1. The starting points of the second portions are adjustable, so they are adjusted until the R- squared values are above 0.99. In this way, these starting points are regarded as the beginning of the constant-rate period.

37 Figure 4.14. Fast-Rate Period and Constant-Rate Period of Field Core Modulus Gradient Third, following the same approach above, the slopes of the straight lines in the second and third portions are obtained for the baseline modulus, surface modulus, and aging exponent of the field cores. These slopes are the rates of the change of the modulus or aging exponent with age in the constant-rate period. The absolute values of these slopes are also known as the constant-rate reaction constants. Thus they are used to determine the constant-rate aging activation energy and pre-exponential factor. As shown in Figure 4.15, the plot gives a straight line with a slope of acsE and an intercept of ln csA , or acnE and ln cnA . Similarly, the values of all constant-rate aging activation energies and pre-exponential factors are obtained for each field core in the two air void content groups. The results are shown in Table 4.3.

38 Figure 4.15. Examples of Arrhenius Plot of Constant-Rate Reaction Constant for Field Cores Aged at 28 and 16˚C Fourth, after using the second and third portions for the constant-rate period, the first portion is used to determine the aging activation energy and pre-exponential factor for the fast- rate period. By minimizing the error between the predicted and measured values, the two unknown variables in each prediction equation (i.e. fast-rate aging activation energy and pre- exponential factor) are obtained using the Excel Solver tool. The results of the fast-rate aging parameters are given in Table 4.3. It is found that the constant-rate aging activation energies of all the field cores correlate well with its corresponding fast-rate aging activation energies, as shown in Figure 4.16. Figure 4.17 presents examples of the comparison between the prediction and measurements. The predicted values match the measured ones very well. To further examine the validity of the aging prediction model and associated parameters for the field modulus gradient, the aging information of the same field core binders (125) is presented in Table 4.4. The extracted and recovered binders are from the same batch of field cores. Both the lab and field binder samples are tested by the Fourier Transform Infrared (FTIR) Spectroscopy to measure the CA level, the change of which indicates the aging speed of the binder. Table 4.4 suggests that the asphalt binder with Evotherm has the highest activation energy and lowest aging speed. The other two types of binders have similar activation energies and aging speeds. For the mixture field cores with the same binders, similar trends are observed. Compared to the laboratory test results of asphalt binders, the results of field cores reflect the aging gradient in an asphalt pavement. In both air void content groups, the surface moduli have lower aging activation energies and much higher aging speed in terms of csk and cbk at 25°C than the baseline moduli. Besides, the comparison reveals that asphalt mixtures have different aging activation energies from their constituent binders. This is because aging of a mixture is influenced by the interaction between the binder and aggregates, and the air voids also play an important role in this process. The field cores with larger air void contents have lower aging activation energies and age faster than those with smaller air void contents for baseline moduli, surface moduli, and aging exponents.

39 Figure 4.16. Goodness of Representation by Aging Prediction Model for Field Aging Gradient Figure 4.17. Goodness of Representation by Aging Prediction Model for Field Aging Gradient

40 Table 4.3 Results of Aging Activation Energies and Pre-Exponential Factors of Field Cores Type of Field Cores Baseline Modulus AV<7% AV>7% afbE (kJ/ mol) fbA acbE (kJ/ mol) cbA cbk at 25 °C afbE (kJ/ mol) fbA acbE (kJ/ mol) cbA cbk at 25 °C HMA 29.01 1.00E4 48.74 3.24E8 0.92 25.55 2.26E3 43.87 7.93E7 1.61 FWMA 28.41 7.76E3 48.53 3.01E8 0.93 26.60 3.46E3 45.45 1.52E8 1.63 EWMA 31.00 2.22E4 49.19 3.77E8 0.89 26.95 4.07E3 46.35 1.93E8 1.44 Type of Field Cores Surface Modulus AV<7% AV>7% afsE (kJ/ mol) fsA acsE (kJ/ mol) csA csk at 25 °C afsE (kJ/ mol) fsA acsE (kJ/ mol) csA csk at 25 °C HMA 23.67 9.73E2 40.44 6.05E7 4.90 22.56 5.81E2 38.44 3.33E7 6.07 FWMA 23.76 9.78E2 40.55 5.98E7 4.64 21.66 3.93E2 38.69 3.34E7 5.50 EWMA 24.70 1.49E3 41.98 9.49E7 4.14 22.99 6.85E2 39.81 4.76E7 4.98 Type of Field Cores n AV<7% AV>7% afnE (kJ/ mol) fnA acnE (kJ/ mol) cnA cnk at 25 °C afnE (kJ/ mol) fnA acnE (kJ/ mol) cnA cnk at 25 °C HMA 34.03 8.87E4 52.97 1.95E6 1.01E-3 33.04 5.83E4 51.72 1.31E6 1.12E-3 FWMA 34.04 8.85E4 52.78 1.74E6 9.66E-4 33.56 7.26E4 51.66 1.19E6 1.04E-3 EWMA 35.02 1.34E5 53.21 1.82E6 8.52E-4 34.04 8.88E4 52.77 1.76E6 9.86E-4 Table 4.4 Results of Aging Activation Energies and Pre-Exponential Factors of Field Core Binder (125) Type of Field Core Binder Laboratory Prepared Binder Sample Extracted and Recovered Binder from Field Cores afE (kJ/mol) fA acE (kJ/mol) cA CA level 1 month CA level 8 months CA level 14 months PG 70-22 49.8 3.45E6 64.1 4.49E7 N/A 0.87 0.90 PG 70-22 foamed 47.1 2.45E6 65.8 8.22E7 0.72 0.86 0.89 PG 70-22 with Evotherm 51.4 1.19E7 72.1 6.47E8 0.74 0.83 0.83 Prediction of Field Aging Using Deflection Data The work presented above is about using asphalt field cores collected from a field project to determine aging characteristics. A further step is taken herein considering the limitations associated with obtaining field cores (e.g. limited number and short aging time). The Falling Weight Deflectometer (FWD) data is a good alternative, which provides a wide range of field

41 moduli of asphalt mixtures at different ages. The FWD is a widely used technique for nondestructive evaluation of pavements. This device is designed to simulate deflection of a pavement under moving traffic. The FWD generates a load pulse that is imparted to the pavement and records the associated deflection. Then the modulus of the pavement layer can be estimated through various computational methods. The FWD is an essential element of the pavement performance monitoring in the LTPP program. Currently the LTPP database contains more than 2,500 asphalt and concrete pavement sections. Most of the asphalt pavements have the FWD data over a long time. Therefore, the LTPP database is a valuable and convenient source to obtain the field moduli at different ages. This method of measuring the aging characteristics of asphalt mixtures aims at using these field moduli over a wide aging time to characterize field aging through the kinetics-based modeling. As mentioned above, the rate of aging is large at the beginning, and then decreases with time. This indicates that the modulus of asphalt mixtures demonstrates an obvious two-stage feature (fast-rate period and constant-rate period) as shown in Figure 4.18. The fast-rate period is nonlinear, while the constant-rate period is almost linear. Figure 4.18. Two-Stage Modulus of Aged FWD Modulus Collection of FWD Data to Determine Rheological and Aging Properties Eight asphalt pavement sections are selected from the four climate zones in the LTPP database, two in each climate zone as shown in Table 4.5. For each pavement section, the following data are collected from the LTPP database:  Field moduli backcalculated from the FWD data. In the LTPP database, the path to locate this information is: “Data”, “Data Selection and Download”, “Performance”, “Deflection”, and finally “Deflection-Backcal Data”. From the “Deflection-Backcal Data”, three kinds of data are collected: o Average modulus: backcalculated modulus values averaged for each FWD pass, from the worksheet of “BAKCAL_MODULUS_SECTION_LAYER”. o Individual modulus: backcalculated modulus values for each measured deflection basin, from the worksheet of “BAKCAL_MODULUS_BASIN_LAYER”. o Surface temperature: the average temperature at the mid-depth of the surface layer used in the backcalculation process, from the worksheet of “BAKCAL_BASIN”.

42  Mixture properties, including the air void content of the mixture, asphalt binder content, and the aggregate gradation. When the information of the air void content is not available, the bulk specific gravity and maximum specific gravity are collected instead to calculate the air void content. In the LTPP database, the path to locate this information is: “Data”, “Data Selection and Download”, “Structure”, “Material-Layer Properties and Field Sampling”, and finally “AC”. Note that the backcalculated modulus from the FWD is at the in-situ field temperature with a loading time of approximately 25 to 35 milliseconds. The modulus values documented in the “Deflection-Backcal Data” are these backcalculated moduli converted to the standard laboratory condition at a 77°F (25°C) temperature and a 100 millisecond loading time (134). The temperature normalization is accomplished using the relationship suggested in the EVERSERIES User’s Guide (134):  24log 6.4721 1.47362 10 pE T   (4.20) where E is the asphalt mixture modulus; and pT is the pavement temperature (°F). Therefore, the collected average modulus and individual modulus are the values at 77°F (25°C), and the surface temperature is regarded as the in-situ field temperature. It is worth mentioning that the FWD data only provides one type of modulus rather than the modulus gradient. Thus one set of aging prediction equations is enough for the FWD data, for example, using Equations 4.13 to 4.15 and replacing the surface modulus is applied to the FWD modulus. Table 4.6 presents an example of the data collected for the pavement section 19-0102 details of which are given in Table 4.5. The air void content is calculated from the bulk specific gravity and maximum specific gravity. The results of the rheological properties of the pavement sections are shown in Figure 4.19. The data forms of these sections are similar, so the data given in Table 4.6 are employed as an example to illustrate the results. After obtaining the binder viscosity for each individual modulus, the values of the viscosity are plotted versus the surface temperature for each test date. According to Equation 4.19, the plot of ln versus  1 RT gives a straight line with a slope of arE and an intercept of ln rA. Note that the data of the individual modulus at the test date of February 16, 2001 in Table 4.6a are too large and therefore not reasonable, so they cannot produce valid results. Therefore, the data of February 16, 2001 are not included in the calculations. This kind of situation exists for the FWD backcalculated modulus in the “Deflection-Backcal Data” from the LTPP database. This is mainly due to the variations in the FWD measurements or during the process of modulus backcalculation. The rheological activation energy and pre-exponential factor are calculated for each pavement section in Table 4.5. Figure 4.19 shows the changes of the rheological activation energy in the four climate zones as the age increases. The magnitude and the rate of change of the rheological activation energy are different between different climate zones. The increase of the rheological activation energy almost exhibits a linear trend with the increase of aging time. This increasing trend is similar to the results obtained from the viscosity of original and aged asphalt binders measured by a rotational viscometer (135). The aged asphalt binders have higher rheological activation energies than original ones. Aging increases the number of polar molecules in the asphalt binder, which leads to higher intermolecular forces and stronger interactions. Thus compared to original asphalt binders, the aged ones possess higher resistance to flow, consequently a higher rheological activation energy.

43 Table 4.5 Information of LTPP Pavement Sections LTPP Sections Date when FWD Data Collected Age of Aging (Days) Type of Asphalt Binder** County, State Climate Zone*** Date of Construction 19-0102* 5/19/1993 199 Penetration Grade 40- 50 Lee, Iowa WF Nov 1, 1992 9/12/1995 1045 5/14/1997 1655 10/8/1999 2532 2/16/2001 3029 19-0110 5/18/1993 198 9/11/1995 1044 10/7/1999 2531 2/15/2001 3028 48-0117 11/13/1997 135 AC-20 Hidalgo, Texas WNF July 1, 1997 5/9/2000 1043 2/12/2002 1687 12/4/2002 1982 48-0167 8/19/1999 779 5/9/2000 1043 2/13/2002 1688 12/5/2002 1983 30-0116 6/16/1999 258 PG 70-28 Cascade, Montana DF October 1, 1998 7/16/2001 1019 7/10/2003 1743 7/11/2005 2475 7/12/2010 4302 30-0124 6/16/1999 258 7/17/2001 1020 7/14/2003 1747 7/13/2005 2477 7/13/2010 4303 35-0102 6/3/1996 215 AC-20 Dona Ana, New Mexico DNF November 1, 1995 5/22/2000 1664 4/30/2002 2372 3/18/2004 3060 4/12/2005 3450 35-0108 6/3/1996 215 5/23/2000 1665 5/1/2002 2373 3/19/2004 3061 4/13/2005 3451 *: “19” is LTPP state code, indicating Iowa; “0102” is test section identification number assigned by LTPP. **: Type of asphalt binder is indicated using the grade of asphalt cement available in the LTPP ***: WF=wet, freeze; WNF=wet, non-freeze; DF=dry, freeze; DNF=dry, non-freeze

44 Table 4.6 Examples of Modulus and Mixture Property Data Collected from the LTPP Database (a) Modulus Data ID Test date FWD Pass Average Modulus* (MPa) @ 77˚F (25˚C) Deflection Basin Number Individual Modulus* (MPa) @ 77˚F (25˚C) Surface Temperature (˚F) 19-0102 5/19/199 3 4 2612 24 2165 79.3 34 2999 86.2 9/12/199 5 14 5813 12 3689 69.6 33 4902 77.2 Date of Construction 5/14/199 7 22 6778 4 5047 62.2 34 5516 65.1 11/1/1992 10/8/199 9 25 8170 29 6667 66.6 17 6688 66.7 2/16/200 1 36 12489 41 22705 35.2 14 10459 35.6 *: Average modulus is not the average of individual modulus; they are from two different sources (b) Mixture Property Data ID Asphalt Binder Content Bulk Specific Gravity Aggregate Gradation 19-0102 5.2 2.243 Sieve Size Percent Passing in mm 1" 25 100 3/4" 19 100 1/2" 12.5 89 Maximum Specific Gravity 3/8" 9.5 78 2.473 No. 4 4.75 59 No. 10 2 35 No. 40 0.425 13 No. 80 0.177 5 No. 200 0.075 4

45 (a) Climate Zones WF and WNF (b) Climate Zones DF and DNF Figure 4.19. Calculated Rheological Activation Energy from FWD Data at Different Ages Modeling Pavement Temperature to Determine Field Aging Temperature The approach adopted herein is the pavement temperature model developed by the group at Texas A&M University (71) to better predict temperature variations over time and depth in pavements. It is a one-dimensional model based on heat transfer fundamentals and coupled with the methods to obtain required climate data from available databases. The sources of heat transfer at the pavement surface include: 1) Solar radiation and reflection of solar radiation at the surface;

46 2) Absorption of atmospheric downwelling long-wave radiation and emission of long-wave radiation; and 3) Convective heat transfer between pavement surface and air, enhanced by wind. Accordingly, the climate data required to describe this heat transfer process include: solar radiation, air temperature, and wind speed. The temperature model consists of a thermal diffusion function for heat transfer in the pavement, a surface boundary condition, and a bottom boundary condition. The surface boundary condition of interest is given as follows: fcrass s QQQQQQ t TxC     ~ 2 (4.21) where C is volumetric heat capacity of the pavement; sT is pavement surface temperature; x is the depth below the pavement surface; sQ is the heat flux due to solar radiation; ~ is the albedo of the pavement surface, the fraction of reflected solar radiation; aQ is the downwelling long-wave radiation heat flux from the atmosphere; rQ is the outgoing long-wave radiation heat flux from the pavement surface; cQ is the convective heat flux between the surface and the air; and fQ is the conduction from the surface into the pavement. The inputs required by the pavement temperature model above include hourly climatic data and site-specific pavement parameters. The hourly climatic data contain hourly solar radiation, hourly air temperature, and hourly wind speed. They are collected from the National Solar Radiation Database (NSRDB), National Climatic Data Center (NCDC), and LTPP database. The site-specific pavement parameters contain albedo, emissivity, absorption coefficient, thermal diffusivity, etc. With these inputs, the model is solved numerically by a finite difference approximation method. The pavement temperature variations are calculated for each pavement section listed in Table 4.5. Take the section 19-0102 as an example. First, all of the necessary inputs are collected for each day from the date of construction (November 1, 1992) to the last test date (February 16, 2001) in Table 4.6a. In each day, the data are collected in an hourly form for 24 hours. The pavement temperature model is programmed as a MATLAB script file to automatically import data and export results. The outputs of the program are the hourly pavement temperature for the 3,029 days from November 1, 1992 to February 16, 2001. Figure 4.20 shows the calculated pavement temperature for the entire aging period and for three arbitrary days. It is clear that the summer temperature pattern is different from those in the winter. The differences in the pavement temperature between four climate zones are presented for an arbitrary winter day and a summer day in Figures 4.21a and 4.21b, respectively. After calculating the variations of the pavement temperature, an appropriate value needs to be chosen as the aging temperature in the aging prediction model. The harmonic mean of the temperatures is proposed for this purpose because the aging temperature is in the equations following the Arrhenius kinetics. The harmonic mean provides the truest average especially in the case involving rates and ratios (136). In the Excel, the harmonic mean is calculated by the HARMEAN function.

47 (a) Hourly Pavement Temperature for Entire Aging Time (b) Hourly Pavement Temperature for a Single Day Figure 4.20. Predicted Pavement Temperature for the LTPP Section 19-0102 (a) One Winter Day (12/22/1999)

48 (b) One Summer Day (8/18/2000) Figure 4.21. Predicted Pavement Temperature at Different Climate Zones Calculation of Aging Properties The pavement section 19-0102 in Table 4.5 is employed as an example to explain the procedures and present the results. This section has five test dates, the first four are used to obtain the coefficients in the aging prediction model and the fifth for model validation. The key to determining the model coefficients is to divide the modulus versus aging time curve into two phases: fast-rate period and constant-rate period, and then to divide the constant-rate period into several segments according to the characteristic of this period. The constant-rate period should have a constant rate which is visualized as a straight line in the modulus versus aging time curve. The approaches are listed below: 1. Plot the average modulus (i.e. FWD backcalculated modulus) in Table 4.6a versus the aging days as shown in Figure 4.22. Fit this curve with a power model and use it to calculate the fitted modulus as listed in Table 4.7. 2. Determine the boundary of the last segment in the fitted modulus versus aging time curve, as shown in Figure 4.23. In this case, the last one is Segment 9 as in Table 3. Select three data points in Segment 9: starting, middle, and end. a. Assign the last aging time as the end of Segment 9. b. Fit the fitted modulus curve in this segment with a linear model. c. Adjust the starting point until the R-squared value of the linear model is around 0.999. 3. After obtaining the starting point of Segment 9, which is also the end of Segment 8, repeat step 2 to determine the starting point of Segment 8. 4. Repeat steps 2 and 3 to obtain Segments 7 to 2. The time interval monotonically decreases from Segment 9 to Segment 2 given in Table 4.7. The minimum time interval of a segment selected in this study is 30 days (i.e. 1 month). If the time interval is equal to or less than 30 days, this segment is regarded as being in the fast-rate period. In this case, Segments 9 to 2 are in the constant-rate period.

49 5. Use the starting point of Segment 2 as the end of Segment 1. Assign the aging time of 1 day as the starting point of Segment 1. Segment 1 is the fast-rate period. 6. Calculate the field aging temperature for each segment, which is given in Table 4.7. 7. Plot the slopes of the linear models in Segments 9 to 2 versus the reciprocal of the field aging temperatures. These slopes are the rates of the change of the modulus with age in the constant-rate period, also known as the constant-rate reaction constants. The constant- rate aging activation energy and pre-exponential factor are determined from this plot. 8. Use the values of the fitted modulus and field aging temperature of the fast-rate period to determine the fast-rate aging activation energy and pre-exponential factor. A plot of the fast-rate and constant-rate aging activation energies is given in Figure 4.24 for each pavement section in Table 4.5. Table 4.7. Examples of Calculated Modulus and Field Aging Temperature in Each Aging Segment ID Aging Phases Aging Segments Aging Time for Fitting (Day) Fitted Modulus (MPa) Time Interval (Day) Field Aging Temperature (K) 19-0102 Fast-Rate Period 1 1 244 30 281 30 1128 276 80 1754 30 269 110 2025 272 Constant- Rate Period 2 110 2025 60 291 140 2257 170 2463 3 80 296 210 2709 250 2930 4 100 297 300 3181 350 3409 5 170 283 435 3760 520 4074 6 230 294 635 4458 750 4804 7 350 287 925 5280 1100 5708 8 600 283 1400 6363 1700 6944 9 832 280 2116 7663 2532 8308

50 Figure 4.22. Fitted FWD Backcalculated Modulus Curve Figure 4.23. Example of Fast-Rate Period and Constant-Rate Period of FWD Backcalculated Modulus for the LTPP Section 19-0102

51 Figure 4.24. Calculated Aging Activation Energy from FWD Data and Climate Data in Different Climate Zones Validation of Aging Prediction Model After determining all of the coefficients in the aging prediction model, this model can be used to predict the modulus at any age. An example of the comparison between the predicted moduli and measured moduli from FWD at each test date is shown in Figure 4.25a. Except for the fourth data points at 10/8/1999, the match of other pairs is satisfactory. To further validate this aging prediction model, the modulus at the last test date is predicted for each pavement section in Table 4.5 using the model coefficients obtained above from the moduli at other test dates. In other words, the aging prediction model along with the associated coefficients is extrapolated from the previous test dates to the last one. The predicted moduli are plotted versus the FWD backcalculated moduli from the LTPP database for all of the pavement sections. The result is given in Figure 4.25b. Considering the variations that exist in the FWD backcalculation and field climate modeling, the validation result in Figure 4.25b is acceptable. (a) Example of Moduli at All FWD Test Dates for Section 19-0102

52 (b) Extrapolated Moduli of All LTPP Sections at Last Test Date Figure 4.25. Comparison of Predicted Moduli and Measured FWD Moduli Pseudo J-Integral Based Paris’ Law for Crack Growth Prediction A physically-based approach to predict crack growth refers to the incorporation of mechanics into the experimental investigation of the cracking resistance of asphalt materials. More specifically, the test data are translated to fracture properties by introducing fracture mechanics or damage indexes through continuum damage mechanics. Fracture mechanics is a branch of mechanics that especially describes the behavior of cracks in solids. The Paris’ law is by far the most widely used fracture mechanics tool to characterize crack growth by fatigue:  ndc A K dN   or  ndc A JdN   (4.22) where c is the crack length; N is the number of load repetitions; K is the range of stress intensity factor; J is the J-integral range; and A and n are fracture coefficients associated with K or J . The increase of the crack length with the number of load repetitions captures the physical process of fatigue cracking. The physical significance of K is that it quantifies the stress state near the crack tip. The physical significance of J is that it is the energy release rate that quantifies the rate of the change of the potential energy due to the crack growth. There have been numerous applications of the Paris’ law in the field of asphalt pavements (17, 104, 108, 137). Besides fracture mechanics, continuum damage mechanics is another tool to study cracking, which treats material damage within the framework of continuum mechanics and targets the dispersed nature of cracking. The continuum damage mechanics employs damage state variables to represent material deterioration at the macroscale (e.g. stiffness degradation) due to damage evolution. It can also account for the evolving microstructure when damage progresses in a material. Applications of continuum damage mechanics in asphalt material characterization and damage modeling have given rise to many analytical solutions (138-139), and also facilitated numerical simulations of damage development (140-141). In order to appropriately and conveniently implement mechanics in top-down cracking modeling, this project proposes two techniques in this respect. The first technique aims at

53 determining a rational and appropriate reference modulus for the use in both fracture mechanics and continuum damage mechanics models by making use of the widely used material property (i.e., dynamic modulus). The second technique discussed subsequently aims at making use of the dynamic modulus and mixture design information (e.g., air void content, aggregate gradation, binder content, etc.) to determine the fracture parameters and eventually allow the crack growth prediction without performing fatigue tests. Quasi-Elastic Simulation of Viscoelasticity The concept of “quasi-elastic simulation of viscoelasticity” is proposed to enable accurate viscoelastic-to-elastic conversion using readily available material properties of asphalt mixtures, like the dynamic modulus. The dynamic modulus is commonly used in asphalt pavement design to account for the frequency and temperature dependence of asphalt surface materials. Standard tests are available to measure the dynamic modulus under a haversine load. In the latest mechanistic-empirical pavement design methods (e.g. the Pavement ME Design), the dynamic modulus is utilized in the multi-layer elastic analysis to determine the primary responses in asphalt pavements. However, a recent study (142) brings up a question about such a usage: the standard method to calculate the dynamic modulus actually characterizes the material response to a sinusoidal load rather than a haversine load. A haversine load can be decomposed into a sinusoidal loading portion and a constant loading portion. To correctly represent the material response to a haversine load in the multi-layer elastic analysis, a combined compliance by the dynamic compliance and the creep compliance is proposed to characterize the relationship between the haversine stress amplitude and the resulting strain amplitude (142). Similarly, a combined modulus, as the average of the dynamic response function and the time-dependent response function, is proposed in this project, called representative elastic compliance or representative elastic modulus. The expression of the representative elastic modulus is given: * 1 1 2 2 p p re f t t E E E t              (4.23) where reE is the representative elastic modulus; *E is the dynamic modulus; and  E t is the relaxation modulus; f is the frequency of a load pulse; and pt is the pulse time of a load. The representative elastic modulus is originally intended to be used in the multi-layer elastic analysis for pavement design. It performs a similar function as the elastic modulus but depends on the loading frequency. If the representative elastic modulus can be used as the reference modulus to compute the pseudo strain, it has practical implications for making use of the prevalent material property in asphalt pavement design, and avoid extra testing efforts devoted to determining the endurance limit (96). The research team explores this application to see whether the resulting pseudo strain has the same physical significance. Using the representative elastic modulus to calculate the pseudo strain is called quasi-elastic simulation of viscoelasticity. In addition, the experimental validation is performed by comparing the calculated representative elastic modulus to the reference modulus measured from the test that resembles repeated traffic loading. The details of the derivation and validation of the representative elastic modulus is presented in Appendix C.

54 Modified Paris’ Law with Application of Quasi-Elastic Simulation The modified Paris’ law is early proposed as follows (103):  nR d A J dN   (4.24) where  is the damage density; RJ is the pseudo J-integral; and A and n  are modified Paris’ law parameters associated with the evolution of the damage density. The damage density is the ratio of the lost area to the total cross-sectional area, its initial value is percent air. It replaces the crack length to account for the fact that there are a multitude of cracks in asphalt mixtures in the early stage of fatigue cracking. The cracking history of asphalt mixtures usually has three stages from the beginning of fracture to failure: 1) the formative phase, in which more and more microcracks are generated when cracking is initiated; 2) the coalescent phase, in which microcracks merge and coalesce to form larger cracks; and 3) the unitary phase, in which a few macrocracks are formed to dominate the failure of the material (96). The first two stages are modeled using Equation 4.24. The unitary phase is modeled by replacing the damage density by the crack length:  nR dc A J dN  (4.25) The pseudo J-integral is defined by: DPW (crack surface area)R J   (4.26) where “DPW” stands for the dissipated pseudo work, which is calculated by:    2 1 DPW t R t V d t t dtdV dt     (4.27) in which  t is the stress applied to the material; R is the pseudo strain; 1t and 2t is the start and end times of a loading period, and V is the volume of the material. For a complete loading and unloading cycle, the amount of DPW is manifested by the area enclosed in a hysteresis loop of the load versus the pseudo displacement. The DPW represents the work consumed due to damage formation in an asphalt material. Thus the physical significance of RJ is the pseudo energy release rate that quantifies the rate of the change of the energy solely responsible for crack growth. The “crack surface area” refers to the area of the crack surfaces on a cross section of the material, which is calculated by: 2 formative and coalescent phases Crack surface area 2 unitary phase S cw     (4.28) in which the factor “2” is needed because each crack has two surfaces; S is the cross-sectional area of the material; and w is the thickness of the material. With the application of the quasi-elastic simulation, the calculation of RJ becomes fairly simple. One way is to compute the R using the reference modulus given by Equation 4.23; then calculate RJ through Equations 4.26 to 4.28. Another option is to convert K to RJ by the following relationship (156):

55   2 2 2 21 1 R I II III R R J K K K E E       (4.29) in which RE is determined as the representative elastic modulus;  is the Poisson’s ratio; IK is the Mode I (opening) stress intensity factor; IIK is the Mode II (in-plane shear) stress intensity factor; and IIIK is the Mode III (out of plane shear) stress intensity factor. Accordingly, the application of the modified Paris’ law to quantify the amount of cracking also has two options. One way is to first simulate the change of the DPW with the increase of the number of load cycles by a power function as: DPW baN (4.30) in which a and b are regression coefficients of the curve of the DPW versus N. Then substitute Equations 4.25 and 4.26 into Equations 4.28 and perform a series of algebraic manipulations, which give the following two similar expressions: 1 11 1 1 0 1 2 1 n bnn n nab nA N S bn                     (4.31) 1 11 1 1 0 1 2 1 n bnn n nab nc A N c w bn                   (4.32) where 0 is the initial damage density; and 0c is the initial crack size. Estimation of Fracture Coefficients by Performance-Related Properties The primary consideration of estimating the modified Paris’ law coefficients by performance-related properties is to allow fatigue prediction when experimental investigations are not available. It can also act as a quick and cost-effective alternative to check fatigue resistance of asphalt mixtures. To ensure the accuracy and reliability of the estimation, the following steps are undertaken:  Step 1. Collect test data and determine A and n  ;  Step 2. Select performance-related material properties;  Step 3. Develop prediction models of A and n  through selected performance-related material properties. Step 1. Collect test data and determine A and n  As the first step, collecting sufficient accurate data of A and n  is of prime importance. Three data sources are identified for this purpose:  Jacobs (47)’s dynamic uniaxial tensile test data;  Luo et al. (103)’s controlled-strain RDT test data; and  Gu et al. (143)’s overlay test data. The details of the collected data and the method to determine A’ and n’ are presented in Appendix D. From the three studies above, there are altogether 89 different pairs of A and n  . The plot of  log A versus n  is shown in Figure 4.26. When a straight line is fitted to the data in Figure 4.26, the R-squared of the linear function is 0.892. This suggests a strong correlation

56 between the modified Paris’ law coefficients. Once one of them is known, the other coefficient can be estimated by:  1 .2 46 3 .61510 nA    (4.33) Figure 4.26. Relationship between Fracture Coefficients A’ and n’ Step 2. Select performance-related material properties The material properties selected to estimate A and n  need to be directly related to the cracking performance of asphalt materials. More importantly, these properties should be readily available for most asphalt pavement design. After extensive screening and evaluation, the following material properties are identified:  Relaxation modulus: a simple model is proposed here for the relaxation modulus in order to reduce the variables in the prediction:   1 mE t Et (4.34) where  E t is the relaxation modulus of asphalt mixtures; and 1E and m are the relaxation modulus parameters.  Air void content: air voids are indispensable components of an asphalt mixture, which also act as initial flaws from which fracture occurs in the material. Therefore, the air void content directly affects the cracking performance of asphalt materials and must be formulated in the prediction model.  Asphalt binder content: the main impact that the asphalt binder exerts on the cracking performance is its aging and increase of brittleness. A higher asphalt binder content results in more aging, which makes the mixture more brittle and less cracking-resistant.

57  Aggregate gradation characteristics: the particle size distribution, or gradation, of aggregates affects almost every important property of asphalt mixtures, such as stiffness, durability, permeability, fatigue resistance, frictional resistance and moisture susceptibility (144). As a result, the aggregate gradation characteristics must be taken into account. The specific procedure proposed to obtain the aggregate gradation characteristic parameters is as follows: 1) Obtain the aggregate gradation information from the mixture design. 2) Model the curve of the cumulative percent passing versus the sieve size using an appropriate mathematical function, as shown in Figure 4.27 for the two aggregate gradations used in Luo et al. (103):  f x x (4.35) where x is the sieve size;  is the gradation aggregate scale parameter; and  is the gradation aggregate shape parameter. Table 4.8 presents examples of the material properties listed above that are used to develop prediction models in the next step, as well as the values of A and n  from the data sources mentioned above. General model Power: f(x) = *x^ Coefficients of Texas limestone (with 95% confidence bounds):  = 32.65 (25.06, 40.23)  = 0.3699 (0.2836, 0.4562) Goodness of fit: R-square: 0.9738 Coefficients of Hanson limestone (with 95% confidence bounds)  = 26.08 (19.12, 33.05)  = 0.4399 (0.3431, 0.5366) Goodness of fit: R-square: 0.9799 Figure 4.27. Determination of Aggregate Gradation Characteristic Parameters 0 5 10 15 20 25 10 20 30 40 50 60 70 80 90 100 Sieve Size (mm) C um ul at iv e Pe rc en t P as si ng (% ) Texas limestone Fitted curve for Texas limestone Hanson limestone Fitted curve for Hanson limestone

58 Table 4.8. Examples of Values of Fracture Coefficients and Performance-Related Material Properties Segment Data Source and Test Type Fracture Coefficients Performance-Related Material Properties A n  1E (MPa) m Air void content (%) Asphalt binder content (%)  Jacobs (1995) Dynamic uniaxial tensile test 4.24 x 10-7 2.55 412 0.181 2.0 6.0 0.399 3.29 x 10-6 1.75 289 0.177 3.0 7.0 0.424 5.18 x 10-7 2.05 228 0.184 8.3 10.0 0.269 3.61 x 10-8 3.05 312 0.177 7.9 6.8 0.387 2.34 x 10-6 1.65 358 0.173 5.1 6.2 0.429 Luo et al. (2013d) Controlled- strain RDT test 3.15 x 10-16 9.80 6555 0.328 4.0 4.4 0.440 1.68 x 10-8 3.47 35390 0.947 7.0 4.4 0.440 2.13 x 10-12 7.42 4722 0.239 4.0 4.4 0.370 1.32 x 10-12 7.36 1956 0.286 7.0 4.4 0.370 Gu et al. (2014) Overlay test 1.35 x 10-13 5.64 502 0.320 7.0 5.2 0.536 4.85 x 10-11 4.90 448 0.310 7.0 5.2 0.536 1.01 x 10-12 5.44 427 0.350 7.0 5.2 0.536 1.38 x 10-14 5.90 481 0.290 7.0 5.2 0.536 Step 3. Develop prediction models of A and n  The multiple regression analysis is conducted to identify the relationship between n  and the selected performance-related material properties above. In order to develop a reliable and concise model for n  , two important issues must be addressed: 1) the number of the variables; and 2) the formulation of the variables. The variables contained in the prediction model must be significant, and the formulation needs to ensure the highest R-squared value. An effective method to increase the R-squared value is to perform some transformations to certain variables. Based on these principles, the final expression of the prediction model of n  is:   1 2 1 116.052 0.135 % 6.500ln % 8.147 5.512 81.515 with 0.943 m n AV AB m E R                 (4.36) where %AV is the air void content, in %; %AB is the asphalt binder content by weight of mixture, in %;  is the aggregate gradation shape parameter; and 1E and m are relaxation modulus parameters, 1E in MPa. The summary output of the multiple regression analysis is given in Table 4.9. The p-values of all the variables are less than 0.05, which indicates that they are significant at the 95 percent confidence level. Note that Equation 4.36 only contains one

59 parameter,  , from the aggregate gradation model in Equation 4.35. This is because when both  and  are used to perform the multiple regression analysis, the p-value of  is larger than 0.05, so the aggregate gradation scale parameter is not a significant variable and excluded from the prediction model. After obtaining n  by Equation 4.36, the other fracture coefficient A is given by Equation 4.33. Figure 4.28 compares the predicted values by Equations 4.33 and 4.36 with the measured values. The R-squared values are satisfactory, so the proposed performance- related material properties as well as the prediction models can predict the fracture coefficients at acceptable accuracy. (a) n’

60 (b) A’ Figure 4.28. Comparison of Predicted and Measured Fracture Coefficients Table 4.9. Summary Output of Multiple Regression Analysis of n’ Using Performance- Related Material Properties Regression Statistics ANOVA Multiple R 0.971 df SS MS F Significance F R Square 0.943 Regression 5 509.986 101.997 274.077 4.879E-50 Standard Error 0.610 Residual 83 30.888 0.372 Observations 89 Total 88 540.875 Variables Coefficients Standard Error t Stat P-value Lower 95% Upper 95.0% Intercept -16.052 2.049 -7.284 1.70E-10 -18.998 -10.848 %AV 0.135 0.033 3.911 1.87E-04 0.063 0.192  ln %AB 6.500 0.890 6.887 1.01E-09 4.359 7.899  8.147 1.416 5.425 5.60E-07 4.866 10.498 1 m 5.512 0.267 19.488 1.03E-32 4.667 5.728  11 mE -81.515 3.821 -20.117 1.16E-33 -84.460 -69.261 Pseudo J-Integral Based Paris’ Law for Crack Initiation Prediction The top-down cracking in asphalt pavements normally consists of two phases: crack initiation and crack propagation. The crack initiation phase starts from microcracks which grow from the air voids and then coalesce to form a visible macro-crack. The embedded air voids of an

61 in-place asphalt pavement are the initial damages which cause the local stress concentrations to allow microcracks to grow so that asphalt pavements suffer fatigue from the first loading cycle. Development of Crack Initiation Model Derivation of Crack Initiation Model for a Load Level The pseudo J-integral based Paris’ law is used to derive the number of loading cycles for top-down crack initiation.  nR d A J dN   (4.37) where  is damage density of the cross-sectional area; RJ is the pseudo J-integral; A and n are Paris’ law coefficients; and N is the number of loading cycles for a load level. The damage density is used in the Paris’ law instead of a single crack due to the existence of multiple microcracks in the top-down crack initiation phase. The damage density is the ratio of the “lost area” to the total cross-sectional area. Initially, the “lost area” is the percent air voids in the asphalt mixtures. Similar to the stress intensity factor used in the linear elastic fracture mechanics, the J-integral is more capable of analyzing the nonlinear feature of viscoelastic materials such as asphalt mixtures (109). The J-integral is an energy release rate parameter. Based on the definition, the pseudo J-integral is determined in Equation 4.38 as the change of the dissipated pseudo work with respect to the change of crack surface area: ( ) R R dW dNJ d CSA dN                 (4.38) where RW is the load induced pseudo-displacement work; and CSA is the cracked surface area. For a haversine load such as the moving traffic, the pseudo-displacement work is proportional to 2 /P S in which P is the load magnitude and s is the pseudo stiffness of the material. For an in-situ asphalt pavement, there are multiple levels of magnitudes of load and numbers of axles, which are characterized using the load spectra model detailed in the next section. Because of this, the pseudo-displacement work and pseudo J-integral are different for different load levels. As the loading level increases, multiple distributed microcracks increase in size, decrease in number, and coalesce. The pseudo displacement work RiW at the loading cycle N of the load level i is determined as follows: ib Ri iW a N (4.39) where ia and ib are the energy parameters of the pseudo displacement work, respectively. Based on the results in the previous study (151) and the sensitivity calculations in this study, the

62 value of bi is determined to be 0.12. Since WRi is proportional to 2 iP s , ai can be related to the load levels as follows: 2 0 0 i i Pa a P        (4.40) where 0P is the load level of 3000 lb of single axle, which is the minimum load level defined in the load spectra; and 0a is the fracture energy induced by a 3000 lb single axle load. The CSA is the cracked area in the cross section, which can be determined as a function of damage density and cross-sectional area, as shown in Equation 4.41. 02CSA A (4.41) where 0A is the critical cross-sectional area. Substitute Equations 4.38 to 4.40 into Equation 4.37, which gives the following result:     ' ' 12 ' ' 0 2 0 02 i n n b Ri i id W a P b Nd A A ddN d CSA A P dN                     (4.42) Integrating Equation 4.42 produces the number of cycles needed for top-down crack initiation at loading level i in Equation 4.43: 1 1 0 1 2 1 01 2 0 0 1 2 1 i n b n c i n n i in i N a P b nA A P b n                                      (4.43) where iN is the number of loading cycles for crack initiation of load level i; 0 is the initial damage density equal to the percentage of the air voids at the pavement surface; and c is the critical damage density corresponding to the critical cracked percentage at which multiple distributed cracks coalesce into a single crack (“top-down crack initiation”) and the single crack then propagates (“top-down crack propagation”). The critical damage density c is related to the percentage of asphalt mastic volume, which can be determined as Equation 4.44:   200% % % % 1 % % %c m b bAV V AV V V AV V           (4.44) where %AV is the air void content, %; mV is the volume of asphalt mastic; bV is the volume of asphalt binder; and 200V is the volume of aggregate passing # 200 sieve.

63 As can be seen, a higher binder content and percentage of passing # 200 sieve with a smaller air void content produces a higher volume of asphalt mastic, which improves the crack resistance in Equation 4.43. It is noted that for different load levels, Ni are different. The fracture parameters A’ and n’ are the same for the same asphalt pavement since they are the material properties of the asphalt mixtures. Simulation of Air Void Distribution It is shown from Equation 4.43 that as the air void content increases, the number of load cycles for crack initiation decreases. The air void content is not uniformly distributed within the pavement depth. The X-ray CT is used to measure the air voids distribution, which is detailed in the reference (152) and not repeated here. An example of air void distribution by cutting a field core into slices is given in Figure 4.29. As seen, the distribution usually follows a C-shaped curve within pavement depth, which indicates the highest air void contents are located at the surface and bottom of an asphalt layer. The higher air void content at the surface of a pavement means more possibilities of generating a larger number of microcracks. It also indicates that the binder oxidative aging is more severe at the pavement surface due to the high oxygen availability and solar radiation. The air void distribution is considered as one of the main causes for the top- down crack initiation. Figure 4.29. Air Void Distribution in Pavement Depth The air void content in the depth of asphalt layer is modeled as follows:    2max minmin 2 2 a aa z a h z h     (4.45) where  a z is the air void distribution at pavement depth z; maxa is the maximum air void content; mina is the minimum air void content; and h is the thickness of the asphalt layer. The average air void content a , which is an mixture design parameter and available in the LTPP database, is calculated as 0 10 20 30 40 50 60 70 80 0% 2% 4% 6% 8% 10% 12% Pa ve m en t D ep th  (m m ) Air Void Content Measured Calculated

64    2max minmin min max20 0 1 1 2 12 3 3 z h z h z z a aa a z dz a h z dz a a h h h               (4.46) Based on the X-ray CT measurement, maxa is about two times as large as mina , the following relationship can be achieved: min 3 4 a a , max 3 2 a a (4.47) Once the average air void content a is known, the air void content at the pavement surface can be computed using Equation 4.47. Determination of Initiation for Multiple Load Levels Asphalt pavement is subjected to different loading cycles at a variety of load levels caused by the moving traffic. Miner’s hypothesis (153) is used to characterize the contribution of the number of different loading cycles for different load levels for the micro-crack damage in top-down crack initiation: 0 1 1 m ij m ij n t N    (4.48) where ijn is annual number of axle loads at load level i of axle category j calculated using the weight-in-motion (WIM) data or predicted using the AADTT and default axle load distribution, which are the two input load levels in this study; and 0t is TDC initiation time (year) that is defined as the time when longitudinal wheelpath cracking first appears at the pavement surface after construction. Characterization of LTPP Data Modeling of Traffic Data Traffic load is one of the most essential inputs in pavement design and analysis. The traffic load including the average WIM data and AADTT data is collected from the LTPP database for each pavement section, if both of the types of data are available. The traffic load is characterized using the load spectrum model to account for different axle types (single, tandem, tridem and quadrem axles), traffic categories (from 1 to 8), vehicle classes (from 4 to 13) and the number of tires (single and dual tires). The load spectra model has advantages over the equivalent single axle loads (ESALs) due to its high accuracy. When the WIM data is not available, the traffic load can also be presented using the AADTT with a default traffic load distribution for the number of axle loads in four axle types. Therefore, the annual traffic load is modeled in two levels. Level one is based on the WIM data, which can be used to characterize the actual annual axle load distribution for each traffic category. The AADTT is utilized as the level two for the traffic modeling. It is noted that the AADTT is the average number of total trucks so that it is not appropriate to use the number

65 only. The load distribution with the AADTT should be estimated. Since the vehicle classes 1 to 3 are regular light passenger vehicle types, the truck traffic for vehicle classes 4 to 13 are taken into account, which contribute significantly to the top-down crack development. The axle load distribution is defined as the number of loads in an interval of a given axle load. The number of loads are measured and recorded from the WIM sensors installed in pavements. A total of four different axle types are taken into account and the corresponding axle load limits and intervals are different, as shown in Table 4.10. Each axle load is composed of vehicle classes 4 to 13. Table 4.10. Characteristics of LTPP Axle Type Axle Type Minimum Load (lb.) Maximum Load (lb.) Load Interval (lb.) Single Axle 3,000 40,000 1,000 Tandem Axle 6,000 80,000 2,000 Tridem Axle 12,000 102,000 3,000 Quadrem Axle 12,000 102,000 3,000 Equation 4.49 is employed to characterize the cumulative axle load distribution for each axle load.    Pije i j c P e    (4.49) where  ic P is the cumulative axle load distribution factor at the load level iP ; iP is the loading level i listed in Table 4.10; and  and  are the scale parameter and shape parameter of the cumulative distribution curve, respectively.  and  are calculated for eight traffic categories of each section. An example of axle load distribution of single axle/single tire is given in Figure 4.30, which is corresponding to traffic category 1. The number of axles for each vehicle class and vehicle distribution factors are calculated for each pavement section using Equation 4.49. Figure 4.30. Example of Axle Load Distribution 0 0.2 0.4 0.6 0.8 1 3000 13000 23000 33000 43000Cu m ul at iv e  Ax le  L oa d  Di st rib ut io n Axle Load (lb) LTPP Data Modeled Data

66 For some cases that the WIM data is not available, the AADTT is used to predict the traffic load distribution as the level two input. The default distribution of a major multi-trailer truck route in the Pavement ME Design shown in Table 4.11 is adopted to calculate the number of vehicles in each truck class distribution. The default number of axles per truck is presented in Table 4.12 (123). Table 4.11 Distribution of Vehicle Classes (123) Vehicle Class Distribution Percentage (%) Vehicle Class Distribution Percentage (%) 4 1.8 9 31.3 5 24.6 10 9.8 6 7.6 11 0.8 7 0.5 12 3.3 8 5.0 13 15.3 Table 4.12 Average Number of Axle for Each Vehicle Class (123) Vehicle Class Single Axle Tandem Axle Tridem Axle Quadrem Axle 4 1.62 0.39 0.00 0.00 5 2.00 0.00 0.00 0.00 6 1.02 0.99 0.00 0.00 7 1.00 0.26 0.83 0.00 8 2.38 0.67 0.00 0.00 9 1.13 1.93 0.00 0.00 10 1.19 1.09 0.89 0.00 11 4.29 0.26 0.06 0.00 12 3.52 1.14 0.06 0.00 13 2.15 2.13 0.35 0.00 Modeling of Top-Down Crack Performance Data The associated complete longitudinal wheelpath cracking historical data are collected for each section starting from the construction time. Based on the definition in the LTPP manual (154), there are three longitudinal cracking severity levels: low, medium and high. For the longitudinal wheelpath cracking, the low severity level is defined as a crack with a mean crack width greater than 3 mm and less than 6 mm, the medium severity level is defined as a crack with a mean crack width greater than 6 mm but less than 19 mm, and the high severity level is named for a crack with a mean crack width greater than 19 mm. The length of longitudinal cracking is collected from the database and then is plotted against the service time, which normally follows an S-shape curve, as shown in Figure 4.31. Equation 4.50 is used to characterize the distress curve for the three severity levels. It is noted that when longitudinal cracks in the wheelpath reach a medium severity level, they are commonly lumped together with alligator cracking. Because of this, there is very little top-down cracking data in the LTPP database that has either a medium or high severity distress level. The longitudinal crack initiation time t0, distress shape parameter  and distress scale parameter  are calculated based on the

67 distress data points. To determine the initiation time t0, only the data of the low severity level is used. Since the length of a typical LTPP section is 150 m, the maximum longitudinal crack length is selected as 300 m for each pavement section. 0( ) o t tl t l e        (4.50) where ( )l t is the longitudinal crack length; 0l is the maximum longitudinal crack length (i.e., 300 m);  and  are the scale parameter and shape parameter of the distress curve, respectively; 0t is the crack initiation time (days); and t is the pavement service time (days). For some pavement sections with no more than two distress points, the first longitudinal wheelpath cracking appearance time is used as the top-down crack initiation time. Figure 4.31. Determination of Top-Down Crack Initiation Time Modeling of Asphalt Layer Properties Dynamic modulus at various temperatures and frequencies of the asphalt mixtures in the surface layer are obtained from the LTPP database for each section to construct the dynamic modulus master curve. It is worth mentioning that the dynamic modulus test results used in this study are chosen only from the unaged or slightly aged field cores. The purpose is to assure that only high quality and reliable data are used in this model and to avoid the complications that arise from the field aging effect on the viscoelastic properties of asphalt mixtures. The dynamic modulus master curve is then converted to the relaxation modulus using the Prony series model to obtain the E1 and m-value of the relaxation modulus in Equation 4.51 which are used as the level of stiffness and relaxation ability, respectively. 1( ) mE t Et (4.51) 0 50 100 150 200 0 1000 2000 3000 4000 5000 To p‐ do w n  Cr ac k  Le ng th  (m ) Service Time (day) Field Data Model Prediction

68 where ( )Et is the relaxation modulus of asphalt mixtures; and E1 and m are the relaxation modulus parameters. The average air voids content in mix design for the selected LTPP sections are also collected, since it affects the fatigue resistance as described above. The average air voids content is used to calculate the percent air at the pavement surface by Equation 4.47. The binder content and aggregate gradation (percentage passing # 200 sieve) are also collected to calculate the volume of mastic. Modeling of Pavement Structures and Climatic Data Total pavement thickness and asphalt layer thickness are obtained for each pavement section, since they are major structural factors influencing the stress and strain distribution of pavements. The climatic data are also collected from the LTPP database including the average annual number of days above 32.2°C and below 0°C. These two environmental factors are used to evaluate the aging and high temperature effects and low temperature effect on the top-down crack initiation time. These two factors are believed to affect the crack initiation time as explained in previous section entitled “Derivation of Crack Initiation Model for a Load Level”. Statistical Analysis for Top-Down Initiation Time Prediction of Energy Parameter Statistical analysis is conducted to predict the values of the energy parameter 0 02 a A determined following the procedure above. The variables used in the statistical analysis include the m-value, , E1, and thickness of asphalt layer. The statistical programming language R is used in this study. It is necessary to check the multicollinearity between the input variables prior to the analysis to avoid issues such as overfitting and inaccurate prediction for new input data. The condition number  (Kappa) is used to check the multicollinearity.  is defined as the square root of the ratio of the largest eigenvalue max to the smallest eigenvalue min , as shown in Equation 4.52. max min    (4.52) If the calculated value of  is greater than 30, it indicates there is multicollinearity between input variables. The result of  is computed to be 11.1, which is much less than the threshold value. It means that little multicollinearity is observed between variables, and a linear regression model is suitable. This result is also reasonable from the engineering experience that the listed input variables are highly independent. Regression analysis is then conducted to develop the statistical model based on the collected data. Equation 4.53 shows the regression equations for 0 02 a A for four climate zones as level one input and for all pavement sections as level two input, respectively. WLF: 5 40 1 0 0.1609 2.056 10 0.677 2.079 10 2 a a E m H A        (4.53a)

69 WF: 5 30 1 0 0.3978 2.301 10 1.725 5.91 10 2 a a E m H A        (4.53b) DNF: 6 40 1 0 0.1820 7.420 10 0.665 7.860 10 2 a a E m H A        (4.53c) DF: 5 30 1 0 0.2689 3.5968 10 1.002 7.356 10 2 a a E m H A        (4.53d) All: 5 40 1 0 0.1796 1.5 10 0.690 7.169 10 2 a a E m H A        (4.53e) where E1 (MPa) and m are the relaxation modulus parameter in Equation 4.51; Ha is the total thickness of the asphalt layer (inch). Figure 4.32 illustrates the prediction accuracy of Equation 4.53. The prediction equation has a R2 value of 0.80, which suggests a fairly strong correlation between the variables including material properties and pavement structure and the energy parameter 0 02 a A . The values of 0 02 a A are approximately 5 to 15 times smaller than the J-integral in top-down crack propagation determined from the finite element method due to the nature of healing that mainly occurs in the top-down crack initiation phase. The effect of each variable agrees well with literature and understanding. For thicker asphalt layers, the induced stress can be effectively distributed, which delays TDC initiation. Modulus increase indicates the increase of damage density as low modulus provides more flexibility than high modulus. A lower m-value indicates a lower relaxation rate, it also decreases with aging, which is detrimental for pavement performance. DNF Zone R² = 0.8389 0.03 0.04 0.05 0.06 0.07 0.03 0.04 0.05 0.06 0.07 Pr ed ic te d  In iti at io n  En er gy  P ar am et er  (N /m ) Calculated Initaiton Energy Parameter (N/m)

70 DF Zone WNF Zone R² = 0.8594 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Pr ed ic te d  In iti at io n  En er gy  P ar am et er  (N /m ) Calculated Initation Energy Parameter (N/m) R² = 0.8178 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 Pr ed ic te d  In iti at io n  En er gy  P ar am et er  (N /m ) Calculated Initiation Enegy Paramter (N/m)

71 WF Zone Figure 4.32. Comparison of Predicted and Calculated Energy Coefficient 0 02 a A Prediction of Top-Down Crack Initiation Time and Model Validation Once 0 02 a A is determined with the known variables in Equation 4.53, the result can be inserted back to Equations 4.43 to 4.48 to back-calculate the crack initiation time t0. On the other hand, for the purpose of practical use of this method, a regression model based on the available data is preferred. Using the similar statistical method for Equation 4.53, a linear regression model shown in Equation 4.54 is developed to serve as a practical approach to predict t0 using the field data:  0 0 0 100 log 21 aB C HT D LT AADTT A At e              (4.54) where A, B, C and D are calibration coefficients. HT and LT are annual number of days above 32°C and 0°C, respectively. The calibration coefficients are obtained based on the minimization of the sum of squared errors: A = 65462227, B = 0.618986, C = 0.000169, D= 2.97953E-05, and E= 2.360746. Figure 4.33 presents the prediction accuracy of top-down crack initiation time, which has a R2 value of 0.826. From Equation 4.54, it is shown that as energy parameter, annual numbers of days of high temperature and low temperature and AADTT increase, the crack initiation time reduces, which is consistent with common understanding. In addition, four regression equations for the crack initiation time are developed to account for different climatic zones, as shown in Appendix I. R² = 0.8899 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 Pr ed ic te d  In iti at io n  En er gy  P ar am et er  (N /m ) Calculated Initiation Energy Parameter (N/m)

72 To validate Equation 4.54, a total of 10 pavement sections in different states for a variety of traffic levels, pavement structures and material properties are selected as the validation pavement sections, which are not used in the model development procedure. The predicted top- down crack initiation time and observed top-down crack initiation time are presented in Figure 4.34. It indicates that the model is generally able to predict top-down crack initiation time considering the fact of field variability. Figure 4.33. Comparison of Predicted and Calculated Top-Down Crack Initiation Time Figure 4.34. Validation LTPP Pavement Sections for Top-Down Crack Initiation Time R² = 0.8258 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 M ea su re d  cr ac k  in iti at io n  tim e  (d ay ) Predicted crack initiation time (day) 0 500 1000 1500 2000 2500 3000 48‐0113 4‐A903 35‐0112 12‐0107 36‐1011 29‐0603 53‐1008 8‐0559 1‐0111 39‐0902 TD C  In iti at io n  TI m e  (d ay s) Observed Predicted

73 Numerical Modeling and Artificial Neural Network (ANN) for Predicting J-Integral In this project, the fracture parameter J-integral for the top-down cracking propagation at the crack tips for different crack depths, material properties and pavement structures is determined using numerical model. The modulus gradient in the asphalt layer induced by aging and temperature is an important factor that is not included in the previous studies of top-down cracking. The tire-pavement interactions are simulated based on the previous measurements in South Africa (34). The effects of different crack depths, modulus for each layer and thickness of the asphalt and base layers are also included in the model developments. In addition to the FEM, statistical analysis is an efficient tool to develop a prediction model in pavement engineering. The ANN is one of the most accurate modeling approaches to predict the material and fracture properties such as the dynamic modulus, the SIF and the distress of asphalt pavements (145-147). The reason for choosing the ANN is that it has some unique advantages including the convenience that the mathematical interacting relationships between various factors do not need to be specified, the capability of learning and training itself from the input and output data, and the high prediction accuracy. Three-Dimensional Finite Element Modeling A 3D FEM analytical models is developed using the commercial program ABAQUS. A longitudinal top-down crack is inserted in the middle part of the pavement model to avoid interference due to the boundary condition. The crack is located in the direction of traffic flow with a 1 m length. The J-integral at the crack tips is calculated using different material properties, crack depths and pavement structures. Material Properties of Pavement System As indicated previously, the modulus gradient plays a key role in the propagation of top- down cracking. It has been shown that layered analysis with uniform moduli in each layer is not accurate compared to the continuous gradient analysis and the most significant difference is found at the surface of the asphalt layer (148), which is non-negligible for the analysis of top- down cracking. A user-defined material subroutine (UMAT) is developed using FORTRAN to model the smooth modulus gradient. Two schematic plots for the typical modulus gradient curves as affected by aging and high surface temperature are shown in Figures 4.35a and 4.35b, respectively. Figure 4.35a shows the modulus gradient for a long-term aging pavement and Figure 4.35b demonstrates another type of modulus gradient for a high air temperature pavement. The viscoelasticity is not considered in the UMAT due to the impractical fact of running a large number of calculations and changing the Prony series parameters of the relaxation modulus in the FEM. However, the temperature dependency and frequency dependency of the asphalt mixtures are taken into account as the surface modulus E0 and the parameters of n and k can be systematically adjusted in the FEM calculations with the aid of the UMAT.

74 (a) Aging Effect (b) Temperature Effect Figure 4.35. Modulus Gradient Curves in Asphalt Layers Tire-Pavement Contact Stresses A typical 315/80R22.5 dual tire static loading with a 0.35 m tire spacing is selected. The tire inflation pressure and the constant load on tires are assumed to be 689.5 kPa and 40 kN, respectively. The corresponding contact stresses are decomposed into vertical stress, longitudinal stress and transverse stress, which are nonuniformly distributed. The contact stress patterns of the three types of stresses are shown in Figure 4.36, which are simplified versions of the measurements by De Beer et al. (34). The measured maximum contact stress Pc in Figure 4.36 is related to the inflation pressure and tire load through a multiple linear regression analysis, as shown in Equation 4.55 (34). 1 2 3C I LP k k P k P   (4.55) where CP is maximum contact stress in kPa; IP is the inflation pressure in kPa; LP is the tire load in kN; and 1k , 2k , 3k = regression coefficients (see Table 4.13).

75 Figure 4.36. Simplified Patterns of 3D Vertical, Longitudinal and Transverse Contact Stresses Table 4.13. Regression Analysis of Tire–Pavement Contact Stress Tire Type Data Samples Contact Stress Regression Constants and Statistics k1 (kPa) k2 k3 R2 315/80R 22.5 89 Vertical 80.4490 0.9021 16.1207 0.92 Longitudinal 106.2760 0.0129 3.2960 0.84 Transverse 210.5240 -0.2003 3.0316 0.83 Using Equation 4.55, with the known inflation pressure and tire load, the maximum contact stresses for the vertical, transverse, and longitudinal components are calculated as 1350.7 kPa, 247 kPa, and 193.7 kPa, respectively. The shapes of tire contact areas are assumed to be rectangular rather than circular shapes, according to Lytton et al. (70). The tire width is 0.315 m, and the tire length is calculated based on Equation 4.56:

76 L I Pl P w   (4.56) where w is the tire width and l is the tire length. With set values of tire width and inflation pressure, the tire length is determined solely by the load level. It should be noted that there are three lengths for the dual tire loading and three different lengths of the single tire loading in the numerical modeling. For both single and dual tires, the three tire patch lengths define the relationship between the load level and tire patch length, which is illustrated in Figure 4.37a. Figure 4.37b shows an example of accumulative axle load distribution. Based on the calculation results, the three stresses determined in one loading area are shown in Figures 4.38a to 4.38c, respectively. The combination of the three stresses used in the FEM is illustrated in Figure 4.38d. (a) Axle Load Level versus Tire Length (b) Accumulative Load Distribution versus Tire Length Figure 4.37. Example of Load and Tire Length Relationship 0 5000 10000 15000 20000 25000 30000 35000 40000 0 5 10 15 20 Ax le  L oa d  (lb ) Tire Length (inch) Single Tire below 6000 lb Single Tire above 6000 lb Dual Tire below 6000 lb Dual Tire above 6000 lb 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 Cu m ul at iv e  Ax le  L oa d  Di st rib ut io n Tire Length (inch)

77 (a) Vertical Contact Stress (b) Transverse Contact Stress (c) Longitudinal Contact Stress (d) Combination of Contact Stresses Figure 4.38. Three Components of Tire-pavement Contact Stress in ABAQUS Pavement Structure, Mesh and Boundary Conditions The length and width of the 3D model are 6 m and 3 m, respectively. The thicknesses of the asphalt and base layers are adjusted for all of the possible ranges in order to take into account, the thickness effect on top-down cracking. The boundaries for the four surrounding

78 sides of the 3D model are fixed in the x and z directions and the bottom side is fully fixed. Sliding is not allowed between layers. Different ratios of crack depth to asphalt layer thickness (top-down cracking ratio) are considered in the FEM. Four crack locations relative to the tire loading are analyzed to determine the most critical one. The first location is at the edge of the tire but outside of the two load patches, the second one is at the edge of the load between the two load patches, the third one is in the center between the load patches and the fourth location is in the center of the load. Table 4.14 shows all the inputs including the pavement structures and material properties that are used in the FEM analysis. Table 4.14. Materials and Structures Inputs in the FEM Layer Type Thickness (mm) Top- down crack Ratio Top- down crack Location Poisson’s Ratio Layer Modulus (MPa) Asphalt Layer 25,75, 125, 200,300, 500 1/10, 3/10, 1/2, 7/10, 9/10 4 0.35 Modulus Gradient n k Surface modulus 0.5, 2, 5 0.5, 1.2, 2.5 500, 2000, 8000, 30000 Base 150,300, 500 na na 0.40 Elastic Modulus 150, 300, 600, 7000, 14000 Subgrade na na na 0.40 30, 150 NOTE: na= not applicable. Figure 4.39a shows the 3D FEM model with meshes developed in this study. The meshes for the crack and loading area are shown in Figure 4.39b with four crack locations, and the meshes for the crack front are magnified and shown in Figure 4.39c. Eight-node linear brick continuum elements with reduced integration (C3D8R) are used in the model. Fine meshes are used in the vicinity of the crack, loading area and asphalt layer whereas coarse meshes are used far away from these areas. To consider the 1/√ singularity of the crack tips, 1/4 is used as the midside node parameter, and the degenerate element at crack tips has a single node with a collapsed side element. Sensitivity analysis of the mesh density and size is conducted to assure both the accuracy and efficiency of the FEM model. A total of 35 nodes are assigned along the crack length; meanwhile, 6 nodes in width and 11 nodes in length are assigned in each of the two rectangular loading areas. The element thickness for the top asphalt layer and the crack area are 15 mm and 2.5 mm, respectively. The area of an element in the crack and loading area is 30 mm  30 mm and there are gradient changes of the meshes from the crack and loading area to other areas. The corresponding single bias mesh for the other parts are 80 mm to 500 mm beginning from the crack location.

79 (a) 3D FE Pavement Model (b) Magnified Mesh View of Crack Front (c) Meshes in Crack and Loading Areas Figure 4.39. 3D FEM Model and Mesh Details for Top-Down Cracking and Loading Numerical Simulation Results The criterion for the most critical location is to identify the highest J-integral value, which is the output of the FEM. The crack location 4 is found to be the most critical one and the J-integral of location 2 has relatively smaller values but is greater than those of locations 1 and 3. This means that when the load is applied on the surface, the crack at the center or at the edge of the tire between the two load patterns is more prone to propagate. Therefore, location 4 is chosen as the default location to calculate the maximum J-integral in this study. After determining the critical location for the top-down crack to develop, the analysis based on the different combinations of pavement materials properties, pavement structures, traffic loading, and crack depths are conducted. Examples of the J-integral within pavement

80 depth are shown in Figures 4.40 (a) to 4.40(c) for different modulus gradient parameters of n, k and asphalt layer thickness. The thickness of asphalt layer is 200 mm in Figures 4.40 (a) and 4.40 (b), and the values for n and k are 3 and 2 in Figure 4.40 (c). It can be seen that the J-integral is not uniformly distributed within the asphalt layer. The shape and magnitude of the J-integral curve is highly dependent on the thickness of the asphalt layer and the modulus gradient parameters n and k. As seen in Figure 4.40 (a), the J-integral sharply increases from the surface to approximately one third of the layer thickness, which means the crack can propagate easily near the surface. After the peak point, the J-integral decreases, which indicates that more energy is needed to continue to propagate the crack until about two thirds of total thickness is reached. In this part, the crack propagation rate is relatively slow. For the last one-third part, the pavement becomes prone to crack again since the J-integral starts to increase. These simulation results match the field observations of top-down cracking, which indicates that most of the top-down cracks are observed in the approximately the top one third of the asphalt layer (1). As seen from Figure 4.40 (b), a larger value of k results in a smaller J-integral, which is consistent with the definition of the J-integral. For a thicker asphalt layer in Figure 4.40 (c), the ratio of crack depth to asphalt layer thickness for the critical crack location is smaller, which means that the critical crack locations are relatively concentrated near the surface of the asphalt layers for all of the conditions and less related to the thickness of the asphalt layer. These analyses suggest that increasing the thickness of asphalt layer is helpful to reduce the propagations of pavement cracking, however, after the long-term aging period and traffic loading, the surface layer becomes more brittle, and more cracks initiate at the surface rather than at the bottom. It is also found that the shapes of modulus gradient (n and k) within the asphalt layer have a significant effect on the distribution of the J-integral and it also confirms that assuming a uniform asphalt layer modulus is not appropriate for accurately predicting top-down cracking. Based on the different combinations of all of the variables in the FEM, there are 194,400 cases needed to be determined. Because of the large number of calculations, the technology from SA-CrackPro is utilized to change the values of all the variables in the FEM model automatically. The details of this method are not reported here. However, interested readers can consult this reference (149). The calculation results for 194,400 cases are used in the ANN modeling for predicting the J-integral at any given pavement depth and pavement condition.

81 (a) Different n Values (b) Different k Values (b) Different Asphalt Layer Thickness Figure 4.40. J-Integral in Pavement Depth with Various Values of n, k and Asphalt Layer Thickness 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 J‐ in te gr al  (P a. m ) Top‐down Crack Ratio n=2, k=2 n=3, k=2 n=4, k=2 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 J‐ in te gr al  (P a. m ) Top‐down Crack Ratio n=3, k=2 n=3, k=0.75 n=3, k=1.5 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 J‐ in te gr al  (P a. m ) Top‐down Crack Ratio Hac=200 Hac=100 Hac=300

82 Artificial Neural Network Modeling of J-Integral Background and Preparation of ANN The task after the finite element modeling is to construct the ANN models for accurately predicting the J-integral of top-down cracks. The ANN models are developed using the commercial program MATLAB. The independent variables are the pavement structures (i.e., thickness of asphalt layer Hac and thickness of base layer Hbase), modulus gradient parameters (i.e., n and k), material properties (i.e., moduli of the surface of asphalt layer Eac, base layer Ebase and subgrade layer Esubgrade), the loading conditions (i.e., tire lengths and tire types) and crack depths. The tire lengths and tire types are used to divide the numerical calculation results into different sets. In all, there are 8 variables that are used as the inputs in the ANN model and the J- integral is the output of the ANN model. In the database, three sets are combined for the single tire loading including three different tire lengths and another three sets are grouped for the dual tires loading with another three tire lengths. For each of the six datasets for the ANN modeling, all of the input variables are normalized to the range of -2 to 2 which are the components in the input layer, and the output variable is normalized to the range of -1 to 1, which is the component in the output layer. The data are randomly divided into a training dataset and a validation dataset before conducting the analysis. The ratio of the training dataset to validation dataset is 4:1. The training dataset is used to develop and train the ANN models and the validation dataset is used to test the statistical accuracy and avoid the overfitting of the ANN models. The architecture of the ANN model used in this study is shown in Figure 4.41. Development and Algorithm of ANN In the ANN algorithm, there are no specific rules to determine the types of activation functions for the hidden layers and output layer, the numbers of hidden layers, and the number of neurons in each layer. Based on the literature and required accuracy of the ANN model (146- 147), a two hidden layers system (1-2-1) is employed. It should be mentioned that for some other cases with a much smaller database, the 1-1-1 system is sufficient to save the computational effects (24). The input layer contains the eight independent variables, the two hidden layers include the corresponding weights, bias terms and activation functions, and an output layer includes the output and the corresponding weights and bias terms. The output of the ANN model is calculated using Equation 4.57. Based on the converged mean square error (MSE) and the computation time compared with different structures and activation functions, a total of 60 neurons are used in each hidden layer. The activation functions for the output layer and two hidden layers are pure linear functions and log-sigmoid functions, which are shown in Equations 4.58 to 4.60. 0 2 1 1 2 0 0 1 1 1 { [ ( ) ] } q q q h h h h k hk jk hj ij i j k k k k J A W A W A W P b b b          (4.57) 0A x (4.58) 1 1hj x A e   (4.59)

83 1 1hk x A e   (4.60) where i , j and k are the subscripts for the input layer, first hidden layer and second hidden layer, respectively; m, n and q are the numbers of inputs (i.e., 8), nodes in the first hidden layer (i.e., 60) and the nodes in the second hidden layer (i.e., 60) , respectively; 0A , hjA and hkA are the activation functions for the output layer (i.e., pure linear), the first hidden layer (i.e., log- sigmoid); and the second hidden layer (i.e., log-sigmoid); 1hijW , 2h jkW and 0kW are the weight factors for the first hidden layer, the second hidden layer and the output layer; 1hjb , 2hkb and 0b are the bias factors for the first hidden layer, second hidden layer and the output layer; iP are the input variables, J is the output of the J-integral. As a widely used supervised learning algorithm, the backpropagation ANN is chosen in this study , which means that once the network is trained the signal will come back to update the initial weights and bias to reduce the calculation errors, as shown in Figure 4.41. There are several backpropagation algorithms available in MATLAB. After evaluating the convergence of each algorithm, only the Levenberg-Marquardt backpropagation (trainlm function in MATLAB) is suitable to obtain the desired prediction results. The learning function also needs to be determined. In general, different learning functions have similar error performance, but the differences of computation times between the learning functions are relatively large. Based on comprehensive comparisons, the gradient descent weight/bias learning function (learngd function in MATLAB) is employed as the learning function in the ANN model. To ensure the convergence and performance of the ANN model, several other model parameters are also considered in the training. Based on the literature and trial and error, the following parameters including the epochs between displays, learning rate, the maximum number of epochs for training, and the performance goal are determined as 0.05, 0.9, 400 and 1e-4, respectively. A total of six ANN models for different traffic loading conditions are successfully developed based on the above technique. The models are further evaluated using the determination coefficient R2. As seen from Figures 4.42 to 4.47, all the R2 values are above 0.99, indicating the accuracy of the model predictions. Once the ANN models are determined, the important parameters including the normalized inputs and output, weights information and network information are extracted from the ANN models. With these parameters, users can predict the J-integral under any given pavement and loading conditions.

84 Figure 4.41. Structure of Artificial Neural Network Figure 4.42. Measured and Predicted J-Integral for Training, Validation, and Overall Datasets for Dual Tire Loadings with Dual Tire Length of 19 mm

85 Figure 4.43. Measured and Predicted J-Integral for Training, Validation, and Overall Datasets for Dual Tire Loadings with Dual Tire Length of 185 mm Figure 4.44. Measured and Predicted J-Integral for Training, Validation, and Overall Datasets for Dual Tire Loadings with Dual Tire Length of 229 mm

86 Figure 4.45. Measured and Predicted J-Integral for Training, Validation, and Overall Datasets for Single Tire Loadings with Single Tire Length of 64 mm Figure 4.46. Measured and Predicted J-Integral for Training, Validation, and Overall Datasets for Single Tire Loadings with Single Tire Length of 305 mm

87 Figure 4.47. Measured and Predicted J-Integral for Training, Validation, and Overall Datasets for Single Tire Loadings with Single Tire Length of 406 mm Prediction of Crack Growth under Thermal Loading On the basis of the existing models on pavement temperature, thermal stress, aging, and crack propagation, a top-down cracking propagation model due to thermal loading is developed. The model utilizes the climate data to capture field pavement temperatures, and uses readily available properties of asphalt mixtures and pavement design information in the LTPP database to determine crack growth characteristics. Theoretical Models for Top-Down Cracking under Thermal Loading Prediction of Temperature in an Asphalt Pavement The pavement temperature model developed by the research group from Texas A&M University (71) is adopted in this study to better predict temperature variations over time and depth in asphalt pavement layers. This pavement temperature model is a new one-dimensional model developed based on energy balance between radiation and conduction energy. A variety of sources of heat transfer at the pavement surfaces are required for implementing the pavement temperature model, including solar radiation, absorption of atmospheric downwelling long-wave radiation by the pavement surface, emission by long-wave radiation to the atmosphere, convective heat transfer that is enhanced by wind, leading to the following surface condition equation. 2 s s s a r c f TxpC Q Q Q Q Q Q t           (4.61) where ρC is the volumetric heat capacity of the pavement; Ts is the pavement surface temperature; x is the depth below the pavement surface; is the (differential) pavement thickness for the energy balance; Qs is the heat flux due to solar radiation; is the albedo of

88 pavement surface, the fraction of reflected solar radiation; Qa is the downwelling long-wave radiation heat flux from the atmosphere; Qr is the outgoing long-wave radiation heat flux from the pavement surface; Qc is the convective heat flux between the surface and the air; and Qf is the heat flux within the pavement at the pavement surface. Prediction of Viscoelastic Thermal Stress The thermal stress in the pavement structure is the result of temperature variation. The pavement temperature model described above contributes to the computation of thermal stress in the pavement structure. In order to determine the viscoelastic thermal stress, the dynamic modulus is used to determine the relaxation modulus and shift factor at specific time for computation of the thermal stress in asphalt pavement layers. The dynamic modulus for the selected pavement sections in each climate zone are collected from the LTPP database to develop the master curves for different temperatures. The dynamic modulus master curve is constructed using the sigmoidal function described as follows. 3 4 2 1 loglog ( ) 1 rr c c t cE t c e     (4.62) where E(tr) is the dynamic modulus; c1 is the minimum value of the logarithm of the dynamic modulus; (c1 +c2) is the maximum value of the logarithm of the dynamic modulus; c3 and c4 are the parameters describing the shape of the sigmoidal function; is the reduced time. In addition, the relaxation modulus data are determined by converting the dynamic modulus data. The Prony series form of a relaxation modulus is as in Equation 4.63. The coefficients of the Prony series are determined by fitting the constructed dynamic modulus master curve. The Equations 4.63 and 4.64 are used to accomplish this conversion between the relaxation modulus and the dynamic modulus.   1 i tN i i E t E e     (4.63)   2 2 * 2 2 2 2 1 11 1 N N j j j j j jj j E E E                            (4.64) where E(t) is the relaxation modulus at reduced time t; Ei, λi are the Prony series parameters for the relaxation modulus curve; E*(ω) is the dynamic modulus at the angular frequency ω; and Ei, λi are the Prony series parameters. The thermal stress in a pavement is computed according to the constitutive equation 4.65, which is the Boltzmann’s Superposition Principle for linear viscoelastic materials. The time- temperature dependent relaxation modulus of the asphalt mixture is required to compute the thermal stress. A finite difference solution has been developed (63, 155) by using the Prony series representation of E(ξ). This finite difference solution is presented in Equation 4.66.    ' '' 0 dE d d          (4.65)      / /1i iii i it e t t E e             (4.66)

89 where σ (ξ) is the thermal stress at reduced time ξ; E(ξ- ξ’) is the relaxation modulus at reduced time ξ- ξ’;  is the strain at reduced time ξ(=α(T(ξ’)-T)); α is the linear coefficient of thermal contraction; T(ξ’) is the pavement temperature at reduced time ξ’; T0 is the pavement temperature when σ=0; ξ’ is the variable of integration; Δε is the change in strain; and Δξ is the changes in reduced time; Prediction of Viscoelastic Thermal Cracking The viscoelastic thermal crack model is developed in this study based on the Paris’ law for crack propagation as follows:  nRc A J   (4.67) where Δc is the change in crack depth due a cooling cycle; is the pseudo J-integral; and A’, n’ are the fracture parameters for the asphalt mixture; Prediction of Thermal J-Integral Using ANN Model The pseudo J-integral is a significant input to predict crack growth. FEM is created to simulate the propagation of top-down cracking due to temperature variation by using the SA- CrackPro (149). Around 36,400 cases are computed in the FEM model at given inputs including thickness and modulus of asphalt concrete and base layers, coefficient of thermal expansion, corresponding crack depth and thermal stress and generating the output of thermal-J integral. Table 4.15 presents total inputs including pavement structure and conditions used in the FEM models. Based on the combination of all parameters inputs and generated outputs of the FEM models, the ANN model is developed using MATLAB. Table 4.15 Pavement Structure and Inputs in the FEM Layer Type Thickness (mm) CTE * Crack Length (mm) Thermal Stress (MPa) Modulus (MPa ) Asphalt 25, 75, 125,200,300, 500 0.00001 2.5,7.5,12.5, 17.5,20,22.5, 30,37.5,50, 52.5, 60,62.5,67.5, 87.5,90,100, 112.5,140,15 0,180,210,25 0, 270,350,450, 0-80 700,4000, 20000 Base 150,300,500 150,300,600, 7000,14000  * Coefficient of thermal expansion. The ANN network contains three layers and neurons in the hidden layers for determining weights and bias. There is no determined rule for selection of the number of hidden layers and types of regression equation used in the network and the number of neurons. For the purpose of short time computation, three layers are used in the ANN model including one input layer, one

90 hidden layer and one output layer. On the basis of the converged mean square error and computation time, 40 neurons are used in the hidden layer. The log-sigmoidal function is used to create the network. Once the network is configured, the network is trained in the progress of developing the ANN model and repeatedly updated weights and bias to reduce the calculation errors using the Levenberg-Marquardt convergence algorithm in the MATLAB program. The parameters containing the learning rate, maximum number of epochs to train and the performance goal are identical to those used in the ANN model for traffic loading. The network is validated successfully following the procedure of developing the ANN model and the required network structure. As shown in Figure 4.48, the R2 of the ANN model is above 0.99, showing that this ANN approach is accurate and satisfactory. The parameters contain inputs and outputs, network structure information and weights for the determined regression equation were determined from the ANN model to predict the thermal J-integral. The input provides the pavement structure and the performance information and is extracted from the coded MATLAB program into the Visual Studio program. The C# language is used as a calling method for the complete crack program to estimate the top-down crack propagation under thermal loading. Figure 4.48 Results of ANN Modeling for Measuring and Predicting J-Integral for Training and Validation.

91 Prediction of Aged Modulus of Asphalt Mixtures The reference modulus is used as one input of the asphalt concrete layer to the ANN model in Equation 4.68. This reference modulus is the representative elastic modulus (109). The dynamic modulus and relaxation modulus data are required to determine the reference temperature. Dynamic modulus | ∗| is based on the constructed master curve (where 1 Hz). The relaxation modulus E(tp) was developed as a simple power law model (where 1 s). The power law coefficients E, and m as in Equation 4.51 are determined by fitting the relaxation modulus data that is generated with the Prony series as in Equation 4.63. Furthermore, in the consideration of aging in the asphalt mixture, the modulus in the asphalt concrete layer will increase with time. The kinetics-based aging prediction of the asphalt mixtures was proposed above and used for the calculation of aged modulus. The aged modulus has been modeled using the following equations. * 1 1 2 2 p p re f t t E E E t              (4.68)   1 fk ti o i cE E E E e k t     (4.69) af a E RT f fk A e   (4.70) ac a E RT c ck A e   (4.71) where is the reference modulus (representative elastic modulus); | ∗| is the dynamic modulus at the frequency of f; E is the relaxation modulus at the time of /2; is the initial modulus; is the intercept of the constant-rate line of modulus; is the fast-rate aging activation energy for modulus; is the constant-rate aging activation energy for modulus; is the fast-rate reaction constant for modulus; is the fast-rate pre-exponential factor for modulus; is the constant-rate reaction constant for modulus; is the constant-rate pre-exponential factor for modulus; R is the universal gas constant value; and is the aging absolute temperature, K. Determination of Fracture Parameters by Pavement Performance-related Properties The Paris’ law parameters A and n depend on the performance-related material properties. The relaxation modulus, air voids content, asphalt binder content, aggregate gradation characteristics are commonly used for asphalt pavement design and related to the asphalt pavement performance. Thus, these performance-related material properties are used to determine the fracture parameters, A and n, for thermal loading. Assembly of Thermal Top-Down Computer Program A computer program is developed in Visual Studio software using the C# language to predict top-down crack growth in asphalt pavement layers under thermal loads. This computer program is divided into four subprograms, including pavement temperature, thermal stress, thermal pseudo J-integral and thermal crack growth. The inputs for the four subprograms are as follows.

92 1. Climate data at a specific pavement section. 2. Coefficients for relaxation modulus model and pavement temperature. 3. Pavement structure and condition, coefficients of thermal expansion, crack depth and thermal stress. 4. Thermal pseudo J-integral, material properties, and pavement performance. The method for computing of thermal-induced top-down cracking is schematically depicted in Figure 4.49 and elaborated as follows. Figure 4.49 Flow Chart of the Process of the Thermal Crack Growth Computations 1) Collect the required climate data from the LTPP and National Solar Radiation Database (NSRDB) database, including daily maximum and minimum air temperature, daily average temperature, daily wind speed, hourly solar radiation. 2) Interpolating the site-specific pavement temperature model parameters with collected data from National Climate Data Center (NCDC) or NSRDB database, including the albedo, emissivity, thermal diffusivity, absorption coefficient, and the pavement heat transfer parameters (a and d). 3) Calculate the hourly pavement temperature based on the pavement temperature model.

93 4) Obtain the material properties data from the LTPP database, including the dynamic modulus data, air void content, binder content, and aggregate gradation. 5) Calculate the reference modulus and relaxation modulus, which are related to the dynamic modulus collected from the LTPP database. 6) Calculate the thermal stress over time for each selected pavement section using the finite difference solution, Equation 4.66. 7) Calculate the J-integral using the developed ANN model for the given pavement structure and condition data. 8) Calculate the fracture parameters, A and n using performance-based material properties, as in Equations 4.33 and 4.36. 9) The crack depth at the crack initiation time is assumed to be 7.5mm. 10) The crack increment due to the temperature variation (1 cycle = 1 hour) is calculated using Paris’ law. Case studies for estimating the top-down cracking in asphalt pavement layers under thermal loading are elaborated below. Ten pavement sections from the LTPP database and four different climatic zones were taken as examples for this mechanistic-empirical prediction of thermally induced top-down cracking. Results of Pavement Temperature The pavement temperature model is implemented numerically by using a finite different approximation method. The model input data include climate data and site-specific pavement parameters. The climate data includes hourly solar radiation, hourly air temperature, and daily wind speed. The model parameters are albedo, emissivity, thermal diffusivity, absorption coefficient, and heat transfer coefficient parameters (a and d). Hourly solar radiation is obtained from the NSRDB. Daily average wind speed can be obtained from the LTPP database and NCDC database or the other meteorological network in each state. Even though the accurate site- specific daily average wind speed is preferred, the interpolation between close locations is acceptable if site-specific daily average wind speed data cannot be obtained in the meteorological database. That is because the pavement temperature model is not sensitive to the wind speed and the daily average values are adequate. Hourly air temperature data can be collected from NCDC database. In order to determine an accurate estimate of pavement temperature based on the temperature mode, numerical values need to be interpolated from the collected site-specific pavement parameters from the NCDC or NSRDB database, including albedo, emissivity, thermal diffusivity, absorption coefficient, and model parameters (a and d). Moreover, the climatic data of the selected pavement sections have been collected from the online database within a period of 26 years from 1984 to 2010. Based on the collected climatic data and site-specific thermal coefficients, the pavement temperature of each of four sections is determined by using the pavement temperature model as implemented in the C# programming language. The results of the calculated pavement temperature over a one-year period are shown in Figure 4.50. The figure shows that pavement sections suffered an extremely severe winter when the lowest pavement temperature is lower than −20 . It illustrates that low temperature can be a contributing cause of top-down cracking.

94 Figure 4.50. Pavement Temperature versus Time (29-1005, 27-1003, 19-102, and 2-1002 are 4 LTPP pavement sections as listed in Table 4.16) Results of Thermal Stress The thermal stress in asphalt pavement layers is computed by Equation 4.66. A program is developed in Visual Studio software using the C# computer language to calculate the hourly thermal stress based on computed hourly pavement temperature, shift factors and Prony series coefficients. The computed hourly longitudinal thermal stress over a 4-year period is shown in the Figure 4.51. It can be observed from the Figure 4.51 that the hourly thermal stress exhibits a high value in the winter and a minor value in the summer. The transverse thermal stress which contributes to top-down cracking is always smaller than the longitudinal thermal stress because the pavement width can contract and relieve some of the transverse stress. Consequently, the transverse thermal stress which contributes to top-down cracks is never larger than 10 psi in the developed Program in Visual Studio to predict the thermal J-integral.

95 Figure 4.51. Longitudinal Thermal Stress versus Time Results of Thermal-Related Top-Down Cracking Prior to computing the thermally related top-down crack growth with time, the fracture parameters are determined using the given material properties. Table 4.16 provides the selected section information, pavement structure and corresponding material properties, including the thickness of each layer, air void content, asphalt content, and aggregate gradation parameter obtained from the LTPP database. The aggregate gradation is used to model the curve of the cumulative percent passing versus the sieve size using an appropriate power law function. The aggregate gradation shape parameter ψ as in Equation 4.36 is presented in Table 4.16. Table 4.16 Section Information and Material Properties. Pavement Section AC Thickness (in) Base Thickness (in) Air Void (%) Asphalt Content (%) AGG Parameter (ψ) Climate Zone 6-0502 8.4 4.8 4.96 4.5 0.46 DNF 32-1030 10.3 1.8 7.07 3.8 0.49 2-1002 3.1 12 5.22 4.9 0.48 DF 8-6013 4.2 24.1 5.78 5.8 0.57 2-1002 5.4 13.5 5.22 4.9 0.48 WF 19-102 7.7 8 5.53 5.2 0.6 27-1003 9.5 0 4.3 5.5 0.5 29-1005 8.9 3.9 1.74 4.2 0.5 1-4129 4.5 5.6 9.28 4.1 0.47 WNF 22-3056 10.2 7.9 7.22 3.5 0.42

96 The other two parameters used to compute the fracture parameters are E1 and m in the power law function for the relaxation modulus. The fracture parameters A and n are calculated using the regression functions in Equations 4.33 and 4.36, as presented in Table 4.17. The longitudinal thermal crack growth of the selected pavement sections listed in Table 4.17 are calculated, the program results show that only 4 of the 10 sections suffered top-down longitudinal cracks under thermal loading. These are illustrated in Figure 4.53. Table 4.17 Fracture Parameters Pavement Section E1 m Ere A n Years (crack depth>15 mm ) Climate Zone 6-0502 4513.62 0.29 6986.57 7.19E-17 10.05 - DNF 32-1030 367.64 0.24 862.60 2.53E-05 0.79 - 2-1002 845.03 0.15 1774.68 9.07E-12 5.96 - DF 8-6013 2843.07 0.65 5379.58 2.41E-15 8.83 - 2-1002 845.03 0.15 1774.68 9.07E-12 5.96 17.50 WF 19-102 1486.60 0.27 2735.39 4.96E-16 9.38 9.50 27-1003 556.73 0.32 1454.71 5.29E-12 6.15 11.50 29-1005 5737.84 0.43 8756.10 6.92E-15 8.46 6.50 1-4129 1536.04 0.26 2182.76 1.75E-13 7.34 - WNF 22-3056 3751.30 0.30 5880.65 3.01E-14 7.95 - The thermal J-integral was predicted by using an ANN model with pavement structure inputs and coefficient of thermal expansion. One of inputs to this ANN model is modulus of the asphalt concrete pavement layers including the effect of aging in the modulus determination. The aged modulus with aging time is illustrated in the Figure 4.52. The curve shape is similar to the shape of the pavement temperature with time, demonstrating that the modulus has a similar variation as does the pavement temperatures.

97 Figure 4.52. Aged Modulus versus Time After obtaining the pavement temperature and thermal stress, the J-integral is determined using the ANN model. The Paris’ law is used to characterize the crack growth. The initial crack length is set at 7.5 mm (6). Thus, the result of crack depth over time is shown in the Figure 4.53 for the selected pavement sections. It is observed that under the influence of temperature variations the crack in the ten selected pavement sections needs approximate 5-18 years to propagate from surface to the bottom of the asphalt pavement layers for the selected pavement sections. Beside influence of temperature variation, the crack growth rate strongly depends on the fracture coefficients, material properties, and pavement structure and conditions. Figure 4.53 Crack Depth versus Time

98 Computation of Top-Down Cracking Calibration Coefficients In the top-down crack propagation calculation, there are two numbers of days calculated including the number of days for traffic load solely and number of days for thermal load solely to reach a standard depth within an asphalt pavement layer. The standard depth is 15 mm, which is the boundary line between low and medium severity. The distress shape parameter  and distress scale parameter  in Equation (4.50) are calculated based on the distress data points, which are used for the calibration purpose. The values of  and  are documented in Appendix H. The two number of days are combined to model the values of  and  , as shown in Equations 4.72 and 4.73. 0 1 TRi i TRi THi NN N           (4.72) 0 1 1 TRi i TRi THi N N N           (4.73) where TRiN and THiN are numbers of days for traffic and thermal load to reach the standard depth, respectively; 0 , 1 , 0 and 1 are calibration coefficients. However, as shown in Table 4.17, only four pavement sections in the wet-freeze zone are determined to have the limited fatigue lives under thermal loading due to the boundary condition of transverse thermal stress. Therefore, two independent equations are used to predict the values of  and  for the rest pavement sections: 1 2i TRiN    (4.74)   21 logi TRiN   (4.75) where 1 , 2 , 1 and 2 are calibration coefficients. Table 4.18 presents all of the calibration coefficients for different climatic zones. Note that due to the limited number of independent LTPP pavement sections that have longitudinal wheelpath cracking in the two freeze zones, the pavement sections in wet-freeze zone and dry- freeze zones are combined with wet non-freeze zone and dry non-freeze zone, respectively. The detailed results of the calibration coefficients for climatic zones are presented in Chapter 5.

99 Table 4.18 Summary of Calibration Coefficients for Four Climatic Zones Climatic Zone 1 2 1 2 Wet 631. 04 2269.8 0.4201 -1.097 Dry 1617.6 -1705.3 0.4201 -1.097

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A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers Get This Book
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TRB's National Cooperative Highway Research Program (NCHRP) Web-Only Document 257: A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers develops a calibrated mechanistic-empirical (ME) model for predicting the load-related top-down cracking in the asphalt layer of flexible pavements. Recent studies have determined that some load-related fatigue cracks in asphalt pavement layers can be initiated at the pavement surface and propagate downward through the asphalt layer. However, this form of distress cannot entirely be explained by fatigue mechanisms used to explain cracking that initiates at the bottom of the pavement. This research explores top-down cracking to develop a calibrated, validated mechanistic-empirical model for incorporation into pavement design procedures.

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