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176 aPPENDIX E Review of Current Permanent Deformation Models Constitutive relationships often need to be developed to properly describe permanent deformation accumulation in unbound granular materials with number of load applica- tions. In this section, a summary is given of the different models proposed by many researchers to predict permanent strain as a function of load and material property related factors. Barksdale Model Barksdale (1972) analyzed standard repeated load triaxial data to propose a linear relationship between permanent axial strain and the logarithm of number of load applications as shown below: ( )Îµ = + log (E-1)a b Np Where ep is the axial permanent strain; N is the number of load applications; a and b are model parameter estimates from linear regression of laboratory experimental data. Phenomenological Model Monismith et al. (1975) proposed a log-log relationship between axial permanent strain and the number of load applications as shown in Equation (E-2). This model, also known as the phenomenological model, is used widely to present permanent deformation test results from laboratory experiments. Îµ = (E-2)ANp b where the definitions of ep, N, A (or a) and b are the same as given above. Note that researchers (Monismith et al. 1975; Maree 1978) have proposed a value less than unity (1.0) for the regression parameter âbâ for stress conditions significantly below the shear strength of the material [30, 35]. However, a value of âbâ that is less than unity (1.0) would imply a permanent deformation accumulation rate of infinity (â) for the first load application (N = 1), and zero for large values of N. This also implies that the âAâ parameter represents an asymptote for the accumulated permanent deformation for large values of N. Note that asymptotic permanent deformation response is typical of unbound aggregate behavior in the âplastic shake- downâ range (Werkmeister 2003). Therefore, the phenomeno- logical model can predict material behavior accurately only for stress levels below the plastic shakedown limit. Thompson and Nauman (1993) observed that the âAâ term in the phenomenological model was significantly affected by stress states (âAâ values typically increased with increasing stress levels), whereas the âbâ parameter varied in the range between 0.12 and 0.20 for different granular material types. Strain Rate Model El-Mitiny (1980) and Khedr (1985) proposed the strain rate model, which was related to the phenomenological model. The strain rate model inversely correlates the rate of permanent axial strain to the logarithm of the number of load repetitions as follows: Îµ = â (E-3)N aN p b where the definitions of ep, N, A (or a) and b are the same as given above. Tseng and Lytton Model Tseng and Lytton (1989) presented a three-parameter per- manent deformation model to predict the accumulation of permanent deformation through material testing. The parameters were developed from the laboratory established relationship between permanent strains and the number of load applications. The curve relationship is expressed as follows: Îµ = Îµ ( )â Ï Î² (E-4)0ea N Where ea is the axial permanent strain; N is the number of load applications, e0, b, and r are material parameters that are dif- ferent for each sample, and are determined based on the water content, resilient modulus, and stress states for base aggregate and subgrade soils through multiple regression analyses. Wolff Model Wolff (1992) developed the following model to predict per- manent strain accumulation in aggregate base and subbase layers from Heavy Vehicle Simulator (HVS) test data. ( )( )Îµ = + â â1 (E-5)mN a ep bN
177 where ep is the axial permanent strain; N is the number of load application; and a, b, and m are model parameters. The primary feature of Wolffâs model is that it accounts for the initial rapid increase in permanent deformation followed by a linear phase in which the permanent deformation increases at a steady rate. Upon differentiating the above expression to study the rate of accumulation of permanent strain ( )âÎµ âNp , one can see that the incremental permanent deformation is equal to (a Ã b) for N = 0, and approaches âmâ as Nââ. Rutting Rate Model Thompson and Nauman (1993) proposed a practical applica- tion of the above model. They used rut depths obtained from field measurements instead of the permanent axial strain term as follows: = = (E-6)RR RDN aN b where RR = Rutting rate; RD = Rut depth; N = Number of load applications; a, b = Model Parameters. Thompson and Nauman (1993) successfully applied their rutting rate model to prediction of the AASHO Road Test section rutting performances. Van Niekerk and Huurman Model Note that none of the above discussed models accounted for the effects of stress states on the accumulation of per- manent deformation in unbound aggregates. Accordingly, van Niekerk and Huurman (1995) proposed the following relationship between plastic strain and the number of load repetitions for unbound granular materials: ( )Îµ = ÏÏï£«ï£ï£¬ ï£¶ï£¸ï£· ÏÏï£«ï£ï£¬ ï£¶ï£¸ï£·1000 (E-7)1 11, ,2 1 1 1 2 a N p f a b f b where ep is the permanent or plastic strain; N is the number of load applications; s1 is the major principal stress, s1,f is the major principal stress at failure; and a1, a2, b1, and b2 are model parameter estimates. This model accounts for the effect stress state on permanent deformation by incorporat- ing the ratio between applied deviator stress, and the deviator stress at failure for a triaxial specimen. Note that as the ratio ( )Ï Ï1, f is kept constant for a particular test, the above equa- tion is essentially the same as the phenomenological model is Equation (E-2). Paute Model Paute et al. (1996) suggested the following relationship between the number of load applications and the accumula- tion of permanent deformation after 100 cycles, considering the maximum permanent axial strain possible, depicted as âaâ in the model: ( )Îµ = âï£«ï£ï£¬ ï£¶ï£¸ï£·â1 100 (E-8)a Np b The model parameter definitions are the same as above. Note that Pauteâs model excluded the rapid rate part of permanent deformation accumulation between the 1st and 100th cycle. This is in accordance with the difficulty of predicting the permanent deformation development within the first 100 cycles which often corresponds to the rapid reorientation of individual particles in the aggregate matrix. Huurman Model Huurman (1997) combined stress level and number of load applications into one expression to predict the accumulation of permanent deformation in unbound granular materials. Equation shows the model proposed by Huurman. Îµ p B A N C D N= ï£«ï£ï£¬ ï£¶ï£¸ï£· + ï£«ï£ï£¬ ï£¶ï£¸ï£· âï£«ï£ï£¬ ï£¶ ï£¸ï£·1000 1000 1exp (E-9) Where the parameters A, B, C, and D account for the stress dependency of permanent strains as shown below: = Ï Ï ï£«ï£ï£¬ ï£¶ï£¸ï£· (E-10)1 11, 2 X x f x where X is a variable representing each parameter A, B, C, or D in Equation E-9; x1 and x2 are variables representing related coefficients a1, a2, b1, b2, c1, c2, d1, and d2, respectively. Ullidtz Model Ullidtz (1997) proposed a stress related permanent model which expresses the accumulated permanent strain in terms of the applied deviator stress and the number of load applica- tions for a triaxial specimen. Equation (E-11) shows the model proposed by Ullidtz. Îµ = Ïï£«ï£ï£¬ ï£¶ï£¸ï£· (E-11)0a p Np d b c where sd is the axial deviator stress, p0 is the normalizing reference stress (often p0 = 1 psi or 1 kPa), a, b, and c are model parameter estimates obtained from regression analy- ses of experimental data.
178 Lekarp and Dawson Model Lekarp and Dawson (1998) used the shakedown concept to investigate the effect of stress state on permanent deforma- tion development, and proposed the following model relat- ing the permanent strain accumulation to the maximum shear stress ratio and the length of the stress path: Îµ p bN L p a q p ref E- ( ) ï£«ï£ ï£¶ï£¸ = ï£« ï£ï£¬ ï£¶ ï£¸ï£· 0 12 max ( ) where: ep (Nref) is the permanent axial strain at a given reference number of cycles Nref, where Nref > 100; L is the length of stress path, p is the mean normal stress ( )( )= Ï + Ï + Ï 31 2 3p ; q is the deviatoric stress (q = s1 - s3); (q/p)max is the maximum stress ratio; p0 is the normalizing reference stress; N is the number of load applications; a and b are model parameter estimates. Although this model included several load related variables, it did not consider the stress path-direction and loading slope which can influence the permanent deformation accumulation. REFERENCES Barksdale, R.D., âLaboratory Evaluation of Rutting in Base Course Materials,â Proceedings of the 3rd International Conference on Asphalt Pavements, University of Michi- gan, Ann Arbor, 1972, pp. 161â174. El-Mitiny, M.R., Material Characterization for Studying Flexible Pavement Behavior in Fatigue and Permanent Deformation, PhD Thesis, Ohio State University, Colum- bus, 1980. Huurman, M., Permanent Deformation in Concrete Block Pavements, Ph.D. Thesis, Delft University of Technology, Delft, the Netherlands, 1997. Khedr, S., âDeformation Characteristics of Granular Base Course in Flexible Pavement,â Transportation Research Record 1043, Transportation Research Board, National Research Council, Washington, D.C., 1985, pp. 131â138. Lekarp, F. and A. Dawson, âModeling Permanent Deforma- tion Behavior of Unbound Granular Materials,â Construc- tion and Building Materials, Vol. 12, No. 1, 1998, pp. 9â18. Monismith, C.L., N. Ogawa, and C.R. Freeme, âPermanent Deformation Characteristics of Subgrade Soils Due to Repeated Loading,â Transportation Research Record 537, Transportation Research Board, National Research Council, Washington, D.C., 1975, pp. 1â17. Maree, J.H., Design Parameters for Crushed Stone in Pave- ments, MS Thesis, University of Pretoria, South Africa, 1978. Paute, J.L., P. Hornych, and J.P. Benaben, âRepeated Load Triaxial Testing of Granular Materials in the French Net- work of Laboratoires des Ponts et ChaussÃ©es,â Flexible Pavements, Proceedings of the European Symposium Euroflex 1993, A.G. Correia, Ed., Balkema, Rotterdam, the Netherlands, 1996, pp. 53â64. Thompson, M.R. and D. Nauman, âRutting Rate Analyses of the AASHO Road Test Flexible Pavements,â Transpor- tation Research Record 1384, Transportation Research Board, National Research Council, Washington, D.C., 1993, pp. 36â48. Tseng, K.-H. and R.L. Lytton, âPrediction of Permanent Deformation In Flexible Pavement Materials,â Impli- cation of Aggregates In the Design, Construction, and Performance of Flexible Pavements, ASTM STP 1016, ASTM, Philadelphia, Pa., 1989, pp. 154â172. Ullidtz, P., âModeling of Granular Materials Using the Dis- crete Element Method,â Proceedings of the 8th Interna- tional Conference on Asphalt Pavements, Seattle, Wash., 1997, pp. 757â769. Van Niekerk, A.A. and M. Huurman, Establishing Complex Behavior of Unbound Road Building Materials from Simple Material Testing, Technical Report No. 7-95-200-16, Delft University of Technology, Delft, the Netherlands, 1995. Werkmeister, S., Permanent Deformation Behaviour of Unbound Granular Materials in Pavement Construc- tions, PhD Thesis, Dresden University of Technology, Dresden, Germany, 2003. Wolff, H., Elasto-Plastic Behavior of Granular Pavement Layers in South Africa, Ph.D. Thesis, University of Preto- ria, Pretoria, Republic of South Africa, 1992.