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Defonnation arid Fracture of Rock JOSEPH B. WALSH Rock is not elastic In the usual sense of the word. Cracks, which are found in nearly all rocks, close under compressive stresses and make the sample stiffer. One side of a crack slips over the opposite side under differential stress, and energy is lost to friction. As a consequence, stress-stra~n curves are non- linear, and hysteresis occurs in complete loadin~unloading cycles. The tensile strength of rock is much less than the compressive strength. Strength is not affected appreciably by moderate changes In temperature or by the rate at which the load is applied. The cracks and pores in rock form a continuous, interconnected network, and rock under natural conditions is usually saturated with fluid. Him pressure or chemical activity of these pore fluids decreases strength. The strength of rock increases dramatically as confining pressure increases. The deformation of most materials used in construction is described for each by a stress-strain curve showing how strain (deformation per unit length) varies with stress tIoad per unit area). The stres~strain curve is approximately linear with most solid materials for small changes in stress, and the well-known Hooke's law applies. Materials for which Joseph B. Walsh is Senior Research Scientist, Department of Earth and Planetary Sci- ences, Massachusetts Institute of Technology, Cam budge. This work was supported by the U.S. Geological Survey under Contract- No. 14-08- 0001-18212 and by the Army Research Office, Durham, under Contract No. D^AG29- 79-C-0032. 87

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88 CONSERVATION OF HISTORIC STONE BUILDINGS stress is proportional to strain until fracture occurs are called linearly elastic; the stres~strain curve for glass, which is such a matenal, is shown in Figure 1. Elastic matenals have the characteristic that strain is a single-valued function of stress; Figure 1 shows that the strain at any specific stress when the Toad is being increased is the same as the strain when the load is being decreased. Some matenals Rubber, for example) are descnbed as nonlinearly elastic; in such cases, strain is a single-valued function of stress, but the stres~strain curve is not a straight line. Elastic / cry in oh 111 en . _, / ~ Elastic L UP Plastic ~ f STRAIN FIGURE 1 Deformation can be divided into elastic and plastic. The stress-strain curve for an elastic material like glass is the same during the loading and unloading portions of a complete cycle. Duc- tile materials like many metals may be elastic until a yield stress, oryx is reached. Plastic flow occurs at higher stresses, and the com- plete curve describes a hysteresis loop with permanent strain sp.

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Deformation and Fracture of Rock 89 The deformation of a large class of materials Most metals, for ex- ample) is elastic for stresses less than the yield stress, and further increases in load result in plastic deformation. In plastic deformation the movement of dislocations allows whole planes of atoms to translate relative to the adjacent plane, with the consequence that relatively large strains occur for small changes in stress. The atomic planes do not translate in the opposite sense when the Toad is lowered, and so the stress-strain curve in the loading cycle is not the same as that during the unloading cycle. A stress-strain curve for a material de- fo~'ed into the plastic region is shown in Figure 1; note that the specimen has undergone a permanent change in length. The type of deformation that a sample undergoes depends on the type of stresses applied, as well as on their magnitudes. The stresses acting on a body can be considered to be the sum of two stress sys- tems the hydrostatic component and the shear or deviatoric com- ponent. Most materials, even ductile materials, are found to behave elastically when only hydrostatic stresses are applied. Elastic materials are those that behave elastically under both hydrostatic and deviatoric stresses, whereas ductile materials undergo plastic deformation at de- viatoric stresses greater than the yield stress. When the load is increased to a sufficiently high level, a specimen eventually fractures. Fracture also can be divided into subcategories that include most engineering materials. "Brittle" failure is said to have occurred when the fractured pieces can be fitted back together that is, when the sample has undergone a negligible arno~,nt of per- manent deformation before and during the fracture process. Glass, of course, is a common example of a brittle material. A fracture is called "ductile" when the sample undergoes an appreciable amount of plastic flow before fracture. A relatively large amount of energy is absorbed by materials that fail in this way, and so such materials are used where toughness is important. The type of fracture that occurs does not depend completely on the material involved. The configuration of the specimen is important: The presence of cracks and sharp notches tends to inhibit ductility and, in extreme cases, ductile material can fail by brittle fracture. The loading rate Once the temperature can also affect the type of fracture that occurs, with high loading rates and low temperatures enhancing the likelihood of brittle failure. This brief discussion of deformation and fracture is intended to pro- vide only a framework for describing the behavior of rock. Deformation and fracture are well-developed scientific fields, and the paragraphs

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90 CONSERVATION OF HISTORIC STONE BUILDINGS here do not provide even an elementary introduction to these complex processes. A comprehensive introduction to the subject can be found in Mechanica] Behavior of Materialist PROPERTIES OF ROCK The pressures and temperatures of interest in building construction are at the low end of the broad range of conditions that has been studied in geology. Even in this restricted range, rock is found to have unusual properties. The deformation cannot be characterized as either elastic or plastic, and fracture involves elements of both brittle and ductile behavior. Most of the minerals that make up rock are elastic, hard, and brittle. Rock-forming minerals are hard and brittle because of the low symmetry of their crystallographic structures. Dislocations cannot move easily in these complicated structures, so plastic flow is almost completely inhibited. One common mineral, calcite, of which marble and limestone are composed, is exceptional in that it can be deformed plastically along one crystallographic plane. However, calcite crystals of most orientations cleave when stressed, and the overall behavior must be classified as brittle. The question, then, is how can rocks that are composed of elastic, brittle crystals exhibit anything but elastic, brittle behavior? DEFORMATION OF ROCK The measurement of -the compressibility of a sample of rock is a stan- dard test that-provides basic information about the rock's internal structure. Typically, a specimen is sheathed with an impermeable jacket and put in a pressure vessel. Strain gauges on the jacket, or various other devices, are used to measure the decrease, TV, in the volume, V, of the sample as the pressure, p, in the vessel is increased. A curve of volumetric strain t^V/V) as a function of pressure is de- veloped from these data, and compressibility, A, is calculated from the inverse slope (^V/Vp) of the curve. An example of a volumetric strain-pressure curve for a granite from Westerly, R.I., is shown in Figure 2. Note that the curve is nonlinear. At low pressure, the slope is relatively low; the slope increases with increasing pressure until at a pressure of approximately 2 kbar (about 30,000 psi) the curve becomes linear. Note also that the curve relating strain and pressure when pressure is decreased is virtually the same as the curve when pressure is increased. Hysteresis is negligible, and behavior is nonlinearly elastic.

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Deformation and Fracture of Rock 6 5 4 Q CC an ~ 3 UJ cr cot / 2 1 _ / O ,/ 1 1 1 1 1 O -1 -2 -3 ~ -5 LINEAR STRAIN, 10-3 FIGURE 2 Volumetnc strain of a sample of Westerly granite as a function of pressure. Note that this rock is nonlinearly elastic under hydrostatic pressured 91 The generally accepted explanation for this behavior, which is ob- served for virtually all types of rock, is that rock contains cracks. At low pressure the cracks are open, and the rock is compliant. Cracks close as the pressure is~increased, and the sample becomes stiffer. At sufficiently high pressures, typically 2 kbar, all cracks are closed, and the sample behaves like its constituent minerals, which are linearly elastic. The cracks found in rock arise from several natural causes. The cooling of a rock after it has solidified creates thermal stresses that are sufficient to fracture the crystalline framework. Likewise, micro- cracking can result from the decrease in stress that a sample experi- ences when it moves from its origin at depth to the surface or from the increases in stress that occur during tectonic deformation.

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92 CONSERVATION OF HISTORIC STONE BUILDINGS The exsolution of gases dunng solidification or the imperfect sin- tenng of grains, for example, also increases porosity. Pores are cavities that are not cracklike (i.e., one dimension is not much smaller than the others) and their effect on deformation is entirely different. Figure 3a shows a rock sample, containing both cracks and pores, under low pressure. When sufficiently high pressure is applied, as in Figure 3b, the cracks close, but the pores do not. Therefore, the nonlinearity in the pressure-strain curve is caused by the cracks, not by the pores. Effects of Cracks and Pores Some progress has been made in evaluating quantitatively the relative effects of cracks and pores on the elastic properties of rock (for a recent review, see Walsh).3 Using techniques in the theory of elasticity, we find that cracks have a very much greater effect on compliance than do pores of equivalent volume. For example, spherical pores that con- stitute a few percent of the volume of a sample cause an increase in compressibility of a few percent. A crack is found theoretically to have (a) O.' D .. a..... 1~'../ ~ .. . . . .- olN~ o O ~ V -_ Cracks AT ZERO PRESSURE Grain Boundaries Pores (b) . . a.. to -../ ::/ . . . . - \ / \ \ \ UNDER CONFINING PRESSURE FIGURE 3 A schematic description of the effect of confining pressure on cracks and pores. Note that cracks close under pressure, whereas pores do not.

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Deformation and Fracture of Rock 4.2 Q u, 0.8 u' us A: - tn in UJ ~ 0.4 o cow o +1.0 ~ 1 1 // //11 / o Ll NEAR STRAI N. 10-3 -1 .0 -2.0 FIGURE 4 Longitudinal and lateral strain of Westerly granite under uniaxial compressions Note that the stres~strain curves are non- linear and nonelastic. 93 nearly the same effect on compressibility as a pore of the same di- ameter. The porosity associated with such a crack is negligibly small compared with the pore, of course. A rule of thumb is that the effect of cracks on elastic properties depends on the total surface area of the crack phase bind is independent of its volume), whereas the effect of pores is in direct proportion to the volume of the pore phase. The compressibility of the Westerly granite sample in Figure 2, for example, is 8.3 mbar- ~ at atmospheric pressure and 1.87 mbar- ~ at high pressure. Cracks in this rock, which account for a porosity of only 0.5 percent, have increased compressibility by a factor of nearly 6. The effect of the pores in the sample, which account for a porosity of approximately 1 percent, is negligible. Cracks also have a pronounced effect on the deformation of rock when stresses other than hydrostatic pressure are applied. Figure 4 shows the longitudinal and lateral strain caused by increasing the uniaxial compressive load on a cylindrical sample of Westerly granite. Compare the longitudinal strain-stress curve in Figure 4 with the volumetric strain-pressure curve in Figure 2. The slope of both curves

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94 CONSERVATION OF HISTORIC STONE BUILDINGS 1 _. ~ J , ~ FIGURE 5 A schematic description of the effect of sliding of crack faces against friction. Note that slip does not occur immediately as the stress is lowered. A hysteresis loop is formed, and the system undergoes permanent displacement. increases with increasing stress. The reason for this is the same in both cases: Increasing compressive stress or pressure causes cracks to close, and the specimen becomes stiffer. Eventually all, or nearly all, cracks close, and strain is a linear function of stress. However, note in Figure 4 that the curve when the applied stress is being decreased is not the same as when the stress is being increased, in contrast to the response to hydrostatic-pressure changes in Figure 2. This behavior is typical of most rocksthat is, rocks are nonlinearly elastic under hydrostatic-pressure changes and are nonelastic under changes in deviatoric stress. Cracks are responsible for the nonelasticity as well as for the non- linearity of rocks. Cracks under changes in hydrostatic pressure merely open and close. However, one side of a suitably oriented crack can slide relative to the other side when the applied stresses contain a

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Deformation and Fracture of Rock 95 deviatoric. component. This relative motion means that the longitu- dinal strain of the sample is somewhat greater than the strain from compression of the solid matrix; as a consequence, yo~,ng's modulus (the slope clcr/~) of the ascending portion of the curve in Figure 4 is less then Young's modulus for the mineral components. The process is Illustrated schematically in Figure 5. Slip between the faces of a microcrack in rock is physically similar to sliding a block against friction on a plane. The spring in Figure 5 represents the elastic element in the deformation owing to the mineral grains. Note that the block does not irnrnediately begin to move in the opposite sense when the load is decreased, much as a door forced against a wedge must be yanked in the opposite direction to dislodge it. Consider the situation at the highest stresses in Figure 5, where all cracks are closed and those cracks that can slide are sliding. Using the analogy of a block sliding on a plane, we see that all these cracks will be jarnrned when the Toad initially is Towered;. The inference is that the initial slope of the unloading curve in Figure 4 represents the response of only the mineral components. The response of the minerals in the rock under conditions in which cracks do not affect behavior can be measured in other ways. We find that, indeed, the initial slope of the unloading curve in Figure 4 represents an adequate approximation of the behavior of intact rock: Young's modulus for intact rock is 730 kbar, and the value from Figure 4 is 710 kbar. Effects of Water and Temperature The deformation of rock is influenced by factors in addition to stress. The porosity in most rocks forms a phase that is largely interconnected. As a result, rock in a wet environment becomes permeated with water. The response to uniaxial stress of two sandstone samples, one of them saturated with water and the other one dry, is shown in Figure 6. The wet sample is more compliant than the dry one. The analogy of a block sliding on a plain offers a plausible explanation for this behavior: Water lubricates the microcracks, slip on cracks is facilitated, and conse- quently compliance is increased. Relatively small changes in temperature do not affect the defor- mational characteristics of rock appreciably as long as the temperature variations do not change the degree of fluid saturation. The elastic properties of the constituent minerals do not vary with temperature to an appreciable extent until the temperature reaches a level of ap- proximately half the melting temperature (in keIvins); these temper- atures are well above those that most building stone is subjected to

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96 FIGURE 6 Stres~strain curves for dry and wet sandstone sam- ples. The wet sandstone is more compliant than the dry.4 CONSERVATION OF HISTORIC STONE BUILDINGS 500 400 N ~ 300 - cn LL cr In 200 100 o 0 1 2 f 1 f f Dry Wet 5 3 4 - STRAIN (x10-~) under normal circumstances. Temperature changes of 20 C to 30 C, however, are sufficient to cause measurable thermal cracking in com- petent granites,4 thereby changing their compliance. Presumably, how- ever, most building materials aIrea`dy will have been subjected to sea- sonal changes of that magnitude during their history. Although excursions in temperature beyond the range that the sample has ex-

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Deformation and Fracture of Rock 97 perienced may cause cracking, repeated cycling to some given level does not appear to have any cumulative effect. FRACTURE OF RO CK A specimen eventually fractures when the stress acting on it is raised to sufficiently high levels. The stress at which fracture occurs depends not only on the type of rock and its previous history but also on factors such as the types of stresses that are applied, the degree of saturation and pressure of the pore fluid, and the rate at which the load is applied. Temperature change over the range experienced in building construc- tion is not a factor of major importance in the discussion here. Fracture research has benefited greatly in recent years from the de- velopment of the scanning electron microscope. Photographs of the interior of two samples of Westerly granite that were taken on a scan- ning electron microscope are shown in Figure 7. Figure 7a shows cracks that were produced by the applied stress at a level before the specimen was completely fractured. The photomicrograph in Figure 7b illustrates the damage produced by the applied stress approximately at the instant of fracture. Note that fracture at the microscopic scale is a brittle process. The crack surface in Figure 7a could be rejoined with virtually no gaps remaining; the microshards in Figure 7b are angular, with none of the rounded or stretched features typical of ductile fracture, ant] presumably the tiny blocks could be rejoined, with sufficient patience, to re-create the original state. The cracks in these photographs have fractured individual mineral crystals to produce the fracture pattern that we see. The fracture strength of the rock, one might suppose, must be directly related to the strengths of the minerals involved. The strengths of single crystals have been investigated theoretically using several different approaches. In aD cases the fracture strength, (a, is given approximately by the relationship: ~ _ cat, O cry ~ E/10, (1) where E is Young's modulus. Young's modulus for quartz, for example, to an order of magnitude, is 106 tears, 7 so the theoretical strength is 105 bars. The fracture strength of a quartz crystal is less than 104 bars in compression and less than 103 bars in tension, and the values for quartzite {polycrystalline quartz rock) are lower by a factor of 10.8 Why is the measured strength so low? Griffith was the first to explain the discrepancy.9 i0 He reasoned that no material was perfect and that flaws in the form of tiny cracks could

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98 CONSERVATION OF HISTORIC STONE BUILDINGS FIGURE 7a A photornicrograph of Westerly granite at a stress well below the fracture. strengths Note that cracks have begun to extend and that the. fracture is- brittle.

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Deformation and Fracture of Rock 50~ FIGURE 7b A photomicrograph of Westerly granite on the verge of fractured Note that fracture on the microscale is brittle, although the stres~strain curve has the characteristics of a ductile material. 99

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100 CONSERVATION OF HISTORIC STONE BUILDINGS be found in solid samples of any size. Using a theoretical analysis of the stresses around cracks, he showed that the maximum stress, which is found at the tip of a crack, is very much greater than the applied stress. The stress, cry, required to break a specimen is therefore very much less than the theoretical strength, cry, given by equation 1, needed to extend the crack. Griffith's analysis showed that, in fact, the actual strength should be Tower for materials having longer cracks. In an elegant experiment using samples of smaller and smaller size, he ex- trapolated to a sample of zero size (and zero crack length) and found the theoretical strength to be in agreement with the value given by equation 1. Griffith's first analysis was restricted to fracture under tensile load- ing, but he generalized the results in his second analysis to loading under compressive states of stress. Griffith's theory showed that the strength of brittle materials like rock in compression should be about 10 times the tensile strength. Further, his theory showed that the compressive strength should rise dramatically when confining pressure is superposed. These theoretical predictions have been verified, at least in a general way, for rock and masonry materials by subsequent ex- perimentation. - Griffith's theory, although it describes in broad outline how the strength of rock depends on the types of stresses applied, does not accurately describe the fracture process itself.6 ~2 Griffith visualized that fracture in compression occurs as it does in tension: Cracks are inactive until- the fracture stress is reached, when the largest crack grows across the specimen, separating it into two pieces. The stress- strain curve for a rock sample in Figure 8 shows that this mode! cannot be correct. Figure 8 shows that the slope of the stress-strain curve first increases as cracks close and then begins to decrease. The decrease in slope is due to the growth of cracks in the sample. A careful examination of stress-strain curves like that in Figure 8 shows that cracks begin to extend when the applied stress is about one-~ird of the fracture strength. The number and length of cracks increase as the applied load is in- creased, and gradually the specimen is weakened until finally a fault is formed, fracturing the sample. Fracture in compression, then, is a complex process involving the growth and coalescence of many cracks. The total nonelastic deformation that can be attributed to the growth of cracks (and also the fracture strength) is found to increase with increasing confining pressure. At sufficiently high confining pressures, the stress-strain curves and the shape of the specimen cannot be dis- tinguished from those for truly ductile materials, although the mech-

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Deformation and Fracture of Rock 2.5 2.0 Q 1.5 _ In In UJ 1.0 _ 0.5 o 'a? / / / 1 1 1 .ooo , 1 STRAI N FIGURE 8 A complete stres~stra~n curve for Westerly grarute under un~- ax~al compression. 101 anism on the microscale in one is brittle failure and, in the other, plastic flow. Effects of Fluids The pressure of fluid in the pores in the rock has a direct effect on fracture strength. Theoretical analysis and laboratory experiments show that pore pressure can be handed effectively by using the "law of effective stress." The effective stress for fracture is given by the applied stress less the pore pressure. The law of effective stress requires that all combinations of applied stress and pore pressure that produce the same effective stress must have the same effect on fracture. As a consequence, the fracture strength for any value of pore pressure and confining pressure can be found once the fracture strength has been established for one set of conditions. An example of how strength and effective stress are related can be found in Figure 9.

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102 _~0 5 ~ 20 15 1 10 ~0 o CONSERVATION OF HISTORIC STONE BUILDINGS O Pp= 100b O Pp = 300b O Pp= 7.00b o 2 4 AsPp. kb 6 FIGURE 9 Compressive strengths for samples with different pore pressures all fall on the same curve when plotted in terms of effective stress.l1 to

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Deformation and Fracture of Rock 103 Pore fluids con also affect strength indirectly. Some fluids, including water, have a corrosive effect on rock-forming minerals. These fluids weaken rock in two ways. The corrosive action in situations where fluid can circulate through the pore and crack network gradually erodes the passages and weakens the matrix. This mechanism is not partic- ularly important when fluids of natural origin are involved, except where the fluids are unusually corrosive to the rock minerals or the fluid is unusually hot. Fluids are also found to affect strength through a mechanism involving surface tension. The fracture strength of sam- ples of a quartzitic shale is shown in Figure 10. Saturating rock with a fluid having low surface tension increases its Toad-ca~rying capacity. Another indirect way in which fluids affect strength involves the permeability of the rock, the size of the sample, and the rate at which the load is applied. Consider a large sample of a relatively impermeable rock that is saturated with water, and assume that the load in the sample is increased very quickly. Fluid in the rock, though it has access to the outside, is trapped in the pores and cracks because of the low permeability and the long path. The pore pressure rises, and the strength decreases because of the law of effective pressure. Appreciable de- creases in strength have been observed in the laboratory on samples of granite as small as a few centimeters.~ Size of Samples Griffith's theory shows that specimens with long cracks have low strength. This suggests that large samples may be weaker than small samples because the probability that a sample contains a long crack is greater for larger samples. The effect of size, which is a matter of concern in manufacturing engineering practice where the range of sizes is relatively small, could be a major factor where the strength of rock is important because the range of interest can be enormous. A number of studies on specimens having a range of sizes (of the order of 1 to 10 cm) suitable for experiments in the laboratory have demonstrated that bigger samples of rock are indeed weaker than smaller ones. These studies show that, in general, the weaker the rock, the larger the effect of size on strength. The relationship between size and strength deter- mined in these studies is usually expressed in the form cry ~(size) -a' /,2) where a for strong rocks like competent marbles and granites is near O.l-and, for a weaker rock like coal, is 0.5.

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04 CONSERVATION OF HISTORIC STONE BUILDINGS 1 0,000 _ . _ . a, - 1 8,000 C) cry 6,000 in > en in at: 4,000 o Cal X 2.000 he , o . . o . C) A w x I c ._ .O A) o - O ~ 0 0.04 0.08 0.12 0.16 POUN DALS PE R FOOT 1 1 1 1 I I I ~ I I 1 1 0 10 20 30 40 50 60 70 DYNES PER CENTIMETER SURFACE TENSION ~ OF IMMERSION LIQUIDS AT 20 C (68 F ~ FIGURE 10 Compressive strengths of samples saturated with var- ious fluids. Note that strength is high for fluids with low surface tensions The range of sizes used in these laboratory studies is barely in the range of interest in building construction. Studies in the field on very large specimens are rare because they are expensive and very difficult to do. The very limited number of observations that have been made on the strength of large samples suggests that equation 2 is valid only for samples less than a critical size, and the strength of larger samples is nearly the same. An example is shown in Figure 11. Increasing the confining pressure tends to mitigate the effect of size. Griffith's theory shows that weak rocks have large cracks, and the above discussion of rock deformation demonstrates that cracks close

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Deformation and Fracture of Rock ]50 100 70 50 30 20 15 ~ 10 at: z 7 5 4 Iron Ore b ~ Diorite -\ ~ ~ O\ \ O\ ~ ~ Jahns (1966} o Pratt et al. (1972) ,& Bieniawski (1968) 0 0.5 1.0 1.5 2.0 SPECIMEN SIDE LENGTH, m 105 2.5 3.0 FIGURE 11 The compressive strength of rock is found to be smaller for larger sam- ples 15,16,17 under compressive stress. A large crack in a large sample that is closed under confining pressure acts like a collection of small cracks. There- fore, the strength of large samples and small samples under confining pressure is closer than one would predict from an examination of their microstructures at atmospheric pressure. SUMMARY I have tried here to describe in a concise way the major elements affecting the deformation and fracture of rock. Rock is found to be an unusual material in that it does not fall into one of the traditional classifications of elastic and plastic, describing deformation, or brittle and ductile, describing fracture. Behavior is governed to a large extent by the porosity found in nearly all rock. The opening and closing of cracks causes nonlinearity in stress-strain curves, and sliding between the faces of cracks against friction is the source of the nonelastic behavior observed under deviatoric stresses. Likewise, cracks weaken

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106 CONSERVATION OF HISTORIC STONE BUILDINGS rock and determine how strength vanes with applied stresses rock is weak In tension once relatively strong In compression, and com- pressive strength increases dramatically with confining pressure. REFERENCES 1. F.A. McClintock and A.S. Argon, Mechanical Behavior of Matenals, Addison- Wesley (Reading, Mass.), 1966. 2. J.B. Walsh, The effect of cracks on Poisson's ratio of rocks, I. Geophys. Res., 70~20), 5249-5258 (1965~. 3. J.B. Walsh, Static deformation of rock, [. Eng. Mech. Div., Proc. Amer. Soc. Civil Eng., 106 (EMS), 1005-1019 ( 1980~. (1957~. 4. M. Nishihara, Stres~strain relation of rocks, Dosh~sha Kogaka Kaiser, 8, 32-54 5. T.-F. Wong and W.F. Brace, Thermal expansion of rocks: Some measurements at high pressure, Tectonophysics, (57) 95-1 17, 1979. 6. T.-F. Wong, Post-failure behavior of Westerly granite at elevated temperatures, Ph.D. thesis, Massachusetts Institute of Technology, 1980, submitted to Int. I. Rock Mech. Min. Sci., 1980. 7. F. Birch, Compressibility; elastic constants, Handbook of Physical Constants (S.P. Clark, Jr., ed.), 97-173, 1966 8. J. Handin, Strength and ductility, Handbook of Physical Constants (S.P. Clark, Jr., ed.J, 223-300, 1966. 9. A.A. Griffith, The phenomenon of flow and rupture in solids, Phil. Trans. R. Soc. London, A221, 163-198, 1921. 10. A.A. Griffith, Theory of rupture, Proc. 1st Intl. Congr. Appl. Mech., Delft, 55-63, 1924. 11. W.F. Brace, personal communication, 1980. 12. F.A. McClintock and J.B. Walsh, Friction on Griffith cracks in rocks under pres- sure, Proc. 4th U.S. Congr. Appl. Mech., 1015-1021, 1962. 13. P.S.B. Colback and B.L. Wild, The influence of moisture content in the com- pressive strength of rock, Rock Mech. Sympos. 3rd), Univ. of Toronto, 65-83, 1965. 14. W.F. Brace and R.J. Martin m, A test of the law of effective stress for crystalline rocks of low porosity, Int. J. Rock Mech. Min. Sci., 5, 415-426, 1968. 15. Z.T. Bieniawski, The effect of specimen size on the compressive strength of coal, Int. I. Rock Mech. Min. Sci., 5, 325-335, 1968. 16. H. Jahns, Measunng the strength of rock in-situ at an increasing scale [in German], Proc. 1st Congress, Int. Soc. Rock Mech., Lisbon, 1, 477-482, 1966. 17. H.R. Pratt, A.D. Black, W.S. Brown, and W.F. Brace, The effect of specimen size on the mechanical properties of unjointed diorite, Int. I. Rock Mech. Min. Sci. 9, 51 529, 1972.

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Deformation and Fracture of Rock BIBLIO GRAPHY lOJ This short review does not do justice to these fields. Rock deformation and fracture have received considerable attention in recent years, and the literature describing ex- perimental and theoretical studies is too voluminous to be listed here. For those who need to read further in the subject, however, the following texts will provide a good starting point: Jaeger, J.C., Elasticity' Fracture and Flow, Methuen (London and John Wiley {New York), 1956. Jaeger, J.C. and N.G.W. Cook, Fundamentals of Rock Mechanics, Methuen, London, 1969. McClintock, F.A. and A.S. Argon, Mechanical Behavior of Materials, Addison-Wesley, Reading, Mass., 1966. Stagg, K.G. and C).C. Zienkiewicz, Rock Mechanics in Engineering Practice, John Wiley, London, 1968.