9.3 Applications in Geomechanics: Elasticity and Small Strains 161 and the Second Law is written in the form derived from the Clausius-Duhem inequality: , , 0 kk kk q sq T T t T (9.8) The mechanical dissipation can now be split into the dissipation related to the deformation of the soil skeleton and the part that results from the flow of pore water. It is clear that we can identify the term , w kk wu c as the input power that is dissipated in the flow of pore water, so we can write ,, w kk k k dsq dwu c T (9.9) where d is the dissipation associated with deformation of the soil skeleton. The Second Law becomes , , 0 kk w kk q dwu T c t T (9.10) So we see the close analogy between the energy dissipated by the heat flux k q and the water flux k w . Although the Second Law requires only that the sum of the three terms in inequality (9.10) be non-negative, we again assume that each individual term is non-negative and therefore that 0d t . From Equation (9.9), we obtain ,, w kk kk wu q s d c T , so that the First Law becomes ij ij usd V H T (9.11) and the analysis proceeds exactly as in Section 4.1, but with ij V replacing ij V and d replacing d throughout. Thus we see that the development in Chapter 4, which was in terms of total stresses (in the absence of pore pressure), is equally applicable in terms of the conventionally defined effective stresses for a satu- rated granular material under the usual assumption of incompressible grains and pore fluid. Houlsby (1997) developed expressions for the power input to an unsaturated granular material (i. e. one composed of grains, liquid, and gas), subject to some simplifying assumptions. An analysis similar to that presented above can be applied to the unsaturated case. Appropriate definitions for effective stress can be made so that the development in earlier chapters is applicable, but additional terms appear in the equations, related to the degree of saturation and the differ- ence between pore water and pore air pressures. The application to unsaturated materials is not, however, straightforward, and further work is required in this area. A more detailed study of the processes of soil deformation and flow in porous media is presented in Chapter 12. 162 9 Dependence of Stiffness on Pressure In the remainder of this chapter and in Chapter 10, we work entirely in terms of effective stresses, and for brevity we use ij V and d instead of ij V and d . 9.4 Dependence of Stiffness on Pressure In recent years, a considerable amount of experimental research has been car- ried out to investigate the mechanical behaviour of soils undergoing very small strains, for which the response is usually assumed to be reversible. The interpre- tation of the data shows that the initial soil stiffness (or small strain stiffness) is a non-linear function of the stress (specifically the mean effective stress). The stiffness is also affected by other variables, such as the voids ratio, and/or the preconsolidation pressure, which we address in Section 9.4.3. The small strain tangent stiffness depends on the stress level, and typically the elastic moduli vary as power functions of the mean stress. Simple elastic or hypoelastic models of this non-linearity can result in behaviour that violates the laws of thermodynamics. To ensure that an elasticity model is thermodynami- cally acceptable, we use the hyperelastic approach here to derive models which allow for variation of elastic moduli as power functions of mean stress. Analysis of many geotechnical problems depends on a realistic representation of the non-linear dependence of the initial stiffness on stress, and we first ex- plore how this has often been achieved in the past. The usual approach is to adopt “hypoelastic” formulations (Fung, 1965) in elastic-plastic models, in which varying tangent moduli are defined. For instance, it is common to adopt the following procedure to calculate elastic moduli for the modified Cam-clay model. The bulk modulus K is usually defined through the pressure-dependent expression 1 p e K c N , and the shear modulus G is then obtained by assum- ing a constant Poisson ratio v. Such a model leads to a non-conservative “elas- tic” response (Zytynski et al., 1978). Instead we adopt the “hyperelastic” ap- proach, which naturally leads to a conservative elastic response, guaranteed to obey the first law of thermodynamics. It is worth remarking here that it is well-recognised that the soil stiffness is also significantly dependent on the strain amplitude. This raises more difficult prob- lems of hysteresis and energy loss and is addressed later in Sections 9.5 and 9.6. Several models have been developed to reproduce the reversible behaviour of soils, and these are reviewed by Houlsby et al. (2005), who also review briefly some typical experimental observations of the small strain stiffness of soils and their semi-empirical interpretations. We present here a hyperelastic, isotropic energy potential capable of accounting for the non-linear dependence of elastic stiffness on stress. We first develop this for triaxial conditions and later for more general stresses. 9.4 Applications in Geomechanics: Elasticity and Small Strains 163 9.4.1 Linear and Non-linear Isotropic Hyperelasticity Experimental evidence suggests that the small strain bulk and shear stiffnesses of soils can be well represented as power functions of the mean effective stress, and we write them in the following forms: n rr p K = k pp §· ¨¸ ¨¸ ©¹ (9.12) n s rr p G = g pp §· ¨¸ ¨¸ ©¹ (9.13) where k and s g are dimensionless constants; r p is a reference pressure, often conveniently taken as atmospheric pressure; and n is an exponent 01ndd . The possibility that the exponent n could be different for the bulk and shear moduli might be considered, but is not pursued further here, as this leads to consider- able additional complexity in the mathematical development. In the following, we will focus our attention on deriving stress-dependent stiffness from potentials that are expressed as functions of invariants either of the strain or of the stress tensor, so that the material behaviour described will be fundamentally isotropic, although it will be seen below that “stress-induced” anisotropy is predicted under certain conditions. Expressed in terms of the triaxial variables, the Helmholtz free energy f (also equal to the elastic strain energy) is written as a function of the strains, i. e. , vs f = HH . It then follows that v f p = w wH s f q = w wH (9.14) and further that the tangent bulk and shear moduli are defined by 2 2 v v p f K = = ww wH wH 2 2 3 s s qf G = = ww wH wH (9.15) Furthermore, it can be shown that off-diagonal terms may in general appear in the incremental stiffness matrix, v s d dp K J = d dq J 3G H ª º ªºª º « » «»« » H ¬¼¬ ¼ ¬ ¼ (9.16) 164 9 Dependence of Stiffness on Pressure where 2 svvs p qf J = = = ww w wH wH wH wH (9.17) When J is non-zero, the material behaves incrementally in an anisotropic manner, even though f is an isotropic function of the strains. This is a case of stress-induced anisotropy. Although elastic behaviour can be derived by differentiation, as in Equations (9.15) and (9.17), this has certain disadvantages. The resulting expressions for the moduli are in terms of the strains, which can be inconvenient, because moduli expressed as functions of stress are usually of more practical use. There- fore, it is useful to use instead the Gibbs free energy function g (which in the context of elasticity theory is also the negative complementary energy): vs g =f p q HH (9.18) When g is expressed as a function of the stresses , g gpq , the strains may be derived as v g = p w H w (9.19) s g = q w H w (9.20) and the terms in the compliance matrix, 13 32 v s dcc dp = dcc dq H ªºª º ªº «»« » «» H ¬¼ ¬¼¬ ¼ (9.21) are 2 1 22 3 3 v g G c = = = p KG Jp wH w w w (9.22) 2 2 22 3 s g K c= = = q KG Jq wH w w w (9.23) 2 3 2 3 vs g J c = = = = qp pq KG J wH wH w ww ww (9.24) The Helmholtz and Gibbs free energy expressions for linear elasticity (which corresponds to 0n in (9.6) and (9.7)) are each quadratic in form: 22 3 22 s vs r g k f = p + §· HH ¨¸ ©¹ (9.25) 22 11 1 26 rs g = p + q pk g §· ¨¸ ©¹ (9.26) 9.4 Applications in Geomechanics: Elasticity and Small Strains 165 with k and s g dimensionless constants. From the above, it is straightforward to derive v r p = k p H , 3 ss r q = g p H , r K = kp , s r G = g p , and 0 J = . The expressions that give non-linear elasticity ( i. e. n Kpv ) under purely iso- tropic stress conditions ( i. e. without q and s H terms) can also be established unambiguously. For 1n z , the expressions for f and g must be 21 1 2 nn r v p f = k n k n H (9.27) 2 1 12 n n r p g= p kn n (9.28) From either of them, one can derive 1 1 n v r p kn = p §· H ¨¸ ¨¸ ©¹ (9.29) and 1 1 n nn v rr Kp = k = k k n pp §· H ¨¸ ¨¸ ©¹ (9.30) For 1n = , the above expressions become singular. A difficulty also arises when 1n = that, if the volumetric strain is taken as zero at 0 p , then it is infi- nite for all finite stresses. This problem can be avoided by shifting the reference point for zero volumetric strain from the origin ( 0 p ) to r p p . This is achieved by changing (9.27) and (9.28) to 21 * 1 2 nn a v p f = k n k n H (9.31) where * 1 1 vv kn H H and 2 1 1 12 n n r pp g = kn pk n n (9.32) This modifies (9.29) to 1 * 11 n v r p kn = p §· H ¨¸ ¨¸ ©¹ (9.33) and (9.30) to 1 * 1 n nn v rr Kp = k = k k n pp §· H ¨¸ ¨¸ ©¹ (9.34) 166 9 Dependence of Stiffness on Pressure but note that this does not effect the expression for stiffness in terms of pres- sure. The asymptotic expressions for 1n = are exp r v p f = k k H (9.35) ln 1 r pp g = kp §· §· ¨¸ ¨¸ ¨¸ ¨¸ ©¹ ©¹ (9.36) From either of them, one can derive: exp v r p = k p H (9.37) and K = kp (9.38) Equations (9.25) and (9.26) apply for 0n for any triaxial stress states, whilst (9.31) and (9.32) [or (9.35) and (9.36) for 1n ] apply when 0n z but only on the isotropic axis. It is our purpose in the following to obtain more general ex- pressions which apply both to any triaxial stress states and for 0n z , and reduce to each of the above equations in the appropriate special cases. This generalisation can be done in a variety of ways. We first consider (for simplicity) the case where the reference point for volumetric strain is at 0 p . Three possible ways of generalising Equations (9.27) and (9.28) are as follows: (a) f is of the form, 22m vvs f = A + BHHH (9.39) where A, B and m are constants. It can be shown that no simple form of g exists for this case. (b) f is of the form, 22 m vs f = A + BHH (9.40) which results in g of the form 21 22 mm g = Cp + Dq (9.41) (c) g is of the form, 22m g = p Ap + Bq (9.42) It can be shown that no simple form of f exists in this case. Einav and Puzrin (2004a) pursue this form of energy function and show [following Houlsby (1985)] how it also has the disadvantage that it introduces a limiting stress 9.4 Applications in Geomechanics: Elasticity and Small Strains 167 ratio, implying that some stress states are unattainable. Both the inability to derive the f function and the inaccessibility of certain stress states are signifi- cant drawbacks of this form of function. All the forms described above exhibit a constant Poisson's ratio under iso- tropic stress conditions. At least at present, available experimental data are in- sufficient to distinguish definitively between the above three approaches, so the selection is based on the simplicity of the mathematics. The approach that proves most versatile is (b), so this is adopted in the following. 9.4.2 Proposed Hyperelastic Potential Triaxial Formulation Following approach (b) above, the generalisation of Equations (9.25) and (9.31) that we seek for the function f must consist of a quadratic function of * v H and s H , raised to an appropriate power. Inspection of the forms of Equations (9.25) and (9.31), followed by some calculation to determine some appropriate con- stant factors, shows that the required general expression is 222 2 21 *2 21 * 3 1 21 1 2 nn nn rss v nn r vo pg fkn kn kn p kn kn §· H H ¨¸ ¨¸ ©¹ H (9.43) where 2 *2 *2 3 1 ss vo v g kn H H H . Note that * v H is used instead of v H to move the origin for volumetric strain to r p p , for consistency with the 1n case. From the above, the stresses and moduli may be obtained by differentiation as in Equa- tions (9.14)–(9.17). The resulting expressions for the moduli are in terms of the strains. It can be shown (after some manipulation) that the Gibbs free energy expres- sion which is the Legendre transform of the expression in Equation (9.43) is 22 22 1 2 1 1 1 31 12 1 12 n n s r n o n r kn p g p + q g kn p k n n p p kn p k n n §· ¨¸ ©¹ (9.44) where 2 22 1 3 o s knq p = p + g . 168 9 Dependence of Stiffness on Pressure Although Equations (9.43) and (9.44) may appear complex, it can be seen that they have the basic structure of (9.40) and (9.41) (allowing for the shift of origin for strain). The particular forms of the functions are chosen so that, after differ- entiation, the moduli reduce to simple expressions. It follows from the above that 1 1 1 1 v nn ro p = kn p p §· ¨¸ H ¨¸ ©¹ (9.45) 1 3 s nn s ro q = p gp H (9.46) and 2 1 12 1 1 1 nn o ro np c = knp p p §· ¨¸ ¨¸ ©¹ (9.47) 2 2 12 1 1 1 33 nn ro o nk n q c= gp p gp §· ¨¸ ¨¸ ©¹ (9.48) 3 1 2 3 n n s o r npq c= gp p (9.49) Some of the above expressions (9.46)–(9.49), are valid for 1n = as well as 1n z , but this is not the case for (9.43)–(9.45). Noting that for 1n , o p p , the asymptotic values of the compliances for 1n are 2 1 2 1 1 3 s kq c kp g p §· ¨¸ ¨¸ ©¹ , 2 1 3 s c g p , and 3 2 3 s q c g p . The asymptotic expressions for 1n = replacing (9.43)–(9.45) are 2 3 exp 2 rss v p gk f= k k §· H H ¨¸ ¨¸ ©¹ (9.50) 2 ln 1 6 s r p pq g = k p g p §· §· ¨¸ ¨¸ ¨¸ ¨¸ ©¹ ©¹ (9.51) 2 2 1 ln 6 v r s p q = kp g p §· H ¨¸ ¨¸ ©¹ (9.52) Finally we can note that the stiffnesses are given by 2 KcD , 1 3GcD , and 3 J cD , where 2 2 3 n sr o r Dkgppp . The implications of the above choice of free energy (and hence complemen- tary energy) are as follows: 9.4 Applications in Geomechanics: Elasticity and Small Strains 169 On the isotropic axis simple modulus expressions may be obtained, and 0 J = . On this axis, it is also possible to define Poisson’s ratio 32 62 ss = k g k + gQ [alternatively expressed in the form 31 2 21 s g k = + Q Q]. For more general stress points, the expressions for moduli are more complex, but the most important feature is that the moduli are still power functions of the mean stress (although they also depend on the stress ratio). The fact that 0 J z for general stress states implies stress-induced anisot- ropic elastic behaviour. The shapes of shear strain and volumetric strain contours are given directly by Equations (9.45), (9.46), and (9.51). Within the range of stress ratios of inter- est, the volumetric strain contours are similar (but not identical to) parabolae symmetrical at about the p -axis and convex toward the origin. Shear strain contours are curves, convex upward in the region of the , p q plot accessible for reasonable soil properties. For 1n = , the shear strain contours become straight lines radiating from the origin. Some undesirable features of the model proposed by Houlsby (1985) and ex- tended by Borja et al. (1997), in particular the crossing of volumetric strain con- tours, are absent. Figure 9.1 shows contours of shear and volumetric strains for 0.5n . Note that the volumetric strain contours correspond to undrained stress paths for elastic behaviour. For 0n z , the approximately parabolic undrained stress paths indicate that (other than on the isotropic axis) the response of the soil is incrementally anisotropic. This stress-induced anisotropy arises as a natural consequence of the hyperelastic formulation, and corresponds well to observations of soil behaviour. 0 200 400 600 800 p' 0 200 400 600 800 q Figure 9.1. Example of volumetric (solid lines) and shear (dashed lines) strain contours for n = 0.5 170 9 Dependence of Stiffness on Pressure Many studies have presented evidence that the stiffness of sands can be expressed as a power function of stress level, but a special feature of the hyperelastic ap- proach adopted here is that it predicts the related effect of the curvature of strain contours. Figure 9.2 is reproduced from Shaw and Brown (1988), who follow the approach of Pappin and Brown (1980) in plotting “resilient” shear and volumetric strain contours derived from an extensive series of cyclic tests on granular mate- rial. Importantly, they show that the volumetric strain contours are approximately parabolic and curved approximately as in Figure 9.1. The shear strain contours (which Shaw and Brown assume to be straight) are also very similar to those in Figure 9.1. Comparable data for clays were presented by Borja et al. (1997). General Stress Formulation The results described above can be generalised to other than triaxial stress states, if the free energy f is written as a function of the strains ij H and the com- plementary energy g as a function of the stresses ij V . In this case, the normal expressions for the stresses and strains, as derivatives of the free energies, are used. The stiffness matrix is ij ijkl kl ij kl f = = d w V w www HHH (9.53) and the compliance matrix is Figure 9.2. Shear and volumetric strain contours presented by Shaw and Brown (1988), based on experimental data on crushed limestone [...]... small and intermediate strain behaviour of overconsolidated clays The purpose of the model was to 9.5 Applications in Geomechanics: Elasticity and Small Strains 177 reproduce the characteristic S-shaped curve in a plot of normalised secant shear stiffness G Go against shear strain log and to generalise this variation of stiffness to any stress path The model was based on the concepts of plasticity theory... (9 .85 ) 0 where are Macaulay brackets and ˆ ˆ is defined from 0 ˆ 2h B ˆ a 2 (9 .86 ) 180 9 Small Strain Plasticity, Non-linearity, and Anisotropy The plastic strains are updated using Equations (9 .82 ) and (9 .86 ): 2 ˆ ˆ B 0 ˆ (9 .87 ) Then the “back stresses” are updated using Equation (9 .81 ) 9.5.3 Behaviour of the Model During Initial Proportional Loading Elasticity with Cross-coupling During initial loading... 181 aa ae , so that aa a a aL ae Then, substituting Equation (9 .89 ) in (9 .86 ) and the result in (9 .84 ) and solving differential Equation (9 .84 ) for ˆ , 0; a , we obtain where a ˆ a aa 1 (9.93) 0 Substitution of (9 .81 ) and (9.93) in expression (9 .82 ) yields a 1 0 D 1 *ˆ 3G h a aa d D 1 0 (9.94) Defining L as the stress point at which the stress path reaches the SSR boundary, we then define a normalised... stress y such that y L 0 0 It also follows that y aa aL We define also a normalised strain x such that xD 1 L (9.95) 0 The normalisation is chosen so that x y within the linear elastic region At the boundary of the LER, y x xe ae aL Equation (9.94) may be rewritten in normalised form as a x y 0 D y *ˆ 3G h a aL (9.96) d aa ae y xe is itself a function of y aL ae 1 xe Double differentiation of (9.96) with... and Burland (19 98) suggest that the value of the parameter m J 3G * should be inferred directly from the slope of the effective stress path in an undrained test 184 9 Small Strain Plasticity, Non-linearity, and Anisotropy 9.5.4 Behaviour of the Model During Proportional Cyclic Loading Consider proportional unloading, taking place after stress reversal at a stress state r reached during initial loading... third region vanishes, and the normalized stress-strain behaviour is again described by the initial logarithmic normalized stress-strain curve given in (9.96) and (9. 98) This curve is rotated by 180 °, and initiates from the origin Generally, it can be shown that for proportional cyclic loading, a cycle of a larger amplitude wipes out any memory of all preceding events of a smaller magnitude If another stress... inevitably produces a non-associated flow rule Therefore, even when the elastic-plastic coupling has been accounted for correctly by using Equations (9.62) and (9.69) or (9.70), using associated plasticity to derive plastic strains may lead to violation of thermomechanical principles 9.5 Small Strain Plasticity, Non-linearity, and Anisotropy Puzrin and Burland (19 98) presented a model for the small and... incremental Equations (9 .84 )–(9 .86 ) and subsequent integration of these equations along any straight effective stress path results in a normalized stress-strain where X is defined by the relationship 9.5 Applications in Geomechanics: Elasticity and Small Strains 183 curve identical to that given by Equation (9. 98) used by Puzrin and Burland (19 98) in their model The model requires seven independent parameters:... and Houlsby (1 983 ) and used by Puzrin and Burland (19 98) to describe the stress-strain behaviour within the LER Logarithmic Normalized Stress-strain Curve ˆ The functional form of h can now be defined following the development of Section 8. 10 When aa ae ; aL , it follows from (9.79) that ˆ yg 0 ˆ , 0 ˆg 0 ˆ , 0 y 0; a a ;1 (9.92) 9.5 Applications in Geomechanics: Elasticity and Small Strains 181 aa... Puzrin and Burland (1996 and 19 98) found that the following logarithmic function fitted experimental data extremely well It is expressed as a normalized stress-strain curve describing both 182 9 Small Strain Plasticity, Non-linearity, and Anisotropy Figure 9.4 Normalized stress-strain curve deviatoric and volumetric behaviour during any initial proportional loading (Figure 8. 2) Note, however, that no particular . also has the disadvantage that it introduces a limiting stress 9.4 Applications in Geomechanics: Elasticity and Small Strains 167 ratio, implying that some stress states are unattainable linked to another important effect that the elastic strains may have on the plastic behaviour of coupled materials. Collins (2002) demonstrated that for coupled materials, derivation of an elastic-plastic. material (i. e. one composed of grains, liquid, and gas), subject to some simplifying assumptions. An analysis similar to that presented above can be applied to the unsaturated case. Appropriate