Chapter 11 Rate Effects 11.1 Theoretical Background 11.1.1 Preliminaries So far we have considered only materials that are rate-independent, that is to say, the response is the same irrespective of the strain rate. This feature of the material behaviour is due to the fact that the dissipation function is chosen to be a homogeneous first-order function of the internal variable rate. Mathemati- cally, this can be expressed through Euler’s equation as ij ij d d w D wD   (11.1) Several features of the behaviour of rate-independent materials follow from this special form of the dissipation function. In particular, the special case of the Legendre transform of a homogeneous first-order dissipation function (see Appendix C, Section C.6) gives rise to the existence of a yield surface. Many materials, whilst primarily rate-independent, do show some small de- pendence on the strain rate. Typically, the yield stress may be observed as in- creasing marginally with the strain rate. Creep under sustained stresses and relaxation of stress at fixed strain are processes that are also related to strain rate effects. Most geotechnical materials, for instance, exhibit these types of behav- iour to a certain extent. This type of response is often modelled semi-empirically, frequently with dif- ferent (and not always consistent) theories used to model the rate-dependence of strength and the processes of creep and relaxation. Properly, however, all these phenomena should be encompassed within a single approach that ex- plains all rate-dependent processes. This can be achieved within the frame- work described in this book by considering dissipation functions that are not 212 11 Rate Effects homogeneous, first-order functions of the internal variable rate, so that Equa- tion (11.1) does not apply. As stated before, the formalism adopted in this book is based on the method used by Ziegler (1977). He describes principally materials that are rate-depen- dent (concentrating mainly on linear viscous materials, for which the dissipation function is quadratic), and devotes relatively little attention to the special case of rate-independence. Here, we have taken the inverse approach. Having started with the rate-independent case, we shall now examine briefly the implications of departures from it. This approach offers a different, and we believe useful, in- sight into the treatment of rate-dependent materials. Following the concepts behind Ziegler’s method, but adopting a slightly dif- ferent terminology and sequence to the argument, first we define: ij ij d d Q w D wD   (11.2) Comparing with Equation (11.1), we see that 1Q for a rate-independent material. For any dissipation function d which is a homogeneous function of degree n in ij D  , it follows from Euler’s theorem that Q is simply a constant equal to 1 n . One possibility is that we generalise the definition of the dissipative general- ised stress to ij ij dw F Q wD  (11.3) so that ij ij ij ij d d w FD Q D wD   , which can be compared with Equation (4.10). In this case, it follows once again that, by comparison with Equation (4.8),  0 ij ij ij FF D  . Then, following exactly the same argument as before, we adopt the constitutive hypothesis (equivalent to Ziegler’s orthogonality condi- tion) that ij ij F F. This form of the equations corresponds exactly to Ziegler’s original approach (although expressed slightly differently). The presence of the factor Q in the above formalism is not too much of an in- convenience when the dissipation is a homogenous function of the internal vari- able rate, because in this case Q is simply a constant. However, if the dissipation is not a homogeneous function of the rates, as proves useful to describe materi- als with a weak rate-dependence, then the presence of the factor Q in Equation (11.3) is a significant inconvenience. Specifically in this case, d is said to be a pseudo-potential rather than a true potential, and it is not possible to take the Legendre transform of d to interchange ij F and ij D  as dependent and inde- pendent variables. 11.1 Theoretical Background 213 11.1.2 The Force Potential and the Flow Potential If, however, d can be written in the form ij ij z d w D wD   , where  ,, ij ij ij zzx DD  , then certain advantages, explored below, can be gained. First we can note that if d is homogeneous and first order in ij D  , then zd{ . Next, it is clear from Euler’s equation that if d is homogeneous and of order n, then zdn . Rather than (11.3), we prefer to adopt the following definition of the dissipa- tive generalised stress: ij ij zw F wD  (11.4) From Equation (11.4), it follows that ij ij dFD  as before, so that once again  0 ij ij ij FF D  , and we make the constitutive assumption ij ij F F . The principal advantage of using the function z is that, unlike the dissipation function d (which is a pseudo-potential), the function z serves as a true potential for ij F . The function z could properly be defined as the generalised stress poten- tial, but for brevity, we shall simply refer to it as the force potential. Because it serves as a potential for the generalised stresses, a simple Legendre transforma- tion can be made:  ,, ij ij ij ij ij wx z d zDF FD   (11.5) such that ij ij ww D wF  (11.6) The function w has a clear analogy with the yield function in the rate- independent case, but because z is not homogeneous and first order in the rates, the Legendre transform is no longer the degenerate special case; so although 0 y for the rate-independent case, in general, the condition 0w does not apply. The function w could properly be called the internal variable rate poten- tial, but again for brevity, we shall simply call it the flow potential. Note that the sum of the force and flow potentials is equal to the dissipation function zwd . The potentials z and w have been defined elsewhere [see, for example, Maugin (1999)], but previous authors seem to have concentrated almost exclu- sively on the linear viscous case where both are quadratic. Confusingly, both z and w have been referred to in the literature as dissipation potentials, a termi- nology we deliberately avoid here. We shall now explore how the potential z can be defined if the dissipation function d is known. Since z is obtained by integration from d (see below), it is determined only to within an additive constant. To make the definition of z precise, we shall specify that 0z when the internal variable rates are all zero. 214 11 Rate Effects Since 0d in this case, and w = d – z, it also follows that w = 0 when the rates are zero. As noted above, if d is a homogeneous function of order n in the internal vari- able rates, we can choose zdn , which is also of order n, so that ij ij z nz d w D wD   . It follows that if d can be represented in the form, 1 N k k dd ¦ (11.7) where each of the N functions k d is itself homogeneous and of order k n in the internal variable rates, then z can be chosen as 1 N k k k d z n ¦ (11.8) We call functions that can be expressed in the form of Equation (11.7) pseudo-homogeneous. For the one-dimensional case  dd D  , then it is straightforward to re- arrange  dz d d D FD D D    as  d dz d D DD   and then integrate to give   0 o o d zd D D D D D ³     (11.9) For more general cases, we can proceed as follows. Consider the definition   0 o ij oij d zd W WD WD W W ³   (11.10) Differentiation with respect to o W gives    oij oij ij o oij zdwWD WD D W wWD    , and setting 1 o W yields the result   ij ij ij ij ij ij z d wD D FD D wD    . This demonstrates that application of the definition (11.10) leads to a potential with the required prop- erty that ij ij zF wwD  satisfies ij ij dFD  . Thus by setting 1 o W in (11.10), we obtain   1 0 ij ij d zd WD D W W ³   (11.11) Each of the cases in Equations (11.8) and (11.9) are special cases of the rela- tionship in (11.11). 11.1 Theoretical Background 215 A simple change of variable to lnx W gives an alternative expression to (11.11) which may be more convenient in some cases:   0 x ij ij zdedx f D D ³  (11.12) Therefore, we have demonstrated that, if the dissipation exists as a function of the internal variable rates, then it is possible to derive the force potential by integration. 11.1.3 Incremental Response In the rate-independent case, we found that the incremental response could always be derived if the model was specified by one of the energy functions ( u, f, g, or h) and the yield function y. Alternatively, if the dissipation function is de- fined then, although the incremental response can be derived by applying vari- ous ad hoc procedures for particular models, it has not proved possible to do this by a completely general and automated procedure. We find that there is a parallel position for the rate-dependent materials. If w is specified, then the incremental response can be obtained automatically, whereas if z is specified, then ad hoc procedures again have to be used for each model. Assuming that the model is specified by the Gibbs free energy g and the flow potential w, we can define the following differential relationships: 22 ij kl kl ij kl ij kl ggww H  V D wV wV wV wD   (11.13) 22 ij kl kl ij kl ij kl ggww F  V D wD wV wD wD    (11.14) ij ij ww D wF  (11.6), bis which can simply be combined to give the incremental stress-strain response 22 ij kl ij kl ij kl ij gg w ww w H  V wV wV wV wD wF   (11.15) Note that for a rate-dependent material, the time increment in a calculation has a real physical meaning, whilst for a rate-independent material, the time increment is artificial. For a real increment of time dt, (11.15) can be rewritten 22 ij kl ij kl ij kl ij gg w dd dt ww w H  V wV wV wV wD wF (11.16) 216 11 Rate Effects As well as the stresses and strains, it is necessary to update ij D and ij F be- cause the differentials in Equation (11.15) are functions of these variables. The updating is achieved by using Equations (11.6) and (11.14). 11.2 Examples 11.2.1 One-dimensional Model with Additive Viscous Term The simplest of one-dimensional elastic-plastic models is defined by the func- tions: 2 2 g E V  VD (11.17) dk D  (11.18) where E is the elastic stiffness and k is the strength. On the other hand, an equivalent viscoelastic model would be defined by (11.17) together with 2 d PD  (11.19) where P is the viscosity. A simple elastic-viscoplastic model is obtained by combining the two dissipa- tion functions: 2 dk DPD  (11.20) A schematic representation of the model represented by Equations (11.17) and (11.20) is given in Figure 11.1, in which it can be seen that the plastic and viscous elements act in parallel. Figure 11.1. Schematic representation of elastic-viscoplastic model 11.2 Examples 217 Noting that the dissipation function is now non-homogeneous and using any of (11.8), (11.9), or (11.11), one can derive 2 2 zk P DD  (11.21) The differentials of the potentials now give g E w V H  D wV (11.22) g w F  V wD (11.23)  S z k w F D PD wD   (11.24) Noting that if 0D  , then  S D  is undefined, we consider first the case when 0Dz  . For this case, it follows that (since V F F), Equation (11.24) gives    SkV PD D  (11.25) so that kV! , and the signs of V and D  must always be the same. A corollary is that if kVd , then 0D  and EH V   , i. e. incrementally elastic behaviour oc- curs. Thus the elastic region is bounded by a “yield surface” in stress space 0kV . The complete response for both 0D  and 0Dz  can therefore be expressed by rearranging (11.25) as  S kV D V P  (11.26) where are Macaulay brackets, 0, 0xx  , ,0xxx t . The above result can be obtained in a mathematically more rigorous and concise way by using the terminology of convex analysis; see Chapter 13 and Appendix D. It now follows that, differentiating (11.22) and substituting (11.26),  S k E V V H  V P   (11.27) Considering tests at a constant strain rate H  , the viscoplastic part of the re- sponse described by (11.27) can be rearranged as  SEk dE E d V VV   HPH PH  (11.28) which integrates to  exp S E Ak §· H V  PH D ¨¸ PH ©¹   (11.29) 218 11 Rate Effects where A is a constant of integration. For the first loading, 0D!  and kV at kEH , so that for this case, we can obtain the solution for the viscoplastic part of the curve as 1exp kE k §· §· H V PH   ¨¸ ¨¸ PH ©¹ ©¹   (11.30) or in normalised form, 1exp 1 1 kE kk k ½ §· VPH H §·    ®¾ ¨¸ ¨¸ PH ©¹ ©¹ ¯¿   (11.31) Figure 11.2 shows the normalised stress-strain curves for 0.0,kPH  0.1, 0.2, and 0.3 (the first is equivalent to the model without viscosity). It can be seen that this model provides a satisfactory starting point for describing rate-dependent plastic behaviour, in which there is a linear increase in strength with strain rate. We have obtained the above response from the specification of the force po- tential. Alternatively, we can subtract (11.21) from (11.20) to obtain the flow potential 2 2wdz  PD  . Substituting (11.26), but writing F instead of V, we can express the flow potential as a function of F: 2 2 k w F P (11.32) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 012345 3.0 HP k  0.0 1.0 2.0 k V kEH Figure 11.2. Normalised stress-strain curves for different constant strain rates 11.2 Examples 219 We could have taken Equation (11.32) as our starting point rather than (11.21). Differentiating (11.32) gives  Sk w F F w D wF P  (11.33) which [using (11.23)], immediately leads to (11.26), so that the derivation of a response based on the flow potential w is briefer than the derivation using the force potential z. 11.2.2 A Non-linear Viscosity Model An alternative approach to modelling viscous effects is to modify (11.18) to 1 n n dk r  D  (11.34) where the constant r, with the dimensions of strain rate, has been introduced to maintain the dimension of stress for the constant k. It immediately follows that 1 n n k zr n  D  (11.35) So that instead of (11.24), we obtain  1 1 S n n z kr   w F D D wD   (11.36) There is now no purely elastic region, and non-linear viscous behaviour oc- curs whenever the stress is non-zero. Noting that Equation (11.36) implies that the signs of V and D  must be the same, it can be rearranged to  1 1 S n r k  V §· D V ¨¸ ©¹  (11.37) so that, differentiating (11.22) and substituting, we obtain  1 1 S n r Ek  V §· V H  V ¨¸ ©¹   (11.38) Considering again tests at constant strain rate H  , (11.38) can be rearranged as  1 1 1S n dr E dk  §· V §· ¨¸ V V ¨¸ ¨¸ HH ©¹ ¨¸ ©¹  (11.39) Equation (11.39) cannot be integrated analytically without recourse to special functions, but Figure 11.3 shows stress-strain curves for 1.1n and different strain rates obtained by numerical integration (using forward differences with 220 11 Rate Effects an interval of 0.1EkH ). The curves shown are for 1,rH  2, 3, and 4. A test at infinitesimal strain rate would simply give 0V . Figure 11.4 shows the stress-strain curves (again produced by numerical inte- gration) for 1rH  for n values of 2, 1.5, 1.2, and 1.1. It shows how the elastic- plastic response is approached asymptotically, for a given strain rate, as 1n o . Although the curves in Figures 11.2 and 11.3 are at first sight remarkably similar, there are a number of important differences in the character of the re- sponse. Firstly, the curves shown are not necessarily for comparable strain rates. Also, for the second case, infinitesimally slow straining results in zero stress, whilst in the first case, it gives the elastic-plastic response. Once straining is stopped, the stress in the first model relaxes to kV , but in the second model, it relaxes to 0V . Clearly, the models represented by Equations (11.21) and (11.35) could both be expressed within a more general model defined by 1 2 1 n n k zk r n  D D  (11.40) in which the first model is obtained by setting 2n and 2 kr P , and the sec- ond model simply by setting 1 0k . This serves as a good example of the way this approach to the formulation of constitutive models allows to be set them within a hierarchy, where simpler models are subsets of more complex models. The model defined by Equation (11.40) is used as the basis for a continuum model in Section 11.2.4. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 012345 4 HP k  1 2 3 k V kEH Figure 11.3. Normalised stress-strain curves for non-linear viscosity model [...]... calibration of the * E model It follows therefore that ˆ k d 1H f and, Equa- 0 tion (11.98) can be rearranged as H H E f (11.99) The parameter H here does not appear separately from the parameter and cannot be calibrated independently against the experimental data Therefore correspondingly For any we adopt H E and calibrate the parameter 11.5 Visco-hyperplastic Model for Undrained Behaviour of Clay... experimental data obtained by Vaid and Campanella (1977) for the behaviour of a natural clay Figure 11. 7a shows the data they obtained in undrained triaxial compression tests at different rates Figure 11.7b shows the equivalent calculations using the above model with the parameters E 1c 160 , a 1.5 , k0 1c 0.545 , k1 1c 0.3 , r 0.008 min 1, and 1c 2.8 min All parameters involving stress have been normalised... stress-controlled loading [specified as t ], Equation (11.99) is a linear first-order differential equation and can be integrated to give t e H Ht E 0 eH f d (11 .100 ) For strain-controlled loading [specified as t ], Equation (11.99) is in general a non-linear first-order differential equation and cannot always be integrated analytically For example, for a constant strain rate loading and the hyf k E k perbolic backbone... rate-dependent models As a stepping stone to the continuous models, we first consider multiple internal variables 11.3 Models with Multiple Internal Variables 225 11.3.1 Multiple Internal Variables In the previous sections, we considered materials characterised by a single kinematic internal variable, ij , which typically was a scalar or a second-order tensor The kinematic internal variable can often... softening can be achieved by adopting the following expression: 1 k ˆ k0 k1 ˆ d (11 .102 ) 0 where k0 and k1 are parameters calibrated to fit experimental data (Note that clearly the model is valid only for the limited range of plastic strains 234 11 Rate Effects 1 ˆd k0 k1 and only for monotonic loading For non-monotonic loading, 0 hardening should be introduced via accumulated plastic strain using a. .. the logarithm of the strain rate This represents quite realistically the behaviour of a number of materials, particularly geotechnical materials Alternatively, if a formulation based on the flow potential is required, then we can note that w d z r w 2 r2 r 2 cosh r , which can be expressed as k r 1 Note that if this form is used, then the expression for strain rate follow very simply by differentiation... consider again first monotonic loading with constant strain rate c (so H n , so that S that H n 1 , integration of c t ) Assuming again the differential Equation (9.71) gives N* H kn aN * n 1 2 c aN * C N * exp E c aN * (11.72) 228 11 Rate Effects where an E En H and CN * is the integration constant which takes a different value for each N * 1 N The value of C1 is obtained from the initial k1 ; E... Undrained Behaviour of Clay 11.5.1 Formulation As discussed above in Section 11.2.3, many rate-dependent processes can be regarded as thermally activated processes The resulting approach is known as rate process theory In Section 11.2.3, we considered a single yield surface model with this type of rate-dependent behaviour The flow potential of Equation 11.2.3 can be easily generalised for internal functions... knowledge of the energy functional and the flow potential functional, it is possible to derive the entire incremental response for an elastic-viscoplastic material This is of particular importance because non-linear material models are frequently implemented in finite element codes for which an incremental response is required Double differentiation of the energy functional gives the rates of the variables:... that the rate x of the process depends on the driving force q in the following way: x (11.41) A sinh Bq where A and B are functions of the temperature We shall consider fixed temperature here, and treat A and B as constants Mitchell (1976) gives a useful discussion of the theory in the context of the mechanics of soils and q with , we are expecting a relationship of the Identifying x with 1 , so that . strain rate, whilst at high values of rH  , it increases linearly with the logarithm of the strain rate. This represents quite realistically the behaviour of a number of materials, particularly. Each of the cases in Equations (11.8) and (11.9) are special cases of the rela- tionship in (11.11). 11.1 Theoretical Background 215 A simple change of variable to lnx W gives an alternative. the strain rate. Creep under sustained stresses and relaxation of stress at fixed strain are processes that are also related to strain rate effects. Most geotechnical materials, for instance,