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236 11 Rate Effects Figure 11.8. Variation of undrained strength with strain rate: comparison between theory and data from Vaid and Campanella (1977) not captured accurately. Figure 11.8 shows the variation of peak strength with strain rate as observed by Vaid and Campanella, with the theoretical results ( open circles) superimposed. The figure shows the characteristic response of the rate process theory: a linear increase in strength with strain rate at low strain rate (appearing as almost a constant strength on the logarithmic plot) and a linear Figure 11.9 Undrained stress-strain curves at different strain rates: (a) data from Vaid and Campan- ella (1977) and (b) theoretical curves 11.5 Visco-hyperplastic Model for Undrained Behaviour of Clay 237 increase with the logarithm of the strain rate at high strain rates. The strain rate at which the transition occurs is characterised by the parameter r, and the inter- section of the two straight line sections indicated on Figure 11.8 occurs at 2rH . The slope of the section at high strain rate is approximately lo g 10 e rP . Figure 11.9a shows detail from tests at two strain rates, together with a test in which the strain rate was suddenly changed from the lower to the higher rate at a strain of about 0.8%. Figure 11.9b shows the equivalent theoretical calculation. The model is clearly able to capture accurately the transition, which appears as the line connecting the two main curves. Figure 11.10a shows the results of constant stress creep tests, in which strain is plotted against time for different constant stress values, and Figure 11.10b Figure 11.10. Comparison of (a) creep data from Vaid and Campanella (1977) and (b) the theoreti- cal curves Figure 11.11. Creep data in terms of strain rate: (a) data from Vaid and Campanella (1977) and (b) the theoretical model 238 11 Rate Effects shows the equivalent calculations. Apart from the detail of the curve at 1 0.5 c c VV , the agreement is very close. Figures 11.11a and 11.11b show the same data presented in terms of strain rate against time. Above a certain stress level, the phenomenon of creep rupture occurs, and Figure 11.12 shows Vaid and Campanella’s data for the time to rupture against the stress level. The super- imposed open circles show the theoretical calculations. No creep rupture is pre- dicted at stress ratios lower than 0.485. It is remarkable that a model entirely encapsulated by the two potential func- tions, defined in Equations (11.101)–(11.103) and using only six material pa- rameters (each of which can be given a clearly defined physical interpretation), is able to capture the diversity of behaviour shown in Figures 11.7–11.12. 11.5.4 Extension of the Model to Three Dimensions The above model may be extended to three dimensions by generalizing Equa- tions (11.101)–(11.103) to 11 2 00 3 ˆˆ ˆ 2 ˆ cosh 1 ij ij ij g k wwdr d r ½ §· cc FF D K °° ¨¸ °° ¨¸ P K ®¾ ¨¸ P °° ¨¸ °°¨¸ ©¹ ¯¿ ³³ K (11.110) 1 01 0 3 ˆˆˆ 2 ij ij ij kkk dD DDK ³ (11.111) 11 00 ˆ ˆˆˆ 18 4 ii jj ij ij ij ij ij ij gdhd KG cc VV VV V D K D D K ³³ (11.112) Figure 11.12. Comparison of theoretical results for rupture life with data from Vaid and Campanella (1977) 11.6 Advantages of the Rate-dependent Formulation 239 where K and G are the bulk and shear moduli and the other parameters retain their original meanings (except that ˆ h now bears the same relationship to G as it previously did to E). A prime is used to indicate the deviator of a tensor. Dif- ferentiation of (11.112) gives 1 0 ˆ 92 kk ij ij ij ij ij g d KG c VG V w H D K wV ³ (11.113) Differentiation of (11.110) gives 3 3 ˆˆ ˆ ˆ ˆ 2 2 ˆ sinh ˆ 3 ˆˆ 2 kl kl ij ij g ij ij kl kl k w r r §· cc FF D K ¨¸ c F w ¨¸ D ¨¸ wF P ¨¸ cc FF ¨¸ ©¹ (11.114) Further differentiation of (11.113) with respect to time and substitution of (11.114) then leads to the incremental stress-strain relationship. 11.6 Advantages of the Rate-dependent Formulation The benefits of extending the rate-independent framework to include rate de- pendency for materials that exhibit any significant rate effects are obvious. There is an additional benefit, however, which is worth a separate discussion. It is well known that when rate-independent behaviour is described as the limiting case of rate-dependent behaviour, significant simplifications in calculations can be achieved. Finite element calculations of purely plastic behaviour can, for instance, be efficiently carried out using a viscoplastic algorithm with an artifi- cial very small viscosity; see, for instance, Owen and Hinton (1980). A compari- son between the incremental response expressions (11.85) and similar expres- sions for the rate-independent case developed in Chapter 8 is a good illustration of the greater simplicity of the rate-dependent equations. The source of the higher complexity of the rate-independent expressions is the necessity to satisfy the consistency condition for yield. In other words, it is necessary to make sure that the stress state in plastic loading always stays on the yield surface. This may also cause significant numerical difficulties, requiring treatment by special pro- cedures. The rate-dependent framework is free from these limitations. Chapter 12 Behaviour of Porous Continua 12.1 Introduction In previous chapters, we have developed a theory for plastic materials in which the entire constitutive response is determined by specification of two potential functions. There are many other areas of continuum mechanics where similar approaches have been made. For instance, Ziegler develops theories for viscous materials. Many authors treat flow processes within a thermodynamic context and frequently use a dissipation function. The special features of rate-indepen- dent materials have been the reason for a slightly different emphasis here, from that in most treatments of the subject. We now explore how the hyperplasticity approach can be generalised and set within the context of a wider variety of types of material behaviour. In particu- lar, we shall continue to emphasize the use of two potential functions and the use of Legendre transformations to obtain alternative formulations. When more complex materials are considered, there are two classes of dissi- pative behaviour. The first is associated with fluxes, for instance flow in a porous medium or the flow of electrical current. In these cases, the dissipation is associ- ated with the spatial gradient of some variable (e. g. the hydraulic head for flow in a porous medium, the voltage for an electrical problem). Constitutive behav- iour is usually described by a linear relationship between the flux and the spatial gradient. The second type of dissipation is associated with the temporal variation of in- ternal variables. The plasticity problems treated in earlier chapters are of this character. Viscous behaviour can also be described in this way. Most texts that treat the thermodynamics of dissipative continua concentrate either on fluxes or on rates of change of internal variables. However, whilst the two problems have much in common, they also have important differences. Most obviously, one involves a spatial variation and the other a temporal varia- tion. It is tempting to treat both in the same way, and many texts adopt this 242 12 Behaviour of Porous Continua approach, using, for instance, “generalised forces” and “generalised fluxes”. Here, we adopt a slightly different approach, keeping separate those variables associated with fluxes and those associated with internal variables. In this way, the different ways that the two types of process appear in the relevant equations can be made clearer. Rather than considering the possibility of abstract, unspecified fluxes, we find it more useful to consider a concrete example. The case that we consider is a very important problem in geomechanics and other fields, namely, flow in a porous medium. This is a useful example because the flux itself has mass, which introduces a number of features to the problem that need careful treat- ment. The porous medium has to be treated as consisting of two phases, and there is a partition of the extensive quantities (e. g. internal energy, entropy) between the solid skeleton and fluid phases. In previous chapters, we adopted a small strain formulation. The problem of coupled fluid and skeleton behaviour cannot be treated rigorously within the small strain framework, because there is a coupling between strains, fluid flow, and density changes. In the small strain formulation, the density is treated as a constant. In the following therefore, it is necessary to move to a large strain formulation. There is a choice between adopting a Lagrangian approach, in which the problem is formulated in terms of initial coordinates, and an Eulerian approach, in which it is formulated within the current coordinates. We adopt the Eulerian approach for much of the following development because this allows a more direct interpretation of the variables. It will prove necessary, however, to transform to Lagrangian variables for part of the analysis. In the small strain approach, for convenience, all extensive quantities were defined per unit volume. Since the density was in effect constant, this is equiva- lent to using extensive quantities per unit mass, but avoids a factor of the den- sity appearing throughout the equations. In large strain analysis, it is necessary to use extensive quantities per unit mass, as is usual in thermodynamics, and we adopt this approach below. 12.2 Thermomechanical Framework As mentioned above, we adopt here an Eulerian approach to describe a material undergoing large strain, i. e. the description of the material is based on the cur- rent coordinate system. In this, it will be necessary to distinguish between the time differential of a variable x at a particular point in space, which we shall denote by xt xww , and the material or convective derivative, which represents the rate of change of an element of the material, which has a current velocity i v . We denote the material derivative by ,ii dx dt x x x v . From this definition, it follows that the chain rule applies to the convective derivative, e. g. the mate- rial derivative of xy is xy xy . 12.2 Thermomechanical Framework 243 12.2.1 Density Definitions, Velocities, and Balance Laws Consider a volume V fixed in space bounded by a surface S. The unit outward normal to the boundary is i n . The volume contains porous material with a skeleton material of density s U and with a porosity n (volume of voids divided by total volume). Thus the mass of skeleton per unit total volume is 1 s nU U . We should also note that U is the “dry density” in the terminology of soil mechanics. The velocity of the skeleton at any point is i v , so that the mass flux of the skeleton per unit area is i vU , and the outward mass flux per unit area from V is ii vnU . For conservation of mass, we can write that the rate of increase of mass within the volume, plus the outward mass flux is zero: 0 ii VS dV v n dSUU ³³ (12.1) Applying Gauss’s divergence theorem 1 , we can write , 0 i i V vdVU U ³ . Then noting that V is arbitrary, we can write this in local form: ,, , , 0 iiiiiii i vvvvU U UU U UU (12.2) which establishes the link between the material rate of change of dry density and the dilatation rate. A comment is relevant here about the importance of the assumption that the volume V is arbitrary. This is only justified provided that V is large enough so that averaged values of stresses, strains, etc., over the volume element are mean- ingful. Such an element is said to be a “representative volume element”. In the context of the mechanics of granular materials, this will typically require that the element contains many thousands of particles. At the same time, the element must be sufficiently small so that changes of stresses, etc., across the element are small. This requirement conflicts with the first, and there are classes of problems for which both criteria cannot be satisfied simultaneously. Such problems (e. g. those involving strong localisation) are not amenable to treatment by conven- tional continuum mechanics. We now allow for the possibility of fluxes of a pore fluid. We shall consider a pore fluid, the amount of which is specified by the parameter w defined as mass of fluid per unit mass of skeleton material (i. e. the water content in the terminology of soil mechanics). Note that in the study of the mechanics of granular media, a wide variety of different quantities are used to define the amount of fluid in the porous medium. The flux of the fluid mass is i m per unit area relative to the skeleton. The total flux vector of the fluid is therefore 1 In the above terminology, Gauss’s divergence theorem states that for any variable x that is con- tinuous and differentiable in V, ,ii SV xn dS x dV ³³ . 244 12 Behaviour of Porous Continua ii mwvU , and the outward flux of the fluid across the boundary S follows as iii mwvnU . We note that the mass of fluid per unit volume of skeleton is Uw. It follows that w wnU U (12.3) where w U is the density of the fluid. The mass flux vector i m can also be written as www ii ii mwnvv U U (12.4) where i w is the Darcy artificial seepage velocity and w i v is the average absolute velocity of the fluid. Noting that the mass of the fluid is conserved, there is a balance equation analogous to (12.1) of the form: 0 iii t VS wdV m wv ndS w w UU ³³ (12.5) which we can rewrite in local form by using the divergence theorem of Gauss to obtain the local conservation law: ,, , , 0 ii i i i i ii wwm wv wv wvU U U U U (12.6) or ,, 0 ii ii wwm wvUU U (12.7) By virtue of the skeleton mass conservation, Equation (12.2), this becomes , 0 ii wmU (12.8) It is convenient to obtain a combined continuity equation for flow of the skeleton and pore fluid. First, we can note that 1 ss nnU U U , so that we can rewrite the mass continuity equation as , 110 sss ii nn nvUUU (12.9) By manipulation of (12.7), we can also obtain , , 0 ww w w iii i nn wnvUUU U (12.10) Finally dividing (12.9) by s U and (12.10) by w U and adding, we obtain , ,, 10 w ws i ii ii i ww s vww n n U UU UU U (12.11) 12.2 Thermomechanical Framework 245 If both the soil grains and the pore fluid are incompressible, then this reduces to the simple form ,, 0 ii ii vw . Introducing 1 ww v U and 1 ss v U, the con- tinuity equation can also be written ,, , wws ii ii i i vwmv wv v U U , where 1 sw vv wv U (12.12) 12.2.2 Tractions, Stresses, Work, and Energy The tractions (forces per unit area) on the skeleton on the fraction 1 n of the boundary S are i t , and the pressure in the pore fluid is p which acts on a fraction n of the boundary. The work done per unit area by the surroundings against the tractions on S is therefore 1 ii ntv , and that done against the pore pressure is w ii npn v . There are also body forces arising from the gravitational field of strength i g . The work done per unit volume by the body forces on the skeleton is ii vgU and on the fluid is w ii wv gU . The heat flux per unit area is i q , so that the outward heat flux from S per unit area is ii qn . As an extensive quantity, the kinetic energy of all the matter enclosed in vol- ume V may be written as the sum of the kinetic energies of the skeleton and of the fluid: 2 2 11 22 w ii VV K v dV w v dV U U ³³ (12.13) At this stage, we are neglecting tortuosity effects, which are due to the fact that the pore fluid must take a tortuous path between the skeleton particles, so that the average speed of the water particles is higher than the magnitude of the average velocity. We shall, however, show how the results can be modified later to take this into account. Now consider the rate of change in kinetic energy in the volume V, which can be written 22 22 ww ii i i V ww w ii i i jj j j SS vv wv v KdV t vv wv v v n dS v n dS §· UU w ¨¸ ¨¸ w ©¹ §· UU §· ¨¸ ¨¸ ¨¸ ©¹ ©¹ ³ ³³ (12.14) The volume integral reflects changes in kinetic energy with time in the vol- ume, and the surface integrals account for the kinetic energy brought into the volume due to the skeleton and pore fluid movement through the surface. Ap- plying the theorem of Gauss and grouping the resulting terms, 246 12 Behaviour of Porous Continua ,,, , ,, , 2 2 ii ii ijj jj jj VV ww ww ii ijj V ww www ii jj j j j j V vv KvvvvdV v v dV wv v v v dV vv wwwv wv wv dV §· U UU U ¨¸ ©¹ U §· U U U U U ¨¸ ¨¸ ©¹ ³³ ³ ³ (12.15) Recalling the mass balance equations for the skeleton and for the fluid, (12.2) and (12.6), respectively, we note that the second and fourth integrals vanish. We introduce also the definitions of the accelerations of the skeleton and fluid parti- cles, respectively: ,iiiijj avvvv (12.16) , wwwww iiiijj avvvv (12.17) where the material derivative with respect to a fluid particle is denoted by , w ii xxv x . The expression for the rate of change of kinetic energy becomes ww ii i i V ww iii ii VV KvawvadV a wa v dV m a dV U U UU ³ ³³ (12.18) 12.2.3 The First Law The First Law of Thermodynamics states that there is a variable, called specific internal energy, such that the rate of increase of internal energy in the volume plus the rate of change of the kinetic energy in this volume is equal to the sum of the rates of energy input at the boundaries plus the rate of work of the body forces in the volume. We attribute a specific internal energy s u to the skeleton and w u to the pore fluid. The first law therefore becomes 1 sw s ww iii t VS ww iii ii VV ww ii ii i i i ii SVS uwudV uvwuvndS awavdVmadV ntv npnv dS v wv gdV qn dS w w UU U U UU ªº UU ¬¼ ³³ ³³ ³³³ (12.19) [...]... function: that it is a potential for stresses and temperature The basic form of these relationships are well known from hyperelasticity, but the particular expressions here deserve some comment First, note that the intensive quantities of pore pressure and temperature each appear as a partial derivative of the internal energy of the skeleton and of the pore fluid The fact that both derivatives are related... the same value of the intensive variable reflects an assumption of intimate mixing of the two phases The temperature of the solids and fluid is assumed to be the same, and the pore pressure acts equally on the solids and the fluid Later we will find it convenient to use separate variables for the two phases, but we shall assume that the values are equal (effectively making the assumption of intimate... usually referred to as that of complementary shear stresses From the former, it follows that ij ,i 1 w gj aj waw j 0 (12.27) which can be recognised as the equations of motion (or the static equilibrium equations in the case of zero acceleration) Equation (12.27) expresses the momentum balance for the porous medium considered as a whole and has been derived as part of a formulation rather than postulated... physical meanings: K s and K w are the isothermal bulk moduli of the skeleton particles and fluid, respectively; 3 s and 3 w are the volumetric thermal expansion coefficients of the skeleton particles and fluid, respectively; 12.6 Example 259 c s and c w are the mass heat capacity at constant pressure po of the skeleton p p particles and fluid, respectively; K and G are the isothermal bulk and shear moduli... right-hand side of (12.35) includes three types of term The first involves the strain rate The second type involves material differentials, and the third involves fluxes The presence of the strain rate poses a problem within the Eulerian formulation because it is not possible to express the strain rate as a material derivative of any observable quantity This problem can be avoided by adopting a Lagrangian... Dissipation Function and Force Potential We also postulate that dissipation is a function of the same state variables, and also of ij (rate of change of internal variables) and of the fluxes, i e., dt dt ij , ij , v s , s s , w, v w , sw , ij , mi , i (12.40) We can either derive the force potential z, from the dissipation, using the procedure described in Section 11. 1.2, or we can assume the form of z and... Formulation 255 longer realistic We address below, however, some of the advantages that follow from adopting the more restrictive approach It is our belief that a very wide variety of material response can be described within this framework Furthermore, we are not aware of any specific counterexamples from the physical world that provide clear evidence that the restrictions imposed above are invalid 12.3 The... is also necessary to introduce the variable pL p det Pij Pik1Pik1 which is the transformation of the pore pres- sure to the Lagrangian coordinate system We note that in Lagrangian coordinates, no distinction is necessary between the time and material derivatives so that ij ij t d ij dt ij In principle, it would be possible to transform all other variables to Lagrangian coordinates, too, but this has... 12.2.4 Equations of Motion No change in internal energy should, however, be caused by either a rigid body translation or rotation, so that we can conclude that 1 w gj aj waw v j 0 and ij ji 0 for all v j and ji These ij ,i j are the virtual work forms of the direct and rotational equilibrium conditions From the latter, it follows that the antisymmetric part of ij must be zero, i e ij is symmetrical This... formulation We can rewrite 1 ij p ij dij 1 ij p L ij ij (12.36) o where ij is the Piola-Kirchhoff stress tensor and ij is the Green-Lagrange xi X j and xi and Xi are the strain defined by 2 ij Pki Pkj ij where Pij current (Eulerian) and initial (Lagrangian) coordinates of a material point measured in a Cartesian system The initial dry density is o It can be shown 1 1 that ij det Pij Pik kl Pjl and ij . terms of strain rate: (a) data from Vaid and Campanella (1977) and (b) the theoretical model 238 11 Rate Effects shows the equivalent calculations. Apart from the detail of the curve at 1 0.5 c c VV. because this allows a more direct interpretation of the variables. It will prove necessary, however, to transform to Lagrangian variables for part of the analysis. In the small strain approach,. 236 11 Rate Effects Figure 11. 8. Variation of undrained strength with strain rate: comparison between theory and data from Vaid and Campanella (1977) not captured accurately. Figure 11. 8
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