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+
+# Chapter 8 Elasto-viscoplastic problems in two dimensions
+
+# 8.1 Introduction
+
+In all inelastic deformations time rate effects are always present to some degree. Whether or not their exclusion has a significant influence on the prediction of the material behaviour depends upon several factors. In the study of structural components under static loading conditions at normal temperatures it is accepted that time rate effects are generally not important and the conventional theory of plasticity, as described in Chapter 7, then models the behaviour adequately. However metals, especially under high temperatures, exhibit simultaneously the phenomena of creep and visco-plasticity. The former is essentially a redistribution of stress and/or strains with time under elastic material response while the latter is a time dependent plastic deformation. Experimental observations cannot distinguish between the two phenomena and their separation has been largely an analytical convenience rather than a physical requirement. Numerical processes, as described in this chapter, allow the simultaneous description of both effects.
+
+A further situation in which time rate effects are important is in the dynamic transient loading of structures. For example, it can be experimentally demonstrated that the instantaneous yield stress of materials under high strain rates can be significantly greater than the corresponding quasi-static value. This class of problem is dealt with in Chapter 10.
+
+In this chapter we utilise the theory of viscoplasticity to provide a unified approach to problems of creep and plasticity. As well as providing solutions to time-dependent situations the viscoplastic algorithm can provide economic solution for classic elasto-plastic problems since it can be readily shown that the steady-state solution of the viscoplastic problem is identical to the corresponding conventional static elasto-plastic solution. Furthermore, by reducing the yield stress of the material to zero, elastic creep problems can be solved.
+
+The concept of ‘overlay models’ is also introduced in this chapter. In this, the solid is assumed, for mathematical convenience only, to be composed of several layers or overlays each of which undergo the same deformation. By assigning different properties to each overlay a composite behaviour can be
+
+<!-- source-page: 282 -->
+
+obtained which exhibits all the essential characteristics of the visco-elastic-plastic response of many real materials.
+
+The basic one-dimensional rheological model developed in Chapter 4 is now extended to the case of a general continuum and the essential steps employed in the numerical solution algorithm are discussed. Since most of the matrix expressions involved in viscoplastic analysis are common to conventional elasto-plastic theory, the majority of the subroutines developed in Chapter 7 can be again used with little or no change. The additional subroutines required are then constructed and assembled to form a working program. Finally it is briefly demonstrated how the overlay principle can be used to simulate a complex material response.
+
+# 8.2 Theory of elasto-viscoplastic solids
+
+# 8.2.1 Basic expressions
+
+In the usual manner for nonlinear continua problems it is assumed that the total strain, $\epsilon$ , can be separated into elastic, $\epsilon_{e}$ , and viscoplastic, $\epsilon_{vp}$ , components, so that the total strain rate can be expressed as $^{(1-3)}$
+
+$$
+\dot {\epsilon} = \dot {\epsilon} _ {e} + \dot {\epsilon} _ {v p}, \tag {8.1}
+$$
+
+where ( $\cdot$ ) represents differentiation with respect to time. The total stress rate depends on the elastic strain rate according to
+
+$$
+\dot {\sigma} = D \dot {\epsilon} _ {e}, \tag {8.2}
+$$
+
+where D is the elasticity matrix. The onset of viscoplastic behaviour is governed by a scalar yield condition of the form
+
+$$
+F (\sigma , \epsilon_ {v p}) - F _ {0} = 0, \tag {8.3}
+$$
+
+in which $F_{0}$ is the uniaxial yield stress which may itself be a function of a hardening parameter, $\kappa$ . For frictional materials $F_{0}$ is the equivalent yield stress as given by Column 4, Table 7.2. It is assumed that viscoplastic flow occurs for values of $F > F_{0}$ only.
+
+It is now necessary to choose a specific law defining the viscoplastic strains. The simplest option is one in which the viscoplastic strain rate depends only on the current stresses, so that
+
+$$
+\dot {\epsilon} _ {v p} = f (\sigma). \tag {8.4}
+$$
+
+This relationship can be generalised to include strain hardening and temperature dependence and the influence of state dependent variables, such as damage parameters for rupture theories, can also be considered.
+
+<!-- source-page: 283 -->
+
+One explicit form of (8.4) which has wide applicability, is offered by the following viscoplastic flow rule. $^{(4)}$
+
+$$
+\dot {\epsilon} _ {v p} = \gamma \langle \Phi (F) \rangle \frac {\partial Q}{\partial \sigma}, \tag {8.5}
+$$
+
+in which $Q = Q(\sigma, \epsilon_{vp}, \kappa)$ is a 'plastic' potential and $\gamma$ is a fluidity parameter controlling the plastic flow rate. The term $\Phi(x)$ is a positive monotonic increasing function for $x > 0$ and the notation $\langle \rangle$ implies
+
+$$
+\begin{array}{l} \langle \Phi (x) \rangle = \Phi (x) \text {   for   } x > 0 \\ \langle \Phi (x) \rangle = 0 \quad x \leqslant 0. \tag {8.6} \\ \end{array}
+$$
+
+Comparison of (8.5) with (7.28) shows an analogy between the flow rule of conventional non-associated plasticity and the present definition of viscoplastic flow rate. If, once again, we restrict ourselves to associated plasticity situations, in which case $F \equiv Q$ , expression (8.5) reduces to
+
+$$
+\dot {\epsilon} _ {v p} = \gamma \langle \Phi (F) \rangle \frac {\partial F}{\partial \sigma} = \gamma \langle \Phi \rangle a, \tag {8.7}
+$$
+
+where the same definition of the flow vector a is employed as in (7.42). Different choices have been recommended $^{(5)}$ for the function $\Phi$ . The two most common versions are
+
+$$
+\Phi (F) = e ^ {M \left(\frac {F - F _ {0}}{F _ {0}}\right)} - 1, \tag {8.8}
+$$
+
+and
+
+$$
+\Phi (F) = \left(\frac {F - F _ {0}}{F _ {0}}\right) ^ {N}, \tag {8.9}
+$$
+
+in which M and N are arbitrary prescribed constants. The latter option, when employed in (8.7) can be made to model the Norton power law of metallic creep by assigning the threshold uniaxial yield value, $F_{0}$ , to zero (or to an arbitrarily small value for numerical convenience).
+
+# 8.2.2 The viscoplastic strain increment
+
+With the strain rate law expressed by (8.7) we can define a strain increment $\Delta\epsilon_{vp}^{n}$ occurring in a time interval $\Delta t_{n}=t_{n+1}-t_{n}$ using an implicit time stepping scheme, as $^{(6)}$
+
+$$
+\Delta \epsilon_ {v p} ^ {n} = \Delta t _ {n} \left[ (1 - \Theta) \dot {\epsilon} _ {v p} ^ {n} + \Theta \dot {\epsilon} _ {v p} ^ {n + 1} \right]. \tag {8.10}
+$$
+
+For $\Theta = 0$ we obtain the Euler time integration scheme which is also referred to as 'fully explicit' (or forward difference method) since the strain increment is completely determined from conditions existing at time, $t_n$ . On the other
+
+<!-- source-page: 284 -->
+
+hand $\Theta = 1$ gives a 'fully implicit' (or backward difference) scheme with the strain increment being determined from the strain rate corresponding to the end of the time interval. The case $\Theta = \frac{1}{2}$ results in the so-called 'implicit trapezoidal' scheme which is also known generally as the Crank-Nicolson rule in the context of linear equations.
+
+To define $\dot{\epsilon}_{vp}^{n+1}$ in (8.10) we can use a limited Taylor series expansion and write
+
+$$
+\dot {\epsilon} _ {v p} ^ {n + 1} = \dot {\epsilon} _ {v p} ^ {n} + H ^ {n} \Delta \sigma^ {n}, \tag {8.11}
+$$
+
+where
+
+$$
+\boldsymbol {H} ^ {n} = \left(\frac {\partial \dot {\epsilon} _ {v p}}{\partial \sigma}\right) ^ {n} = \boldsymbol {H} ^ {n} (\sigma^ {n}), \tag {8.12}
+$$
+
+and $\Delta\sigma^{n}$ is the stress change occurring in the time interval $\Delta t_{n}=t_{n+1}-t_{n}$ . Thus (8.10) can be rewritten as
+
+$$
+\Delta \epsilon_ {v p} ^ {n} = \dot {\epsilon} _ {v p} ^ {n} \Delta t _ {n} + C ^ {n} \Delta \sigma^ {n}, \tag {8.13}
+$$
+
+where
+
+$$
+\boldsymbol {C} ^ {n} = \Theta \Delta t _ {n} \boldsymbol {H} ^ {n}. \tag {8.14}
+$$
+
+We draw the attention of the reader to the fact that the matrix H defined in (8.12) is the matrix whose eigenvalues determine the limiting time step length, $\Delta t_{n}$ which can be employed in the explicit integration schemes. The matrix H depends on the stress level and no difficulty arises in its evaluation and specific forms will be developed in Section 8.5.
+
+# 8.2.3 Stress increments
+
+Using the incremental form of (8.2) we obtain
+
+$$
+\Delta \sigma^ {n} = D \Delta \epsilon_ {e} ^ {n} = D (\Delta \epsilon^ {n} - \Delta \epsilon_ {v p} ^ {n}). \tag {8.15}
+$$
+
+Or expressing the total strain increment in terms of the displacement increment as
+
+$$
+\Delta \epsilon^ {n} = B ^ {n} \Delta d ^ {n}, \tag {8.16}
+$$
+
+and substituting for $\Delta \epsilon_{vp}^n$ from (8.13), then (8.15) becomes
+
+$$
+\Delta \sigma^ {n} = \hat {D} ^ {n} (B ^ {n} \Delta d ^ {n} - \dot {\epsilon} _ {v p} ^ {n} \Delta t _ {n}), \tag {8.17}
+$$
+
+where
+
+$$
+\hat {\boldsymbol {D}} ^ {n} = (\boldsymbol {I} + \boldsymbol {D} \boldsymbol {C} ^ {n}) ^ {- 1} \boldsymbol {D} = (\boldsymbol {D} ^ {- 1} + \boldsymbol {C} ^ {n}) ^ {- 1}. \tag {8.18}
+$$
+
+In (8.16) and (8.17) the notation $B^{n}$ is employed to denote the possibility that the strain matrix may not be constant throughout the solution. For example, if large deformations are to be considered, the strain matrix for a Lagrangian formulation is nonlinear and can be written
+
+<!-- source-page: 285 -->
+
+$$
+\boldsymbol {B} ^ {n} = \boldsymbol {B} _ {0} + \boldsymbol {B} _ {N L} ^ {n}, \tag {8.19}
+$$
+
+where $B_{0}$ represents the standard linear terms which do not vary during solution and $B_{NL}^{n}$ contains the nonlinear quadratic terms. These latter expressions are dependent on the current displacements and therefore vary throughout the solution process.
+
+The matrix $D^{n}$ is a symmetric matrix when the visco-plastic law is associative. For the non-associated case, the matrix $C^{n}$ is unsymmetric, requiring unsymmetric equation solvers for analysis.
+
+For the solution of linear elastic problems by the explicit scheme ( $\Theta = 0$ ), equation (8.17) simplifies considerably to give
+
+$$
+\Delta \sigma^ {n} = D (B \Delta d ^ {n} - \dot {\epsilon} _ {v p} ^ {n} \Delta t _ {n}). \tag {8.20}
+$$
+
+# 8.2.4 Equations of equilibrium
+
+The equations of equilibrium to be satisfied at any instant of time, $t_n$ , are
+
+$$
+\int_ {\Omega} [ \boldsymbol {B} ^ {n} ] ^ {T} \boldsymbol {\sigma} ^ {n} d \Omega + \boldsymbol {f} ^ {n} = \mathbf {0}, \tag {8.21}
+$$
+
+where $f^{n}$ is the vector of equivalent nodal loads due to applied surface tractions, body forces, thermal loads, etc. During a time increment the equilibrium equations which must be satisfied are given by the incremental form of (8.21) to be
+
+$$
+\int_ {\Omega} [ \boldsymbol {B} ^ {n} ] ^ {T} \Delta \sigma^ {n} d \Omega + \Delta \boldsymbol {f} ^ {n} = \mathbf {0}, \tag {8.22}
+$$
+
+in which $\Delta f^{n}$ represents the change in loads during the time interval $\Delta t_{n}$ . In the majority of problems encountered in engineering the load increments are applied as discrete steps and thus $\Delta f^{n} = 0$ for all time steps other than the first within an increment.
+
+Using (8.13) and (8.20) the displacement increment occurring during time step $\Delta t_{n}$ can be calculated as
+
+$$
+\Delta d ^ {n} = \left[ K _ {T} ^ {n} \right] ^ {- 1} \Delta V ^ {n}
+$$
+
+$$
+\Delta V ^ {n} = \int_ {\Omega} [ B ^ {n} ] ^ {T} \hat {D} ^ {n} \dot {\epsilon} _ {v p} ^ {n} \Delta t _ {n} d \Omega + \Delta f ^ {n}, \tag {8.23}
+$$
+
+where $K_{T}^{n}$ is the tangential stiffness matrix with the following form
+
+$$
+\boldsymbol {K} _ {T} ^ {n} = \int_ {\Omega} [ \boldsymbol {B} ^ {n} ] ^ {T} \hat {\boldsymbol {D}} ^ {n} \boldsymbol {B} ^ {n} d \Omega , \tag {8.24}
+$$
+
+and $\Delta V^{n}$ are termed the incremental pseudo-loads. The displacement increments, $\Delta d^{n}$ , when substituted back into (8.20) give the stress increments
+
+<!-- source-page: 286 -->
+
+$\Delta \sigma^n$ and thus
+
+$$
+\sigma^ {n + 1} = \sigma^ {n} + \Delta \sigma^ {n}
+$$
+
+$$
+\boldsymbol {d} ^ {n + 1} = \boldsymbol {d} ^ {n} + \Delta \boldsymbol {d} ^ {n}. \tag {8.25}
+$$
+
+Use of (8.15) and (8.16) gives
+
+$$
+\Delta \epsilon_ {v p} ^ {n} = B ^ {n} \Delta d ^ {n} - D ^ {- 1} \Delta \sigma^ {n}, \tag {8.26}
+$$
+
+and then
+
+$$
+\epsilon_ {v p} ^ {n + 1} = \epsilon_ {v p} ^ {n} + \Delta \epsilon_ {v p} ^ {n}. \tag {8.27}
+$$
+
+Arrival at stationary or steady state conditions can be monitored by examination of the strain rates. In particular $\dot{\epsilon}_{vp}$ , as given by (8.7), is calculated at each time interval and the time marching process halted as soon as this quantity becomes tolerably small.
+
+# 8.2.5 Equilibrium correction
+
+The stress increment calculation is based on a linearised form of the incremental equilibrium equations (8.22). Therefore the total stresses, $\sigma^{n+1}$ , obtained by accumulating all such stress increments are not strictly correct and will not exactly satisfy the equations of equilibrium, (8.21). There are several solution procedures available for applying the necessary correction and Reference 7 discusses the relative merits of various options. The simplest approach is to evaluate $\sigma^{n+1}$ according to (8.20) and (8.25) and then compute the residual, or out-of-balance, forces, $\psi$ , as
+
+$$
+\psi^ {n + 1} = \int_ {\Omega} [ B ^ {n + 1} ] ^ {T} \sigma^ {n + 1} d \Omega + f ^ {n + 1} \neq 0, \tag {8.28}
+$$
+
+noting, for geometrically nonlinear problems, that $B^{n+1}$ is evaluated for a displacement state $d^{n+1}$ . This residual force is then added to the applied force increment at the next time step. Such a technique avoids an iteration process and at the same time achieves a reduction in error.
+
+# 8.3 Selection of the time step length
+
+It can be shown $^{(14)}$ that the time integration scheme formally represented by (8.10) is unconditionally stable for values of $\Theta\geqslant\frac{1}{2}$ . This implies that the time marching scheme is numerically stable but does not guarantee the accuracy of the solution at any stage; so that in practice even for values of $\Theta\geqslant\frac{1}{2}$ limits must be placed on the time step length in order to achieve a valid solution.
+
+For $\Theta<\frac{1}{2}$ the integration process is only conditionally stable and numerical time integration can only proceed for values of $\Delta t_{n}$ less than some critical value. We now proceed to establish rules for choosing the time step length for computation.
+
+<!-- source-page: 287 -->
+
+Schemes can be employed in which the time step length can be either constant or vary for each time interval. In the variable scheme the magnitude of the time step is controlled by a factor $\tau$ which limits the maximum effective viscoplastic strain increment, $\Delta \bar{\epsilon}_{vp^n}$ as a fraction of the total effective strain, $\bar{\epsilon}^n$ , so that
+
+$$
+\Delta \tilde {\epsilon} _ {v p} ^ {n} = (\sqrt {\frac {2}{3}}) \left\{\dot {\epsilon} _ {i j} ^ {n}\right) _ {v p} \left(\dot {\epsilon} _ {i j} ^ {n}\right) _ {v p} \} ^ {1 / 2} \Delta t _ {n} \leqslant \tau \bar {\epsilon} ^ {n}. \tag {8.29}
+$$
+
+For isoparametric elements, all strains are evaluated at the Gaussian integration points. Therefore $\Delta t_{n}$ must be computed to satisfy (8.29) at each such point and the least value taken for analysis. A variant on the above is to limit the time step length according to
+
+$$
+\left\{\dot {\epsilon} _ {i i} ^ {n} \right\} ^ {\frac {1}{2}} v p \Delta t _ {n} \leqslant \tau \left\{\epsilon_ {i i} ^ {n} \right\} ^ {\frac {1}{2}}, \tag {8.30}
+$$
+
+in which $\epsilon_{tt}^{n}$ is the first total strain invariant and $(\dot{\epsilon}_{ii}^{n})_{vp}$ is the first viscoplastic strain rate invariant. Thus $\Delta t_{n}$ can be formally written for this case as
+
+$$
+\Delta t _ {n} \leqslant \tau \left[ \epsilon_ {i i} ^ {n} / \left(\dot {\epsilon} _ {i i} ^ {n}\right) _ {v p} \right] ^ {\frac {1}{2}} \min. \tag {8.31}
+$$
+
+The minimum in (8.31) is that taken over all integrating points in the solid. The value of the time increment parameter $\tau$ must be specified by the user and for explicit time marching schemes accurate results have been obtained $^{(4,8)}$ in the range $0.01 < \tau < 0.15$ . For implicit schemes, values of $\tau$ up to 10 have been found to be stable though the accuracy deteriorates.
+
+Another useful limit can be imposed while using the variable time stepping scheme. The change in the time step length between any two intervals is limited according to
+
+$$
+\Delta t _ {n + 1} \leqslant k \Delta t _ {n}, \tag {8.32}
+$$
+
+where k is a specified constant. Experience suggests a value of $k = 1 \cdot 5$ to be suitable although there are no fixed criteria for its specification.
+
+The above time step limiting values are basically empirical. Theoretical restrictions on the time step length have been provided by Cormeau $^{(9)}$ for specific forms of the viscoplastic flow rule and for explicit time integration only. In particular, for associated viscoplasticity $Q \equiv F$ and a linear function $\Phi(F) = F$ we have the following limits on the time step length.
+
+$$
+\Delta t \leqslant \frac {(1 + \nu) F _ {0}}{\gamma E} \quad \text { for   Tresca   materials }
+$$
+
+$$
+\Delta t \leqslant \frac {4 (1 + \nu) F _ {0}}{3 \gamma E} \quad \text { Von   Mises }
+$$
+
+$$
+\Delta t \leqslant \frac {4 (1 + \nu) (1 - 2 \nu) F _ {0}}{\gamma (1 - 2 \nu + \sin^ {2} \phi) E} \quad \text { Mohr   -   Coulomb }, \tag {8.33}
+$$
+
+<!-- source-page: 288 -->
+
+where $\gamma$ is the fluidity parameter and $\phi$ is the angle of internal friction. The term $F_{0}$ is the uniaxial yield stress for Tresca and Von Mises solids and is the equivalent value ( $c \cos \phi$ ) for Mohr–Coulomb materials where c is the cohesion. No simple expression exists for the limiting time step length in Drucker–Prager solids.
+
+# 8.4 Computational procedure
+
+The essential steps in the solution process can be summarised as follows. Solution to the problem must begin from the known initial conditions at time t = 0, which are, of course, the solution of the static elastic situation. At this stage $d^{0}$ , $F^{0}$ , $\epsilon^{0}$ , $\sigma^{0}$ are known and $\epsilon_{vp^{0}} = 0$ . The time marching scheme described in Section 8.2.4 can then be employed to advance the solution by one timestep at a time. The solution sequence adopted is as follows.
+
+Stage 1 Suppose at time $t = t_{n}$ we have an equilibrium situation and $d^{n}, \sigma^{n}$ , $\epsilon^{n}, \epsilon_{vp}^{n}$ , $F^{n}$ are known. The following quantities are assembled:
+
+(a) $\pmb{B}^n = \pmb{B}_0 + \pmb{B}_{NL}(\pmb{d}^n),$
+
+(b) $C^n = C^n (\sigma^n,\Delta t_n),$
+
+(c) $\hat{D}^n = (D^{-1} + C^n)^{-1},$
+
+(d) $K_{T^n} = \int_{\Omega} [B^n]^T \hat{D}^n B^n d\Omega,$
+
+(e) $\dot{\epsilon}_{vp}^{n} = \gamma \langle \Phi \rangle a^{n}$ .
+
+Stage 2 i) Compute the displacement increments $\Delta d^n$ according to (8.23) as
+
+$$
+\Delta d ^ {n} = [ K _ {T} ^ {n} ] ^ {- 1} \Delta V ^ {n},
+$$
+
+where
+
+$$
+\Delta V ^ {n} = \int_ {\Omega} [ B ^ {n} ] ^ {T} \hat {D} ^ {n} \dot {\epsilon} _ {v p} ^ {n} \Delta t _ {n} d \Omega + \Delta f ^ {n}.
+$$
+
+ii) Calculate the stress increment $\Delta\sigma^{n}$ as
+
+$$
+\Delta \sigma^ {n} = \hat {D} ^ {n} (B ^ {n} \Delta d ^ {n} - \dot {\epsilon} _ {v p} ^ {n} \Delta t _ {n}).
+$$
+
+Stage 3 Determine the total displacements and stresses
+
+$$
+\boldsymbol {d} ^ {n + 1} = \boldsymbol {d} ^ {n} + \Delta \boldsymbol {d} ^ {n}
+$$
+
+$$
+\sigma^ {n + 1} = \sigma^ {n} + \Delta \sigma^ {n}.
+$$
+
+Stage 4 Calculate the viscoplastic strain rate
+
+$$
+\dot {\epsilon} _ {v p} ^ {n + 1} = \gamma \langle \Phi \rangle a ^ {n + 1}.
+$$
+
+Stage 5 Apply the equilibrium correction. First calculate $B^{n+1}$ using dis-
+
+<!-- source-page: 289 -->
+
+placements $d^{n+1}$ . Substitute stresses $\sigma^{n+1}$ into the equilibrium equations and evaluate the residual forces $\psi^{n+1}$ as
+
+$$
+\psi^ {n + 1} = \int_ {\Omega} [ B ^ {n + 1} ] ^ {T} \sigma^ {n + 1} d \Omega + f ^ {n + 1}.
+$$
+
+Add these to the vector of incremental pseudo loads for use in the next time step
+
+$$
+\Delta \boldsymbol {V} ^ {n + 1} = \int_ {\Omega} [ \boldsymbol {B} ^ {n + 1} ] ^ {T} \hat {\boldsymbol {D}} ^ {n + 1} \dot {\boldsymbol {\epsilon}} _ {v p} ^ {n + 1} \Delta t _ {n + 1} d \Omega + \Delta \boldsymbol {f} ^ {n + 1} + \boldsymbol {\psi} ^ {n + 1}. \tag {8.34}
+$$
+
+Stage 6 Check to see if the viscoplastic strain rate $\dot{\epsilon}_{vp}^{n+1}$ is acceptably close to zero at each Gaussian integrating point throughout the structure (i.e. to within a specified tolerance).
+
+If so, steady state conditions are deemed to have been achieved and the solution is either terminated or the next load increment is applied. If $\dot{\epsilon}_{vp}^{n+1}$ is non-zero return to Stage 1 and repeat the entire procedure for the next time step.
+
+The above algorithm can be employed with either a constant or variable time step length. For the variable time step option the interval length $\Delta t_{n+1}$ , for the next time step must be calculated according to (8.29) or (8.31) subject to the restriction of (8.32).
+
+# 8.5 Evaluation of matrix, H
+
+For solution by the fully implicit or semi-implicit (trapezoidal) time stepping scheme, matrix $C^{n}$ is required which in turn can be expressed in terms of $H^{n}$ as indicated in (8.14). Matrix $H^{n}$ must be explicitly determined for the yield criterion assumed for material behaviour. From (8.7) and (8.12) we have
+
+$$
+\boldsymbol {H} = \frac {\partial \dot {\boldsymbol {\epsilon}} _ {v p}}{\partial \boldsymbol {\sigma} ^ {n}} = \gamma \left\{\Phi \frac {\partial \boldsymbol {a} ^ {T}}{\partial \boldsymbol {\sigma}} + \frac {d \Phi}{d F} \boldsymbol {a} \boldsymbol {a} ^ {T} \right\}, \tag {8.35}
+$$
+
+where the symbols $\langle\rangle$ on $\Phi$ and the superscript n are dropped for convenience. Restricting discussion to the Von Mises yield criterion we have, from (7.64),
+
+$$
+\boldsymbol {a} ^ {\prime} = \frac {\partial F}{\partial \sigma} = \frac {\partial [ (\sqrt {3}) (J _ {2} ^ {\prime}) ^ {1 / 2} ]}{\partial \sigma}, \tag {8.36}
+$$
+
+or
+
+$$
+\boldsymbol {a} ^ {\prime} = \frac {\partial F}{\partial J _ {2} ^ {\prime}} \frac {\partial J _ {2} ^ {\prime}}{\partial \sigma} = \frac {\sqrt {3}}{2 \left(J _ {2} ^ {\prime}\right) ^ {1 / 2}} \left\{\sigma_ {x} ^ {\prime}, \sigma_ {y} ^ {\prime}, \sigma_ {z} ^ {\prime}, 2 \tau_ {y z}, 2 \tau_ {z x}, 2 \tau_ {x y} \right\}, \tag {8.37}
+$$
+
+<!-- source-page: 290 -->
+
+for a three dimensional situation. Thus
+
+$$
+\boldsymbol {a} \boldsymbol {a} ^ {T} = \frac {3}{4 J _ {2} ^ {\prime}} \boldsymbol {M} _ {2}, \tag {8.38}
+$$
+
+where
+
+$$
+\boldsymbol {M} _ {2} = \left[ \begin{array}{c c c c c c} \left(\sigma_ {x} ^ {\prime}\right) ^ {2} & \sigma_ {x} ^ {\prime} \sigma_ {y} ^ {\prime} & \sigma_ {x} ^ {\prime} \sigma_ {z} ^ {\prime} & 2 \sigma_ {x} ^ {\prime} \tau_ {y z} & 2 \sigma_ {x} ^ {\prime} \tau_ {z x} & 2 \sigma_ {x} ^ {\prime} \tau_ {x y} \\ & \left(\sigma_ {y} ^ {\prime}\right) ^ {2} & \sigma_ {y} ^ {\prime} \sigma_ {z} ^ {\prime} & 2 \sigma_ {y} ^ {\prime} \tau_ {y z} & 2 \sigma_ {y} ^ {\prime} \tau_ {z x} & 2 \sigma_ {y} ^ {\prime} \tau_ {x y} \\ & & \left(\sigma_ {z} ^ {\prime}\right) ^ {2} & 2 \sigma_ {z} ^ {\prime} \tau_ {y z} & 2 \sigma_ {z} ^ {\prime} \tau_ {z x} & 2 \sigma_ {z} ^ {\prime} \tau_ {x y} \\ & & & 4 \left(\tau_ {y z}\right) ^ {2} & 4 \tau_ {y z} \tau_ {z x} & 4 \tau_ {y z} \tau_ {x y} \\ & \text {Symmetric} & & & 4 \left(\tau_ {z x}\right) ^ {2} & 4 \tau_ {z x} \tau_ {x y} \\ & & & & & 4 \left(\tau_ {x y}\right) ^ {2} \end{array} \right]. \tag {8.39}
+$$
+
+Also from (8.37)
+
+$$
+\frac {\partial \boldsymbol {a} ^ {T}}{\partial \sigma} = \frac {\sqrt {3}}{2 \left(J _ {2} ^ {\prime}\right) ^ {1 / 2}} \boldsymbol {M} _ {1} - \frac {\sqrt {3}}{4 \left(J _ {2} ^ {\prime}\right) ^ {3 / 2}} \boldsymbol {M} _ {2}, \tag {8.40}
+$$
+
+where
+
+$$
+\boldsymbol {M} _ {1} = \left[ \begin{array}{c c c c c c} \frac {2}{3} & - \frac {1}{3} & - \frac {1}{3} & 0 & 0 & 0 \\ & \frac {2}{3} & - \frac {1}{3} & 0 & 0 & 0 \\ & & \frac {2}{3} & 0 & 0 & 0 \\ & & & 2 & 0 & 0 \\ & & \text { Symmetric } & 2 & 0 \\ & & & & 2 \end{array} \right]. \tag {8.41}
+$$
+
+Substituting from (8.38) and (8.40) into (8.35), and restoring the symbols $\langle \rangle$ , we have finally
+
+$$
+\boldsymbol {H} = p _ {1} \boldsymbol {M} _ {1} + p _ {2} \boldsymbol {M} _ {2}, \tag {8.42}
+$$
+
+where
+
+$$
+p _ {1} = \gamma \left\langle \frac {\sqrt {3}}{2 \left(J _ {2} ^ {\prime}\right) ^ {1 / 2}}. \Phi \right\rangle
+$$
+
+$$
+p _ {2} = \gamma \left\langle \frac {3}{4 J _ {2} ^ {\prime}} \frac {d \Phi}{d F} - \frac {(\sqrt {3}) \Phi}{4 \left(J _ {2} ^ {\prime}\right) ^ {3 / 2}} \right\rangle . \tag {8.43}
+$$
+
+The form of $d\Phi/dF$ depends on the explicit form of $\Phi$ employed, examples of which were given in (8.8) and (8.9). Matrix $H^{n}$ is then obtained by using stresses $\sigma^{n}$ to evaluate $J_{2}^{\prime}$ and $M_{2}$ .
+
+For two-dimensional situations (plane stress, plane strain and axial symmetry) the only relevant stress terms are given in (7.72). In this case $M_{1}$ and $M_{2}$ reduce, on deletion of the appropriate terms, to