Issue 29

N.A. Nodargi et alii, Frattura ed Integrità Strutturale, 29 (2014) 111-127; DOI: 10.3221/IGF-ESIS.29.11 113 The model is supplemented by the evolution law of plastic strain p ε and strain-like internal variables α . With the assumption of normal-dissipative or standard material behavior, i.e. of associative plastic flow, it is customary to introduce a scalar dissipation function defined as the support function of the elastic domain       p p , , sup K D      σ q ε α σ ε q α     (4) and to express the evolution law in the form:     p p p , , , D D   α ε σ ε α q ε α       (5) It is worth noting that the dissipation function is convex, non-negative, zero only at the origin and positively homogeneous of degree one; moreover it is nondifferentiable at the origin, where its sub-differential set coincides with the elastic domain [7]. Substituting the decomposition (1) in the constitutive law (2) and adding the result to (5), the constitutive differential equations are obtained:         p p p p p p , , , , D D           ε ε α α ε ε α ε α 0 ε ε α ε α 0         (6) often referred to as Biot’s equation of standard dissipative systems, see [22, 24, 33] for instance. We now proceed with the numerical approximation of the rate Eq. (6). In particular, we adopt a backward Euler integration scheme and assume that plastic strain p ε and strain-like internal variables α vary linearly in each time step 1 ] [ , n n t t  . Consequently, by exploiting the degree-one positive homogeneity of the dissipation function, the incremental form of (6) follows:         p p p p p 1 p p p 1 , , , , n n n n n n D D                             ε ε α α ε ε ε α α ε α 0 ε ε ε α α ε α 0   (7) with the notation     • • | n t t n   ,     1 1 • • | n t t n     ,       1 • • • n n     . Eq. (7) represent the Euler-Lagrange first order conditions associated to the incremental minimization problem         p e,trial p trial p 1 1 , inf , , n n D              ε α ε ε α α ε α  (8) where e,trial p 1 1 n n n     ε ε ε , trial 1 n n   α α define the elastic trial state, i.e. the material state corresponding to no plastic evolution. Therefore, the incremental minimization problem (8) determines the internal state of the material for finite increments of time. We remark that the present formulation is similar to the one proposed in [24], under the assumption of rate- independent evolution and linear variation in time of plastic strain p ε and strain-like internal variables α . COMPUTATION OF THE DISSIPATION FUNCTION inematic and isotropic hardening are distinguished by assuming that the yield function can be represented as   k i , f q  σ q , where k q [resp., i q ] is the stress-like kinematical [resp., isotropic] hardening variable. Accordingly, the dissipation function is given by         k i k i p p k i k k i i , , , 0 , , sup q f q D q                σ q σ q ε α σ ε q α (9) where k  α [resp., i   ] is the strain-like kinematical [resp., isotropic] hardening variable. Setting  k   σ σ q , it results        k i i p p p k i k k i i { , , } ( , ) 0 , , sup ( ) q f q D q                 σ q σ ε α σ ε q ε α (10) K