Issue 29

N.A. Nodargi et alii, Frattura ed Integrità Strutturale, 29 (2014) 111-127; DOI: 10.3221/IGF-ESIS.29.11 112 solution by a Newton-Raphson scheme. That strategy appears to be attractive because of the quadratic rate of convergence of Newton method, preserved by the adoption of an algorithmically consistent tangent operator. Details on the subject can be found in several recent textbooks, e.g. [7-11]. However, in many cases of practical interest, the local- convergence characteristic of Newton schemes does not guarantee global convergence [12-14]. For instance, convergence may not be attained for elastic trial stress in the neighborhood of high curvature points of the yield surface [15]. In the literature, various techniques have been proposed to avoid this drawback, as the use of line search methods, sub-stepping, adaptive sub-stepping, or other ad-hoc integration algorithms, see for example [16-21]. An alternative approach is to set the elastic-plastic problem in the framework of normal-dissipative standard media in the sense of Halphen and Nguyen [22]. The evolution of internal variables is assumed to be governed by a convex scalar dissipation function which is the support function of the elastic domain (e.g., [23]). Consequently, the flow law results in a statement of the classical postulate of maximum plastic dissipation [7]. Exploiting this formulation, several state update algorithms have been proposed, either at local level [24-26] or at finite element level [27, 28]. Conceptually, the underlying basic idea is that the evolution of internal variables in a finite time step incrementally minimizes a suitable convex functional, given by the sum of the internal energy and the dissipation function. This paper focuses on the incremental energy formulation, for elastic-plastic hardening materials characterized by isotropic deviatoric yield function and associative flow law, in the framework of infinitesimal deformation. A two-step algorithm is proposed to perform the material state update. In the first step, an elastic prediction of the updated material state is carried out. In case it is not plastically admissible, the Newton-Raphson method is adopted to solve the constitutive variational problem. An efficient strategy to compute the dissipation function is proposed. In particular, adopting the Haigh-Westergaard representation (e.g. see [29-31]), the problem is reduced to a non-linear scalar equation. Moreover closed-form expressions of the gradient and of the Hessian of Haigh-Westergaard coordinates, as well as of the dissipation function, are presented. With the aim of proving the robustness and stability of the proposed algorithm, numerical tests on a single integration point and FE simulations are provided. This numerical approach appears to be complementary to the classical return map strategy, because no convergence difficulties arise if the stress is close to points of the yield surface with high curvature. The paper is organized as follows: a first part is concerned with a brief discussion on the incremental energy minimization in hardening elastoplasticity. Then the computation algorithm of dissipation function and the proposed state update algorithm are extensively treated. Finally a selected number of numerical tests are reported. INCREMENTAL ENERGY MINIMIZATION FORMULATION et n   ε  be the total strain at a fixed point  x  of a solid dim n    , in practice a Gauss point in a typical finite element solution. In the case of infinitesimal deformation, the total strain ε , i.e. the symmetric part of the displacement gradient, is additively decomposed as: e p   ε ε ε (1) where e ε and p ε are the elastic and plastic parts, respectively. The kinematic description of the material state is completed by a set of strain-like internal variables n   α  . The corresponding conjugated stress-like variables are the stress n   σ  and the stress-like internal variables n   q  , respectively. By standard thermodynamic arguments, the stress-like variables are derived from an Helmholtz free energy function   e , .  ε α Assuming the latter to be strictly convex, the constitutive equations follow     e e e , , ,     α ε σ ε α q ε α (2) where (•)  is the sub-differential operator of non-smooth convex functions (see [22, 32] and references therein). In the context of elastic-plastic materials, the generalized stresses σ , q are constrained to belong to an admissible set, denoted as elastic domain and introduced by means of the yield function   , f σ q :       , : , 0 n n f K       σ q σ q   (3) Hereafter we suppose the yield function to be convex; consequently the elastic domain is convex itself. L