Anisotropic damage coupled with plasticity - model development and comparison with experiments (INVITED)

Abstract eng:
In the present contribution, a new anisotropic and coupled damage-plasticity model is developed. It is based on a second order damage tensor and applies two potentials, one for the plasticity and one for the damage part. The formulation includes only very few material parameters and can be easily transferred to large deformations. Example computations and comparison with experimental results show that the model yields physically reasonable results whereas it is numerically still feasible. Basic modeling steps in the context of small deformation In contrast to isotropic damage based on a scalar damage variable D, the modeling of anisotropic damage requires to work either with a second-order or a fourth-order damage tensor. Approaches of the first kind can be found e.g. in [1,2]. Models based on a fourth-order damage tensor are presented e.g. in [3,4]. In the present work, we start the derivation by formulating the Helmholtz free energy function ! as isotropic function of a second-order damage tensor D and the elastic strain tensor !e. As such we let depend ! on the ten invariants of D and !e (J1, ..., J10). Inserting the ansatz into the ClausiusDuhem inequality we finally come to defining two potentials, one for the plasticity and one for the damage part. Using these, the evolution equations for the plastic strain tensor !p, the damage tensor D and the two hardening variables (plastic and damage part) are formulated in an associated form. The damage potential includes two functions fd and qd (hardening part) which allow to adjust the damage evolution over time as well as the stress-strain curve quite well. A good correlation with experiments can be achieved. Interestingly, the model can be well adapted to metallic materials but also to non-metallic materials such as concrete or composites. The concrete choice of the Helmholtz free energy function is leaned on the work of Lemaitre (see e.g. [5,6]) where we, however, slightly modify the suggestion of the mentioned authors in such a way that ! is a function of the invariants J1, ..., J10. In this way, the material tensor obtains with respect to the orthotropic frame an orthotropic structure which can be well exploited to determine the material parameters hidden in !. Further important is the fact that we can introduce one or several "drivers" for the damage. Suitable drivers are the strain tensor, the plastic strain tensor, the continuum mechanical stress tensor or the effective stress tensor. Tension-compression asymmetry is included by considering only the positive eigenvalues of these drivers (Macaulay bracket). Numerical issues At the Gauss point level of the finite element implementation, several challenging numerical problems have to be tackled. First of all, we need to robustly manage the case differentiation into (1) elastic, (2) only damage, (3) only plastic, (4) coupled plastic damage behaviour. For the integration of the sophisticated evolution equations suitable algorithms shall be used, e.g. the backward Euler algorithm. This leads to a set of highly non-linear equations the solution of which can be e.g. performed by means of Newton-Raphson's method. The associated tangent is here computed numerically. Further, the algorithm has to recognize non-physical cases, e.g. DA (A = 1,2,3) (eigenvalues of D) <0 or >1. Since anisotropic damage modeling is per se elaborate it is important to keep the numerical effort at the finite element level as low as possible. In the present contribution a special reduced integration finite element technology is suggested which works with only one Gauss point. An important issue is the hourglass stabilization which has to be developed in such a way that the anisotropic material behaviour is taken into account. Some example computations, striving for a comparison of algorithms (fully implicit and partially explicit) are shown below.

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