Abstract
Material models exhibiting softening effects due to damage or localization share the problem of leading to ill-posed boundary value problems that lead to physically meaningless, mesh-dependent finite element results. It is thus necessary to apply regularization techniques that couple local behavior, described, e.g., by internal variables, at a spatial level. The common way to do this is to take into account higher gradients of the field variables, thus introducing an internal length scale. In this paper, we suggest a different approach to regularization that does not make use of any nonlocal enhancement like the inclusion of higher gradients or integration over local sub-domains nor of any classical viscous effects. Instead we perform an appropriate relaxation of the (condensed) free energy in a time-incremental setting which leads to a modified energy that is coercive and satisfies quasiconvexity in an approximate way. Thus, in every time increment a regular boundary value problem is solved. The proposed approach holds the same advantage as other methods, but with less numerical effort. We start with the theoretical derivation, discuss a rate-independent version of the proposed model and present details of the numerical treatment. Finally, we give finite element results that demonstrate the efficiency of this new approach.
Highlights
In the past decades, softening phenomena and failure mechanisms due to localized deformation in engineering materials have been intensively studied in the field of continuum mechanics of materials and computational methods such as the finite element method
Mesh-dependence is a direct result of ill-posedness for such models. This effect becomes apparent in both the spatial distribution of the damage parameter and the global behavior of the system: The force/displacement curves depart for different finite element discretizations, and even worse, they do not converge to a final result when the mesh size is decreased
A plot of the material behavior due to damage processes is shown in Fig. 1: The stress/strain diagram is given on the left-hand side and the corresponding energy/strain diagram is plotted on the right-hand side
Summary
In the past decades, softening phenomena and failure mechanisms due to localized deformation in engineering materials have been intensively studied in the field of continuum mechanics of materials and computational methods such as the finite element method. Numerous approaches have been developed to remove the limitations of classical continuum theories when dealing with softening phenomena and strain localization problems. Some of these are briefly discussed in the chapter. We give an overview of continuum damage models and associated regularization techniques in Sect. 6 the failing of those to be quasiconvex 10, we will introduce a global parameter related to the energy release rate of the system which allows to render the approach rateindependent This will allow to establish a distinct relation to the parameters of Griffith theory and perform a comparison with a model employing gradient-regularization.
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