Three regularization concepts are assessed regarding their variational structure and interference with the predicted physics of (quasi-)brittle damage: the fracture energy concept, viscous regularization and micromorphic regularization. They are first introduced in a unified variational framework, depicting how they distinctively evolve from incremental energy minimization. The analysis of a certain time interval of a one-dimensional example is used to show how viscous and micromorphic regularization retains well-posedness within the softening regime. By way of contrast, the fracture energy concept is characterized by ill-posedness—as known from previous non-variational analyses. Numerical examples finally demonstrate the limitations and capabilities of each concept. The ill-posed local fracture energy concept leads by its design to a spatially constant fracture energy—in line with Griffith’s theory. The viscous regularization, in turn, yields a well-posed problem but artificial viscosity can add a bias to unloading and fracture thickness. Furthermore, and even more important, a viscous regularization does not predict a spatially constant fracture energy due to locally heterogeneous loading rates. The well-posed micromorphic regularization is in line with the underlying physics and does not show this undesired dependency. However, it requires the largest numerical efforts, since it is based on a coupled two-field formulation.
A model framework for the analysis of isotropic quasi-brittle damage, was recently presented in [1]. Within this paper, the model in [1] is significantly improved. To be more precise, and in contrast to [1], the novel model: (i) eliminates unnecessary model parameters, (ii) can be better interpreted from a physics point of view, (iii) can capture a fully softened state (zero stresses) and (iv) is characterized by a very simple evolution equation which (v) can be integrated fully implicitly and (vi) the resulting time discrete evolution equation can be solved analytically providing a numerically efficient closed form solution, cf.[2].In line with [3], damage models can often be rewritten into a variational framework. To be more precise, an incremental potential can be defined whose stationary conditions define the underlying model. Following [4], the rate potentialĖ is given byĖwhere the superimposed dot denotes the material time derivative. The rate potential depends on the velocity fieldu and on the rates of internal variablesα. The quantity ρ denotes the density, b are mass-specific forces and t * prescribed tractions acting on the Neumann boundary ∂B N . The local rate potential I is defined aṡwhere D is the dissipation function and Ψ the Helmholtz free energy depending on the engineering strains ε = ∇ sym u and the internal variables. Stationary of potential (1) with respect to the velocity field yields balance of linear momentum and stationary with respect to the rates of internal variables results in Biot's equation which implies the constitutive evolution equation, cf.[5]. Prototype modelFollowing the framework of incremental energy minimization, cf.[4], the model is defined by postulating the Helmholtz free energy and the dissipation function. A common set for modeling of isotropic quasi-brittle material degradation is given byin which E is the fourth-order elasticity tensor, α an internal variable and α u an additional material parameter regulating the evolution speed of damage variable d. Stationary of potential (1) with respect toα gives the evolution equation. By considering the time discrete counterpart and the second law of thermodynamics (healing has to be excluded, i.e.α < 0) both equations can be combined to α n+1 = max(α n , Ψ 0 ), i.e., the internal variable stores the maximum Helmholtz free energy of the undamaged material. Normalized results of this prototype model are shown in Fig 1. Fig. 1a shows the normalized stressstrain-diagram, which has a typical softening behavior. Fig. 1b visualizes the normalized determinant of the corresponding tangent modulus. It can be seen that -as expected -the tangent determinant becomes negative at the onset of softening. Hence, the type of the partial differential equation changes, and thus, the boundary value problem becomes ill-posed, cf.[6].3 Rate-dependent damage modelIn this subsection the relaxation-based damage model, presented in [1], will be improved, cf. [2]. For that purpose, the local rate potential is defined as
It is well known, that modeling of material softening behavior can lead to ill-posed boundary value problems. This, in turn, leads to meshdependent results as far as the finite-element-method is concerned [1]. Several solution strategies in order to regularize the aforementioned problem have been proposed in the literature, cf.[2]. However, these strategies often involve high implementational effort. An approach which is very efficient from an implementational point of view is the so-called micromorphic approach by [3,4]. This regularization technique includes gradients of internal variables implicitly into the framework, while preserving the original structure of the underlying local constitutive model. However, it is shown that a straightforward implementation of the micromorphic approach does not work for single-surface ductile damage models. By analyzing the respective equations, a modification of the micromorphic approach is proposed -first for a scalar internal variable, i.e., isotropic damage. Subsequently, the novel regularization method is extended to tensorvalued damage, i.e., anisotropic material degradation. *
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