Gradients Of Internal Variables Research Articles

Regularizational approach for modeling ductile damage

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 froman 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 tensor valued damage, i.e., anisotropic material degradation.

Strategies for the Computation of Configurational Forces in Dissipative Media

AbstractConfigurational forces can be interpreted as driving forces on material inhomogeneities such as crack tips. In dissipative media the total configurational force on an inhomogeneity consists of an elastic contribution and a contribution due to the dissipative processes in the material. For the computation of discrete configurational forces acting at the nodes of a finite element mesh, the elastic and dissipative contributions must be evaluated at integration point level. While the evaluation of the elastic contribution is straightforward, the evaluation of the dissipative part is faced with certain difficulties. This is because gradients of internal variables are necessary in order to compute the dissipative part of the configurational force. For the sake of efficiency, these internal variables are usually treated as local history data at integration point level in finite element (FE) implementations. Thus, the history data needs to be projected to the nodes of the FE mesh in order to compute the gradients by means of shape function interpolations of nodal data as it is standard practice. However, this is a rather cumbersome method which does not easily integrate into standard finite element frameworks. An alternative approach which facilitates the computation of gradients of local history data is investigated in this work. This approach is based on the definition of subelements within the elements of the FE mesh and allows for a straightforward integration of the configurational force computation into standard finite element software. The suitability and the numerical accuracy of different projection approaches and the subelement technique are discussed and analyzed exemplarily within the context of a crystal plasticity model. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Localization properties of strain-softening gradient plasticity models. Part II: Theories with gradients of internal variables

The second part of this paper compares and evaluates enhancements of the conventional plasticity theory by gradients of internal variables. Attention is focused on their performance as localization limiters. Both explicit and implicit gradient formulations are considered. It is shown that certain models suffer by serious mathematical deficiencies that would complicate their numerical implementation. Some other models are appropriate only at early stages of the softening process but later exhibit locking accompanied by a spurious expansion of the localized plastic zone. The comparative study indicates that a convenient and robust tool for regularized modeling of the entire localization process is provided by the implicit gradient approach combined with a suitable form of the hardening/softening law.

Localization properties of strain-softening gradient plasticity models. Part I: Strain-gradient theories

This paper compares and evaluates strain-gradient extensions of the conventional plasticity theory. Attention is focused on the ability of individual formulations to act as localization limiters, i.e., to regularize the boundary value problem in the presence of softening and to prevent localization of plastic strain increments into a set of zero measure. To keep the presentation simple and to highlight the essential properties of the investigated models, only the static, rate-independent response in the small-strain range and in the one-dimensional setting is considered. These restrictions permit an analytical or semi-analytical treatment of the problem, while the basic characteristics of the solutions remain valid in the general, multi-dimensional case. The onset of localization is characterized as a bifurcation from a uniform state. The subsequent evolution of the localized process zone and of the shape of the strain profile is studied numerically. It is shown that certain pathologies, e.g., expansion of the plastic region accompanied by stress locking, may arise at later stages of localization. A similar analysis of models with gradients of internal variables is presented in a companion paper.

Gradient constitutive relations: numerical aspects and application to gradient damage

The description of localisation and structural failure through local strain-softening constitutive laws leads to ill-posed problems. To retrieve a well-posed problem, the material state is described through the gradients of internal variables in addition to the strain and the internal variables themselves. Under some assumptions, it is shown that these gradient laws can be cast into a variational formulation. After time discretisation, the evolution of the internal variable field is characterised by a minimum principle. A specific algorithm is proposed to perform the minimisation in the context of finite elements. Finally, it is applied to a gradient brittle damage model.