Regularization of continuum damage mechanics models for 3-D brittle materials using implicit gradient enhancement

Abstract

A well-known problem which arises in local damage models is that they suffer from mesh dependence and strain localization. In this study, we employ the non-local implicit gradient damage formulation for mesh sensitivity analysis of brittle materials. The basic concept of the non-local implicit gradient model is that the strain at a given point depends not only on the strain at that point but also on the nearby strain field. The local equivalent strain is replaced with a non-local equivalent strain, which is constructed by solving a system of partial differential equations (PDEs). We solve this system of PDEs using finite element method (FEM) and split-operator method which has been applied to 3-D microplane damage models. We extend this approach to continuum damage models and apply it to brittle material for three different loading experiments. In these examples, we demonstrate that the implementation of the implicit gradient damage model using the split-operator method can successfully be applied to complex, 3-D geometries requiring large-scale unstructured FEM meshes, and eliminate mesh dependence. In not all cases the simulations were able to obtain the desired fracture widths as this would require meshes at even smaller resolutions which is beyond our current computing capacity.

Introduction

It is widely acknowledged in the literature that local continuum damage models lead to localization issues [32], [23]. This means that the predictions of the finite element simulations change significantly when changing the size and orientation of the mesh used for discretization. The growth of damage tends to localize in the smallest band that can be captured by the spatial discretization resulting in damage localization at the smallest length scale set by the length of a single element in the finite element mesh. Consequently, the solutions for a case with a progressively damaged zone converge to a solution with a localization zone of zero volume when successive mesh refinements are applied. In the end, the numerical prediction fails to converge to a physically meaningful result. Therefore, the global response of the system shows a strong dependence on the spatial discretization. The causes and effects of localization have been discussed in detail in Borst et al. [4].

The introduction of non-locality in the constitutive model is one of the popular methods to regularize the localization of damage in order to eliminate mesh dependence. In a non-local damage formulation, the state of stress (or strain) at a point does not only depend on the state of strain and its history at that particular point but also depends on the state of the media in a finite neighbourhood of that point. There are two families of non-local formulations, namely, on the one hand, integral formulations using a kernel to regularize the stress or strain [1], [25], [2] and on the other hand, gradient-based formulations [8], [5], [27], [42], [21]. Non-local approaches introduce a length scale to the mathematical representation of the physical system and thus prevent the localization problem. The concept of non-locality for elastic deformations was introduced by Kröner [20] and later extended to hardening plasticity by Mühlhaus and Alfantis [27]. A breakage diffusion model for strength softening rock is studied by Mühlhaus and Gross [28]. Lasry et al. [22] introduced the concept of a non-local constitutive model as a localization limiter for a strain-softening material. These localization limiters [3], [19] force the damage to grow in a zone with a finite width that is independent of the spatial finite element discretization.

There are two forms of non-local gradient formulation namely explicit and implicit, where the non-local equivalent strain is related to the local equivalent strain through partial differential equations [44]. Explicit gradient enhancement is be considered as weakly non-local only, as it fails to regularize the stress (or strain) under some circumstances [55]. On the other hand, implicit gradient enhancement has the advantage of being strongly non-local. It can be shown that they are largely equivalent to the integral type [43], see also in Section 2.

The idea of implicit gradient enhancement is to introduce a second differential equation in addition to the quasi-static equilibrium equation [43]. The equation takes the form of a Helmholtz equation to calculate the non-local field as the counterpart of a local equivalent strain or another local internal variable. The damage evolution law now depends on the regularized equivalent strain rather than the local version. The regularized equivalent strain is obtained by solving the Helmholtz equation. Consequently, the momentum and Helmholtz equations form a non-linear, coupled system of partial differential equations (PDEs). We use the split-operator method [13], [15] and small time steps to solve the two equations sequentially by alternating between them while increasing loading. Section 2.4 details the equations solved using the split-operator approach. The split-operator algorithm is a robust scheme commonly used in implementations of gradient damage [26] and phase-field models [29]. The method is also used in 2-D nonlinear elastic brittle damage by Pires-Dominuges et al. [46] and for 3-D non-local microplane damage models by Zreid et al. [55].

In this paper we extend this implementation to the general 3-D case without any assumption on damage pattern. Our implementation support massive parallel computation which is crucial for the solution of problems with very large finite element meshes required to resolve the targeted spatial length scales of the damage zone. The implementation is applied to the case of a 3-D rectangular rock specimens with a pre-existing surface flaw under uniaxial compression. Local damage models have been investigated to simulate this experimental setup using unstructured meshes in 3-D where results have extensively been compared with a rich set of experimental observations [31]. In order to be able to represent the flaw the finite element method with an unstructured mesh is applied. The rock heterogeneity is modeled through random sampling at a fixed grid resolution which is then interpolated to the simulation meshes with varying mesh sizes. The allows an investigation on mesh dependence for the damage model results without changing material properties for meshes with different element sizes. To our knowledge, this paper presents the first non-local damage model applied to this experimental set-up.

The paper is organized as follows: In Section 2 we present the ingredients of the non-local model including the governing equations and the damage model based on the implicit gradient approach and the provision of heterogeneous material properties. Section 2.4 introduces the operator splitting and its implementation. Results of simulation of the damage propagation for three different application cases are presented in Section 3. The effect of localization length scale is discussed in Section 4. In Section 5 the mesh sensitivity analysis for all three application cases is discussed. An evaluation of the results is given in Section 6.