A gradient model for finite strain elastoplasticity coupled with damage (original) (raw)

A strain gradient plasticity based damage model for quasibrittle materials

2016

Boundary value problems for a softening material suer from loss of uniqueness in the post-peak regime. Numerical solutions to such problems shows mesh dependency due to lack of internal length scale in the formulation. A regularization method which introduces a characteristic length is required to get mesh independent results. A second gradient model introduces a characteristic length by taking into account the second gradient of the displacement in the principle of virtual work and thus regularizing the solution of the boundary value problem. In this work a second gradient nite element model has been developed. The regularization property of the method has been studied for elastoplastic and damage constitutive laws. It has been shown that mesh independent results can be achieved in this model. Even though unique solution is not achieved a nite number of solutions have been obtained from the proposed model.

Modelling of strain softening materials based on equivalent damage force

Computer Methods in Applied Mechanics and Engineering, 2018

The main aim of the work presented in this paper was treatment of damage and deformation localisation observed in the finite element method (FEM) analysis of strain softening materials combined with local constitutive models where damage is represented using continuum damage mechanics (CDM). The CDM/FEM approach typically suffers from a number of shortcomings, including mathematical (change of the type of partial differential equations leading to ill-posed boundary value problem), numerical (pronounced mesh dependency) and physical (infinitely small softening zone with the zero dissipated energy). The approach proposed here is still based on the local constitutive model including damage, but introduces an alternative representation of damage effects in the system of linear momentum balance equations. The damage effects are included through equivalent damage force (EDF), which contributes to the right-hand side of the momentum balance equations. The main advantages of this approach are that the problem remains well posed, as the type of partial differential equations remains unchanged when the material enters softening; numerical stability is preserved without a need for regularisation measures; and significantly reduced mesh dependency. In addition, the EDF approach can be used in combination with existing local CDM damage models and does not violate symmetry of the material stiffness tensor. The EDF approach is applicable to modelling of strain softening typically observed in damaged quasi brittle materials such as fibre reinforced composites and concrete. The EDF model was implemented in the in-house developed coupled FEM-SPH code, where an explicit FEM code is coupled with a stable Total-Lagrange form of SPH. Its performance is demonstrated in the analysis of a dynamic one dimensional (1D) stress wave propagation problem, which was analytically solved by Bazant and Belytschko in 1985. For a range of loading rates that correspond to the material softening regime, the numerical results shown nonlocal character with a finite size of the damaged zone, controlled with the damage characteristic length, which can be experimentally determined and is an input parameter independent of the discretisation density.

Coupled damage-plasticity constitutive model and direct stress interpolation

Computational Mechanics, 2008

In this paper we develop the governing equations of the coupled damage-plasticity model, which is capable of representing the main mechanisms of inelastic behavior including irreversible plastic deformation, change of elastic response and the localized failure. We show in particular how such model should be implemented within the stress-based variational formulation, providing an important advantage for local computation of the internal variables, which thus remains very robust and even non-iterative for the case of linear hardening model. Several simple examples are presented in order to illustrate the kind of response the model can represent. Keywords coupled damage-plasticity ¡ stress interpolation ¡ cyclic loading

Damage model for brittle elastic solids with unequal tensile and compressive strengths

Engineering Fracture Mechanics, 1994

The paper presents a rate-type constitutive analysis of damage, applicable to brittle materials whose elastic properties degrade during a deformation process. Different tensile and compressive material responses are modeled incorporating positive and negative projections of the stress or strain tensors. Proposed evolution laws for the rate of compliance tensors are consistent with some of the prominent features of brittle material response. A new structure of the damage surface is introduced for a more accurate account of the effects of the hydrostatic states of stress on the overall response. Derived rate constitutive equations provide the explicit representation of the tangent compliance tensor. The proposed model is applied to uniaxial tension and compression to illustrate nonlinear relationships between stress and longitudinal, lateral, and volumetric strains. The proposed model is compared with some of the existing theories.

ScienceDirect Modelling of strain softening materials based on equivalent damage force

The main aim of the work presented in this paper was treatment of damage and deformation localisation observed in the finite element method (FEM) analysis of strain softening materials combined with local constitutive models where damage is represented using continuum damage mechanics (CDM). The CDM/FEM approach typically suffers from a number of shortcomings, including mathematical (change of the type of partial differential equations leading to ill-posed boundary value problem), numerical (pronounced mesh dependency) and physical (infinitely small softening zone with the zero dissipated energy). The approach proposed here is still based on the local constitutive model including damage, but introduces an alternative representation of damage effects in the system of linear momentum balance equations. The damage effects are included through equivalent damage force (EDF), which contributes to the right-hand side of the momentum balance equations. The main advantages of this approach are that the problem remains well posed, as the type of partial differential equations remains unchanged when the material enters softening; numerical stability is preserved without a need for regularisation measures; and significantly reduced mesh dependency. In addition, the EDF approach can be used in combination with existing local CDM damage models and does not violate symmetry of the material stiffness tensor.

Isotropic Elastoplasticity Fully Coupled with Non-Local Damage

Engineering, 2010

This paper presents a simple damage-gradient based elastoplastic model with non linear isotropic hardening in order to regularize the associated initial and boundary value problem (IBVP). Using the total energy equivalence hypothesis, fully coupled constitutive equations are used to describe the non local damage induced softening leading to a mesh independent solution. An additional partial differential equation governing the evolution of the non local isotropic damage is added to the classical equilibrium equations and associated weak forms derived. This leads to discretized IBVP governed by two algebric systems. The first one, associated with equilibrium equations, is highly non linear and can be solved by an iterative Newton Raphson method. The second one, related to the non local damage, is a linear algebric system and can be solved directly to compute the non local damage variable at each load increment. Two fields, linear interpolation triangular element with additional degree of freedom is terms of the non local damage variable D is constructed. The non local damage variable D is then transferred from mesh nodes to the quadrature (or Gauss) points to affect strongly the elastoplastic behavior. Two simple 2D examples are worked out in order to investigate the ability of proposed approach to deliver a mesh independent solution in the softening stage.

A Gradient-Dependent Constitutive Model to Simulate Impact Damage Problem

Conventional continuum mechanics models of inelastic deformation processes are size scale independent since they do not possess intrinsic length scales in their constitutive description. In contrast, there is considerable experimental evidence that inelastic flow in crystalline materials is size-dependent. As soon as material failure dominates a deformation process, the material increasingly displays strain softening (localization) and the finite element computation is considerably affected by the mesh size and alignment and gives non-physical descriptions of the localized regions. Gradient-enhanced constitutive viscoplastic and viscodamage equations that include explicit and implicit micro-structural length scale measures are presented in this work. The governing equations are appropriate for polycrystalline metals. Numerical simulations are performed to study the effect of including these material lengths on the dynamic localization of plastic flow in shear bands for impact-damage related problems. It is shown that the inherent material length scale predictions agree well with the width of the shear bands in ductile metals as compared to the experimental results.

A classification of higher-order strain-gradient models in damage mechanics

Archive of Applied Mechanics (Ingenieur Archiv), 2003

The use of higher-order strain-gradient models in mechanics is studied. First, existing second-gradient models from the literature are investigated analytically. In general, two classes of second-order strain-gradient models can be distinguished: one class of models has a direct link with the underlying microstructure, but reveals instability for deformation patterns of a relatively short wave length, while the other class of models does not have a direct link with the microstructure, but stability is unconditionally guaranteed. To combine the advantageous properties of the two classes of second-gradient models, a new, fourth-order strain-gradient model, which is unconditionally stable, is derived from a discrete microstructure. The fourthgradient model and the second-gradient models are compared under static and dynamic loading conditions. A numerical approach is followed, whereby the element-free Galerkin method is used. For the second-gradient model that has been derived from the microstructure, it is found that the model becomes unstable for a limited number of wave lengths, while in dynamics, instabilities are encountered for all shorter wave lengths. Contrarily, the secondgradient model without a direct link to the microstructure behaves in a stable manner, although physically unrealistic results are obtained in dynamics. The fourth-gradient model, with a microstructural basis, gives stable and realistic results in statics as well as in dynamics.