Gradient-enhanced Damage Models Research Articles

Analytical solution of a gradient-enhanced damage model for quasi-brittle failure

This paper presents an approach for analytically solving gradient-enhanced damage (GED) models. The proposed approach provides rigorous proofs for essential phenomena observed in the traditional GED model, including damage widening, characteristic length sensitivity, and stress-locking. The derived cohesive law is a useful technique for accurately determining the material parameters of the GED model, significantly reducing the sensitivity to characteristic length. Furthermore, this study presents an isotropic damage model that considers both tensile and shear failures and can capture complex crack paths. Finally, a series of numerical examples is shown to demonstrate the efficacy of the suggested method.

Finite element modelling of refractories fracture process zone with gradient enhanced damage models

This study investigates the numerical simulation of fracture behaviour in quasi-brittle materials like magnesia spinel refractories using the Gradient-Enhanced Damage (GED) model. It focuses on the complex modelling of these materials non-linear responses and compares conventional and variant GED models through a wedge splitting test. The results demonstrate that all GED models show a good fit to experimental data. However, the conventional GED model falls short in accurately depicting the fracture process zone. In contrast, the localizing GED model more accurately represents the fracture process zone, limiting spurious damage distribution, but requires finer meshing, elevating computational demands. The stress-based variant reduces spurious damage but is less effective comparatively. The study also assesses the role of heterogeneous strength distribution in replicating realistic crack patterns as observed in experiments.

A novel thermo-mechanical local damage model for quasi-brittle fracture analysis

This paper is devoted to numerical investigation of quasi-brittle fracture under thermal-elastic loading condition using a novel thermal-mechanical local damage model associated with the enhanced bi-energy norm based equivalent strain. In contrast to the non-local or gradient-enhanced damage models, a local damage counterpart generally requires less computational effort. The common mesh-dependent issue encountered in the local approaches is here mitigated by incorporation of fracture energy and element characteristic length into the calculation of damage evolution. Equivalent strain is derived based on the so-called bi-energy norm concept and the recent Mazars’ criterion. Here, the damage evolution is induced by both the mechanical and thermal loads, in which both stress and thermal conductance capacity are reduced across the damage zone. In this work, we present an efficient staggered scheme to solve the coupled thermal-mechanical damage equations. The performance and accuracy of the developed model are validated via several numerical examples of quasi-static crack growth problems, in which comparisons between the computed results and reference solutions from other numerical approaches and/or experiments are presented.

On topology optimization with gradient-enhanced damage: An alternative formulation based on linear physics

Numerous topology optimization formulations have been proposed in order to enhance structural resistance to material failure. A clear line can often be drawn between those methods which attempt to constrain local failure criteria and those that explicitly model the failure physics during the optimization process. In this work the former method is extended in a manner inspired by the mathematical form of typical gradient-enhanced damage models. Importantly, the proposed formulation relies on linear physics during the optimization procedure, which greatly increases its speed and robustness, both of which are essential in industrial applications where large numerical models are typically used. The size effect introduced by using such a numerical model is further investigated and select observations are provided, such as spurious “fin-like” patterns that emerge depending on the type of structure and loading conditions. Finally, the load capacity is verified for each optimized design through a post-optimization verification procedure which is unaffected by the density-based design parameterization and associated material interpolation schemes.

An anisotropic damage formulation for composite materials based on a gradient-enhanced approach: Formulation and implementation at small strain

Classical gradient enhanced damage models have been intensively studied and demonstrated to provide reliable predictions of structural failure. Singularities due to local softening have been properly overcome to obtain stable and converged numerical solutions. As expected, mesh-dependency has been avoided as well. Motivated by experiments, several damage patterns of fiber reinforced material in practical applications are comprehensively characterized as anisotropic properties, which are governed by both the elastic response and the damage behavior simultaneously. The present work describes an anisotropic damage evolution by incorporating a gradient enhanced model for composite materials. Three independent non-local variables are introduced to govern the damage evolution along three orthogonal directions, which subsequently lead to independent degradation of the components in material stress and the consistent tangent. The constitutive law of the coupled problem is derived and, subsequently, is implemented into the context of the Finite Element framework. To appropriately demonstrate the capabilities of this approach, representative benchmark problems are studied and compared to corresponding experiments. Consequently, related findings and potential perspectives are summarized to provide further context for future applications.

A comparative study and ABAQUS implementation of conventional and localizing gradient enhanced damage models

The conventional gradient models based on continuum damage mechanics are found to exhibit inconsistencies in terms of damage initiation and growth. With the decrease in non-local interactions, localizing gradient damage model describes the failure phenomena in a physically meaningful manner. In the present study, both conventional and localizing gradient damage models are implemented in the commercial finite element package ABAQUS through user subroutines. The motivation behind the implementation is the popularity and computational efficiency of the ABAQUS. It is demonstrated through various numerical examples that the localizing gradient damage model is more effective and efficient than the conventional gradient damage model. A detailed stress oscillation study is carried out to provide more insight on the localizing gradient damage model. A new approach is proposed for an implicit representation of initial notch/crack in the gradient damage models. The enhanced capability of the present implementation is demonstrated by solving the problems with complicated geometry under complex loading. A detailed description of the working of the user subroutines (UEL and UMAT) and other associated activities is elucidated. The source codes of the present implementation are provided for better understanding and further development.

On the implementation of finite deformation gradient-enhanced damage models

We introduce a comprehensive framework for the efficient implementation of finite deformation gradient-regularised damage formulations in existing finite element codes. The numerical implementation is established within a thermo-mechanically fully coupled finite element formulation, where the heat equation solution capabilities are utilised for the damage regularisation. The variationally consistent, gradient-extended and geometrically non-linear damage formulation is based on an overall free energy function, where the standard local free energy contribution is additively extended by two non-local terms. The first additional term basically contains the referential gradient of the non-local damage variable. Secondly, a penalty term is added to couple the local damage variable—the evolution of which is governed by an ordinary differential equation—and the non-local damage field variable that is governed by an additional balance equation of elliptic type.

Phase field and gradient enhanced damage models for quasi-brittle failure: A numerical comparative study

This paper presents a comparative study of the gradient-enhanced damage models (GED) of Peerlings et al. (1996), Vandoren and Simone (2018) and the phase field damage/fracture model (PFM) of Wu (2017), Wu and Nguyen (2018) within the context of the computational modeling of the fracture of quasi-brittle materials (concrete, ceramic, rock, ice, etc.). Being continuous damage/fracture models, these two models enjoy the simplicity of modeling the fracture process on a fixed finite element mesh. The similarities and differences of the two models are discussed by examining governing equations and conducting numerical simulations of some mode I and mixed-mode fracture benchmark tests. The most worthy findings are: (i) both classes of models can handle the initiation and propagation of cohesive cracks, (ii) they are totally different–PFM behaves like a cohesive zone model (a sub-class of fracture mechanics) when the length scale is sufficiently small and the response is insensitive to this length scale whereas GED is a non-local damage model (a sub-class of continuum damage mechanics) of which response obviously depends on the length scale.

An efficient FE-implementation of implicit gradient-enhanced damage models to simulate ductile failure

We present a strategy which allows to implement a fully coupled, gradient-enhanced damage law into commercial FEM-codes with little effort. This enables a robust simulation of practical engineering problems avoiding spurious mesh dependency due to damage influence. The method is applied to a model for shear dominated, ductile damage of metals, which is used by engineers and often available in its local formulation within FEM-programs; e.g., ABAQUS. The model consists of a damage initiation criterion, followed by a damage evolution law which is coupled to the elastic-plastic constitutive equations. Convergence studies show the applicability of the implementation for 2D and 3D boundary value problems. Crack growth simulations of a benchmark problem show reasonable results compared to similar approaches from literature.

Modeling and simulation of quasi-brittle failure with continuous anisotropic stress-based gradient-enhanced damage models

Two anisotropic stress-based gradient-enhanced damage models are proposed to address the issue of spurious damage growth typical of continuous standard gradient-enhanced damage models. Both models are based on a decreasing interaction length upon decreasing stresses and do not require additional model parameters or extra degrees of freedom when compared to standard gradient-enhanced models. It is observed that with the proposed models damage spreading is significantly reduced due to the occurrence of non-physical oscillations in the nonlocal strain field near the strain localization band. Model improvements to eliminate these strain oscillations upon vanishing length scale values are proposed. The capability of the models and their patched versions to correctly simulate damage initiation and propagation is investigated by means of mode-I failure, shear band and four-point bending tests.

Gradient damage vs phase-field approaches for fracture: Similarities and differences

Gradient-enhanced damage models and phase-field models are seemingly very disparate approaches to fracture. Whereas gradient-enhanced damage models find their roots in damage mechanics, which is a smeared approach from the onset, and gradients were added to restore well-posedness beyond a critical strain level, the phase-field approach to brittle fracture departs from a discontinuous description of failure, where the distribution function is regularised, leading to the inclusion of spatial gradients as well. Herein, we will consider both approaches, and discuss their similarities and differences. The averaging (diffusion) equations for the averaging field and the phase-field will be compared, and it is shown that the diffusion equation for the phase-field can be conceived as a special case of the averaging equation of a gradient-damage model where the damage is averaged. Further, the role of the driving force is examined, and it is shown that subtle differences in the degradation functions commonly adopted in damage and phase-field approaches are key to the observation that, different from damage mechanics, the fracture process zone does not broaden in the wake of the crack tip.

A simplified implementation of a gradient-enhanced damage model with transient length scale effects

Gradient-enhanced damage models with constant gradient activity suffer from spurious damage growth at high deformation levels. This issue was resolved by Geers et al. (Comput Methods Appl Mech Eng 160(1---2):133---153, 1998) by expressing the gradient activity parameter as a function of the local equivalent strain at the expense of adding one set of degrees of freedom to those of the standard model. In this contribution, a new formulation of the gradient-enhanced damage model with variable length scale is presented which eliminates the need for the extra set of degrees of freedom. The merits of the proposed formulation are demonstrated, and the choice of the damage evolution law and its impact on the model performance are discussed.

A multiscale mean-field homogenization method for fiber-reinforced composites with gradient-enhanced damage models

In this work, a gradient-enhanced homogenization procedure is proposed for fiber reinforced materials. In this approach, the fibers are assumed to remain linear elastic while the matrix material is modeled as elasto-plastic coupled with a damage law described by a non-local constitutive model. Toward this end, the mean-field homogenization is based on the knowledge of the macroscopic deformation tensors, internal variables and their gradients, which are applied to a micro-structural representative volume element (RVE). The macro-stress is then obtained from a homogenization procedure. The methodology holds for 2-phase composites with moderate fiber volume ratios, and for which, at the RVE size, the matrix can be considered as homogeneous isotropic and the ellipsoidal fibers can be considered as homogeneous transversely isotropic. Under these assumptions, the method is successfully applied to simulate the damage process occurring in unidirectional carbon-fiber reinforced epoxy composites submitted to different loading conditions.

Stability of Homogeneous States with Gradient Damage Models: Size Effects and Shape Effects in the Three-Dimensional Setting

Considering a family of gradient-enhanced damage models and taking advantage of its variational formulation, we study the stability of homogeneous states in a full three-dimensional context. We show that gradient terms have a stabilizing effect, but also how those terms induce structural effects. We emphasize the great importance of the type of boundary conditions, the size and the shape of the body on the stability properties of such states.

Study of the New Leon model for concrete failure

The New Leon model (NLM) is based on a combination of the flow theory of plasticity and damage mechanics. First, the damage evolution law of the model, which yields finite element mesh size-dependent results, was regularized. The slope of the softening curve predicted by the model was slightly too flat compared with uniaxial tensile test results; thus, the evolution law of the damage variable was enhanced to correct defects. The yield function of the New Leon model had two vertices at the intersection with the hydrostatic axis. To avoid this problem, the stress state was assumed to be projected onto the hydrostatic axis in the first vertex and along the hydrostatic axis in the second. The New Leon model was then compared with the Extended Leon model in a first experiment by Hurlbut and a second experiment by Imran. Finally, the New Leon, extended finite element, and gradient enhanced damage models were applied to plain concrete tests to simulate the cracking of concrete, and crack mouth opening displacement results were obtained. Compared with other models, the New Leon model yielded values that were generally in better agreement with the experimental results.

Gradient-enhanced damage modeling of cracked bodies by reproducing kernel element method

In crack modeling by extended finite element method, the enrichment by the asymptotic analytical solution works excellent for linear elastic fracture mechanics. However for the more sophisticated condition of a gradient-enhanced damage model, firstly the analytical solution is not available and secondly higher order continuity of the shape functions is required. In this study, the extended finite element model of the cracked body is augmented with shape functions of higher order continuity and higher order consistency generated by reproducing kernel element method. They are used in a region near the crack tip to have an appropriate interpolation of the damaged zone while for the rest of the solution domain standard shape functions are used to reduce the computational cost. No new nodes are needed when defining the higher order shape functions by reproducing kernel element method and thus remeshing is avoided during the crack growth. After describing the formulations, several numerical examples are implemented to investigate the ability of the proposed method, for both the linear elastic fracture mechanics and the gradient-enhanced damage models.

Initiation of damage with implicit gradient-enhanced damage models

Through a mixed local and nonlocal formulation, a continuous transition from a local to a nonlocal behavior is obtained. Initially driven by the local action, the damage driving quantity at a material point in non-damaged materials is later influenced by the nonlocal effect only if the occurrence of damage is effectively found in the surroundings. Qualitative resemblance is hence expected in the prediction of damage initiation by the non-regularized and regularized damage models. Numerical simulations with simple tests and comparisons with existing nonlocal damage models show the qualitative improvement of the mixed formulation in the prediction of damage initiation for 1D and 2D damaged structures.

Explicit and implicit gradient-enhanced damage models

Explicit and implicit gradient-enhanced damage models

Explicit and implicit gradient-enhanced damage models

ABSTRACT This article provides a brief overview of nonlocal differential damage models. The basic concepts of nonlocal averaging are briefly recalled and used as basis for the derivation of implicit and explicit gradient-enhanced models. Some applications to typical localization problems highlight the merits and demerits of this class of models.

Dispersion analysis and element-free Galerkin solutions of second- and fourth-order gradient-enhanced damage models

Gradient-dependent damage formulations incorporate higher-order derivatives of state variables in the constitutive equations. Different formulations have been derived for this gradient enhancement, comparison of which is difficult in a finite element context due to higher-order continuity requirements for certain formulations. On the other hand, the higher-order continuity requirements are met naturally by element-free Galerkin (EFG) shape functions. Thus, the EFG method provides a suitable tool for the assessment of gradient enhanced continuum models. Dispersion analyses have been carried out to compare different gradient enhanced models with the non-local damage model. The formulation of the additional boundary conditions is addressed. Numerical examples show the objectivity with respect to the discretization and the differences between various gradient formulations with second- and fourth-order derivatives. It is shown that with the same underlying internal length scale, very different results can be obtained. Copyright © 2000 John Wiley & Sons, Ltd.