Anisotropic Continuum Damage Mechanics Research Articles

A Thermodynamically Consistent Anisotropic Continuum Damage Mechanics Model With Damage-Dependent Equivalent Strain

Abstract This paper proposes a new thermodynamically consistent anisotropic continuum damage mechanics model that the degradation of material is described by a set of damage variables. We adopt a set of novel equivalent strains which are functions of both the strain and the damage state variables as well. A viscous stabilization method is considered to improve the convergence during the material softening. The evolution of damage state variables is updated through a Newton–Raphson iterative process. The expression of the consistent tangent stiffness matrix is also derived. A smoothed exponential damage evolution shape function with rescaling regularization method is proposed to describe the material softening process and the mesh dependency phenomenon is relieved. Several sets of numerical examples including three-point-bending of a notched beam, uniaxial tension of unnotched and notched plate of composite material with multiple layups are presented to verify the anisotropic damage models.

Simulating the processes controlling ice-shelf rift paths using damage mechanics

Abstract Rifts are full-thickness fractures that propagate laterally across an ice shelf. They cause ice-shelf weakening and calving of tabular icebergs, and control the initial size of calved icebergs. Here, we present a joint inverse and forward computational modeling framework to capture rifting by combining the vertically integrated momentum balance and anisotropic continuum damage mechanics formulations. We incorporate rift–flank boundary processes to investigate how the rift path is influenced by the pressure on rift–flank walls from seawater, contact between flanks, and ice mélange that may also transmit stress between flanks. To illustrate the viability of the framework, we simulate the final 2 years of rift propagation associated with the calving of tabular iceberg A68 in 2017. We find that the rift path can change with varying ice mélange conditions and the extent of contact between rift flanks. Combinations of parameters associated with slower rift widening rates yield simulated rift paths that best match observations. Our modeling framework lays the foundation for robust simulation of rifting and tabular calving processes, which can enable future studies on ice-sheet–climate interactions, and the effects of ice-shelf buttressing on land ice flow.

A convex anisotropic damage model based on the compliance tensor

This work is devoted to anisotropic continuum-damage mechanics in the quasi-static, isothermal, small-strain setting. We propose a framework for anisotropic damage evolution based on the compliance tensor as primary damage variable, in the context of generalized standard models for dissipative solids. Based on the observation that the Hookean strain energy density of linear elasticity is jointly convex in the strain and the compliance tensor, we design thermodynamically consistent anisotropic damage models that satisfy Wulfinghoff’s damage-growth criterion and feature a convex free energy. The latter property permits obtaining mesh-independent results on component scale without the necessity of introducing gradients of the damage field. We introduce the concepts of stress-extraction tensors and damage-hardening functions, implicitly describing a rigorous damage-analogue of yield surfaces in elastoplasticity. These damage surfaces may be combined in a modular fashion and give rise to complex damage-degradation behavior. We discuss how to efficiently integrate Biot’s equation implicitly, and show how to design specific stress-extraction tensors and damage-hardening functions based on Puck’s anisotropic failure criteria. Last but not least we demonstrate the versatility of our proposed model and the efficiency of the integration procedure for a variety of examples of interest.

Coupled continuum damage mechanics and crystal plasticity model and its application in damage evolution in polycrystalline aggregates

In the present study, damage initiation and growth in a polycrystalline aggregate are investigated. In this regard, an anisotropic continuum damage mechanics coupled with rate-dependent crystal plasticity theory is employed. Using a thermodynamically consistent procedure, a finite deformation formulation is derived. For this purpose, the damage tensor is incorporated in the crystal plasticity formulation for a cubic single crystal. The damage evolution is considered to be dependent on the history of damage, equivalent plastic strain, stress triaxiality, and Lode parameters. This material model is implemented in the commercial finite-element code Abaqus/Standard by developing a user material subroutine (UMAT). Using the available experimental tests of 316L single crystal in the literature, the crystal plasticity hardening and damage parameters are calibrated considering the stress–strain curve before and after necking, respectively. The damage sites in a single-phase polycrystalline aggregate are also considered using a polycrystalline model consisting of grains with random sizes and orientations. The results show that the damage arises at the grain boundaries and triple junctions. Moreover, growth of the damage mostly occurs in the grains with higher Schmid factor compared to the neighboring grains. The presented model manifests capacity for determination of damage initiation sites and damage evolution in polycrystalline models.

Application of anisotropic continuum damage mechanics in ratcheting characterization

In this study, a combination of Chaboche’s nonlinear kinematic hardening model and Lemaitre’s anisotropic continuum damage mechanics (CDM) model is used to predict ratcheting strain of different loading paths. The fictitious continuum domain is considered, and the consistent relations are developed between real and fictitious materials. The return mapping algorithm is developed to obtain the plastic strain of different loading paths. Non-proportional loading paths are simulated by the combined model. The ratcheting strain of the stress-strain controlled loading paths is obtained numerically and compared with the experimental results of the cyclic loading tests of Corona et al.

Anisotropic continuum damage analysis of thin-walled pressure vessels under cyclic thermo-mechanical loading

The present study intends to analyze damage in thin-walled steel cylinders undergoing constant internal pressure and thermal cycles through use of anisotropic continuum damage mechanics (CDM) model coupled with nonlinear kinematic hardening rule of Chaboche. Materials damage in each direction was defined based on plastic strain and its direction. Stress and strain distribution over wall-thickness was described based on the CDM model and the return mapping algorithm was employed based on the consistency condition. Plastic zone expansion across the wall thickness of cylinders was noticeably affected with change in internal pressure and temperature gradients. Expansion of plastic zone over wall-thickness at inner and outer surfaces and their boundaries demarking elastic and plastic regions was attributed to the magnitude of damage induced over thermo-mechanical cycles on the thin-walled samples tested at various pressure stresses.

XFEM to couple nonlocal micromechanics damage with discrete mode I cohesive fracture

A computational tool is developed to simulate the propagation of a discrete fracture within a continuum damage process zone. Microcrack initiation and propagation prior to coalescence are represented by a nonlocal anisotropic Continuum Damage Mechanics (CDM) model in which the crack density is calculated explicitly. A damage threshold is defined to mark the beginning of crack coalescence. When that threshold is reached, a cohesive segment is inserted in the mesh to replace a portion of the damage process zone by a segment of discrete fracture. Discretization is done with the extended Finite Element Method (XFEM), which makes it possible to simulate fracture propagation without assigning the fracture path a priori. Rigorous calibration procedures are established for the cohesive strength (related to the damage threshold) and for the cohesive energy release rate, to ensure the balance of energy dissipated at the micro and macro scales. The XFEM-based tool is implemented into an open source object-oriented numerical package (OOFEM), and used to simulate wedge splitting and three-point bending tests. Results demonstrate that the proposed numerical method captures the entire failure process in mode I, from a mesh-independent diffuse damage zone to a localized fracture. Future work will investigate mixed mode fracture propagation.

Shakedown-ratcheting analysis of Bree's problem by anisotropic continuum damage mechanics coupled with nonlinear kinematic hardening model

In the present research, Bree's diagram, which has been used for ratcheting assessment of pressurized equipment in ASME III NH, has been developed when the pressure vessel contains discontinuities such as voids and microcracks. Nature of these defects leads to an anisotropic damage. Anisotropic Continuum Damage Mechanics (CDM) is considered to account effects of these discontinuities on the behavior of the structure. Shakedown – ratcheting response of a hollow sphere under constant internal pressure and cyclic thermal loadings is studied by using anisotropic CDM theory coupled with nonlinear kinematic hardening of Chaboche. Return mapping method is used to solve numerically the developed relations. Elastic, elastic shakedown, plastic shakedown and ratcheting regions are illustrated in the modified Bree's diagram. Influence of anisotropic damage due to the plastic deformation, is studied and it was shown that the plastic shakedown region is diminished because of the developed damage.

Using ultrasonic investigations to develop anisotropic damage models for initially transverse isotropic materials undergoing damage to remain transverse isotropic

A severe limitation imposed by many continuum damage mechanics models is the assumption of initial isotropy in many anisotropic damage models. This may place unrealistic assumptions about the material being modelled or restrict the application of continuum damage mechanics to materials without significant anisotropy. It remains a challenge to use anisotropic continuum damage mechanics to model common rocks and materials with significant initial anisotropy, for example sedimentary rocks or brittle composite materials. We show how ultrasonic investigations in experiments where an initially transverse isotropic material undergoes damage-induced anisotropy can be used to guide the development of transverse isotropic damage models. We provide a robust way to validate and advance models of general anisotropic damage evolution based on continuum damage mechanics.

Damage interaction and angle effects on the erosion behavior of soda-lime-silica glass

The present contribution aims to examine the erosion behavior of soda-lime-silica glass in connection with damage interaction and angle effects. Experimental observations are reported on glass plates subjected to sandblasting process using alumina abrasive particles for different sandblasting durations and impact angles. The damage and erosion mechanisms are computed through a numerical model of the sandblasted glass plate. The glass internal stiffness degradation due to impact process is accounted for by an anisotropic stress-based continuum damage mechanics model. The glass erosion is simulated by means of a vanishing element technique using the critical values of damage components as failure criterion. A parametric numerical study is carried out to bring insights into damage interaction and angle effects on the material loss.

Anisotropic (Continuum Damage Mechanics)-based multi-mechanism model for semi-crystalline polymer

In this work, the anisotropic damage of semi-crystalline polymers is investigated. The model, developed within a thermodynamic framework, includes the following features: (i) the degree of crystallinity; (ii) the hydrostatic pressure effect; and (iii) the damage anisotropy. The adopted tensorial damage variable is based on the Continuum Damage Mechanics approach under the energy equivalence assumption. For the quantification of the anisotropy, a parameter called “shape factor” is defined as the ratio between the void mean diameter and the void mean height. This parameter is linked to the main axial and the main radial damage components. Experimental data taken from the recent literature using the tomography technique were selected to assess the model capability. Finite element simulations of notched round bar specimens subjected to tensile test stopped at three key loading stages are systematically compared with experimental data. The proposed model was able not only to accurately simulate the macroscopic response of the material, but more interestingly, to reproduce the spatial distribution of the shape factor. This demonstrates the anisotropy effects of the material under study induced at different stages of the deformation.

Microdamage modelling of crack initiation and propagation in FCC single crystals under complex loading conditions

A regularised anisotropic continuum damage mechanics model is presented that couples crystal plasticity with damage induced on crystallographic planes. Two and three-dimensional finite element simulations of crack initiation and propagation in single crystal specimens under complex multiaxial loading with dominating shear are presented for the first time. The crack path in a four-point double notched shear specimen is predicted, in accordance with experimental testing of single crystal nickel-based superalloy AM1. The initiation, bifurcation and propagation of a crack in a shifted double notched specimen under creep conditions is then simulated and compared to the experimental response of nickel-based superalloy MC2 at high temperature. The attention is focused on the influence of coupling between viscoplasticity and damage on the crack path in these single crystals.

CONTINUUM DAMAGE MECHANICS MODEL FOR LOW-VELOCITY IMPACT DAMAGE ANALYSIS OF COMPOSITE LAMINATES

A three-dimensional anisotropic continuum damage mechanics (CDM) model, including damage characterization, damage initiation criterion and damage evolution law, was presented to analyze low-velocity impact damage of composite laminates. With considering affects of the fracture plane angle, material constitutive relation of damage states in the fracture plane coordinates was established by introducing damage state variable matrix in the material principal coordinates. The onset of damage was evaluated by the Puck criterion and the evolution of damage was controlled by equivalent strain on the fracture plane. Based on the viewpoint of strain energy release, the material was assumed to exhibit linear strain-softening behavior. Fiber fracture (FF) and inter-fiber fracture (IFF) were simulated within the lamina. Due to the experimental evidence which indicated that multiple intralaminar cracks developed in a composite laminate and coalesced in to one interface crack (delamination) under impact load, a intralaminar matrix crack density parameter at saturation was introduced to scale interlaminar delamination. The relevant low velocity damage response parameters of the laminates [03/45/-45]S and [45/0/-45/90]4S were predicted with the proposed model at various impact energies. A good agreement was achieved with experimentally obtained data and showed the validity of CDM model in this paper. Furthermore, results for different mesh-densities indicated that the approach by introducing a characteristic length of the element could alleviate the mesh dependency at the stage of material damage evolution.

Fracture of a borosilicate glass under triaxial tension

The growth of 3D star-like cracks in a porous borosilicate glass was induced by “cold-to-hot” thermal shocks inducing triaxial tension in cylindrical specimens. As a consequence of the dynamic character of their propagation, the cracks had a “mirror-mist-hackle” aspect. Damage in unbroken specimens was characterized either by a modification of their vibration modes or by direct observations of sections. Pores were identified as the crack initiation sites. Thermo-mechanical simulations were done to analyze the stress field and its consequences, using either linear thermo-elasticity or an anisotropic continuum damage mechanics approach. Crack initiation was predicted at mid-height, at the center of the specimens, in accordance with the observations. The pore-induced stress concentration was shown to depend on the local stress triaxiality, as well as the stress intensity factor for an annular crack initiated from a pore. The capacity of the anisotropic continuum damage approach to predict qualitatively the influence of the shape of the specimen, as well as damage-induced modifications of vibration modes and the total fracture surface was demonstrated.

Anisotropic density growth of bone—A computational micro-sphere approach

Bones are able to adapt their local density when exposed to mechanical loading. Such growth processes result in densification of the bone in regions of high loading levels and in resorption of the material in regions of low loading levels. This evolution and optimisation process generates heterogeneous distributions of bone density accompanied by pronounced anisotropic mechanical properties. While several constitutive models reported in the literature assume the growth process to be purely isotropic, only few studies focus on the modelling and simulation of anisotropic functional adaptation we can observe in vivo. Some of these few computational models for anisotropic growth characterise the evolution of anisotropy by analogy to anisotropic continuum damage mechanics while others include anisotropic growth but assume isotropic elastic properties.The objective of this work is to generalise a well-established framework of energy-driven isotropic functional adaptation to anisotropic microstructural growth and density evolution. We adopt the so-called micro-sphere concept, which proves to be extremely versatile and flexible to extend sophisticated one-dimensional constitutive relations to the three-dimensional case. In this work we apply this framework to the modelling and simulation of anisotropic functional adaptation by means of a directional density distribution, which evolves in time and in response to the mechanical loading condition. Several numerical studies highlight the characteristics and properties of the anisotropic growth model we establish. The formulation is embedded into an iterative finite element algorithm to solve complex boundary value problems. In particular, we consider the finite-element-simulation of a subject-specific proximal tibia bone and a comparison to experimental measurements. The proposed model is able to appropriately represent the heterogeneous bone density distribution. As an advantage over several other computational growth models proposed in the literature, a pronounced local anisotropy evolution is identified and illustrated by means of orientation-distribution-type density plots.

Simulation of Crack Propagation in Three-Point Bending Piezoelectric Beam Based on Three-Dimensional Anisotropic Piezoelectric Damage Mechanics

ABSTRACTA finite element method combined with three-dimensional anisotropic piezoelectric continuum damage mechanics is presented to simulate quasi-static crack propagation behavior in piezoelectric ceramics in this paper. In this method, the three-dimensional anisotropic piezoelectric damage constitutive model is utilized for characterizing the effects of mechanical and electrical damages on the fields near the crack tip, the combined-damage from the dominant mechanical and electrical damage components is regarded as the fracture criterion, and the gradient of combined-damage is assumed to control crack growth direction. A set of numerical simulations of the midspan crack propagation in a three-point bending PZT-4 beam are performed in various loading conditions. After the numerical results are validated by comparison with the corresponding experimental ones, the effects of mechanical and electrical loads on the cracking be havior are respectively evaluated. It is found from the obtained results that mechanical and electrical loads influence on the damage fields in the vicinity of the crack-tip, as well as the crack growth rate, in a significant way. With the increment in mechanical loading, the crack growth rate obviously increases. This means that positive and negative electric fields enhance and inhibit crack propagation, respectively.

How cracks affect the contact characteristics during impact of solid particles on glass surfaces: A computational study using anisotropic continuum damage mechanics

Finite element analyses are used to simulate the response of glass impacted by solid particles. The Sun-Khaleel stress-based continuum damage mechanics (CDM) model is used to describe the constitutive behavior of glass. The contribution of different cracking systems in the strength degradation of glass is explicitly taken into account by the anisotropic CDM model which is enhanced by coupling with a vanishing element technique. Focusing on the initiation and propagation of the ring/cone crack, a combined numerical and experimental approach is proposed to estimate the CDM stress limits. The changes in contact characteristics due to erosion damage mechanism in glass are then studied using the developed predictive tool. In accordance with available experimental observations, it is found that the impact-induced damage reduces the energy restitution coefficient and increases the contact time.

Eshelby stress tensor in large strain anisotropic damage mechanics

The paper deals with a framework for anisotropically damaged elastic–plastic solids and the definition of the Eshelby stress tensor in anisotropic continuum damage mechanics. The proposed anisotropic elastic–plastic-damage model is based on a generalized macroscopic theory within the context of nonlinear continuum damage mechanics taking into account kinematic description of damage. It employs the consideration of damaged as well as fictitious undamaged configurations related via metric transformations which allow for the interpretation of damage tensors. A macroscopic yield condition is used to adequately describe the plastic flow properties of ductile metals and a damage criterion takes into account isotropic as well as anisotropic effects providing a realistic physical representation of material degradation. Furthermore, the unbalance of pseudo-momentum is established based on the second law of thermodynamics formulated with respect to the damaged configurations. Evaluation of a strain energy function and assuming the existence of pseudo-potentials of plastic as well as of damage dissipation leads to the definition of the Eshelby stress tensor, inhomogeneity force and material dissipation force.

A theoretical and computational framework for anisotropic continuum damage mechanics at large strains

Abstract The main objective of this work is the formulation and algorithmic treatment of anisotropic continuum damage mechanics at large strains. Based on the concept of a fictitious, isotropic, undamaged configuration an additional linear tangent map is introduced which allows the interpretation as a damage deformation gradient. Then, the corresponding Finger tensor – denoted as damage metric – constructs a second order, internal variable. Due to the principle of strain energy equivalence with respect to the fictitious, effective space and the standard reference configuration, the free energy function can be computed via push-forward operations within the nominal setting. Referring to the framework of standard dissipative materials, associated evolution equations are constructed which substantially affect the anisotropic nature of the damage formulation. The numerical integration of these ordinary differential equations is highlighted whereby two different schemes and higher order methods are taken into account. Finally, some numerical examples demonstrate the applicability of the proposed framework.

On the theory and computation of anisotropic damage at large strains

The goal of this contribution is the formulation and algorithmic treatment of anisotropic continuum damage mechanics at large strains. Based on the concept of an isotropic, fictitious and undamaged configuration a Finger—type damage metric tensor in terms of the fictitious linear tangent map is introduced. Referring to the principle of strain energy equivalence with respect to the fictitious, effective space and the standard reference configuration the free Helmholtz energy can be formulated within the standard reference configuration and treated in the spirit of standard dissipative materials. Thereby, the introduced damage potential substantially affects the anisotropic nature of the damage formulation. Finally, some numerical examples demonstrate the applicability of the proposed framework.