Damage Tensor Research Articles

A computational approach with surface-based cohesive contact for meso-scale interface damage simulation in 3D braided composites

The yarn/yarn and yarn/matrix interface debonding has been recognized as a vital failure mode of 3 D braided composites. We present in this paper a meso-scale finite element (FE) model, which considers yarn/yarn and yarn/matrix interface debonding, for modeling progressive damage evolution of 3 D braided composites under typical tensile and shear loadings. In this setting, the damage state of braiding yarns and matrix is described through a continuum damage model (CDM) coupled with Murakami damage tensor; a bilinear traction-separation description is employed to govern the yarn/yarn and yarn/matrix interface behavior modeled by surface-based cohesive contact. We thus develop a user-material subroutine VUMAT (ABAQUS/Explicit) for our progressive damage simulation, including stress analysis, failure analysis and material properties degradation scheme. The mechanical properties of 3 D braided composites, and more importantly the damage evolution of interface debonding are thoroughly analyzed. The proposed FE modeling strategy provides a new perspective for the interface response study of other textile composites.

A three‐dimensional multiscale damage‐poroelasticity model for fractured porous media

AbstractThe paper investigates the failure of brittle rocks within a multiscale/multiphysics computational modeling framework so as to explicitly incorporate both microstructural and hydromechanical aspects in their overall (nonlinear) fracture behavior. Herein, the rock is idealized as a microfissured porous medium via a representative elementary volume (REV) containing distributed oblate spheroidal open microcracks and spheroidal nanopores at the lower scales. A two‐scale analytical homogenization procedure is then performed on the REV, assuming a saturated pore space across the scales. The end result is a microstructurally enriched continuum damage‐poroelasticity constitutive model within the generalized Biot's framework that inherits the underlying small‐scale characteristics. A set of damage tensors is derived based on the directional density distribution of microcracks that operate on both the poromechanical and hydraulic properties. Microstructural evolution is modeled following changes in microcrack length, aperture, and number density, including nanoporosity. To this end, a robust localization procedure is used that accurately captures the macroscopic softening due to microcracking events, leading to a nonlinear (but path‐independent) model. To investigate the baseline features of the model, including the salient effects of microcracking on induced anisotropy and deterioration of poroelastic properties, numerical results of material point‐based computations are discussed in detail. Finally, predictions of the current model are validated against experiments on Fontainebleau sandstone.

Strength anisotropy of jointed rock slope considering mining damage: a numerical analysis method

A combination of Hoek–Brown strength criterion and geological strength index (GSI) rock mass classification is a common method to determine rock mass strength. However, the conventional method involving the determination of disturbance coefficient (D) cannot reflect the anisotropic characteristics of strength of jointed rock mass when considering mining damage. A numerical analysis method is proposed to determine the value of D using jointed rock damage tensor and mining damage evolution simulation. The anisotropy of rock mass strength at different mining depths for a high jointed rock slope in Dagushan open-pit mine was analyzed by mining damage simulation. Additionally, the Rosenblueth moment point estimation principle was combined with Monte-Carlo method to obtain the random strength of a jointed rock mass considering the effects of mining probability damage. The results show that the influence of the unloading due to excavation to stress state of rock mass is relatively obvious within 100 m away from the slope surface, however, the area of drastic change of damage is limited to a range of ∼50 m. Moreover, the reliability of local slope, analyzed using limit equilibrium method (Janbu), shows that consideration of spatiotemporal anisotropy of rock mass with mining damage is closer to reality.

Using structural tensors for inelastic material modeling in the finite strain regime – A novel approach to anisotropic damage

The idea of the paper is to apply the well established structural tensors approach to complex inelastic material modeling in the finite strain regime. Using this concept, a very general and new anisotropic damage model is derived which is the major novel contribution of the paper. In this framework, a second order damage tensor plays the role of a structural tensor. The damage model is coupled to finite elasto-plasticity such that its first purpose is to model damage-induced anisotropy in ductile materials, in particular metals. Further structural tensors could be introduced to represent initial anisotropy of materials such as soft tissues, composites or thermoplastics. The main results of the derivation are the suitable choice of the Helmholtz free energy function, the development of yield and damage potentials as well as the formulation of physically reasonable evolution equations for the plastic deformation, the damage tensor as well as plastic and damage hardening. It is important to mention that all tensorial internal variables are symmetric and referred to the undeformed configuration. To the knowledge of the authors, the interpretation of a structural tensor of the undeformed configuration as damage tensor is new. Two-point tensors such as the plastic part of the deformation gradient show up in the theoretical derivation but are never computed. The plastic spin remains undetermined which has the advantage that a statement about its evolution is not needed. Further, the evolution equation for the plastic deformation includes only six non-linear scalar equations instead of nine. The numerical implementation of the model is straightforward. The number of material parameters remains moderate. A strategy to identify them by means of standard experiments is discussed. Nevertheless, the validation of the model by means of own experiments or experimental results from the literature is left for further work.

Stress distribution model of hard and brittle mud shale based on hydration damage

Hard and brittle shale developed in the middle and deep layers of the Bohai Sea cause severe challenges in drilling. During the drilling process, drilling fluid easily penetrates the rock along the microcracks, which leads to the deterioration of the mechanics and strength parameters of the rock. This consequently changes the stress distribution in the borehole wall. In addition, the fluid seepage hydrates the rock, which affects the relationship between the change in water content at any position and at any time in the rock mass is solved. The hydration equivalent of the elastic modulus is regarded as damage evolution, and a damage tensor considering hydration is introduced. A constitutive relationship of hard and brittle mud shale considering the effect of hydration damage is established, and the stress distribution around the well of the hard and brittle mudstone under non-uniform stress is solved by a semi-numerical and semi-analytical method. In addition, the influence law of hydration damage on the stress distribution around the well is analyzed. Taking an 8-1/2-inch section of an exploratory well in the Bozhong 19-6 gas field as an example, the stress distribution and collapse pressure analysis of the well bore revealed that after the section of the well was drilled for 100 h, the formation stiffness caused by hydration damage significantly decreased and borehole instability occurred inside the borehole wall. The risk of collapse was greater in the direction of the smallest in situ stress. The effect of hydration expansion on the stress distribution was limited, showing only a slight effect on the tangential stress near the well wall. The collapse of hard and brittle mud shale is mainly related to deterioration of the rock stiffness performance and strength parameters caused by hydration damage. The section incurred a short period of collapse, in which the maximum diameter expansion rate was equivalent to 5.9%, and the density of drilling fluid used on site was relatively low. Therefore, the density of the drilling fluid should be appropriately increased to enhance its sealing properties. The model reveals the hydration collapse mechanism of hard and brittle shale to a certain extent, which provides a theoretical basis for the safe and efficient drilling in similar formations.

Non-linear anisotropic damage rheology model: Theory and experimental verification

We extend the isotropic non-linear damage rheology model with a scalar damage parameter to a more complex formulation that accounts for anisotropic damage growth under true triaxial loading. The model takes account of both the anisotropy of elastic properties (associated with textural rock structure) and the stress- and damage-induced anisotropy (associated with loading). The scalar, isotropic model is modified by assuming orthotropic symmetry and introducing a second-order damage tensor, the principal values of which describe damage in three orthogonal directions associated with the orientations of the principal loading axes. Different damage components, accumulated under true triaxial loading conditions, allows us to reproduce both stress-strain curves and damage- and stress-induced seismic wave velocity anisotropy. The suggested model generalization includes a non-classical energy term similar to the isotropic non-linear scalar damage model, which allows accounting for the abrupt change in the effective elastic moduli upon stress reversal.For calibration and verification of the model parameters, we use experimental stress-strain curves from deformation of dry sandstone under both conventional and true triaxial stress conditions. Cubic samples were deformed in three orthogonal directions with independently controlled stress paths. To characterize crack damage, changes in ultrasonic P-wave velocities in the three principal directions were measured, together with the bulk acoustic emission output. The parameters of the developed model were constrained using the conventional triaxial test data, and provided good fits to the stress-strain curves and P-wave velocity variations in the three orthogonal directions. Numerical simulation of the true triaxial test data demonstrates that the anisotropic damage rheology model adequately describes both non-linear stress-strain behavior and P-wave velocity variations in the tested Darley Dale sandstone.

Experimental investigation of anisotropic fatigue characteristics of shale under uniaxial cyclic loading

The influence of combining the inherent anisotropy and fatigue loading on shale’s mechanical behaviors is rarely investigated. To study the anisotropic fatigue behaviors of shale, uniaxial fatigue tests under different cyclic stress limits and loading frequencies are conducted using shale specimens with 7 different bedding orientations. The anisotropic characteristics of the fatigue failure mechanism, stress-strain behavior, cycle numbers, irreversible strain and energy evolution of each hysteresis loop are analyzed. The combined effects of bedding and fatigue loading can cause crack branching and coalescence, which generates a crack network. Maximum and minimum fatigue cycle numbers are obtained when the bedding orientations are 90° and 75°, respectively; correspondingly, the maximum and minimum energies of each hysteresis loop are obtained. The irreversible strain and energy of each hysteresis loop decrease, become approximately constant and then increase before failure when the bedding orientations are 0°, 15°, 30°, 45° and 90°; this rate is approximately constant for the bedding dip angles of 60° and 75°. Increasing either the loading frequency or cyclic stress limit increases the energy and irreversible strain of each hysteresis loop as well as decreases the cycle number. Fatigue damage indicators defined by the secant modulus, irreversible strain, dissipated energy and accumulated hysteresis loop area cannot reflect the effects of shale’s inherent anisotropy. Multiscalar or tensorial damage factors, which can describe the anisotropic fatigue damage features of shale, need be further studied.

Estimation of dynamic thermo viscoelastic moduli of short fiber‐reinforced polymers based on a micromechanical model considering interphases/interfaces conditions

AbstractA novel three‐dimensional micromechanical model describing frequency‐ and temperature‐dependent viscoelastic behavior of short fiber‐reinforced polymers (SFRPs) with graded interphase and slightly weakened interfaces are established. Elastic‐viscoelastic correspondence approach, time‐temperature superposition principle and Rao and Dai's (Compos. Struct. 2017; 168: 440) modification on the calculation of interface damage tensors are applied for dynamic thermo viscoelasticity prediction of the composite with the consideration of interphase/interface conditions. Certain parameters like filler content, interphase property, the aspect ratio of inclusions, frequency, and temperature were thoroughly analyzed to see their effect on the dynamic behavior of unidirectional SFRPs. Some valuable conclusions are made, which constitutes a valuable contribution to the theoretical prediction of SFRPs' mechanical properties.

Computational simulation of damage accumulation processes in cracked bodies by the UMAT procedure of SIMULIA Abaqus

The paper presents the experience of using the user subroutine UMAT for finite element package SIMULIA Abaqus/CAE for damage accumulation processes in the vicinity of the crack. A continuum damage mechanics model based on the constitutive relations of linear elastic isotropic materials with the incorporated damage tensor components is used to describe the material behavior. The material nonlinearity arising from the deformation process is modeled by introducing an anisotropic damage tensor of the second rank into the constitutive equation. The material model is described by means of user procedure UMAT of SIMULIA Abaqus. The finite element (FE) mechanical constitutive model is implemented in Abaqus/Standard via a UMAT routine. Numerical experiments for a large series of cracked specimens have been performed. Computed stress and damage tensor components were found. It is shown that they are not dependent on the FE mesh refinement. Distributions of the damage tensor components in the vicinity of the crack tip in cracked specimens of different configurations under mixed mode loading in a wide range of mixed mode loadings are found. The configurations of active damage accumulation process zone in the cracked specimens are obtained. It is shown that the damage accumulation process has substantial influence on the stress-strain state in the vicinity of the crack tip and leads to decrease of the stress concentration in cracked specimens.

An anisotropic brittle damage model with a damage tensor of second order using a micromorphic approach

AbstractThe work presents a gradient‐extended ansisotropic damage model for brittle materials utilizing a second order damage tensor. In order to take into account tension compression asymmetry (TCA), the strain tensor is split into a tension and a compression related part by means of a spectral decomposition. The formulation is based on a global incremental potential allowing for large time/load steps. The gradient extension is accomplished using the micromorphic approach. Although using a damage tensor of second order, only one scalar additional degree of freedom (micromorphic variable) is introduced. First, the effects of TCA are illustrated with a simple uniaxial loading case at integration point level. Afterwards, the effect of the additionally introduced artificial viscosity, used in order to circumvent snapback situations, is shown for a structural example (without TCA). Finally, a structural example considering TCA is presented for which spurious failure in compression can be avoided.

Extended finite element method and anisotropic damage plasticity for modelling crack propagation in concrete

In this article, a new algorithm is developed for predicting crack initiation and growth direction in 3D solid concrete. The algorithm combines the anisotropic damage-plasticity model used for simulation of concrete material degradation (crack) and irreversible plastic deformation and the eXtended Finite Element Method (X-FEM). The Voyiadjis local anisotropic damage-plasticity model is expressed in a non-local form to prevent strain and damage localization in damaged points. Also, a new crack propagation direction criterion based on second order damage tensor of anisotropic damage, containing different material degradation components in different directions is proposed for 3D problems. Computational algorithms with an elastic predictor and plastic and damage corrector returning map are also developed. Finally, three numerical examples are solved and results of crack propagation paths predicted by the proposed algorithm are compared with available experimental data and those of other numerical simulations. It is shown that the results from the proposed method are in better agreement with experimental data compared to other reported numerical methods.

Non-proportional stress and stress-strain controlled paths cyclic loading modeling by using anisotropic continuum damage model

In this study, the effect of developed anisotropic damage on the ratcheting is studied. Damage occurs during extra loading in an anisotropic manner. A modified anisotropic continuum damage model of Murakami is developed in this paper and combined with nonlinear kinematic hardening model of Chaboche. The constitutive relations for anisotropic damage of elastoplastic materials are developed based on irreversible thermodynamics. 2nd rank symmetric damage tensor, the kinematic hardening tensor and tensor of damage potential surface movement are specified according to the internal state manifold which is consistent to the thermodynamic state of solids. The symmetrization of effective stress tensor is carried out by the proposed method of Cordebois and Sidoroff (1982). The proposed yield surface distortion model consists of equivalent effective stress and it distorts in the stress space by evolution of damage parameter. Thus, the induced anisotropy is due to plastic deformation and damage evolution. Combination of the nonlinear kinematic hardening model of Chaboche and the model of anisotropic continuum damage is applied to cyclic loading modeling and the ratcheting strains are predicted under various loading paths.

Multi-scale damage framework for textile composites: Application to plain woven composite

A damage mechanics-based progressive damage analysis (PDA) of plain woven textile composite (PWTC) is performed in this work. The primary objective is to identify the micro-scale and meso-scale failure modes, and compute the ultimate strength of PWTC under in-plane loadings. Multi-scale PDA is performed on an equivalent cross-ply laminate (ECPL) model by maximum stress theory to predict the meso-scale failure modes of PWTC. The damaged stiffness is computed by 3 different ways: 1) hypothesis of strain energy equivalence by proposed 4th order anisotropic damage tensor, 2) applying isotropic damage law, and 3) eliminating the stiffness terms. Novel multi-scale PDA approaches, applying Mori-Tanaka (MT) theory on ECPL model and on the quarter portion of representative volume element of PWTC, are performed to predict micro-scale failure modes. The macro-scale constitutive behaviour of PWTC is theoretically predicted and the corresponding strengths are computed. A user material (UMAT) subroutine is developed to validate the proposed constitutive behaviour of PWTC solving macro-scale boundary value problems in ABAQUS®. The PDA simulation results of 3D PWTC bar model are found to be reasonably matching with the in-house experiments.

Numerical modeling of fracturing in permeable rocks via a micromechanical continuum model

SummaryThe paper presents a micromechanical approach to describe the failure of low‐permeability brittle rocks as a multiscale fracturing process based on a poroelastic microcrack‐damage model. Failure is formulated deep down at the fine pore scale as a material degradation phenomenon driven by microcrack growth that also impacts upon hydromechanical properties. A set of damage tensors describes the effect of dual‐scale porosities (nanopores and microcracks) on both the hydraulic and poroelastic rock properties. Essentially, the multiscale model reconstructs the coupling effect of hydromechanical forces at the continuum level from the ground up through the upscaling of multiphase interactions at the fundamental structural level of the material. As a result, many macroscopic characteristics emerge naturally such as friction angle, fracture properties, and most importantly, Biot's coefficient taking on a tensorial form that is generally anisotropic. The model is validated within the framework of finite elements to illustrate various baseline constitutive features such as the effect of microcrack growth on the nonlinear stress‐strain response and the induced anisotropy in the context of lab experimental tests and boundary value problems. Heterogeneities of the rock samples were incorporated by choosing material properties to be stochastic following Weibull and lognormal distributions. Numerical results appropriately replicated typical experimental observations where fracture localization and propagation are shown to be a multiscale phenomenon emerging from microcrack growth and coalescence at the microscale, with concomitant enhancement in fluid conductivity.

Efficient algorithmic incorporation of tension compression asymmetry into an anisotropic damage model

In order to exclude spurious failure in compression the consideration of tension compression asymmetry (TCA) in damage models is of great interest in research as well as in industry. This paper presents an efficient and unifying algorithmic procedure for the incorporation of TCA into damage models. With the presented approach the implementation effort is drastically reduced since the equations and subroutines of the model without TCA can be reused. The strategy is first introduced and illustrated for an isotropic damage model. Afterwards, the procedure is applied to a gradient-extended anisotropic damage model with a second order damage tensor recently developed by the authors. For this model studies at integration point level as well as structural examples are presented. Two different aspects are included in the presented TCA approach and are demonstrated with the results of the simulations: (i) different damage evolution in tension and compression and (ii) stiffness recovery (crack closure) under compression. Due to the utilized gradient-extended formulation, being very efficient since only one additional degree of freedom is introduced, the finite element computations are shown to deliver mesh-objective results.

Towards stochastic continuum damage mechanics

In classical continuum damage mechanics, the distribution of cracks over differently oriented planes is an even deterministic function defined on the unit sphere. The coefficients of its Fourier expansion are completely symmetric and completely traceless tensors of even rank, the so-called fabric or damage tensors. We propose a stochastic generalisation of the above described mathematical model, where damage tensors are mean-square continuous wide-sense homogeneous and isotropic random fields.

The response of a partially loaded composite half-space weakened by local defects

The elastic field in a fiber-reinforced half-space which consists of distinct constituents and localized internal defects is predicted by employing a multiscale analysis. A normal distributed uniform loading is applied over a rectangular region of the half-space surface, while the other parts are kept traction-free. The effect of internal defects and the half-space traction-free boundary conditions are accounted to by incorporating a damage tensor in the constitutive relations which describe the half-space phases. In the micro level of the micro-to-macro analysis, the effective behavior of the composite half-space is determined by employing a micromechanics analysis. This is followed by a macromechanical analysis in which the triple discrete Fourier transform is applied in the domain of which the formulated problem is solved. The damaged composite half-space response is obtained by inverting the Fourier transform, in conjunction with an iterative procedure. The predicted elastic field in the half-space which is based on the present approach is validated by comparisons with exact and finite element solutions. The offered method is applied to generate the elastic field in a fiber-reinforced half-space with a broken fiber, lost fiber, debonded fiber, missing fiber and a matrix void. Applications of numerous defects of these and other types are discussed.

Gradient-extended anisotropic brittle damage modeling using a second order damage tensor – Theory, implementation and numerical examples

The paper presents a general gradient-extended continuum mechanical framework for materials with internal variables based on additional generalized balance equations. The framework is applied to the case of anisotropic brittle damage where damage is modeled by a second order damage tensor. Although using a second order damage tensor the proposed efficient formulation being implemented into finite elements uses only one scalar additional nodal degree of freedom. Based on the damage growth criterion a specific form of the elastic strain energy is proposed for initially isotropic materials such that artificial stiffening effects are excluded a priori. Special focus is placed on the numerical implementation at the integration point level: Within the concept of generalized standard materials a regularized dissipation potential is used to cope with different inequality constraints, leading to the introduction of penalty viscosity parameters which are chosen sufficiently large such that the occurring errors remain negligibly small. Furthermore, a novel additional damage hardening is suggested which ensures that the eigenvalues of the damage tensor do not exceed the value one. By means of several numerical examples it is demonstrated that the model delivers mesh-independent results and is able to represent (i) localized damage (fracture) and (ii) diffuse (distributed) damage. Finally, isotropic damage (which can be shown to be a special case of the model) and anisotropic damage are compared considering two numerical examples where the occurrence of either localized or diffuse damage will be shown to be crucial.

Parametrically homogenized continuum damage mechanics (PHCDM) models for composites from micromechanical analysis

This paper develops a parametrically homogenized continuum damage mechanics (PHCDM) model for unidirectional fiber reinforced epoxy composites undergoing progressive damage. The PHCDM models are designed to overcome limitations of prohibitive computational overhead associated with many homogenization methods. They are thermodynamically consistent, reduced order continuum models with explicit representation of microstructural morphology through strategically determined parameters. The PHCDM model is derived from detailed micromechanics simulations of the representative volume element (RVE) using energy equivalence principles. Micromechanical failure is due to fiber–matrix interface debonding and matrix cracking. The macroscopic PHCDM models represent damage anisotropy through a second order damage tensor that contributes to the evolution of a damage surface in the space of damage work conjugate. The damage surface characterizes the initiation and evolution of damage in the composite. The constitutive relation between damage and its work conjugate is represented by an anisotropic fourth order damage surface tensor Pijkl, whose components are expressed as functions of current damage state. These are calibrated and validated from homogenized micromechanics (HMM) simulations. The PHCDM model is incorporated in a commercial finite element code and structural analysis of macroscopic composite components are executed for understanding concurrent damage and failure at multiple scales.

Micromechanical Modeling of a Generator Stator Electrical Insulation System

AbstractA gradient‐extended damage theory is shortly presented. The focus is on the internal thermodynamic forces and their extension to gradient‐extended models. This continuum mechanical framework is used for an application to an anisotropic damage model with a 2nd order damage tensor.