Second-order Damage Tensor Research Articles

Gradient-enhanced modelling of deformation-induced anisotropic damage in metallic glasses

Recent experimental and computational studies at different scales reveal an apparent flow-induced anisotropy of the inelastic deformation behaviour in metallic glasses (MGs). However, the anisotropic damage behaviour accompanied by the formation of shear bands is not adequately described in the previous constitutive modelling work. In this study, we develop a thermodynamically-consistent, anisotropic damage model incorporating tension–compression asymmetry (TCA) to describe the unique deformation and damage behaviours. The elastic–plastic response, including the normal stress sensitivity and the plastic dilatancy, is captured by the extended Mohr–Coulomb criterion. A second-order damage tensor is adopted for describing the anisotropic damage behaviour of MGs. By augmenting the Helmholtz free energy function to be dependent on the gradient of the free volume concentration and the gradient of the nonlocal damage parameter, the kinetic equations for the corresponding internal variables are derived within the framework of finite deformation continuum thermodynamics. Different material degradations under tension and compression are considered by decomposing the elastic strain tensor into a positive mode and a negative mode in the free energy function, where the tension-strain-induced damage is fully switched on while the compression-strain-induced damage is only partially activated. The influence of the TCA parameter on the damage initiation, crack propagation and shear bands density, as well as the influence of the gradient regularisation parameter on the width of the damage zone, are systematically studied. The proposed damage model is validated by experiments carried on unnotched and notched microcantilever beams. The simulation results show that both the displacement–load curves and the shear band patterns are in good agreement with the experimental observations.

A new anisotropic elasto-plastic-damage model for quasi-brittle materials using strain energy equivalence

An anisotropic elasto-plastic-damage model is developed for quasi-brittle materials within the thermodynamics framework. The anisotropic damage is characterized by second-order damage tensors that are different under tensile and compressive loadings. The hypothesis of strain energy equivalence is adopted with two components, including the hypothesis of elastic strain energy equivalence and the hypothesis of plastic strain energy equivalence. The hypothesis of plastic strain energy equivalence is extended from metals to quasi-brittle materials so as to derive the relations of hardening stresses, which are previously assumed between the undamaged and damaged configurations for quasi-brittle materials. A plastic yield criterion is adopted simultaneously with a damage criterion to account for the coupled elasto-plastic-damage behavior under different loadings. The damage criterion can capture larger axial damage in uniaxial tension and smaller axial damage in uniaxial compression as compared with the lateral damage. The Helmholtz free energy functions are derived for three components: elastic, plastic, and damage. The constitutive laws are derived for three scenarios: the elastoplastic behavior without further damage evolution, the damage behavior without further plasticity evolution, and the coupled elasto-plastic-damage behavior. The anisotropic elasto-plastic-damage model is implemented in ABAQUS as a user material (UMAT) subroutine. The coupled constitutive laws in the continuum domain are implemented in an uncoupled way in the discrete domain with three steps, elastic predictor, plastic corrector, and damage corrector. The UMAT is validated in different conditions, including uniaxial, biaxial, and three-point bending tests. The advantage of considering the plastic free energy in anisotropic damage is illustrated through the strength envelope in biaxial loadings. The numerical results generally agree with experimental results, including the stress-strain curves in uniaxial tests, strength envelope in biaxial tests, and load-deflection curves in three-point bending tests.

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.

Fatigue life predictions of metal structures based on a low-cycle, multiaxial fatigue damage model

A modeling strategy is developed accounting for the simulation of fatigue in metallic structures subjected to multiaxial, cyclic loading conditions. Cyclic plastic deformations are assumed to cause microcrack nucleation and growth, which in turn, lead to a deterioration of the mechanical and thermal properties. Microcrack nucleation and growth are modeled by a continuum damage mechanics formulation which accounts for a progressive loss of material integrity. The formulation utilizes a critical plane method for the prediction of microcrack nucleation. The emerging microcracks are assumed to be aligned with the critical plane orientation causing an anisotropic damage behavior. A second-order damage tensor is defined to approximate the damage behavior by an orthotropic material response. Damage evolution caused by microcrack growth is obtained from an empirical energy criterion based on the inelastic strain energy density. This way, the isotropic material response of a pristine material becomes orthotropic during damage evolution causing a direction dependent loss of material integrity. The continuum damage mechanics formulation in combination with the nonlinear Finite Element Method enables the simulation of structural deterioration caused by a propagating region of material failure. The approach is implemented into the Finite Element Method and extended by a cycle jump technique to reduce the computational time. The modeling strategy is exemplified on micrometer-sized cantilever beams under low-cycle fatigue conditions.

Development of an elastothermoviscoplasticity damage model for injection molded short fiber reinforced thermoplastics with anisotropic damage evolutions

ABSTRACTThis paper presents a finite deformation anisotropic and temperature-dependent constitutive model, which is developed for predicting elastoviscoplasticity and progressive damage behavior of injection molded short fiber reinforced thermoplastics (SFRPs) by the usage of the internal state variable (ISV) theory. In the present model, the SFRP is considered as an anisotropic single lamina and the viscoplasticity of the SFRP is captured with the help of three inelastic ISVs. A second-order damage tensor is introduced to model the evolution of the micro cracks/voids in the SFRP composites while also reflecting the anisotropic nature of the damage. The constitutive model developed herein arises employing standard postulates of continuum mechanics with the kinematics, thermodynamics, and kinetics being internally consistent. The developed model is then calibrated to the injection molded 35 wt% short glass fiber reinforced polyamide 6,6 (designation PA66GF-35) with all material constants obtained by fitting the model to the experimental data. A parametric study is conducted on the model to numerically investigate the effects of strain rate, temperature, and fiber orientation effects on the elastoviscoplasticity and damage performance of such material.

Micromechanics based framework with second-order damage tensors

The harmonic product of tensors---leading to the concept of harmonic factorization---has been defined in a previous work (Olive et al, 2017). In the practical case of 3D crack density measurements on thin or thick walled structures, this mathematical tool allows us to factorize the harmonic (irreducible) part of the fourth-order damage tensor as an harmonic square: an exact harmonic square in 2D, an harmonic square over the set of so-called mechanically accessible directions for measurements in the 3D case. The corresponding micro-mechanics framework based on second---instead of fourth---order damage tensors is derived. An illustrating example is provided showing how the proposed framework allows for the modeling of the so-called hydrostatic sensitivity up to high damage levels.

Constitutive equations of a tensorial model for strain-induced damage of metals based on three invariants

On the basis of the physical concepts of void formation, nucleation, and growth, generalized constitutive equations are formulated for a tensorial model of plastic damage in metals based on three invariants. The multiplicative decomposition of the metric transformation tensor and a thermodynamically consistent formulation of constitutive relations leads to a symmetric second-order damage tensor with a clear physical meaning. Its first invariant determines the damage related to plastic dilatation of the material due to growth of the voids. The second invariant of the deviatoric damage tensor is related to the change in void shape. The third invariant of the deviatoric tensor describes the impact of the stress state on damage (Lode angle), including the effect of rotating the principal axes of the stress tensor (Lode angle change). The introduction of three measures with related physical meaning allows for the description of kinetic processes of strain-induced damage with an equivalent parameter in a three-dimensional vector space, including the critical condition of ductile failure. Calculations were performed by using experimentally determined material functions for plastic dilatation and deviatoric strain at the mesoscale, as well as three-dimensional graphs for plastic damage of steel DC01. The constitutive parameter was determined from tests in tension, compression, and shear by using scanning electron microscopy, which allowed to vary the Lode angle over the full range of its values Open image in new window. In order to construct the three-dimensional plastic damage curve for a range of triaxiality parameters \(-1 \le ST \le 1\) and of Lode angles Open image in new window, we used our own, as well as systematized published experimental data. A comparison of calculations shows a significant effect of the third invariant (Lode angle) on equivalent damage. The measure of plastic damage, based on three invariants, can be useful for assessing the quality of metal mesostructure produced during metal forming processes. In many processes of metal sheet forming the material experiences, a non-proportional loading accompanied by rotating the principal axes of the stress tensor and a corresponding change of Lode angle.

Stochastic modeling of damage evolution in composites under environmental ageing

A mathematical model is proposed for predicting damage evolution and stiffness degradation in composite materials under environmental ageing. Environmental parameters such as moisture, temperature, and ultraviolet (UV) radiation cause hygrothermal loads on the structure, which leads to damage evolution. The present work establishes a constitutive model for treating the damage density as a random variable. A forward stochastic differential equation (FSDE) is proposed to predict the damage density evolution. Damage nucleation and annihilation rates are taken into consideration in terms of Brownian motions. A second-order damage tensor is developed using the solution of the FSDE. The damage tensor is incorporated into the constitutive model for predicting the elastic moduli. Finally, the proposed model is verified against experimental observations under certain hygrothermal conditions.

A temperature dependent creep damage model for polycrystalline ice

We present a continuum damage model for the temperature dependent creep response of polycrystalline ice under a multiaxial state of stress, suited for ice in polar regions. The proposed model is based on a thermo-viscoelastic constitutive law for ice creep and a local orthotropic damage accumulation law for tension, compression and shear loadings. Orthotropic damage is represented by a symmetric second-order damage tensor and its effect on creep is incorporated through the effective stress concept. The unknown model parameters are first calibrated using published experimental data from constant uniaxial stress tests and then predictions are made for constant strain rate and multiaxial loadings. The predicted results are in good agreement with both experimental and numerical results in the literature illustrating the viability of the proposed model. The model is mainly intended for studying the failure mechanisms of polar ice at low deformation rates with depth varying temperature profiles.

An Elastoplastic-anisotropic Damage Model for Concrete

An elastoplastic-anisotropic damage constitutive model for the description of nonlinear behavior of concrete is presented. The yield surface is developed in effective stress spaces, which takes into account the hardening effect and better match the experimental data. The stiffness degradation and softening effect are considered in the framework of continuum damage mechanics formulation. The second-order damage tensor is used to characterize the anisotropy induced by the orientation of microcracks. In order to simulate the unilateral effect, the elastic Helmholtz free energy is decomposed into a volumetric part and a deviatoric part. The different behavior under tensile and compressive loadings is modeled by using different variables in effective stress and damage tensor. Numerical results of the model accord well with experimental results at the material and structural levels.

Application of Damage Model in Stability Analysis of ShuangjiangKou Underground Caverns

A damage constitutive model was established to evaluate the stability of jointed rock masses around the underground caverns of Shuangjiangkou Hydropower Station. In this model, a second-order damage tensor was formulated to describe the initial geometric imperfection of jointed rock masses. A failure criterion for crack coalescence and a damage evolution equation were proposed by considering the secondary cracks induced by excavations according to the fracture mechanics principles. A subprogram under the framework of FLAC-3D was developed for the damage constitutive model and the damage evolution equation. The model and program were then applied in the stability assessment for the Shuangjiangkou underground cavern group.

A novel constitutive model of skeletal muscle taking into account anisotropic damage

The purpose of this study is to develop a constitutive model of skeletal muscle that describes material anisotropy, viscoelasticity and damage of muscle tissue. A free energy function is described as the sum of volumetric elastic, isochoric elastic and isochoric viscoelastic parts. The isochoric elastic part is divided into two types of shear response and the response in the fiber direction. To represent the dependence of the mechanical properties on muscle activity, we incorporate a contractile element into the model. The viscoelasticity of muscle is modeled as a three-dimensional model constructed by extending the one-dimensional generalized Maxwell model. Based on the framework of continuum damage mechanics, the anisotropic damage of muscle tissue is expressed by a second-order damage tensor. The evolution of the damage is assumed to depend on the current strain and damage. The evolution equation is formulated using the representation theorem of tensor functions. The proposed model is applied to the experimental data on tensile mechanical properties in the fiber direction and the compression properties in the fiber and cross-fiber directions in literature. The model can predict non-linear mechanical properties and breaking points.

Development and use of an anisotropic damage model for the high-temperature life assessment of a P91 weldment

This paper describes the development of a finite element (FE) continuum damage mechanics methodology to deal with anisotropic creep and anisotropic damage for a P91 weld metal. The method employs a second-order damage tensor, together with a novel rupture stress approach to define the evolution of this tensor. The method takes advantage of the transverse isotropic nature of the weld metal, to achieve a reduction in the number of material constants required from test data (and hence tests) to define the damage evolution. Hill's anisotropy potential theory is employed to model the secondary creep. An FE implementation of the model is applied to the creep life assessment of an internally pressurized axisymmetric P91 pipe weldment. The study also considered the effect of various material mismatch ratios in the various zones of the weldment. Here material mismatch is considered to be a combination of the difference in uniaxial minimum strain rates and in the uniaxial rupture times.

Development of damage state in porous ceramics under compression

This article describes a double-scale approach to modelling of damage growth in the porous polycrystalline ceramics under compression. A theoretical way of determination of the compliance tensor components for porous ceramics and their experimental verification was proposed. The mesomechanical approach, described in details in [T. Sadowski, S. Samborski, Comput. Mater. Sci. 28 (2003) 512–517], gives an insight into physical phenomena taking place in the material structure under loading. The phenomenological model contains an analysis of compliance changes in macroscale, due to defects growth. A decomposition of the fourth-order compliance tensor S ijkl was assumed. The partial tensors stand for: elastic matrix compliance, pore influence and crack influence on material behaviour. A loading history called loading–unloading–reloading Procedure (LURP) was applied during experimental testing. A second-order damage tensor D ij in phenomenological model describes damage history and anisotropy. Its components were estimated by an analysis of the current material deformation state and Young’s moduli changes. Results of experiments for two polycrystalline porous ceramics (magnesia and alumina) were compared with the numerical results given by mesomechanical model, which is based on an analysis of the Representative Surface Element (RSE) and needs some additional data to describe polycrystalline structure. They were obtained by Scanning Electron Microscopy (SEM). Numerical results demonstrated that the proposed method is a good approximation of the real ceramics behaviour.

Effective Stress and Vector-valued Orientational Distribution Functions

The original Kachanov—Rabotnov damage variable is inherently microplane-based and should be expressed by a scalar-valued orientation distribution function (ODF), and the corresponding effective stress is a vector-valued ODF. The analysis of vector-valued ODFs is established in this article in the spirit of the analysis of scalar-valued ODFs by Kanatani, K. (1984a). Distribution of Directional Data and Fabric Tensors, International Journal of Engineering Science, 22: 149—164 and Kanatani, K. (1984b). Stereological Determination of Structural Anisotropy, International Journal of Engineering Science, 22: 531—546. Explicit expansions of vector-valued ODFs up to the sixth-order have been developed, and the relationship of fabric tensors of different order is addressed. The fabric tensors and expansion of the Kachanov—Rabotnov effective stress vector can be fully determined by the scalar-valued ODF characterizing the microplane damage. The second-order effective stress and damage tensors of Murakami, S. (1988). Mechanical Modeling of Material Damage, ASME Journal of Engineering Materials and Technology, 55: 280—286 are related to the second-order expansion of the Kachanov—Rabotnov effective stress vector. Therefore, the analysis of vector-valued ODFs furnishes a rigorous and unified mathematical basis for the damage theory of Kachanov, L.M. (1958). Time of the Rupture Process under Creep Conditions, Izv. Akad. Nauk., USSR. Otd. Tekhn. Nauk. 8: 26—31, Rabotnov, I.N. (1963). On the Equations of State for Creep, Progress in Applied Mechanics, the Prager Anniversary, 8: 307—315, and Murakami, S. (1988). Mechanical Modeling of Material Damage, ASME Journal of Engineering Materials and Technology, 55: 280—286.

Modeling of anisotropic damage and creep deformation in brittle rocks

A new constitutive model is proposed for the description of induced anisotropic damage in brittle rocks. The formulation of the model is based on relevant results from micromechanics consideration. The distribution of microcracks is approximated by a second-order damage tensor. The effective elastic properties of damaged material are derived from the free enthalpy function. The evolution of damage is directly related to the growth of microcracks in different space orientations. The volumetric dilatancy due to sliding crack opening is taken into account. The model is extended to the description of creep deformation in brittle rocks. The time dependent deformation is seen as a consequence of the sub-critical propagation of microcracks due to stress corrosion process. The proposed model is applied to a typical brittle rock, the Lac du Bonnet granite. A general good agreement is obtained between numerical simulations and experimental data.

Uniqueness and minimum theorems for a multifield model of brittle solids

Some minimum theorems potentially useful to construct numerical schemes related to quasi-static evolution of damage in brittle elastic solids are proposed. The approach is that of multifield theories, with a second-order damage tensor describing the microcrack density. The use of damage entropy flux and damage pseudo-potential are both investigated.

Coupling between anisotropic damage and permeability variation in brittle rocks

In this paper, a coupled constitutive model is proposed for anisotropic damage and permeability variation in brittle rocks under deviatoric compressive stresses. The formulation of the model is based on experimental evidences and main physical mechanisms involved in the scale of microcracks are taken into account. The proposed model is expressed in the macroscopic framework and can be easily implemented for engineering application. The macroscopic free enthalpy of cracked solid is first determined by approximating crack distribution by a second-order damage tensor. The effective elastic properties of damaged material are then derived from the free enthalpy function. The damage evolution is related to the crack growth in multiple orientations. A pragmatic approach inspired from fracture mechanics is used for the formulation of the crack propagation criterion. Compressive stress induced crack opening is taken into account and leads to macroscopic volumetric dilatancy and permeability variation. The overall permeability tensor of cracked material is determined using a micro–macro averaging procedure. Darcy's law is used for fluid flow at the macroscopic scale whereas laminar flow is assumed at the microcrack scale. Hydraulic connectivity of cracks increases with crack growth. The proposed model is applied to the Lac du Bonnet granite. Generally, good agreement is observed between numerical simulations and experimental data. Copyright © 2005 John Wiley & Sons, Ltd.

Damage mechanics and its application for the design of an underground theater

This paper presents a damage mechanics method for the evaluation of the elastic damage of jointed rock mass and its application in predicting damage distribution in the jointed rock mass in the excavation for an underground structure in the framework of continuum damage mechanics. The damage variable is a second-order damage tensor which is assumed to describe the damage effect of jointed rock mass. The elasticity tensor defined in the model is verified by employing laboratory cement mortar specimens with various cases of joint distribution. The same specimen is analyzed using the developed damage mechanics method as well as the normal FE method, which employs joint elements to simulate jointed cracks. In order to apply the developed damage elasticity theory to three-dimensional damage analysis in underground structure excavation, a three-dimensional grouted rockbolt element is introduced, which considers the resistance of the grout to the relative shear displacement between the bolt and the rock mass around it in the axial direction. Plastic behaviors of the steel bar and the grout of the grouted bolt are considered in the model by using various constitutive laws. Finally, an engineering example for damage analysis is presented, namely an underground theater assumed to be excavated in a geological environment of jointed rock.

Dual orthotropic damage–effect tensors with complementary structures

This paper deals with secant constitutive relations of orthotropic elastic damage based on the so-called damage–effect tensors, namely the fourth-order operators that define the linear transformations between nominal and effective stress and strain quantities. The damage–effect tensors are expressed by orthotropic representations in terms of symmetric second-order damage tensor variables. The paper provides a set of new dual orthotropic damage–effect tensors that possess complementary structures in the dual compliance- and stiffness-based derivations. More specifically, each orthotropic damage–effect tensor of the solution set possesses an inverse (its dual counterpart) that displays the structure of the (major) transpose of the tensor obtained by replacing the adopted second-order damage tensor variables with their inverses.