Damage Tensor Research Articles

Formulation and verification of an anisotropic damage plasticity constitutive model for plain concrete

This paper presents a new constitutive model based on the combination of plasticity and anisotropic damage mechanics to predict the nonlinear response of plain concrete. The aim is to overcome the deficiencies of the previous anisotropic damage-plasticity models in simulating concrete failure under multiaxial loadings. To effectively combine plasticity and damage, a decoupled algorithm and consequently a strain equivalence hypothesis are employed. A stress-based yield criterion and a non-associative flow rule are used in the plasticity formulation. The stress tensor is decomposed into positive and negative parts to consider the unilateral effect of concrete damage. Consequently, two sets of damage criteria and two anisotropic damage tensors are defined, which leads to automatically accounting for the stiffness recovery in transition from tensile to compressive stress. The viscous model of Duvaut–Lions is employed to improve mesh dependency. Moreover, the formulation is regularized to capture large crack opening and closing when the material has experienced large amounts of strain. The numerical implementation of the proposed model is described in detail. A special in-house finite element program incorporating the proposed approach is developed. The efficiency of the model is verified by comparing numerical results and experimental data for different benchmark problems such as monotonic and cyclic uniaxial tests, monotonic biaxial test, and mixed-mode multidimensional structural tests.

A comparison of micromorphic gradient‐extensions for anisotropic damage

AbstractThe modeling of inelastic phenomena with tensor‐valued internal variables requires a regularization to counteract mesh dependence. Here, we consider the regularization of anisotropic damage at finite strains and seek for an efficient formulation based on a reduced number of nonlocal degrees of freedom. We, thus, equip the brittle version of an anisotropic model with different gradient‐extensions in the micromorphic framework using a full and a reduced regularization of the damage tensor. The models are compared in the structural simulation of an asymmetrically notched specimen with a perforated shear band section. High agreement is observed between the full and the reduced regularization with two nonlocal degrees of freedom. Furthermore, we present a novel study of the evolution of the reduced micromorphic field variables.

An anisotropic full-network model with damage surface for the Mullins effect in filled rubbers

The Mullins effect is a highly anisotropic damage phenomenon exhibited by filled rubbers among other soft materials. When filled rubbers are subjected to uniaxial tension, their apparent stiffness drops in the direction of stretching but is essentially unaltered in the transverse directions. However, micromechanical full-network models where Mullins softening is described at the level of individual chains often predict that uniaxial deformations induce transverse softening in addition to softening in the stretching direction. Moreover, these approaches typically require the storage of damage state variables for each chain, which is computationally expensive. Taking an alternative approach, we present a full-network model for the Mullins effect where the damage state is described by a single macroscopic damage tensor from which the damage state in each direction can be calculated. The evolution of damage is specified through damage surfaces and damage flow rules, which depend on the directions of principal stretches. The model is shown to reproduce experimental data for filled rubbers sequentially subjected to uniaxial tension in different directions. The model is also implemented in the finite element software ABAQUS as a user subroutine UMAT to illustrate the suitability of the model to simulate non-homogeneous deformation states.

A tensorial energy-release-rate based anisotropic damage-plasticity model for concrete

This paper proposes an anisotropic damage model for concrete to capture its plasticity and damage behavior effectively. Utilizing the framework of continuum damage mechanics, the model considers the influence of plastic Helmholtz free energy on damage evolution. This approach explicitly deduces damage-related thermodynamic forces without resorting to assumptions such as strain equivalence or energy equivalence. The proposed model employs tensile and compressive damage tensors to describe the development of anisotropic damage in concrete. The solution for plastic strain utilizes the empirical plasticity model. Furthermore, a unified evolutionary criterion for isotropic and anisotropic damage is also established, incorporating damage yield functions and orthogonal flow laws. Numerical simulations are conducted to validate the model’s ability to reproduce typical nonlinear behaviors of concrete structures under various load conditions.

Azimuthal pore pressure response to teleseismic waves: effects of damage and stress anisotropy

SUMMARY Pore pressure oscillations induced by stress variations, including propagating seismic waves from remote earthquakes, have been widely observed in various groundwater systems. The monitored pressure change in wells shows significant water-level oscillations to volumetric strain as well as to S and Love waves. Recent observations demonstrated azimuthal dependence of the pore pressure oscillations with respect to stress indicators and fault zone orientation. Within the fault zone, damage-induced anisotropy is the result of the alignment and orientation of cracks and other internal flaws within the rock. In this work, we provide a complete quantitative description of the pore pressure changes induced by passing seismic waves associated with different orientations and values of principal stress and damage tensor components. The model quantifies the azimuthal dependence of the pore pressure response by a non-dimensional ratio defined as the amplitude of the pressure oscillations induced by a shear strain normalized to the volumetric strain. Three angles and two values are needed to calculate the azimuthal dependence of the pore pressure response: the angle between the directions of the maximum horizontal stress and the seismic event; fault zone orientation; microcrack orientation within the fault zone; and damage and stress values. The model predicts that maximum pore pressure response occurs when microcracks and maximum horizontal stress are in the same orientation, high damage and high stress anisotropy. By adjusting these quantities, we recalculate results of recent seismological studies in the Arbuckle disposal well, Osage County, Oklahoma. The presented model successfully predicts the observed azimuthal dependence in wave-induced fluid pressure response and relates the anisotropic response to tectonic indicators such as the orientations of the maximum horizontal stress, fault zone, and microfractures.

Isogeometric smooth crack-band model (isCBM) using spress–sprain relations adapted to microplane theory

Recent optimum fitting of the data from eleven different types of distinctive fracture tests of quasibrittle materials, including the gap tests and the size effect tests, revealed that the crack band model (CBM) with the M7 damage constitutive law outperforms all the other existing computational fracture models for concrete. This is attributed to physically correct boundary conditions, especially those at the crack faces,to the crack front of finite width which allows using a triaxial tensorial damage constitutive law, and to choosing for this law a realistic material behavior description, the M7. Nevertheless the CBM has limitations—the width of the crack band cannot be varied, the statistically smooth damage distribution across the band cannot be resolved, and the propagation direction is biased by the mesh orientation when a regular mesh is used. These limitations have been overcome by the smooth crack band model (sCBM), which uses unconventional homogenization of damage mechanics to yield an additional Helmholtz free energy density, called the “sprain energy”. This energy is a function of the curvature tensor (or a hessian) of the vectorial displacement field, called the “sprain curvature tensor” (or sprain tensor for brevity), which is a sum of the strain gradient tensor and, importantly, the rotation gradient tensor. But, in a previous study, difficulties arose in computational implementation—the use of constant-strain elements required the curvature to be resisted by sprain forces that represented the derivatives of sprain energy with respect to nodal displacements and had to be applied on the nodes of adjacent elements. The need for adjacent nodes is here avoided by a variational derivation of the Isogeometric Smooth Crack-Band Model (isCBM). The microplane damage constitutive model M7 for concrete is adopted, which is made possible by establishing the relations of the sprain stress (or spress, for brevity) and sprain on microplanes of arbitrary orientations. Using Isogeometric Analysis (IGA) based on non-uniform rational B-Splines (NURBS), the excessive displacement curvature and its resistance (spress) can be described by the shape functions with C1-continuity. Computational results show that the orientation bias of regular meshes is eliminated while keeping the constitutive law intact, which is important for compatibility with the existing material laws in general. In addition, comparisons with experimental data reveal that isCBM with M7 can reproduce the correspondence between the size of the fracture process zone and the fracture energy, as indicated by gap tests. Finally, the isCBM does not compromise the strain-path-dependence of material response.

An anisotropic damage model for finite strains with full and reduced regularization of the damage tensor

AbstractThe modeling of damage as an anisotropic phenomenon enables the consideration of arbitrarily oriented microcracks at the material point level. Yet, the incorporation of material softening into structural simulations still requires a regularization of for example, the degrading variable. There exist different possibilities for a regularization in case of anisotropic damage with varying numbers of nonlocal degrees of freedom corresponding to for example, the symmetric integrity tensor , the principal traces of the damage tensor or a scalar damage hardening variable. Here, we propose a finite strain formulation with a symmetric second order damage tensor of which all six independent components are regularized with a corresponding nonlocal degree of freedom. Due to the significant increase in computational cost caused by the full regularization of the damage tensor, alternative approaches for a reduced regularization with fewer nonlocal degrees of freedom are discussed. Thereafter, the results of a numerical example using the model with full regularization are presented.

Anisotropic viscoelasticity/damage coupled thermodynamic model for transparent polymer

Transparent polymer such as PMMA has been used for manufacturing viewport windows of deep-sea manned submersible. This paper develops an anisotropic viscoelasticity/ damage coupled thermodynamic model under finite-deformation framework for creep analysis of PMMA polymer. By introducing the strain equivalence principle with the effective stress concept, a four-order damage tensor that is composed of a two-order damage tensor conjugated to the two-order thermodynamic force, is imposed on the elastic tensors of two springs to describe mechanical degradation of material, and a damage potential function is proposed to derive the rate of two-order damage tensor. After the Drucker-Prager’s type failure criterion is met, damage induces anisotropy, which also accelerates damage evolution conversely. The developed model and numerical algorithm are first used to study the creep behaviors of a dog-done PMMA tensile specimen by comparing with the creep experimental results of material, and then the creep behaviors of the PMMA conical viewport window under constant water pressure by comparing with the creep experimental results of window well.

A thermodynamically consistent anisotropic damage model for metallic glasses

AbstractThe recent success in manufacturing large‐size, also called bulk metallic glasses (BMGs) using 3D‐printing based on laser powder bed fusion (LPBF) opens an avenue for the broad application of this material class. To explore the great potential of both as‐cast and 3D‐printed BMGs, a comprehensive understanding and an accurate prediction of the plastic deformation and damage behaviour of this material class are indispensable. In this study, we develop a thermodynamically consistent anisotropic damage model incorporating tension‐compression asymmetry (TCA) to describe the unique inelastic deformation and damage behaviours of metallic glasses. The widely observed normal stress sensitivity and plastic dilatancy in metallic glasses are considered by using an extended Mohr‐Coulomb criterion in the constitutive description. Furthermore, a second‐order damage tensor is adopted for describing the anisotropic damage behaviour. 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 governing equations for the corresponding internal variables are derived within the framework of finite deformation. The damage localisation and mesh dependency are correspondingly alleviated. The simulation result shows that the shear band patterns are in good agreement with the experimental observations from literature.

A stress-state-dependent damage criterion for metals with plastic anisotropy

The paper discusses the effect of stress state and of loading direction on the onset and evolution of damage in anisotropic ductile metals. A series of experiments with uniaxially and biaxially loaded specimens covering a wide range of stress states and different loading directions is used in combination with corresponding numerical simulations to develop damage criteria. The underlying continuum damage model is based on kinematic definition of damage tensors. The strain rate tensor is additively decomposed into elastic, plastic and damage parts. The anisotropic plastic behavior of the investigated aluminum alloy sheets is governed by the Hoffman yield condition taking into account the strength-differential effect revealed by uniaxial tension and compression tests. Based on this yield criterion generalized anisotropic stress invariants as well as the generalized stress triaxiality and the generalized Lode parameter are defined characterizing the stress state in the anisotropic ductile metal. A damage criterion formulated in terms of these anisotropic stress invariants is proposed and damage mode parameters allow adequate consideration and combination of different damage processes on the micro-level. At the onset of damage the anisotropic stress parameters are determined. With these experimental-numerical data the damage mode parameters are identified depending on stress state and loading direction.

Determining the influence of irradiation swelling on creep and damage in elements with orthotropic material properties

This paper reports a method that numerically models the deformation and accumulation of hidden damage in structural elements that are in an inhomogeneous thermal field and are exposed to radiation. Materials considered are those exhibiting orthotropy (transversal isotropy) of long-term properties. The problem is stated as an boundary – initial value one. To solve it, the finite element method and the initial-difference method of time integration are used. To simulate the anisotropy of the process of accumulation of hidden damage, a damage tensor is applied. The development of irradiation swelling strains is described using the equation for a limited temperature range and a specific fluence value. The results of numerical modeling of creep and damage in plates in tension with circular notches, which are in an inhomogeneous temperature field, are considered. The material of the plates is a titanium alloy VT1-0. It was found that the effect of irradiation significantly, up to 6‒7 %, increases the level of deformation in the plate. Radiation significantly, by almost 4 times, reduces the time until the completion of the hidden fracture of the plates. It was found that orthotropy of radiation swelling properties leads to redistribution of areas with significant strains and damage values. It has been established that the effect of irradiation swelling also qualitatively changes the nature of the distribution of maximum damage in the plate, which extends to a fairly large area. Such results are due to the additional effect of irradiation swelling strains on the rate of general irreversible deformation and redistribution of stresses

Crack-parallel stress effect on fracture energy of plastic hardening polycrystalline metal identified from gap test scaling

The gap test is a new type of fracture test developed in 2020, in which the end supports of a notched beam are installed with gaps that close only after the elasto-plastic pads next to notch introduce a desired constant crack-parallel compression σxx (also called the T-stress). The test uses the size effect method to identify how such a compression alters the material fracture energy, Gf, and the characteristic size cf of the fracture process zone (FPZ). In 2020, experiments showed that a moderate σxx doubled the Gf of a quasibrittle material (concrete) and a high σxx reduced its Gf to almost zero. A preliminary study by Nguyen et al. (2021) showed that the gap test can be extended to plastic-hardening polycrystalline metals. A generalized scaling law with an intermediate asymptote for large-scale yielding in small structures was derived, and limited tests of aluminum alloy showed its applicability. In this study, geometrically scaled gap tests of notched three-point bend fracture specimens of aluminum are conducted at three different levels of σxx. An extended structural strength scaling law that captures the transition from the micrometer-scale FPZ through millimeter-scale yielding zone (YZ) to large-scale structures which follow linear elastic fracture mechanics (LEFM) is derived and then applied to analyze the effect of σxx. Presented here are the gap tests of aluminum alloy, in which three different levels of σxx are applied to scaled notched four-point-bend beams of depths D = 12, 24, 48 and 96 mm. Using an extended size effect law for plastic-hardening metals, it is found that, at crack-parallel stress σxx≈−40% of the yield strength, the critical J-integral value gets roughly quadrupled, not only because of the well-known enlargement of the hardening YZ whose width is of millimeter scale, but also because of the increase of the FPZ width of micrometer scale. These results can be reproduced neither by line crack models, including the LEFM, cohesive crack and phase-field models, nor by peridynamic and various nonlocal models that ignore the tensorial nature of the material stress at the crack tip. The crack band models, being able to represent an FPZ of finite width and a YZ whose size evolves depending on σxx, can capture the effect of crack-parallel stresses provided that a realistic 3D tensorial damage constitutive model is used. Here, Bai–Wierzbicki’s model is shown to capture the σxx effect on the Gf and Jcr qualitatively.

An anisotropic damage–permeability model for hydraulic fracturing in hard rock

Hydraulic fracturing is the most efficient method to exploit valued geothermal energy trapped in low-permeable hard rock, e.g. granite. Most research on the hydraulic fracturing has focused on its application in shale gas and oil. However, the hydraulic fracturing performs differently in geothermal reservoir, as the rock properties are quite different. In this work, an anisotropic damage–permeability model is developed on the fundament of continuum theory to study the hydraulic fracturing of hard rock in geothermal reservoir. The plastic-hardening and damage-softening behaviours are considered in this model. A cubic law is adopted to characterize the damage enhanced permeability. Its directional information is converted from damage tensor, while the effect of compression stress on permeability is isotropic and characterized by an impact factor. The newly developed model is calibrated and validated by a series of stress–strain curve, damage and axial permeability from triaxial tests on granite. In the application to cyclic fracturing test at Aspö Hard Rock Laboratory, the capacity of newly developed model is proven by good matching of measured injection pressure, permeability, etc. The results show clearly that the fracture is mostly activated by tensile failure in this case. Moreover, the stimulated fracture will be closed during flow back and re-activated in subsequent re-fracturing. If the fracture from previous fracturing is not re-activated completely, no new fractures will be created in current re-fracturing, and the damage amasses continuously due to repeated re-activation of closed fracture during re-fracturing.

Criterion of the limit state of composites materials

The work proposes and substantiates a type of phenomenal criterion of composite material destruction at the stage of macro crack origination. It considers damageability and two mechanisms of destruction: tearing and slice. It describes that as parameter of damageability it can be taken specific energy of extra stresses or specific energy of diffusion. For proportionate and complex processes of metallic materials loading they result equally in practice. Methodologies of respective energetic parameters search are described. It is set threshold value of stress in the element of construction during the reach of it it is necessary to consider an accumulation of damage. It is described the methodology that describes interconnection between anisotropy factors and components of damage tensor.

ON THE CONVEXITY OF POTENTIAL OF NONLINEAR ELASTIC MEDIA MODEL WITH TENSOR DAMAGE PARAMETER

For solving problems of the theory of elasticity using nonlinear models, the question of the convexity of the potential used and the proof of the uniqueness of the solution are of decisive importance. The present work is devoted to the determination of conditions for the local strict convexity of the potential for the nonlinear elasticity model, which ensure the uniqueness of the solution to the problem in a sufficiently small neighborhood of the desired solution. The considered model is a generalization of a scalar nonlinear rheological model of brittle solid deformation for the case of the tensor damage parameter, the principal values of which describe the reduction in the cross-sectional area of the material in three orthogonal directions. An additional term of the second order in deformation in the elastic potential makes it possible to describe the dependence of the elastic moduli on the type of the stress-strain state, the dilatancy of the material under shear deformation, as well as the nonlinear deformation response even at low loads. The introduced damage tensor of the second rank makes it possible to describe the damage-induced anisotropy of the elastic properties of the material. Conditions of local strict convexity in the principal axes of the strain tensor are obtained in this work for the general case of misaligned strain and damage tensors. To illustrate the obtained convexity conditions, two special cases of the damage tensor type are considered: a transversely isotropic fractured medium with coaxial strain and damage tensors, and a transversely isotropic medium with obliquely oriented fracturing. For both cases, the dependence of the limiting values of damage on the parameter of the degree of anisotropy is shown. It is shown that in the case of weak damage anisotropy, the potential convexity conditions for the scalar damage parameter give a minorant estimate of the maximum allowable damage for various types of stress-strain state. For obliquely oriented fracturing, the dependences of the maximum permissible damage on the degree of anisotropy, the angle of inclination and the type of stress-strain state are plotted.

An energy-based elastoplastic damage model for concrete at high temperatures

Although structural concrete is widely used in civil engineering, the proper modeling of its thermo-mechanical behavior remains a challenging issue for the academic and engineering communities due to the complexity involved in the underlying micro-cracking process. In this work, the previous thermodynamically consistent energy-based elastoplastic damage model is extended to concrete at high temperatures, with the aim to describe the material nonlinear behavior and provide numerical predictions of concrete structures. Both the thermal expansion and transient creep strain are incorporated in the kinematics. Within the framework of continuum damage mechanics, two mechanical damage variables that lead to a fourth-order damage tensor are introduced to describe stiffness degradation of concrete under predominant tension and compression, respectively. Moreover, temperature-dependent mechanical properties are adopted to describe the thermal degradation and the thermo-mechanically coupled behavior of concrete at high temperatures. The plastic flow is described by the effective stress space plasticity and the evolution laws for the mechanical damage variables are driven by the conjugate elastoplastic damage energy release rates. The proposed model is validated by several benchmark experiments of plain concrete and then applied to reinforced concrete (RC) structures. The well agreement between the numerical predictions and experimental results indicates that the proposed model is promising in quantifying the global responses and assessing the safety of plain concrete and RC structures at high temperatures.

The Gap test – Effects of crack parallel compression on fracture in carbon fiber composites

This paper explores the global Mode I fracture energy of a carbon fiber composite subject to a biaxial stress state at the crack tip, specifically in which one stress component is compressive and parallel to the crack. Based on an experimental technique previously coined as The Gap Test and Bažant’s Type II Size Effect Law, it is found that there is a monotonic decrease in the Mode I fracture energy as the crack parallel compressive stress increases. Compared to the nominal value of fracture energy, where no crack parallel compression is applied, the fracture energy is observed to decrease by up to 37% for a compressive stress equal to 44% of the compressive failure limit of the composite. This weakening effect is attributed to splitting cracks that are induced at the crack tip due to the crack parallel compression, which are identified via crack tip photomicroscopy. This is a novel result that challenges the century old hypothesis of fracture energy being a constant material property and further, shows for the first time that crack parallel compression leads to a composite structure being dangerously weaker than expected.The experimental campaign is also buttressed with a computational campaign that provides a framework capable of capturing the effects of crack parallel compression. Through the use of the crack band model, which correctly characterizes the fracture process zone tensorially, coupled with a fully tensorial damage law, the simulated results provide satisfactory agreement with the experimental data. Conversely, when a reduced tensorial damage law defines the crack band it is shown that the structural strength and fracture energy are dangerously overpredicted. This emphasizes the importance of using a crack band model coupled with a fully tensorial damage law to accurately predict fracture in composites.

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.

Generalization of anisotropic damage mechanics modeling in the material point method

AbstractAnisotropic damage mechanics is derived by redefining its fourth‐ranked damage tensor, , not by its effect on stiffness reduction, but as a tensor that partitions total strain into bulk material strain and an cracking strain associated with crack‐opening displacement. This recharacterization of is irrelevant for 1D modeling, but significantly clarifies 3D algorithms. The new 3D derivation starts with three damage parameters associated with three independent strength models for three components of crack traction. By postulating a traction failure surface dependent on current damage state and requiring that all traction components simultaneously decay to zero at failure, the three damage parameters naturally couple to a single parameter. Many prior methods assume evolving strength depends only on damage. In real materials, strength often depends on other variables such as temperature, pressure, strain rate, and so forth. This article proposes a new general theory that extends prior methods to properly account for such external variables. The general damage mechanics methods must account for extra variables during damage initiation, damage evolution, and elastic loading and unloading. Several examples focus on the new concepts for coupling damage parameters and for using general methods to model materials with pressure‐dependent strength properties.

An Anisotropic Continuum Damage Mechanics Model for the Simulation of Yield Surface Evolution and Ratcheting

A new anisotropic continuum damage model was presented on the basis of irreversible thermodynamics. Second rank symmetric damage tensor, kinematic hardening tensor, and damage potential surface were introduced. The yield surface in this study has been modeled to distort in the stress space by the evolution of the damage tensor. The new model was applied in yield surface characterization and led to a well description of the subsequent yield surfaces. This model is then applied to predict experimental results of the cyclic nonproportional loading paths obtained by Kang et al. [2004], Hassan et al. [2002] and Taleb and Cailletaud [2011]. It is demonstrated that the new model can predict the yield surface distortion and the ratcheting strains of multiaxial nonproportional cyclic loading with acceptable accuracy and by using less material constants with respect to the previous models.