Order Damage Tensor Research Articles

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.

A two-surface gradient-extended anisotropic damage model using a second order damage tensor coupled to additive plasticity in the logarithmic strain space

The objective of the present paper is to develop a thermodynamically consistent coupled damage-plasticity model at large deformations, which accounts for damage anisotropy. Moreover, a ‘two-surface’ approach allows modeling plasticity and damage independently. Thus, both phenomena are treated as separate dissipative mechanisms, making the model attractive for application to both brittle and ductile materials. The framework is based on Continuum Damage Mechanics. Furthermore, logarithmic strain measures – also known as Hencky strain – are considered for the kinematics, while the decomposition of the total deformation into elastic and plastic parts is based on the additive split. Hence, the derivation of the model and its conjugated quantities takes place in the logarithmic strain space, but these are subsequently transformed to their Lagrangian counterparts to be applicable in standard finite element formulations. Consequently, the transformation of constitutively dependent quantities such as stresses, but also the various associated material sensitivities, are addressed here. Another main aspect of this work is the gradient-extension of the presented model in order to cure mesh sensitivity in case of material softening. To this end, a novel gradient extension is derived using the invariants of the second order damage tensor, which is based on the micromorphic approach. In addition to the theoretical framework, special attention is paid to the finite element implementation, the formulation of the local residuals, and additionally the computation of the material tangents to achieve quadratic convergence rate within the Newton–Raphson scheme. Single element studies as well as representative structural examples investigate the model’s response to various loading scenarios, the effect of damage anisotropy and further highlight its ability to provide mesh-independent results while undergoing large deformations.

Laboratory characterisation of anisotropic and heterogeneous damage of rock sample using acoustic emission and ultrasonic monitoring technologies

Rock damage characterisation of Gosford sandstone samples under triaxial loading was achieved using an integrated approach, combing passive acoustic emission (AE) and active ultrasonic wave monitoring. The location of AE events is firstly obtained using isotropic wave velocity field measured by continuous ultrasonic wave. Then, passive AE tomography is conducted to progressively-update the anisotropy and heterogenous P-wave velocity field of rock samples. According to the variance in P-wave velocity, damage induced anisotropy and heterogeneity can be identified. The results suggest that the inhomogeneous damage evolution highly dependent on the loading stages and can well reflect the microcrack behaviour. Anisotropic damage variable along the whole loading process is also analysed, by a second order damage tensor with a transversely anisotropy assumption. It is observed that the lateral component of damage is much higher than the axial component. This is mainly due to the dilation behaviour of rock sample. This paper will cast more light on anisotropic and heterogenous damage characterisation using acoustic/ultrasonic lab experiments.

Anisotropic damage model for concrete and other quasi-brittle materials

A thermodynamically consistent formulation to model anisotropic damage for quasi-brittle materials is proposed. The model is based on proper expressions for the specific Gibbs free energy and the complementary form of the dissipation potential. Damaging of the material is described by a symmetric positive definite second order damage tensor. Especially, the failure surface is formulated in such a way that it will mimic the behaviour of the well known Ottosen’s four parameter failure surface. While testing the model against the experimental results found in literature, the results were in good agreement in uniaxial tensile and compressive loadings as well as in biaxial compression. Besides the correct failure stress states, the model predicts the correct failure modes of concrete: axial splitting along the direction of uniaxial compression and tensile damaging normal to the direction of tension.

Parametrically homogenized continuum damage mechanics (PHCDM) models for unidirectional composites with nonuniform microstructural distributions

This paper develops a parametrically homogenized continuum damage mechanics (PHCDM) model for multiscale analysis of damage and failure in composite structures with nonuniform microstructures. Unidirectional glass fibers are nonuniformly dispersed in the microstructures of these epoxy matrix composites. The PHCDM models are thermodynamically consistent, reduced order constitutive models with coefficients that are explicit functions of microstructural descriptors and evolving material variables. Damage anisotropy is represented through a second order damage tensor that contributes to the evolution of a damage surface in the space of damage work conjugate, which characterizes the initiation and evolution of damage. The nonuniform microstructural morphology descriptors are optimally expressed in terms of representative aggregated microstructural parameters or RAMPs for incorporation in the PHCDM coefficients. Optimal expressions for RAMPs are determined through principal component analysis of the two-point correlation functions. The functional forms of RAMPs in PHCDM coefficients are determined using machine learning tools operating on data generated by micromechanical analysis. It is shown that PHCDM models accounting for the fiber distribution information yield a much higher accuracy than those only accounting for fiber volume fractions. The developed PHCDM model is incorporated in a commercial finite element code and structural analysis of structural composites is executed for understanding concurrent damage and failure at multiple scales. The paper successfully demonstrates the accuracy and significant efficiency of the resulting PHCDM model in analyzing deformation and damage in nonuniform composites across length scales for various loading conditions.

Reliability Analysis of Notched Plates under Anisotropic Damage Based on Uniaxial Loading Using Continuum Damage Mechanics Approach

Extensive recent researches have been underway to model the fracture mechanics degradation based on continuum damage mechanics (CDM) technique. CDM theory is a powerful tool for solving problems such as large plastic deformations that the fracture mechanics is unable to solve. This model is derived by means of the thermodynamics internal variable theory and based on the experimental results on material properties. In this paper, the reliability of rectangular plates containing a central circular hole under static tensile load using the CDM approach for ductile fracture has been studied. To investigate the initiation and evolution of damages, anisotropic damage expressed by second order damage tensor is used to derive constitutive equations. Then, these relationships together with material constants are implemented with subroutine in ABAQUS software. The reliability assessment has been investigated using first order reliability method (FORM) and second order reliability method (SORM). Based on the FORM and SORM, the limit state functions and random variables have been obtained according to the energy density release rate. The probability of failure of each plate with different hole sizes is estimated based on the anisotropic damage theory, and the results are compared with the isotropic damage model. Finally, the sensitivity analysis of the coefficient of variation is performed.

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.

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.

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.

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.

Quantifying the orthotropic damage tensor for composites undergoing damage-induced anisotropy using ultrasonic investigations

We quantify the evolution of the general fourth order damage tensor for initially orthotropic composites undergoing damage-induced anisotropy using ultrasonic investigations. Two of the most challenging problems which arise in continuum damage mechanics are firstly the selection of variables to describe the internal damage and secondly the difficulty in modelling materials with significant initial anisotropy such as composites. This research helps advance models of anisotropic damage to overcome both these challenges. We demonstrate how to identify the directionality and magnitude of the introduced damage using experimental ultrasonic measurements of damaged elastic moduli for initially anisotropic materials. This analysis provides a robust way to validate and advance models based on continuum damage mechanics and develop phenomenological models of anisotropic damage evolution for initially orthotropic materials.

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.

Anisotropic damage of titanium plates under uniaxial tension after reverse bending

Influence of low-cycle reverse bending and the crystallographic texture on damage anisotropy of commercial titanium sheets during the subsequent uniaxial tensile tests was studied. The fourth-rank order damage tensor D was used at the analysis of damage anisotropy of titanium sheets. Only the sole component D=1−E/E0 of this tensor is not equal to zero for uniaxial tension (E0 and E is the elastic modulus of intact material and current modulus determined from the uniaxial tensile tests, respectively). Damage caused by stresses of proof strength and ultimate strength is estimated. The damage increases with increasing of number of reverse bending cycles. The correlations of damage and mechanical properties anisotropy with the crystallographic texture are found.

A tensorial based progressive damage model for fiber reinforced polymers

Progressive damage simulation is of major importance for predicting the damage behavior of composite laminates. This work faces the challenge to apply a progressive damage law for oblique fracture planes. Based on Puck’s or Camanho’s failure condition the damage initiation point and the fracture plane are determined. The presented damage evolution techniques for degradation of the elasticity tensor and their deficiencies are introduced. A new degradation approach based on an eighth order damage tensor is derived in this work. For efficient application in a finite element simulation a mapping of the fourth order material tensor to the fracture plane coordinates is conducted. The degradation is applied in the fracture plane system. Improved numerical stability and physically plausible behavior can be achieved by this approach. Both is demonstrated by simulation of a composite cube under out-of-plane compression load and a low velocity impact simulation. For evaluation and validation, the achieved results are compared with experimental data.

On Anisotropic Damage Theories based on 2nd Order Damage Tensors

AbstractIsotropic and anisotropic damage theories are shortly reviewed concerning different aspects like thermodynamical consistency, effective undamaged configurations, crack‐closure effects and the interpretation of the damage variable. A recently proposed damage growth criterion for second order damage tensors is discussed in integral as well as incremental form. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Phase field based nonlocal anisotropic damage mechanics model

A nonlocal anisotropic damage theory is developed in this work through the phase field method to address the anisotropic damage evolution in materials. The anisotropic damage is discussed and appropriate nonconserved order parameters in three mutually perpendicular directions are defined to find the growth of the components of a second order diagonal damage tensor corresponding to the principal directions of a general second order damage tensor. In contrast to the previous models, two new tensors are proposed to act as interpolation and potential functions along with the Allen–Cahn equation in order to obtain the evolution of the order parameters, which is the basis of the definition of the damage rate. The tensor formulation of the growth of the components of the damage tensor is proposed for the first time. It is shown that, by introducing a set of material parameters including a length scale parameter due to damage, there is a robust and simplified way to model the nonlocal behavior of damage and predict the corresponding material behavior as components of a second order diagonal damage tensor.

A multiaxial constitutive model for concrete in the fire situation: Theoretical formulation

This paper aims to develop a multiaxial concrete model for implementation in finite element software dedicated to the analysis of structures in fire. The need for proper concrete model remains a challenging task in structural fire engineering because of the complexity of the concrete mechanical behavior characterization and the severe requirements for the material models raised by the development of performance-based design. A fully three-dimensional model is developed based on the combination of elastoplasticity and damage theories. The state of damage in concrete, assumed isotropic, is modeled by means of a fourth order damage tensor to capture the unilateral effect. The concrete model comprises a limited number of parameters that can be identified by three simple tests at ambient temperature. At high temperatures, a generic transient creep model is included to take into account explicitly the effect of transient creep strain. The numerical implementation of the concrete model in a finite element software is presented and a series of numerical simulations are conducted for validation. The concrete behavior is accurately captured in a large range of temperature and stress states. A limitation appears when modeling the concrete post-peak behavior in highly confined stress states, due to the coupling assumption between damage and plasticity, but the considered levels of triaxial confinement are unusual stress states in structural concrete.