Isotropic Damage Model Research Articles

Fatigue damage evaluation of adhesively bonded butt joints with a rubber-modified epoxy adhesive

A damage evolution of adhesively bonded butt joints with a rubber-modified adhesive has been investigated under cyclic loading. An isotropic continuum damage model coupled with a kinetic law of damage evolution was applied to the butt joint. To solve the kinetic law, analytic and numerical methods were tried: the former solution was derived with some simplifications and the latter one was derived rigorously without simplications. On comparing the analytic solutions with the numerical ones, it was confirmed that differences in the two solutions were small. Furthermore, the estimated S-N curves based on the analytic equation agreed well with experimental data.

Creep-damage predictions in thin-walled structures by use of isotropic and anisotropic damage models

Two typical material models describing isotropic and anisotropic creep-damage processes in metals and alloys are applied to the simulation of the mechanical behaviour of transversely loaded thin plates. Numerical solutions of the initial boundary value problem have been obtained by use of both the solid- and the shell-type finite elements. The simulation results for a rectangular plate show the sensitivity of long-term predictions to the material models, the element types as well as the boundary conditions. The influence of these factors on the creep lifetime predictions is discussed.

Simulation of localization failure with strain-gradient-enhanced damage mechanics

Strain gradient implies an important characteristic in localized damage deformation, which can be observed in the softening state of brittle materials, and strain gradients constitute the basic behaviours of localization failure area of the materials. The most important point in strain gradient is its damaging function including an internal length scale, which can be used to express the scale effects of mechanical responses of brittle rock mass. By extending the strain gradient theory and introducing an intrinsic material length scale into the constitutive law, the authors develop an isotropic damage model as well as a micro-crack-based anisotropic damage model for rock-like materials in this paper. The proposed models were used to simulate the damage localization under uniaxial tension and plain strain compression, respectively. The simulated results well illustrated the potential of these models in dealing with the well-known mesh-sensitivity problem in FEM. In the computation, elements with C1 continuity have been implemented to incorporate the proposed models for failure localization. When regular rectangle elements are encountered, the coupling between finite difference method (FDM) and conventional finite element method (FEM) is used to avoid large modification to the existing FEM code, and to obtain relatively higher efficiency and reasonably good accuracy. Application of the anisotropic model to the 3D-non-linear FEM analysis of Ertan arch dam has been conducted and the results of its numerical simulation coincide well with those from the failure behaviours obtained by Ertan geophysical model test. In this paper, new applications of gradient theories and models for a feasible approach to simulate localized damage in brittle materials are presented. Copyright © 2002 John Wiley & Sons, Ltd.

General expressions of constitutive equations for isotropic elastic damaged materials

The general expressions of constitutive equations for isotropic elastic damaged materials were derived directly from the basic law of irreversible thermodynamics. The limitations of the classical damage constitutive equation based on the well-known strain equivalence hypothesis were overcome. The relationships between the two elastic isotropic damage models (i. e. single and double scalar damage models) were revealed. When a single scalar damage variable defined according to the microscopic geometry of a damaged material is used to describle the isotropic damage state, the constitutive equations contain two “damage effect functions”, which describe the different influences of damage on the two independent elastic constants. The classical damage constitutive equation based on the strain equivalence hypothesis is only the first-order approximation of the general expression. It may be unduly simplified and may fail to describe satisfactorily the damage phenomena of practical materials.

A plastic nonlocal damage model

In the present paper, a plastic nonlocal damage model is proposed for studying the mechanical response of structural elements made of cementitious materials. A new isotropic damage model, which is able to describe the behavior of a wide class of cementitious materials, is presented. A regularization technique, based on the introduction of the damage Laplacian in the damage limit function, is adopted to overcome the analytical and computational problems induced by the softening constitutive law. A Drucker–Prager type of plastic limit function is proposed considering isotropic hardening. A numerical procedure, based on an implicit `backward-Euler' technique for the time integration of the plastic and damage evolution equations, is presented. To solve each nonlinear step, a predictor–corrector iterative method is developed within the splitting method. In particular, the damage evolution is determined solving a constrained minimization problem of a convex functional. The proposed algorithm is implemented in a finite element code and it is used to study the structural behavior of elements made of masonry materials.

Characterization of the Torsional Behavior of Photosensitive Polymer Obtained by Stereolithography

An experimental and numerical mixed method based on the torsion test with and without warping of cross sections has been developed in order to characterize the shear behavior of a polymer material obtained layer by layer from a photosensitive resin (CiBA SL5170) polymerized by laser beam by the stereolithography technique. In this article, first the stereolithography technique is briefly presented, as well as the torsion device with and without warping of cross sections. Experimental and numerical results in static torsion of the polymer SL5170 are presented: shear modulus and strength of the material. From test results, a modeling of the superposition ratio of tensile and torsion stresses is proposed. It shows that the torsion of beams is complex. Torsion stresses and the tensile stress owed to the warping of section are superposed. In small-deformations state, material behavior behaves in a linear elastic way; experimental and numerical results are quantitatively and qualitatively coherent and conform to predictions of the elastic torsion theory. From experimental analysis of fatigue and elastoplastic damage behavior of the polymer, a phenomenological approach of the isotropic damage model is proposed. Based on the thermodynamics of irreversible processes, this local approach starts from a variant of the Chaboche-Lemaître model coupling plasticity and damage. This formulation is justified by the necessity to take into account the real stress state produced in the polymer during the torsion test.

Fracture energy based bi-dissipative damage model for concrete

An isotropic damage model for concrete is presented. The main features of the model are: limited number of constitutive parameters required; independent modelling of tension and compression behaviour by means of two damage variables and two separate activation criteria (bi-dissipative model); independent definition of tension and compression fracture energies by means of constitutive parameters which do not affect the pre-peak behaviour; consistent modelling of the unilateral effect upon transition from tension to compression. To overcome the problem of mesh dependence in the softening regime, the effectiveness of a fracture energy based regularisation strategy is shown and the conditions for its correct application are defined. Comparisons of numerical simulations with experimental tests reveal the good accuracy and the robustness of the proposed model.

Isotropic damage model with different tensile–compressive response for brittle materials

Incremental constitutive equations for brittle materials are formulated on the grounds of a frictional microcracked elastic model. The hypothesis of isotropic damage and the description of normal and tangential contact tractions on the crack faces by means of two second-order tensors provide a constitutive model with a reduced number of internal variables. The evolution equations of the latter ones are deduced from frictional and damage limit states and corresponding flow rules, from which the different behaviour to tensile and compressive stress states and dissipation at constant damage are represented. The model response is analysed for different stress states and limit strength domains are derived for monotonically increasing biaxial and triaxial stress states. Comparisons of the theoretical results with experimental data from literature related to concrete and cast-iron corroborate the proposed approach.

Weak Forms of Shakedown for Elastic–Plastic Structures Exhibiting Ductile Damage

A special weak-form shakedown is studied for elastic–plastic internal-variable material models with nonlinear hardening, damageable elastic moduli and damageable yield surface, in the hypothesis of ductile damage, (i.e. damage induced by plastic strains), but the precise evolutive law of damage being left unspecified. Sufficient weak-form shakedown theorems are presented, one static and another kinematic, each assessing whether eventually plastic deformations cease together with their consequences, including ductile damage. A two-sided delimitation is provided, within which the weak-form shakedown safety factor can be located. An upper bound to the post-transient damage for a particular isotropic damage model is also proposed. A simple numerical application is presented.

A non-local model with tension and compression damage mechanisms

A non-local isotropic damage model is proposed which can be used to predict the behaviour of rock-like materials up to failure. Two isotropic damage variables account for the progressive degradation of mechanical properties under stress states of prevailing tension and compression and two internal lengths, one for tension and the other for compression, are introduced as localisation limiters. A linear bifurcation analysis highlights the regularisation properties of the non-local model. An iterative scheme for the numerical solution of the finite-step problem consisting of a linear global predictor, an averaging phase and a non-linear local corrector is presented. Some illustrative examples of tension and splitting tests show the effectiveness of the proposed model.

On the form of free energy and specific heat in coupled thermo-elasticity with isotropic damage

This paper concerns the form of the free energy function in coupled thermo-mechanical problems with isotropic damage, particularly where damage accrues through both mechanical and thermal strains, and when the material is exposed to elevated temperatures. We show that with the normal assumption of a constant specific heat coefficient, mechanical dissipation is negative and so the second law of thermodynamics is violated. This is true even for a general isotropic damage model that allows independent damage on the Young’s modulus and the bulk modulus. Our approach is to make specific heat damage dependent, and under these conditions, we show positive dissipation for a range of problems; for the case of concrete, at least, there is material evidence supporting this model. For elevated temperatures, we use the logarithmic form of thermal energy, again showing positive dissipation. Comparisons between the forms show a notable difference in the energy transformed to heat, which is significant when reintroduced into the computations.

Damage and crack modeling in single-edge and double-edge notched concrete beams

The numerical modeling of damage and crack propagation in concrete and concrete structures has evolved considerably in the past years. In this contribution, a higher order continuum model is used to model the failure behavior of single-edge notched (SEN) and double-edge notched (DEN) concrete beams loaded in four-point-shear. Different types of boundary conditions, i.e. with freely rotating, fixed or constrained loading supports, are investigated and the experimentally observed curved crack paths are compared with the numerical simulations. The influence of the ratio of the compressive strength and the tensile strength is scrutinized and its relation with the failure mechanism is investigated. It is shown that an isotropic gradient-enhanced damage model permits to obtain a good agreement between experimental results and numerical simulations.

Fuzzy-random probabilistic analysis of rock mass responses to explosive loads

This paper addresses the effects of randomness of initial damage in a rock mass and the critical tensile strain of the rock material on its dynamic responses and damage under explosive loads. A fuzzy definition is proposed to describe the fuzzy nature of failure phenomenon in a rock mass. The initial damage of the rock mass is estimated using the longitudinal and transverse elastic wave velocities. By using statistical analysis, the initial damage of the rock mass is found having the Beta distribution. The statistical estimation of a damage state and properties of randomly damaged rock mass are evaluated by the Rosenbluth's point estimate method. In numerical calculation, an isotropic continuum damage model with the initial damage and the cumulative damage dependent on an equivalent tensile strain is suggested to model the rock mass behavior under blast loads. A Beta distribution is proposed to represent the probabilistic distribution of the damage variable of the rock mass under explosive loads. Several types of membership functions are suggested to represent the fuzziness of material failure. Based on the fuzzy–random probabilistic theory, a model including both the effects of randomness and fuzziness is proposed for the failure analysis of rock mass under explosive loads. The suggested models are coded and linked with an available computer program AUTODYN2D through its user's subroutine capacity. The fuzzy failure probability and dynamic responses of the rock mass are calculated. Numerical results are compared with those obtained from independent field tests.

Simulation of strain localization with gradient enhanced damage models

The incorporation of higher order strain gradients into the constitutive equations of continuum damage mechanics is presented. Thereby, not only scalar-valued isotropic damage models but also anisotropic damage models allowing for directional dependent stiffness degradation are elaborated. An elegant possibility of describing anisotropic material behavior based on the microplane theory is demonstrated. Its conceptual simplicity originates from the idea of modeling the material behavior through uniaxial stress–strain laws on several individual material planes. For each plane individual damage loading functions are introduced allowing for different failure modes. In order to account for long ranging microstructural mechanisms, second-order gradients of the strains are incorporated in each of these damage loading functions. The overall response can be determined by an integration of the resulting microplane laws over the solid angle. The features of gradient enhanced continuum damage are demonstrated by means of several selected examples.

An isotropic damage model for concrete

The studies on modeling of concrete fracture by employing finite element methods have been carried out for the last thirty years. Among these constitutive models, isotropic damage concepts have been found more advantageous for its numerical practicability on modeling of concrete fracture compared to smeared crack applications. In this paper, anew damage function is proposed by simplifying the calculated curves for the damage function in a previous study. A new relation for the fracture energy of concrete is obtained by using this new damage function and the suitability of the recommended equation with the previously proposed relations is verified. Besides that, a realistic damage bounding surface for concrete elements subjected to biaxial stress states is proposed. Finally, the successful results of the applications of the damage model to a plain concrete specimen and a shear panel are illustrated.

Parameter identification of gradient enhanced damage models with the finite element method

In this contribution an algorithm for parameter identification of gradient enhanced damage models is proposed, in which non-uniform distributions of the state variables such as stresses, strains and damage variables are taken into account. To this end a least-squares functional consisting of experimental data and simulated data is minimized, whereby the latter are obtained with the Finite-Element-Method. In order to improve the efficiency of the minimization process, a gradient-based optimization algorithm is applied, and therefore the corresponding sensitivity analysis for the coupled variational problem is described in a systematic manner. For illustrative purpose, the performance of the algorithm is demonstrated for a square panel under tension, in which an isotropic gradient damage model is used.

Spurious Bifurcations in Transient Analysis with Isotropic Damage

In the setting of thermomechanical transient analysis of concrete using isotropic damage, the writers noted the consistent occurrence of spurious bifurcations of the deformation field, for thermal and mechanical loading of panel specimens. The cause of the problem is discussed, noting that bifurcation occurs in a simple uniaxial problem, in tension or compression, when there is any (small) difference in material properties. With a transient analysis there are always small momentum effects, and these generate slight variations in strain. For a strain-based isotropic damage model, this results in slight variations in damage, hence stiffness, and so bifurcation is inevitable in the symmetric homogeneous problem. A simple cure to the problem is adopted and the model is tested on three point bend specimens. Apart from its known deficiencies, the local damage model is shown to be very sensitive to the choice of model parameters, mesh size and structure, and the level of numerical error. A nonlocal formulation, ...

Numerical Simulation of Ductile Fracture Based on Local Approach Concept

In this paper a finite-element analysis on ductile fracture in two-dimensional quasi-static state is performed by using the local approach concept in continuum damage mechanics. An isotropic damage model based on the generalized concept of effective stress is proposed. Crack propagation is achieved by removing critically damaged elements. The finite-element approximation of a largely deforming body based on the incremental total Lagrangian concept is carried out. As numerical examples, the mesh size sensitivity analysis and the simulation of the shearing mode failure in plane strain state are carried out to verify the present formulation qualitatively. For an edge cracked plate under plane stress state, load-displacement curves and successively fractured shapes are shown. It can be concluded that the proposed model may be stated as a reasonable tool to explain ductile fracture initiation and crack propagation.

Dynamic response analysis of rock mass with stochastic properties subjected to explosive loads

Abstract This paper carries out probabilistic analysis of dynamic responses of rock mass under blast loads. Statistical analysis of the initial damage and strength constants of the rock mass is performed by using both the field and laboratory test data. The initial damage of the rock mass is found having the beta distribution, while the critical tensile strain has the normal distribution. These statistical properties are incorporated into the constitutive law and cumulative damage model for rock mass. The statistical estimation of stress wave propagation in the rock mass due to underground explosion is evaluated by Rosenblueth's point estimate method. In numerical calculation, an isotropic continuum damage model considering both the initial damage and the cumulative damage dependent on an equivalent tensile strain is suggested to model rock mass behaviour under blast loads. The suggested models are programmed and linked to an available computer program Autodyn2D through its user's subroutine capability. U...

Computational modelling of gradient‐enhanced damage in quasi‐brittle materials

To avoid the well-known drawbacks of the classical continuum damage theory when localization occurs, an isotropic gradient-enchanced damage model is proposed in which the loading function not only depends on the damage value, but also on its Laplacian. The initial boundary value problem obtained adopting this model is considered both in statics and in dynamics. In the dynamic context the finite-step problem is formulated according to a Newmark scheme; the constitutive law is integrated by the backward difference rule. An iterative procedure for the finite-step solution is discussed. Finite elements space discretization is carried out in terms of generalized variables on the basis of two variational principles pertinent to the two phases of the iterative process. One- and two-dimensional numerical tests show the regularizing effect of the gradient term and the effectiveness of the proposed discretization technique. Copyright © 1999 John Wiley & Sons, Ltd.