Rodrigue Desmorat

Anisotropic damage law of evolution

A formulation for anisotropic damage is established in the framework of the principle of strain equivalence. The damage variable is still related to the surface density of microcracks and microvoids and, as its evolution is governed by the plastic strain, it is represented by a second order tensor and is orthotropic. The coupling of damage with elasticity is written through a tensor on the deviatoric part of the energy and through a scalar taken as its trace on the hydrostatic part. The kinetic law of damage evolution is an extension of the isotropic case. Here, the principal components of the damage rate tensor are proportional to the absolute value of principal components of the plastic strain rate tensor and are a nonlinear function of the effective elastic strain energy. The proposed damage evolution law does not introduce any other material parameter. Several series of experiments on metals give a good validation of this theory. The coupling of damage with plasticity and the quasi-unilateral conditions of partial closure of microcracks naturally derive from the concept of effective stress. Finally, a study of strain localization makes it possible to determine the critical value of the damage at mesocrack initiation.

A two scale damage concept applied to fatigue

The ductile type of damage is a phenomenon now well understood. Once the fully coupled set of constitutive equations is identified, Damage Mechanics is a powerful tool to predict failure. Brittle materials do not exhibit such a damageable macroscopic behavior. Nevertheless, they still fail. On the idea that damage is localized at the microscopic scale, a scale smaller than the mesoscopic one of the Representative Volume Element (RVE), we propose a three-dimensional failure modeling for monotonic as well as for fatigue loading. We develop a two scale model of what we shall call brittle damage: at the microscopic scale, micro-cracks or micro-voids exhibit a damageable plastic-like behavior with no effect on the global (mesoscopic) elastic behavior. Microscopic failure is assumed to coincide with the RVE failure. This model turns out to represent quite well physical phenomena related to high cycle fatigue such as the mean stress effect, the nonlinear accumulation of damage, initial strain hardening or damage effect and the nonproportional loading effect for bi-axial fatigue. The model has been implemented as a post-processor computer code. A simplified identification procedure for the determination of the material properties is given.

Modeling Microdefects Closure Effect with Isotropic/Anisotropic Damage

Continuum damage mechanics (CDM) for metals is often written in terms of an isotropic (scalar) damage. In this case, solutions have been proposed to represent the differences of behavior in tension and in compression also called quasi-unilateral (QU) conditions or microdefects closure effect. A recent anisotropic damage model has been developed to take into account the damage orthotropy induced by plasticity (Lemaitre, J., Demorat R. and Sauzay, M. (2000). Anisotropic Damage Law of Evolution, Eur. J. Mech. A/Solids, 19: 513—524). The purposes here are then two. First, a unified framework for isotropic and anisotropic damage is proposed. Then, it is to extend Ladevèze and Lemaitres framework (Ladevèze, P. and Lemaitre, J. (1984). Damage Effective Stress in Quasi Unilateral Conditions, In: Proceedings of the 16th International Congress of Theoretical and Applied Mechanics, Lyngby, Denmark) for the QU conditions to anisotropic damage induced by plasticity. Yield surfaces and damage versus accumulated plastic strain curves, drawn for different loading, illustrate the effect of the QU conditions on the damage evolution.

Nonstandard Thermodynamics Framework for Robust Computations with Induced Anisotropic Damage

Many anisotropic damage models have been proposed for different materials, ductile as well as quasi-brittle. The main drawback of the corresponding analyzes is that a large number of material parameters is often introduced, leading to identification difficulties and also to model complexity and associated numerical difficulties. It is also sometimes difficult to ensure the continuity of the stresses if the quasi-unilateral effect of microcracks closure and the dissymmetry tension/ compression are represented. In order to solve those difficulties, one proposes to write the damage models in a specific nonstandard thermodynamics framework. The damage states are represented by a symmetric second-order tensor and the damage rate is assumed governed by a positive second-order tensor having a clear meaning: the absolute or the positive value of the plastic strain rate tensor for ductile materials, the positive part of the total strain tensor in quasi-brittle materials. Such a nonstandard feature makes the proof of the the positivity of the intrinsic dissipation necessary. This important proof is given in the considered framework for any damage law ensuring (anisotropic) damage increase and for any case, 3D, proportional or nonproportional. This extends then to induced anisotropy the isotropic case property of a positive damage rate as a sufficient condition for the thermodynamics second principle to be fulfilled. Altogether with the fact that the thermodynamics potential can be continuously differentiated, the example of an anisotropic damage model for concrete (build in this framework) is given. It allows for robust finite element implementation. Both space (classical nonlocal with internal length, nonlocal with internal time) and time regularizations (visco- or delay-damage) are used and applied to quasi-static and dynamic cases. Examples on concrete and reinforced concrete structures are given.

Singularities in bi-materials : parametric study of an isotropic/anisotropic joint

Problems in fracture mechanics are frequently solved in terms of crack tip singularities. Geometries other than cracks also exhibit singular stresses at points such as corners, edges and interfacial joints. Corners occuring in monolayers or multilayered media have been studied under the assumption that each layer is isotropic. For general elastic plane problems, the present study extends the earlier results to anisotropy. For orthotropic joints, generalized Dundurs parameters are introduced. Isotropic results are a limiting case of the present analysis. In the vicinity of a singular point, the displacements and stresses may be expressed as a function of the polar coordinates r − θ by: u=hrδg(φ), σ=hrδ−1F(φ) where h is the intensity factor and δ the singularity exponent (0 < δ < 1). The values of h and δ reduce to the stress intensity factor k and the complex exponent 0.5 + ie for the limiting case of cracks at the interface of dissimilar media. Using an anisotropic complex potential method, the present analysis gives δ as the solution of an eigenvalue problem and as the root of a nonlinear equation det A(δ) = 0. It leads to a closed-form expression for g and F. The matrix A depends on the number of layers at the singular point, their relative elastic properties and the boundary conditions such as free surface or bonded interface close to the singularity corner. A closed-form expression is derived for A which depends on 3 generalized Dundurs parameters for a metal matrix composite isotropic metal interface joint. This compares to the 2 Dundurs parameters needed for joints with isotropic layers. The intensity factor h of any singularity is determined from a path independent integral, using an extraction function which is more singular than that defining the actual stress state.

Anisotropic 3D delay-damage model to simulate concrete structures

Dynamic loadings lead to material degradation and structural failure. This is even more the case for concrete structures where the parts initially in compression break in tension due to waves propagation and reflection. The dissymmetry (mainly due to damage induced anisotropy) of the material behavior plays a major role in such cases. Loading induced damage is often anisotropic and one proposes here to take advantage of such a feature to build a damage model for concrete, dissymmetric in tension and in compression, 3D, suitable for dynamic computations. A single 2nd order tensorial damage variable D is considered with a damage law ensuring a damage rate proportional to the square of the positive part of the strain tensor. One focus in the present work on viscous regularizations for the anisotropic damage model proposed. Numerical examples illustrate the efficiency of the model to deal with 3D structures.

Fast estimation of localized plasticity and damage by energetic methods

Abstract Structural failure often follows the initiation of cracks occurring at corners, free edges or interfaces. Continuum damage mechanics gives quantitative information about such cracking. But when used in a fully coupled manner (with elasticity and plasticity), it leads to costly computations. In order to obtain helpful results for a fine and fast design, we propose to determine localized plasticity and damage by use of local post-calculations, which follow a simple elastic finite element computation. Energetic methods such as Neubers, such as the strain energy density or as the complementary energy density methods, are justified for small scale yielding by use of path-independent integrals. They are extended to cyclic loading inducing fatigue and the case of thermal stresses is considered. For plane problems, these methods are completed by the analytical determination of the stress triaxiality along free edges or rigid inclusions. The crack initiation conditions are then quickly estimated by the time-integration of Lemaitres damage law. Calculations made for a holed plate (plane strain) and for a bi-axial testing specimen (plane stress) validate the methods.

Dissymétrie de comportement élastique anisotrope couplé ou non à l'endommagement

Resume. Un comportement elastique anisotrope different en traction et en compression est modelise dans un cadre thermodynamique. La decomposition de Kelvin du tenseur souplesse elastique est utilisee et permet de traiter tout cas de chargement 3D. Le couplage avec l’endommagement est ecrit pour des variables d’endommagement tensorielles d’ordre 4 ou d’ordre 2. La formulation proposee assure naturellement la continuite du tenseur des contraintes et du taux de restitution d’energie. Le cas particulier d’une elasticite isotrope initiale est traite.

Reinitiation of a crack reaching an interface

We consider here a bi-material made of two layers bonded together by an interface. The specimen is loaded in tension parallel to the interface and the existence of a mode I crack is assumed. The crack initiated in just one layer reaches the interface normally. We then study the second of the two possible cases: the crack crosses the interface and goes straight into the second layer, in mode I also; or the crack debonds the interface before reinitiating in the second layer at the debond tip.In the present study the conditions of the reinitiation of the crack in the second layer after debonding of the interface are presented. The maximum debond distance is calculated by means of a Shear Lag analysis associated with a damage constitutive equation.Qualitative rules for design are pointed out to make the interface a location of crack arrest or at least of crack growth delay. These rules are mainly: small thickness of the possibly cracked layer, strong interface and tough substrate.

Anisotropic damage modeling of concrete materials

An anisotropic damage model is proposed for concrete materials. As required by thermodynamics a single damage variable, tensorial, is considered for any loading: as a state variable it represents the micro-cracking pattern whatever the loading sign. Damage anisotropy is used to model the strong dissymmetry tension/compression.The Ladevèze damage variable H = (1 − D)−1/2 is introduced within a deviatoric/hydrostatic split. An original shear-bulk coupling is derived, in accordance with numerical discrete element computations. The sought property of gradual stress softening, with a tail in stress–strain diagram, is obtained. Stress triaxiality is used to enhance the performance of Mazars criterion and therefore of the full anisotropic damage model in bicompression.