Jean Lemaitre

A Course on damage mechanics

1 Phenomenological Aspects of Damage.- 1.1 Physical Nature of the Solid State and Damage.- 1.1.1 Atoms, Elasticity and Damage.- 1.1.2 Slips, Plasticity and Irreversible Strains.- 1.1.3 Scale of the Phenomena of Strain and Damage.- 1.1.4 Different Manifestations of Damage.- 1.1.5 Exercise on Micrographic Observations.- 1.2 Mechanical Representation of Damage.- 1.2.1 One-Dimensional Surface Damage Variable.- 1.2.2 Effective Stress Concept.- 1.2.3 Strain Equivalence Principle.- 1.2.4 Coupling Between Strains and Damage Rupture Criterion Damage Threshold.- 1.2.5 Exercise on the Micromechanics of the Effective Damage Area.- 1.3 Measurement of Damage.- 1.3.1 Direct Measurements.- 1.3.2 Variation of the Elasticity Modulus.- 1.3.3 Variation of the Microhardness.- 1.3.4 Other Methods.- 1.3.5 Exercise on Measurement of Damage by the Stress Amplitude Drop.- 2 Thermodynamics and Micromechanics of Damage.- 2.1 Three-Dimensional Analysis of Isotropic Damage.- 2.1.1 Thermodynamical Variables, State Potential.- 2.1.2 Damage Equivalent Stress Criterion.- 2.1.3 Potential of Dissipation.- 2.1.4 Strain-Damage Coupled Constitutive Equations.- 2.1.5 Exercise on the Identification of Material Parameters.- 2.2 Analysis of Anisotropic Damage.- 2.2.1 Geometrical Definition of a Second-Order Damage Tensor.- 2.2.2 Thermodynamical Definition of a Fourth-Order Damage Tensor.- 2.2.3 Energetic Definition of a Double Scalar Variable.- 2.2.4 Exercise on Anisotropic Damage in Proportional Loading.- 2.3 Micromechanics of Damage.- 2.3.1 Brittle Isotropie Damage.- 2.3.2 Ductile Isotropie Damage.- 2.3.3 Anisotropie Damage.- 2.3.4 Microcrack Closure Effect, Unilateral Conditions.- 2.3.5 Damage Localization and Instability.- 2.3.6 Exercise on the Fiber Bundle System.- 3 Kinetic Laws of Damage Evolution.- 3.1 Unified Formulation of Damage Laws.- 3.1.1 General Properties and Formulation.- 3.1.2 Stored Energy Damage Threshold.- 3.1.3 Three-Dimensional Rupture Criterion.- 3.1.4 Case of Elastic-Perfectly Plastic and Damageable Materials.- 3.1.5 Identification of the Material Parameters.- 3.1.6 Exercise on Identification by a Low Cycle Test.- 3.2 Brittle Damage of Metals, Ceramics, Composites and Concrete.- 3.2.1 Pure Brittle Damage.- 3.2.2 Quasi-brittle Damage.- 3.2.3 Exercise on the Influence of the Triaxiality on Rupture.- 3.3 Ductile and Creep Damage of Metals and Polymers.- 3.3.1 Ductile Damage.- 3.3.2 Exercises on the Fracture Limits in Metal Forming.- 3.3.3 Creep Damage.- 3.3.4 Exercise on Isochronous Creep Damage Curves.- 3.4 Fatigue Damage.- 3.4.1 Low Cycle Fatigue.- 3.4.2 Exercise on Creep Fatigue Interaction.- 3.4.3 High Cycle Fatigue.- 3.4.4 Exercise on Damage Accumulation.- 3.5 Damage of Interfaces.- 3.5.1 Continuity of the Stress and Strain Vectors.- 3.5.2 Strain Surface Energy Release Rate.- 3.5.3 Kinetic Law of Debonding Damage Evolution.- 3.5.4 Simplified Model.- 3.5.5 Exercise on a Debonding Criterion for Interfaces.- 3.6 Table of Material Parameters.- 4 Analysis of Crack Initiation in Structures.- 4.1 Stress-Strain Analysis.- 4.1.1 Stress Concentrations.- 4.1.2 Neuters Method.- 4.1.3 Finite Element Method.- 4.1.4 Exercise on the Stress Concentration Near a Hole.- 4.2 Uncoupled Analysis of Crack Initiation.- 4.2.1 Determination of the Critical Point(s).- 4.2.2 Integration of the Kinetic Damage Law.- 4.2.3 Exercise on Fatigue Crack Initiation Near a Hole.- 4.3 Locally Coupled Analysis.- 4.3.1 Localization of Damage.- 4.3.2 Postprocessing of Damage Growth.- 4.3.3 Description and Listing of the Postprocessor DAMAGE 90.- 4.3.4 Exercises Using the DAMAGE 90 Postprocessor.- 4.4 Fully Coupled Analysis.- 4.4.1 Initial Strain Hardening and Damage.- 4.4.2 Example of a Calculation Using the Finite Element Method.- 4.4.3 Growth of Damaged Zones and Macrocracks.- 4.4.4 Exercise on Damage Zone at a Crack Tip.- 4.5 Statistical Analysis with Microdefects.- 4.5.1 Initial Defects.- 4.5.2 Case of Brittle Materials.- 4.5.3 Case of Quasi Brittle Materials.- 4.5.4 Case of Ductile Materials.- 4.5.5 Volume Effect.- 4.5.6 Effect of Stress Heterogeneity.- 4.5.7 Exercise on Bending Fatigue of a Beam.- History of International Damage Mechanics Conferences.- Authors and Subject Index.

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.

Formulation and Identification of Damage Kinetic Constitutive Equations

The damage kinetic constitutive equations are derived from the thermodynamics of irreversible process in which physical considerations and experimental results are introduced in order to choose the proper variables and the analytical forms of the potentials. A general damage model is formulated and then applied to ductile damage, low cycle fatigue, high cycle fatigue and creep damage in order to identified particular kinetic laws to be introduced in structure calculations to predict the initiation and the growth of cracks.

Continuum damage mechanics and local approach to fracture: Numerical procedures

Abstract In this paper, continuum damage mechanics is applied to the prediction of the failure of structures. The numerical implementation of this theory within the framework of the finite element method is described in details for both initiation and propagation problems. Practical examples are given to demonstrate the usefulness of this so-called ‘local approach to fracture’ in the case of creep and ductile damages.

Damage 90: a post processor for crack initiation

A post processor is fully described which allows the calculation of the crack initiation conditions from the history of strain components taken as the output of a finite element calculation. It is based upon damage mechanics using coupled strain damage constitutive equations for linear isotropic elasticity, perfect plasticity and a unified kinetic law of damage evolution. The localization of damage allows this coupling to be considered only for the damaging point for which the input strain history is taken from a classical structure calculation in elasticity or elastoplasticity. The listing of the code, a ‘friendly’ code, with less than 600 FORTRAN instructions is given and some examples show its ability to model ductile failure in one or multi dimensions, brittle failure, low and high cycle fatigue with the non-linear accumulation, and multi-axial fatigue. During the’past decades, the problem of macrocrack growth up to failure of structures has received much attention through the tremendous development of fracture mechanics. Crack initiation, which requires a knowledge of what happens before a mesocrack breaks the representative volume element (RVE), cannot be treated by classical fracture mechanics dealing with pre-existing meso or macro cracks. The usual engineering method of predicting the crack initiation consists of using some phenomenological or empirical criteria or relations to define the conditions of local rupture of which the following are examples: - a critical value of the maximum principal stress; - the von Mises equivalent stress or better the damage equivalent stress equal to some critical value for

Application of Continuous Damage Mechanics to Strain and Fracture Behavior of Concrete

Elasticity and damage by microcracking constitute the essential phases of the mechanical behavior of concrete.

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.

The mechanical behavior of an alumina carbon/epoxy laminate

An experimental study has been made of a laminate consisting of monolithic thin alumina plates alternating with unidirectional carbon/epoxy (C/E) prepreg tapes. The main advantages of this system over the traditional means of reinforcing ceramics, are the avoidance of large flaws due to processing, which occur in fiber reinforced brittle matrix composites, and the nearly isotropic behavior under biaxial loading. In addition, the multiple fracture mechanism occurring in the system gives rise to pseudo ductile behavior and enhanced strain energy dissipation. The mechanical behavior of the laminate is explored. The effects of the number of layers, volume fraction and transverse properties are also investigated. The loss of stiffness with increase of the applied strain is estimated using a simple shear lag theory, which includes the plastic behavior of the interface.

SECTION 6.15 – A Two-Scale Model for Quasi-Brittle and Fatigue Damage

A two-scale model for quasi-brittle and fatigue damage describes the progressive deterioration of solid materials up to a mesocrack initiation. It is used to predict the state of damage and the conditions of crack initiation in mechanical components subjected to mechanical and thermal loadings. It applies to the quasi-brittle type of damage as brittle failure and high cycle fatigue. In both cases the damage is always localized at a microscale in the vicinity of a defect considered as a weak inclusion. Two scales are considered: the mesoscale or scale of the representative volume element (RVE) of continuum mechanics, and the microscale or scale of a microdefect (microvoids, microcracks) embedded in the RVE. The model is written for isotropy, but anisotropic damage is easily incorporated. An initial elastic structure calculation is needed to define the stress and strain fields at the mesoscale. The time histories of the stresses and strains at the most loaded point(s) of the structure are then the inputs of the model. The main output is the evolution of the damage as a function of loading, time, or of the number of cycles. The present analysis may also model failure with (visco-) plasticity at mesoscale.