Homogenization and damage for composite structures

Abstract

Groupe Modell.sotioti rt c'al~~ul. Lohorotoirr d'Acou.\tiqur rf Mi~canlyue. C'.!V.R.S rf L'tiweryite P . ef IM. Curie. Purls. Fruticc SUMMARYAfter presenting shortly the main resufts of the homogenization theory for periodic media, we give two applications related to damage evaluation and simulation-for composite .materials: (i) simulation of the evolution of damage by fibre rupture in a unidirectional composite, involving parameters defined on the microscale; and (ii) prcdiction of debonding near an unloaded boundary in a stratified structure, using boundary layer asymptotic expansions. . INTRODUCTIONComposite materials are more and more frequently used in the realization of structures of extreme mechanical performance. The consequence is an imperious need of predicting the appearance of damage and the resulting behaviour, This damage -ruptures of fibres, matrix fissuration. debonding--comes from the fact that the great heterogeneity of the materials generates local overstresses at the microstructure level. In materials wherc the structure is periodic or quasiperiodic, homogeniLation theory developed in the last fifteen years'. 8 . ''. 2 2 -2 5 provides a systematic and mathematically justified method for the computation of those overstresses that appear on the microscale and which are usely called microstresses despite the fact that they are not small. After presenting shortly the main results of this homogenization theory, we give two applications in the domain of damage evaluation in composite materials: (i) simulation of the evolution of damage by fibre rupture in a unidirectional composite; and (ii) prediction of debonding near a free edge in a stratified structure. HOMOGENIZATION M E T H O D I. Formulation qf the problemLet us consider an elastic body which occupics a region R related to a system of orthonormal axes (0, x,, x2. x3). This body is subjected to a system of body forces f and surface forces F on a portion rF of the boundary dR. The other portion of the boundary is To, on which H zero displacement condition is imposed.Furthermore, the material is elastic with fine periodic structure, i.e. R is covered by a set of identical periods of rectangular, hexagonal o r more complicated shape.

An asymptotic method for the prediction of the anisotropic effective elastic properties of the cortical vein: superior sagittal sinus junction embedded within a homogenized cell element

“…If U (X, Y ) is the displacement field, it may be expressed in terms of infinite expansion series, as proposed in [Sanchez-Palencia 1974;Ladeveze et al 1985;Dumont et al 1987;Devries et al 1989]: The elastic deformation written here in small deformations is decomposed as a function of slow and rapid variables (it can be developed for large deformation):…”

Numerical Study of Damage Growth in Particulate Composites

“…There has been an extensive work in this area so that all the references can not be compiled here. A few of them are given in the references (Devries et al, 1989;Felice and Rizzi, 1997;Siboni and Benveniste, 1991;Gibson, 1994). One of the techniques is called the unit cell method (Aboudi, 1987(Aboudi, , 1989.…”

Asymptotic homogenization for modeling of wingbox structures

“…One technique that can meet these objectives is that of asymptotic homogenization and the mathematical details can be found in [7][8][9][10]. Many problems of different length-scale in elasticity, thermoelasticity, piezoelasticity, fracture mechanics, design and optimization as they pertain to composite beams, plates and shells have been analyzed using asymptotic homogenization [11], [12] and can be found in the works of [12][13][14][15][16][17][18][19][20].…”