Karam Sab

Asymptotic analysis of heterogeneous Cosserat media

Abstract The present work deals with the development of homogenization procedures for periodic heterogeneous linear elastic Cosserat media. It is resorted to asymptotic methods classically used in periodic homogenization. It is shown that the nature of the homogeneous equivalent medium depends on the hierarchy of three characteristic lengths: the size l of the heterogeneities, the Cosserat intrinsic lengths l c of the constituents and the typical size L of the considered structure. When l and l c are comparable and much smaller than L , the effective medium is proved to be a Cauchy continuum with volume couples, whereas the case l c ∼ L leads to a Cosserat effective medium. Finite element simulations are provided in the case of a fiber-matrix composite for a large range of characteristic lengths l c and for two different volume fractions. Reference calculations involving every heterogeneity are compared to the response obtained using a homogeneous equivalent medium. The results confirm the predicted hierarchy of models and also show that a Cosserat effective medium still provide a good estimation when all characteristic lengths have the same order of magnitude.

Foam mechanics: nonlinear response of an elastic 3D-periodic microstructure

The compressive response of a 3D open-cell foam with periodic tetrakaidecahedral cells is studied through combined theoretical and numerical efforts. Under compressive loading the response is characterized by an extended load plateau following the relatively sharp rise to a maximum load. Several processes of loading have been simulated numerically using appropriately nonlinear kinematics. The onset of failure under macroscopic loading conditions is shown to be the reason of the load plateau. A failure surface is defined in macroscopic stress space by the onset of the first buckling-type instability encountered along proportional load paths. The analysis is carried out through two methods. The first one consists in increasing specimen size with periodic boundary conditions leading to the termed microfailure surface. The second one consists in considering both periodic and nonperiodic displacements variations on a minimum unit cell. The resulting failure surfaces are shown to coincide. Moreover, the postbuckling analysis has been carried out for two particular loadings: the uniaxial compression and the uniaxial deformation.

Out of plane model for heterogeneous periodic materials: the case of masonry

Abstract The aim of this paper is to propose a 3D model to study masonry walls subject to in plane and out of plane actions through a rigorous homogenization procedure. 2D rigorous models in several perturbative parameters have been already developed – by Cecchi–Di Marco (European J. Mech. A Solids 19 (2000) 535–546), Cecchi–Rizzi (Internat. J. Solids Structures 38/1 (2001) 29–36) and Cecchi–Sab (European J. Mech. A Solids 21 (2002) 249–268) – so as to study the behaviour of masonry walls subject to actions parallel to the middle plane. By comparison with previous models, in this paper the study of masonry takes into account the formulation of a 3D model where masonry is assumed periodic in the middle plane, i.e. in the orthogonal directions to the thickness. The size of the thickness is comparable with the one of the period. The asymptotic model that has been developed allows the identification of the 3D solid with a 2D Love–Kirchoff plate, in which the anisotropy is connected with the arrangement of blocks. The obtained results give the values of homogenized elastic plate constants (Caillerie, Math. Meth. Appl. Sci. 6 (1984) 159–191).

Yield design of thin periodic plates by a homogenization technique and an application to masonry walls

Abstract A homogenization method for determining overall yield strength properties of thin periodic plates from their local strength properties is proposed within the framework of the yield design theory. The proposed method is applied to the determination of the in-plane and out of plane strength criterion for masonry described as a regular assemblage of infinitely resistant bricks separated by Coulomb interfaces. To cite this article: K. Sab, C. R. Mecanique 331 (2003).

Cosserat modelling of elastic periodic lattice structures

Abstract A generalized theory for linking a periodic discrete medium, with translations and rotations (for instance, fully-joint beam assembly), to an equivalent Cosserat continuum is presented. This theory will be applied to honeycomb plane networks. Homogenized behaviour for rotations will be exhibited. Then, forecasts of discrete and continuous models will be confronted on the same plane simple shear problem.

On the choice of parameters in the phase field method for simulating crack initiation with experimental validation

The phase field method is a versatile simulation framework for studying initiation and propagation of complex crack networks without dependence to the finite element mesh. In this paper, we discuss the influence of parameters in the method and provide experimental validations of crack initiation and propagation in plaster specimens. More specifically, we show by theoretical and experimental analyses that the regularization length should be interpreted as a material parameter, and identified experimentally as it. Qualitative and quantitative comparisons between numerical predictions and experimental data are provided. We show that the phase field method can predict accurately crack initiation and propagation in plaster specimens in compression with respect to experiments, when the material parameters, including the characteristic length are identified by other simple experimental tests.

Shear Correction Factors for Functionally Graded Plates

The Reissner-Mindlin plate model for calculation of functionally graded materials has been proposed in literature by using shear correction coefficient of homogeneous model. However, this use is a priori not appropriate for the gradient material. Identification of the transverse shear factors is thus investigated in this paper. The transverse shear stresses are derived by using energy considerations from the expression of membrane stresses. Using the obtained transverse shear factor, a numerical analysis is performed on a simply supported FG square plate whose elastic properties are isotropic at each point and vary through the thickness according to a power law distribution. The numerical results of a static analysis are compared with available solutions from previous studies.

Finite-deformation second-order micromorphic theory and its relations to strain and stress gradient models

Germain’s general micromorphic theory of order n is extended to fully non-symmetric higher-order tensor degrees of freedom. An interpretation of the microdeformation kinematic variables as relaxed higher-order gradients of the displacement field is proposed. Dynamical balance laws and hyperelastic constitutive equations are derived within the finite deformation framework. Internal constraints are enforced to recover strain gradient theories of grade n . An extension to finite deformations of a recently developed stress gradient continuum theory is then presented, together with its relation to the second-order micromorphic model. The linearization of the combination of stress and strain gradient models is then shown to deliver formulations related to Eringen’s and Aifantis’s well-known gradient models involving the Laplacians of stress and strain tensors. Finally, the structures of the dynamical equations are given for strain and stress gradient media, showing fundamental differences in the dynamical behaviour of these two classes of generalized continua.

Justification of the Bending-Gradient Theory Through Asymptotic Expansions

In a recent work, a new plate theory for thick plates was suggested where the static unknowns are those of the Kirchhoff-Love theory, to which six components are added representing the gradient of the bending moment [1]. This theory, called the Bending-Gradient theory, is the extension to multilayered plates of the Reissner-Mindlin theory which appears as a special case when the plate is homogeneous. This theory was derived following the ideas from Reissner [2] without assuming a homogeneous plate. However, it is also possible to give a justification through asymptotic expansions. In the present paper, the latter are applied one order higher than the leading order to a laminated plate following monoclinic symmetry. Using variational arguments, it is possible to derive the Bending-Gradient theory. This could explain the convergence when the thickness is small of the Bending-Gradient theory to the exact solution illustrated in [3]. However, the question of the edge-effects and boundary conditions remains open.

A stochastic nonlocal damage model

Deterministic nonlocal damage models permit avoiding spurious mesh sensitivity and predict ‘structure’ size effect which is in accordance with experimental observations on notched specimens made of concrete. However, these deterministic models are unable to predict ‘volume’ size effect exhibited by unnotched specimens in direct tension because the effect of heterogeneity has been introduced via a deterministic strain-softening behaviour.The main idea of our model is to consider two scales: the meso-scale (the representative volume) of size lr and the micro-scale, of size d (d ⩽ lr). The plane is subdivided in square cells of size d. Each cell is supposed to be homogeneous and behaves in a purely brittle manner. The heterogeneity of the material properties are introduced by attributing to each cell a random threshold value picked according to a power law distribution function of parameter m. The breaking criterion is nonlocal.In finite element calculations, each cell is discretized in finite element meshes which have the same threshold value. Each finite element mesh behaves in a purely brittle manner and its breaking criterion is the ratio of the volume average of the elastic energy over a cube of size lr and the threshold value.Numerical simulations on notched and unnotched specimens will show that the proposed model yields mesh insensitive calculations and predicts both ‘volume’ and ‘structure’ size effects which are qualitatively in accordance with experimental observations.