Françoise Krasucki

Mathematical Analysis of a Bonded Joint with a Soft Thin Adhesive

This paper considers the problem of two adherents joined by a soft thin adhesive along their common surface. Using the asymptotic expansion method, the authors obtain a simplified model in which the adhesive is treated as a material surface and is replaced by returning springs. The authors show weak and strong convergence of the exact solution toward the solution of the limit problem. The singularities of the limit problem are analyzed, and it is shown that typically they are logarithmic. Furthermore, the authors investigate the phenomenon of boundary layer by studying the correctors, the extra terms, which must be added to the classical asymptotic expansion to verify the boundary conditions. The correctors show that, contrary to the adherents, in the adhesive there are power-type singularities, which are at the base of the failure of the assemblage.

A NEW DUALITY APPROACH TO ELASTICITY

The displacement-traction problem of three-dimensional linearized elasticity can be posed as three different minimization problems, depending on whether the displacement vector field, or the stress tensor field, or the strain tensor field, is the unknown. The objective of this paper is to put these three different formulations of the same problem in a new perspective, by means of Legendre-Fenchel duality theory. More specifically, we show that both the displacement and strain formulations can be viewed as Legendre-Fenchel dual problems to the stress formulation. We also show that each corresponding Lagrangian has a saddle-point, thus fully justifying this new duality approach to elasticity.

Multi-materials with strong interface: Variational modelings

We introduce a simplified model for a multi-material made up of two elastic bodies connected by a strong thin material layer whose stiffness grows as 1/e. The model is obtained by identifying the Γ-limit of the stored strain energy functionalof the physical problem when the thickness e of the intermediate layer tends to zero. The intermediate layer behaves as a stiffening elastic membrane. Furthermore, in the linear anisotropic case, we establish the strong convergence of the exactsolution toward the solution of the limit problem.

On the existence of the Airy function in Lipschitz domains. Application to the traces of H2

Abstract We prove the existence of the Airy function w corresponding to the stress tensor S in a plane domain Ω connected, eventually not simply connected, with Lipschitz boundary Γ. In order to understand the relations between the boundary values of w and S · n , we study the tangential derivative from H1/2(Γ) into H−1/2(Γ). With the help of these results we characterize the traces of a function in H 2 (Ω) extending the previous results of Necas and Grisvard.

A DOMAIN DECOMPOSITION METHOD FOR A BONDED STRUCTURE

We show that the transmission conditions through two elastic bodies bonded by a thin adhesive layer can be written as Robin-type conditions, well-suited for using a domain decomposition algorithm for which we prove convergence. Numerical approximation by means of finite element methods is also presented and analyzed. Convergence of the discrete algorithm is proven as well as optimal error estimates.

Analysis of interfaces of variable stiffness

The effects of an interface of variable stiffness joining two elastic half-planes have been investigated under the hypothesis that the load is constituted by two equals and opposites concentrate forces applied at a certain distance from the interface. The integro-differential equation governing the problem has been determined by superposition principle and making use of the classical solution for concentrate force in an elastic plane. By applying the complex variable methods and the results of Muskhelishvili, the problem is reduced to that of two ordinary differential equations which have been easily integrated. The closed-form solution has been obtained for an arbitrary distribution of stiffness and without restrictions on the position of the loads. Successively, the specific cases of a constant and parabolic distribution of stiffness have been discussed in detail, and it has been shown how the general solution can be simplified in these examples. These cases deserve an interest in practical applications, the former because permits to compute the distribution of interface stress, the latter because allows to detect the effects of the lost of interface stiffness due, for example, to a damage or to a defect. The proposed solution can be used as a Green function to solve problems with arbitrary, but symmetric, distributions of loads.

Yield design of bonded joints

This paper deals with the problem of yield design of two bodies, the adherents, joined along their common surface by a thin layer of a third material, the adhesive. Since the computation of the solution of this problem is usually very difficult, a simplified model is developed, which also permits us to facilitate the qualitative analysis of the joint behaviour and the determination of its strength. The model is obtained by considering the limit case of adhesive of zero thickness, and the general properties of this limit problem are discussed in detail. The strength of the limit interface is governed by a limit convex which can be determined on the basis of the original convex of the adhesive. It is shown that the limit convex is susceptible to an outstanding geometrical interpretation, which largely simplifies its determination and the study of its properties. Yield criteria which are isotropic or do not depend on the mean stress are explicitly considered, and some practical examples of determination of the limit convex end the paper.

Numerical simulation of debonding of adhesively bonded joint

Abstract This paper describes a numerical method to simulate the debonding of adhesively bonded joints. Assuming that the adhesive thickness and the adhesive Young’s modulus are small with respect to the characteristic length of the joint and to the Young’s modulus of the adherents, a simplified model is derived in the case of large displacements using the asymptotic expansion technique. Then, the problem of the crack growth is stated, in the case of a stable growth, as the search of the local minima of the total energy of the joint, sum of the mechanical energy and the Griffith’s fracture energy. This is made using the Newton’s method. To this end, the expressions of the first and second derivatives of the mechanical energy with respect to a crack front displacement are derived analytically. Finally, numerical examples are presented, highlighting the unstable character of the crack growth at initiation.

Modeling of smart materials with thermal effects: Dynamic and quasi-static evolution

We present a mathematical model for linear magneto-electro-thermo-elastic continua, as sensors and actuators can be thought of, and prove the well-posedness of the dynamic and quasi-static problems. The two proofs are accomplished, respectively, by means of the Hille–Yosida theory and of the Faedo–Galerkin method. A validation of the quasi-static hypothesis is provided by a nondimensionalization of the dynamic problem equations. We also hint at the study of the convergence of the solution to the dynamic problem to that to the quasi-static problem as a small parameter — the ratio of the largest propagation speed for an elastic wave in the body to the speed of light — tends to zero.

Matched asymptotic expansion method for an homogenized interface model

Our aim is to demonstrate the effectiveness of the matched asymptotic expansion method in obtaining a simpli ed model for the influence of small identical heterogeneities periodically distributed on an internal surface on the overall response of a linearly elastic body. The results of some numerical experiments corroborate the precise identi cation of the di fferent steps, in particular of the outer/inner regions with their normalized coordinate systems and the scale separation, leading to the model.