In an attempt to simulate buckling of nonuniform coatings, we consider the problem of an embedded crack in a functionally graded coating bonded to a homogeneous substrate subjected to a compressive loading. The coating is graded in the thickness direction and the material gradient is orthogonal to the crack direction which is parallel with the free surface. The loading consists of a uniform compressive strain applied away from the crack region. The graded coating is modeled as a nonhomogeneous medium with an isotropic stress-strain law. Using a nonlinear continuum theory and a suitable perturbation technique, the plane strain problem is reduced to an eigenvalue problem describing the onset of buckling. Using integral transforms, the resulting plane elasticity equations are converted analytically into singular integral equations which are solved numerically to give the critical buckling strain and the corresponding crack opening displacement shapes. The main objective of the paper is to study the influence of
The space T poly (R d ) of all tensor fields on R d , equipped with the Schouten bracket is a Lie algebra. The subspace of ascending tensors is a Lie subalgebra of T poly (R d ).In this paper, we compute the cohomology of the adjoint representations of this algebra (in itself and T poly (R d )), when we restrict ourselves to cochains defined by aerial Kontsevitch's graphs like in our previous work (Pacific J of Math, vol 229, no 2, (2007) 257-292). As in the vectorial graphs case, the cohomology is freely generated by all the products of odd wheels.
Résumé.L'espace des tenseurs ascendants est une sous algèbre de Lie de l'algèbre de Lie (graduée) T poly (R d ) des champs de tenseurs sur R d muni du crochet de Schouten.Dans cet article, on calcule la cohomologie des représentations adjointes de cette sous algèbre de Lie, en se restreignant à des cochaînes définies par des graphes de Kontsevich aériens comme dans [AAC1] et [AAC2]. On retrouve un résultat analogue à celui de la cohomologie des graphes vectoriels et linéaires: elle est librement engendrée par des produits de roues de longueurs impaires.
Let T poly (R d ) denote the space of skew-symmetric polyvector fields on R d , turned into a graded Lie algebra by means of the Schouten bracket. Our aim is to explore the cohomology of this Lie algebra, with coefficients in the adjoint representation, arising from cochains defined by linear combination of aerial Kontsevich graphs. We prove that this cohomology is localized at the space of graphs without any isolated vertex, any "hand" or any "foot". As an application, we explicitly compute the cohomology of the "ascending graphs" quotient complex.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.