Generalized von Kármán equations

AbstractIn a previous work, the first author has identified three-dimensional boundary conditions “of von Kármán's type” that lead, through a formal asymptotic analysis of the three-dimensional solution, to the classical von Kármán equations, when they are applied to the entire lateral face of a nonlinearly elastic plate.In this paper, we consider the more general situation where only a portion of the lateral face is subjected to boundary conditions of von Kármán's type, while the remaining portion is subjected to boundary conditions of free edge. We then show that the asymptotic analysis of the three-dimensional solution still leads in this case to a two-dimensional boundary value problem that is analogous to, but is more general than, the von Kármán equations. In particular, it is remarkable that the boundary conditions for the Airy function can still be determined solely from the data

A nonlinear Korn inequality on a surface

AbstractLet ω be a domain in R2 and let θ:ω¯→R3 be a smooth immersion. The main purpose of this paper is to establish a “nonlinear Korn inequality on the surface θ(ω¯)”, asserting that, under ad hoc assumptions, the H1(ω)-distance between the surface θ(ω¯) and a deformed surface is “controlled” by the L1(ω)-distance between their fundamental forms. Naturally, the H1(ω)-distance between the two surfaces is only measured up to proper isometries of R3.This inequality implies in particular the following interesting per se sequential continuity property for a sequence of surfaces. Let θk:ω→R3, k⩾1, be mappings with the following properties: They belong to the space H1(ω); the vector fields normal to the surfaces θk(ω), k⩾1, are well defined a.e. in ω and they also belong to the space H1(ω); the principal radii of curvature of the surfaces θk(ω), k⩾1, stay uniformly away from zero; and finally, the fundamental forms of the surfaces θk(ω) converge in L1(ω) toward the fundamental forms of the surface θ(ω¯) as k→∞. Then, up to proper isometries of R3, the surfaces θk(ω) converge in H1(ω) toward the surface θ(ω¯) as k→∞.Such results have potential applications to nonlinear shell theory, the surface θ(ω¯) being then the middle surface of the reference configuration of a nonlinearly elastic shell

New compatibility conditions for the fundamental theorem of surface theory

International audienceThe fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices (a_{αβ}) of order two and a field of symmetric matrices (b_{αβ}) of order two together satisfy the Gauss and Codazzi–Mainardi equations in a connected and simply-connected open subset ω of R^2, then there exists an immersion θ : ω → R^3 such that these fields are the first and second fundamental forms of the surface θ(ω) and this surface is unique up to proper isometries in R^3. In this Note, we identify new compatibility conditions, expressed again in terms of the functions a_{αβ} and b_{αβ}, that likewise lead to a similar existence and uniqueness theorem. These conditions take the form ∂_1A_2−∂_2A_1+A_1A_2−A_2A_1=0 in ω,where A_1 and A_2 are antisymmetric matrix fields of order three that are functions of the fields (a_{αβ}) and (b_{αβ}), the field (a_{αβ}) appearing in particular through its square root. The unknown immersion θ : ω → R^3 is found in the present approach in function spaces ‘with little regularity’, viz., W^{2,p}_loc(ω;R^3), p > 2

A generalization of the classical Cesàro–Volterra path integral formula

International audienceIf a symmetric matrix field e of order three satisfies the Saint Venant compatibility conditions in a simply-connected domain Ω in R^3, there then exists a displacement field u of Ω such that e = (∇u^T + ∇u)/2 in Ω. If the field e is sufficiently smooth, the displacement u(x) at any point x ∈ Ω can be explicitly computed as a function of e and CURL e by means of a Cesàro–Volterra path integral formula inside Ω with endpoint x. We assume here that the components of the field e are only in L^2(Ω), in which case the classical path integral formula of Cesàro and Volterra becomes meaningless. We then establish the existence of a “Cesàro–Volterra formula with little regularity”, which again provides an explicit solution u to the equation e = (∇u^T + ∇u)/2 in this case

Continuity in H1 {H}^{1}-norms of surfaces in terms of the L1 {L}^{1}-norms of their fundamental forms

International audienceThe main purpose of this Note is to show how a ‘nonlinear Korn’s inequality on a surface’ can be established. This inequality implies in particular the following interesting per se sequential continuity property for a sequence of surfaces. Let ω be a domain in R2, let θ :closure(ω) → R^3 be a smooth immersion, and let θ_k :closure(ω) → R^3, k=1,2,..., be mappings with the following properties: They belong to the space H^1(ω); the vector fields normal to the surfaces θ_k(ω), k=1,2,..., are well defined a.e. in ω and they also belong to the space H^1(ω); the principal radii of curvature of the surfaces θ_k(ω) stay uniformly away from zero; and finally, the three fundamental forms of the surfaces θ_k(ω) converge in L^1(ω) toward the three fundamental forms of the surface θ(ω) as k → ∞. Then, up to proper isometries of R^3, the surfaces θ_k(ω) converge in H^1(ω) toward the surface θ(ω) as k → ∞

Recovery of a displacement field on a surface from its linearized change of metric and change of curvature tensors

International audienceWe establish that the components of the linearized change of metric and change of curvature tensors associated with a displacement field of a surface in R^3 must satisfy compatibility conditions, which are the analogues ‘on a surface’ of the Saint Venant equations in three-dimensional elasticity. We next show that, conversely, if two symmetric matrix fields of order two satisfy these compatibility conditions over a simply-connected surface S ⊂ R^3, then they are the linearized change of metric and change of curvature tensors associated with a displacement field of the surface S

New formulations of linearized elasticity problems, based on extensions of Donati's theorem

The classical Donati theorem is used for characterizing smooth matrix fields as linearized strain tensor fields. In this Note, we give several generalizations of this theorem, notably to matrix fields whose components are only in H-1. We then show that our extensions of Donati's theorem allow to reformulate in a novel fashion linearized three-dimensional elasticity problems as quadratic minimization problems with the strains as the primary unknowns