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 new duality approach to elasticity

International audienceThe 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

Nonlinear Donati compatibility conditions for the nonlinear Kirchhoff -von K arm an-Love plate theory

International audienceLinear Donati compatibility conditions guarantee that the components of symmetric tensor fields are those of linearized change of metric or linearized change of curvature tensor fields associated with the displacement vector field arising in a linearly elastic structure when it is subjected to applied forces. These compatibility conditions take the form of variational equations with divergence-free tensor fields as test-functions, by contrast with Saint-Venant compatibility conditions, which take the form of systems of partial differential equations. In this paper, we identify and justify nonlinear Donati compatibility conditions that apply to a nonlinearly elastic plate modeled by the Kirchhoff-von K'arm'an-Love theory

Nonlinear Donati compatibility conditions and the intrinsic approach for nonlinearly elastic plates

Linear Donati compatibility conditions guarantee that the components of symmetric tensor fields are those of linearized change of metric or linearized change of curvature tensor fields associated with the displacement vector field arising in a linearly elastic structure when it is subjected to applied forces. These compatibility conditions take the form of variational equations with divergence-free tensor fields as test-functions, by contrast with Saint-Venant compatibility conditions, which take the form of systems of partial differential equations. In this paper, we identify and justify nonlinear Donati compatibility conditions that apply to a nonlinearly elastic plate modeled by the Kirchhoff-von K'arm'an-Love theory. These conditions, which to the authors' best knowledge constitute a first example of nonlinear Donati compatibility conditions, in turn allow to recast the classical approach to this nonlinear plate theory, where the unknown is the position of the deformed middle surface of the plate, into the intrinsic approach, where the change of metric and change of curvature tensor fields of the deformed middle surface of the plate are the only unknowns. The intrinsic approach thus provides a direct way to compute the stress resultants and the stress couples inside the deformed plate, often the unknowns of major interest in computational mechanics

Weak vector and scalar potentials. Applications to Poincaré's theorem and Korn's inequality in Sobolev spaces with negative exponents.

19 pagesInternational audienceIn this paper, we present several results concerning vector potentials and scalar potentials with data in Sobolev spaces with negative exponents, in a not necessarily simply-connected, three-dimensional domain. We then apply these results to Poincaré's theorem and to Korn's inequality

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

Expression of Dirichlet boundary conditions in terms of the Cauchy–Green tensor field

International audienceIn a previous work, it was shown how the Cauchy–Green tensor field C := ∇Φ^T ∇Φ ∈ W^{2,s}(Ω; S^3_>), s > 3/2, canbe consideredasthesoleunknowninthehomogeneous Dirichlet problem of nonlinear elasticity posed over a domain Ω ⊂ R^3, instead of the deformation Φ ∈ W^{3,s}(Ω; R^3) in the usual approach. The purpose of this Note is to show that the same approach applies as well to the Dirichlet–Neumann problem. To this end, we show how the boundary condition Φ = Φ_0 on a portion Γ_0 of the boundary of Ω can be recast, again as boundary conditions on Γ_0, but this time expressed only in terms of the new unknown C ∈ W^{2,s}(Ω; S^3_>)