Analytical Solutions to General Anti-Plane Shear Problems In Finite Elasticity

This paper presents a pure complementary energy variational method for solving anti-plane shear problem in finite elasticity. Based on the canonical duality-triality theory developed by the author, the nonlinear/nonconex partial differential equation for the large deformation problem is converted into an algebraic equation in dual space, which can, in principle, be solved to obtain a complete set of stress solutions. Therefore, a general analytical solution form of the deformation is obtained subjected to a compatibility condition. Applications are illustrated by examples with both convex and nonconvex stored strain energies governed by quadratic-exponential and power-law material models, respectively. Results show that the nonconvex variational problem could have multiple solutions at each material point, the complementary gap function and the triality theory can be used to identify both global and local extremal solutions, while the popular (poly-, quasi-, and rank-one) convexities provide only local minimal criteria, the Legendre-Hadamard condition does not guarantee uniqueness of solutions. This paper demonstrates again that the pure complementary energy principle and the triality theory play important roles in finite deformation theory and nonconvex analysis.Comment: 23 pages, 4 figures. Mathematics and Mechanics of Solids, 201

Frequency jumps in the planar vibrations of an elastic beam

The small amplitude transverse vibrations of an elastic beam clamped at both extremities are studied. The beam is modeled as an extensible, shearable planar Kirchhoff elastic rod under large displacements and rotations, and the vibration frequencies are computed both analytically and numerically as a function of the loading. Of particular interest is the variation of mode frequencies as the load is increased through the buckling threshold. While for some modes there is no qualitative changes in the mode frequencies, other modes experience rapid variations after the buckling threshold. For slender beams, these variations become stiffer, eventually resulting in a discontinuous jump of frequency at buckling, in the limit of inextensible, unshearable beams

Hyperbolic-parabolic singular perturbation for Kirchhoff equations with weak dissipation

We consider Kirchhoff equations with a small parameter epsilon in front of the second-order time-derivative, and a dissipative term whose coefficient may tend to 0 as t -> + infinity (weak dissipation). In this note we present some recent results concerning existence of global solutions, and their asymptotic behavior both as t -> + infinity and as epsilon -> 0. Since the limit equation is of parabolic type, this is usually referred to as a hyperbolic-parabolic singular perturbation problem. We show in particular that the equation exhibits hyperbolic or parabolic behavior depending on the values of the parameters.Comment: 20 pages, 2 tables, 1 figure, conference paper (7th ISAAC congress, London 2009

Theoretical tools for atom laser beam propagation

We present a theoretical model for the propagation of non self-interacting atom laser beams. We start from a general propagation integral equation, and we use the same approximations as in photon optics to derive tools to calculate the atom laser beam propagation. We discuss the approximations that allow to reduce the general equation whether to a Fresnel-Kirchhoff integral calculated by using the stationary phase method, or to the eikonal. Within the paraxial approximation, we also introduce the ABCD matrices formalism and the beam quality factor. As an example, we apply these tools to analyse the recent experiment by Riou et al. [Phys. Rev. Lett. 96, 070404 (2006)]

Morphoelastic rods Part 1: A single growing elastic rod

A theory for the dynamics and statics of growing elastic rods is presented. First, a single growing rod is considered and the formalism of three-dimensional multiplicative decomposition of morphoelasticity is used to describe the bulk growth of Kirchhoff elastic rods. Possible constitutive laws for growth are discussed and analysed. Second, a rod constrained or glued to a rigid substrate is considered, with the mismatch between the attachment site and the growing rod inducing stress. This stress can eventually lead to instability, bifurcation, and buckling

Convergence of the Crank-Nicolson-Galerkin finite element method for a class of nonlocal parabolic systems with moving boundaries

The aim of this paper is to establish the convergence and error bounds to the fully discrete solution for a class of nonlinear systems of reaction-diffusion nonlocal type with moving boundaries, using a linearized Crank-Nicolson-Galerkin finite element method with polynomial approximations of any degree. A coordinate transformation which fixes the boundaries is used. Some numerical tests to compare our Matlab code with some existing moving finite elements methods are investigated