A nonlinear Korn inequality based on the Green-Saint Venant strain tensor

A nonlinear Korn inequality based on the Green-Saint Venant strain tensor is proved, whenever the displacement is in the Sobolev space $W^{1,p}$, $p\geq 2$, under Dirichlet conditions on a part of the boundary. The inequality can be useful in proving the coercivity of a nonlinear elastic energy

Nonlinear free fall of one-dimensional rigid bodies in hyperviscous fluids

We consider the free fall of slender rigid bodies in a viscous incompressible fluid. We show that the dimensional reduction (DR), performed by substituting the slender bodies with one-dimensional rigid objects, together with a hyperviscous regularization (HR) of the Navier--Stokes equation for the three-dimensional fluid lead to a well-posed fluid-structure interaction problem. In contrast to what can be achieved within a classical framework, the hyperviscous term permits a sound definition of the viscous force acting on the one-dimensional immersed body. Those results show that the DR/HR procedure can be effectively employed for the mathematical modeling of the free fall problem in the slender-body limit.Comment: arXiv admin note: substantial text overlap with arXiv:1305.070

TWO-SCALE HOMOGENIZATION FOR A MODEL IN STRAIN GRADIENT PLASTICITY

Abstract. Using the tool of two-scale convergence, we provide a rigorous mathematical setting for the homogenization result obtained by Fleck and Willis (J. Mech. Phys. Solids, 2004) concerning the effective plastic behaviour of a strain gradient composite material. Moreover, moving from deformation theory to flow theory, we prove a convergence result for the homogenization of quasistatic evolutions in the presence of isotropic linear hardening

Balance Laws and Weak Boundary Conditions in Continuum Mechanics

A weak formulation of the stress boundary conditions in Continuum Mechanics is proposed. This condition has the form of a balance law, allows also singular measure data and is consistent with the regular case. An application to the Flamant solution in linear elasticity is shown