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Let Ω ⊂ R 3 be an open and bounded set with Lipschitz boundary and outward unit normal ν . For 1 < p < ∞ we establish an improved version of the generalized L p -Korn inequality for incompatible tensor fields P in the new Banach space W 0 1 , p , r (dev sym Curl ; Ω , R 3 × 3) = { P ∈ L p (Ω ; R 3 × 3) ∣ dev sym Curl P ∈ L r (Ω ; R 3 × 3) , dev sym (P × ν) = 0 on ∂ Ω } where r ∈ [ 1 , ∞) , 1 r ≤ 1 p + 1 3 , r > 1 if p = 3 2. Specifically, there exists a constant c = c (p , Ω , r) > 0 such that the inequality ‖ P ‖ L p (Ω , R 3 × 3) ≤ c ‖ sym P ‖ L p (Ω , R 3 × 3) + ‖ dev sym Curl P ‖ L r (Ω , R 3 × 3) holds for all tensor fields P ∈ W 0 1 , p , r (dev sym Curl ; Ω , R 3 × 3) . Here, dev X : = X - 1 3 tr (X) 1 denotes the deviatoric (trace-free) part of a 3 × 3 matrix X and the boundary condition is understood in a suitable weak sense. This estimate also holds true if the boundary condition is only satisfied on a relatively open, non-empty subset Γ ⊂ ∂ Ω . If no boundary conditions are imposed then the estimate holds after taking the quotient with the finite-dimensional space K S , d S C which is determined by the conditions sym P = 0 and dev sym Curl P = 0 . In that case one can replace ‖ dev sym Curl P ‖ L r (Ω , R 3 × 3) by ‖ dev sym Curl P ‖ W - 1 , p (Ω , R 3 × 3) . The new L p -estimate implies a classical Korn's inequality with weak boundary conditions by choosing P = D u and a deviatoric-symmetric generalization of Poincaré's inequality by choosing P = A ∈ so (3) . The proof relies on a representation of the third derivatives D 3 P in terms of D 2 dev sym Curl P combined with the Lions lemma and the Nečas estimate. We also discuss applications of the new inequality to the relaxed micromorphic model, to Cosserat models with the weakest form of the curvature energy, to gradient plasticity with plastic spin and to incompatible linear elasticity. [ABSTRACT FROM AUTHOR]

Geometric rigidity states that a gradient field which is $$L^p$$ -close to the set of proper rotations is necessarily $$L^p$$ -close to a fixed rotation, and is one key estimate in nonlinear elasticity. In several applications, as for example in the theory of plasticity, energy densities with mixed growth appear. We show here that geometric rigidity holds also in $$L^p+L^q$$ and in $$L^{p,q}$$ interpolation spaces. As a first step we prove the corresponding linear inequality, which generalizes Korn's inequality to these spaces. [ABSTRACT FROM AUTHOR]

When one slightly pushes a thin elastic sheet at its center into a hollow cylinder, the sheet forms (to a high degree of approximation) a developable cone, or d-cone for short. Here we investigate one particular aspect of d-cones, namely the scaling of the elastic energy with the sheet thickness $$h$$. Following recent work of Brandman et al. () we study the Dirichlet problem of finding the configuration of minimal elastic energy when the boundary values are given by an exact d-cone. We improve their result for the energy scaling. In particular, we show that the deviation from the logarithmic energy scaling is bounded by a constant times the double logarithm of the thickness. [ABSTRACT FROM AUTHOR]

Starting from three-dimensional elasticity we derive a rod theory for biphase materials with a prescribed dislocation at the interface. The stored energy density is assumed to be non-negative and to vanish on a set consisting of two copies of SO(3). First, we rigorously justify the assumption of dislocations at the interface. Then, we consider the typical scaling of multiphase materials and we perform an asymptotic study of the rescaled energy, as the diameter of the rod goes to zero, in the framework of Γ-convergence. [ABSTRACT FROM AUTHOR]

The asymptotic behaviour of solutions of three-dimensional nonlinear elastodynamics in a thin plate is studied, as the thickness h of the plate tends to zero. Under appropriate scalings of the applied force and of the initial values in terms of h, it is shown that three-dimensional solutions of the nonlinear elastodynamic equation converge to solutions of the time-dependent von Kármán plate equation. [ABSTRACT FROM AUTHOR]

We provide existence theorems in nonlinear elasticity for minimum problems modeling the deformations of a crystal with a given dislocation. A key technical difficulty is that due to the presence of a the dislocation the elastic deformation gradient cannot be in L 2. Thus one needs to consider elastic energies with slow growth, to which the original results of Ball cannot be applied directly. [ABSTRACT FROM AUTHOR]

A rigorous derivation is given of a rod theory for a multiphase material, starting from three-dimensional nonlinear elasticity. The stored energy density is supposed to be nonnegative and to vanish exactly on a set consisting of two copies of the group of rotations SO(3). The two potential wells correspond to the two crystalline configurations preferred by the material. We find the optimal scaling of the energy in terms of the diameter of the rod and we identify the limit, as the diameter goes to zero, in the sense of Γ-convergence. [ABSTRACT FROM AUTHOR]

Let Σ ⊂ R 3 be a smooth compact connected surface without boundary. Denote by A its second fundamental form and by Å the tensor A−(tr A/2)Id. In [4] we proved that, if ‖ Å‖ L 2 (Σ) is small, then Σ is W 2,2-close to a round sphere. In this note we show that, in addition, the metric of Σ is C 0–close to the standard metric of S 2. [ABSTRACT FROM AUTHOR]

We study properties of generalized convex hulls of the set $K={\rm SO}3\cup{\rm SO}3 H$ with ${\rm det}\, H>0$. If K contains no rank-1 connection we show that the quasiconvex hull of K is trivial if H belongs to a certain (large) neighbourhood of the identity. We also show that the polyconvex hull of K can be nontrivial if H is sufficiently far from the identity, while the (functional) rank-1 convex hull is always trivial. If the second well is replaced by a point then the polyconvex hull is trivial provided that there are no rank-1 connections. [ABSTRACT FROM AUTHOR]

A class of stored energy densities that includes functions of the form $W(\vec F) = a|\vec F|^p + g(\vec F,{\rm adj}\vec F) +h({\rm det} \vec F)$ with $a>0$, g and h convex and smooth, and $20$ such that, if a 3 by 3 matrix $\vec F_0$ satisfies $h'({\rm det} \vec F_0) |\vec F_0|^{3-p} \le k$, then W is $W^{1,p}$-quasiconvex at $\vec F_0$ on the restricted set of deformations $\vec u$ that satisfy condition (INV) and ${\rm det} \nabla\vec u >0$ a.e. (and hence that are one-to-one a.e.). Condition (INV) is (essentially) the requirement that $\vec u$ be monotone in the sense of Lebesgue and that holes created in one part of the material not be filled by material from other parts. The key ingredient in the proof is an isoperimetric estimate that bounds the integral of the difference of the Jacobians of $\vec F_0\vec x$ and $\vec u$ by the $L^p$-norm of the difference of their gradients. These results have application to the determination of lower bounds on critical cavitation loads in elastic solids. [ABSTRACT FROM AUTHOR]

Minimizers of functionals like $$\int_0^1 { \in ^2 u^2 _{xx} } + (u_x^2 - 1)^2 + u^2 dx$$ subject to periodic (or Dirichlet) boundary conditions are investigated. While for ε=0 the infimum is not attained it is shown that for sufficiently small ε > 0, all minimizers are periodic with period ∼ ε. Connections with solid-solid phase transformations are indicated. [ABSTRACT FROM AUTHOR]