42 problems

Let an inclusion be embedded in an infinite isotropic elastic medium, and prescribe a uniform eigenstrain (or eigenstress) in the inclusion. The Eshelby uniformity conjecture. The…

Let ${\cal S}$ be an $m$-dimensional sheet smoothly and isometrically embedded in a $d$-dimensional Euclidean space, with $d<2m$, and let $p$ be any point of ${\cal S}$. A subsheet…

A deformation is a mapping describing the motion of a body, and its principal invariants are the scalar invariants associated with the eigenvalues of its deformation tensor. A defo…

Let $\Omega\subset\mathbb{R}^3$ be a compact domain, and let $\epsilon\in C^2(\mathcal{S}^2,\Omega)$ be an elastic strain field in an isotropic medium that satisfies equilibrium an…

Let $n\geq 3$ and let $P$ be a slowness polynomial corresponding to a positive definite stiffness tensor. A polynomial contains a square if it has a nonconstant factor with exponen…

Let a deformation have constant principal invariants of its deformation tensor. Ericksen's conjecture. Such a deformation must be homogeneous. Ericksen's conjecture concerns univer…

Pressure-independence conjecture.

Let $K$ be a real quadratic field with discriminant $ath?]$ and let $\mathcal{O}_f=\mathbb{Z}+f\mathcal{O}_K$ be the order of conductor $f$. Call $f$ split-free if it is not divisi…

Zero-background-modulus conjecture. The situation of a free-standing elastic sphere is simply found by setting the elastic modulus of the supporting background to zero.

Let $\Omega$ be an axially symmetric planar convex domain centered at the origin, and let the fail points be the points where the stress $|\nabla u|$ is maximal, with $u$ the torsi…

Let $\Omega$ be a convex planar domain symmetric about two coordinate axes, and call such a domain axially symmetric. The maximal stress is the maximum of $|\nabla u|$, where $u$ i…

Let $\Omega$ be a bounded convex domain, and let the load $f \in \mathcal{M}(\overline{\Omega};\mathbb{R})$ be finitely supported. An optimal string system consists of a pair…

Let $E(3)$ be the space of three-dimensional stiffness tensors, let $F$ map a stiffness tensor to its slowness polynomial, and write the target as $\mathbb{R}[p_1,p_2,p_3]$. The sl…

Let a three-dimensional stiffness tensor be orthorhombic, monoclinic, or triclinic according to its anisotropy symmetry, and let its slowness surface encode the propagation kinemat…

Chapman–Gotti–Gotti conjecture. For every non-integer rational $\alpha>1$, the set of elasticities of $\mathbb{N}_0[\alpha]$ is

The paper studies Piola transformations for non-standard finite elements in $H(\operatorname{div})$ and $H(\operatorname{div};\mathbb{S})$, including Arnold–Winther elements, and d…

Let $\Omega_0$ be a fixed open set containing the disclination point $o$, and consider the stress components $\sigma_{11}$, $\sigma_{22}$, and $\sigma_{12}$ as the ratio $E/D$ tend…

Let an inclusion be a bounded region in an elastic medium, and let a polynomial eigenstrain of degree $n$ be prescribed in it. The inclusion has the polynomial conservation propert…

Let $k$ denote the wave number and let $\omega(k)$ denote the corresponding wave frequency in Cosserat elasticity. For the material discussed in the source, the wave speed $v(k)$ s…

Let the constitutive coefficients of an isotropic Cosserat elastic half space satisfy … The secular equation has a solution, namely a Rayleigh wave speed. Uniqueness conjecture. Fo…

Let $\nu$, $\gamma$, and $\beta$ be the parameters of the static system for self-gravitating polytropic elastic balls, and let $\mathcal{O}$ denote the domain of parameter values f…

Consider an elastic particle moving towards a rigid wall in an incompressible Stokes fluid with no-slip boundary conditions. Let internal mechanical energy storage refer to energy…

A physically meaningful rebound is a rebound that persists in the vanishing-viscosity limit, for an elastic body moving towards a wall in an incompressible fluid with no-slip bound…

Let $h\to0$ be the film-thickness parameter, let $\alpha$ and $\gamma$ be the exponents specifying the prestrain regime, and let cases (i), (ii), and (iii) denote the corresponding…

Homeomorphism conjecture. Every minimizer $h_\circ\in\mathscr M^p(\overline{\mathbb{X}},\overline{\mathbb{Y}})$ is a homeomorphism.