On the relative energies of the Rivlin and cavitation instabilities for compressible materials

In 1948, Rivlin showed that if a cube of incompressible neo-Hookean material is subjected to a sufficiently large uniform normal dead-load on its boundary, then an asymmetric deformation is the minimizer of the energy in the class of homogeneous deformations. Ball showed in 1982 that, for classes of compressible elastic materials, if a ball of the material is subjected to a sufficiently large uniform radial dead-load, then a deformation forming a cavity is the minimizer of the energy in the class of radial deformations. In this paper we consider compressible hyperelastic materials and show that under such dead-loading, if a local minimizer of the radial energy forms a cavity, then there necessarily exists an asymmetric homogeneous deformation with less energy. Our approach extends and generalizes previous results of Abeyaratne and Hou for the incompressible case. </jats:p

On the Uniqueness of Energy Minimizers in Finite Elasticity

The uniqueness of absolute minimizers of the energy of a compressible, hyperelastic body subject to a variety of dead-load boundary conditions in two and three dimensions is herein considered. Hypotheses under which a given solution of the corresponding equilibrium equations is the unique absolute minimizer of the energy are obtained. The hypotheses involve uniform polyconvexity and pointwise bounds on derivatives of the stored-energy density when evaluated on the given equilibrium solution. In particular, an elementary proof of the uniqueness result of Fritz John (Commun. Pure Appl. Math. 25:617–634, 1972) is obtained for uniformly polyconvex stored-energy densities.</p

Diffeomorphic approximation of Sobolev homeomorphisms

Every homeomorphism h : X -> Y between planar open sets that belongs to the Sobolev class W^{1,p}(X,Y), 1<p<\infty, can be approximated in the Sobolev norm by diffeomorphisms.Comment: 21 pages, 5 figure

Whirl mappings on generalised annuli and the incompressible symmetric equilibria of the dirichlet energy

In this paper we show a striking contrast in the symmetries of equilibria and extremisers of the total elastic energy of a hyperelastic incompressible annulus subject to pure displacement boundary conditions.Indeed upon considering the equilibrium equations, here, the nonlinear second order elliptic system formulated for the deformation u=(u1,…,uN) : EL[u,X]=⎧⎩⎨⎪⎪Δu=div(P(x)cof∇u)det∇u=1u≡φinX,inX,on∂X, where X is a finite, open, symmetric N -annulus (with N≥2 ), P=P(x) is an unknown hydrostatic pressure field and φ is the identity mapping, we prove that, despite the inherent rotational symmetry in the system, when N=3 , the problem possesses no non-trivial symmetric equilibria whereas in sharp contrast, when N=2 , the problem possesses an infinite family of symmetric and topologically distinct equilibria. We extend and prove the counterparts of these results in higher dimensions by way of showing that a similar dichotomy persists between all odd vs. even dimensions N≥4 and discuss a number of closely related issues

Quasiconvexity at the boundary and the nucleation of austenite

Motivated by experimental observations of H. Seiner et al., we study the nucleation of austenite in a single crystal of a CuAlNi shape-memory alloy stabilized as a single variant of martensite. In the experiments the nucleation process was induced by localized heating and it was observed that, regardless of where the localized heating was applied, the nucleation points were always located at one of the corners of the sample - a rectangular parallelepiped in the austenite. Using a simplified nonlinear elasticity model, we propose an explanation for the location of the nucleation points by showing that the martensite is a local minimizer of the energy with respect to localized variations in the interior, on faces and edges of the sample, but not at some corners, where a localized microstructure, involving austenite and a simple laminate of martensite, can lower the energy. The result for the interior, faces and edges is established by showing that the free-energy function satisfies a set of quasiconvexity conditions at the stabilized variant in the interior, faces and edges, respectively, provided the specimen is suitably cut

Thrombotic risk in COVID-19: a case series and case-control study

Background: A possible association between COVID-19 infection and thrombosis, either as a direct consequence of the virus or as a complication of inflammation, is emerging in the literature. Data on the incidence of venous thromboembolism (VTE) is extremely limited. Methods: We describe 3 cases of thromboembolism refractory to heparin treatment, the incidence of VTE in an inpatient cohort, and a case-control study to identify risk factors associated with VTE. Results: We identified 274 confirmed (208) or probable (66) COVID-19 patients. 21 (7.7%) were diagnosed with VTE. D-dimer was elevated in both cases (confirmed VTE) and controls (no confirmed VTE) but higher levels were seen in confirmed VTE cases (4.1vs 1.2 µg/mL P <0.001). Conclusion: Incidence of VTE is high in patients hospitalised with COVID-19. Urgent clinical trials are needed to evaluate the role of anticoagulation in COVID-19. Monitoring of D-dimer and anti-factor Xa levels may be beneficial in guiding management