Twists and shear maps in nonlinear elasticity: explicit solutions and vanishing Jacobians

Abstract

In this paper we study constrained variational problems that are principally motivated by nonlinear elasticity theory. We examine in particular the relationship between the positivity of the Jacobian det ∇u and the uniqueness and regularity of energy minimizers u that are either twist maps or shear maps. We exhibit explicit twist maps, defined on two-dimensional annuli, that are stationary points of an appropriate energy functional and whose Jacobian vanishes on a set of positive measure in the annulus. Within the class of shear maps we precisely characterize the unique global energy minimizer u σ : Ω → R 2 in a model, two-dimensional case. The shear map minimizer has the properties that (i) det ∇u σ is strictly positive on one part of the domain Ω, (ii) det ∇u σ = 0 necessarily holds on the rest of Ω, and (iii) properties (i) and (ii) combine to ensure that ∇u σ is not continuous on the whole domain.

A continuously perturbed Dirichlet energy with area-preserving stationary points that ‘buckle’ and occur in equal-energy pairs

“…4) shows that the global minimizer is not the identity. It is worth pointing out that pure displacement variational problems with infinitely many equilibria were first proposed by F. John in [16] and later confirmed rigorously in [19], with further analysis in [6]. The domain in these cases was an annulus on which the identity boundary condition was imposed; solutions are distinguished by the integer number of times the outer boundary is rotated relative to the inner boundary.…”

“…2 The problems treated are of mixed displacement-traction or pure traction type. Non-uniqueness in the context of pure displacement problems is considered in [6,16,19], and we discuss this point further below in relation to the pure displacement problem dealt with in this paper. There is also a significant body of work on the question of uniqueness of equilibria in finite elasticity which includes but is not limited to [5,16,17,22,23,29,30].…”

“…and noting that W (x, 1; ) = + 1 , it follows, after an integration by parts, that 6) and the class C by…”

On the assembly of $C^1-$stationary points of a polyconvex functional and finite BOP-theory

“…We start our discussion by recalling the ELE as given in (6). Plugging u = re M R ∈ A M r and test functions of the form ϕ(x) = ge N R where g ∈ C ∞ c ((0, 1)), N ∈ N \ {0} into (6) yields the BVP (7). The first statement makes this rigorous and establishes first simple results of the radial part r.…”

“…The latter assembly of the counterexample is important since, we follow this part very closely. For more regularity results in infinite nonlinear elasticity see [7,10,22].…”