arXiv subjects

Jonathan J. Bevan

Publications and source records attributed to Jonathan J. Bevan.

5 recordsLinked to original sources

arXiv2022

A uniqueness criterion and a counterexample to regularity in an incompressible variational problem

In this paper we consider the problem of minimizing functionals of the form $E(u)=\int_B f(x,\nabla u) \,dx$ in a suitably prepared class of incompressible, planar maps $u: B \rightarrow \mathbb{R}^2$. Here, $B$ is the unit disk and $f(x,ξ)$ is quadratic and convex in $ξ$. It is shown that if $u$ is a stationary point of $E$ in a sense that is made clear in the paper, then $u$ is a unique global minimizer of $E(u)$ provided the gradient of the corresponding pressure satisfies a suitable smallness condition. We apply this result to construct a non-autonomous, uniformly convex functional $f(x,ξ)$, depending smoothly on $ξ$ but discontinuously on $x$, whose unique global minimizer is the so-called $N-$covering map, which is Lipschitz but not $C^1$.

math.AP↗

arXiv2017

A calibration method for estimating critical cavitation loads from below in 3D nonlinear elasticity

In this paper we give an explicit sufficient condition for the affine map $u_λ(x):=λx$ to be the global energy minimizer of a general class of elastic stored-energy functionals $I(u)=\int_Ω W(\nabla u)\,dx$ in three space dimensions, where $W$ is a polyconvex function of $3 \times 3$ matrices. The function space setting is such that cavitating (i.e., discontinuous) deformations are admissible. In the language of the calculus of variations, the condition ensures the quasiconvexity of $I(\cdot)$ at $λ\mathbf{1}$, where $\mathbf{1}$ is the $3 \times 3$ identity matrix. Our approach relies on arguments involving null Lagrangians (in this case, affine combinations of the minors of $3 \times 3$ matrices), on the previous work Bevan & Zeppieri, 2015, and on a careful numerical treatment to make the calculation of certain constants tractable. We also derive a new condition, which seems to depend heavily on the smallest singular value $λ_1(\nabla u)$ of a competing deformation $u$, that is necessary for the inequality $I(u) < I(u_λ)$, and which, in particular, does not exclude the possibility of cavitation.

math.AP↗

arXiv2016

A condition for the Holder regularity of strong local minimizers of a nonlinear elastic energy in two dimensions

We prove the local Hölder continuity of strong local minimizers of the stored energy functional \[E(u)=\int_{\om}λ|\nabla u|^{2}+h(\det \nabla u) \,dx\] subject to a condition of `positive twist'. The latter turns out to be equivalent to requiring that $u$ maps circles to suitably star-shaped sets. The convex function $h(s)$ grows logarithmically as $s\to 0+$, linearly as $s \to +\infty$, and satisfies $h(s)=+\infty$ if $s \leq 0$. These properties encode a constitutive condition which ensures that material does not interpenetrate during a deformation and is one of the principal obstacles to proving the regularity of local or global minimizers. The main innovation is to prove that if a local minimizer has positive twist a.e\frenchspacing. on a ball then an Euler-Lagrange type inequality holds and a Caccioppoli inequality can be derived from it. The claimed Hölder continuity then follows by adapting some well-known elliptic regularity theory. We also demonstrate the regularizing effect that the term $\int_{\om} h(\det \nabla u)\,dx$ can have by analysing the regularity of local minimizers in the class of `shear maps'. In this setting a more easily verifiable condition than that of positive twist is imposed, with the result that local minimizers are Hölder continuous.

math.AP↗

arXiv2016

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

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 $\det \nabla u$ and the uniqueness and regularity of energy minimizers $u$ that are either twist maps or shear maps. We exhibit \emph{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 $u_σ: Ω\to \mathbb{R}^2$ in a model, two-dimensional case. The shear map minimizer has the properties that (i) $\det \nabla u_σ$ is strictly positive on one part of the domain $Ω$, (ii) $\det \nabla u_σ = 0$ necessarily holds on the rest of $Ω$, and (iii) properties (i) and (ii) combine to ensure that $\nabla u_σ$ is not continuous on the whole domain.

math.AP↗

arXiv2015

A simple sufficient condition for the quasiconvexity of elastic stored-energy functions in spaces which allow for cavitation

In this note we formulate a sufficient condition for the quasiconvexity at $x \mapsto λx$ of certain functionals $I(u)$ which model the stored-energy of elastic materials subject to a deformation $u$. The materials we consider may cavitate, and so we impose the well-known technical condition (INV), due to Müller and Spector, on admissible deformations. Deformations obey the condition $u(x)= λx$ whenever $x$ belongs to the boundary of the domain initially occupied by the material. In terms of the parameters of the models, our analysis provides an explicit upper bound on those $λ>0$ such that $I(u) \geq I(u_λ)$ for all admissible $u$, where $u_λ$ is the linear map $x \mapsto λx$ applied across the entire domain. This is the quasiconvexity condition referred to above.

math.CA↗