Gradient polyconvex material models and their numerical treatment

$\ au_N$-构型与多凸梯度流

The $τ_N$-configurations and polyconvex gradient flows

本文推广$T_N$-构型为$\ au_N$-构型，通过凸积分构造多凸函数支持满足开性条件的构型族，证明多凸梯度流初边值问题存在非$C^1$的Lipschitz弱解，并在维数≥4时简化验证。

Minimizers of Nonlocal Polyconvex Energies in Nonlocal Hyperelasticity

We develop a theory of existence of minimizers of energy functionals in vectorial problems based on a nonlocal gradient under Dirichlet boundary conditions. The model shares many features with the peridynamics model and is also applicable to nonlocal solid mechanics, especially nonlinear elasticity. This nonlocal gradient was introduced in an earlier work, inspired by Riesz' fractional gradient, but suitable for bounded domains. The main assumption on the integrand of the energy is polyconvexity. Thus, we adapt the corresponding results of the classical case to this nonlocal context, notably, …

不规则扩散与聚凸梯度流中的正则性丧失

Irregular Diffusions and Loss of Regularity in Polyconvex Gradient Flows

本文通过构造新的几何结构，揭示了在聚凸梯度流中，即使初始数据光滑，扩散方程也可能产生无处光滑的弱解，从而导致正则性失效。

$Γ$-convergence of polyconvex functionals involving s-fractional gradients to their local counterparts

In this paper we study localization properties of the Riesz $s$-fractional gradient $D^s u$ of a vectorial function $u$ as $s \nearrow 1$. The natural space to work with $s$-fractional gradients is the Bessel space $H^{s,p}$ for $0 < s < 1$ and $1 < p < \infty$. This space converges, in a precise sense, to the Sobolev space $W^{1,p}$ when $s \nearrow 1$. We prove that the $s$-fractional gradient $D^s u$ of a function $u$ in $W^{1,p}$ converges strongly to the classical gradient $Du$. We also show a weak compactness result in $W^{1,p}$ for sequences of functions $u_s$ with bounded $L^p$ norm of…

A polyconvex extension of the logarithmic Hencky strain energy

Adapting a method introduced by Ball, Muite, Schryvers and Tirry, we construct a polyconvex isotropic energy function $W\colon\operatorname{GL^+}(n)\to\mathbb{R}$ which is equal to the classical Hencky strain energy \[ W_{\mathrm{H}}(F) = μ\,\lVert\operatorname{dev}_n\log U\rVert^2+\fracκ{2}\,[\operatorname{tr}(\log U)]^2 = μ\,\lVert\log U\rVert^2+\fracΛ{2}\,[\operatorname{tr}(\log U)]^2 \] in a neighborhood of the identity matrix; here, $\operatorname{GL^+}(n)$ denotes the set of $n\times n$-matrices with positive determinant, $F\in\operatorname{GL^+}(n)$ denotes the deformation gradient, $U=…

A sharp interface evolutionary model for shape memory alloys

We show the existence of an energetic solution to a quasistatic evolutionary model of shape memory alloys. Elastic behavior of each material phase/variant is described by polyconvex energy density. Additionally, to every phase boundary, there is an interface-polyconvex energy assigned, introduced by M. Šilhavý. The model considers internal variables describing the evolving spatial arrangement of the material phases and a deformation mapping with its first-order gradients. It allows for injectivity and orientation-preservation of deformations. Moreover, the resulting material microstructures ha…

On the existence of minimisers for strain-gradient single-crystal plasticity

We prove the existence of minimisers for a family of models related to the single-slip-to-single-plane relaxation of single-crystal, strain-gradient elastoplasticity with $L^p$-hardening penalty. In these relaxed models, where only one slip-plane normal can be activated at each material point, the main challenge is to show that the energy of geometrically necessary dislocations is lower-semicontinuous along bounded-energy sequences which satisfy the single-plane condition, meaning precisely that this side condition should be preserved in the weak $L^p$-limit. This is done with the aid of an 'e…

由离散位错流驱动的率无关大应变弹塑性演化的能量解

Energetic solutions to rate-independent large-strain elasto-plastic evolutions driven by discrete dislocation flow

本文针对由离散位错驱动的单晶大应变弹塑性演化模型，构造了率无关流动法则下的能量解，并借助滑移轨迹、非线性Gronwall估计和时间重标定证明了存在性。