Elastoplasticity of gradient-polyconvex materials

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。

Gradient polyconvex material models and their numerical treatment

Gradient polyconvex materials are nonsimple materials where we do not assume smoothness of the elastic strain but instead regularity of minors of the strain is required. This allows for a larger class of admissible deformations than in the case of second-grade materials. We describe a possible implementation of gradient polyconvex elastic energies. Besides, a new geometric interpretation of gradient-polyconvexity is given and it is compared with standard second-grade materials. Finally, we demonstrate application of the proposed approach using two different models, namely, a St.-Venant Kirchho…

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, …

$Γ$-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…

Cosserat非线性弹性与非线性偶应力理论的多凸性

Polyconvexity for Cosserat nonlinear elasticity and nonlinear couple-stress theory

该研究针对未被标准多凸理论覆盖的非线性弹性能量，引入微旋转与相对拉伸变量，在两种情形下证明解的存在性，得到含极因子曲率的偶应力型非线性弹性模型。

Relaxation of nonlinear elastic energies involving deformed configuration and applications to nematic elastomers

We start from a variational model for nematic elastomers that involves two energies: mechanical and nematic. The first one consists of a nonlinear elastic energy which is influenced by the orientation of the molecules of the nematic elastomer. The nematic energy is an Oseen--Frank energy in the deformed configuration. The constraint of the positivity of the determinant of the deformation gradient is imposed. The functionals are not assumed to have the usual polyconvexity or quasiconvexity assumptions to be lower semicontinuous. We instead compute its relaxation, that is, the lower semicontinuo…

率无关弹塑性弹簧网络模型与非局部随机场的均匀化

Homogenization of rate-independent elastoplastic spring network models with non-local random fields

该研究建立了含随机长程相互作用的弹塑性弹簧网络的离散到连续极限，证明演化解收敛于耦合分数阶非局部能量的连续模型，为纤维增强复合材料动态模拟奠定数学基础。

大应变磁弹性力学中的存在性结果

Existence results in large-strain magnetoelasticity

本文研究大应变磁弹性力学静态与准静态变分问题，采用混合欧拉-拉格朗日表述，证明静态极小化元存在性及准静态增量问题可解性，并通过梯度多凸性正则化建立能量解的存在性。

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。