AI 中文总结
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。
AI 中文摘要
我们研究一类非线性弹性能量，其本构结构自然以拉伸变量表示，但未被基于变形梯度的标准多凸理论直接覆盖。该问题通过引入独立微旋转$\overline R \in {\rm SO}(3)$和相对拉伸$\overline U = \overline R ^T{\rm D}\varphi$得到解决。曲率变量$\overline R ^T\operatorname{Curl} \overline R$控制旋转场的完整一阶变分，并使极小化序列具有强紧性。恒等式$\operatorname{Cof}( \overline R^T{\rm D}\varphi) = \overline R^T\operatorname{Cof}{\rm D}\varphi$和$\det( \overline R^T{\rm D}\varphi)=\det{\rm D}\varphi$可通过变形梯度对应子式的弱连续性，识别提升后拉伸的子式。我们在两种情形下证明存在性：第一种情形允许对$( \overline U,\operatorname{Cof} \overline U,\det \overline U )$的凸依赖，并假设独立的余子式强制条件；第二种情形仅依赖$( \overline U,\det \overline U )$，且无需独立的余子式界。对于约束模型，弱闭条件$\overline R ^T{\rm D}\varphi\in\operatorname{Sym}^{+}(3)$结合$\det{\rm D}\varphi>0$，对每个容许对均意味着$\overline R =R:=\operatorname{polar}({\rm D}\varphi)$，因此提升后的问题等价于包含极因子曲率的变形问题，所得模型是选择性旋转正则化的偶应力型非线性弹性模型，而非纯一阶梯度的Biot模型。
英文摘要
We study a class of nonlinear elastic energies whose constitutive structure is naturally expressed in terms of a stretch variable but is not directly covered by the standard polyconvex theory formulated in the deformation gradient. The problem is lifted by introducing an independent microrotation $ \overline R \in {\rm SO}(3)$ and the relative stretch $ \overline U = \overline R ^T{\rm D}φ$. The curvature variable $ \overline R ^T\operatorname{Curl} \overline R $ controls the full first-order variation of the rotation field and yields strong compactness of minimizing sequences. The identities $\operatorname{Cof}( \overline R^T{\rm D}φ) = \overline R^T\operatorname{Cof}{\rm D}φ$ and $\det( \overline R^T{\rm D}φ)=\det{\rm D}φ$ then allow the minors of the lifted stretches to be identified through the weak continuity of the corresponding minors of the deformation gradients. We prove existence in two regimes. The first allows convex dependence on $( \overline U ,\operatorname{Cof} \overline U ,\det \overline U )$ and assumes separate cofactor coercivity. The second depends only on $( \overline U ,\det \overline U )$ and requires no independent cofactor bound. For the constrained model, the weakly closed condition $ \overline R ^T{\rm D}φ\in\operatorname{Sym}^{+}(3)$, together with $\det{\rm D}φ>0$, implies $ \overline R =R:=\operatorname{polar}({\rm D}φ)$ for every admissible pair. The lifted problem is therefore equivalent to a deformation problem containing the curvature of the polar factor. The resulting model is a selectively rotationally regularized nonlinear elastic model of couple-stress type, rather than a purely first-gradient Biot model.
