论文标题
不可压缩杆的理论：通过$γ$ -Convergence的严格推导
Theories for incompressible rods: a rigorous derivation via $Γ$-convergence
论文作者
论文摘要
我们使用变异收敛来得出一维杆理论的层次结构，从非线性弹性中的三维模型开始，但受局部体积预测的约束。由此产生的$γ$限制的密度取决于最小化问题，其痕量约束是由不可压缩性决定性条件的线性化产生的。虽然下限的证据取决于适当的约束正规化，但上限需要仔细，明确的局部保留恢复序列。在借助无差分扩展的帮助下，将横截面变量解耦后，我们应用一个内部扰动参数来强制执行所需的非凸层决定符约束。为了说明我们的发现，我们讨论了各向同性材料的特殊情况。
We use variational convergence to derive a hierarchy of one-dimensional rod theories, starting out from three-dimensional models in nonlinear elasticity subject to local volume-preservation. The densities of the resulting $Γ$-limits are determined by minimization problems with a trace constraint that arises from the linearization of the determinant condition of incompressibility. While the proofs of the lower bounds rely on suitable constraint regularization, the upper bounds require a careful, explicit construction of locally volume-preserving recovery sequences. After decoupling the cross-section variables with the help of divergence-free extensions, we apply an inner perturbation argument to enforce the desired non-convex determinant constraint. To illustrate our findings, we discuss the special case of isotropic materials.
