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 thatarises 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.

中文翻译：

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不可压缩杆的理论：通过Γ收敛的严格推导

我们使用变分收敛来导出一维杆理论的层次结构，它是从非线性弹性中的三维模型开始的，该三维模型受局部体积保留的影响。最终Γ极限的密度由具有跟踪约束的最小化问题确定，该跟踪约束源自不可压缩行列式条件的线性化。虽然下限的证明依赖于适当的约束正则化，但上限需要仔细，明确地构造本地保留体积的恢复序列。在借助无散度扩展解耦横截面变量之后，我们应用内部微扰参数来强制执行所需的非凸行列式约束。为了说明我们的发现，我们讨论各向同性材料的特殊情况。