We study the equilibrium of hyperelastic solids subjected to kinematic constraints on many small regions, which we call perforations. Such constraints on the displacement u are given in the quite general form u(x)∈Fx, where Fx is a closed set, and thus comprise confinement conditions, unilateral constraints, controlled displacement conditions, etc., both in the bulk and on the boundary of the body. The regions in which such conditions are active are assumed to be so small that they do not produce an overall rigid constraint, but still large enough so as to produce a non-trivial effect on the behaviour of the body. Mathematically, this is translated in an asymptotic analysis by introducing two small parameters: ɛ, describing the distance between the elements of the perforation, and δ, the size of the element of the perforation. We find the critical scale at which the effect of the perforation is non-trivial and express it in terms of a Γ-limit in which the constraints are relaxed so that, in their place, a penalization term appears in the form of an integral of a function φ(x,u). This function is determined by a blow-up procedure close to the perforation and depends on the shape of the perforation, the constraint Fx, and the asymptotic behaviour at infinity of the strain energy density σ. We give a concise proof of the mathematical result and numerical studies for some simple yet meaningful geometries.

中文翻译：

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在许多小区域上具有运动学约束的弹性体


我们研究了在许多小区域（我们称为穿孔）上受到运动学约束的超弹性固体的平衡。这种对位移 u 的约束以非常一般的形式给出 u（x）∈Fx，其中 Fx 是一个封闭集，因此包括本体和物体边界上的约束条件、单边约束、受控位移条件等。假设这些条件处于活动状态的区域非常小，以至于它们不会产生整体的刚性约束，但仍然足够大，以便对身体的行为产生非平凡的影响。在数学上，通过引入两个小参数将其转换为渐近分析： ɛ，描述射孔元件之间的距离，以及 δ，射孔元件的大小。我们发现穿孔效应非同小异的临界尺度，并用Γ极限来表示，其中约束被放宽，因此，在它们的位置上，惩罚项以函数 φ（x，u） 的积分形式出现。该函数由靠近射孔的放大程序确定，并取决于射孔的形状、约束 Fx 和应变能密度σ无穷远处的渐近行为。我们给出了数学结果的简明证明，并为一些简单而有意义的几何进行了数值研究。