Abstract

The paper considers a boundary-value problem of nonlinear elasticity for a mapping (deformation) in two weak formulations: in the form of a variational equilibrium equation and in the form of minimization of a multidimensional integral energy functional. From a mathematical point of view, both formulations refer to problems of functional analysis, in terms of which their mathematical correctness is discussed. Using the methods of variational calculus, with the example of two simple problems, it is proved that, for some nonlinear elastic models, in the corresponding boundary-value problems, there can be mappings with slip-type discontinuities and there can be a limit load, i.e., a value of external forces above which the boundary-value problem has no solution. Among such models there are elastic potentials with linear growth with respect to the mapping’s gradient, e.g., the well-known statistical Bartenev–Khazanovich model and the phenomenological Treloar model. The relation between these effects is discussed. It is also noted that the results obtained should be taken into account in the practical use of linear-growth elastic potentials. With the example of the problem of the axisymmetric torsion or tension of a circular cylinder, lower estimates for the limit load are constructed analytically using the methods of variational calculus and optimization theory. An analysis of the relations obtained shows that a polynomial growth of order p is characterized by polynomial hardening of order p – 1. With a linear increase in the specific strain energy with the strain gradient, a saturation effect is observed, which corresponds to the limit load. This behavior is typical of boundary-value problems of deformation plasticity, where a limit load at zero hardening also exists, i.e., of ideal elastoplasticity.

中文翻译：

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非线性弹性边值问题中的不连续映射与极限载荷

摘要

本文考虑了两个弱公式中映射（变形）的非线性弹性的边值问题：以变分平衡方程的形式和以多维积分能量泛函的最小化形式。从数学的角度来看，这两个公式都涉及泛函分析问题，并据此讨论了它们的数学正确性。运用变分学的方法，以两个简单问题为例，证明了对于某些非线性弹性模型，在相应的边值问题中，可以存在带滑移型不连续点的映射，并且可以存在极限载荷，即，边值问题无解的外力值。在这些模型中，存在相对于映射梯度呈线性增长的弹性势能，例如著名的统计 Bartenev–Khazanovich 模型和唯象 Treloar 模型。讨论了这些影响之间的关系。还应注意，在线性增长弹性势的实际应用中应考虑所获得的结果。以圆柱体的轴对称扭转或拉伸问题为例，运用变分法和最优化理论的方法，分析构造了极限载荷的下限估计值。对获得的关系的分析表明，阶数的多项式增长 还应注意，在线性增长弹性势的实际应用中应考虑所获得的结果。以圆柱体的轴对称扭转或拉伸问题为例，运用变分法和最优化理论的方法，分析构造了极限载荷的下限估计值。对获得的关系的分析表明，阶数的多项式增长 还应注意，在线性增长弹性势的实际应用中应考虑所获得的结果。以圆柱体的轴对称扭转或拉伸问题为例，运用变分法和最优化理论的方法，分析构造了极限载荷的下限估计值。对获得的关系的分析表明，阶数的多项式增长p以p – 1阶多项式硬化为特征。随着特定应变能随应变梯度线性增加，观察到饱和效应，这对应于极限载荷。这种行为是典型的变形塑性边界值问题，其中也存在零硬化极限载荷，即理想弹塑性。