Mathematical analysis of nonlinear elasticity for elucidation of fracture phenomena

考虑到断裂现象的应用，我们研究了下列非线性问题的数学分析，并得到了结果。首先，在具有裂纹且应力应变关系服从幂律的非线性弹性力学中，我们证明了边值问题弱解的唯一可解性。其次，我们证明了在非线性条件（库仑摩擦定律和非穿透条件）下，两个粘结的不同的线性弹性介质之间的界面裂纹模型的边值问题的解的可解性。进一步研究了裂纹尖端附近解的行为。

With application to fracture phenomena in mind, we studied mathematical analyses for the following nonlinear problems and obtained the results. First, in nonlinear elasticity which has a crack and the stress-strain relation is governed by power law, we showed a unique solvability of the weak solution for the boundary value problem. Second, we proved a solvability of the solution of a boundary value problem as a model of interfacial crack under the nonlinear condition (Coulomb friction law and non-penetration condition) between two bonded dissimilar linearized elastic media. Further, we investigated the behavior of the solution near the crack tip.