标题： 一种从变分断裂力学中的非线性模型推导出的薄脆梁的Griffith-Euler-Bernoulli理论 显示英文标题

标题： A Griffith-Euler-Bernoulli theory for thin brittle beams derived from nonlinear models in variational fracture mechanics

摘要： 我们研究一个平面薄脆梁，其受到弹性变形和裂纹的影响，这些裂纹通过一个非线性Griffith能量泛函来描述，该泛函作用于梁的$SBV$变形。 特别是我们考虑弹性体积贡献由于梁的有限弯曲与使梁完全断裂成几个大块所需的表面能相当时的情况。 在纵横比趋于零的极限情况下，我们严格推导出一个有效的Griffith-Euler-Bernoulli泛函，该泛函作用于表示梁中线的分段$W^{2,2}$正则曲线。 该泛函的弹性部分是薄梁在弯曲主导状态下经典的Euler-Bernoulli泛函，其以曲线的曲率表示。 此外，还出现了一个与曲线及其一阶导数的不连续点数量成比例的断裂项。 显示英文摘要

摘要： We study a planar thin brittle beam subject to elastic deformations and cracks described in terms of a nonlinear Griffith energy functional acting on $SBV$ deformations of the beam. In particular we consider the case in which elastic bulk contributions due to finite bending of the beam are comparable to the surface energy which is necessary to completely break the beam into several large pieces. In the limit of vanishing aspect ratio we rigorously derive an effective Griffith-Euler-Bernoulli functional which acts on piecewise $W^{2,2}$ regular curves representing the midline of the beam. The elastic part of this functional is the classical Euler-Bernoulli functional for thin beams in the bending dominated regime in terms of the curve's curvature. In addition there also emerges a fracture term proportional to the number of discontinuities of the curve and its first derivative.