On the Motion of Curved Dislocations in Three Dimensions: Simplified Linearized Elasticity
On the Motion of Curved Dislocations in Three Dimensions: Simplified Linearized Elasticity
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DOI：
10.1137/20m1325654
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发表时间：
2020-03
期刊：
影响因子：
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通讯作者：
I. Fonseca;Janusz Ginster;Stephan Wojtowytsch
中科院分区：
文献类型：
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作者：
I. Fonseca;Janusz Ginster;Stephan Wojtowytsch
结果表明，在核-半径截断正则化简化弹性中（弹性能量取决于完整的位移梯度而不是对称的位移梯度），当截断半径收敛于零时，由弹性能量负梯度作用于位错曲线上的力渐近于曲线的平均曲率。在Holder空间中提供了严格的错误界限。作为应用，在H^1$型耗散给出运动规律时，建立了弹性能梯度流运动的位错收敛到圆弧泛函梯度流运动的位错，并在通常的L^2$耗散下收敛到协维2$的曲线缩短流。在第二种情况下，假设存在性和规律性，而$H^1$梯度流被完全一般地（短时间内）处理。这里发展的方法是一个蓝图，更物理设置的线性各向同性弹性。
It is shown that in core-radius cutoff regularized simplified elasticity (where the elastic energy depends quadratically on the full displacement gradient rather than its symmetrized version), the force on a dislocation curve by the negative gradient of the elastic energy asymptotically approaches the mean curvature of the curve as the cutoff radius converges to zero. Rigorous error bounds in Holder spaces are provided. As an application, convergence of dislocations moving by the gradient flow of the elastic energy to dislocations moving by the gradient flow of the arclength functional, when the motion law is given by an $H^1$-type dissipation, and convergence to curve shortening flow in co-dimension $2$ for the usual $L^2$-dissipation is established. In the second scenario, existence and regularity are assumed while the $H^1$-gradient flow is treated in full generality (for short time). The methods developed here are a blueprint for the more physical setting of linearized isotropic elasticity.