We consider a semi-linear integro-differential equation in dimension one associated to the half Laplacian whose solution represents the atom dislocation in a crystal. The equation comprises the evolutive version of the classical Peierls–Nabarro model. We show that for a large number of dislocations, the solution, properly rescaled, converges to the solution of a well known equation called by Head (1972) ”the equation of motion of the dislocation continuum”. The limit equation is a model for the macroscopic crystal plasticity with density of dislocations. In particular, we recover the so called Orowan’s law which states that dislocations move at a velocity proportional to the effective stress.

我们考虑一个半线性积分微分方程，该方程与半拉普拉斯算子有关，该半拉普拉斯算子的解表示晶体中的原子位错。该方程式包括经典Peierls–Nabarro模型的演化形式。我们表明，对于大量的位错，该解决方案经过适当地重新缩放，可以收敛到由Head（1972）称为“位错连续体的运动方程”的众所周知的方程式的解。极限方程是具有位错密度的宏观晶体可塑性的模型。特别是，我们恢复了所谓的Orowan定律，该定律指出，位错以与有效应力成比例的速度运动。