Travelling wave solutions for a quasilinear model of Field Dislocation Mechanics
Travelling wave solutions for a quasilinear model of Field Dislocation Mechanics
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DOI：
10.1016/j.jmps.2010.09.008
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发表时间：
2010-12
影响因子：
5.3
通讯作者：
A. Acharya;Karsten Matthies;J. Zimmer
中科院分区：
文献类型：
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作者：
A. Acharya;Karsten Matthies;J. Zimmer
我们考虑一个精确的减少场位错力学模型的标量问题在一个空间维度和调查存在的静态和缓慢，刚性移动的单一或集合的平面螺旋位错墙在这种设置。两类阻力系数函数被认为是，即那些与线性增长的原点附近和那些恒定或更一般的次线性增长。给出了这些螺旋壁微结构的所有可能平衡的数学表征。我们还证明了行波解的存在性线性阻力系数函数在低波速度和排除存在非常数有界行波解的次线性阻力系数函数。结果表明，在这种标量情况下，适当的解的概念是粘性解。在静态情况下的控制方程是不适当的，它表明，没有比较原理成立。研究结果表明，短期性质的应力场的个别位错墙，这表明，在模型中存在的非线性可能有稳定的效果。我们预测理想化的无位错细胞几乎任意大小的偶极位错壁微结构穿插作为我们的模型，与相应的金斯堡-朗道相场型梯度流模型的可能的非单调平衡的预测形成鲜明对比的特征的容许平衡。
We consider an exact reduction of a model of Field Dislocation Mechanics to a scalar problem in one spatial dimension and investigate the existence of static and slow, rigidly moving single or collections of planar screw dislocation walls in this setting. Two classes of drag coefficient functions are considered, namely those with linear growth near the origin and those with constant or more generally sublinear growth there. A mathematical characterisation of all possible equilibria of these screw wall microstructures is given. We also prove the existence of travelling wave solutions for linear drag coefficient functions at low wave speeds and rule out the existence of nonconstant bounded travelling wave solutions for sublinear drag coefficients functions. It turns out that the appropriate concept of a solution in this scalar case is that of a viscosity solution. The governing equation in the static case is not proper and it is shown that no comparison principle holds. The findings indicate a short-range nature of the stress field of the individual dislocation walls, which indicates that the nonlinearity present in the model may have a stabilising effect. We predict idealised dislocation-free cells of almost arbitrary size interspersed with dipolar dislocation wall microstructures as admissible equilibria of our model, a feature in sharp contrast with predictions of the possible non-monotone equilibria of the corresponding Ginzburg–Landau phase field type gradient flow model.