Chebakov R, Kaplunov J, Rogerson G A
School of Computing and Mathematics , Keele University , Keele ST5 5BG, UK.
Proc Math Phys Eng Sci. 2016 Feb;472(2186):20150800. doi: 10.1098/rspa.2015.0800.
The dynamic response of a homogeneous half-space, with a traction-free surface, is considered within the framework of non-local elasticity. The focus is on the dominant effect of the boundary layer on overall behaviour. A typical wavelength is assumed to considerably exceed the associated internal lengthscale. The leading-order long-wave approximation is shown to coincide formally with the 'local' problem for a half-space with a vertical inhomogeneity localized near the surface. Subsequent asymptotic analysis of the inhomogeneity results in an explicit correction to the classical boundary conditions on the surface. The order of the correction is greater than the order of the better-known correction to the governing differential equations. The refined boundary conditions enable us to evaluate the interior solution outside a narrow boundary layer localized near the surface. As an illustration, the effect of non-local elastic phenomena on the Rayleigh wave speed is investigated.
在非局部弹性框架内，考虑具有无牵引表面的均匀半空间的动态响应。重点是边界层对整体行为的主导作用。假设一个典型波长大大超过相关的内部长度尺度。结果表明，领先阶长波近似在形式上与具有垂直不均匀性且局限于表面附近的半空间的“局部”问题一致。对不均匀性的后续渐近分析导致对表面经典边界条件的显式修正。该修正的阶数大于对控制微分方程的更知名修正的阶数。改进后的边界条件使我们能够评估靠近表面的狭窄边界层之外的内部解。作为一个例子，研究了非局部弹性现象对瑞利波速度的影响。
