FS-FS-RThe-Plane from Elastic displacements and viscous hydrodynamic flows in wedge-shaped geometries with a straight edge: Green’s functions for parallel forces

For homogeneous and isotropic linearly elastic solids and for incompressible fluids under low-Reynolds-number conditions, the fundamental solutions of the associated continuum equations were derived a long time ago for bulk systems. That is, the corresponding Green’s functions are available in infinitely extended systems, where boundaries do not play any role. However, introducing boundaries renders the situation significantly more complex. Here, we derive the corresponding Green’s functions for a linearly elastic homogeneous and isotropic material in a wedge-shaped geometry. Two flat boundaries confine the material and meet at a straight edge. No-slip (NOS) and free-slip (FRS) conditions are considered. The force is oriented in a direction parallel to the straight edge of the wedge. Assuming incompressibility, our expressions also apply to the situation of low-Reynolds-number hydrodynamic viscous fluid flows. Thus, they may be used, for instance, to describe the motion of self-propelled objects guided by an edge or the distortion of soft elastic actuators in wedge-shaped environments of operation.

针对均质各向同性线弹性固体，以及低雷诺数条件下的不可压缩流体，相关连续介质方程的基本解早在很久以前就已针对体相系统完成推导。换言之，在无边界的无限大系统中，对应的格林函数（Green’s functions）已被完全确定，此时边界不会产生任何影响。然而，引入边界后，问题的复杂度会显著提升。本文针对楔形几何构型下的均质各向同性线弹性材料，推导了其对应的格林函数。该材料由两个平直边界约束，二者交汇于一条直棱。本文考虑了无滑移（No-slip, NOS）与自由滑移（Free-slip, FRS）边界条件，且作用力的方向平行于该楔形结构的直棱。在假设不可压缩的前提下，本文所得表达式同样适用于低雷诺数的流体动力粘性流场景。因此，该表达式可用于诸多实际场景，例如描述受直棱引导的自驱动物体的运动，或是楔形工作环境中柔性弹性驱动器的形变。