Small amplitude sinusoidal fluctuation

An isotropic elastic solid with a nominally flat, traction-free surface is subjected to an initial equilibrium stress field. Suppose that the shape of the free surface S in the undeformed reference configuration of the material is not actually a plane, but that it is slightly wavy. The nominally flat surface coincides with the plane y = 0 and the position of the actual surface varies with respect to y = 0 in the x—direction. At time t, the position of the surface at coordinate x is given hy y = h x,t). For the discussion in this section, it is assumed that the slope of the surface is small everywhere, that is, h,x C 1 at all points on the surface. The boundary condition which must be enforced on the wavy surface is that the traction is zero. 

If Oij is the stress field evaluated at a point on the surface and nj is the outward unit normal vector there then CTijUj = 0 at that point. This condition must be enforced pointwise on the wavy surface, and this will be done for the case of small amplitude surface slope. A sketch of a small por- 

Prior to perturbation of surface shape, an equilibrium state of stress 

The total stress field for the perturbed shape of the surface is of the form 

Then (8.44) indicates that the boundary conditions on the additional stress field a x,y) due to perturbation of the shape of the free surface to y = h x, t) will be 

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