Xx cos (x, ν) + Xy cos (y, ν) + Zx cos (z, ν) = Xν,
(2)
where ν denotes the direction of the outward-drawn normal to the bounding surface, and Xν denotes the x-component of the applied surface traction. The relations between stress-components and strain-components are expressed by either of the sets of equations (1) or (3) of § 26. The relations between strain-components and displacement are the equations (1) of § 11, or the equivalent conditions of compatibility expressed in equations (1) and (2) of § 16.
39. We may proceed by either of two methods. In one method we eliminate the stress-components and the strain-components and retain only the components of displacement. This method leads (with notation already used) to three partial differential equations of the type
| (λ + μ) | ∂ | ( | ∂u | + | ∂v | + | ∂w | ) + μ ( | ∂²u | + | ∂²u | + | ∂²u | ) + ρX = 0, |
| ∂x | ∂x | ∂y | ∂z | ∂x² | ∂y² | ∂z² |
(3)
and three boundary conditions of the type
| λ cos (x, ν) ( | ∂u | + | ∂v | + | ∂w | ) + μ { 2 cos (x, ν) | ∂u | + cos (y, ν) ( | ∂v | + | ∂u | ) |
| ∂x | ∂y | ∂z | ∂x | ∂x | ∂y |
| + cos (z, ν) ( | ∂u | + | ∂w | ) } = Xν. |
| ∂z | ∂x |
(4)