2.6 Boundary Conditions
27
1
G
∂ 2 τ xy
∂x∂y
=
1
E
∂ 2 σ x
∂y 2 +
∂ 2 σ y
∂x 2 − ν
∂ 2 σ x
∂x 2
+
∂ 2 σ y
∂y 2 +
∂ 2 σ z
∂x 2 +
∂ 2 σ z
∂y 2
.
(2.11)
By circular permutation of indexes of coordinate axes, we obtain four more
formulas of a similar kind. This system of six Eqs. (2.10) and (2.11) is called
compatibility conditions.
2.6 Boundary Conditions
Let us separate an infinitely small triangle on the surface of a body and assume that
the element endures the action of only normal stresses σ (Fig. 2.2). Angles between
the vector σ and axes Ox, Oy, Oz are designated, respectively, by
nx,
ny,
nz.
Assume the separated element to be the basis of a three-face pyramid whose faces
are located in coordinate planes. The areas of these faces will be
S xy =
1
2
dxdy, S xz =
1
2
dxdz, S yz =
1
2
dydz.
(2.12)
Here, the values S xy , S xz , S yz are the areas of projections of the pyramid basis
onto coordinate planes. They can be expressed using the area S of the body surface
element:
S xy = S cos
nz, S yz = S cos
nx, S zx = S cos
ny.
(2.13)
By zeroing the sum of projections onto the axis Ox of all forces acting on the
separated pyramid and using formulas (2.13), we find
S(σ x cos
nx + τ xy cos
ny + τ xz cos
nz − σ cos
nx) = 0.
Fig. 2.2 Equilibrium of a
boundary element