Page 207 - Introduction to Continuum Mechanics
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192 Stress
4.0 Boundary Condition for the Stress Tensor
If on the boundary of some body there are applied distributive forces, we call them surface
tractions. We wish to find the relation between the surface tractions and the stress field that
is defined within the body.
If we consider an infinitesimal tetrahedron cut from the boundary of a body with its inclined
face coinciding with the boundary surface (Fig. 4.9), then as in Section 4.1, we obtain
where n is the unit outward normal vector to the boundary, T is the stress tensor evaluated at
the boundary and t is the force vector per unit area on the boundary. Equation (4.9.1) is called
the stress boundary condition.
Fig. 4.9
Example 4.9.1
Given that the stress field in a thick wall elastic cylinder is
where A and B are constants.
(a) Verify that the given state of stress satisfies the equations of equilibrium in the absence of
body forces.
(b) Find the stress vector on a cylindrical surface r = a.
