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1.16 Boundary Conditions in Terms of Surface Forces

We now consider the relationship between the stress components and the given surface forces acting on the boundary of a body. The equations of equilibrium that must be satisfied within a body are derived in Section 1.8. The distribution of stress in a body must also be such as to accommodate the conditions of equilibrium with respect to externally applied forces. The external forces may thus be regarded as a continuation of the internal stress distribution.

Consider the equilibrium of the forces acting on the tetrahedron shown in Fig. 1.19b, and assume that oblique face ABC is coincident with the surface of the body. The components of the stress resultant p are thus now the surface forces per unit area, or surface tractions, px , py , and pz . The equations of equilibrium for this element, representing boundary conditions, are, from Eqs. (1.26),

Equation 1.48


For example, if the boundary is a plane with an x-directed surface normal, Eqs. (1.48) give px = s x , py = t xy , and pz = t xz ; under these circumstances, the applied surface force components px , py , and pz are balanced by s x , t xy , and t xz , respectively.

It is of interest to note that, instead of prescribing the distribution of surface forces on the boundary, the boundary conditions of a body may also be given in terms of displacement components. Furthermore, we may be given boundary conditions that prescribe surface forces on one part of the boundary and displacements on another. When displacement boundary conditions are given, the equations of equilibrium express the situation in terms of strain, through the use of Hooke's law and subsequently in terms of the displacements by means of strain–displacement relations (Sec. 2.3). It is usual in engineering problems, however, to specify the boundary conditions in terms of surface forces, as in Eq. (1.48), rather than surface displacements. This practice is adhered to in this text.

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