Page 173 - Analysis and Design of Energy Geostructures

146 Analysis and Design of Energy Geostructures

Figure 4.5 Typical force fields applied to a material: (A) continuum and (B) sectional
representation.

the vector b i , which act on volumes of the material; and contact forces, herein associ-
ated with the vectors t i and p i , which act on surfaces of the material (cf. Fig. 4.5A).
Forces acting on materials are called stresses when appropriately defined on a unit basis.
Body forces result from influences from the outside of the material, that is they
act on S from the exterior of C. These forces include, for example, mass forces.
Contact forces result from influences from the inside of the material in the case of the
functions t i , that is they act on the boundary @S of the portion S from the interior of
C, while they result from influences from the outside of the material in the case of
the functions p i , that is they act on the boundary @S of the portion S from the exte-
rior of C. These forces include, for example, forces caused by the application of a
uniform or nonuniform temperature variation to a material whose deformation is
partly or entirely restrained, as well as forces caused by the application of a nonuni-
form temperature variation [different than a temperature field varying linearly with a
set of rectangular Cartesian coordinates (Boley and Weiner, 1997)] to a material that
is free to deform.
The system of internal forces t i applied at a given point H on any cross-sectional
surface π i , which passes through H and describes a portion of a continuum material
subjected to a given force field, ensures equilibrium of the considered portion of mate-
rial. This equilibrium against the considered force field applied to the material would
not be satisfied if these forces were not present (cf. Fig. 4.5B). This system of forces is
associated with the specific surface that is chosen at any given point of the material
and, as this surface tends to zero, it constitutes the components of the so-called stress
vector t i (cf. Fig. 4.6).
The stress vector depends on the orientation of surface according to the following
linear mapping (Cauchy, 1823)