Page 168 - Analysis and Design of Energy Geostructures

Deformation in the context of energy geostructures 141

Figure 4.2 Typical variations of configurations.

phenomena describe deformations and involve a relative variation in the position of
the material points. The latter phenomenon describes a rigid-body motion because
there is no variation in the relative position of the material points. Deformations of
materials are called strains when they are interpreted on a unit basis.
Strains can be analysed in terms of geometrical configuration as well as in analytical
terms independently from the causes that produce them [for the full development,
see, e.g. Timoshenko and Goodier (1970)]. In general, they are related to the spatial
derivative of the displacement. Displacements are the fundamental kinematic variable
of most problems involved in mechanics (Carpinteri, 1995). In continuum mechanics,
this function is often considered with reference to the initial coordinates and time
through a so-called Lagrangian description. The same approach is considered here.

4.3.2 Strain displacement relations

Strains can be described through a finite or infinitesimal approach through the
strain displacement relations. Strains are defined infinitesimal if the components of
displacement and the gradient of displacement are quantities of the first order. The
above assumption involves considering displacements that do not to vary too abruptly
from point to point, so that considering only first-order quantitates is appropriate.
While simplifying the mathematical description of the problem, this assumption can
be widely accepted for problems involving energy geostructures. Strain displacement
relations ensure geometrical consistency for the material.
The strain of an infinitesimal material element can be expressed through the strain
tensor, ε ij , which is a symmetric second-order tensor (i and j take integral values 1, 2
and 3). The infinitesimal strain tensor is characterised by nine components that in rect-
angular Cartesian coordinates read