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Deformation in the context of energy geostructures 145

Figure 4.4 Concept of compatibility. Modified after Davis, R.O., Selvadurai, A.P., 1996. Elasticity and
Geomechanics. Cambridge University Press.

the latter case, this implies that additional conditions, known as Saint Venant’s com-
patibility equations (Saint Venant, 1870), need to be defined. Compatibility equations
are six differential equations that are both necessary and sufficient conditions to guar-
antee the existence of a single-valued displacement field [for the full development, see,
e.g. Davis and Selvadurai (1996)]. In rectangular Cartesian coordinates they read
2 2 2
2 @ ε xy 5 @ ε xx 1 @ ε yy
2 2
@x@y @ y @ x
2 2 2
2 @ ε yz 5 @ ε yy 1 @ ε zz
2 2
@y@z @ z @ y
2 2 2
2 @ ε zx 5 @ ε zz 1 @ ε xx
2 2
@z@x @ x @ z
! ð4:9Þ
2 @
5 2 @ε yz 1 @ε zx 1 @ε xy
@ ε xx
@y@z @x @x @y @z
!
2 @
5 @ε yz 2 @ε zx 1 @ε xy
@ ε yy
@z@x @y @x @y @z
!
2 @
5 @ε yz 1 @ε zx 2 @ε xy
@ ε zz
@x@y @z @x @y @z

4.5 Stress
4.5.1 Concepts of strength and stress
The strength of materials is associated with the influence of forces and the interpretation
of the deformations caused by such forces. Two kinds of forces can be distinguished to
act on any portion S of a continuum material C: body forces, herein associated with