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Chapter 2. Fundamentals of’mechanics ofs0lid.s 41

If strains E,, E,, E, and yrc, yc. 7,- satisfy Eqs. (2.35), we can find rotation angles
Q,, w,. o, integrating Eqs. (2.34) and then determine displacements u,, u,, u,
integrating Eqs. (2.32).
Six compatibility equations, Eqs. (2.35), derived formally as compatibility
conditions for Eqs. (2.32) have a simple physical meaning. Assume that we have a
continuous solid shown in Fig. 2.1 and divide it into a set of pieces that perfectly
match each other. Now, apply some strains to each of these pieces. Obviously, for
arbitrary strains, the deformed pieces cannot be assembled into a continuous
deformed solid. This will happen only under the condition that the strains satisfy
Eqs. (2.35). However. even if the strains do not satisfy Eqs. (2.35), we can assume
that the solid is continuous but in a more general Riemannian (curved) space rather
than in traditional Euclidean space in which the solid existed before the deformation
(Vasiliev and Gurdal, 1999). Then, six quantities k and r in Eqs. (2.36), being
nonzero, specify curvatures of the Riemannian space caused by small strains E
and y. Compatibility equations, Eqs. (2.35), require these curvatures to be equal to
zero which means that the solid should remain in the Euclidean space under
deformation.

2.8. Admissible static and kinematic fields

In solid mechanics we introduce static field variables which are stresses and
kinematic field variables which are displacements and strains.
The static field is said to be statically admissible if the stresses satisfy equilibrium
equations, Eqs. (2.5), and are in equilibrium with surface tractions on the body
surface S, where these tractions are given (see Fig. 2.1), i.e., if Eqs. (2.2) are
satisfied on S,.
The kinematic field is referred to as kinematically admissible if displacements and
strains are linked by strain4isplacement equations, Eqs. (2.22), and displacements
satisfy kinematic boundary conditions on the surface S,, where displacements are
prescribed (see Fig. 2.1).
Actual stresses and displacements belong, naturally, to the corresponding
admissible fields though actual stresses must in addition provide admissible
displacements, while actual displacements should be associated with admissible
stresses. Mutual correspondence between static and kinematic variables is estab-
lished through the so-called constitutive equations that are considered in the next
section.

2.9. Constitutive equations for an elastic solid

Consider a solid loaded with body and surface forces as in Fig. 2.1. These
forces induce some stresses, displacements, and strains that compose the fields of
actual static and kinematic variables. Introduce some infinitesimal additional
displacements du,, du,,, and du, such that they belong to a kinematically