Page 64 - Mechanics Analysis Composite Materials

Chapter 2. Ftindanlentn1.v of mechanics of solids 49
integrated if strains satisfy six compatibility equations, Eqs. (2.35). We can write
these equations in terms of stresses using constitutive equations, Eq. (2.48). Thus,
the stress formulation of the problem is reduced to a set of nine equations consisting
of three equilibrium equations and six compatibility equations in terms of stresses.
At first glance it looks like this set is not consistent because it includes only six
unknown stresses. However, this is not the case because of special properties of
compatibility equations. As was noted in Section 2.7, these equations provide the
existence of Euclidean space inside the deformed body. But this space automatically
exists if strains can be expressed in terms of three continuous displacements as in
Eqs. (2.22). Indeed, substituting strains, Eqs. (2.22), into compatibility equations,
Eqs. (2.35), we can readily see that they are identically satisfied for any
three functions u,, uV, and u,. This means that the solution of six Eqs. (2.35)
including six strains is not unique. The uniqueness is ensured by three equilibrium
equations.

2.11. Variational principles

Equations of Solid Mechanics considered in the previous sections can be also
derived from variational principles that establish the energy criteria according to
which the actual state of the body under loading can be singled out of a system of
admissible states (see Section 2.8).
Consider a linear elastic solid and introduce two mutually independent fields of
variables: statically admissible stress field ut,01.. u;,T:), T:, t:_ and kinematically
admissible field characterized by displacements u:, u:I, uy and corresponding strains
/I
L, ,E,, E,, y,, ,y:=, yyc. To construct the energy criteria allowing us to distinguish the
I/
I/
I/
actual variables from admissible ones, consider the following integral similar to the
energy integral in Eqs. (2.51) and (2.52)
(2.58)

Here, in accordance with the definition of a kinematically admissible field (see
Section 2.8).

(2.59)

Substituting Eqs. (2.59) into Eq. (2.58) and using the following evident relation-
ships between the derivatives