Page 68 - Mechanics Analysis Composite Materials

Chapter 2. Fundamentals of mechanics qf solids 53

According to the initial principle, Eq. (2.66), STL = 0. Variation of displacements
yields, as earlier equilibrium equations, variation of stresses results in strain-
displacement equations, and variation of strains gives constitutive equations (V:
should be expressed in terms of strains).
The second form of the mixed variational principle can be derived from the
principle of minimum strain energy discussed in Section 2.11.2. Again expand the
class of admissible static fields and introduce stresses that satisfy force boundary
conditions but are not linked with equilibrium equations, Eqs. (2.5). Then we can
apply the principle of minimum strain energy if we construct an augmented
functional adding Eqs. (2.5) as additional constraints. Using displacements as
Lagrange’s multipliers we obtain

According to the original principle, Eq. (2.67), ~WL= 0. Variation with respect to
stresses (W, should be expressed in terms of stresses) yields constitutive equations in
which strains are expressed in terms of displacements via strain-displacement
equations, Eqs. (2.22), while variation of displacements gives equilibrium equations.
Equations and principles considered in this chapter will be used in the following
chapters of the book for analysis of composite materials.

2.12. References

Vasiliev. V.V. and Gurdal. 2. (1999).Optimal structural design. In Optimal Design (V.V. Vasiliev and
Z. Gurdal eds.). Technomic, Lancaster, pp. 1-29.