Page 65 - Mechanics Analysis Composite Materials

50 Mechanics and analysis of composite materials
we arrive at

Applying now Green’s integral transformation, Eq. (2.4), to the first three terms
under the integral and taking into account that statically admissible stresses should
satisfy equilibrium equations, Eqs. (2.5), (2.6), and force boundary conditions,
Eqs. (2.2), we obtain from Eqs. (2.58) and (2.60)

For actual stresses, strains and displacements, Eq. (2.61) reduces to the following
equation

(2.62)

known as Clapeyron’s theorem.

2.11.1. Principle of minimum total potential energy

This principle allows us to distinguish the actual displacement field of the body
from kinematically admissible fields. To derive it, assume that the stresses in
Eq. (2.61) are actual stresses, i.e., c’= 6, z’ = z, while the displacements and
the corresponding strains differ from the actual values by small kinematically
admissible variations, Le., u” = u + 6u, E” = E + 88, y” = y + 6y. Substituting these
expressions into Eq. (2.61) and subtracting Eq. (2.62) from the resulting equation
we arrive at