Page 66 - Mechanics Analysis Composite Materials

Chapter 2. Fundamentals of mechanics of solids 51

Assume that under small variation of displacements and strains belonging to the
kinematically admissible fields surface tractions and body forces do not change.
Then, we can write the foregoing result in the following form

6K:-hA=O. (2.63)

Here

,6y,,)dV (2.64)

is the variation of the strain energy (internal potential energy of an elastic solid)
associated with small kinematically admissible variations of strains and

(2.65)

can be formally treated as work performed by surface tractions and body forces on
the actual displacements. Expressing stresses in Eq. (2.64) in terms of strains with
the aid of constitutive equations, Eq. (2.46), and integrating, we can determine w:
which is the body strain energy written in terms of strains. Quantity T = K; -A is
referred to as the total potential energy of the body. This name historically came
from problems in which external forces had a potential function F = -A so that
T = &;+ F was the sum of internal and external potentials, i.e. the total potential
function. Then, condition in Eq. (2.63) reduces to

6T=O (2.66)
which means that T has a stationary (actually, minimum) value under small
admissible variation of displacements in the vicinity of actual displacements. Thus,
we arrive at the following variational principle of minimum total potential energy:
the actual displacement field, in contrast to all kinematically admissible fields,
delivers the minimum value of the body total potential energy. This principle is a
variational form of the displacement formulation of the problem discussed in
Section 2.10. As can be shown, variational equations ensuring the minimum value of
the total potential energy of the body coincide with equilibrium equations written in
terms of displacements.