Page 67 - Mechanics Analysis Composite Materials

52 Mechanics and analysis of composite materials
2.11.2. Principle of minimum strain energy

This principle is valid for a linear elastic body and establishes the criterion
according to which the actual stress field can be singled out of all statically
admissible fields. Assume that displacementsand strains in Eq. (2.61) are actual, i.e.
u” = u, E” = E, yN = y, while stresses differ from the actual values by small statically
admissible variations, Le., O‘ = o +60, z’ = z +67. Substituting these expressions in
Eq. (2.61) and subtracting Eq. (2.62) for the actual state we get
SW, = 0 , (2.67)

where

is the variation of the strain energy associated with variation of stresses. Expressing
strains in terms of stresses with the aid of constitutive equations, Eq. (2.48), and
integrating, we can determine W,, which is the body strain energy written in terms of
stresses. As earlier, Eq. (2.67) indicates that strain energy, W,, has a stationary (in
fact, minimum) value under admissible variation of stresses. As a result we arrive at
the following variational principle of minimum strain energy: the actual stress field,
in contrast to all statically admissible fields, delivers the minimum value of the
body strain energy. This principle is a variational form of the stress formulation of
the problem considered in Section 2.10. As can be shown, variational equations
providing the minimum value of the strain energy are compatibility equations
written in terms of stresses. It is important that stress variation in Eq. (2.68) should
be performed within the statically admissible field, Le., within stresses that satisfy
equilibrium equations and force boundary conditions.

2.11.3. Mixed variational principles
Two variational principles described above imply variations with respect to
displacements only or to stresses only. There exist also the so-called mixed
variational principles in which variation is performed with respect to both kinematic
and static variables. The first principle from this group follows from the principle of
minimum total potential energy considered in Section 2.1 1.1. Let us expand the class
of admissible kinematic variables and introduce displacements that are continuous
functions satisfying displacement boundary conditions and strains that are not
linked with these displacements by strain-displacement equations, Eqs. (2.22).
Then we can apply the principle of minimum total potential energy performing a
conditional minimization of the total potential energy and introduce Eqs. (2.22)
as additional constraints imposed on strains and displacements with the aid of
Lagrange’s multipliers. Using stresses as these multipliers we can construct the
following augmented functional