Page 63 - Mechanics Analysis Composite Materials

48 Mechanics and analysis of composite materials
Here 61 = T, 03 = -T and all the other stresses are equal to zero. The strain energy,
Eq. (2.51), can be presented now in the following form

1 l+v 7
u = -(a1q + QQ) = --- . (2.56)
2 E
Because Eqs. (2.55) and (2.56) specify one and the same quantity, we get
E
G=- (2.57)
2(1 -I-v)

Thus, an isotropic material is characterized within the linear elastic model by two
independent elastic constants - E and v.

2.10. Formulations of the problem
The problem of Solid Mechanics is reduced, as follows from the foregoing
derivation, to a set of 15 equations, i.e., three equilibrium equations, Eqs. (2.5),
six strain-displacement equations, Eqs. (2.22), and six constitutive equations,
Eq. (2.46) or (2.48). This set of equations is complete, Le., it contains 15 unknown
functions among which there are six stresses, six strains, and three displacements.
Solution of a particular problem should satisfy three boundary conditions that can
be written at any point of the body surface. Static or force boundary conditions
have the form of Eqs. (2.2), while kinematic or displacement boundary conditions
are imposed on three displacement functions.
There exist two classical formulations of the problem -displacement formulation
and stress formulation.
According to displacement formulation, we first determine displacements u,~,uY,
and u, from three equilibrium equations, Eqs. (2.5), written in terms of displace-
ments with the aid of constitutive equations, Eq. (2.46), and strain4isplacement
equations, Eqs. (2.22). Having found displacements we use Eqs. (2.22) and (2.46) to
determine strains and stresses.
Stress formulation is much less straightforward than the displacement one.
Indeed, we have only three equilibrium equations, Eqs. (2.5), for six stresses which
means that the problem of solid mechanics is not, in general, a statically determinate
problem. All possible solutions of the equilibrium equations (obviously, there is an
infinite number of them because the number of equations is less than the number
of unknown stresses) satisfying force boundary conditions (solutions that do not
satisfy them, obviously, do not belong to the problem under study) comprise the
class of statically admissible stress fields (see Section 2.8). Assume that we have one
of these stress fields. Now, we can readily find strains using constitutive equations,
Eq. (2.48), but to determine displacements, we need to integrate a set of six strain-
displacement equations, Eqs. (2.22) which having only three unknown displace-
ments are, in general, not compatible. As shown in Scction 2.7, this set can be