Page 249 - Mechanics Analysis Composite Materials

234 Mechanics and anatysis of composite materials

where

Tact = u&,-&t, Tadm = u&, -Aadm
are the total energies of the actual state and some admissible kinematic state
expressed in terms of the strain energy, U, and work A performed by force V, on
displacement A (see Fig. 5.7). For both states

Aact =Aadm = V,A
and condition (5.23) reduces to

u& < u:dm . (5.24)

For the actual state, with due regard to Eq. (5.20) we get

1
1
Uact =-KY, =+Y, 2 . (5.25)
2
For the admissible state. we should use the following general equation:
I P I

and admit some approximation for yxz. The simplest one is yz = yx, so that

(5.26)
-e
Then, Eqs. (5.24H5.26) yield

S
Sssd /G=dz.
-e
Comparing this inequality with Eq. (5.21) we can conclude that this equation
specifies the upper bound for S55.
To determine the lower bound, we should apply the principle of minimum strain
energy according to which

Gc, < U& , (5.27)

where