Page 58 - Mechanics Analysis Composite Materials

Chapter 2. Fundamenta1.y of mechanics of solids 43

The last transformation step was performed with due regard to Eqs. (2.5), (2.6), and
(2.37). Finally, Eq. (2.38) acquires the form

Because the right-hand side of this equation includes only internal variables, i.e.,
stresses and strains, we can conclude that the foregoing formal rearrangement
actually allowed us to transform the work of external forces into the work of
internal forces or into potential energy accumulated in the body. For further
derivation, let us introduce for the sake of brevity new notations for coordinates
and use subscripts I, 2, 3 instead of x,y, z, respectively. We also use the following
notations for stresses and strains

Then, Eq. (2.39) can be written as

(2.40)

where

dU = 0j.jdEij . (2.41)

This form of equation implies summation over repeated subscripts i,j = 1,2.3.
It should be emphasized that by now dU is just a symbol, which does not mean
that there exists function U and that dU is its differential. This meaning dU acquires
if we restrict ourselves to the consideration of an elastic material described in
Section 1.1. For this material, the difference between the body potential energy
corresponding to some initial state A and the energy corresponding to some other
state B does not depend on the way undertaken to transform the body from state A
to state B. In other words, the integral

ajidsi,i= U(B)- U(A)
.4
does not depend on the path of integration. This means that the element of
integration is a complete differential of function U depending on ~;j,i.e., that