Page 341 - Mechanics Analysis Composite Materials

326 Mechanics and analysis of composite materials

This first-order differential equation can be solved for E in the general case. Omitting
rather cumbersome transformations we arrive at the following solution:

This result corresponds to Eq. (7.33) of the hereditary theory with one-term
exponential approximation of the creep compliance in Eq. (7.36), where N = 1.
Taking more terms in Eq. (7.36) we get more flexibility in approximation of
experimental results with exponential functions. However, the main features of
material behavior are, in principal the same that for the one-term approximation
(see Figs. 7.16 and 7.17). Particularly, there exists the long-time modulus that
follows from Eq. (7.34) if we pass to the limit for t + 00, i.e.

o0 E
&(t)+-? El =
El 1 + J," C(8)d8 '

For the exponential approximation in Eq. (7.36),

I = IC(8)d8 = NA

n=l
0
Because integral Z has a finite value, the exponential approximation of the creep
compliance can be used only for materials with limited creep. There exist more
complicated singular approximations, e.g.

A A
C(8) = - C(8) = -ee-P"
8" ' 8"

for which Z + 00 and El = 0. This means that for such materials, creep strain can be
infinitely high.
Useful interpretation of the hereditary theory constitutive equations can be
constructed with the aid of the integral Laplace transformation according to which
function f(t) is associated with its Laplace transform f*(p)as
n
f* (p) = / f(t)ePP' dt
0
For some functions that we need to use for the examples presented below, we have

1 1
f(t)= 1, f*(p) = -, f(t) =e-", .f*(p) =- . (7.42)
P E+P