Page 148 - Mechanics Analysis Composite Materials

Chapter 4. Mechanics of a composite layer 133

be identified by the condition do > 0. Being applied for unloading (i.e., for do < 0),
Eqs. (4.25) correspond to nonlinear elastic material with stress-strain diagram
shown in Fig. 1.2. For elastic-plastic material (see Fig. 1.5), unloading diagram
is linear. So, if we reduce the stresses by some increments AO.~,Abv, AT^?, the
corresponding increments of strains will be

Direct application of nonlinear equations (4.25) substantially hinders the problem
of stress-strain analysis because these equations include function o(0)in Eq. (4.32)
which, in turn, contains secant modulus E,(a). For the power approximation
corresponding to Eq. (4.33), E, can be expressed analytically, i.e.,

1
1
_---+cnon-= .
Es E
However, in many cases E, is given graphically as in Fig. 4.10 or numerically in the
form of a table. Thus, Eqs. (4.25) sometimes cannot be even written in the explicit
analytical form. This implies application of numerical methods in conjunction with
iterative linearization of Eqs. (4.25).
There exist several methods of such linearization that will be demonstrated using
the first equation in Eqs. (4.25), i.e.,

(4.34)

In the method of elastic solutions (Ilyushin, 1948), Eq. (4.34) is used in the following
form:

(4.35)

where s is the number of the iteration step and

For the first step (s = l), we take qo = 0 and solve the problem of linear elasticity
with Eq. (4.35) in the form

(4.36)

Finding the stresses, we calculate yl and write Eq. (4.35) as