Page 150 - Mechanics Analysis Composite Materials

Chapter 4. Mechanics of a composite layer 135

In contrast to Eq. (4.39, stresses d, and v;.in the second term correspond to the
current step rather than to the previous one. This enables us to write Eq. (4.37) in
the form analogous to Hooke's law, i.e.,

(4.38)

where

(4.39)

are elastic variables corresponding to the step with number s - 1. The iteration
procedure is similar to that described above. For the first step we take EO= E and
vo = v in Eq. (4.38). Find e:, et.and 61,determine El, VI, switch to the second step
and so on. Graphical interpretation of the process is presented in Fig. 4.1Ib.
Convergence of this method is by an order higher than that of the method of elastic
solutions. However, elastic variables in the linear constitutive equation of the
method, Eq. (4.38), depend on stresses and hence, on coordinates whence the
method has got its name. This method can be efficiently applied in conjunction with
the finite element method according to which the structure is simulated with the
system of elements with constant stiffness coefficients. Being calculated for each step
with the aid of Eqs. (4.39), these stiffnesses will change only with transition from
one element to another, and it practically does not hinder the finite element method
calculation procedure.
The iteration process having the best convergence is provided by the classical
Newton's method requiring the following form of Eq. (4.34):

(4.40)
&;. = c-1 + c;;'(o-; - a;:') + qg(a;,- CT:') +c?;;yT& - ?;;I) ]
where

Because coefficients c are known from the previous step (s - I), Eq. (4.40) is linear
with respect to stresses and strains corresponding to step number s. Graphical
interpretation of this method is presented in Fig. 4.1 IC. In contrast to the methods
discussed above, Newton's method has no physical interpretation and being
characterized with very high convergence, is rather cumbersome for practical
applications.