Page 154 - Mechanics Analysis Composite Materials

Chapter 4. Mechanics of a composite layer 139

Fig. 4.12. Loading path (OA) in the stress space.

where, stresses with superscript ‘0’ can depend on coordinates only. For such
loading, CT = cop, do = oodp, and Eqs. (4.46) and (4.49) yield

do(a) = BnonP3do = Bno~-2p”-3dp . (4.52)

Consider, for example, the first equation of Eqs. (4.47). Substituting Eqs. (4.51) and
(4.52) we get

This equation can be integrated with respect top. Using again Eqs. (4.5 1) we arrive
at the constitutive equation of the deformation theory

Thus, for a proportional loading, the flow theory reduces to the deformation theory
of plasticity. Unfortunately, before the problem is solved and the stresses are found
we do not know whether the loading is proportional or not and what particular
theory of plasticity should be used. There exists a theorem of proportional loading
(Ilyushin, 1948) according to which the stresses increase proportionally and the
deformation theory can be used if:
(1) external loads increase in proportion to one loading parameter,
(2) material is incompressible and its hardening can be described with the power
law CT = Se”.
In practice, both conditions of this theorem are rarely met. However, existing
experience shows that the second condition is not very important and that the
deformation theory of plasticity can be reliably (but approximately) applied if all
the loads acting on the structure increase in proportion to one parameter.