Page 145 - Mechanics Analysis Composite Materials

130 Mechanics and analysis of composite materials

characteristics associated with the change of material shape under which it
demonstrates the plastic behavior.
In the theory of plasticity, plastic potential Up is assumed to be a function of
stress intensity a, and according to Eqs. (4.18), plastic strains are

(4.23)

Consider further a plane stress state with stresses a,, a),,and zxv in Fig. 4.5. For this
case, Eq. (4.19) acquires the form

(4.24)

Using Eqs, (4.15H4.17) and (4.23), (4.24) we finally arrive at the following
constitutive equations:

(4.25)

where

1 dU,
o(a) =-- . (4.26)
a da
To find ~(a),we need to specify dependence of U, on a. The most simple and
suitable for practical applications is the power approximation

u,=ca”, (4.27)

where C and n are some experimental constants. As a result, Eq. (4.26) yields

To determine coefficients C and n we introduce the basic assumption of the plasticity
theory concerning the existence of the universal stress-strain diagram (master
curve). According to this assumption, for any particular material there exists the
dependence between stress and strain intensities, i.e., a = (P(E) (or E =f(a)), that is
one and the same for all the loading cases. This fact enables us to find coefficients C
and n from the test under uniaxial tension and extend thus obtained results to an
arbitrary state of stress.