Page 143 - Mechanics Analysis Composite Materials

128 Mechanics and analysis of composite materials

Then, Eqs. (4.12) yield the following cubic constitutive law:

The corresponding approximation is shown in Fig. 4.6 with a solid line. Retaining
more higher-order terms in Eq. (4.1l), we can describe nonlinear behavior of any
isotropic polymeric material.
To describe nonlinear elastic-plastic behavior of metal layers, we should use
constitutive equations of the theory of plasticity. As known, there exist two basic
versions of this theory - the deformation theory and the flow theory that are briefly
described below.
According to the deformation theory of plasticity, the strains are decomposed
into two components - elastic strains (with superscript 'e') and plastic strains
(superscript 'p'), i.e.,

E,I = E& + &; . (4.15)
We again use the tensor notations of strains and stresses (Le., cij and ou)introduced
in Section 2.9. Elastic strains are linked with stresses by Hooke's law, Eqs. (4.1),
which can be written with the aid of Eq. (4.10) in the form

(4.16)

where U, is the elastic potential that for the linear elastic solid coincides with
complementary potential U,in Eq. (4.10). Explicit expression for U,can be obtained
from Eq. (2.5 1) if we change strains for stresses with the aid of Hooke's law, i.e.,

(4.17)

Now present plastic strains in Eqs. (4.15) in the form similar to Eq. (4.16):

(4.18)

where Up is the plastic potential. To approximate dependence of Up on stresses,
a special generalized stress characteristic, i.e., the so-called stress intensity 0, is
introduced in classical theory of plasticity as