Page 139 - Mechanics Analysis Composite Materials

124 Mechanics and analysis of composite materials

through volume or bulk modulus
E
K= (4.5)
3(1 -2~)

For v = 1/2,K + 00, = 0, and dK = dV for any stresses. Such materials are
called incompressible-they do not change their volume under deformation and can
change only their shape.
The foregoing equations correspond to the general three-dimensional stressed
state of a layer. However, working as a structural element of a thin-walled
composite laminate a layer is usually loaded with a system of stresses one of which,
namely, transverse normal stress o, is much less than the other stresses. Bearing this
in mind, we can neglect the terms in Eqs. (4.1)that include az and write these
equations in a simplified form:

or

where E =E/( 1 - v2).

4.1.2. Nonlinear models

Materials of metal and polymeric layers considered in this section demonstrate
linear response only under moderate stresses (see Fig. 1.11 and 1.14). Further
loading results in nonlinear behavior to describe which we need to apply one of
nonlinear material models discussed in Section 1. I.
A relatively simple nonlinear constitutive theory suitable for polymeric layers can
be constructed using nonlinear elastic material model (see Fig. 1.2). In the strict
sense, this model can be applied to materials whose stress-strain curves are the same
for active loading and unloading. But normally structural analysis is undertaken
only for active loading. If unloading is not considered, elastic model can be formally
used for materials that are not perfectly elastic.
There exists a number of models developed to describe nonlinear behavior of
highly deformable elastomers like rubber (Green and Adkins, 1960). Polymeric
materials used to form isotropic layers of composite laminates admitting, in
principal, high strains usually do not demonstrate them in composite structures
whose deformation is governed by fibers with relatively low ultimate elongation
(I-3%). So, creating the model we can restrict ourselves to the case of small strains,
i.e., to materials whose typical stressstrain diagram is shown in Fig. 4.6.