Page 20 - Mechanics Analysis Composite Materials

Chapter 1. Introduction 5
Absolute and specific values of mechanical characteristics for typical materials
discussed in this book are listed in Table 1.1.
After some generalization, modulus can be used to describe nonlinear material
behavior of the type shown in Fig. 1.4. For this purpose, the so-called secant, E,,
and tangent, Et, moduli are introduced as

While the slope 01 in Fig. I .4determines modulus E, the slopes p and y determine Es
and E,, respectively.As it can be seen, Es and E,, in contrast to E, depend on the level
of loading, i.e., on IJ or E. For a linear elastic material (see Fig. 1.3), E, = Et = E.
Hooke’s law, Eq. (1.6), describes rather well the initial part of stress-strain
diagram for the majority of structural materials. However, under relatively high
level of stress or strain, materials exhibit nonlinear behavior.
One of the existing models is the nonlinear elastic material model introduced
above (see Fig. 1.2). This model allows us to describe the behavior of highly
deformable rubber-type materials.
Another model developed to describe metals is the so-called elastic-plastic
material model. The corresponding stress-strain diagram is shown in Fig. 1.5. In
contrast to elastic material (see Fig. 1.2), the processes of active loading and
unloading are described with different laws in this case. In addition to elastic strain,
E~,which disappears after the load is taken off, the residual strain (for the bar shown
in Fig. 1.1, it is plastic strain, sp)retains in the material. As for an elastic material,
stress-strain curve in Fig. 1.5 does not depend on the rate of loading (or time of
loading). However, in contrast to an elastic material, the final strain of an elastic-
plastic material can depend on the history of loading, Le., on the law according to
which the final value of stress was reached.
Thus, for elastic or elastic-plastic materials, constitutive equations, Eqs. (1.4), do
not include time. However, under relatively high temperature practically all the
materials demonstrate time-dependent behavior (some of them do it even under
room temperature). If we apply to the bar shown in Fig. 1.1 some force F and keep
it constant, we can see that for a time-sensitive material the strain increases under
constant force. This phenomenon is called the creep of the material.
So, the most general material model that is used in this book can be described
with the constitutive equation of the following type:
E =f(o,t, 0 , (1-9)

where t indicates the time moment, while CJ and T are stress and temperature
corresponding to this moment. In the general case, constitutive equation, Eq. (1.9),
specifies strain that can be decomposed into three constituents corresponding to
elastic, plastic and creep deformation, i.e.

E=&,+Ep+ec . (1.10)