Page 18 - Mechanics Analysis Composite Materials

Chapter 1. Introduction 3
Naturally, for any material, there should exist some interrelation between stress
and strain, i.e.

E =f’(o) or c = (~(8). ( 1.4)
These equations specify the so-called constitutive law and are referred to as
constitutive equations. They allow us to introduce an important concept of the
material model which represents some idealized object possessing only those
features of the real material that are essential for the problem under study. The
point is that performing design or analysis we always operate with models rather
than with real materials. Particularly, for strength and stiffness analysis, this model
is described by constitutive equations, Eqs. (1.4), and is specified by the form of
function /(a) or (P(E).
The simplest is the elastic model which implies that AO) = 0, cp(0) =0 and that
Eqs. (1.4) are the same for the processes of an active loading and an unloading. The
corresponding stress-strain diagram (or curve) is presented in Fig 1.2. Elastic
model (or elastic material) is characterized with two important features. First, the
corresponding constitutive equations, Eqs. (1.4), do not include time as a parameter.
This means that the form of the curve shown in Fig. 1.2 does not depend on the rate
of loading (naturally, it should be low enough to neglect the inertia and dynamic
effects). Second, the work performed by force F is accumulated in the bar as
potential energy, which is also referred to as strain energy or elastic energy.
Consider some infinitesimal elongation dA and calculate elementary work
performed by the force F in Fig 1.1 as d W= F dA. Then, work corresponding to
point 1 of the curve in Fig. 1.2 is

where A, is the elongation of the bar corresponding to point 1 of the curve. The
work W is equal to elastic energy of the bar which is proportional to the bar volume
and can be presented as