Page 144 - Mechanics Analysis Composite Materials

Chapter 4. Mechanics of a composize layer I29

Transforming Eq. (4.19) with the aid of Eqs. (2.13) we can reduce it to the following
form:

This means that 0 is an invariant characteristic of a stress state, i.e., that it does
not depend on position of a coordinate frame. For a unidirectional tension as
in Fig. 1.1, we have only one nonzero stress, e.g., 011.Then Eq. (4.19) yields
rs = 01I. In a similar way, strain intensity E can be introduced as

(4.20)

Strain intensity is also an invariant characteristic. For a uniaxial tension (Fig. 1.1)
with stress CTI I and strain EI I in the loading direction, we have ~22= ~33= -Y,,EI I,
where vp is the elastic-plastic Poisson's ratio which, in general, depends on 01I. For
this case, Eq. (4.20) yields

&=;(I + \'p)Ell . (4.21)

For an incompressible material (see Section 4.1.1), vp = 1/2 and E = EII. Thus,
numerical coefficients in Eqs. (4.19) and (4.20) provide 0 = 011 and E = I:II for
uniaxial tension of an incompressible material. Stress and strain intensities in
Eqs. (4.19) and (4.20) have an important physical meaning. As known from
experiments, metals do not demonstrate plastic properties under loading with
stresses 0, = or= 0: = 00 resulting only in the change of material volume. Under
such loading, materials exhibit only elastic volume deformation specified by
Eq. (4.2). Plastic strains occur in metals if we change material shape. For a linear
elastic material, elastic potential U in Eq. (2.51) can be reduced after rather
cumbersome transformation with the aid of Eqs. (4.3), (4.4) and (4.19), (4.20) to the
following form:

U=:a"&o+~aE. (4.22)
-
The first term in the right-hand side part of this equation is the strain energy
associated with the volume change, while the second term corresponds to the change
of material shape. Thus, CT and E in Eqs. (4.19) and (4.20) are stress and strain