Page 153 - Mechanics Analysis Composite Materials

138 Mechanics and analysis of composite materials

As can be seen, in contrast to the deformation theory, stresses govern the increments
of plastic strains rather than the strains themselves.
In the general case, irrespective of any particular approximation of plastic
potential Up,we can obtain for function do(o) in Eqs. (4.47) the expression similar
to Eq. (4.32). Consider a uniaxial tension for which Eqs. (4.47) yield

Repeating the derivation of Eq. (4.32) we finally get

do(o) = *(- 1 -k) ,
(4.48)
0
where E,(0)= da/de is the tangent modulus introduced in Section 1.1 (see Fig. 1.4).
Dependence of Et on strain for an aluminum alloy is shown in Fig. 4.10. For the
power approximation of plastic potential

Up=Bo" , (4.49)

matching Eqs. (4.46) and (4.48) we arrive at the equation
ds 1
-= - f
do E
Upon integration we get
o Bn
E=-+- o"-' (4.50)
E n-1
This result coincides with Eq. (4.33) within the accuracy of coefficients C and B.
As in the theory of deformation, Eq. (4.50) can be used to approximate the
experimental stress-strain curve and to determine coefficients B and n. Thus,
constitutive equations of the flow theory of plasticity are specified with Eqs. (4.47)
and (4.48).
For a plane stress state, introduce the stress space shown in Fig. 4.12 and referred
to Cartesian coordinate frame with stresses as coordinates. In this space, any
loading can be presented as a curve specified by parametric equations a, = o,(p),
4. = o~,,(p),T.~)= ~,~~(p),where p is the loading parameter. To find strains
corresponding to point A on the curve, we should integrate Eqs. (4.47) along this
curve thus taking into account the whole history of loading. In the general case,
the obtained result will be different from what follows from Eqs. (4.25) of the
deformation theory for point A. However, there exists one loading path (the straight
line OA in Fig. 4.12) that is completely determined by the location of its final point
A. This is the so-called proportional loading during which the stresses increase in
proportion to parameter p, i.e.,