Page 189 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C003.fm Page 169 Monday, May 23, 2005 5:42 PM

Elastic-Plastic Fracture Mechanics 169

By comparing Equation (A3.61) and Equation (A3.63), one sees that the deformation and incre-
mental theories of plasticity coincide only if the latter equation can be integrated to obtain the
former. If the deviatoric stress components are proportional to the effective stress:

S = ωσ e (A3.64)
ij
ij

where w is a constant tensor that does not depend on strain, then the integration of Equation
ij
(A3.63) results in Equation (A3.61). Thus deformation and incremental theories of plasticity are
identical when the loading is proportional in the deviatoric stresses. Note that the total stress
components need not be proportional in order for the two theories to coincide; the flow rule is not
influenced by the hydrostatic portion of the stress tensor.
Proportional loading of the deviatoric components does not necessarily mean that deformation
plasticity theory is rigorously correct; it merely implies that the deformation theory is no more objec-
tionable than the incremental theory. Classical plasticity theory, whether based on incremental strain or
total deformation, contains simplifying assumptions about material behavior. Both Equation (A3.61)
and Equation (A3.63) assume that the yield surface expands symmetrically and that its radius does
not depend on hydrostatic stress. For monotonic loading ahead of a crack in a metal, these
assumptions are probably reasonable; the assumed hardening law is of little consequence for
monotonic loading, and hydrostatic stress effects on the yield surface are relatively small for most
metals.
Budiansky [48] showed that the deformation theory is still acceptable when there are modest
deviations from proportionality. Low work-hardening materials are the least sensitive to nonpro-
portional loading.
Since most of classical fracture mechanics assumes either plane stress or plane strain, it is
useful to examine plastic deformation in the two-dimensional case, and determine under what
conditions the requirement of proportional deviatoric stresses is at least approximately satisfied.
Consider, for example, plane strain. When elastic strains are negligible, the in-plane deviatoric
normal stresses are given by

σ − σ σ − σ
S = xx yy and S = yy xx (A3.65)
xx
2 yy 2
assuming incompressible plastic deformation, where s = (s + s )/2. The expression for von
yy
xx
zz
Mises stress in plane strain reduces to
σ = 1 3 S 2 6 S + x[ 2 ] 12 / (A3.66)
e
2 x x y

where S = t . Alternatively, s can be written in terms of principal normal stresses:
e
xy
xy
σ 3 [ σ = − σ ] where s > s
e
2 1 2 1 2
(A3.67)
= 3S
1

Therefore, the principal deviatoric stresses are proportional to s in the case of plane strain. It can
e
easily be shown that the same is true for plane stress. If the principal axes are fixed, S , S , and
yy
xx