Page 190 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C003.fm Page 170 Monday, May 23, 2005 5:42 PM

170 Fracture Mechanics: Fundamentals and Applications

S must also be proportional to s . If, however, the principal axes rotate during deformation, the
e
xy
deviatoric stress components defined by a fixed coordinate system will not increase in proportion
to one another.
In the case of Mode I loading of a crack, t is always zero on the crack plane, implying
xy
that the principal directions on the crack plane are always parallel to the x-y-z coordinate axes.
Thus, the deformation and incremental plasticity theories should be equally valid on the crack
plane, well inside the plastic zone (where elastic strains are negligible). At finite angles from
the crack plane, the principal axes may rotate with deformation, which will produce nonpro-
portional deviatoric stresses. If this effect is small, the deformation plasticity theory should
be adequate to analyze stresses and strains near the crack tip in either plane stress or plane
strain.
The validity of the deformation plasticity theory does not automatically guarantee that the
crack-tip conditions can be characterized by a single parameter, such as J or K. Single-parameter
11
fracture mechanics requires that the total stress components be proportional near the crack tip, a
much more severe restriction. Proportional total stresses imply that the deviatoric stresses are
proportional, but the reverse is not necessarily true. In both the linear elastic case (Appendix 2.3)
and the nonlinear case (Appendix 3.4) the stresses near the crack tip were derived from a stress
function of the form

Φ=κ s θrf () (A3.68)

where k is a constant. The form of Equation (A3.68) guarantees that all stress components
are proportional to k, and thus proportional to one another. Therefore any monotonic function
of k uniquely characterizes the stress fields in the region where Equation (A3.68) is valid.
Nonproportional loading automatically invalidates Equation (A3.68) and the single-parameter
description that it implies.
As stated earlier, the deviatoric stresses are proportional on the crack plane, well within the
plastic zone. However, the hydrostatic stress may not be proportional to s . For example, the loading
e
is highly nonproportional in the large-strain region, as Figure 3.12 indicates. Consider a material
point at a distance x from the crack tip, where x is in the current large-strain region. At earlier
stages of deformation the loading on this point was proportional, but s reached a peak when the
yy
ratio x s /J was approximately unity, and the normal stress decreased with subsequent deformation.
o
Thus, the most recent loading on this point was nonproportional, but the deviatoric stresses are
still proportional to s .
e
When the crack grows, the material behind the crack tip unloads elastically and the defor-
mation plasticity theory is no longer valid. The deformation theory is also suspect near the elastic-
plastic boundary. Equation (A3.65) to Equation (A3.67) were derived assuming the elastic strains
were negligible, which implies s = 0.5(s + s ) in plane strain. At the onset of yielding,
yy
xx
zz
however, s = n (s + s ), and the proportionality constants between s and the deviatoric
yy
e
zz
xx
stress components are different than for the fully plastic case. Thus when elastic and plastic
strains are of comparable magnitude, the deviatoric stresses are nonproportional, as w (Equation
ij
(A3.64)) varies from its elastic value to the fully plastic limit. The errors in the deformation
theory that may arise from the transition from elastic to plastic behavior should not be appreciable
in crack problems, because the strain gradient ahead of the crack tip is relatively steep, and the
transition zone is small.
11 The proportional loading region need not extend all the way to the crack tip, but the nonproportional zone at the tip
must be embedded within the proportional zone in order for a single loading parameter to characterize crack-tip
conditions.