Chapter 9
Modeling Material Failure
9.5 Plastic fracture mechanics
Thus far we have avoided discussing the
complicated material behavior in the process zone near the crack tip. This is acceptable as long as the process
zone is small compared with the specimen dimensions, and a clear zone of K
dominance is established around the crack tip.
In some structures, however, the materials are so tough and ductile that
the plastic zone near the crack tip is huge and comparable to specimen dimensions. Linear elastic fracture mechanics cannot be
used under these conditions. Instead, we
adopt a framework based on plastic
solutions to crack tip fields.
In this section, we address three issues:
1. The size of the plastic zone at the crack tip is
estimated
2. The asymptotic fields near the crack tip in a plastic
material are calculated
3. A phenomenological framework for predicting fracture
in plastic solids is outlined.
9.5.1 Dugdale-Barenblatt cohesive zone model of yield at a crack tip
The simplest estimate of the size of the
plastic zone at a crack tip can be obtained using Dugdale & Barenblatt’s
cohesive zone model, which gives the plastic zone size at the tip of a crack in
a thin sheet (deforming under conditions of plane stress) as
where is the crack tip stress intensity factor and Y is the material yield stress.

|
This estimate is derived as follows. Consider a crack of length 2a in an
elastic-perfectly plastic material with elastic constants and yield stress Y. We assume that the specimen is a thin sheet,
with thickness much less than crack length, so that a state of plane stress is
developed in the solid. We anticipate
that there will be a region near each crack tip where the material deforms
plastically. The Mises equivalent stress
should not exceed yield in this region. It’s hard to find a solution with stresses
at yield everywhere in the plastic zone, but we can easily construct an
approximate solution where the stress along the line of the crack satisfies the
yield condition, using the ‘cohesive zone’ model illustrated in the
picture.
Let denote the length of the cohesive zone at each
crack tip. To construct an appropriate
solution we extend the crack in both directions to put fictitious crack tips at
,
and distribute tractions of magnitude over the crack flanks from to ,
and similarly at the other crack tip..
Evidently, the stress then satisfies along the line of the crack just ahead of each
crack.
We can use point force solution given in the
table in Section 9.3.3 to compute the stress intensity factor at the fictitious
crack tip. Omitting the tedious details of evaluating the integral, we find
that
The * on the stress intensity factor is
introduced to emphasize that this is not the true crack tip stress intensity
factor (which is of course ), but the stress intensity factor at the
fictitious crack tip. The stresses must remain bounded just ahead of the
fictitious crack tip, so that must be chosen to satisfy . This gives
Its more sensible to express this in terms of stress intensity factor
This estimate turns out to be remarkably accurate for plane stress
conditions, where more detailed calculations give
For plane strain the plastic zone is smaller: detailed calculations show
that the plastic zone size is
9.5.2 Hutchinson-Rice-Rosengren (HRR) crack tip fields for stationary
crack in a power law hardening solid
The HRR fields are an exact solution to the
stress, strain and displacement fields near a crack tip in a power-law strain
hardening, rigid plastic material, which is subjected to monotonically
increasing stress at infinity. The model
is based on the following assumptions:
1. The solid is infinitely
large, and contains an infinitely long crack with its tip at the origin
2. The material is a rigid
plastic, strain hardening solid with uniaxial stress -v- strain curve
where are material properties, with n>1.
The HRR solution shows that the stress,
strain and displacement fields at a point in the solid can be calculated from functions
of the form
where are dimensionless functions of the angle and the hardening index n only, and J
is the value of the (path independent) J integral
where
with . W can
be interpreted as the total work done in loading the material up to a stress under monotonically increasing, proportional
loading.
These results are important for two reasons:
1. They show that the
magnitudes of the stress, strain and displacement near the crack tip are
characterized by J. Thus, in highly plastic materials, J can replace K as the fracture criterion.
2. They illustrate the nature
of the stress and strain fields near the crack tip. In particular, they show that the stress has
a singularity for n=1 (a linear stress-strain curve) we recover the square
root singularity found in elastic materials; while for a perfectly plastic
solid ( ) the stress is constant. In contrast, the strains have a square root
singularity for n=1 and an singularity for
Derivation The HRR
solution is derived by solving the following governing equations for
displacements ,
strains and stresses
Strain-displacement relation
Stress
equilibrium
Boundary
conditions on
The
stress-strain relation for a power-law hardening rigid plastic material
subjected to monotonically increasing, proportional loading (this means that
material particles are subjected to stresses and strains whose principal axes
don’t rotate during loading) can be expressed as
where are material constants and is the deviatoric stress and is the Von Mises effective stress. Of course, we don’t know a priori that material elements ahead of a crack tip experience proportional
loading, but this can be verified after the solution has been found. It is helpful to note that under proportional
loading, the rigid plastic material is indistinguishable from an elastic
material with strain energy potential
The J integral must then be
path independent.
The equilibrium condition may be satisfied
through an Airy stress function ,
generating stresses in the usual way as
The solution can be derived from an Airy function that has a separable
form
where the power and are to be determined.
The strength of the singularity can be determined using the J integral.
Evaluating the integral around a circular contour radius r enclosing the
crack tip we obtain
For the J integral to be path
independent, it must be independent of r and therefore W must be
of order . The Airy function gives stresses of order ,
and the corresponding strain energy density would have order . Consequently, for a path independent J,
we must have ,
so
Note for a linear material (n=1), we find ,
which corresponds to the expected square-root stress singularity.
We can now scale the governing equations as
discussed in Section 7.2.13. To this
end, define normalized length, displacement, strain, stress and Airy function
as
With these definitions the governing equations reduce to
Strain-displacement relation
Stress
equilibrium
Constitutive
equation
In addition, the stresses are related to the normalized Airy function by
while the expression
for the J integral becomes
where . The only material parameter appearing in the
scaled equations is n. In addition, note that J has been eliminated from the equations, so the solution is
independent of J.
The stresses can be derived from an Airy
function
The scaling of displacements, strain and
stress with load and material properties then follows directly from the
definition of the normalized quantities.
To compute the full expression for and hence to determine is a tedious and not especially
straightforward exercise. The governing
equation for is obtained from the condition that the strain
field must be compatible. This requires
Computing the stresses from the Airy
function, deducing the strains using the constitutive law and substituting the
results into this equation yields a fourth order nonlinear ODE for f,
which must be solved subject to appropriate symmetry and boundary
conditions. The solution must be found
numerically details are given in Hutchinson Journal of the
Mechanics and Physics of Solids, 16 13 (1968)
and Rice and Rosengren ibid, 31.
9.5.3. Plastic fracture mechanics based on J
There are many situations (e.g. in design of
pressure vessels, pipelines, etc) where the structure is purposely made from a
tough, ductile material. Usually, one cannot apply LEFM to these structures,
because a large plastic zone forms at the crack tip (the plastic zone is
comparable to specimen dimensions, and there is no K dominant zone). Some other approach is needed to design
against fracture in these applications.
Two related approaches are used one is based on the HRR crack tip field and
uses J as a fracture criterion; the other uses the crack tip opening
displacements as a fracture criterion.
Only the J based approach will be discussed here.
The most important conclusion from the HRR
crack tip field is that the amplitude of stresses, strains and displacements
near a crack tip in a plastically deforming solid scale in a predictable way
with J. Just as stress intensity factors quantify the
stress and strain magnitudes in a linear elastic solid, J can be used as a parameter
to quantify the state of stress in a plastic solid.
Phenomenological J
based fracture mechanics is based on the same reasoning that is used to justify
K based LEFM. We postulate that we will
observe three distinct regions in a plastically deforming specimen containing a
crack,
1. A process zone near the crack tip, with finite
deformations and extensive material damge, where the asymptotic HRR field is
not accurate
2. A J dominant zone, outside the process zone, but small
compared with specimen dimensions, where the HRR field accurately describes the
deformation
3. The remainder, where stress and strain fields are
controlled by specimen geometry and loading.
As for LEFM, we hope that the process zone is controlled by the
surrounding J dominant zone, so that crack tip loading conditions can be
characterized by J.
J based fracture mechanics is applied in much the same way as
LEFM. We assume that crack growth starts
when J reaches a critical value (for mode I plane strain loading this
value is denoted ). The
critical value must be measured experimentally for a given material, using
standard test specimens. To assess the
safety of a structure or component containing a crack, one must calculate J
and compare the predicted value to - if the structure is safe.

|
Practical application of J based fracture mechanics is somewhat
more involved than LEFM. Tests to
measure are performed using standard test specimens deeply cracked 3 or 4 point bend bars are
often used. Calibrations for the latter
case are available in J. R. Rice, P. C. Paris and J. G. Merkle,. Progress in
Flaw Growth and Fracture Toughness Testing, ASTM STP 536,. 231 (1973).
Calculating J for a specimen or component usually requires a
full field FEM analysis. Cataloging
solutions to standard problems is much more difficult than for LEFM, because
the results depend on the stress-strain behavior of the material. Specifically, for a power-law solid
containing a crack of length a and subjected to stress ,
we expect that
For example, a slit crack of length 2a
subjected to mode I loading with stress has (approximately see He & Hutchinson J. Appl. Mech 48 830 1981)
Finally, to apply the theory it is necessary
to ensure that both test specimen and component satisfy conditions necessary
for J dominance. As a rough rule of
thumb, if all characteristic specimen dimensions (crack length, etc) exceed J dominance is likely to be satisfied.