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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  Â and comparable to specimen dimensions. Â Linear elastic fracture mechanics cannot be
used under these conditions. Instead,
we adopt a framework based on plastic
solutions to crack tip fields.Â
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.Â
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. Â
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..Â
Evidently, the stress then satisfies  along the line of the crack just ahead of
each crack. Â
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 Â
 Â 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.Â
Â
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.Â
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  ).Â
The critical value must be measured experimentally for a given
material, using standard test specimens.Â
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.Â
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.
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