HRR Integral

1
HRR Integral
Recall the material equation, Now the
singularity, unlike varies as a function of
n and state of stress.
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HRR Integral
Recall with is the integration
constant, as shown in the Figure below.
Recall the material equation, Now the
singularity, unlike , varies as a function of
n and state of stress.
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HRR Integral, cont.
Angular variation of dimensionless stress for n3
and n13.
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HRR Integral, cont.
Note the singularity is of the strenth .
For the specific case of n1 (linearly elastic),
we have singularity. Note also that the HRR
singularity still assumes that the strain is
infinitesimal, i.e.,
, and not the finite strain
. Near the tip where the
strain is finite, (typically when ),
one needs to use the strain measure .  

• Some consequences of HRR singularity
• In elastic-plastic materials, the singular field
  is given by
• 
  (with n1 it is
  LEFM)
• stress is still infinite at .
• the crack tip were to be blunt then
  since it is now a free surface. This
  is not the case in HRR field.
• HRR is based on small strain theory and is not
  thus applicable in a region very close to the
  crack tip.

5
HRR Integral, cont.
Large-strain crack tip finite element results of
McMeeking and Parks. Blunting causes the stresses
to deviate from the HRR solution close to the
crack tip.
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HRR Integral, cont.
Some additional notes on CTOD and
J-integral   CTOD is based on Irwins
formulation Or based on strip yield
(Dugdales model)
CTOD is generally determined by three point bend
test.
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HRR Integral, cont.
J-Integral

• Derived for non-linear elastic material and
  hence is not valid directly for elasto-plastic
  material during unloading part.
• Assumes deformation theory, and hence needs
  proportional loading.
• Assumes small strain formulation (no large
  strain, no large rotation admissible).
• J is path independent when there is no traction
  or displacement in the crack faces.
• J cannot be evaluated at the tip where there is
  singularity since the theory assumes that inside
  the domain the region should be analytic.

8
HRR Integral, cont.
Large Strain Zone HRR singularity still predicts
infinite stresses near the crack tip. But when
the crack blunts, the singularity reduces. In
fact at for a blunt
crack. The following is a comparison when you
consider the finite strain and crack blunting. In
the figure, FEM results are used as the basis for
comparison.
The peak occurs at and decreases as
. This corresponds to approximately twice
the width of CTOD. Hence within this region, HRR
singularity is not valid.
Large-strain crack tip finite element results of
McMeeking and Parks. Blunting causes the stresses
to deviate from the HRR solution close to the
crack tip.
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HRR Integral, cont.
Evaluation of J Early works of Landes and
Begley were based on the definition

• Obtain curve for different crack
  sizes .
• Compute from the previous results.
• Plot (see Fig.3-13 below).
• The method is not useful there are too many
  specimens and tests.

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HRR Integral, cont.
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HRR Integral, cont.
Evaluation of J using DENT (Double edge notched
tension) Let the deep notch
where H
represents a function. Now,
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HRR Integral, cont.
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HRR Integral, cont.
Evaluation of J using Edge Cracked Plate in
bending
It can be shown that In general for any
configuration where is a dimensionless
constant.
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HRR Integral, cont.
Relationship between J and CTOD   In LEFM
. However in the case of small scale yielding
(SSY), where m is a
dimensionless constant that depends on the state
of stress and the material properties.
Contour along the boundary of the strip yield
zone ahead of a crack tip.
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HRR Integral, cont.
Note that in the figure is a contour along
the strip yield zone. If the cohesive zone extent
is considerably larger than the cohesive zone
width, i.e., then and the
traction on the crack face
. Then,

. For a fixed value of ,
If the cohesive zone length is small compared
to other dimensions, then Since
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HRR Integral, cont.
For the strip yield model, since The above
relationship is valid for plane stress with
elastic-perfectly plastic materials.  
More rigorous analysis based on HRR
Integral   Shih showed that when HRR integral can
be assumed to be valid, then where is a
dimensionless constant. For non-hardening
materials, . Thus in summary, there is a
unique relationship between J and CTOD. Both the
quantities are valid for elastic-plastic
materials under certain assumptions.
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HRR Integral, cont.
3.4 Crack Growth Resistant Curves
Schematic J resistance curve for a ductile
material.
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HRR Integral, cont.
Both J and CTOD increase as the crack begins to
grow and display a rising R-curve. R represents
the resistance of the material to crack growth.
Initially there is a stable crack growth and then
it becomes unstable. Note that and critical
CTOD indicate the initiation of crack growth. The
initiation cannot be precisely defined in
practice. For generic materials, R-curve may
provide a better indication of the toughness of
the material rather than just . If we define
a tearing modulus, then
Stability and R-curve for elastic materials   We
know from the Griffith theory, in perfectly
brittle elastic materials crack extension starts
when where is the surface
energy per side of the crack face. Since the
crack size a extends after initiation, the strain
energy release rate (or crack driving force) G
increases leading to an unstable crack growth in
ideally elastic brittle material. However if the
material is not ideal elastic, then the
resistance to crack growth R increases as shown
in the following figure.
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HRR Integral, cont.
Schematic driving force R-curve diagrams
The left figure shows a material with a flat
R-curve, while the right shows that of rising
R-curve. In the flat type, while the crack is
stable at stress level , it becomes unstable
at a stress of . On the rising curve, the
crack continues to grow from with
stress levels . Thus the conditions
for stable crack growth is
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HRR Integral, cont.

• Unstable crack growth occurs when
• Some notes on R-curve for elastic materials
• For a flat R-curve, one can define a critical
  value of .
• For a rising R-curve, it is difficult to
  estimate the precise value of G at which crack
  starts to grow. This provides no information on
  the growth process or the R-curve.

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HRR Integral, cont.

Reasons for R-curve Resistance to fracture may
arise from the energy needed to create new
surfaces, additional inelastic irrecoverable
dissipative energy needed. For a flat R-curve,
one can define a critical value of If
and is a constant, then R-curve is flat
(only in ideal brittle material). For ductile
fracture in metals, plastic zone size increases
with crack growth and hence R-curve rises. For
infinitely large specimen, plastic zone size
eventually reaches a steady state R-curve then
reaches a steady state. Falling R-curve is
possible if strain-rate effect reduces the size
of plastic zone with crack growth. State of
stress alters the shape of R-curve. Plane stress
has steeper R-curve compared to plane strain.
R-curve may be affected by free boundary effects.
Displacement control produces more stable than
load control.
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HRR Integral, cont.
Schematic J resistance curve for a ductile
material.
The driving force is expressed in terms of
applied tearing modulus Condition for stable
crack growth
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HRR Integral, cont.
Unstable crack growth when
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HRR Integral, cont.

J-controlled fracture J-controlled fracture
established crack tip conditions for J as well as
CTOD.
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HRR Integral, cont.
L---characteristic length scale uncrack ligment
in SSY, K and J characterize crack tip. r is
where singularity. For monotonic,
quasistatic loading J-dominated zone occurs in
the plastic zone. Well inside the plastic zone
(J-controlled), elastic singularity is not valid.
However inside the plastic zone, HRR solution is
valid and the stresses . Now finite
strain occurs within 2 from tip. Here HRR
field is not valid.
In SSY, K characterizes crack tip, but
singularity is not valid upto the tip. Here J is
valid for elastic-plasticity, but the
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HRR Integral, cont.

• In elastic-plastic conditions, J is still
  approximately valid. Here K-field is not valid.
• As plastic zoone increases in size, K-dominated
  zone disappears but J-dominated zone persists.
• Here K has no meaning j-integral and CTOD are
  valid parameters.

Large scale yielding

• Size of finite strain zone increases.
• J is not unique.
• Single parameter fracture mechanics does not
  exist.
• Critical J exhibit a size and geometry
  dependence.

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HRR Integral, cont.

J-controlled crack growth
Note that in elastic unloading, non-linear
elastic modeling (J-integral) is not valid. For
J-controlled fracture, non proportional
plasticity must be embedded within J-controlled
region.
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HRR Integral, cont.
SSY
controlled crack growth