ElasticPlastic Fracture Mechanics

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Elastic-Plastic Fracture Mechanics

• Introduction
• When does one need to use LEFM and EPFM?
• What is the concept of small-scale and
  large-scale yielding?

• Background Knowledge
• Theory of Plasticity (Yield criteria, Hardening
  rules)
• Concept of K, G and K-dominated regions
• Plastic zone size due to Irwin and Dugdal

• Contents of this Chapter
• The basics of the two criteria used in EPFM COD
  (CTOD), and J-Integral (with H-R-R)
• Concept of K- and J-dominated regions, plastic
  zones
• Measurement methods of COD and J-integral
• Effect of Geometry

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LEFM and EPFM

• LEFM
• In LEFM, the crack tip stress and displacement
  field can be uniquely characterized by K, the
  stress intensity factor. It is neither the
  magnitude of stress or strain, but a unique
  parameter that describes the effect of loading at
  the crack tip region and the resistance of the
  material. K filed is valid for a small region
  around the crack tip. It depends on both the
  values of stress and crack size.

We noted that when a far field stress acts on an
edge crack of width a then for mode I, plane
strain case
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LEFM cont.
For 0
Singularity dominated region
LEFM concepts are valid if the plastic zone is
much smaller than the singularity zones. Irwin
estimates Dugdale strip yield model
ASTM a,B, W-a 2.5 , i.e.
of specimen dimension.
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• EPFM
• In EPFM, the crack tip undergoes significant
  plasticity as seen in the following diagram.

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EPFM cont.

• EPFM applies to elastoc-rate-independent
  materials, generally in the large-scale plastic
  deformation.
• Two parameters are generally used
• Crack opening displacement (COD) or crack tip
  opening displacement (CTOD).
• J-integral.
• Both these parameters give geometry independent
  measure of fracture toughness.

y
x
Sharp crack
Blunting crack
ds
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EPFM cont.

• Wells discovered that Kic measurements in
  structural steels required very large thicknesses
  for LEFM condition.
• --- Crack face moved away prior to fracture.
• --- Plastic deformation blunted the sharp crack.

Note
since
Sharp crack
Blunting crack

• Irwin showed that crack tip plasticity makes the
  crack behave as if it were longer, say from size
  a to a rp
• 
  -----plane stress
• From Table 2.2,
• Set ,

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CTOD and strain-energy release rate

• Equation
  relates CTOD ( ) to G for small-scale yielding.
  Wells proved that
• Can valid even for large scale yielding, and is
  later shown to be related to J.
• can also be analyzed using Dugdales strip
  yield model. If is the opening at the end
  of the strip.

Consider an infinite plate with a image crack
subject to a Expanding in an infinite
series,
If
, and can be given as
In general,
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Alternative definition of CTOD
Blunting crack
Sharp crack
Blunting crack
Displacement at the original crack tip
Displacement at 900 line intersection, suggested
by Rice
CTOD measurement using three-point bend specimen
displacement
Vp
expanding
'
'
'
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Elastic-plastic analysis of three-point bend
specimen
V,P
Where is rotational factor, which equates
0.44 for SENT specimen.

• Specified by ASTM E1290-89
• --- can be done by both compact tension, and
  SENT specimen
• Cross section can be rectangular or W2B square
  WB
• KI is given by

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CTOD analysis using ASTM standards
Figure (a). Fracture mechanism is purely
cleavage, and critical CTOD lt0.2mm, stable
crack growth, (lower transition). Figure (b).
--- CTOD corresponding to initiation of stable
crack growth. ---
Stable crack growth prior to fracture.(upper
transition of fracture steels). Figure (c)
and then ---CTOD at the maximum load
plateau (case of raising R-curve).

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More on CTOD
The derivative is based on Dugdales strip yield
model. For Strain hardening materials, based on
HRR singular field.
By setting 0 and n the strain hardening index
based on
Definition of COD is arbitrary since
A function as the tip is
approached
Based on another definition, COD is the distance
between upper and lower crack faces between two
45o lines from the tip. With this Definition
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Where ranging from
0.3 to 0.8 as n is varied from 3 to 13 (Shih,
1981)
Condition of quasi-static fracture can be
stated as the Reaches a critical value
. The major advantage is that this
provides the missing length scale in relating
microscopic failure processes to macroscopic
fracture toughness.

• In fatigue loading, continues to vary
  with load and is a
• function of
• Load variation
• Roughness of fracture surface (mechanisms
  related)
• Corrosion
• Failure of nearby zones altering the local
  stiffness response

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3.2 J-contour Integral

• By idealizing elastic-plastic deformation as
  non-linear elastic, Rice proposed J-integral, for
  regions
• beyond LEFM.
• In loading path elastic-plastic can be modeled
  as non-linear elastic but not in unloading part.
• Also J-integral uses deformation plasticity. It
  states that the stress state can be determined
  knowing
• the initial and final configuration. The
  plastic strain is loading-path independent. True
  in proportional
• load, i.e.
• under the above conditions, J-integral
  characterizes the crack tip stress and crack tip
  strain and
• energy release rate uniquely.
• J-integral is numerically equivalent to G for
  linear elastic material. It is a path-independent
  integral.
• When the above conditions are not satisfied, J
  becomes path dependent and does not relates to
  any
• physical quantities

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3.2 J-contour Integral, cont.
y
x
ds
Consider an arbitrary path ( ) around the crack
tip. J-integral is defined as
where w is strain energy density, Ti is component
of traction vector normal to contour.
It can be shown that J is path independent and
represents energy release rate for a material
where is a monotonically increasing with
Proof Consider a closed contour Using
divergence theorem
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Evaluation of J Integral ---1
Evaluate
Note is only valid if such a
potential function exists Again,
Since
Recall
(equilibrium) leads to
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Evaluation of J Integral ---2
Hence, Thus for any closed contour
Now consider
1
2
3
4
Recall
On crack face, (no traction
and y-displacement), thus ,
leaving behind Thus any counter-clockwise path
around the crack tip will yield J J is path
independent.
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Evaluation of J Integral ---3
y
a
x
2D body bounded by
In the absence of body force, potential energy
Suppose the crack has a vertical extension, then
(1)
Note the integration is now over
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Evaluation of J Integral ---4
Noting that
(2)
Using principle of virtual work, for
equilibrium, then from eq.(1), we have
Thus,
Using divergence theorem and multiplying by -1
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Evaluation of J Integral ---5
Therefore, J is energy release rate , for
linear or non-linear elastic material
In general Potential energy Ustrain
energy stored Fwork done by external force and
A is the crack area.
p
u
-dP
a
Displacement
p
Complementary strain energy
0
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Evaluation of J-Integral
For Load Control
For Displacement Control
The Difference in the two cases is
and hence J for both load
Displacement controls are same
JG and is more general description of energy
release rate
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More on J Dominance

• J integral provides a unique measure of the
  strength of the singular
• fields in nonlinear fracture. However there are
  a few important
• Limitations, (Hutchinson, 1993)
• Deformation theory of plasticity should be valid
  with small strain
• behavior with monotonic loading
• (2) If finite strain effects dominate and
  microscopic failures occur, then
• this region should be much smaller compared
  to J dominated region
• Again based on the HRR singularity

Based on the condition (2), we would like to
evaluate the inner radius ro of J dominance. Let
R be the radius where the J solutions are
satisfied within 10 of complete solution. FEM
shows that
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• However we need ro should be greater than the
  forces zone
• (e.g. grain size in intergranular fracture,
  mean spacing of voids)
• Numerical simulations show that HRR singular
  solutions hold
• good for about 20-25 of plastic zone in
  mode I under SSY
• Hence we need a large crack size (a/w gt0.5) .
  Then finite strain
• region is , minimum ligament
  size for valis JIC is
• For J Controlled growth elastic unloading/non
  proportional loading
• should be well within the region of J
  dominance
• Note that near tip strain distribution for a
  growing crack has a
• logarithmic singularity which is weaker then
  1/r singularity for a
• stationary crack

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Williams solution to fracture problem
Williams in 1957 proposed Airys stress
function As a solution to the biharmonic
equation For the crack problem the boundary
conditions are Note will have singularity
at the crack tip but is single valued Note that
both p and q satisfy Laplace equations such
that
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Now, for the present problem.
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Williams Singularity3
Applying boundary conditions,
Case (i)
or,
Case (ii)
Since the problem is linear, any linear
combination of the above two will also be
acceptable. Thus Though all values are
mathematically fine, from the physics point of
view, since
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Williams Singularity4
Since U should be provided for any annular rising
behavior and R ,
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Williams Singularity4
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Williams Singularity5
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Williams Singularity6
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HRR Singularity1
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HRR Singularity2