Fracture Behavior of Bulk Crystalline Materials
Rice’s J-Integral As A Fracture Parameter Limitations
Ductile-to-Brittle Transition Impact Fracture Testing
Fatigue The S-N Curve Fatigue Strength
Creep
Rice’s J-Integral
Parameter which characterizes fracture under elastic-plastic and fully plastic conditions Similar to the K parameter in fully elastic fracture
Rice defined the J-integral for a cracked body as follows:
W = elastic strain energy densityT = traction vectoru = displacement vector = counter clockwise contour beginning on the lower crack
surface and ending on any point on the upper crack surface
dsx
uTWnJ ii
11
Rice’s J-Integral
Rice’s J-Integral
Relation between J and Potential Energy under linear elastic conditions, J
becomes the Griffith’s crack extension force.
Relation is also critical because some derivations of J rely on this concept.
For a body of thickness B:da
dU
BJ
1
The J-Integral as a Fracture Parameter
JIc and J - a curves relationship between J and a, ductile crack
length extension, was hypothesized. also proposed a physical ductile tearing
process during different stages of fracture. J was only used to specify the onset of ductile
tearing, point 3 in the figure.this point was defined as JIc, the critical J in mode I at
the onset of ductile tearing.
The J-Integral as a Fracture Parameter
JIc is defined at the intersection of the crack blunting line and the line which defines the J- a curve.
crack blunting line is described by:
this construction is necessary because it is quite difficult to define this parameter with physical detection to a high degree of consistency.
aJ o 2
The J-Integral as a Fracture Parameter
The J-Integral as a Fracture Parameter
J-dominance crack tip conditions are equal for all geometries
and they are all controlled by the magnitude of J.large deformation zone (zone of intense deformation)
can be expected to extend one CTOD distance beyond the crack tip
this zone is surrounded by a larger zone where J dominance applies.
in order for J to be a valid fracture parameter, all pertinent length parameters (crack size, ligament size, and thickness) all exceed several times t
20 ,,
cJ
cBaWa
Example Calculation of the J-Parameter
http://risc.mse.vt.edu/~farkas/cmsms/public_html/jint/cav6.gif picture not on website!!
Limitations of the J-Integral
nonlinear elasticity or deformation theory of plasticity only applies to elastic-plastic materials under monotonic loading no unloading is permitted
small deformation theory was used in developing: path independence of J relationship of J with potential energy, crack tip
stress fields and CTOD stresses cannot exceed 10% or ductility will occur.
Ductile-to-Brittle Transition
Ductile-to-Brittle Transition
Materials may transition from ductile to brittle behavior This phenomenon most often occurs in BCC
and HCP alloys due to a decrease in temperature.
At low temperatures, materials which experience this transition become brittle. This can lead to rapid, catastrophic failure, with little or no warning.
Ductile-to-Brittle Transition
Curve A represents this transition in a steel specimen
The range of temperatures over which this occurs as shown in the next slide is approximately 20 to 80C
Impact Fracture Testing
This temperature range is determined through two standardized testing methods: Charpy impact testing Izod impact testing
These tests measure impact energy through the mechanism shown on the next page The energy expended is computed from the
difference between h and h’, giving the impact energy
Impact Fracture Testing
Impact Fracture Testing
Impact Fracture Testing
Energy per unit length crack growth
Fatigue
Occurs when a material experiences lengthy periods of cyclic or repeated stresses which can lead to failure at stress levels much lower than the tensile or yield strength of the material. Fatigue is estimated to be responsible for
approximately 90% of all metallic failures Failure occurs rapidly and without warning.
The stresses acting repeatedly upon the material may be due to tension-compression type stresses bending or twisting type stresses
Fatigue
The average mean stress, or maximum and minimum stress values are given by:
Stress amplitude is given by:
r being the range of stress.
And the stress ratio of the maximum and minimum stress amplitudes:
Note that tensile stresses are positive while compressive stresses are always negative
2minmax
m
22minmax
ra
max
min
R
The S-N Curve
Data from the tests are plotted as stress S versus the logarithm of the number of cycles to failure, N.
When the curve becomes horizontal, the specimen has reached its fatigue limit This value is the maximum stress which can be
applied over an infinite number of cycles The fatigue limit for steel is typically 35 to 60%
of the tensile strength of the material
The S-N Curve
Fatigue testing is performed using a rotating-bending testing apparatus shown below. Figure 8.18.
Specimens are subjected to relatively high cyclic stresses up to about two thirds of the tensile strength of the material.
Fatigue data contains considerable scatter, the S-N curves shown are “best fit” curves.
Fatigue Strength
Fatigue strength is a term applied for nonferrous alloys (Al, Cu, Mg) which do not have a fatigue limit. The fatigue strength is the stress level the
material will fail at after a specified number of cycles (e.g. 107 cycles).
In these cases, the S-N curve does not flatten out.
Fatigue life Nf, is the number of cycles that will cause failure at a constant stress level.
Creep
Permanent deformation under a constant stress occurring over time Three stages of creep:
PrimarySteady-statetertiary
Testing performed at constant stress and temperatureDeformation is plotted as a function of time
Creep