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:

J

  G  

Wn

1 

T i

 

u x i

1 

ds

 W = elastic strain energy density  T = traction vector  u = displacement vector  G = counter clockwise contour beginning on the lower crack surface and ending on any point on the upper crack surface

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:

J

  1

B dU da

The J-Integral as a Fracture Parameter

 J Ic and J D a curves  relationship between J and D 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 J Ic , the critical J in mode I at the onset of ductile tearing.

The J-Integral as a Fracture Parameter

 J Ic is defined at the intersection of the crack blunting line and the line which defines the J D a curve.

 crack blunting line is described by:

J

 2 

o

D

a

 this construction is necessary because it is quite difficult to define this parameter with physical detection to a high degree of consistency.

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

a

,

W



a

,

B



c J

  

c

 20 d t

Example Calculation of the J-Parameter

 http://risc.mse.vt.edu/~farkas/cmsms/pu blic_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 80  C

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: 2 Stress amplitude is given by:   r 

a

 

m

 

r

  max   max  min  2 being the range of stress.

2  min And the stress ratio of the maximum and minimum stress amplitudes:

R

   min max  Note that tensile stresses are positive while compressive stresses are always negative

The S-N Curve

 Data from the tests are plotted as stress versus the logarithm of the number of cycles to failure, N .

S  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. 10 7 cycles).  In these cases, the S-N curve does not flatten out.

 Fatigue life N f , 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:  Primary  Steady-state  tertiary  Testing performed at constant stress and temperature  Deformation is plotted as a function of time

Creep