Rice’s J-Integral zParameter which characterizes fracture under elastic-plastic and fully plastic conditions ySimilar to the K parameter in fully elastic fracture zRice defined the J-integral for a cracked body as follows: xW = elastic strain energy density xT = traction vector xu = displacement vector = counter clockwise contour beginning on the lower crack surface and ending on any point on the upper crack surface
zRelation between J and Potential Energy yunder linear elastic conditions, J becomes the Griffith’s crack extension force. yRelation is also critical because some derivations of J rely on this concept. zFor a body of thickness B:
The J-Integral as a Fracture Parameter J Ic and J - a curves relationship between J and a, ductile crack length extension, was hypothesized. yalso proposed a physical ductile tearing process during different stages of fracture. yJ was only used to specify the onset of ductile tearing, point 3 in the figure. xthis 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- a curve. xcrack blunting line is described by: ythis 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
zJ-dominance ycrack tip conditions are equal for all geometries and they are all controlled by the magnitude of J. xlarge deformation zone (zone of intense deformation) can be expected to extend one CTOD distance beyond the crack tip xthis 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
Example Calculation of the J-Parameter zhttp://risc.mse.vt.edu/~farkas/cmsms/pu blic_html/jint/cav6.gif ypicture not on website!!
Limitations of the J- Integral znonlinear elasticity or deformation theory of plasticity only applies to elastic-plastic materials under monotonic loading yno unloading is permitted zsmall deformation theory was used in developing: ypath independence of J yrelationship of J with potential energy, crack tip stress fields and CTOD ystresses cannot exceed 10% or ductility will occur.
zMaterials may transition from ductile to brittle behavior yThis phenomenon most often occurs in BCC and HCP alloys due to a decrease in temperature. yAt 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 zCurve 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 zThis temperature range is determined through two standardized testing methods: yCharpy impact testing yIzod impact testing zThese tests measure impact energy through the mechanism shown on the next page yThe energy expended is computed from the difference between h and h’, giving the impact energy
Impact Fracture Testing
zEnergy per unit length crack growth
Fatigue zOccurs 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. yFatigue is estimated to be responsible for approximately 90% of all metallic failures yFailure occurs rapidly and without warning. zThe stresses acting repeatedly upon the material may be due to ytension-compression type stresses ybending or twisting type stresses
Fatigue zThe average mean stress, or maximum and minimum stress values are given by: zStress amplitude is given by: r being the range of stress. zAnd the stress ratio of the maximum and minimum stress amplitudes: yNote that tensile stresses are positive while compressive stresses are always negative
The S-N Curve zData from the tests are plotted as stress S versus the logarithm of the number of cycles to failure, N. zWhen the curve becomes horizontal, the specimen has reached its fatigue limit yThis value is the maximum stress which can be applied over an infinite number of cycles yThe fatigue limit for steel is typically 35 to 60% of the tensile strength of the material
The S-N Curve zFatigue testing is performed using a rotating-bending testing apparatus shown below. Figure 8.18. zSpecimens are subjected to relatively high cyclic stresses up to about two thirds of the tensile strength of the material. zFatigue data contains considerable scatter, the S-N curves shown are “best fit” curves.
Fatigue Strength zFatigue strength is a term applied for nonferrous alloys (Al, Cu, Mg) which do not have a fatigue limit. yThe fatigue strength is the stress level the material will fail at after a specified number of cycles (e.g. 10 7 cycles). yIn these cases, the S-N curve does not flatten out. zFatigue life N f, is the number of cycles that will cause failure at a constant stress level.
Creep zPermanent deformation under a constant stress occurring over time yThree stages of creep: xPrimary xSteady-state xtertiary yTesting performed at constant stress and temperature xDeformation is plotted as a function of time