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ch6-EPFM2 - HRR Integral Recall with I n 1 n 1 EJ ij = 0 n...

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HRR Integral Recall with is the integration constant. ( 29 1 1 0 2 0 , n ij n EJ n I r σ σ σ θ ασ + = % ( 29 1 0 2 0 , n n ij n EJ n E I r ασ ε ε θ ασ + = % n I Recall the material equation, Now the singularity, unlike varies as a function of n and state of stress. 0 0 0 n ε σ σ α ε σ σ = + I K
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HRR Integral Recall with is the integration constant, as shown in the Figure below. ( 29 1 1 0 2 0 , n ij n EJ n I r σ σ σ θ ασ + = % ( 29 1 0 2 0 , n n ij n EJ n E I r ασ ε ε θ ασ + = % n I Recall the material equation, Now the singularity, unlike , varies as a function of n and state of stress. 0 0 0 n ε σ σ α ε σ σ = + I K
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HRR Integral, cont. Angular variation of dimensionless stress for n=3 and n=13.
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HRR Integral, cont. Note the singularity is of the strenth . For the specific case of n=1 (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 . ( 29 1 1 1 n r + 1 r ( 29 1 , , 2 ij i j j i u u ε = + ( 29 1 , , , , 2 ij i j j i k i k j E u u u u = + + 0.1 ij ε < E Some consequences of HRR singularity In elastic-plastic materials, the singular field is given by (with n=1 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. 1 1 1 1 1 2 n ij n ij J k r J k r σ ε + + = = 0 r = 0 at 0 xx r σ = =
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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 Irwin’s formulation Or based on strip yield (Dugdale’s model) 2 1 2 2 1 2 2 4 4 2 I p YS p y I I y YS YS K r r K u K K G u E π σ μ π δ π σ π σ = + = = = = 8 ln sec 2 ys ys ys a E G σ π σ δ π σ σ = = CTOD is generally determined by three point bend test.
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HRR Integral, cont. 0 ( ) with and ij x i ij ij i ij j u J wdy t ds x w dV t n ε σ ε σ Γ = - = = J-Integral Derived for non-linear elastic material and hence is not valid directly for elasto- plastic material during unloading part.
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