3/18/2002, Monday
Energy Approach to Fracture
Double Cantilever Beam (DCB)
If the crack extends by an amount da, the necessary additional surface energy is obtained from the work done by the external body forces Pdu and the release of strain energy dW.
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G=Pduda
−dWda
W is the strain energy of DCB
Body Stiffness M
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M =Pu
Crack extension would result in a decrease of plate stiffness
Energy Release Rate
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u=PM
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G=PduBda
−dWBda
=12B
P2∂(1M)∂a
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W =P 2
2M
Force Deflection Relation
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1M
=8a3
EBD3
From Mechanics of Materials, Elementary simple beam problem,
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u=8a3
EBD3 P
Fixed Loading
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∂G∂a
>0
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G=12P2a2
EB2D3
Fixed Grip
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G=3
16Bu2EBD3 1
a3
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M =EBD3
8a3
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G=PduBda
−dWBda
=−1
2Bu2∂M
∂a
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∂G∂a
<0
Energy Release Rate
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G=ddA
t T⋅udsΓ
∫ − wdΩΩ∫
⎡
⎣ ⎢ ⎤
⎦ ⎥
J-Integral
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J = T ⋅∂u∂x
dsΓ
∫ − wdyΩ∫
Reversibility of stress-strain relationship
Path Independent
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J =0
For a closed contour:
Crack Tip Mesh