3/18/2002, monday energy approach to fracture. double cantilever beam (dcb) if the crack extends by...
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3/18/2002, Monday
Energy Approach to Fracture
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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
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Body Stiffness M
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M =Pu
Crack extension would result in a decrease of plate stiffness
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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
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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
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Fixed Loading
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∂G∂a
>0
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G=12P2a2
EB2D3
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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
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Energy Release Rate
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G=ddA
t T⋅udsΓ
∫ − wdΩΩ∫
⎡
⎣ ⎢ ⎤
⎦ ⎥
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J-Integral
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J = T ⋅∂u∂x
dsΓ
∫ − wdyΩ∫
Reversibility of stress-strain relationship
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Path Independent
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J =0
For a closed contour:
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Crack Tip Mesh