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