EN0175-12

# EN0175-12 - EN0175 10 12 06 Announcements Mid-term(Tuesday...

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Unformatted text preview: EN0175 10 / 12 / 06 Announcements: Mid-term (Tuesday, Oct 31) (1 page double sided notes allowed) Proposal/team for ABAQUS project (Thursday, Nov 9) Review of strain concepts x v y v 1 3 2 deform ( ) t x y y , v v v = , ( ) i i i i u x t x x x y y + = = , , , 3 2 1 Deformation gradient: x F y v v d d = , j i ij x y F ∂ ∂ = Cauchy-Green strains: F F C T = , kj ki ij F F C = T F F B = , jk ik ij F F B = Stretch tensors: C U = , B V = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = 2 2 2 III II I C λ λ λ , C U III II I = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = λ λ λ Polar decomposition: R V U R F = = 1 EN0175 10 / 12 / 06 Let us see how the principal values/directions of C-G strains are related: The principal values/directions of C and U are m m U U v v λ = , m m C U v v 2 λ = Multiply the above equation by R , m R m U R U v v λ = Since F R V U R = = , we have ( ) ( ) m R m R V U v v λ = Comparing this with the principal value equations for V and B , n n V V v v λ = (and n n B V v v 2 λ = ) we can see that λ λ λ = = V U , m R n v v = Therefore, we have one set of principle values ( ) III II I λ λ λ , , and two sets of principle...
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EN0175-12 - EN0175 10 12 06 Announcements Mid-term(Tuesday...

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