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 Kevin A. Goodheart September 3, 2002 10:31

Tensor Notation

Here is the problem:

T_ij = S_ij - .3*S_kk + S_ik*S_kj-.3*S_kl*S_kl

The first term is i=1,j=1,k=1

T_11 = S_11 -.3(S_11+S_22+S_33) + S_11*S_11 - .3*(?)

What do I do with this kl. I am trying to implement the explicit algebraic stress model, one could argue that if I can't figure out the tensor notation then I shouldn't be trying to implement an EASM model but that is just one of many problems.

Thank You, Kevin

 sylvain September 3, 2002 11:10

Re: Tensor Notation

(?) = S11*S11+S12*S12+S13*S13

+ S21*S21+S22*S22+S23*S23

+ S31*S31+S32*S32+S33*S33

It's the same for any ij. Was not it written S_kl*S_lk ? Which is the same if S is symetric,

 h.jordi September 3, 2002 11:13

Re: Tensor Notation

hi kevin.

T_ij = S_ij - (S_kk) + [S_ik*S_kj] - {S_kl*S_kl}

you understood the first bracketed term ( ) was an implied sumation.

well the second bracketed term [ ] is also an implied summation on k. so you get =

S_i1*S_1j + S_i2*S_2j + S_i3*S_3j

and, the third term { }. this has a double repeated index, l and k and so you sum on both. giving =

S_11*S_11 + S_12*S_12 + S_13*S_13 + S_21*S_21 + S_22*S_22 + S_23*S_32 + S_31*S_31 + S_32*S_32 + S_33*S_33

H.

 Kalyan September 3, 2002 12:30

Re: Tensor Notation

Kevin,

The expression

T_ij = S_ij - .3*S_kk + S_ik*S_kj-.3*S_kl*S_kl

is not tensorially consistent because there are scalars (2ns and 4th terms on the right) mixed in with tensors (other terms). A correct form would be

T_ij = S_ij - (.3*S_kk) delta_ij + S_ik*S_kj - (.3*S_kl*S_kl) delta_ij

Terms involving repeated indices (k and l) have an implicit double summation.

 Kevin A. Goodheart September 4, 2002 02:27

Re: Tensor Notation

Thank You all for you help with my Tensor Notation and also catching my two mistakes, not summing the second term and leaving out the delta_ij function.

Now time to have fun solving this equaiton.

Kevin

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