Question about calculationg dev[b] in uniaxial tension.

[yzsong]

Hello,Jorgen
I have a question about Arruda-Boyce 8-chain model. This question
confused me a lot.In Arrduda-Boyce 8-model the cauthy stress

T=A*dev[B]

here A is the expression that I cannot type it in notepad.exe.

in uniaxial tension of incompressible material the stress

T=A*(λ*λ-1/λ)

However, I cannot calculate this value. My calculation is listed as follows.
Is there any wrong in calculating dev[B]


λ*λ
B= 1/λ
1/λ
dev[B]=[B]-1/3trace[B]I

2λ*λ-2/λ
c=dev[B]=[B]-1/3*(λ^2+2/λ)I=1/3* 1/λ-λ*λ
1/λ-λ*λ
In my thought c11 should be equal to λ*λ-1/λ and c22=c33=0, but I cannot get that
result. Could you please point out my mistake?

thanks a lot.

[sq]

This is an easy mistake- I've made it myself.

c22, c33 should not equal zero. The zero stress condition in the 22 and 33 directions comes on application of the zero-traction boundary conditions.

[yzsong]

Dear sq.
Thanks for your reply.
There is a coefficient of 2/3, how to get rid of it? I noticed that there was not 2/3 in the formula of the literature.

[sq]

Okay, good question, so here's the trick: application of the boundary condition stress22 = stress33 =0 means that to complete the uniaxial formulation you have to subtract 1/lambda - lambda^2 from all three diagonal elements in the result. Give it a try!