Hi all,

I am looking for a general definition of the derivative of a tenorial product (e.g. when the expression for Stress contains nonlinear terms in deformation gradient, F ).

I start with a very simple example:

∂F_pq/∂F_mn = δ_pm δ_qn , i.e. Kronecker delta with first index of F_pq and first index of F_mn, and second Kronecker delta for second pair of indices q & n.

However the problem arises when we have a product of two or more tensors. Is the following (using chain rule) a valid derivative?

∂ (A_ij B_jk) /∂F_mn = [∂A_ij / ∂F_mn ] B_jk + A_ij [∂B_jk / ∂F_mn]

I used this method of using indicies, and found that results are sometimes very different than what is written (with any explaination) in textbooks.

E.g. if A=F^-1 and B=F then

∂(F^-1_ij F_jk) /∂F_mn = [∂F^-1_ij / ∂F_mn ] F_jk + F^-1_ij [∂F_jk / ∂F_mn] = 0 (because F-1 • F = I and ∂(I)/∂F = 0

Therefore, [∂F^-1_ij / ∂F_mn ] F_jk = - F^-1_ij [∂F_jk / ∂F_mn]

and [∂F^-1_ij / ∂F_mn ] = - F^-1_ij δ_jm δ_kn F^-1_jk

hence [∂F-1_ij / ∂F_mn ] = - F-1_im F-1_jn

However, in most books, Matrix Cook Book, and the Wikipedia article for Tensor derivatives, I found the second term in result is different (the indices have interchanged their positions), i.e.

[∂F^-1_ij / ∂F_mn ] = - F^-1_im F-^1_nj

I will be very thankful if anyone can kindly explain what went wrong in the derivative (above in red) which I have calculated

Best regards,

Mubeen

Hello all,

I am trying to investigate(as a part of my internship work) 2-3 nonlinear material models of visoelasticity. As usual, free energy has been divided in to two parts, viscous energy function and elastic energy function. In one model, I have elastic energy as a function of strain invariants,viscous energy as function of C(Right Cauchy strain tensor) and C_v(Right Cauchy viscous strain tensor) and a evolution equation of C_v with respect to time. To find stress tensor(of some kind) I have to differentiate it with C or deformation gradient but, I am unable to differentiate C_v by C. Initially, I was assuming that \frac{dC_v}{dC} will be zero but later I realised it is wrong. I have attached a pdf containing my functions and all other information.

P.S.- This is my first question(or blog) if you find something wrong in the way I have written question or the way I asked. Please notify me.

Hi all,

I am looking for an algorithm to get the inverse of a 4th rank tensor (e.g. the compliance tensor S_(ijkl) from elastic stiffness tensor C_(ijkl)) S_(ijkl)=C_(ijkl)^(-1)

I am programming in FORTRAN, and for this purpose I wasn't able to find neither any algorithm nor any existing subroutine.

If anyone at this forum has any idea about this inversion, kindly guide me.

Best regards,

Mubeen.