Statement of this problem:
In the textbook, the following differential equilibrium equations can be expressed by tensors:
Using Einstein's summation convention, the formula in the figure above can be abbreviated as follows:
In addition, the strain coordination equations in the figure below can be abbreviated as:
I would like to know how to implement the above summation convention with the help of MMA's tensor operator.
My problem is slightly different from this one because I have involved derivation operations and I need to use notation like $div(σ)+F=0$ to memorize differential equilibrium equations to reduce the burden of memorizing deformable compatibility equations.
Objectives to be addressed of this question:
I want to find a universal tensor operation function to express the equations expressed by various tensors in elasticity as shown in the figure below (thank you very much for xzczd's answer, which has made a good demonstration meeting my requirements).
If I could, I would like to find a way to express the deformation compatibility equation expressed by strain or stress in a similar way as $div(σ)+F=0$ represents the equilibrium differential equation, so as to reduce the burden of memory.
That's the main purpose of this question. I hope I can solve this problem with your help.
What needs further explanation in the comments:
The tensor operation I mentioned mainly refers to the tensor with derivative in the textbook. For example,
kl
after the comma in the lower corner of $e_{ij,kl}$ represents the second derivative of $e_{ij}$. This is different from the usual tensor description.Part of my question can also be expressed as "can I have a function that convert $σji,j+Fi=0$ to ".
The textbook I used didn't specify the specific meaning of the first two
ee
, but I saw the relevant answers, I think it should meanLeviCivitaTensor
.
Div
; it works also on matrices. Actually, a basis-free notation for the above would be $\operatorname{div}(\sigma) + F = 0$, wo why struggling with indices? $\endgroup$ – Henrik Schumacher Mar 22 '20 at 10:53LeviCivitaTensor
, and the last two $e$ denote strain. (Once again, we still need OP's clarification, of course. ) $\endgroup$ – xzczd Mar 22 '20 at 12:11