I am trying to make sure I understand your question. So you have the equilibrium equation (rho d^2U/dt - \nabla \sigma == F) and you would like to explain the relation to the function SolidMechanicsStress. Is this correct? I have the feeling that I am misunderstanding something.
In fact what I am trying to understand is how to write the stress operator form of the -nabla sigma epsilon.
The form it is put in i.e; SolidMechanicsPDEComponent[{{u[x, y, z], v[x, y, z], w[x, y, z]}, {x, y, z}}, isotropicPars] // MatrixForm
when one runs it gives an unevaluated form of the equilibrium equations.
The thing that I don't quite get is how one goes about creating this operator. I see now in 13.0 with the new solid mechanics package it is already there, but say if I wanted to create my own version of an operator, how do I go about doing this
Perhaps I am just thinking the wrong way.
Take the image of the output of the SolidMechanics PDE Component
Here I see that the top row would be the equivalent of the nabla sigma term (in the x-coordinate), which is the sum of terms in w, v and u.
Each of these terms consists of different 3x3 matrices (which are all sub-matrices of the elasticity matrix), multiplied by the gradients in the displacements (u,v,w). My question is how are these sub-matrices created?
Maybe things are becoming clearer (especially with the new solid mechanics tutorial)
Perhaps the way for me to understand it is as follows:
the first term in the top row consists of a 3x3 matrix multiplied by the gradient of w
the gradient of w is a list of (gamma xz, gamma yz and epsilon zz)
Ah, I just saw below in the solid mechanics text the following
SolidMechanicsPDEComponent[{{u[x, y, z], v[x, y, z], w[x, y, z]}, {x, y, z}}, anisotropicPars] // MatrixForm
this is quite helpful. Does this mean say the first term in the top line i.e.{{-c15,-c14,-c13},{}{}}.nabla w gives sigma xx, tau xz and tau xy ?
i.e., {{-c15,-c14,-c13},{}{}}.nabla w = {sigma xx, tau xz, tau xy}
Ah, think I have worked it out now. I was getting completely confused by the Voigt notation and forgetting that we are dealing with tensor multiplication. If I understand right, each of the three x three matrices is one of the "submatrices" of the full elasticity tensor.
Yes, exactly right. I am working on the background for an answer. Please allow for some time before I get back to you. The problem is that I do not mention what the relation of the equilibrium equation is to the out put get from SolidMechanicsPDEComponent. I'll get back to you.
@Dunlop, I have notebook that gives an explanation and some background. I have uploaded that to wolframcloud.com/obj/ruebenko/Published/… if you can not access that, please let me know your email such that I can send it to you. You can reach me at ruebenko AT wolfram DOT com
Have a look at this an let me know if this is useful, for you or your students. If so, I'll include this in a future version of the solid mechanics monograph.
