Hi,
I’m still pretty new at Fenics and i want to solve a coupled field problem. I’m using large deformations and I was looking to some examples in the documentation. I found this: 9. Hyperelasticity — FEniCS Project
The right Cauchy-Green tensor C is given by: C = F.T * F
However, i thought that you have to use C = dot(F.T, F), because it’s the dot-product C_{KL} = F_{Km} F_{mL}.
I need to use the dot- and inner-product very often, so i would like to know, if there is any difference between these two formulations dot() and ()*().
thank you for your quick response.
I have a further question, even if it’s not about the mentioned topic:
In the example of hyperelasticity, the deformation gradient is given by: F = I + grad(u)
I thought, that grad(u) is defined as u_i,j; but in my opinion, it should be like that : F_{iJ} = I_{iJ} + \frac{\partial u_i}{\partial X_J}. So it should be the derivation of the displacement with respect to the reference configuration.
So the question is: If I use large displacements, is the gradient defined with respect to the reference configuration? \nabla( \, ) = \frac{\partial (\,)}{\partial X_J}
The grad operator differentiates with respect to the physical coordinates of the mesh. Everything in the hyperelasticity demo is mathematically consistent with the mesh being defined in the reference configuration. I think your source of confusion may be coming from the fact that much of the problem-independent API and UFL documentation uses a lower-case “x” for mesh coordinates, e.g., defining dx as the integration measure over the mesh, or using x as a generic coordinate in Expressions. However, the physical interpretation of that coordinate in a specific application depends on how you define your variational forms.
thank you very much for your detailed explanation. It helped me alot.
I’m just a little bit confused right now, because I try to compute a coupled field problem and it does not converge. And at the moment I have absolutely no idea what the mistake is. But thanks a lot!