Tensor operations in micromechanical modeling (UMAT-Abaqus)

[Laspalas]

Hello,

I am programming a micromechanical model for a composite material in Abaqus through the UMAT. This model involves tensorial operations with four order tensors 3x3x3x3 (Cijkl -Compliance, Sijkl- Eshelby, Eijkl- stiffness, ....).

For example for calculating the concentration tensor for the matrix I use the following expresion:

A = ((1-Vf)I + Vf(I + SCm(Ei-Em))^-1)^-1

where A: concentration tensor
Vf: volume fraction
I: identity tensor
S: Eshelby tensor
Ei, Em: Stiffness tensor of the matrix and the inclusion
Cm: Stiffnes tensor of the matrix

When I use contracted notation of these tensors to represent them as a 6x6 matrices, I think I have to introduce extra operations to account for the symmetries of the stress and strain 2nd rank tensors and for the relationship between the strain componets and the engineering strains.

But when I do this my final Stiffness matrix for the composite is not symmetric. So this indicates to me that something is wrong.

I am thinking in performing the operations directly on the tensors instead of on the matrices but I do not have clear how to invert a four rank tensor.

Could somebody give some insight in how to perform the tensor inversion? Or alternatively, how to transform correctly the tensor operations into matrices operations?

Best regards

[Jorgen]

Those are good questions. I am not quite clear on your model formulation, but you mention that the stiffness matrix is not symmetric. If by "stiffness matrix" you mean the Jacobian stiffness matrix then could it not be the case that your model has an unsymmetric Jacobian? Are you sure it should be symmetric?

- Jorgen

[hhspiny]

Hello,

I am programming a micromechanical model for a composite material in Abaqus through the UMAT. This model involves tensorial operations with four order tensors 3x3x3x3 (Cijkl -Compliance, Sijkl- Eshelby, Eijkl- stiffness, ....).

For example for calculating the concentration tensor for the matrix I use the following expresion:

A = ((1-Vf)I + Vf(I + SCm(Ei-Em))^-1)^-1

where A: concentration tensor
Vf: volume fraction
I: identity tensor
S: Eshelby tensor
Ei, Em: Stiffness tensor of the matrix and the inclusion
Cm: Stiffnes tensor of the matrix

When I use contracted notation of these tensors to represent them as a 6x6 matrices, I think I have to introduce extra operations to account for the symmetries of the stress and strain 2nd rank tensors and for the relationship between the strain componets and the engineering strains.

But when I do this my final Stiffness matrix for the composite is not symmetric. So this indicates to me that something is wrong.

I am thinking in performing the operations directly on the tensors instead of on the matrices but I do not have clear how to invert a four rank tensor.

Could somebody give some insight in how to perform the tensor inversion? Or alternatively, how to transform correctly the tensor operations into matrices operations?

Best regards


the so called C matrix should always be symmetric for all materials if you convert the tensor to 6x6 matrix (otherwise, you won't be able to do the conversion). the trick is the 2x conversion between tensor strain and engineer strain.

best way is to find a composite material introduction book, any of them you will find a full set of equation allows you to do all the conversion. it is pretty long equations.

for more generic materia, which does not possess the 3 symmetric planes, then you can not convert to 6x6 matrix.