Dear Rhushikesh,

DDSDDE in ABAQUS UMAT has a size of n*n where n is the sum of the number of direct and shear stress components and depends the way the problem is modeled (3D/2D/1D). For a 3D problem with 3 direct and 3 shear stress components, this leads to a 6*6 matrix. As far as I am aware this dimension has to be maintained while defining DDSDDE.

Eventhough the stiffness matrix (Cijkl) is a fourth order tensor with 81 components, it can be represented by 36 components due to minor diagonal symmetries i.e symmetry of stress and strain tensor ( Cijkl = Cjikl, Cijkl = Cijlk). This leads the the stiffness tensor being written in matrix form or the Voigt notation with 36 components. Major diagonal symmetry ( Cijkl=Cklij) further reduces the number of unique components of the stiffness tensor to 21 (For a general anisotropic material).

Regards,

Abhilash

Thanks for the answer.

But actually there are 6 shear components in 3D (ofcourse they are symmetric) so we can represent the stress matrix with 9 components which should lead to 9*9 arrangement of DDS and DDSDDE. What is your thought on this?

Regards,

Rhushikesh

Dear Rhushikesh,

Mechanical constitutive relations can be represented either in tensor notation (9 stress/strain components, 81 stiffness components) or in matrix notation/Voigt notation ( 6 stress/strain components, 36 stiffness components) using the symmetry property of stress and strain. Both are equivalent and the matrix notation is a simplified form with the redundant relations arising due to symmetry of stress and strain eliminated.

Since ABAQUS uses the Voigt notation, the stiffness matrix will be 6*6 for 3D elements. This is the dimensions of the DDSDDE array ABAQUS needs users to follow in a UMAT.

Hope this clarifies your question.

Regards,

Abhilash