UMAT - definition of DDSDDE 1

[jojo998]

In a UMAT calculation I have to return the STRESS and the tangent DDSDDE, I am not sure about the correct definition of the tangent.

(1) The STRESS is the Cauchy stress.

(2) The Green-Lagrange strain E is calculated via the deformation gradient F=DFGRD1 as E = 1.0/2.0 * ( transpose(F)*F - 1)

(2) Let's suppose we have a strain energy function W, which should be a function of the Green-Lagrange strain E.
The second Piola-Kirchhoff stress is yielded as S=dW/dE.
A push-forward operation and division by J=det[F] yields the Cauchy stress sigma=(1/J)*F*S*transpose(F).

(3) The tangent C can be calculated as C=d2W/dE2, so as the second derivative of the strain energy function with respect to the Green-Lagrange strain.
This gives dot(S) = C : dot(E).
This can be transformed to the spatial configuration to the Truesdell rate of Cauchy stress circ(sigma) and rate of deformation tensor d.
This gives circ(sigma) = c : d, with c=(1/J)*F*F*C*transpose(F)*transpose(F).
A possible spatial tangent c is defined.

The Abaqus Documentation says that DDSDDE is defined as partial(Delta(sigma))/partial(Delta(epsilon)). The documentation says that Delta(sigma) are the stress increments (I reckon Cauchy stress) and Delta(epsilon) are the strain increments.

i) What exactly is epsilon in a finite strain simulation? The logarithmic strains? How can epsilon be transformed to d=sym[dot(F)inv(F)]?

ii) How can I transform the given spatial tangent c to a correct DDSDDE?

 

[TERIO]

1. I looked in the Analysis users manual and saw this in section 1.2.2:

The default “integrated” strain measure, E, output by ABAQUS/Standard to the data (.dat) and results (.fil) files for all elements that can handle finite strain is obtained by integrating the strain rate numerically in a material frame:

epsilon_(n+1)=dR*epsilon_*dR^T+depsilon

where

depsilon=integral from t_ to t_(n+1) of D dt

So it looks like they rotate the strain with the material. It appears that the increment is D=sym(L) integrated over time.


2. I'm not sure how you would transform the tangent. On the bright side it doesn't have to be exactly right anyway, it will just converge slower if it is wrong.

 

[jojo998]

Thank you very much for you answer.

However, I cannot believe that I am the only one facing this problem (I have not found an answer in forums), continuum mechanics is described via several strain measures, Green-Lagrange strains, Euler-Almansi strains, Rate of deformation tensor, let's even include Hencky strains etc.

What exactly does Abaqus demand? How can a material model given in classical textbooks be integrated?

Regards.

 

[xerf]

I believe you are not the only one, but in general most people looking for answers on the forums are not experts at writing UMATs.

I recommend the book by Dunne and Petrinic - Introduction to Computational Plasticity which describes computational plasticity (UMAT) in context of Abaqus. The book by Simo and Hughes on Computational Inelasticity is very good also but more general.

I think the general approach to obtain the material stiffness matrix is to linearize the constitutive response.

There is a J2 plasticity UMAT example somewhere in the Abaqus documentation (either Example or Verification manual). Please note that some UMAT can get quite complicated, I have seen a couple research UMATs having thousands of code lines (as compared to linear elasticity which can be described with several code lines).Some people spent an entire Phd program to come up with a UMAT for a new material, therefore writing UMAT can be non-trivial.

I recommend you take a look at the Theory manual also, the chapter on material models.