thanks to this forum, I already found a lot about the correct definition of the tangent modulus in Umat subroutines. I am testing my subroutines on a simple cube and unfortunately, my subroutines only work for normal tension. It seems that I am still doing some systematic error.
Following the user subroutine reference manual, the material jacobian matrix is
DDSDDE = (delta sigma)/(delta epsilon)
So this is Cauchy stress derived to logarithmic strain. On the same site, there is also the definition: For rate form constitutive laws the tangent moduli is given as
DDSDDE = 1/J (partial delta (J sigma)) / (partial delta epsilon)
with J = Det(F). So this is the Kirchhoff stress derived to the logarithmic strain. Further I found out that Abaqus needs the tangent modulus with Jaumann stress rate.
As I am doing large strain analysis, I do not use the logarithmic strain, but the deformation gradient DFGRD1 to compute strain and stresses at the end of the increment. Second my constitutive law is pure elastic, so far I do not use a rate formulation. The material is defined by a strain energy potential Psi(C), with Cauchy Green deformation tensor C = DFGRD1transpose DFGRD1.
I tried two approaches so far to compute DDSDDE:
Tangent modulus in spatial frame: DDSDDEspatial = 4 dd Psi(C) / dC dC
Then push forward to material frame: DDSDDEmaterial = 1/J DFGRD1 DFGRD1 DDSDDEspatial DFGRD1transpose DFGRD1transpose
Convert to Jaumann: DDSDDE = DDSDDEmaterial + h 
I implemented a numerical approximation of DDSDDE as proposed in . The algorithm estimates the tangent moduli by a forward differentiation of the stress.
Unfortunately, both approaches do only work for normal tension, shear or combined loading fails.
What does Abaqus need? Cauchy or Kirchhoff tangent moduli?
Is there anything in the .inp file I need to consider? Element types etc.
Thanks in advance any hints how to get my UMAT working!
 Belytschko 2001: Nonlinear Finite Elements for Continua and Structures
 Sun 2008: Numerical approximation of tangent moduli for finite element implementations of nonlinear hyperelastic material models