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# STRESS and DDSDDE in ABAQUS UMAT

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(@sjimenez)
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Joined: 16 years ago

Hi, all:

Im in the process of learning UMAT. I have some questions. Hope you can help.

(1) My understanding is that UMAT requires calculation of Cauchy stress and store them in STRESS. Is it correct?

(2) My strain energy function, W, is a function of Green strain, E. So, S=dW/dE yields the 2nd Piola-Kirchhoff stress. I should push forward the P-K stress to get Cauchy stress via sigma=(1/J)*F*S*transpose(F). Is is correct?

(3) Since logarithmic strain, not the Green strain, is passed into UMAT through STRAN, in order to calculate the Cauchy stress, sigma, I need to obtain the Green strain from the deformation gradient F. Is it correct?

(4) Finally, in an earlier threat ([url] http://polymerfem.com/forums/archive/index.php/t-374.html [/url]), Jorgen indicated that DDSDDE is the partial derivative of the increment in Kirchhoff stress with respect to the increment in logarithmic strain. However, in an ABAQUS training material, it is stated as follows:

For small-deformation problems (e.g., linear elasticity) or

large-deformation problems with small volume changes (e.g., metal

plasticity), the consistent Jacobian is

C=partial derivative(delta_sigma)/partial derivative(delta_epsilon)

where delta_sigma is the increment in (Cauchy) stress and delta_epsilon is the increment in strain. (In finite-strain problems, delta_epsiolon is an

approximation to the logarithmic strain.)

Im really confused now:confused:. Shouldnt DDSDDE be calculated as the partial derivative of the increment in Cauchy stress with respect to the increment in Almansi strain, because they are a conjugate pair?

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(@jorgen)
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Joined: 5 years ago

(1) Yes

(2) If your model is purely hyperelastic you might want to use a UHYPER subroutine. Your approach otherwise seems OK - you need the Cauchy stress.

(3) Sounds right

(4) I dont have the Abaqus training materials, and I still believe that the definition that I quoted is valid (based on the Abaqus docs).

- Jorgen

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Topic starter
(@sjimenez)
New Member
Joined: 16 years ago

Since the deformation gradient tensor passed into UMAT is in a modified form,
F_bar=F*(J_bar/J)^(1/n)
Can I still use this modified deformation gradient tensor to calculate the Green strain,
E=0.5*[transpose(F_bar)*F_bar-I]?

If so, should I use DFGRD0 or DFGRD1?

Thanks.

Posts: 3998
(@jorgen)
Member
Joined: 5 years ago

For practical purposes you can consider DFGRD0 and DFGRD1 as the true deformation gradients. You can use them to calculate the the greens train at t0 and t0+dt.

- Jorgen

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