stress and stiffness quantities to be used in TANG and FORM

[fethio]

Dear FEAP users,

There are two commonly encountered measures of stress in the formulation and post-processing of finite element problems. Namely, the Cauchy stress (true stress) and the 2nd Piola-Kirchoff (PK). Energetically these are conjugated by the Almansi and the Green strains, respectively. 

FEAP, reports in its output file the Cauchy stress along with the Almansi strains. In version 8.5, Green strains also appear in the output file, as a result of the STREss command.

During the formulation of the stiffness matrix in a total lagrangian framework, the PK stresses are utilized along with the Green strains. I am somewhat confused about which stress and stiffness measure is used during the TANG and FORM procedures. It seems that the PK stress and its stiffness counterpart are used, but where is this converted into a Cauchy stress during the output phase??

I am working on converting the pfiber element into a contractile one, and I clearly need to understand what should be used for the active stress and stiffness components.

[FEAP_Admin]

Most of FEAP's elements assemble the weak form equations in the current configuration, which effectively uses the Kirchhoff stress.
Have a look at Zienkiewicz, Taylor, and Fox, The finite element method for solid and structural mechanics, Section 5.3.3.

[fethio]

Dear Admin,

Thank you for the reply and the reference. I have an older edition and was able to track down the section that you referred to. I understand how the Kirchoff stress can be interchanged with the 2nd PK as a result of expressing the variational form in the current configuration.

Nevertheless, in the pfiber formulation there is direct reference to the 2nd PK stress (the multiplication of the structure tensor by  two, and the note which says: S_IJ = 2 * dPsi/C_IJ ).  So, should we understand that in this element (pfiber.f) integration is performed in the reference configuration?

[Prof. S. Govindjee]

Stress computations can be done in the code in many different ways, but in almost all elements they are pushed to the current configuration for assembly of the weak form.

[fethio]

Thank you for your reply Prof Govindjee.

You mention that

in almost all elements they are pushed to the current configuration for assembly

So, I guess pfiber.f is one of the few exceptions? Or, is the push forward operation applied afterwards?

The reason I am confused about this is because of lines 198-199 in pfiber.f

Code: [Select]

!       Multiply by 2 (S_IJ = 2 * dPsi/C_IJ)
        mm(:) = 2.d0*mm(:)

Here, it seems things are computed in terms of a referential basis...

[K.Li]

I do not have FEAP 8.5, but in FEAP 8.4,  a factor of 2 was multiplied to tau.

Code: [Select]

c       Multiply by 2 (S_IJ = 2 * dPsi/C_IJ)
        tau    = 2.d0*tau
This factor of 2 is the 2 before dPsi/C_IJ