I strongly recommend book

"Nonlinear Solid Mechanics: A Continuum Approach for Engineering" by Gerhard A. Holzapfel

to study such problems and understand basics about different stress and strain tensors.

Formally, material time derivative of deformation gradient is energy conjugate pair to I PK tensor.

Maybe those documents will help you in developing your FE:

https://www.sintef.no/globalassets/project/evitameeting/2012/kmm-geilo-2012-lecture-11b.pdf

http://kis.tu.kielce.pl//mo/COLORADO_NFEM/colorado/NFEM.Ch09.pdf

Thank you Stanisław Burzyński

I strongly recommend book:

An Introduction to Continuum Mechanics by Morton E. Gurtin ,

a classic in the field of continuum mechanics.

Fabrizio

Thank you Fabrizio Morlando

Not vector, but bivalent tensor...

First one is non-symmetric, it connects forces in deformed stressed configuration to underfomed geometry+mass (initially known volumes, areas, densities), and it is energetically conjugate(* to the motion gradient (commonly mistakenly called “deformation gradient”, despite comprising of rigid rotations). First (sometimes its transpose) is also known as “nominal stress” and “engineering stress”.

[ I denote it as bold T, see picture ]

Second one is symmetric, it connects loads in initial undeformed configuration to initial mass+geometry, and it’s conjugate(* to the right Cauchy–Green deformation tensor (and thus to the Cauchy–Green–Venant measure of deformation).

The first is simplier when you use just motion gradient and is more universal, but the second is simplier when you prefer right Cauchy–Green deformation and its offsprings.

There’s also popular Cauchy stress, which relates forces in deformed configuration to deformed geometry+mass.

*) “energetically conjugate” means that their product is energy, here: elastic (potential) energy per unit of volume

Thank you Vadique Myself

But which one preferred in computation mechanics?

Could you provide examples, please.

Thanks

> But which one [is] preferred in computation mechanics?

It depends on what do you compute (and how). In fluid mechanics, for example, both of them are likely fully useless, because you don’t have (don’t care for) known stiff initial configuration.

Personally, I prefer first Piola–Kirchhoff stress, mostly because I prefer to use motion gradient. But Green’s deformation is much more popular, especially for little displacements (they are larger in rods/beams, and are much smaller for shells and three-dimensional bodies — that’s why motion gradient is popular for one-dimensional modelling). And there’re many examples, they are easily googlable. For second tensor https://adtzlr.github.io/ttb/example_stvenantkirchhoff.html. Plus for sure the source code of Code_Aster does worth diving into it https://bitbucket.org/code_aster/codeaster-src/src

Thank you so much Vadique Myself

But using first Piola–Kirchhoff stress leads to un-symmetric stiffness matrix, as deformation gradient is un-symmetric tensor, is it right?

Nope.

Tetravalent stiffness “ceiiinosssttuv” tensor of nonpolar (momentless) continuum is constant in its symmetry no matter which measure of stress you use. When you choose just motion gradient, then it’s second derivative of elastic energy per volume unit by nabla location vector (transposed motion gradient) and then by transpose of it (which is motion gradient). That’s the same as double derivation by Green’s deformation.

In the question about what’s the first — chicken (deformation) or egg (stress), the answer is — none of them, but the instinct of reproductivity (energy)

(-;

Second Piola-Kirchoff tensor is symmetric but 1st piola is not , which gives advantages in computation.

Thank you Prithivirajan Veerappan

But if I use 1st Pilola tensor, linearization of weak form of virtual work leads finally to symmetric stiffness matrix for conservative system.

Dear Khaled,

The first Piola Kirchoff stress tensor relates the Cauchy stress tensor to the stress in the deformed space. This is not a symmetric tensor and for computational ease, this we use a the second Piola Kirchoff stress which is symmetric in nature (as it is a result of the product between the Finger tensor and Cauchy stress tensor).