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cauchy stress, first piola kirchoff stress, second piola kirchof stress

Some basic questions involving cauchy stress, first piola kirchof stress and second piola kirchof stress:

1) We know that Cauchy stress involved deformed areas-therefore this (Cauchy stress) has an obvious physical interpretation

2)Now, first piola kirchof stress is expressed as:

S = JF^-1 . sigma

where, J is the jacobian of the deformation gradient which physically is the measure of the volume change produced by a deformation.

F is the deformation gradient

sigma is Cauchy stress.

My question is, what is the mathematical justification of S?If I were to derive the expression(S = JF^-1 . sigma) for "S"  how woul I start?

3)Similarly second piola kirhoff stress is:

= JF^-1 . sigma F^-T

In this case, If I were to derive the expression for second piola kirchof stress (that is the expression JF^-1 . sigma F^-T)   how would I start?

arash_yavari's picture

There was a long discussion on stress tensors a while ago. Look at:


Yes, I see the discussion on contravariant and covariant relationship of stress tensors.

However, my question was not on the contravariant and covariant part.

My question was that, what is the physical significance of first piola kirchof stress being:

S = JF^-1 . sigma

Similarly, what is the physical significance of second piola kirchof stress being:

=JF^-1 . sigma F^-T

What is the basis of the above relationships?

arash_yavari's picture

Given a small surface with unit normal "n" in the deformed configuration, Cauchy stress acting on it gives you the traction (force per unit area) on this surface. When the first Piola-Kirchoff stress acts on the unit normal "N" of the undeformed surface it gives you the same traction in the deformed configuration. If you pullback the traction to the underformed configuration using F, the second Piola-Kirchhoff stress gives you this pullback traction when acting on "N". The relation between these stresses is a simple consequence of the two definitions I just mentioned and can be found in any text on continuum mechanics. Look at the one by Spencer.


I'm following the book by Bower.

So, does it mean:

considering the first piola kirchoff stress,

S = JF^-1 . sigma

the reason we multiply by F^-1 is to get back the undeformed configuration?Why does "J" also come into the expression then?


 for ANY dF/dt

 we have the following stress power identity:

    stress power= (\sigma :d)*dv =( \Pi : dF/dt)*dV =(S : dE/dt)*dV,  (1)

 where \sigma is the Cauchy stress, \Pi is the first P-K stress, S is the second P-K stress

 F is deformation gradient tensor, dF/dt  is its material derivative

 d is the velocity gradient tensor (with respect to current configuration)

d=1/2(l+l^\top), l=(dF/dt).(F^-1) 

E is the Green strain tensor and  dF/dt   denotes its material derivative

 dv is the material volume element of current configuration and dV is the

material volume of reference configuration. dv=J*dV 

  Then from the  stress power identity in (1) and the arbitrariness of dF/dt,

   you can derive the expressions of 1st and 2nd P-K stress.

  I think the more natural way to derive the stress measure is from its energy conjugate

 strain measure. That is to say you should first identify what kind of strain measure will

 enter into the stress power indentity, then  you can DERIVE  the corresponding stress

measure expressions by some MATHEMATICAL OPERATIONS.  From my point of view,

 "strain" is more fundamental than "stress" since it is a geometry quantity.

 But some (derived) stress measures also have "(fictitious) physical meanings" as

documented in the continuum mechanics books. 


 best regards

 Xu Guo





ok, fine i got the derivations- I have attached  a pdf file (in the frst post of this thread) which contains extract of bowers chapter 2- setion on internal forces- I have put my question there in regard to cauchy and first piolla kirchhoff stress.Can Arash, Xu or Sir Alam Bower or anyone help with the basic fundamental question?

Hello, here are 2 pages from my disseration explaining why P (the first Piola-Kirchhoff stress) has the form that it does.  Basically, P is chosen such that it satisfies eq I.3.13.  The terms J and F^-1 come in to play when you are comparing areas in the reference configuration to those in the current configuration.  Hope it helps.

The second Piola-Kirchhoff stress is less easy to justify on physical grounds, but it was formed basically because it is symmetric and "lives" entirely in the reference configuration.


Tim Kostka

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