Kinetic Description

The virtual power principle in Virtual Power Principle will state equilibrium in terms of Cauchy true stresses and the conjugate virtual strain rate, the rate of deformation. It is worth noticing that, from the engineer's point of view, the Cauchy true stress is probably the only measure of practical interest because it is a direct measure of the traction being carried per unit area of any internal surface in the body under study. This is the reason why Radioss reports the stress as the Cauchy stress. The second Piola-Kirchhoff stress is, however, introduced here because it is frequently mentioned in standard textbooks.

The relationship between the Piola-Kirchhoff stress and the Cauchy stress is obtained as follows. Starting from the definition of Green's strain as explained in Kinematic Description, Equation 20 ,(1)
E = 1 2 ( F T F − I )
the strain rate is given by:(2)
E ˙ = 1 2 ( F ˙ T F + F T F ˙ )
The power per unit reference volume is:(3)
P = E ˙ S
Where S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaaaa@36CE@ represents the tensor of second Piola-Kirchhoff stresses. On the other hand, for Cauchy stresses:(4)
P = ε ˙ σ | F |
(5)
( F ˙ T F + F T F ) S = ( F ˙ F − 1 + F − T F ˙ T ) σ | F |
You immediately have:(6)
F S F T = σ | F |

Second Piola-Kirchhoff stresses have a simple physical interpretation. They correspond to a decomposition of forces in the frame coordinate systems convected by the deformation of the body. However, the stress measure is still performed with respect to the initial surface.