What is Piola-Kirchhoff stress?

Nominal stress/First Piola–Kirchhoff stress This stress is unsymmetric and is a two-point tensor like the deformation gradient. The asymmetry derives from the fact that, as a tensor, it has one index attached to the reference configuration and one to the deformed configuration.

What is second Piola-Kirchhoff stress tensor?

On the other hand, the second Piola-Kirchhoff stress tensor is defined as the tensor producing a force vector when applied to the undeformed area vector . is sometimes termed the “pull-back” of the force vector .

What is the trace of the stress tensor?

The trace of the stress tensor Tr(σ) is also known as the first invariant J1 of the stress.

What is the deformation gradient?

The deformation gradient F is the derivative of each component of the deformed x vector with respect to each component of the reference X vector.

What is Nansons formula?

Nanson’s formula is an important relation that can be used to go from areas in the current configuration to areas in the reference configuration and vice versa. This formula states that. d a n = J d A F − T ⋅ N {\displaystyle da~\mathbf {n} =J~dA~{\boldsymbol {F}}^{-T}\cdot \mathbf {N} }

Is Cauchy a stress tensor objective?

Thus, both the left Cauchy-Green deformation tensor B and the Eulerian strain tensor e = (I − B−1)/2 are objective, whereas the right Cauchy-Green deformation tensor C and the Lagrangian strain tensor E = (C − I)/2 are nonobjective.

How do you calculate principal stress from a stress tensor?

In 2-D, the principal stress orientation, θP , can be computed by setting τ′xy=0 τ ′ x y = 0 in the above shear equation and solving for θ to get θP , the principal stress angle. Inserting this value for θP back into the equations for the normal stresses gives the principal values.

What is the Green strain tensor?

Green Strain Definition. The Green strain tensor, E , is based on the deformation gradient as follows. E=12(FT⋅F−I) Recall that FT⋅F F T ⋅ F completely eliminates the rigid body rotation, R , from the problem because. FT⋅F=(R⋅U)T⋅(R⋅U)=UT⋅RT⋅R⋅U=UT⋅U.

What is the Cauchy-Green tensor?

Right Cauchy-Green tensor CIJ is one of the deformation tensors which excludes rigid rotation as follows: (2.300) CIJ can be physically described as the change in the square of relative position vector in an undeformed body to that of a deformed one as follows: (2.301)

What is Cauchy-Green deformation tensor?

Physically, the Cauchy–Green tensor gives us the square of local change in distances due to deformation, i.e. Invariants of are often used in the expressions for strain energy density functions. The most commonly used invariants are. where is the determinant of the deformation gradient and.