Lagrangian Stresses Piola Kirchoff Stress Tensors

96 In Sect. 2.2 the discussion of stress applied to the deformed configuration dA (using spatial coordiantes x), that is the one where equilibrium must hold. The deformed configuration being the natural one in which to characterize stress. Hence we had df = tdA (4.147-a)

(note the use of T instead of a). Hence the Cauchy stress tensor was really defined in the Eulerian space.

97 However, there are certain advantages in referring all quantities back to the undeformed configuration (Lagrangian) of the body because often that configuration has geometric features and symmetries that are lost through the deformation.

98 Hence, if we were to define the strain in material coordinates (in terms of X), we need also to express the stress as a function of the material point X in material coordinates.

4.6.1 First

99 The first Piola-Kirchoff stress tensor T0 is defined in the undeformed geometry in such a way that it results in the same total force as the traction in the deformed configuration (where Cauchy's stress tensor was defined). Thus, we define df = t0dA0 (4.148)

where t0 is a pseudo-stress vector in that being based on the undeformed area, it does not describe the actual intensity of the force, however it has the same direction as Cauchy's stress vector t.

100 The first Piola-Kirchoff stress tensor (also known as Lagrangian Stress Tensor) is thus the linear transformation T0 such that t0 = T0n0 (4.149)

and for which df = t0dA0 = tdA ^ t0 = using Eq. 4.147-b and 4.149 the preceding equation becomes

T0n0

ddA0

and using Eq. 4.36 dAn = dA0 (det F) (F n0 we obtain

T0n0 = T(det F) (F-1f n0 the above equation is true for all n0, therefore

101 The first Piola-Kirchoff stress tensor is not symmetric in general, and is not energitically correct. That is multiplying this stress tensor with the Green-Lagrange tensor will not be equal to the product of the Cauchy stress tensor multiplied by the deformation strain tensor.

102 To determine the corresponding stress vector, we solve for T0 first, then for dA0 and n0 from dA0n0 = det FFtn (assuming unit area dA), and finally t0 = T0n0.

4.6.2 Second

103 The second Piola-Kirchoff stress tensor, T is formulated differently. Instead of the actual force df on dA, it gives the force df related to the force df in the same way that a material vector dX at X is related by the deformation to the corresponding spatial vector dx at x. Thus, if we let df = tdA0

where dff is the pseudo differential force which transforms, under the deformation gradient F, the (actual) differential force df at the deformed position (note similarity with dx = FdX). Thus, the pseudo vector t is in general in a differnt direction than that of the Cauchy stress vector t.

104 The second Piola-Kirchoff stress tensor is a linear transformation T such that t = T no

t thus the preceding equations can be combined to yield df = FT n0 dA0

we also have from Eq. 4.148 and 4.149

df = t0dA0 = T0n0ddA0 and comparing the last two equations we note that

which gives the relationship between the first Piola-Kirchoff stress tensor T0 and the second Piola-Kirchoff stress tensor T.

105 Finally the relation between the second Piola-Kirchoff stress tensor and the Cauchy stress tensor can be obtained from the preceding equation and Eq. 4.153

and we note that this second Piola-Kirchoff stress tensor is always symmetric (if the Cauchy stress tensor is symmetric). It can also be shown that it is energitically correct.

106 To determine the corresponding stress vector, we solve for T first, then for dA0 and n0 from dA0n0 = dt F Ftn (assuming unit area dA), and finally t = Tn0.

Example 4-14: Piola-Kirchoff Stress Tensors

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