∂
2 E xx
∂y 2 þ
∂
2 E yy
∂x 2 À 2
∂
2 E xy
∂x∂y
¼ 0
∂
2 E yy
∂z 2 þ
∂
2 E zz
∂y 2 À 2
∂
2 E yz
∂y∂z
¼ 0
∂
2 E zz
∂x 2 þ
∂
2 E xx
∂z 2 À 2
∂
2 E zx
∂z∂x
¼ 0
À
∂
2 E xx
∂y∂z
þ
∂
∂x
À
∂E yz
∂x
þ
∂E zx
∂y
þ
∂E xy
∂z
¼ 0
À
∂
2 E yy
∂z∂x
þ
∂
∂y
∂E yz
∂x
À
∂E zx
∂y
þ
∂E xy
∂z
¼ 0
À
∂
2 E zz
∂x∂y
þ
∂
∂z
∂E yz
∂x
þ
∂E zx
∂y
À
∂E xy
∂z
¼ 0
These six compatibility equations must be satisfied to ensure a compatible strain
field and to ensure that only a single-valued displacement exist at any point.
Therefore, compatibility conditions and sufficient conditions for unique displacement values are necessary.
While there are six compatibility equations, only three of them can be linearly
independent because they are obtained from three independent displacements u, v, w.
2.13 Piola-Kirchhoff Stress Tensors
Cauchy stress tensor is defined in the spatial coordinates (in the deformed configuration). Euler strain is also defined in spatial position in the deformed configuration.
Therefore, using Euler strain definition with Cauchy stress tensor is appropriate.
On the other hand, Lagrangian and material (local) formulations are based on
undeformed configuration or any other reference state configuration that does not
change. In this case, it is also more appropriate to define the Piola-Kirchhoff stress
tensor in local (material) reference coordinates. There are two Piola-Kirchhoff stress
definitions using the undeformed (or any other stationary reference) state as the basis
for formulation.
2.13.1 First Piola-Kirchhoff Stress Tensor σ
0
The first Piola-Kirchhoff stress tensor is also called the Lagrangian stress tensor. The
derivatives are defined with respect to material (local) coordinates. However, this
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2 Stress and Strain in Continuum