A Brief Review of Elasticity and Viscoelasticity for Solids 1 Introduction

H. T. Banks, S. H. Hu and Z. R. Kenz / Adv. Appl. Math. Mech., 3 (2011), pp. 1-51 11
Here
T (n) = ( T (n)
x
, T (n)
y
Cauchy’s formula (2.10) can be written concisely as
, T (n) ) T,
z and n = (nx , n y , n z ) T .
T (n) = σn,
where σ is the Cauchy stress tensor defined in (2.9).
Remark 2.7. In addition to the Cauchy stress tensor, there are other stress tensors
encountered in practice, such as the first Piola-Kirchhoff stress tensor and second Piola-
Kirchhoff stress tensor. The differences between the Cauchy stress tensor and the Piola-
Kirchhoff stress tensors as well as relationships between the tensors can be illustrated
as follows:
1. Cauchy stress tensor: relates forces in the present (deformed/spatial) configuration
to areas in the present configuration. Hence, sometimes the Cauchy stress is also called
the true stress. In addition, the Cauchy stress tensor is symmetric, which is implied
by the fact that the equilibrium of an element requires that the resultant moments
vanish. We will see in Section 2.2.4 that the Cauchy stress tensor is used in the Eulerian
equation of motion. Hence the Cauchy stress tensor is also referred to as the Eulerian
stress tensor.
2. First Piola-Kirchhoff stress tensor (also called the Lagrangian stress tensor in [24,25]):
relates forces in the present configuration with areas in the initial configuration. The
relationship between the first Piola-Kirchhoff stress tensor P and the Cauchy stress
tensor σ is given by
P = |A|σ(A −1 ) T . (2.11)
From the above equation, we see that in general the first Piola-Kirchhoff stress tensor
is not symmetric (its transpose is called the nominal stress tensor or engineering stress
tensor). Hence, the first Piola-Kirchhoff stress tensor will be inconvenient to use in a
stress-strain law in which the strain tensor is always symmetric. In addition, we will
see in Section 2.2.5 that the first Piola-Kirchhoff stress tensor is used in the Lagrangian
equation of motion. As pointed out in [25], the first Piola-Kirchhoff stress tensor is the
most convenient for the reduction of laboratory experimental data.
3. Second Piola-Kirchhoff stress tensor (referred to as Kirchoff stress tensor in [24, 25],
though in some references such as [44] Kirchhoff stress tensor refers to a weighted
Cauchy stress tensor and is defined by |A|σ): relates forces in the initial configuration
to areas in the initial configuration. The relationship between the second Piola-
Kirchhoff stress tensor S and the Cauchy stress tensor σ is given by
S = |A|A −1 σ(A −1 ) T . (2.12)
From the above formula we see that the second Piola-Kirchhoff stress tensor is symmetric.
Hence, the second Piola-Kirchhoff stress tensor is more suitable than the first
Piola-Kirchhoff stress tensor to use in a stress-strain law. In addition, by (2.11) and
(2.12) we find that
S = A −1 P. (2.13)