Page 166 - The Combined Finite-Discrete Element Method

STRESS 149

the particular surfaces of the material element are orientated in the directions of vectors
of the global triad. As these directions coincide with the edges of the material element,
these components also represent normal and shear stress components. In the case of the
first Piola-Kirchhoff stress, the material element is chosen to coincide with the local
triad, which is chosen on the initial configuration. Thus, stress components represent
surface traction per unit area of the initial configuration. The direction of the initial stress
components corresponds with the directions of individual vectors of the global triad. Thus,
when viewed on the initial configuration, they would appear as normal or tangential
to the surfaces of the material element. However, this is not the case. The material
element has deformed, and the surfaces of the material element are no longer orthogonal
to the individual vectors of the global triad. Thus, the first Piola-Kirchhoff stress is
best viewed as a surface traction per unit area of the initial configuration (undeformed
configuration) with individual components expressed in the direction of the local triad.
Thus, the only difference between these two stresses is in the surfaces to which individual
stress components correspond. Traction force over a given surface can be obtained using
the tensor
Tm dA = (det F)TF −T n dA (4.87)
∂P
∂P t
where m is the outward unit normal field for a deformed configuration, expressed in some
global coordinate system, and n is the corresponding outward unit normal field for the
initial configuration expressed in the global coordinate system.
4.5.3 Second Piola-Kirchhoff stress tensor

As explained above, Cauchy stress components represent normal and tangential (shear)
traction forces. This is not the case with the first Piola-Kirchhoff stress, where stress
components are oriented at an arbitrary angle relative to the surface of the deformed
elemental volume. Thus, the second Piola-Kirchhoff stress is introduced. It is defined by
the expression
−1
−1
σ = F S = (det F)F TF −T (4.88)
The same definition can be expressed using tensor matrices
  T   
ˆ ˆ ˆ  ˆ  
˜ i x j ˜ x k x S ˆ ˜x S ˆ ˜y S ˆ ˜z i ˜ S ˆ ˜x S ˆ ˜y S ˆ ˜z
˜
x
x
x
x
x
x
   
 ˆ ˆ    
˜
j
σ = i ˜ j ˜ ˜ ˆ  S ˆ ˜y S ˆ ˜z  =  ˆ  S ˆ ˜y (4.89)
y
y
y
 y y k y   S ˆ ˜x y y    S ˆ ˜x y S ˆ ˜z 
ˆ ˜ ˆ ˜ ˆ ˜ S ˆ ˜x S ˆ ˜y S ˆ ˆ ˜ S ˆ ˜x S ˆ ˜y S ˆ
z
z˜z
z
z
z˜z
z
i z j z k z k
where, for instance, the component
 
x
S ˆ ˜x
! "
ˆ ˜ ˆ ˜ ˆ ˜  S ˆ ˜x (4.90)
i x i y i z y 
S ˆ ˜x
z
˜
represents the traction force components in the i direction. In other words, the second
Piola-Kirchhoff stress is obtained by expressing components of traction force on each sur-
face of the deformed material element in terms of the base vectors of the deformed triad:
(i, j, k) (4.91)
˜ ˜ ˜
as shown in Figure 4.10.