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STRESS 145
t cauchy stress
j xy
component
i
Elemental volume in
current (deformed)
k configuration
Figure 4.7 Cauchy stress tensor components in deformed configuration. Note that the material
element is taken in the directions of the global base vectors.
j
m
s y
s x
i
s z
k
Figure 4.8 Surface traction force.
defined as the linear mapping from one vector space into another vector space. In simple
terms, for a given normal it returns traction force.
4.5.2 First Piola-Kirchhoff stress tensor
Cauchy stress is defined for the deformed body, i.e. stress is defined as traction force per
unit area of the deformed body. Sometimes it is easier to deal with the initial configura-
tion than with the deformed configuration. Thus, stress called ‘Piola-Kirchhoff stress’ is
introduced. The first Piola-Kirchhoff stress is defined by the expression
S 1 = (det F)TF −T (4.70)
To understand the above definition, it is necessary to investigate each element in the
above formula. First, the term (det F) represents the ratio of volume of deformed material
element and volume of undeformed material element (initial configuration). Thus, if a
