Biomedical Engineering Reference
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finite deformation theory, particles in the material are ''tracked'' using a moving
coordinate system called the material coordinate system, with coordinates denoted
by (X 1 , X 2 , X 3 ). This coordinate system moves throughout the deformation to
ensure that the coordinates of each particle remains constant. Another set of
coordinates, the spatial coordinate system, is used as reference. This coordinate
system marks coordinate points, denoted by (x 1 , x 2 , x 3 ) which remain constant
throughout the deformation; the spatial coordinate system therefore remains static
in time. The deformation of the material can be quantified by the changes in
direction and length of the lines that connect two adjacent material particles. It is
calculated from the differences in material and spatial coordinates and specified
using the deformation gradient tensor (F),
F ¼ ox
ð 13 Þ
oX :
Removing the directional components of this tensor, i.e., removing the rotational
dependency of the deformation gradient tenor, leaves the right Cauchy-Green
deformation tensor (C),
C ¼ F T F ;
ð 14 Þ
which can be related to a strain tensor in the form of the Lagrangian finite strain
tensor (E) through the following relationship,
E ¼ 1
2
ð
C I
Þ ;
ð 15 Þ
where I is the identity matrix. The right Cauchy-Green deformation tensor can be
used to specify three invariants (I 1 , I 3 , I 3 ), i.e., quantities which remain constant
under coordinate rotations, which are used for expressing several passive consti-
tutive relationships ( Sect. 4.2 ). The invariants can alternatively be specified in
terms of the principal stretch ratios of C: k 1 , k 2 and k 3 , such that,
I 1 ¼ tr ð C Þ¼ k 1 þ k 2 þ k 3 ;
ð 16 Þ
h
i ¼ k 1 k 2 þ k 2 k 3 þ k 3 k 1 ;
I 2 ¼ 1
2
tr ð C Þ 2 tr ð C 2 Þ
ð 17 Þ
I 3 ¼ det ð C Þ¼ k 1 k 2 k 2 :
ð 18 Þ
Both the force and the unit surface area in a material are required in order to
calculate stress. In a large deformation problem, both the force and area are
different in the undeformed and deformed configurations. There are therefore
various definitions for the stress tensor. An important definition, useful for rep-
resenting passive material behavior, is the second Piola-Kirchhoff stress tensor
(T), which represents the force measured per unit undeformed area, acting on a
local geometric element in the undeformed configuration,
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