Fundamentals - Preventive Biomechanics - page 39

global rotation of the body! Furthermore, F must be invertible such that J ¼
det F 6¼ 0 is fulfilled.

3.2.3.6 Strain Tensors

Following one-dimensional application (see Sect. 3.2.3.1 ), strain measure gener-
ally results from a change in length. If a body is not a simple rod shape, but an
arbitrary three-dimensional shape, this kind of utilization can principally be
applied. Thus, it is reasonable to determine the change in length of the (three)
edges of a volume element dV. Since the volume element is spatially oriented it is
advantageous to assign the edge length dX to the (vectorial) line element dX in the
ICFG and dx in the CCFG. Furthermore, it is advantageous to determine the
modulus or the square (norm) of a vectorial line element dx and dX, respectively.
Using the mapping property ( 3.53 ) and the equality of a vector with its transpo-
sition (for arbitrary vectors a it applies that a ¼ a T

and thus

dx ¼ F dX ¼ dx T ¼ð F dX Þ T ¼ dX T F T ¼ dX F T Þ

the squares of dx and dX are obtained

ð dx Þ 2 ¼ dx dx ¼ð F dX Þð F dX Þ¼ dX ð F T F Þ
|{z}
C

dX

ð dX Þ 2 ¼ dX dX ¼ð F 1 dx Þð F 1 dx Þ¼ð dx F T Þð F 1 dx Þ¼ dx ð F F T Þ 1 dx

|{z}
B

ð 3 : 62 Þ

and thus

ð dx Þ 2 ¼ dx dx ¼ dX C dX ; ð dX Þ 2 ¼ dX dX ¼ dx B 1 dx ð 3 : 63 Þ

where C and B are referred to as (note for further identical representations in
( 3.63 ) the relation ( 3.57 )) right and left C AUCHY -strain tensor

C : ¼ F T F
¼ð I þ H Þ T ð I þ H Þ I þ H þ H T þ H T H
I þ u rþr u þðr u Þð u rÞ
B : ¼ F F T
¼ð I þ H Þð I þ H Þ T I þ H þ H T þ H H T I þ u rþr u þð u rÞðr u Þ
ð 3 : 64 Þ

and since both tensors are symmetrical, it follows