Biomedical Engineering Reference
In-Depth Information
According to ( 3.92 ), each stress vector t k in ( 3.87 ) is yielded by left scalar
multiplication of the C AUCHY stress tensor S with the corresponding indexed normal
vector n k (this can be shown instantaneously by scalar multiplication of S with
e k ¼ n k according to ( 3.89 ) 1 . The stress state at a (material) point X is thus defined
by either the three stress vectors t k or the six stress coordinates r ij of S. This can be
generalized for any arbitrary cutting plane dA with normal vector n which may also
be situated on the surface area (boundary) of the body (cf. Fig. 3.18 ). This leads to
C AUCHY 's lemma
t n ð X ; t; n Þ¼ n S ð X ; t Þ
ð 3 : 93 Þ
whereby the stress vector t n of area element dA is given by scalar multiplication of the
normal vector n with the stress tensor S (S maps n onto t n ). According to ( 3.93 )the
stress vector t n is a linear function of the normal vector n,whereasS is independent of n.
Note: Some prefer right scalar multiplication of n in ( 3.92 ) and ( 3.93 ), which
however, entails that in ( 3.87 ) et seqq. the first index of the stress coordinates r ij
denotes the direction of the stress vector and the second index denotes the
direction of the normal vector of the area element!
First and Second P IOLA -K IRCHHOFF Stress Tensor.IfP I is referred to as the
first P IOLA -K IRCHHOFF stress tensor and t 0n is the stress vector corresponding to area
element dA 0 in the ICFG, analogue to ( 3.93 ) the following C AUCHY lemma applies
t 0n ð X ; t; n 0 Þ¼ n 0 P I ð X ; t Þ
ð 3 : 94 Þ
A correlation between P I and S can thus be established as follows: substitution
of ( 3.93 ) and ( 3.94 )in( 3.85 ) using ( 3.54 ) and a T ¼ T T a (for arbitrary vectors
a and tensors T) and the symmetry of S regarding ( 3.128 ) leads to
Z
t 0n dA 0 ¼ Z
A 0
n 0 P I dA 0 Z
A 0
Z
P I T dA 0 ¼ ! Z
A
P I T ð dA 0 n 0 Þ
|{z}
dA 0
t n dA
A 0
A 0
Z
A
n SdA Z
A
Z
A
S dA Z
A 0
S T ð dAn Þ
|{z}
dA
JS F T dA 0
ð 3 : 95 Þ
Comparison of the underlined integrands in ( 3.95 ) for arbitrarily directed area
elements dA 0 leads, after transposition of both sides to a correlation of the first
P IOLA -K IRCHHOFF stress tensor and the C AUCHY stress tensor as follows
P I ¼ JF 1 S ð CofF Þ T S
S ¼ J 1 F P I ;
and
respectively : ð 3 : 96 Þ
Note: Some specify the first P IOLA -K IRCHHOFF stress tensor by the transposition
of ( 3.96 ). This is a result of the differently defined lemma of C AUCHY (cf. the note
after ( 3.93 )).
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