Biomedical Engineering Reference
In-Depth Information
Fig. 3.18
On C AUCHY 's
lemma
2
4
3
5
S ¼ r ij e i e j ¼ r 11 e 1 e 1 þ r 12 e 1 e 2 þ r 13 e 1 e 3
þ r 21 e 2 e 1 þ r 22 e 2 e 2 þ r 23 e 2 e 3
þ r 31 e 3 e 1 þ r 32 e 3 e 2 þ r 33 e 3 e 3
r 11
r 12
r 13
or ½ S ¼
r 21
r 22
r 23
h e i e i i:
r 31
r 32
r 33
ð 3 : 89 Þ
In ( 3.89 ), e i e j are the basis dyads and the r ij are the nine stress coordinates
defined previously in ( 3.87 ). According to ( 3.89 ) 2 , the (three) normal stress
components r 11 , r 22 and r 33 are arranged on the principal diagonal and the (six)
shear stress components r 12 , r 13 -r 32 are arranged on the secondary diagonals.
C AUCHY ' S Lemma. Comparing the right-hand side of ( 3.87 ) with the expression
in ( 3.89 ) 1 shows visual agreement of the structural arrangement of the stress
coordinates r 11 , r 12 etc. Furthermore, it can be seen that after identical transfor-
mation of the first term in ( 3.89 ) 1 and comparison with ( 3.88 ) the following
relation yields
S ¼ r ij e i e j e i ð r ij e j Þ
|{z}
t i
¼ e i t i
ð 3 : 90 Þ
The right-hand side (underlined) of ( 3.90 ) represents a ''dyadic left multipli-
cation'' of t i with e i . Left scalar multiplication of both underlined terms in ( 3.90 )
with e k yields the following k-th stress vector t k
e k S ¼ð t i e i Þ e k ¼ð e k e i Þ
|{z}
d ki
t i ¼ d ki t i ¼ t k :
ð 3 : 91 Þ
Due to e k ¼ n k (k = 1, 2, 3) (see Fig. 3.18 ) and using ( 3.86 ), the underlined
term in ( 3.91 ) can be expressed as
t k X ; t; n k
ð
Þ n k SX ; ðÞ k ¼ 1 ; 2 ; 3
ð
Þ
ð 3 : 92 Þ
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