Biomedical Engineering Reference
In-Depth Information
r u ¼ A 1 b u1 q a k 1 þ A 2 b u2 q a k 1 þ e o b u3 þ B k b u4 q ;
ð 41 Þ
r z ¼ A 1 b z1 q a k 1 þ A 2 b z2 q a k 1 þ e o b z3 þ B k b z4 q ;
ð 42 Þ
s zu ¼ A 1 b c1 q a k 1 þ A 2 b c2 q a k 1 þ e o b c3 þ B k b c4 q ;
ð 43 Þ
where
b q1 ¼ C g2 þ a k C g3 ;
b q2 ¼ C g2 a k C g3 ;
ð 44 Þ
b q3 ¼ C g1 þð C g2 þ C g3 Þ a 1 ;
b q4 ¼ C g6 þð C g2 þ 2C g3 Þ a 2 ;
ð 45 Þ
with the index q ¼ z ; u ; q ; c corresponding to the g-index values g ¼ 1 ; 2 ; 3 ; 6,
respectively, and with
a k ¼ C 22 = C 33 ;
ð 46 Þ
a 1 ¼ð C 12 C 13 Þ=½ C 33 ð 1 a k Þ;
a 2 ¼ð C 26 2C 36 Þ=½ C 33 ð 4 a k Þ;
ð 47 Þ
C ij denoting the ð i ; j Þ component of the stiffness matrix
k for the kth MLU.
It is worth pointing out that during the deformation path both strain and stress
components, and thereby also the strain-dependent MTU material properties, are
z- and u-independent.
Assuming the MLUs as perfectly bonded layers, the 3N unknown constants
A 1 ; A 2 ; B k (with k ¼ 1 ::: N) are solved by imposing the following 3 ð N 1 Þ
continuity conditions at the MLU's interfaces:
C
u q ð r k Þ¼ u k þ 1
u u ð r k Þ¼ u k þ 1
r r ð r k Þ¼ r k þ 1
ð r k Þ;
ð r k Þ;
ð r k Þ;
ð 48 Þ
q
u
r
and the three equilibrium incremental relationships:
Z r k
2p X
N
q 2 s zu dq ¼ 0 ;
r r ð r i Þ¼ p ;
r r ð r i þ h a Þ¼ 0 ;
ð 49 Þ
r k 1
k ¼ 1
the latter prescribing the average torque related to the tangential stress increment
s zu to be zero.
References
1. van Holde, K.E., Matthews, C.: Biochemistry. Benjamin/Cummings Publishing Company
Inc., Menlo Park (1995)
2. Fratzl, P.: Collagen: Structure and Mechanics. Springer, New York (2008)
3. Bruel, A., Oxlund, H.: Changes in biomechanical properties, composition of collagen and
elastin, and advanced glycation endproducts of the rat aorta in relation to age. Atherosclerosis
127, 155-165 (1996)
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