Biomedical Engineering Reference

In-Depth Information

3. The displacement at each point and in each direction for which traction was

prescribed in the boundary conditions (e.g., the portion
S
u
of the boundary of
V

in Fig.
3.1
).

4. The traction at each point and in each direction for which displacement was

prescribed in the boundary conditions (e.g., the portion
S
t
of the boundary of
V
in

Fig.
3.1
).

These are obtained by solving the following sets of equations at each point

within the model:

1. Strain-displacement equations that define the strain tensor field
«
(
x
) at each

point
x
within the volume
V
T
of the tendon and
V
B
of the bone in terms of the

displacement vector field,
u
(
x
). Here and throughout the chapter, we write these

equations using index notation, in which subscripts are understood to represent

indices ranging from 1 to 3, commas represent partial differentiation, and

repeated indices within a grouping imply summation over the three values of

that index. The strain displacement relations are then:

1

e
ij
¼

2
ðu
i;j
þ u
j;i
Þ

(3.1)

in a

particular coordinate system,
u
i
represents the three components of the displace-

ment vector
u
in that coordinate system, and
u
i;j
¼ @u
i
=@x
j
.

2. Compatibility relations. Since the nominal strain tensor has six independent

components and the displacement vector only three, not every choice of a

spatially varying strain field relates to a displacement field. Strain fields that

do meet this criterion satisfy the compatibility relations:

where
e
ij
represents the 3
3 matrix of components of the strain tensor

«

e
ij;kl
þ e
kl;ij
e
jl;ik
e
ik;jl
¼
0

(3.2)

where the additional subscript following the comma represents a second deriva-

tive with respect to the components of the coordinate system, so that
e
ij;kl
¼ @

2

e
ij
=@x
k
@x
l
.

3. Constitutive relations that predict the strain tensor at a point in terms of the stress

tensor

s

. For the case of linear, isotropic elasticity, these are:

1

E
½ð
1
þ nÞs
ij
nd
ij
s
kk

e
ij
¼

(3.3)

E

ð
1
þ nÞ
e
ij
þ

E

ð
1
þ nÞð
1
2
nÞ
d
ij
e
kk

s
ij
¼

(3.4)

where
d
ij
is Kronecker's delta function that equals 1 when
i ¼ j
and 0 otherwise,

and
e
kk
¼ e
11
þ e
22
þ e
33
.

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