Biomedical Engineering Reference
In-Depth Information
matrix ε
ij
and the change in volume fraction
e
of the matrix material from a
reference value ξ
0
.
The equations of the governing system for this theory are [15]
ε
ij
= (
u
i, j
+
u
j, i
)/2
(2.4)
σ
ij, j
+ γ(ξ
0
+
e
)
b
i
= 0
(2.5)
σ
ij
= (ξ
0
+
e
)
C
ijkm
(
e
)ε
km
(2.6)
ė
=
A
(
e
) +
A
km
(
e
)ε
km
(2.7)
where
u
i
are displacement components, σ
ij
the components of the stress ten-
sor,
b
i
the components of the body force,
C
ijkm
(
e
) components of the fourth-
rank elasticity tensor,
A
(
e
) and
A
ij
(
e
) the material coefficients dependent on
the change in volume fraction
e
of the adaptive elastic material from the
reference volume fraction, and the superimposed dot indicates the material
time derivative.
Equation (2.4) is the strain-displacement relations for small strain; Equation
(2.5) represents the condition of equilibrium in terms of stress. Equation
(2.6) stands for a generalization of Hooke's law; Equation (2.7) represents
the remodeling rate equation, and it specifies the rate of change of the volume
fraction as a function of ξ and ε
ij
.
A positive value of
ė
means that the volume
fraction of elastic material in increasing, whereas a negative value means that
the volume fraction is decreasing. Equation (2.7) is obtained from the conser-
vation of mass and the constitutive assumption that the rate of mass deposi-
tion or absorption is dependent on only the volume fraction
e
and ε
ij
.
The
system of Equations (2.6) and (2.7) is an elementary mathematical model of
Wolff's law. Equation (2.6) is a statement that the moduli occurring in Hooke's
law actually depend on the volume fraction of solid matrix material present.
Equation (2.7) is an evolutionary law for the volume fraction of matrix
material.
The theory just described involves the functions
A
(
e
),
A
ij
(
e
), and
C
ijkm
(
e
),
characterizing the material properties. Cowin [11] mentioned that no data
exist in the literature on the values of the functions
A
(
e
) and
A
ij
(
e
), and the
data on
C
ijkm
(
e
) suggest that it can be approximated as a linear function of
e.
Hegedus and Cowin [15] introduced an approximation scheme that gave
C
ijkm
(
e
) as a linear function of
e.
This scheme involved a series expansion in
which terms of the orders
e
3
,⎮ε
ij
⎮
e
2
, and ⎮ε
ij
⎮
2
e
were ignored and terms of
the orders
e,
⎮ε
ij
⎮, ⎮ε
ij
⎮
e,
and
e
2
retained. The scheme showed that
A
ij
(
e
) were
also linear in
e,
whereas
A
(
e
) was quadratic in
e,
and thus the constitutive
relations (2.6) and (2.7) were approximated by
0
1
σ=ξ
(
CeC
+
)
ε
(2.8)
ij
0
ijkm
ijkm m
