Biomedical Engineering Reference
In-Depth Information
y
y
∂
u/
∂
y
dy
dy
∂
u/
∂
x
g
xy
x
x
dx
dx
(a)
(b)
Figure 6.5
Definition of shear strain: (a) tensor definition
as the average
ε
xy
=
ε
yx
=
(
∂v/∂
x
+
∂
u
/∂
y
)
/
2; (b) engineer-
ing definition as the total
γ
xy
=
∂v/∂
x
+
∂
u
/∂
y
.
and
σ
x
σ
xy
σ
xz
T
σ
ij
=
σ
yx
σ
y
σ
yz
→
σ
={
σ
x
σ
y
σ
z
σ
xy
σ
yz
σ
xz
}
(6.21)
σ
zx
σ
zy
σ
z
and the fourth-order elasticity tensor
C
ijkl
is represented by a square (6
×
6)-matrix
C
. This formalism is closer to actual computer implementations than the tensorial
notation [12]. Furthermore, we use the unit vector
1
={
111000
}
T
and the diagonal
matrix
L
=
111222
, which is not a unit matrix because the strain vector is
expressed using the engineering definition of shear strain (e. g.,
γ
xy
=
2
ε
xy
)rather
than the tensor definition, cf. Figure 6.5.
Now, the generalized Hooke's law for isotropic materials based on Lame's
constants can be written in matrix form as
σ
=
(2
µ
L
−
1
+
λ
1
⊗
1
)
ε
(6.22)
or in explicit form with all components as
σ
x
σ
y
σ
λ
+
2
µλ
λ
000
ε
x
ε
y
ε
λ
λ
+
2
µλ
000
λ
λ
λ
+
2
µ
000
z
z
=
·
(6.23)
σ
0
0
0
µ
00
2
ε
xy
xy
σ
0
0
0
0
µ
0
2
ε
yz
yz
σ
0
0
0
0
0
µ
2
ε
xz
xz
The dyadic product
⊗
used in Eq. (6.22) is in general defined for two vectors
a
and
b
by
...
a
1
b
1
a
1
b
2
a
1
b
n
a
1
a
2
a
n
b
1
b
2
b
n
...
a
2
b
1
a
2
b
2
a
2
b
n
.
.
.
.
.
.
a
⊗
b
=
⊗
=
(6.24)
.
a
n
b
1
. . .
a
n
b
n
T
.
where the same result can be obtained by the matrix multiplication
a
·
b
