Biomedical Engineering Reference
In-Depth Information
of the element and the distributed load (force per unit mass) acting on the element
in the
x
-direction, so successively (also see Chapter
8
):
−
σ
xx
dydz
back plane
dx
dydz
σ
xx
+
∂σ
xx
∂
x
frontal plane
−
σ
xy
dxdz
left plane
dy
dxdz
σ
xy
+
∂σ
xy
∂
right plane
y
−
σ
xz
dxdy
bottom plane
dz
dxdy
σ
xz
+
∂σ
xz
∂
z
top plane
q
x
ρ
dxdydz
volume,
with resultant (in
x
-direction):
∂σ
xx
dV
.
(11.7)
Similarly, the resulting forces in
y
- and
z
-direction can be determined. All external
forces applied to the volume element are stored in a column. Using the gradient
operator
dxdydz
∂σ
xx
∂
x
+
∂σ
xy
∂
y
+
∂σ
xz
∂
x
+
∂σ
xy
∂
y
+
∂σ
xz
∂
z
+
q
x
ρ
=
∂
z
+
q
x
ρ
∼
introduced earlier and the symmetrical Cauchy stress matrix
σ
the
column with external forces can be written as
⎡
⎤
+
∂σ
xy
∂
y
+
∂σ
xz
∂
z
∂σ
xx
∂
x
+
q
x
ρ
⎣
⎦
dV
T
∂σ
xy
∂
x
+
∂σ
yy
∂
y
+
∂σ
yz
∂
z
T
+
q
y
ρ
dV
=
∼
σ
+
ρ
q
∼
(11.8)
+
∂σ
yz
∂
y
∂σ
xz
∂
x
+
∂σ
zz
∂
z
+
q
z
ρ
with the column
q
∼
for the distributed load defined according to
⎡
⎣
⎤
⎦
q
x
q
y
q
z
q
∼
=
.
In vector/tensor notation the following expression can be given for the resulting
force on the element
dV
:
∇·
σ
+
ρ
q
dV
.
For the considered element the time derivative of the momentum equals the
external load, leading to
∇·
σ
+
ρ
q
=
ρ
a
,
(11.9)
