Biomedical Engineering Reference
In-Depth Information
3.3 The State of Stress at a Point
Stress is a mental construct of humans to represent the internal interactions or
internal forces on a material object. The concept of a stress will first be introduced
as a vector. Consider the potato-shaped object shown in Fig.
3.2
. An imaginary
plane
characterized by its normal
n
divides the object into an upper portion
U
and
a lower portion
L
. Considering
L
as a free object, the action of
U
upon
L
is statically
equivalent to a resultant force
f
and couple
m
. Assuming that the interaction is
distributed across
S
S
, each area element
D
A
(i)
of the intersection of
S
and the object
may be considered as transmitting a force
D
f
(i)
and a moment
D
m
(i)
. The average
stress vector,
t
A
ðnÞ
(
P
), at the point
P
acting on the plane whose normal is
n
, is defined
as the ratio
Þ¼
D
f
ð
i
Þ
t
A
ðnÞ
ð
P
A
ð
i
Þ
:
(3.9)
D
The quantity
t
A
ðnÞ
D
f
(i)
is a vector and
D
A
(i)
is a scalar.
is a vector because the force
The average couple stress vector
c
A
at the point
P
acting on the plane whose
normal is
n
is defined analogously by the ratio
ðnÞ
ð
P
Þ
Þ¼
D
m
ð
i
Þ
D
c
A
ðnÞ
ð
P
:
(3.10)
A
ð
i
Þ
These two definitions are illustrated in Fig.
3.2
. The stress vector
t
(
n
)
(
P
) acting at
the point
P
on the plane whose normal is
n
is defined as the limit of the average
Δ
f
(i)
Δ
m
(i)
Δ
f
(i)
/ΔA
(i)
n
U
Δ
m
(i)
/ΔA
(i)
P
ΔA
(i)
Σ
L
Fig. 3.2
A plane
, with normal
n
, is shown passing through a point
P
in a potato-shaped object.
The force and moment acting across the plane at the point
P
are indicated
S
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