Civil Engineering Reference
In-Depth Information
The forces are distinguished by the work they do under deformations.
The body force
f
:
−→ R
3
results in a force
fdV
acting on a volume
element
dV
. Surface forces are specified by a function
t
:
×
S
2
3
−→ R
where
S
2
3
: Let
V
be an arbitrary subdomain of
(with a
sufficiently smooth boundary), and let
dA
be an area element on the surface with
the unit outward-pointing normal vector
n
. Then the area element
dA
contributes
t (x, n)dA
to the force, which also depends on the direction of
n
. The vector
t(x,n)
is called the
Cauchy stress vector.
The main axiom of mechanics asserts that in an equilibrium state, all forces
and all moments add to zero. Here we must take into account both surface forces
and body forces.
denotes the unit sphere in
R
1.2 Axiom of Static Equilibrium.
(Stress principle of Euler and Cauchy)
Let
B
be a (deformed) body in equilibrium. Then there exists a vector field
t
such that in every subdomain
V
of
B
, the (volume) forces
f
and the stresses
t
satisfy
f(x)dx
+
t (x, n)ds
=
0
,
(
1
.
8
)
V
∂V
x
∧
f(x)dx
+
x
∧
t (x, n)ds
=
0
.
(
1
.
9
)
V
∂V
3
.
Once the existence of the Cauchy stress vector is given, its exact dependence
on the normal
n
can be determined. Here and in the sequel, we use the following
sets of matrices:
M
Here the symbol
∧
stands for the vector product in
R
3
, the set of 3
×
3 matrices,
3
+
3
M
, the set of matrices in
M
with positive determinants,
3
O
, the set of orthogonal 3
×
3 matrices,
3
+
3
3
+
O
:
= O
∩M
,
3
S
, the set of symmetric 3
×
3 matrices,
3
>
3
.
S
, the set of positive definite matrices in
S
Let t(
·
,n)
∈
C
1
(B,
R
3
), t (x,
·
)
∈
C
0
(S
2
,
R
3
), and
1.3 Cauchy's Theorem.
3
) be in equilibrium according to 1.2. Then there exists a symmetric
tensor field T
f
∈
C(B,
R
C
1
(B,
3
) with the following properties:
∈
S
S
2
,
t(x,n)
=
T(x)n
,
x
∈
B, n
∈
(
1
.
10
)
div
T(x)
+
f(x)
=
0,
x
∈
B,
(
1
.
11
)
T(x)
=
T
T
(x)
,
x
∈
B.
(
1
.
12
)
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