Notes on stresses for manifolds 201
Using the results of [2], it is shown in [3] that there is a unique section σ of L W, V
m−1
T
∗
called the Cauchy stress such that
t
✂
wv
1
, . . . , v
m−1
= σ wv
1
, . . . , v
m−1
, for any collection of m − 1 vectors v
1
, . . . , v
m−1
∈ T
x
∂
✁
, x ∈ ∂
✁
, where the dependence on x was omitted in order to simplify the notation. Using the notation ι : ∂
✁
→ for the natural
inclusion mapping, so that ι
∗
: V
m−1
T
∗
→ V
m
− 1 T
∗
∂
✁
is the restriction of forms, we
may write t
✂
w = ι
∗
σ w which we will also write as t
✂
= ι
∗
σ —the generalized Cauchy
formula. We will refer to this result as the generalized Cauchy theorem. Assume that x
i
, w
α
are local vector bundle coordinates in a neighborhood π
−1
U ⊂ W , U ⊂
with local basis elements {W
α
e
α
} so a section of W is represented locally by w
α
W
α
e
α
. Then, denoting the dual base vectors by {W
α
e
α
} a stress σ is represented locally by
σ
α 1... ˆk...m
W
α
e
α
⊗ d x
1
∧ . . . ∧ d d x
k
∧ . . . ∧ d x
m
, where a “hat” indicates the omission of an item an index or a factor. The value of σ w is
represented locally by σ
α 1... ˆk...m
w
α
d x
1
∧ . . . ∧ d d x
k
∧ . . . ∧ d x
m
.
3. The revised boundedness postulate
If we substitute the generalized Cauchy formula into the expression for F
✂
w we obtain
F
✂
w = Z
✂
b
✂
w + Z
∂
✂
t
✂
w =
Z
✂
b
✂
w + Z
∂
✂
ι
∗
σ w =
Z
✂
b
✂
w + Z
✂
d σ w ,
where Stokes’ theorem was used in the last line. It is clear form the local expression for σ w that the exterior derivative dσ w depends on the derivative of w an not only on the local value
of w. In other words, F
✂
w is a local linear functional on the first order jet j
1
w .
Using the observation that F
✂
should be a local linear functional on the first jet of w, we replace the boundedness postulate iv by the following
Revised boundedness postulate
There is a section S of L J
1
W , V
m
T
∗
such that F
✂
w =
Z
✂
bw +
Z
∂
✂
t
✂
w ≤
Z
✂
S j
1
w ,
where the absolute value of an m-form θ , S j
1
w in this case, is given as
|θ x| = θ
x if θ x is positively oriented,
−θ x
if θ x is negatively oriented
202 R. Segev
relatively to the orientation chosen on .
It is noted that the revised boundedness postulate may also be written as Z
∂
✂
t
✂
w ≤
Z
✂
S j
1
w ,
for some section S of L J
1
W , V
m
T
∗
. This follows from −
Z
✂
bw
+ Z
∂
✂
t
✂
w ≤
Z
✂
bw +
Z
∂
✂
t
✂
w ≤
Z
✂
S j
1
w so
Z
∂
✂
t
✂
w ≤
Z
✂
S j
1
w +
Z
✂
bw
≤ Z
✂
S j
1
w +
Z
✂
bw
= Z
✂
S j
1
w +
bw
≤ Z
✂
S j
1
w ,
for some S .
For an arbitrary x ∈ we want to show that
t
✂
w = Σ T
x
∂
✁
w = ι
∗
σ w ,
for a unique element of L W
x
, V
m
− 1T
x
, where in the equation above we omitted the dependence on x.
Just as in [3], the proof the generalized Cauchy theorem is based on the following points: a The assertion is local and written in an invariant form and hence it may be proved in any
vector bundle chart. b Using a local basis {W
α
e
α
} for the neighbohood where the vector bundle chart is used,
any vector w ∈ W
x
may be expressed in the form w = w
α
W
α
e
α
, so t
✂
w = w
α
τ
✂
α
, where, τ
✂
α
= t
✂
W
α
e
α
. c For the local vector field W
α
e
α
in the chart neighborhood of x, the scalar valued exten- sive property given by the volume term β
α
= bW
α
e
α
, the flux density term τ
✂
α
=
t
✂
W
α
e
α
, and the source term s
α
= S j
1
W ase
α
satisfies the generalized Cauchy postulates for scalar valued quantities see [2]. In particular, it is noted that if S j
1
w is represented locally by
S j
1
w
α 1...m
d x
1
∧ . . . ∧ d x
m
= S
α 1...m
w
α
+ S
i α
1...m
w
α ,
i
d x
1
∧ . . . ∧ d x
m
the components dual to w
α
and those dual to w
α i
differ in notation only by the number of indices, then, s
α
= | S
α 1...m
| . Hence, by the Cauchy theorem for scalars [2], there is a
unique collection of dim W
x
m − 1-forms σ
α
such that τ
✂
α
= ι
∗
σ
α
. These forms represent σ x ∈ L W
x
, V
m−1
T
x
∂
✁
in the given chart.
Notes on stresses for manifolds 203
4. Variational stress densities
