200 R. Segev
forces by body forces and surface forces in the Cauchy approach is completely equivalent to the representation of forces by variational stress densities in the variational approach.
2. Cauchy’s stress theory for manifolds
Let π : W →
be a vector bundle over the m-dimensional orientable manifold . It is assumed
that a particular orientation is chosen on . The vector bundle is interpreted as the bundle of
generalized velocities over . The manifold
is interpreted as the universal body and the vector bundle is interpreted as the bundle of generalized velocities over
. Cauchy’s stress theory for manifolds, presented in [3], considers for each compact m-dimensional submanifold with
boundary
✁
of linear functionals of the generalized velocity fields containing a volume term
and a boundary term of the form F
✂
w =
Z
✂
b
✂
w +
Z
∂
✂
t
✂
w. Here, using the notation
V
p
T
∗
X for the bundle of p-forms on a manifold X , w is a section of
W , b
✂
, the body force, is a section of L W, V
m
T
∗
✁
and t
✂
the boundary force is a section of L W,
V
m−1
T
∗
∂
✁
. The functional F
✂
is interpreted as the force, or power, functional and the value F
✂
w is classically interpreted as the power of the force for the generalized velocity
field w. Cauchy’s postulates for the force system
{F
✂
= b
✂
, t
✂
} presented in [3] may be sum- marized as follows.
i For every x ∈
and every body
✁
, b
✂
x = bx, that is, the value of the body force at
a point is independent of the body containing it. Accordingly, we will omit the subscript
✁
. ii Let us consider the Grassmann bundle of hyperplanes G
m−1
T →
whose fiber G
m−1
T
x
at any point x ∈
is the Grassmann manifold of hyperplanes, i.e., m − 1-
dimensional subspaces of the tangent space T
x
. Let L W,
m−1
G
m−1
T
∗
→ G
m−1
T be the vector bundle over G
m−1
T whose fiber over a hyperplane H ⊂ T
x
is the vector space of linear mappings L W
x
, V
m−1
H
∗
. Then, the dependence of t
✂
on
✁
is via a smooth section
Σ : G
m−1
T → L W,
m−1
G
m−1
T
∗
,
the Cauchy section, such that t
✂
= ΣH where H = T
x
∂
✁
. iii The Cauchy section Σ is continuous.
iv There is a section ζ of L W, V
m
T
∗
such that F
✂
w =
Z
✂
bw
+ Z
∂
✂
t
✂
w ≤
Z
✂
ζ w for every body
✁
.
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
⊗ dx
1
∧ . . . ∧d d x
k
∧ . . . ∧dx
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
∧ . . . ∧dx
m
.
3. The revised boundedness postulate
