Some Extensions and Analysis of Flux and
Stress Theory
Reuven Segev
Department of Mechanical Engineering
Ben-Gurion University
Structures of the Mechanics of Complex Bodies
October 2007
Centro di Ricerca Matematica, Ennio De Giorgi
Scuola Normale Superiore
R. Segev (Ben-Gurion Univ.)
Flux and Stress Theories
Pisa, Oct. 2007
1 / 17
Generalized Bodies
The Material Structure Induced by an Extensive Property
R. Segev (Ben-Gurion Univ.)
Flux and Stress Theories
Pisa, Oct. 2007
2 / 17
$
'
Organisms
Organisms
• Material points, bodies and subbodies are primitive concepts in
continuum
mechanics.
These notions
are somehow
related
Material
points, bodies
and subbodies
are primitive
concepts
in to the
continuum
mechanics.
These
notions
are
somehow
related
to
the
conservation of mass.
conservation of mass.
• In growing bodies, material points are added and removed from
In growing bodies, material points are added and removed from the
the body.
body.
fingerprints,
birthmarks
are distinguishable.
• Examples:
Examples:
fingerprints,
birthmarks
are distinguishable.
• An organism
a body
structure
although
mass
is not
preserved.Can
An organism
has ahas
body
structure
although
mass
is not
preserved.
formalize
idea? this idea?
Can this
motivate
Assume
we have
an extensive
property.
• Assume
we have
an extensiv
p.
%
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Flux and Stress Theories
Reuven Segev: Geometric Methods, March 200
Pisa, Oct. 2007
3 / 17
The Material Structure Induced by an Extensive Property
The Material Structure Induced by an Extensive Property
In the classical case we have the
flux vector field h. It can be inIn the classical case we have the flux vectegrated to give us a material structor field h. It can be integrated to give us
ture. structure.
a material
A material
is identified
A material
point ispoint
identified
with an
integral
linean(aintegral
flow line).
with
lineThis
(a procedure
flow line).
may induce
material structure
associated
This procedure
may induce
matewith rial
any structure
extensive associated
property, e.g.,
withcolor
any
and energy.
extensive property, e.g., color and
energy.
h
will be the velocity field of the material points.
ρ h
• wewill
be the velocity field of the material points.
Can
ρ generalize the same idea for the general manifold case where the
flow (m − 1)-form replaces the vector field?
• Can we generalize the same idea for the general manifold case where the flow
&
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Flux and Stress Theories
Pisa, Oct. 2007
4 / 17
TheThe
CaseCase
where
a Volume
Element
is Specified
where
a Volume
Element
is Specified
is not
to have
metric
strucIt isnecessary
not necessary
toa have
a metric
structure
in order
that form
the flux
form
re in
order that
the flux
J be
rep-J
w
be represented
a vector field.
sented
by a vectorby
field.
v
Assume that you have a volume eleAssume
that you have a volume element
x
ment θ (m-form) on U . This may be
m-form)
on
U
.
This
may
be
thought
thought of as the density of the propas erty
the pdensity
of the or
property
if it is
if it is positive
anotherppositive
u
sitive
or another
positive property, e.g.,
property,
e.g., mass.
Given J and θ, find a vector v such that for every pair of tangent vectors,
ass.
w, θ, find a vector v such that for every pair of tangent
• Givenu,J and
θ (v, u, w) = J (u, w)
written as
J = vy θ.
vectors, u, w,
θ (v, u, w) = J (u, w)
written as
J = vy θ.
For a given θ there is a unique such vector v—the kinematic flux—a
generalization of the velocity field.
• For a given θ there is a unique such vector v—the kinematic flux—a
The vector field v depends linearly on the flux J.
generalization
of the velocity field.
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Flux and Stress Theories
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5 / 17
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The Flux Bundle
The Flux Bundle
Let us examine how the kinematic
flux v varies as we vary the volume
Let us examine how the kinematic flux v
varieselement.
as we vary the volume element.
Since
the space
m-forms
x is
Since the space
of mof
-forms
at x at
is 1dimensional,
as we vary
the vary
volume
1-dimensional,
as we
theelevolmentume
the resulting
vectors
v
remain
on
a
element the resulting vectors
line (1-D subspace of the tangent space).
v remain on a line (1-D subspace
of the tangent space).
w
u
Another
characterization:
If a surface
element
(say the
by the
• Another
characterization:
If a surface
element
(sayone
thedefined
one defined
by th
vectors
u, w)
the the
line,line,
the the
fluxflux
through
it vanishes.
vectors
u, contains
w) contains
through
it vanishes.
This is analogous to the situation with the velocity field.
• This is analogous to the situation with the velocity field.
A collections of subspaces is referred to as a distribution. This
distribution
is the flux
bundle. is referred to as a distribution. This
• A collections
of subspaces
distribution is the flux bundle.
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Flux and Stress Theories
Pisa, Oct. 2007
6 / 17
'
Generalized Body Points
Generalized Body Points
Integral manifolds of the distribution,
the
1-dimensional
flux bundle
Integral manifolds
of the distribution,
the in
this case,
submanifolds
whose
1-dimensional
flux are
bundle
in this case,
tangent space
a pointspace
is the at
corare submanifolds
whoseattangent
responding
line
of
the
flux
bundle
a point is the corresponding line of the
flux bundle
at that
point.
at that
point.
In general such
integral
manifolds
In general such
integralneed
maninot exist folds
(higher
dimensions),
however
need
not exist (higher
dimenthey always
existhowever
for 1-dimensional
bunsions),
they always
exist
dles as is the case here.
for 1-dimensional bundles as is the
Eachcase
integral
here.line manifold is identified with a body point.
Actual formulation is done on space-time manifold to allow time
• Each integral line manifold is identified with a body point.
dependent fluxes. There β is included in τ and dJ = s.
• Actual formulation is done on space-time manifold to allow time
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Flux and Stress Theories
Reuven Segev: Pisa,
Geometric
Methods,
March
Oct. 2007
7 / 17
Frames in Space-Time
Frames in Space-Time
Cartesian Product
an event e
(t, x )
a frame
Space-Time U
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&
Time
Axis
Space
Flux and Stress Theories
Pisa, Oct. 2007
8 / 17
'
Property-Induced
Property-InducedFibration
Fibration and
and Frame
Frame
No volume element: Fibration
—no real valued time is assigned to events
a worldline
Space-Time
R. Segev (Ben-Gurion Univ.)
A volume element: Integrable vector field
—real valued time is assigned to events
time
axis
non-unique
models of space
Flux and Stress Theories
a worldline
Space-Time
Pisa, Oct. 2007
9 / 17
'
Space Formulation VS. Space-Time Formulation
Space Formulation VS. Space-Time Formulation
Space Formulation
Space
Formulation
dim U
=3
dim U = 3
dim
BZ=
= 33 Z
Z dimB
lance
Balance
Z
ββ++
Z
ττ =
=
Z
Space-Time Formulation
Space-Time
dim E =Formulation
4
dim E = 4
β
dim
R=Z=44
dim
Z R
ss
Z
Z
tt== s s
τ
RR
B
∂R
∂R
BB
∂∂B
B
B
surface
term
2-form
on
a
3-D
manifold
3-form
a 4-D
manifold
rface term
2-form on a 3-D manifold 3-formonon
a 4-D
manifold
source term
3-form on a 3-D manifold
4-form on a 4-D manifold
urce
term
3-form
on a 3-D manifold 4-form
on a 4-D manifold
flux form
J—3 components
J—4 components
x form
J—3
components
J—4
variables
—time
dependent
—fixedcomponents
values at events
field
equation
β
+
dJ
=
s
dJ
=
s
riables
—time dependent
—fixed values at events
ld equation
β + dJ = s
R. Segev (Ben-Gurion Univ.)
dJ = s
Flux and Stress Theories
Pisa, Oct. 2007
10 / 17
Flow Potentials
Although we do not have vector velocity fields, we have material points.
In addition, we have analogs for the flow potentials.
In the case s = 0 we obtain (say the 4-D case) dJ = 0.
Assume that A is any (m − 2)-form on U . Then, J = dA satisfies the
differential balance equation—A is a flow potential. Since in general,
Z
∂M
ι∗ ω =
Z
dω,
M
for every control region B
Z
dJ =
B
R. Segev (Ben-Gurion Univ.)
Z
∂B
ι ∗ (J ) =
Z
∂B
Z
ι∗ (dA) =
ι∗ ι∗ (A) = 0.
∂(∂B )=∅
Flux and Stress Theories
Pisa, Oct. 2007
11 / 17
Summary: The Structure on Space-Time manifold
Associated with an Extensive Property
Balance laws are formulated in terms of forms.
The flux vector field is replaced by a flux (m − 1)-form in the
m-dimensional space.
Flow lines still make sense using the flux bundle.
Generalized body points may be associated with an arbitrary extensive
property—organisms.
A particularly compact formulation in space-time.
A positive extensive property induces a material frame.
R. Segev (Ben-Gurion Univ.)
Flux and Stress Theories
Pisa, Oct. 2007
12 / 17
Stresses for Generalized Bodies
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Flux and Stress Theories
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13 / 17
Forces for Generalized Bodies
Force densities are linear mappings on the values of the generalized
velocities.
In the case where a material structure is induce by an extensive
property and a volume element is given, the induced generalized
velocity w depends linearly on the flux form J.
It would be a natural generalization to replace generalized velocities by
flux forms as fields on which forces operate to produce power.
The physical dimension of forces will not be power per unit velocity but
power per per unit flux of the property p.
Z
For the spacetime formulation FB (J ) =
tB (J ),
B ⊂ E.
∂B
V
V
tB (e) : m−1 Te∗ E → m−1 Te∗ ∂B .
R. Segev (Ben-Gurion Univ.)
Flux and Stress Theories
Pisa, Oct. 2007
14 / 17
(e) .
Stresses for Generalized Bodies
r hand the flux density of energy may be written in
Consider
the energy extensive property. It has a flux density term
R
(e)force
boundary
as tB ( J ). flux form J(e) such that τ (e) = ι∗ ◦ J(e) .
τ
and
a corresponding
∂B
On the other hand the flux density of energy may be written in terms of
eoremtheimplies
that t = ι∗ ◦ σ so the energy flux
boundary force asBtB (J ).
e) = ι∗ ◦ J (e) = ι∗ ◦ σ ( J ). Hence,
Cauchy’s theorem implies that tB = ι∗ ◦ σ so the energy flux density is
τ (e) = ι∗ ◦ J (e) = ι∗ ◦ σ(J ). Hence,
J (e) = σ ( J ) .
J ( e ) = σ ( J ).
is the linear mapping that transforms the flux of the
stress isThe
theCauchy
linearstress
mapping
that transforms the flux of the
property p into the flux of energy.
Vm−energy.
V
1 ∗
o the flux
σe : of
Te E → m−1 Te∗ E . The stress at a point (event) is a linear
transformation
on the space of (m − 1)-forms.
V
1 ∗
→May mbe−applied
Te Eto. “resources”
The stressother
at athen
point
(event) is a linear
energy?
n on the space of (m − 1)-forms.
∗
eE
R. Segev (Ben-Gurion Univ.)
Flux and Stress Theories
Pisa, Oct. 2007
15 / 17
Local Representation of Stress-Tensors
Denote by {êi } the basis of the m-dimensional space of (m − 1)-forms.
Denote its dual basis by {êj }.
Since the stress at a point is a linear transformation on the space of
j
(m − 1)-forms it may be represented in the form σ̂i êj ⊗ êi .
If we had a volume element θ we would have an isomorphism
V m−1 ∗
(T U ) ↔ TU of (m − 1)-forms and vectors, such that J ↔ v are given
by θ (v, u, w) = J (u, w).
Thus, with a volume element and due to the following structure,
V
σ
(T ∗ U ) −−−−−→ m−1 (T ∗ U )
x

i
1
i−
yθ
θ 
V m−1
TU
σ̃
−−−−−→
TU ,
one may represent a stress σ by a linear transformation σ̃ on TU .
Surprisingly, σ̃ is independent of the volume element θ. In fact, you can
construct a natural isomorphism σ ↔ σ̃ without a volume element.
R. Segev (Ben-Gurion Univ.)
Flux and Stress Theories
Pisa, Oct. 2007
16 / 17
Maxwell Stress-Energy Tensor without a Metric
Maxwell 2-form: g, a flow potential for J, i.e., J = dg.
Faraday 2-form: f such that df = 0.
Assume a volume element and set w = iθ (J ) to be the vector field
representing the flux form.
define the stress-energy tensor as the section σ of
V
V
L m−1 (T ∗ U ), m−1 (T ∗ U ) by
σ (J ) = wy g ∧ f − wy f ∧ g.
The power is
dσ(J ) = (wy f) ∧ J + (Lw g) ∧ f − (Lw f) ∧ g.
—a generalization of the Lorentz force (wy f) ∧ J. (L is the Lie
derivative.) The two additional terms cancel in the traditional situation.
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Flux and Stress Theories
Pisa, Oct. 2007
17 / 17