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Geometric Methods in the Formulation of Continuum Mechanics Reuven Segev

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Department of Mechanical Engineering Ben Gurion University, Israel [email protected]

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Reuven Segev: Geometric Methods, March 2001

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Topics

      

Scalar-Valued Extensive Properties, The Material Structure Induced by an Extensive Property, Forces and Cauchy Stresses, Variational Stresses, Stresses for Generalized Bodies,

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The Global Point of View: C1 -Functionals, Locality and Continuity in Constitutive Theory.

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Reuven Segev: Geometric Methods, March 2001

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Scalar-Valued Extensive Properties

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Reuven Segev: Geometric Methods, March 2001

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Volume Elements

An infinitesimal element defined by the tan- v 1 gent vectors v1 ; v2 ; v3 2 Tx U , U —the space (3-dimensional) manifold.

  

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v3 x

v2

For a given property p, ρx (v1 ; v2 ; v3 )—the amount of the property in the element. ρx : (Tx U )3 ! R .

ρx should be linear in each of the three vectors—ρx multi-linear. ρx (v1 ; v2 ; v3 ) should vanish if the three are not linearly independent (flat element). Hence, for example, since ρx (v + u; v2 ; v + u) = 0

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0 = ρx (v; v2 ; v) + ρx (u; v2 ; u) + ρx (v; v2 ; u) + ρx (u; v2 ; v) = ρx (v; v2 ; u) + ρx (u; v2 ; v):

ρx is anti-symmetric (alternating), i.e., ρx (v; v2 ; u) = ρx (u; v2 ; v)!

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Reuven Segev: Geometric Methods, March 2001

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Volume Elements and m-Forms

For a manifold U of dimension m integration for the total quantity of the property p is defined using alternating forms.

 

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Vm

Tx U is the collection of m-alternating multi-linear mappings on Vm  Vm  S (T U ) = x2 Tx U is the bundle of m-multi-linear Tx U . alternating forms on U .

U

Vm

An m-differential form ρ : U ! (T  U ), or a volume element (not Vm  Tx U the infinitesimal elements generated by the vectors), ρ (x) 2 is integrated to give the sum of the contents of the extensive property in the various infinitesimal elements in any region R  U ,

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Z

R

ρ.

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Reuven Segev: Geometric Methods, March 2001

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An (m

1)-Form on the Boundary—Flux Density

An infinitesimal element defined by the tangent vectors v1 ; v2 2 Tx ∂ R , ∂ R —the boundary (say 2dimensional) of a control region R .



x v1

∂

v2

R

For a given property p, we would like to integrate the flux density out of the boundary. Now τx (v1 ; v2 )—the flux through the the element. τx : (Tx ∂ R )2 ! R .

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Since ∂ R is an (m 1)-dimensional manifold, the flux density is a Vm 1  T ∂ R , an (m 1)-form on ∂ R . mapping τ : ∂ R !

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Orientation

The fact that the volume element is anti-symmetric causes a complication. The sign of the evaluation τ (v1 ; v2 ) (or ρ (v1 ; v2 ; v3 )) will change according to the way we order the vectors.

 

v2

v1

v1 v2 v v1 v2 1 v 2 u2 u1

Orientability—the ability to construct the various coordinate systems such that the Jacobian transformation matrix has a positive determinant.

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This is equivalent to the ability to construct a volume element that does not vanish at any point on the manifold.

A choice of such a form, say θ , determines an orientation on the manifold. If θ (v1 ; : : : ;vm ) > 0, the vectors are positively oriented .

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The Balance of an Extensive Property

For an oriented manifold U of dimension m we consider control regions—m-dimensional compact submanifolds with boundary.

 

ρ is time dependent with time-derivative β . For a fixed control region R R in space β is the rate of change of the property inside R .

R

R

For each control region R there is a flux density τ such that is the total flux of the property out of R .



R ∂

R

τR

There is a positive m-form s on U such that for each region R

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Z Z β+ τ

R

∂

R

R 

Z

R

s:

Usually, equality is assume to hold (no absolute value) and s is interpreted as the source density of the property p.

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Review of the Classical Cauchy Postulate and Theorem

Cauchy’s postulate and theorem are concerned with the dependence of τ on R .

R

n

x

R T ∂R ∂

x

 It uses the metric properties of space.  τR (x) is assumed to depend on R only through the unit normal to the

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boundary at x. Generalize this to dependence on Tx ∂ R .



The resulting Cauchy theorem asserts the existence of the flux vector h such that τ (x) = h
n.

R

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The Generalization of Cauchy’s Theorem (m 1)-Forms on an m-Dimensional Manifold

For the 3-dimensional example, we want to measure the flux through any infinitesimal surface element (on the various planes through x), say the one generated by the vectors v; u.

v0

u v

v

v + v0

u u

Denote by J (v; u) the flux through that infinitesimal element.

 

J (v; u) should be linear in both arguments—J is multilinear.

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J (v; u) should vanish it they are not linearly independent—J is alternating.

A 2-form in a 3-dimensional space, or generally, an (m an m-dimensional manifold.

1)-form on

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The Dimension of the Space of m-Forms Say fe1 ; e2 ; e3 g is a base of the tangent space at a fixed point x. The matrix of ρ is ρi jk = ρ (ei ; e j ; ek ).

 

e1 x

e2

However, because it is alternating, ρ has only one independent component, e.g., ρi jk = 0 if any two indices are equal. It is enough to know ρ123 = ρ (e1 ; e2 ; e3 ), the volume of the basic element, to know the amount of property in all other infinitesimal elements.

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e3

In general, the dimension of

Vm

(Tx U )

is 1.

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The Dimension of the Space of (m

1)-Forms

Again, fe1 ; e2 ; e3 g is a base of the tangent space at x. The matrix of the 2-form J is Ji j = J (ei ; e j ). Now, as J is alternating there are 3 different independent components, namely, J (e2 ; e3 ); J (e1 ; e3 ); J (e1 ; e2 ). In general, the dimension of

Vm 1  Tx U is m.

In other words, if we know the flux density through the three basic surface elements we know the flux through any other infinitesimal surface element. J (u; v) = Ji j ui v j .

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The three components of the flux 2form are the generalizations of the three components of the flux vector field.

J13 = J (e1 ; e3 )

e3

e1

J23 = J (e2 ; e3 ) x

J12 = J (e1 ; e2 )

e2

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Cauchy’s Formula and the Restriction of Forms

The (m 1)-form J on U (m components) induces by restriction an (m 1)-form τ on ∂ R .

x u v

—τ is given by

τ (v; u) = J (v; u):

R T ∂R ∂

x

The induced form τ has a single component as it is an (m 1)-form on the (m 1)-dimensional manifold ∂ R . The mapping that assigns τ to J is the restriction and it is denoted as

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τ



= ι (J ):

This equation is the required generalization of Cauchy’s formula.

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The Induced Orientation and Newton’s Third Law Now, R 0 has the same tangent space

at x as R . w is a vector pointing out of R (into R 0 ). The form ι  (J ) is one for both R and R 0 . How do we distinguish the surface flux densities τ and τ 0 ?

R

R

U

w

U

∂

R0

x u v

R T ∂R T ∂ R0 ∂

x x

=

 It was assumed that was oriented so there is a way to tell whether any ordered triplet fu; v; wg is positively or negatively oriented.

R

 This induces an orientation on the boundary of each region. At x 2 ∂ , take any outwards (relative to ) pointing vector w and set fu; vg to be positively oriented on ∂ if fw; u; vg is positively oriented in .

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R

R

R

U

R

 Hence, the orientation on ∂ 0 is opposite to that of ∂ . Thus, if J (u; v) is the flux out of the infinitesimal -positively oriented element fu; vg, the flux out of 0 for the same geometric element is J (v; u) = J (u; v).

R

R

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Stokes’ Theorem and the Differential Balance Law

 The boundary integral in the balance law

Z

β+

Z

τR

=

Z s

R ∂R R of the property p assumes now the form Z Z τR = ι  (J ) ∂R ∂R

:

 Stokes’ theorem (a generalization of the divergence theorem etc.): There is an m-form dJ (having a single component), such that

Z

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 Then, for each

∂

R

ι  (J ) =

Z

R

dJ:

RZ, the balance takes the form Z Z R

β+

R

dJ =

R

s;

hence,

β + dJ = s:

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Reuven Segev: Geometric Methods, March 2001

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Generalized Bodies

The Material Structure Induced by an Extensive Property

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Organisms

Material points, bodies and subbodies are primitive concepts in continuum mechanics. These notions are somehow related to the conservation of mass. In growing bodies, material points are added and removed from the body. Examples: fingerprints, birthmarks are distinguishable. An organism has a body structure although mass is not preserved. Can motivate this idea?

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Assume we have an extensive property.

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The Material Structure Induced by an Extensive Property

In the classical case we have the flux vector field h. It can be integrated to give us a material structure. A material point is identified with an integral line (a flow line). This procedure may induce material structure associated with any extensive property, e.g., color and energy.



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h will be the velocity field of the material points. ρ

Can we generalize the same idea for the general manifold case where the flow (m 1)-form replaces the vector field?

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The Case where a Volume Element is Specified

It is not necessary to have a metric structure in order that the flux form J be represented w by a vector field. v Assume that you have a volume element θ x (m-form) on U . This may be thought of as u the density of the property p if it is positive or another positive property, e.g., mass.  Given J and θ , find a vector v such that for every pair of tangent vectors, u; w,

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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. The vector field v depends linearly on the flux J.

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The Flux Bundle

Let us examine how the kinematic flux v varies as we vary the volume element. Since the space of m-forms at x is 1-dimensional, as we vary the volume element the resulting vectors v remain on a line (1-D subspace of the tangent space).



w u

Another characterization: If a surface element (say the one defined by the vectors u; w) contains the line, the flux through it vanishes.

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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.

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Generalized Body Points

Integral manifolds of the distribution, the 1-dimensional flux bundle in this case, are submanifolds whose tangent space at a point is the corresponding line of the flux bundle at that point. In general such integral manifolds need not exist (higher dimensions), however they always exist for 1dimensional bundles as is the case here.

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Each integral line manifold is identified with a body point. Actual formulation is done on space-time manifold to allow time dependent fluxes. There β is included in τ and dJ = s.

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Frames in Space-Time

Cartesian Product an event e

(t ; x)

a frame

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Space-Time

U

Time Axis Space

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Property-Induced Fibration and Frame

No volume element: Fibration —no real valued time is assigned to events

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A volume element: Integrable vector field —real valued time is assigned to events

time axis a worldline Space-Time

non-unique models of space

a worldline Space-Time

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Space Formulation VS. Space-Time Formulation Space Formulation dim U = 3 dim R = Z Z 3 Z Balance surface term source term flux form variables field equation

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R

β+

∂

R

τ

=

R

Space-Time Formulation dim E = 4 β dim Z R= Z 4 s

2-form on a 3-D manifold 3-form on a 3-D manifold J—3 components —time dependent β + dJ = s

∂

R

t=

R

s

τ

3-form on a 4-D manifold 4-form on a 4-D manifold J—4 components —fixed values at events dJ = s

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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

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∂M

for every control region R Z

Z

R

dJ = ∂

R

ι  (J ) =

Z

∂

R

ι ω

Z =

dω ;

M

Z

ι  (dA) = ∂ (∂

R

?

)=


  ι ι (A) = 0:

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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 m-dimensional space.

Flow lines still make sense using the flux bundle. Generalized body points may be associated with an arbitrary extensive property—organisms.

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1)-form in the

A particularly compact formulation in space-time. A positive extensive property induces a material frame.

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Forces and Cauchy Stresses

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Cauchy Stress Theory on Manifolds

Reminder: The classical Cauchy theory for the existence of stress uses the metric structure of the Euclidean space.

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How would you generalize the notion of stress and Cauchy’s postulate so the theory can be formulated on a general manifold?

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Added Benefit Such a stress object will unify the classical Cauchy stress and Piola-Kirchhoff stress. If you consider a material body as a manifold, all configurations of the body, in particular, the current configuration and any reference configuration, are equivalent charts in terms of the manifold structure of the body. The transformation from the Cauchy stress to the Piola-Kirchhoff stress will be just a transformation rule for two different representations of the stress object.

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In Classical Continuum Mechanics The force on a body R in the material manifold R 3 is given by

R=

F

Z

R

R

Z

b dv + ∂

R

R

t da:

R is the body force on R ; tR is the surface force on R . The force system f(bR tR )g is considered as a set function. b

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;

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Cauchy’s Postulates for the dependence on

R

R.

R (x) = b(x).

The body force b does not depend on the body, i.e., b

The surface force at a point on the boundary of a control volume depends on the normal to the boundary at that point, i.e., t (x) = Σx (n(x)).

R

Σx is assumed to be continuous. There is a vector field s on the material manifold, the ambient force or self force (usually taken as zero), such that

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R=

I

Z

R

R

Z

b dv +

Cauchy’s Theorem: Σx is linear.

∂

R

R

Z

t da =

R

s dv:

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Obstacles to the Generalization to Manifolds:

 

You cannot integrate vector fields on manifolds. You do not have a unit normal if you do not have a Riemannian metric.

Basic modifications:

  

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Use integration of forms on manifolds to integrate scalar fields. Write the force in terms of power expanded for various velocity fields so you integrate a scalar field. Use dependence on the tangent space instead of direction of the normal. Use restriction of forms for Cauchy’s formula.

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Preliminaries for Continuum Mechanics on Manifolds: U

is the material manifold, dim U

R

a body is an m-dimensional submanifold on U .

= m;

M is the physical space manifold, dim M

=

µ.

A configuration of a body R is an embedding κ : R ! M . A velocity is a mapping w : R ! T M such that τ configuration.

M Æ w = κ is a

Alternatively,

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if κ  (τ ) : W = κ  (T M ) ! U is the pullback, a velocity at κ may be regarded as a section w : U ! W .

M

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Velocity Fields

w

R

&

κ

M

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Bundles and Pullbacks E—a bundle (e.g., T

= κ E

M)

Ex projection

π

R a body

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x

U

κ

M space manifold

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Sections of Bundles

= κ E

E—a bundle (e.g., T

section w

M)

Ex projection

π

R a body

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x

U

κ

M space manifold

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Force Densities:

R (w) =

F

Z

R

b

Z

R (w) +

∂

R

R (w)

t

;

for linear m ^

R (x) : Wx ! Tx U Thus, bR is a section of b

L W;

m ^

;

and

R (y) : Wy !

t




m ^




m^1

(T R ) =

T R;W

m^1




;

R

and t is a section of

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L W;



m^1

— W -valued forms.

(T ∂ R ) =

Ty ∂ R :

T ∂ R;W




;

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Vector Valued Forms:


Vk

 γx 2 L Wx (Tx P) , P  U a submanifold, k  dim(P).  γ˜x : (Tx P)n ! Wx , alternating, multi-linear. ;

γ˜x 2



k ^

(Tx U

;



Wx

);

a vector valued form.

The requirement

γ˜x (v1 ; : : : ;vk )(u) = γx (u)(v1 ; : : : ;vk ); for any collection of k vectors v1 ; : : : ;vk , and u 2 Wx generates an isomorphism

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L Wx ;

k ^




(Tx P) =

k ^

(Tx U

;

Wx ):

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What Will Cauchy’s Theorem and Formula Look Like?

For scalars, the flux form was an (m 1)-form J on an mdimensional manifold. By restriction, the Cauchy formula, τ = ι  (J ), induces an (m 1)form on Tx ∂ R .

x u v

R



R

R

R T ∂R ∂

x

For the case of force theory, t (w) is an (m Vm 1  Tx ∂ R . power, where t (x) : Wx !

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1)-form, the flux of

The natural generalization: at each point x there is a linear mapping V σx : Wx ! m 1 Tx U , called the stress at x, such that
 t (w) = ι σ (w) . In other words

R

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R = ι Æ σ

t

;

is the required Cauchy formula.

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The Cauchy Postulates: Notes.

R

The dependence of t (x) on the subbody R through the tangent space to R is assumed to be continuous in the tangent space and point x. This aspect, that we neglected before, should be meaningful.



y

u v

R T ∂R ∂

x

The collection of hyperplanes, Gm 1 (T U )—the Grassmann bundle,
i.e., Gm 1 (T U ) x is the manifold of (m 1)-dimensional subspaces of Tx U .

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x

The mapping that assigns the surface forces to hyperplanes will be referred to as the Cauchy section. At each point it is a mapping

Σx : Gm

1 (Tx U )

! L Wx

m^1 ;

Gm

1 (Tx U )





:

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The Cauchy Postulates: The Cauchy Section More precisely, consider the diagram

π  (W )

 (π ) πG

!

G

x ? ?

W

π

!

Gm


 Vm 1 Gm 1 (T U )

1 (T U )

? ?π y G

U

Then, the Cauchy section is a section

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Σ : Gm 1 (T U ) ! L πG (W );

It is assumed that Σ is smooth.

m^1

Gm

1 (T U )





:

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The Cauchy Postulates: Boundedness

We need the analog of the boundedness assumption Z Z β+ τ

R 

Z

s;

R R ∂R R R  where eventually we get τR = ι (J ) and τR = dJ. R ∂R  We write the scalar boundedness for the power, so β = b(w) and τR = tR (w).  We anticipate that tR = ι  Æ σ . Hence, the bounded expression is

Z Z Z Z Z Z

 b(w) + t (w) = b(w) + ι σ (w) = b(w) + d σ (w) :

& R

∂

R

R

R

∂

R

R

R

Thus, the expression should be bounded by the values of both w and its derivative—the first jet j1 (w).

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Consequences of the (Generalized) Cauchy Theorem
 Since t (w) = ι σ (w) , the total power is given as

R

R (w) =

F

 

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Z

Z

b(w) +

R

∂

R

R (w) =

t

Z

R

Z

b(w) +

R

d σ (w)




:

R (w) depends linearly on the values of w and its

The density of F derivative.

For manifolds, there is no way to separate the value of the derivative of a section from the value of the section. Hence j1 (w)—the first jet of w is a single invariant quantity that contains both the value and the value of the derivative.

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Thus, the expression should be bounded by the values of both w and its derivative—the first jet j1 (w).

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Variational Stresses

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Jets Wx

A jet of a section at x is an invariant quantity containing the values of both the section and its derivative.

j1 (w)x

W

w

R

x

J 1 (W )x —the collection of all possible values of jets at x—the jet space.

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J 1 (W )—the collection of spaces, the jet bundle.

J 1 (W )x

jet J 1 (W )

x

R

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Variational Stresses

We obtained FR (w) =

Z

R




b(w) + d σ (w)

:

 The value of the power density at a point is linear in the jet of w.  Hence, there is a section S, such that Sx : J

1

m ^ (W )x ! Tx U

such that

1

Sx j (w)x




= b(w) + d




σ (w)

:

Vm (T  U )
as a variational

 We will refer to such a section S of L stress density. It produces power from the jets (gradients) of the velocity fields. J 1 (W );

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 Thus, FR (w) =

Z

R




b(w) + d σ (w)

=

Z

R

1




S j (w)

:

Conclusion: A Cauchy stress σ and a body force b induce a variational stress density S.

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Variational Stress Densities:

 

Variational stress densities are sections of the vector bundle
Vm  1 L J (W ); (T U ) . If S is a variational stress density, then the power of the force F it represents over the body R , while the the generalized velocity is w, is given by Z

R

F



(w) =

R

S( j1 (w)):

This expression makes sense as S( j1 (w)), is an m-form whose value
1 at a point x 2 R is S(x) j (w)(x) .

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The local representation of S is through the arrays Sα and Sβj . The

single component of the m-form S

j1 (w)




Sα wα + Sβj wβ; j :

in this representation is

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Reuven Segev: Geometric Methods, March 2001

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Linear Connections vertical Wy

Wx

w

U

&

Tπ

π

π x

horizontal component

vertial component Wx

Tπ

w

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y

Γ x

U

y

W

W

no connection

Γ—the connection mapping

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The Case where a Connection is Given:

 

If a connection is given on the vector bundle W , the jet bundle is
isomorphic with the Whitney sum W  L T R ; W by j1 (w) 7! (w; ∇w), where ∇ denotes covariant derivative.

R

A variational stress may be represented by sections (S0 ; S1 ) of L W;

m ^




 (T U )

R L L T U

;

W

m
^ ;


 (T R )

so the power is given by (see Segev (1986))

R (w) =

F

&

Z

R

Z

S0 (w) +

R

S1 (∇w):


Vm 
We will refer to the section S1 of L L T U ; W ; (T R ) as the

variational stress tensor.



With an appropriate definition of the divergence, a force may be written in terms of a body force and a surface force.

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Problems:

  

Relate the two approaches for stress theory on manifolds in the general case where a connection is not given. In particular, can we extract the generalized Cauchy stress σ from the variational stress S invariantly? Can you write a generalized definition of the divergence that applies even without a connection?

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Reuven Segev: Geometric Methods, March 2001

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$ 51

The Vertical Subbundle of the Jet Bundle:

 

Let π01 : J 1 (W ) ! W be the natural projection on the jet bundle that assign to any 1-jet at x 2 R the value of the corresponding 0-jet, i.e., the value of the section at x. We define VJ 1 (W ), the vertical sub-bundle of J1 (W ), to be the vector bundle over R such that VJ 1 (W ) = (π01 )

1

(0);

where 0 is the zero section of W .

& 

There is a natural isomorphism I : VJ (W ) ! L T U ; W +

1




:

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Reuven Segev: Geometric Methods, March 2001

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The Vertical Component of a Variational Stress:

 

Let ιV : VJ 1 (W ) ! J 1 (W ) be the inclusion mapping of the sub-bundle. Consider the linear injection

ιn = ιV Æ (I



+

)

1

: L T U ;W




! J1 (W )

:

Thus we have a linear surjection

ιn : L J 1 (W );

m ^




 (T R )

! L L TU

W

;

m
^ ;


 (T R )

given by ιn (S) = S Æ ιn .

& 

$ 52

For a variational stress S, we will refer to S = ιn (S) 2 L L T U ; W +

as the vertical component of S.

m
^ ;


 (T R )

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Reuven Segev: Geometric Methods, March 2001

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$ 53

If the variational stress is represented locally by (Sα ; S j ), then, S+ is β

represented locally by i

S+ α



i = Sα :

Clearly, one cannot define invariantly (without a connection) a “horizontal” component to the stress.

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Reuven Segev: Geometric Methods, March 2001

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$ 54

Variational Fluxes



Since the jet of a real valued function ϕ on R can be identified with a pair (ϕ ; d ϕ ) in the trivial case where W = R  R , the jet bundle can be identified with the Whitney sum W  T  U .

 VJ1 (W ) can be identified with T U

R

and the vertical component of
Vm   the variational stress is valued in L T U ; (T R ) . We will refer
Vm   to sections of L T U ; (T R ) as variational fluxes.



There is a natural isomorphism

&

i^ :

m^1

  (T R ) ! L T U ;

given by i^ (ω )(φ ) = φ ^ ω .

m ^


 (T R )

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Reuven Segev: Geometric Methods, March 2001

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The Cauchy Stress Induced by a Variational Stress:



Consider the contraction natural vector bundle morphism c : L L T U ;W

m
^

given by

;






 (T R )

R W ! L T U ;

m ^


 (T R )

c(B; w)(φ ) = B(w φ );


Vm



$ 55


 (T R ) , w 2 W , and φ 2 T  U , where

for B 2 L L T U ; W ; (w φ )(v) = φ (v)w. We also write wy B for c(B; w).


Vm 
For a section S of L L T U ; W ; (T R ) and a vector field w, +

wy S+ is a variational flux.

&

Consider the mapping iσ : L L T U ; W

m
^ ;

 (T R )




!L W

m^1 ;


 (T R )

such that iσ Æ S+ (w) = i^ 1 (wy S+ ). It is linear and injective.

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Reuven Segev: Geometric Methods, March 2001

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$ 56


Vm  Vm  1 pσ = iσ Æ ι : L J (W ); (T R ) ! L W;

1


 (T R ) is a linear

mapping (no longer injective) that gives a Cauchy stress to any given variational stress.

 

Locally, σ is represented by σβ ıˆ such that σβ ıˆwβ is the i-th component of the (m 1)-from σ (w). Locally pσ is given by (x

&

i

;

Sα ; Sβj ) 7! (xi ; σβ ıˆ)

where, i

σβ ıˆ = ( 1)i 1 S+ β ;

(no sum over i):

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Reuven Segev: Geometric Methods, March 2001

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$ 57

The Divergence of a Variational Stress:



For a given variational stress S and a generalized velocity w, consider the difference, an m-form, d pσ (S)(w)





1

S j (w)




:

Locally, the difference is represented by Sαi ;i







α Sα w

This shows that the difference depends only on the values of w and not its derivative. Define the generalized divergence of the variational stress S to be the
Vm  section Div(S) of the vector bundle L W; (T R ) satisfying

& σ

Div(S)(w) = d pσ (S)(w)

=




1

S j (w)




= d σ (w)

pσ (S), for every generalized velocity field w.

1

S j (w)




;

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Reuven Segev: Geometric Methods, March 2001

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$ 58

The Principle of Virtual Power:



Given a variational stress S, the expression for the power is

R (w) =

F



1

R

S j (w)




:

Using the previous constructions and Stokes’ theorem we have

R

F



Z

Z

(w) = ∂

R

 σ (w) ιR




Z

R

Div(S)(w);

where, σ = pσ (S) is the the Cauchy stress induced by the variational stress S, and ι  is the restriction of (m 1)-forms on R to ∂ R .

R



  Thus we have for tR (w) = ιR σ (w) = ιR pσ (S)(w) and Div S + bR = 0, a force for each subbody R of the form

&

R (w) =

F

Z

Z

R (w) + tR (w) R ∂R b

:

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Reuven Segev: Geometric Methods, March 2001

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$ 59

Conclusions:



The mapping relating the values of variational stress fields and Cauchy stresses 1

pσ : L J (W );

m ^


 (T U ) ! L W ;

m^1


 (T U ) ;

is linear, surjective, but not injective.



However, the mapping between the fields p : S 7! (σ ; b);

σ

=

pσ Æ S ;

b=

is injective.

& 

The inverse, p

1 : (σ ; b)

7! S, is given by

S(x)(A) = bx (wx ) + d σ (w)x ;

for any vector field w whose jet at x is A.

Div S;

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Reuven Segev: Geometric Methods, March 2001

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$ 60

Stresses for Generalized Bodies

&

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Reuven Segev: Geometric Methods, March 2001

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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.

& 

$ 61

R

Z

For the spacetime formulation F (J ) = Vm 1  t ( e) : Te E

 R

!

Vm 1  Te ∂ R .

∂

R

R (J)

t

;

R E:

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Reuven Segev: Geometric Methods, March 2001

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Stresses for Generalized Bodies

Consider the energy extensive property. It has a flux density term R (e) τ and a corresponding flux form J(e) such that τ (e) = ι  Æ J (e) .

∂

$ 62

R

On the other hand the flux density of energy may be written in terms of the boundary force as t (J ).

R

Cauchy’s theorem implies that t = ι  Æ σ so the energy flux density is τ (e) = ι  Æ J (e) = ι  Æ σ (J ). Hence,

R

J (e) = σ (J ): The Cauchy stress is the linear mapping that transforms the flux of the property p into the flux of energy.

& 

Vm 1  σe : Te E

!

Vm 1  Te E . The stress at a point (event) is a linear

transformation on the space of (m



1)-forms.

May be applied to “resources” other then energy?

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Reuven Segev: Geometric Methods, March 2001

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$ 63

Local Representation of Stress-Tensors

 Denote by feˆi g the basis of the m-dimensional space of (m Denote its dual basis by feˆ j g.

1)-forms.

 Since the stress at a point is a linear transformation on the space of ˆ j eˆ j eˆi . (m 1)-forms it may be represented in the form σ i  If we had a volume element θ we would have an isomorphism Vm 1 (T  ) $ T 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,

U

U

Vm

&

1

i 1 θ

(T 

x ? ?

T

U

U

)

σ

σ˜

! Vm !

1

(T 

? ? y iθ

T

U

U

)

;

one may represent a stress σ by a linear transformation σ˜ on T

U.

 Surprisingly, σ˜ is independent of the volume element θ . In fact, you can construct a natural isomorphism σ $ σ˜ without a volume element.

%

Reuven Segev: Geometric Methods, March 2001

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The Maxwell Stress-Energy Tensor without a Metric

   

Maxwell 2-form: g, a flow potential for J, i.e., J = d g. Faraday 2-form: f such that d f = 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
Vm 1  Vm 1  (T U ); (T U ) by L


σ (J ) = wy g



$ 64

^f




wy f

^g

:

The power is

&

d σ (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.

%

Reuven Segev: Geometric Methods, March 2001

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$ 65

The Global Point of View

C n-functionals

&

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Reuven Segev: Geometric Methods, March 2001

'

Review of Basic Kinematics and Statics on Manifolds



 

The mechanical system is characterized by its configuration space—a manifold Q.

$ 66

Tκ

Q κ

Velocities are tangent vectors to the manifold— elements of T Q . A Force at the configuration κ is a linear mapping F : Tκ Q ! R .

&

Q

Can we apply this framework to Continuum Mechanics?

%

Reuven Segev: Geometric Methods, March 2001

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$ 67

Problems Associated with the Configuration Space in Continuum Mechanics

  

What is a configuration? Does the configuration space have a structure of a manifold? The configuration space for continuum mechanics is infinite dimensional.

&

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Reuven Segev: Geometric Methods, March 2001

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$ 68

Configurations of Bodies in Space

A mapping of the body into space; material impenetrability—one-to-one; continuous deformation gradient (derivative); do not “crash” volumes—invertible derivative.

κ

&

κ(

A body

R U

Space

R

)

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Reuven Segev: Geometric Methods, March 2001

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$ 69

Manifold Structure for Euclidean Geometry

If the body is a subset of R 3 and space is modeled by R 3 , the collection of differentiable mappings C 1(R ; R 3 ) is a vector space However, the subset of “good” configurations is not a vector space, e.g., κ κ = 0—not one-to-one. We want to make sure that the subset of configurations Q is an open subset of C 1(R ; R 3 ), so it is a trivial manifold. configurations

&

R

C 1( ; R3 ) all differentiable mappings

configurations

R

C 1( ; R3 ) all differentiable mappings

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Reuven Segev: Geometric Methods, March 2001

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$ 70

The C 0-Distance Between Functions

The C 0-distance between functions measures the maximum difference between functions. A configuration is arbitrarily close to a “bad” mapping. Space

&

a configuration solid

“bad mapping” dotted

Body

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Reuven Segev: Geometric Methods, March 2001

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$ 71

The C 1-Distance Between Functions

 The C 1 distance between functions measures the maximum difference between functions and their derivative

ju vjC

1

= sup

fju(x)

v(x)j; jDu(x)

Dv(x)jg:

 A configuration is always a finite distance away from a “bad” mapping. Space

&

a configuration solid

“bad mapping” dotted

Body

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Reuven Segev: Geometric Methods, March 2001

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Conclusions for R 3

If we use the C 1-norm, the configuration space of a continuous body in space is an open subset of C 1(R ; R 3 )-the vector space of all differentiable mapping.

Q is a trivial infinite dimensional manifold and its tangent space at any point may be identified with C 1(R ; R 3 ).

& 

$ 72

A tangent vector is a velocity field.

Rg

κf

u(κ (x)) =

d κ (x) dt

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Reuven Segev: Geometric Methods, March 2001

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For Manifolds

R and space U are differentiable manifolds.  The configuration space is the collection Q Emb R U of the embeddings of the body in space. This is an open submanifold of the infinite dimensional manifold C R U .  The tangent space T Q may be characterized as T Q fw : R ! T Q jτ Æ w κ g or alternatively, T Q C κ  T U  Both the body

=

1(

(

;

)

)

;

κ

κ

$ 73

=

=

κ  (T

M

κ

;

1 (

=

T

)

):

M Tx

M

projection

&

R a body

τ w

x

κ

M space manifold

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Reuven Segev: Geometric Methods, March 2001

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$ 74

Representation of C 0-Functionals by Integrals

 Assume you measure the size of a function using the C0-distance, kwk = supfjw(x)jg.  A linear functional F : w 7! F (w) is continuous with respect to this norm if F (w) ! 0 when max jw(x)j ! 0.  Riesz representation theorem: A continuous linear functional F with respect to the C 0-norm may be represented by a unique measure µ in the form F (w) =

&

F (w) =

δ

R

R

Z

R

w dµ:

wφ dx

F (w) =

δ

wφ dx

force density φ

Velocity w

R

R

F isn’t sensitive to the derivative

force density φ

Body

R

Body

R

Velocity w

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Reuven Segev: Geometric Methods, March 2001

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$ 75

Representation of C 1-Functionals by Integrals

 Now, you measure the size of a function using the C 1-distance, kwk = supfjw(x)j; jDw(x)jg.

 A linear functional F : w 7! F (w) is continuous with respect to this norm if F (w) ! 0 when both max jw(x)j ! 0 and max jDw(x)j ! 0.  Representation theorem: A continuous linear functional F with respect to the C 1-norm may be represented by measures σ0 ; σ1 in the form F (w) =

&

Z

R

w d σ0 +

Z

R

Dw d σ1 :

F is sensitive to the derivative R R F (w) = φ0 wdx + φ1 Dwdx

R

δ

R

“self” force density φ0 Velocity w Body

R

stress density φ1

Velocity gradient Dw

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Reuven Segev: Geometric Methods, March 2001

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$ 76

Non-Uniqueness of C 1-Representation by Integrals

 We had an expression in the form F (w) =

Z

R

w d σ0 +

Z

R

w0 d σ1 :

 If we were allowed to vary w and w0 independently, we could determine σ0 and σ1 uniquely.  This cannot be done because of the condition w0 = Dw. F (w) =

&

δ

R

R

φ0 wdx + φ1 w0 dx R

R

“self” force density φ0 Velocity w Body

R

stress density φ1 w0

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Reuven Segev: Geometric Methods, March 2001

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$ 77

Unique Representation of a Force System

 Assume we have a force system, i.e., a force FP for every subbody

P of R .

 We can approximate pairs of non-compatible functions w and w0 , i.e., w0 6= Dw, by piecewise compatible functions. approximation of w

Calculate

R

R wdσ0 Body

R

&  

R wdσ0

P1 P2 w0

Calculate

R

::::::

approximation of

R w0 dσ1

R

R w0 dσ1

R

Body

R

P1 P2

::::::

This way the two measures are determined uniquely. One needs consistency conditions for the force system.

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Reuven Segev: Geometric Methods, March 2001

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Generalized Cauchy Consistency Conditions

P

 Additivity: FP

$ 78

1

P

1

2

[P2 (wjP1 [P2 ) = FP1 (wjP1 ) + FP2 (wjP2 ):

 Continuity:

P

If i ! A, then FP (wjP ) coni 1 verges and the limit depends on A only.

P

i

A

 Uniform Boundedness: There is a K P and every w,

&

>

0 such that for every subbody

jFP (wjP )  K kwP k

:

Main Tool in Proof: Approximation of measurable sets by bodies with smooth boundaries.

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Reuven Segev: Geometric Methods, March 2001

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$ 79

Generalizations

  

All the above may be formulated and proved for differentiable manifolds. This formulation applies to continuum mechanics of order k > 1 (stress tensors of order k). One should simply use the Ck-norm instead of the C 1-norm. The generalized Cauchy conditions also apply to continuum mechanics of order k > 1. This is the only formulation of Cauchy conditions for higher order continuum mechanics.

&

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Reuven Segev: Geometric Methods, March 2001

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$ 80

Locality and Continuity in Constitutive Theory

&

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Reuven Segev: Geometric Methods, March 2001

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$ 81

Global Constitutive Relations (Elasticity for Simplicity)

 Q, the configuration space of a body R .

0  C R L R R , the collection of all stress fields over the body.

0  Ψ : Q ! C R L R R , a global constitutive relation. ;

3

;

3

3

;

;

3

space

&

stress

configuration κ

Body

R

Ψ Global constitutive relation.

stress field σ = Ψ(κ )

Body

R

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Reuven Segev: Geometric Methods, March 2001

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$ 82

Locality and Materials of Grade-n

Germ Locality: If two configurations κ1 and κ2 are equal on a subbody containing X, then the resulting stress fields are equal at X. space

stress Ψ Ψ(κ1 )

κ1 κ2

Ψ(κ2 ) X

P

Body

X

P

R

Body

R

Material of Grade-n or n-Jet Locality: If the first n derivatives of κ1 and κ2 are equal at X, then, Ψ(κ1 )(X ) = Ψ(κ2 )(X ). (Elastic = grade 1.)

&

space

stress Ψ Ψ(κ1 )

κ1 κ2

Ψ(κ2 ) X

P

Body

X

R

P

Body

R

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Reuven Segev: Geometric Methods, March 2001

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n-Jet Locality and Continuity

Q

R





Basic Theorem: If a constitutive relation Ψ : ! is local and ;L R ;R continuous with respect to the C n-norm, then, it is n-jet local. In particular, if Ψ is continuous with respect to the C 1-topology, the material is elastic. space

C0

space

P

κ2

X

3

Whitney’s extension

κ1 j

κ1

3

space

restriction

κ2 j

P

κ10

κ20

X

Body

R

P

Body

X

P

R

Body

stress

R

Ψ

Ψ

&

$ 83

stress

Ψ(κ10 )

Ψ(κ1 )

Ψ(κ20 )

Ψ(κ2 ) X

P

Body

X

R

P

Body

R

%

Reuven Segev: Geometric Methods, March 2001