Notes on Global Stress and Hyper-Stress Theories
The fundamental ideas and tools of the global geometric formulation of stress and hyper-stress theory of continuum mechanics are introduced. The proposed framework is the infinite dimensional counterpart of statics of systems having finite number of degrees of freedom, as viewed in the geometric approach to analytical mechanics. For continuum mechanics, the configuration space is the manifold of embeddings of a body manifold into the space manifold. Generalized velocity fields are viewed as elements of the tangent bundle of the configuration space and forces are continuous linear functionals defined on tangent vectors, elements of the cotangent bundle. It is shown, in particular, that a natural choice of topology on the configuration space, implies that force functionals may be represented by objects that generalize the stresses of traditional continuum mechanics.
Key words and phrases:Continuum mechanics; Differentiable manifold; Stress; Hyper-stress; Global analysis; Manifold of mappings; de Rham currents.
2000 Mathematics Subject Classification:74A10; 58Z05; 58A25; 53Z05; 57N35
- 1 Introduction
- 2 Notation and Preliminaries
- 3 Banachable Spaces of Sections of Vector Bundles over Compact Manifolds
- 4 The Construction of Charts for the Manifold of Sections
- 5 The -Topology on the Space of Sections of a Fiber Bundle
- 6 The Space of Embeddings
- 7 The General Framework for Global Analytic Stress Theory
8 Duals to Spaces of Differentiable Sections of a Vector Bundle: Localization
of Sections and Functionals
- 8.1 Spaces of differentiable sections over a manifold without boundary and linear functionals
- 8.2 Localization of sections and linear functionals for manifolds without boundaries
- 8.3 Localization of sections and linear functionals for manifolds with corners
- 8.4 Supported sections, static indeterminacy and body forces
- 8.5 Supported functionals
- 8.6 Density dual and smooth functionals
- 8.7 Generalized sections and distributions
- 9 de Rham currents
10 Vector-valued currents
- 10.1 Vector valued forms
- 10.2 Vector valued currents
10.3 Local representation of vector valued currents
- 10.3.1 The inner product of a vector valued current and a vector field
- 10.3.2 The tensor product of a current and a co-vector field
- 10.3.3 Representation by -currents
- 10.3.4 The exterior product of a vector valued current and a multi-vector field
- 10.3.5 The contraction of a vector valued current and a form
- 10.3.6 Representation by -currents
- 11 The Representation of Forces by Hyper-Stresses and Non-Holonomic Stresses
- 12 Simple Forces and Stresses
- 13 Concluding Remarks
These notes provide an introduction to the fundamentals of global analytic continuum mechanics as developed in [ES80, Seg81, MH94, Seg86b, Seg86a, Seg16]. The terminology “global analytic” is used to imply that the formulation is based on the notion of a configuration space of the mechanical system as in analytic classical mechanics. As such, this review is complementary to that of [Seg13], which describes continuum mechanics on differentiable manifolds using a generalization of the Cauchy approach to flux and stress theory.
The setting for the basics of kinematics and statics is quite simple and provides an elegant geometric picture of mechanics. Consider the configuration space containing all admissible configuration of the system. Then, construct a differentiable manifold structure on the configuration space, define generalized (or virtual) velocities as tangent vectors, elements of , and define generalized forces as linear functions defined on the space of generalized velocities, elements of . The result of the action of a generalized force on a generalized velocity is interpreted as mechanical power. Thus, such a structure may be used to encompass both classical mechanics of mass particles and rigid bodies as well as continuum mechanics. The difference is that the configuration space for continuum mechanics and other field theories is infinite dimensional.
It is well known that the transition from the mechanics of mass particles and rigid bodies to continuum mechanics is not straightforward and requires the introduction of new notions and assumptions. The global analytic formulation explains this observation as follows. Linear functions, and forces in particular, are identically continuous when defined on a finite dimensional space. However, in the infinite dimensional situation, one has to specify exactly the topology on the infinite dimensional space of generalized velocities with respect to which forces should be continuous. Then, the properties of force functionals are deduced from the continuity requirement through a representation theorem. In other words, the properties of forces follow directly from the kinematics of the theory.
For continuum mechanics of a body in space , the basic kinematic assumption is traditionally referred to as the axiom of material impenetrability. A configuration of the body in space is specified by a mapping which is assumed to be injective and of full rank at each point—an embedding. Hence, the configuration space for continuum mechanics should be the collection of embeddings of the body manifold into the space manifold. It turns out that the -topology is the natural one to use in order to endow the collection of embeddings with a differentiable structure of a Banach manifold. The -topologies for are admissible also.
It follows that forces are continuous linear functionals on the space of vector fields over the body of class , , equipped with the -topology with a special role for the case . A standard procedure based on the Hahn-Banach theorem leads to a representation theorem for a force functional in terms of vector valued measures.
The measures representing a force generalize the stress and hyper-stress objects of continuum mechanics. On the one hand, as expected, a stress measure is not determined uniquely by a force. This is in accordance with the inherent static indeterminacy of continuum mechanics and it follows directly from the representation procedure. While the case leads to continuum mechanics of order one, the cases are extensions of higher order continuum mechanics. Thus, an existence theorem for hyper-stresses follows naturally. The relation between a force and a representing stress object is a generalization of the principle of virtual work in continuum mechanics and so it is analogous to the equilibrium equations.
The representation of forces by stress measures is significant for two reasons. First, the existence of the stress object as well as the corresponding equilibrium condition are obtained for stress distributions that may be as singular as Radon measures. In addition, while force functionals cannot be restricted to subsets of a body, measures may be restricted to subsets. This reflects a fundamental feature of stress distributions—they induce force systems on bodies. It is emphasized that in no further assumptions of mathematical or physical nature are made.
The framework described above applies to continuum mechanics on general differentiable manifolds without any additional structure such as a Riemannian metric or a connection. The body manifold is assumed here to be a compact manifold with corners. However, as described in [Mic20], it is now possible to extend the applicability of this framework to a wider class of geometric object—Whitney manifold germs.
Starting with the introduction of notation used in the manuscript in Section 2, we continue with the construction of the manifold structure on the space of embeddings. Thus, Section 3 describes the Banachable vector spaces used to construct the infinite dimensional manifold structure on the configuration space and Section 4 is concerned with the Banach manifold structure on the set of -sections of a fiber bundle . This includes, as a special case, the space of -mappings of the body into space and also provides a natural extension to continuum mechanics of generalized media. After describing the topology in in Section 5, we show in Section 6, that the set of embeddings is open in , . As such, it is a Banach manifold also and the tangent bundle is inherited from that of . In Section 7 we outline the framework for the suggested force and stress theory as described roughly above. Sections 8, 9 and 10 introduce relevant spaces of linear functionals on manifolds , and present some of their properties. These include some standard classes of functionals such as de Rham currents and Schwartz distributions on manifolds. The representation theorem of forces by stress measures in considered in Section 11. Section 12 discusses the natural situation of simple forces and stress, that is, the case .
Concluding remarks and references to further studies are made in Section 13.
2. Notation and Preliminaries
2.1. General notation
A collection of indices , will be represented as a multi-index and we will write , the length of the multi-index. In general, multi-indices will be denoted by upper-case roman letters and the associated indices will be denoted by the corresponding lower case letters. Thus, a generic element in a -multilinear mapping is given in terms of the array , . In what follows, we will use the summation convention for repeated indices as well as repeated multi-indices. Whenever the syntax is violated, e.g., when a multi-index appears more than twice in a term, it is understood that summation does not apply.
A multi-index induces a sequence in which is the number of times the index appears in the sequence . Thus, . Multi-indices may be concatenated naturally such that .
In case an array is symmetric, the independent components of the array may be listed as with . A non-decreasing multi-index, that is, amulti-index that satisfies the condition , will be denoted by boldface, upper-case roman characters so that a symmetric tensor is represented by the components , . In particular, for a function , a particular partial derivative of order is written in the form
where is a non-decreasing multi-index with .
The notation , will be used for both the partial derivatives in and for the elements of the basis of the tangent space of a manifold at a point . The corresponding dual basis for will be denoted by .
Greek letters, , , will be used for strictly increasing multi-indices used in the representation of alternating tensors and forms, e.g.,
Given a strictly increasing multi-index with , we will denote the strictly increasing -multi-index that complements to by . In this context, , , etc. will indicate generic increasing multi-indices. The Levi-Civita symbol will be denoted as or , so that for example , where we also set
(Note that we view and as two distinct indices so summation is not implied in a term such as .)
The following identifications will be implied for tensor products of vector spaces and vector bundles
For vector bundles and over a manifold , let be a section of and a section of . The notation is used for the section of given by
For two manifolds and , will denote the collection of -mappings from to . If is a fiber bundle, is the space of -sections .
2.2. Manifolds with corners
Our basic object will be a fiber bundle where is assumed to be an oriented manifold with corners. We recall (e.g., [DH73, Mic80, Mel96, Lee02, MRD08]) that an -dimensional manifold with corners is a manifold whose charts assume values in the -quadrant of , that is, in
In the construction of the manifold structure, it is understood that a function defined on a quadrant is said to be differentiable if it is the restriction to the quadrant of a differentiable function defined on . If is an -dimensional manifold with corners, a subset is defined to be a -dimensional, , submanifold with corners of if for any there is a chart , , such that .
With an appropriate natural definition of the integral of an -form over the boundary of a manifold with corners, Stokes’s theorem holds for manifolds with corners (see [Lee02, pp. 363–370]).
Relevant to the subject at hand is the following result (see [DH73, Mic80, Mel96, Mic20]). Every -dimensional manifold with corners is a submanifold with corners of a manifold without boundary of the same dimension. In addition, if is compact, it can be embedded as a submanifold with corners in a compact manifold without boundary of the same dimension [Mel96, pp. I.24–26]. Furthermore, -forms defined on , may be extended continuously and linearly to forms defined on . Such a manifold is referred to as an extension of . Each smooth vector bundle over extends to a smooth vector bundle over . Each immersion (embedding) of into a smooth manifold without boundary is the restriction of an immersion (embedding) of into .
It is emphasized that manifold with corners do not model some basic geometric shapes such as a pyramid with a rectangular base or a cone. However, much of material presented in this review is valid for a class of much more general objects, Whitney manifold germs as presented in [Mic20].
2.3. Bundles, jets and iterated jets
We will consider a fiber bundle , where is -dimensional and the typical fiber is -dimensional. The projection is represented locally by , , . Let
be the tangent mapping represented locally by
The vertical sub-bundle of is the kernel of . An element is represented locally as . With some abuse of notation, we will write both and . For with and , we may view as an element of . In other words, elements of the vertical sub-bundle are tangent vectors to that are tangent to the fibers.
Let be a section and let
be the pullback of the vertical sub-bundle. Then, we may identify with the restriction of the vertical bundle to .
We will denote by the corresponding -jet bundle of . When no ambiguity may occur, we will often use the simpler notation and refer to a section of as a section of . One has the additional natural projections , , and in particular [Sau89]. The jet extension mapping associates with a -section, , of , a continuous section of the jet bundle .
Let be a section of which is represented locally by
, . Then, denoting the -th derivative of by , a local representative of is of the form
Accordingly, an element is represented locally by the coordinates
2.3.2. Iterated (non-holonomic) jets
Completely non-holonomic jets for the fiber bundle are defined inductively as follows. Firstly, one defines the fiber bundles
Then, we define the iterated -jet bundle as
By induction, is a well defined fiber bundle..
When the projections are used inductively -times, we obtain a projection
Let be a -section of . The iterated jet extension mapping
is naturally defined by
Note that we use here as a generic jet extension mapping, omitting the indication of the domain.
There is a natural inclusion
Let be a vector bundle, then is a vector bundle. Continuing inductively,
is a vector bundle. In this case, the inclusion is linear. Naturally, elements in the image of are referred to as holonomic.
2.3.3. Local representation of iterated jets
The local representatives of iterated jets are also constructed inductively. Hence, at each step, , to which we refer as generation, the number of arrays is multiplied. Hence powers of two are naturally used below. Thus, it is proposed to use multi-indices of the form , where , , etc. are binary numbers that enumerate the various arrays included in the representation. For example, a typical element of , in the form
is written as
and for short
Here, is the generation where the -th array appears and it is given by
where denotes the integer part of . In (2.23,2.24) the generations are separates by semicolons. As indicated in the example above, with each we associate a multi-index as follows. For each binary digit 1 in there is an index , . Thus, the total number of digits 1 in , which is denoted by , is the total number of indices, , in . In other words, the length, , of the induced multi-index satisfies
Note also that the expression , is not a multi-index since we use upper-case letters to denote multi-indices. Here, the subscript serves for the enumeration of the indices. If no ambiguity may arise, we will often make the notation somewhat shorter and write for . Continuing by induction, let a section of be represented locally by , . Then, its -jet extension, a section of , is of the form
where is the binary representation of . It is noted that the array contains the derivatives of the array , and that . Thus indeed, the number of digits that appear in , i.e., , determine the length of the index .
It follows that an element of may be represented in the form
That is, for each with , we have an index such that if and if .
A similar line of reasoning leads to the expression for the local representatives of the iterated jet extension mapping. For a section , the iterated jet extension , a section of , the local representation , , , satisfies
Indeed, if , , with represent , then, is represented locally by
Thus, by induction, any with and , may be written as , , so that .
Let an element be represented by , then is represented by .
The right and left contractions of a -form and a -vector are given respectively by
for every -vector . We will use the notation
for the mappings satisfying
respectively. The left and right contraction differ by a factor of .
For the case , ; as the mappings and are injective, they are invertible. Specifically, consider the mappings
respectively. One can easily verify that these mappings are isomorphisms, and in fact, they are the inverses of the the contraction mappings defined above.
where we view s an element of the double dual. Thus,
3. Banachable Spaces of Sections of Vector Bundles over Compact Manifolds
For a compact manifold , the infinite dimensional Banach manifold of mappings to a manifold and the manifold of sections of the fiber bundle , are modeled by Banachable spaces of sections of vector bundles over , as will be described in the next section. In this section we describe the Banachable structure of such a space of differentiable vector bundle sections and make some related observations. Thus, we consider in this section a vector bundle , where is a smooth compact -dimensional manifold with corners and the typical fiber of is an -dimensional vector space. The space of -sections , , will be denoted by or by if no confusion may arise. A natural real vector space structure is induced on by setting and , .
3.1. Precompact atlases
Let , , be a finite collection of compact subsets whose interiors cover such that for each , is a subset of a domain of a chart on and
is some given vector bundle chart on . Such a covering may always be found by the compactness of (using coordinate balls as, for example, in [Lee02, p. 16] or [Pal68, p. 10]). We will refer to such a structure as a precompact atlas. The same terminology will apply for the case of a fiber bundle.
3.2. The -topology on
For a section of and each , let
be a local representative of .
Such a choice of a vector bundle atlas and subsets makes it possible to define, for a section ,
Palais [Pal68, in particular, Chapter 4] shows that is indeed a norm endowing with a Banach space structure. The dependence of this norm on the particular choice of atlas and sets , makes the resulting space Banachable, rather than a Banach space. Other choices will correspond to different norms. However, norms induced by different choices will induce equivalent topological vector space structures on .
3.3. The jet extension mapping
Next, one observes that the foregoing may be applied, in particular, to the vector space of continuous sections of the -jet bundle of . As a continuous section of is locally of the form
the analogous expression for the norm induced by a choice of a precompact vector bundle atlas is
Once, the topologies of and have been defined, one may consider the jet extension mapping
For a section , with local representatives , is represented by a section , the local representatives of which satisfy,
Clearly, the mapping is injective and linear. Furthermore, it follows that
(Note that since we take the supremum of all partial derivatives, we could replace the non-decreasing multi-index by a regular multi-index .) Thus, in view of Equation (3.4),
and we conclude that is a linear embedding of into . Evidently, is not surjective as a section of need not be compatible, i.e., it need not satisfy (3.8), for some section of . As a result of the above observations, has a continuous right inverse
3.4. The iterated jet extension mapping
In analogy, we now consider the iterated jet extension mapping
Specializing Equation (2.31) for the case of the non-holonomic -jet bundle
a section of is represented locally in the form
Thus, the induced norm on is given by
where the supremum is taken over all , , , with , and with .
Specializing (2.32) for the case of a vector bundle, it follows that if the section of , satisfies , its local representatives satisfy
It follows that in
(where the supremum is taken over all , , , with , and with ), it is sufficient to take simply all derivatives , for . Hence,
where the supremum is taken over all , , , and with It is therefore concluded that
In other words, one has a sequence of linear embeddings
where, is the natural inclusion (2.21) and defined as , for every continuous section of , is the inclusion of sections. These embeddings are not surjective. In particular, sections of need not have the symmetry properties that hold for sections of .
4. The Construction of Charts for the Manifold of Sections
In this section, we outline the construction of charts for the Banach manifold structure on the collection of sections as in [Pal68]. (See a detailed presentation of the subject in this volume [Mic20].)
Let be a -section of . Similarly to the construction of tubular neighborhoods, the basic idea is to identify points in a neighborhood of with vectors at which are tangent to the fibers. This is achieved by defining a second order differential equation, so that a neighboring point in the same fiber as is represented through the solution of the differential equations with the initial condition by . In other words, is the image of under the exponential mapping.
To ensure that the image of the exponential mapping is located on the same fiber, , the spray inducing the second order differential equation is a vector field
which is again tangent to the fiber in the sense that for
This condition, together with the analog of the standard condition for a second order differential equation, namely,
imply that is represented locally in the form
Finally, is a bundle spray so that
Bundle sprays can always be defined on compact manifolds using partitions of unity and the induced exponential mappings have the required properties.
The resulting structure makes it possible to identify an open neighborhood —a vector bundle neighborhood—of in with
(We note that a rescaling is needed if is to be identified with the whole of . Otherwise, only an open neighborhood of the zero section of will be used to parametrize .)
Once the identification of with is available, the collection of sections may be identified with , . Thus, a chart into a Banachable space is constructed, where is identified with the zero section.
The construction of charts on the manifold of sections, implies that curves in in a neighborhood of are represented locally by curves in the Banachable space . Thus, tangent vectors may be identified with elements of . We therefore make the identification
5. The -Topology on the Space of Sections of a Fiber Bundle
The topology on the space of sections of fiber bundles is conveniently described in terms of filters of neighborhoods (e.g., [Trè67]).
5.1. Local representatives of sections
We consider a fiber bundle , where is assumed to be a compact manifold with corners and the typical fiber is a manifold without a boundary. Let , , and , be a precompact (as in Section 3.1) fiber bundle trivialization on . That is, the interiors of cover , and . Let , , be an atlas on so that cover .
Consider a -section . For any , we can set
so that , and so, the local representatives of are
Thus, re-enumerating the subsets and we may assume that we have a precompact trivialization