Covariant Linearization of elasticity
In this paper we derive a general linearized theory for first-order continuum dynamics on manifolds with particular application to incompatible elasticity. We adopt a global approach viewing the equations of motion as a -form on the configuration space which is the Banach manifold of time-dependent embeddings of a body manifold into a space manifold . The linearization is done by differentiating the equations 1-form with respect to an affine connection which we construct and study extensively. We provide detailed coordinate computations for the linearized equations of a large class of problems in continuum dynamics on manifolds.
- 1 introduction
- 2 Geometric preliminaries
- 3 Elastodynamics
- 4 Linearization
The derivation of a linear theory from a nonlinear theorem is a central theme in mathematics, with innumerable applications in the various sciences. In the context of continuum mechanics, and notably in the theory of elasticity, the linear theories actually preceded the nonlinear theories (see Maugin [Mau16]). In fact, the equations of linear elasticity are commonly derived directly from the balance laws (assuming small deformations) (Gurtin [Gur73]), rather than as approximation to the nonlinear theory.
Linear theories of elasticity play several key roles in the analysis of nonlinear theories: (i) they serve as an intermediate step for proving the existence and the uniqueness of solutions for nonlinear theories, (ii) solutions of nonlinear problems can sometimes be obtained as limits of sequences of solutions of linearized problems, and (iii) they serve as a central tool in stability analysis [MH83].
The linearization of nonlinear continuum theories is nowadays a standard, however, its current scope does not fully cover the wealth of systems of current interest. To a large extent, existing linear theories address systems that are geometrically Euclidean. From a mathematical perspective, the state-space in continuum mechanics can be described as the embeddings of a body into a space, both viewed as differentiable manifolds.
For example, in a class of elastic systems dealing with residually-stressed bodies, the body manifold is viewed as a smooth manifold endowed with a Riemannian metric; the metric represents local equilibrium distances and angles between neighboring material elements. A configuration is an embedding of the body manifold into the ambient space, which is usually assumed Euclidean, although, non-Euclidean ambient spaces are of relevance even without recurring to relativistic theories [KOS17a]. When the geometries of the body of the ambient space are incompatible, there is no notion of stress-free reference configuration, hence the very notion of small deformations is not naturally defined as it is when both body and space are assumed Euclidean. Incompatible elasticity is just one example in which complex geometries interact in a non-trivial way with mechanical laws and material properties.
Physical theories in which non-Euclidean geometry plays a central role are best formulated in a covariant manner, i.e., in a way that does not rely on a particular system of coordinates. The classical reference for the covariant linearization of elasticity theories is the book of Marsden and Hughes [MH83]. Their starting point is a general notion of linearization, which we hereby define:
Let be a smooth (possibly infinite-dimensional) vector bundle endowed with a connection . Let be a -section of . The linearization of at is an affine mapping given by
Marsden and Hughes formulate the equations of nonlinear elasticity as a section of an infinite-dimensional vector bundle over the manifold of configurations and compute their linearization for a general class of constitutive relations. In their calculation, however, it is implicitly assumed that the ambient space is Euclidian, hence that the manifold of configurations is a vector space. This assumption is reflected in the linearization of the acceleration vector field and more subtly, in the linearization of the stress tensor. Accounting for a non-Euclidean ambient space is not just a matter of technicalities, which might be overcome, for example, by adopting a local coordinate system. A curved space affects the basic notion of inertia, and may destroy the symmetries that are at the heart of the classical derivation of continuum theories; this lack of symmetries reflects, for example, in the presence of so-called self-forces, which arise from interactions of the body with inhomogeneous geometric incompatibilities.
Other approaches to covariant linearization can be found in Yavari and Ozakin [AA08], where the authors linearize the energy and momentum balance laws, and in [GLM13] where linearization is computed around a normal state.
In this paper, we derive a general linearized theory for first-order continuum dynamics on manifolds, with a particular application to incompatible elasticity. We adopt a global approach, where the space of configurations is the Banach manifold of time-dependent embeddings of a body manifold into a space manifold . In this setting, the equations of motion are a 1-form on the configuration space . The linearization of those equation is in the sense of Definition 1.1, where the connection on the cotangent bundle of the configuration space is induced in a natural way from a given connection on the tangent bundle of the space manifold.
In the global approach to continuum dynamics, the equations of motion can be viewed as a natural generalization to Newton’s laws. Velocity is the time derivative of the configuration; the acceleration is the covariant time-derivative of the velocity field with respect to the connection ; the force field, which is a 1-form , is composed of external loadings and internal forces, where the latter are determined by the material properties through a constitutive relation. The equations of motion is obtained by pairing the acceleration to the force via a Riemannian metric on the configuration space .
Generally, elements of are represented by vector-valued measures. Hence, the linearized equations of motion may be as singular as measures and in particular, assume no local differential form. However, in the case where the loadings and the constituting relations satisfy certain regularity properties, the equations of motion as well as their linearization have local forms. which we derive as well.
The structure of the paper is as follows: In Section 2 we discuss the geometric structure of the space where is a compact smooth manifold and is a smooth manifold without boundary. We first introduce the Banach manifold structure of and its tangent bundle . Next, we construct a metric and connection on . To this end, we assume that a Riemannian metric is given on the target space and that a volume form is prescribed for the source manifold . The connection is induced by a connection for . We discuss the construction of in detail, and show that if is metric with respect to then so is with respect to .
In Section 3, we use the results of Section 2 to formulate Newton’s equations for continuum dynamics. We identify the configuration space of time-dependent -embeddings as an open subset of the manifold . The connection for gives a notion of covariant derivative that defines the acceleration, whereas the metric for pairs the acceleration with force. The force part of the equation is induced by a constitutive relation (which is assumed time-independent) and a loading; the whole equation is viewed as a section of the cotangent bundle of the configuration space.
In Section 4, we derive the linearized form of the nonlinear equations of motion derived in Section 3. We first obtain a general expression for general, time-independent constitutive relations. We then derive a local differential representation for the case of a smooth constitutive relation; the linearized equations are formulated both in a covariant manner and in local coordinates.
2 Geometric preliminaries
In this section we present the geometric foundations for continuum dynamics on manifolds. We start by briefly recalling the notion of jets, which are the covariant constructs for encoding functions along with their derivatives.
2.1 Jet bundles
Let and be smooth manifolds of dimensions and . A -jet from to is an equivalence class of triples , where , is a neighborhood of and . Two triples and are equivalent if
There exists local charts in and , with respect to which the local representatives of and have the same values and first derivatives at .
Equivalently, and are equivalent if
where is the tangent map of at . We denote the -jet of at by
The third condition in the definition of a 1-jet implies that and have the same values at and the same first derivatives at with respect to any local coordinate charts.
We denote by the set of all -jets from to . The set can be given the structure of a smooth manifold of dimension ; it is also a fiber bundle over with respect to the (source) projection map
Let be a smooth vector bundle over . Define
Then is a vector bundle over . The first jet extension,
is a linear immersion.
2.2 The manifold
Let be a smooth, compact, orientable -dimensional manifold, and let be a smooth orientable -dimensional manifold without boundary endowed with a Riemannian metric . Let be the space of mappings . Endow with the Whitney -topology [Mic80], a subbase of which consists of sets of the form
Loosely speaking, the Whitney -topology is the topology of uniform convergence of the function and its first derivative.
The space is not a vector space, since is not a linear space. However, can be given a structure of an infinite-dimensional Banach manifold: a topological space locally homeomorphic to a Banach space and equipped with a smooth structure (see Lang [Lan99]).
Given a mapping , a coordinate chart for at is constructed as follows: Let be the Levi-Civita connection of and let be the corresponding exponential map, where is a neighborhood of the zero section of , such that
is an embedding (i.e., a diffeomorphism onto its image). Let
be the embedding induced by the pullback with , and denote its image by . Then, the canonical chart at
is given by
It’s inverse is given by
The differentiable structure obtained by the atlas
Since is open, is open. Since is locally identified with , it follows that the tangent space is isomorphic to the Banachable space of vector fields along ,
The Banach space structure for may be constructed as follows: Let be a Finsler structure on , that is, for every , is a norm and varies smoothly between the fibers of . Since is compact, a Finsler structure exists, and moreover, any two Finsler structures on are equivalent. We define a complete norm on by
One may verify that the topology induced by the norm on coincides with its Whitney topology. Thus, the canonical chart is indeed a homeomorphisms onto its image.
The tangent bundle may be identified with the bundle
Moreover, for every the mapping
is a trivialisation for along the canonical chart corresponding to the trivialisation for under the bundle equivalence. For details see Eliasson [Eli67]. Note that for
where is the unique Jacobi field along along the geodesic satisfying and .
2.3 Connection and metric for
Following Eliasson [Eli67], we construct a connection for . Let be a (possibly infinite dimensional) fiber bundle over a smooth manifold and let be the vertical bundle defined by . An Ehresmann connection is a splitting of the short exact sequence
satisfying where is the inclusion. is often referred to as the connection form of the Ehresmann connection. The horizontal bundle is then identified with . In case that is a vector bundle we have a canonical identification . Thus, induces a unique mapping which we call a connection map for .
A linear connection should also satisfy, the following condition: for every denote by scalar multiplication by , then for every ,
Suppose that and are modelled over the Banach spaces and respectively. Then has the local form
where is linear in and . The condition implies that is of the form and the linearity condition implies that is linear in . Thus, a linear connection map has the local form
where is a bilinear transformation called the local connector of at .
In the particular case where is finite-dimensional and , the local connector is given by the Christoffel symbols,
Given a connection map for , one can define a covariant derivative on in the following way: For a section , set its covariant derivative as . That is, for and
If a section is represented by , that is, locally , and has a local representation , then a simple computation gives that the coordinate representation of is
where is the coordinate corresponding to .
Turning back to the problem at hand, let and let be the connection map corresponding to the Levi Civita connection on . One can then show (see [Eli67] for details) that induces a connection map
defined by composition,
Denote the corresponding connection by . By definition, for , and ,
Note that on the right-hand side, and , hence, we obtain indeed a map , i.e., an element of . The exponential map for with respect to is given by composition with , thus, the canonical coordinate charts are normal coordinates in the following sense: for every , is a -geodesic [Eli67]. In particular, the local connector of in the canonical coordinate chart vanishes at the zero section (corresponding to ).
We next turn to construct a Riemannian metric for . Assume that a mass form, which is a positive -form on is given. Using the isomorphism
define a metric for by
The mass density of is incorporated in the mass form . Locally,
where is a mass density function. In cases where is endowed with a Reimannian metric , it is often natural to take for mass form the Riemannian volume form , corresponding to the mass destiny .
As always, the metric induces an isometric immersion given by
However, since the manifold is not a Hilbert manifold, is not an isomorphism. For this reason, is often called a weak Riemannian structure (as opposed to a strong Riemannian structure).
2.4 Metricity of the connection
We next show that the connection and the metric for are compatible, namely, for ,
The metricity of the connection will be used in several instances in the mechanical context.
Let , and sufficiently small. Then, for every ,
where is the Jacobi field as defined in Section 2.2.
Define by and . We need to prove that . is a Jacobi field along the geodesic . Since and satisfies the Jacobi equation, we get that
In other words, is also a Jacobi field along . Moerover, and
The result follows from the existence and uniqueness of solutions to ordinary differential equations. ∎
The following lemma is a standard result in the theory of Jacobi fields (see e.g. [DOC92]).
Let , and be normal coordinates centered at , and let () be a radial geodesic emanating from . Then for any given locally by , the Jacobi field along with initial conditions , is given locally by
The connection is metric with respect to . In other words, for every
Let . It suffices to show that vanishes at some coordinate chart at . Let
be the canonical chart around and let
be the corresponding trivialization of along given by (2.1) and (2.2). Since is a normal coordinate chart, the Christoffel symbols (i.e., the local connector) of vanish at . Therefore, it suffices to prove that the derivative of the local representative of vanishes at the zero section (corresponding to ).
The local representative of ,
is given by
where and . More explicitly, using (2.3),
Note that the vector field evaluated at is given by .
Now, Let (so that ), and as before. Then
where in the passage to the third line we interchange integration over and differentiation with respect to time, and the last equality follows from lemma 2.1.
It suffices to show the the integrand vanishes at every . Let and we need to prove that
Since is metric with respect to ,
Let be normal coordinates centred at , and let . Then by lemma 2.2, is given locally by hence, is a constant vector field along and
which completes the proof.
The proof of theorem 2.1 shows in fact, that is metric with respect to whenever is metric with respect to . Note that metricity does not depend on the choice of mass form for .
3.1 The manifold of configurations
A body manifold is a smooth compact and orientable -dimensional manifold. A space manifold is a smooth orientable -dimensional manifold without boundary.
We assume that is equipped with a Riemannian metric and that is equipped with a mass form . The canonical charts for are constructed as in Section 2 using the exponential map induced by the Levi-Civita connection of .
½ the space of -embeddings of in . Let be a closed time interval. The configuration space,
is the space of -paths of embeddings of in .
we may view as an open subset of . therefore inherits the Banach manifold structure of .
Note that there is a natural inclusion , given by
We refer to as the space of stationary configurations.
The tangent bundle is called the bundle of virtual displacements, or generalised velocities. For , an element is called a virtual displacement at . As in the general case, we have the isomorphisms,
where the above inclusion is open; in other words, we view as an open submanifold of .
Denote the restriction of the connection map (see (2.4)) to by , that is,
Denote the corresponding connection by , namely,
The metric for is given by
By Theorem 2.1, is metrically consistent with .
Throughout this paper, points in and are denoted by and respectively. The indices of coordinates in will be denoted by Greek letters, whereas indices of coordinates in will be denoted by Roman letters. A point is represented by .
3.2 Forces and stresses
Let . A force at is an element . The action of a force on a virtual displacement is called a virtual power.
For simplicity, we will focus our attention on forces that are independent of time derivatives; that is, forces of the form
where is a smooth family of elements , and . With a slight abuse of terminology, we refer to elements of as forces as well.
We therefore turn to present the structure of , the space of forces over stationary configurations. First, note that unlike in finite dimensions, the tangent and cotangent bundles and are not isomorphic. In particular, given a stationary configuration , the dual space depends on the topology of . Since the topology of takes into account first derivatives, so do the elements of .
More formally, let , and consider the first jet extension
which is a continuous linear embedding. By the Hahn-Banach theorem, its dual map,
is onto. We conclude that to every force at corresponds a (non-unique) , satisfying
We call a stress at . We say that a stress at represents the force if the relation (3.4) holds. Note however, that for a given force , there may be more than one stress representing it. This reflects the well-known static indeterminacy of continuum mechanics.
In fact, stresses may also be viewed as cotangent vectors of some other manifold; Let be the manifold of -sections . Then for every one has a canonical isomorphism
For more details see [KOS17a].
In general, stresses and forces, which are continuous linear functionals on differentiable sections, may be singular. Locally, and in particular, if can be covered by a single chart, every stress is represented by a collection of measures on ,
by the formula
If the measures are absolutely continuous, we may write
where and . This suggests the following definition (see [Seg86]):
Let . A variational stress density at is a smooth -form valued in the vector bundle