#### Abstract

The thermomechanical behavior of a material is expressed mathematically by means of one or more constitutive equations representing the response of the body to the history of its deformation and temperature. These settings induce a set of connections which can express local properties. We replace two of them by a second order connection and prove that the holonomity of this connection classifies our materials.

#### 1. Introduction

The use of differential geometry in material science is based on 1-jet calculus. This technique is described in, for example, [1, 2]. A material body endowed with a constitutional equation induces naturally a linear connection, and several important physical properties of the material are described by means of its geodesics. The cited books handle one constitutional equation and thus one appropriate linear connection. In case that a material is endowed with more than one constitution equation, that is, by more than one connection, the topic of higher order connections appears. Note that the topic of higher order connections is widely studied; see, for example, [3–5]. Such approach is not established so far in the material science, and this paper thus formulates introductory principles and problems of the theory of materials endowed with more than one constitution equation.

#### 2. Geometric Motivation of Higher Order Connections

To show the compatibility with the geometric concept of a connection, let us now recall its generalization to higher order connections; see [6] for general concepts. The following section is based on [7].

*Definition 1. *A connection on bundle is defined by the structure on a manifold where is *vertical distribution* tangent to the fibers and is *horizontal distribution* complementary to the distribution . The transport of the fibers along the path is realized by the horizontal lifts given by the distribution on the surface . If the bundle is a vector one and the transport of fibers along an arbitrary path is linear, then the connection is called linear.

We will assume that the base manifold is of dimension and the fibers are of dimension . Then

On the neighborhood , let us consider local base and fiber coordinates: Base coordinates are determined by the projection and the coordinates on a neighborhood .

*Definition 2. *On a neighborhood we define a local adapted basis of the structure :

The horizontal distribution is the linear span of the vector fields and the annihilator of the forms ,

*Definition 3. *A classical affine connection on manifold is seen as a linear connection on the bundle . On the tangent bundle one can define the structure . The indices in the formulas are denoted by Latin letters all of them ranging from 1 to . The functions , , and are of the form in the sign is changed to comply with the classical theory:

*Definition 4. *Higher order connections are defined as follows: on tangent bundle the structure is defined where , on the structure is defined where , , and so forth.

#### 3. Jet Prolongation of a Fibered Manifold and Higher Order Connections

To compute with second order connections in an efficient way we have to go deeper in the theory. Structural approach introduced by C. Ehresmann and developed in, for example, [3] reads that th order holonomic prolongation of is a space of -jets of local sections and nonholonomic prolongation of is defined by the following iteration: (1); that is, is a space of 1-jets of sections over the target space ;(2). Clearly, we have an inclusion given by . Further, th semiholonomic prolongation is defined by the following induction. First, by we denote the projection and by the projection , . If we set and assume we have such that the restriction of the projection maps into , we can construct and define Obviously, , and are bundle functors on the category of fibered manifolds with -dimensional bases and -dimensional fibres and locally invertible fiber-preserving mappings.

Alternatively, one can define the th order semiholonomic prolongation by means of natural target projections of nonholonomic jets; see [4]. For let us denote by the target surjection with being the identity on . We note that the restriction of these projections to the subspace of semiholonomic jet prolongations will be denoted by the same symbol. By applying the functor we have also the surjections , and, consequently, the element is semiholonomic if and only if In [4], the proof of this property can be found and the author finds it useful when handling semiholonomic connections and their prolongations.

Finally, the following functorial definition of semiholonomic prolongation of a fibered manifold can be found. Assume that the functor comes equipped with the canonical transformation given by the restriction of jet target projections. Then there are two canonical transformations , and one can define as the equalizer of these two transformations. Then this is equivalent to the definition

To define a higher order connection we start with the definition of general connection; see [3].

*Definition 5. *A general connection on the fibered manifold is a section of the first jet prolongation .

By the substitution of the target space by ,, and , respectively, one obtains definition of th order holonomic, semiholonomic, and nonholonomic connections; that is, a higher order connection is a section of the appropriate jet prolongation of a fibered manifold.

Let us recall that the semiholonomity condition on a higher order connection defined in the geometric way is now transformed into the equality of all projections from Definition 4.

Previous approach to connections is suitable for conceptual considerations and operations with connections, such as prolongations of connections, natural operators, and some classifications. For us the following theorem is quite useful; see [8] for the proof.

Theorem 6. * Second order nonholonomic connections on are in bijection with triples , where are first order connections on and is a tensor field. Connection is semiholonomic if and only if . Connection is holonomic if and only if and .*

Now, one can define the following relation *~* on the space of second order nonholonomic connections.

*Definition 7. *Let the triples represent two second order connections in the sense of Theorem 6. They belong to the same equivalence class of the relation *~* if and only if and .

*Remark 8. *It is easy to see that *~* is an equivalence relation and let us denote as the a class of this relation. Finally, the class consists of semiholonomic connections if and only if for any and holonomic if in addition.

#### 4. Ehresmann Prolongation

Given two higher order connections and , the product of and is the th order connection defined by

Concerning the holonomity, according to [4] if both and are of the first order, then is semiholonomic if and only if and is holonomic if and only if is curvature-free in addition, which corresponds to Theorem 6.

As an example we show the coordinate expression of an arbitrary nonholonomic second order connection and of the product of two first order connections. The coordinate form of is where , , and are arbitrary smooth functions. Further, if the coordinate expressions of two first order connections are then the second order connection has equations For order three see [5]. If we apply the multiplication on just one connection , the second order connection is called the Ehresmann prolongation of connection . By iteration we obtain a connection of an arbitrary order.

In the following proposition we show that concerning order 2 only the choice of Ehresmann prolongation makes sense. We use the notation of [3], where the map is obtained from the natural exchange map as a restriction to the subbundle . Note that while depends on the linear connection on , its restriction is independent of any auxiliary connections. We remark that originally the map was introduced by M. Modugno. We also recall that J. Pradines introduced a natural map with the same coordinate expression.

Now we are ready to recall the following assertion; see [9] for the proof.

Proposition 9. *All natural operators transforming first order connection into second order semiholonomic connection form a one-parameter family:
*

To meet the classical theory mentioned in Section 2 let us note that the corresponding operation is the following; see also [10]. If we apply the tangent functor two times on a projection and a section we obtain respectively. The mappings , and are defined by the sections of fibered manifolds , and .

Let us consider local coordinates on the following manifolds in the form Let us also consider a function defined on a manifold , whose local coordinate form is derived by means of differentials to fit the coordinates on : Furthermore, . We use this notation in the following formulae.

If the section is defined by local functions , then the sections and are defined by its differentials , , and :

The case when the coefficients and in (18) are arbitrary functions corresponds to a nonholonomic connection on the fibered manifold .

The case when , where are arbitrary functions corresponds to a semiholonomic connection on the fibered manifold .

The case when corresponds to a holonomic connection on the fibered manifold .

The functions and define nonholonomic, semiholonomic, or holonomic Ehresmann prolongation of a connection, respectively.

#### 5. Material Connection

Following the books [1, 2], the material body is a trivial manifold without boundary. A coordinate chart is identified as a reference configuration, a configuration of a material body is an embedding Choosing a frame in , we can identify with . Now, one can associate with any given configuration the deformation defined as the composition: In coordinates,

For a simple hyperelastic body, the constitutive equation is of the form: . Two points are materially isomorphic if there exists a non-singular linear map , between their tangent spaces: such that identically for all deformation gradients . A body is materially uniform if, and only if, there exists a material isomorphism from a fixed point to each point .

We shall call the point an archetypal material point, and the material isomorphisms from the archetypal material point to the body points will be referred to as implants. A collection of such implants is a uniformity field.

A material archetype will be defined as a frame at . We will say that two vectors at two different points and of an open set are materially parallel with respect to the given uniformity field, if they have the same components in the respective local bases of the uniformity field.

A material symmetry at a point is material automorphism. A material symmetry at a point can be seen as a transformation such that The collection of all material symmetries in constitutes group called material symmetry group.

In coordinates, let be the natural basis of . By means of the uniformity maps this basis induces a smooth field of bases in , which we will denote by . We now adopt a coordinate system in , which we call , with natural basis ; then The vector field can be expressed in terms of components in either basis, namely: Defining the Christoffel symbols of the local material parallelism as we can write material covariant derivative of the field wit respect to the given material parallelism

If the symmetry group is trivial identity group, the material implants are unique. The local material connection is unique if the symmetry group is discrete (i.e., consisting of a finite number of elements).

Recall, that the body is locally homogeneous if and only if there exists local material connection where Christoffel symbols are symmetric, for each point.

To apply the multiplication of connections on the material connections, we have to modify (12) for linear connections. The rest would be done by substitution of the previous characteristics in the equations. If two linear connections and on the same base manifold are by coordinate formula (11), then is given by (12). Should the connections and be linear, the result would be obtained by substitution in (11), where and are functions of the base manifold coordinates . The equations of would therefore look like

Theorem 10. *Let be a material body and let be a class of second order connections. The constitutive equations are in the same projective class if and only if is semiholonomic. *

*Proof. *The class of nonholonomic connections was introduced in Definition 7. If the element belongs to , then from Theorem 6 the semiholonomity is equivalent to the property . In particular, two constitutive equations determine two projectively equivalent connections of the first order.

*Remark 11. *The projective class of connections shares the same geodesics. In particular, if we describe “least energy deformation” of the material body based on two constitutive equations which lead to second order semiholonomic connection, then it is based on geodesics of one material connection of the first order.

In fact, in our setting there is no extension of our result to connections of higher order than two (for explanation see [11]). This is the reason why the material equipped with two constitutive equations plays an interesting role in the theory of material bodies. Let us finally remark that the reformulation of the whole theory to the concept of infinitesimal connections on Lie groupoids can help; see [4].

#### 6. Conclusions

We showed that if we represent the material properties by means of a second order connection, then its holonomity corresponds to the type of the material. Our ideas were motivated by handling materials with two constitution equations and it occurred that for more than two constitution equations a change of mathematical approach is needed.

#### Acknowledgments

The first author was supported by the Grant GA ČR, Grant no. 201/09/0981, and the second author by Grant no. FSI-S-11-3.