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.

The use of differential geometry in material science is based on 1-jet calculus. This technique is described in, for example, [

To show the compatibility with the geometric concept of a connection, let us now recall its generalization to higher order connections; see [

A connection on bundle

We will assume that the base manifold

On the neighborhood

On a neighborhood

The horizontal distribution

A classical affine connection on manifold

Higher order connections are defined as follows: on tangent bundle

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, [

Alternatively, one can define the

Finally, the following functorial definition of semiholonomic prolongation of a fibered manifold can be found. Assume that the functor

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

A general connection on the fibered manifold

By the substitution of the target space by

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

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 [

Now, one can define the following relation

Let the triples

It is easy to see that

Given two higher order connections

Concerning the holonomity, according to [

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

In the following proposition we show that concerning order 2 only the choice of Ehresmann prolongation makes sense. We use the notation of [

Now we are ready to recall the following assertion; see [

All natural operators transforming first order connection

To meet the classical theory mentioned in Section

Let us consider local coordinates on the following manifolds in the form

If the section

The case when the coefficients

The case when

The case when

The functions

Following the books [

For a simple hyperelastic body, the constitutive equation is of the form:

We shall call the point

A material archetype will be defined as a frame at

A material symmetry at a point

In coordinates, let

If the symmetry group

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 (

Let

The class

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 [

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.

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.