Semiholonomic Second Order Connections Associated with Material Bodies

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 𝜋∶𝑀₁→𝑀 is defined by the structure Δ_ℎ⊕Δ_𝑣 on a manifold 𝑀₁ where Δ_𝑣=ker𝑇𝜋 is vertical distribution tangent to the fibers and Δ_ℎ is horizontal distribution complementary to the distribution Δ_𝑣. The transport of the fibers along the path 𝛾⊂𝑀 is realized by the horizontal lifts given by the distribution Δ_ℎ on the surface 𝜋⁻¹(𝛾). 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 𝑀 is of dimension 𝑛 and the fibers are of dimension 𝑟. Then dimΔ_ℎ=𝑛,dimΔ_𝑣=𝑟. (1)

On the neighborhood 𝑈⊂𝑀₁, let us consider local base and fiber coordinates: 𝑢^𝑖,𝑢^𝛼,𝑖=1,2,…,𝑛;𝛼=𝑛+1,…,𝑛+𝑟. (2)Base coordinates (𝑢^𝑖) are determined by the projection 𝜋 and the coordinates (𝑢^𝑖) on a neighborhood 𝑈=𝜋(𝑈),𝑢^𝑖=𝑢^𝑖∘𝜋.

Definition 2. On a neighborhood 𝑈⊂𝑀₁ we define a local (adapted) basis of the structure Δ_ℎ⊕Δ_𝑣: 𝑋_𝑖𝑋_𝛼=𝜕𝜕𝑢^𝑗𝜕𝜕𝑢^𝛽⋅𝛿^𝑗_𝑖0Γ^𝛽_𝑖𝛿^𝛽_𝛼,𝜔^𝑖𝜔^𝛼=𝛿^𝑖_𝑗0−Γ^𝛼_𝑗𝛿^𝛼_𝛽⋅𝑑𝑢^𝑗𝑑𝑢^𝛽. (3)

The horizontal distribution Δ_ℎis the linear span of the vector fields (𝑋_𝑖) and the annihilator of the forms (𝜔^𝛼), 𝑋_𝑖=𝜕_𝑖+Γ^𝛽_𝑖𝜕_𝛽,𝜔^𝛼=𝑑𝑢^𝛼−Γ^𝛼_𝑖𝑑𝑢^𝑖. (4)

Definition 3. A classical affine connection on manifold 𝑀 is seen as a linear connection on the bundle 𝜋₁∶𝑇𝑀→𝑀. On the tangent bundle 𝑇𝑀→𝑀 one can define the structure Δ_ℎ⊕Δ_𝑣. The indices in the formulas are denoted by Latin letters all of them ranging from 1 to 𝑛. The functions Γ^𝛼_𝑖, 𝑋_𝑖, and 𝜔^𝛼 are of the form (in Γ^𝛼_𝑖 the sign is changed to comply with the classical theory): Γ^𝛼_𝑖⇝−Γ^𝑖_𝑗𝑘𝑢^𝑘₁,𝑋_𝑖=𝜕_𝑖+Γ^𝛼_𝑖𝜕_𝛼⇝𝑋_𝑖=𝜕_𝑖−Γ^𝑘_𝑖𝑗𝑢^𝑖₁𝜕¹_𝑘,𝜔^𝛼=𝑑𝑢^𝛼−Γ^𝛼_𝑖𝑑𝑢^𝑖⇝𝑈^𝑖₁₂=𝑢^𝑖₁₂+Γ^𝑖_𝑗𝑘𝑢^𝑘₁𝑢^𝑗₂. (5)

Definition 4. Higher order connections are defined as follows: on tangent bundle 𝑇𝑀 the structure Δ⊕Δ₁ is defined where ker𝑇𝜌₁=Δ₁, on 𝑇(𝑇𝑀) the structure Δ⊕Δ₁⊕Δ₂⊕Δ₁₂ is defined where ker𝑇𝜌_𝑠=Δ_𝑠⊕Δ₁₂, 𝑠=1,2, 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 𝑟th order holonomic prolongation 𝐽^𝑟𝑌 of 𝑌 is a space of 𝑟-jets of local sections 𝑀→𝑌 and nonholonomic prolongation 𝐽^𝑟̃𝑌 of 𝑌 is defined by the following iteration: (1)𝐽¹̃𝑌=𝐽¹𝑌;   that is,  𝐽¹̃𝑌 is a space of 1-jets of sections 𝑀→𝑌 over the target space 𝑌;(2)𝐽^𝑟̃𝑌=𝐽¹(𝐽̃^𝑟−1𝑌→𝑀). Clearly, we have an inclusion 𝐽^𝑟𝑌⊂𝐽^𝑟̃𝑌 given by 𝑗^𝑟_𝑥𝛾↦𝑗¹_𝑥(𝑗^𝑟−1𝛾). Further, 𝑟th semiholonomic prolongation 𝐽^𝑟𝑌⊂𝐽^𝑟̃𝑌 is defined by the following induction. First, by 𝛽₁=𝛽_𝑌 we denote the projection 𝐽¹𝑌→𝑌 and by 𝛽_𝑟̃𝐽=𝛽^𝑟−1_𝑌 the projection 𝐽^𝑟̃𝑌=𝐽¹𝐽̃^𝑟−1𝑌→𝐽̃^𝑟−1𝑌, 𝑟=2,3,…. If we set 𝐽¹𝑌=𝐽¹𝑌 and assume we have 𝐽^𝑟−1𝑌⊂𝐽̃^𝑟−1𝑌 such that the restriction of the projection 𝛽_𝑟−1∶𝐽̃^𝑟−1𝑌→𝐽̃^𝑟−2𝑌 maps 𝐽^𝑟−1𝑌 into 𝐽^𝑟−2𝑌, we can construct 𝐽¹𝛽_𝑟−1∶𝐽¹𝐽^𝑟−1𝑌→𝐽¹𝐽^𝑟−2𝑌 and define 𝐽^𝑟𝑌=𝐴∈𝐽¹𝐽^𝑟−1𝑌;𝛽_𝑟(𝐴)=𝐽¹𝛽_𝑟−1(𝐴)∈𝐽^𝑟−1𝑌. (6) Obviously, 𝐽^𝑟,𝐽^𝑟, and 𝐽^𝑟̃ are bundle functors on the category ℱℳ_𝑚,𝑛 of fibered manifolds with 𝑚-dimensional bases and 𝑛-dimensional fibres and locally invertible fiber-preserving mappings.

Alternatively, one can define the 𝑟th order semiholonomic prolongation 𝐽^𝑟𝑌 by means of natural target projections of nonholonomic jets; see [4]. For 𝑟≥𝑞≥0 let us denote by 𝜋^𝑟_𝑞 the target surjection 𝜋^𝑟_𝑞∶𝐽^𝑟̃𝑌→𝐽^𝑞̃𝑌 with 𝜋^𝑟_𝑟 being the identity on 𝐽^𝑟̃𝑌. 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 𝐽^𝑘 we have also the surjections 𝐽^𝑘𝜋^𝑟−𝑘_𝑞−𝑘∶𝐽^𝑟̃𝑌→𝐽^𝑞̃𝑌, and, consequently, the element 𝑋∈𝐽^𝑟̃𝑌 is semiholonomic if and only if 𝐽^𝑘𝜋^𝑟−𝑘_𝑞−𝑘(𝑋)=𝜋^𝑟_𝑞(𝑋)foranyintegers1≤𝑘≤𝑞≤𝑟. (7) 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 𝐽^𝑟−1 comes equipped with the canonical transformation 𝐽^𝑟−1→𝐽^𝑟−2 given by the restriction of jet target projections. Then there are two canonical transformations 𝐽¹𝐽^𝑟−1→𝐽¹𝐽^𝑟−2, and one can define 𝐽^𝑟 as the equalizer of these two transformations. Then this is equivalent to the definition 𝐽^𝑟𝑌=𝐽²𝐽^𝑟−2𝑌∩𝐽¹𝐽^𝑟−1𝑌. (8)

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 𝑌→𝑀 is a section Γ∶𝑌→𝐽¹𝑌 of the first jet prolongation 𝐽¹𝑌→𝑌.

By the substitution of the target space by 𝐽^𝑟𝑌,𝐽^𝑟𝑌, and 𝐽^𝑟̃𝑌, respectively, one obtains definition of 𝑟th 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 𝑇𝜌_𝑠 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. (1) Second order nonholonomic connections 𝜃∶𝑌→𝐽²̃𝑌 on 𝑌→𝑀 are in bijection with triples (Γ₁,Γ₂,𝐺), where Γ₁,Γ₂∶𝑌→𝐽¹𝑌 are first order connections on 𝑌→𝑀 and 𝐺∶𝑌→𝑉𝑌⊗⊗²𝑇^∗𝑀 is a tensor field. (2) Connection 𝜃 is semiholonomic if and only if  Γ₁=Γ₂. (3) Connection 𝜃 is holonomic if and only if  Γ₁=Γ₂ and 𝐺∶𝑌→𝑉𝑌⊗𝑆²𝑇^∗𝑀.

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

Definition 7. Let the triples (Γ₁,Γ₂,𝐺),(Γ₁,Γ₂,𝐺) 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 Γ₁=Γ₁ and Γ₂=Γ₂.

Remark 8. It is easy to see that ~ is an equivalence relation and let us denote [𝜃]=[(Γ₁,Γ₂,𝐺)] as the a class of this relation. Finally, the class [𝜃] consists of semiholonomic connections if and only if Γ₁=Γ₂ for any (Γ₁,Γ₂,𝐺)∈[𝜃] and holonomic if 𝐺∶𝑌→𝑉𝑌⊗𝑆²𝑇^∗𝑀 in addition.

4. Ehresmann Prolongation

Given two higher order connections Γ∶𝑌→𝐽^𝑟̃𝑌 and Γ∶𝑌→𝐽^𝑠̃𝑌, the product of Γ and Γ is the (𝑟+𝑠)th order connection Γ∗Γ∶𝑌→𝐽̃^𝑟+𝑠𝑌 defined by Γ∗Γ=𝐽^𝑠̃Γ∘Γ. (9)

Concerning the holonomity, according to [4] if both Γ and Γ are of the first order, then Γ∗Γ∶𝑌→𝐽²̃𝑌 is semiholonomic if and only if Γ=Γ and Γ∗Γ is holonomic if and only if Γ 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 Δ∶𝑌→𝐽²̃𝑌 is 𝑦^𝑝_𝑖=𝐹^𝑝_𝑖𝑥,𝑦,𝑦^𝑝_0𝑖=𝐺^𝑝_𝑖𝑥,𝑦,𝑦^𝑝_𝑖𝑗=𝐻^𝑝_𝑖𝑗𝑥,𝑦, (10) where 𝐹, 𝐺, and 𝐻 are arbitrary smooth functions. Further, if the coordinate expressions of two first order connections Γ,Γ∶𝑌→𝐽¹𝑌 are Γ∶𝑦^𝑝_𝑖=𝐹^𝑝_𝑖𝑥,𝑦,Γ∶𝑦^𝑝_𝑖=𝐺^𝑝_𝑖𝑥,𝑦, (11) then the second order connection Γ∗Γ∶𝑌→𝐽²̃𝑌 has equations 𝑦^𝑝_𝑖=𝐹^𝑝_𝑖,𝑦^𝑝_0𝑖=𝐺^𝑝_𝑖,𝑦^𝑝_𝑖𝑗=𝜕𝐹^𝑝_𝑖𝜕𝑥^𝑗+𝜕𝐹^𝑝_𝑖𝜕𝑦^𝑞𝐺^𝑞_𝑗. (12) For order three see [5]. If we apply the multiplication on just one connection Γ, the second order connection Γ∗Γ is called the Ehresmann prolongation of connection Γ. 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 𝑒∶𝐽²𝑌→𝐽²𝑌 is obtained from the natural exchange map 𝑒_Λ∶𝐽¹𝐽¹𝑌→𝐽¹𝐽¹𝑌 as a restriction to the subbundle 𝐽²𝑌⊂𝐽¹𝐽¹𝑌. Note that while 𝑒_Λ depends on the linear connection Λ on 𝑀, its restriction 𝑒 is independent of any auxiliary connections. We remark that originally the map 𝑒_Λ was introduced by M. Modugno. We also recall that J. Pradines introduced a natural map 𝐽²𝑌→𝐽²𝑌 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 Γ∶𝑌→𝐽¹𝑌 into second order semiholonomic connection 𝑌→𝐽²𝑌 form a one-parameter family: Γ↦𝑘⋅(Γ∗Γ)+(1−𝑘)⋅𝑒(Γ∗Γ),𝑘∈ℝ. (13)

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 𝑇 two times on a projection 𝜋∶𝐸→𝑀 and a section 𝜎∶𝑀→𝐸 we obtain 𝑇𝜋∶𝑇𝐸→𝑇𝑀,𝑇²𝜋∶𝑇²𝐸→𝑇²𝑀,𝑇𝜎∶𝑇𝑀→𝑇𝐸,𝑇²𝜎∶𝑇²𝑀→𝑇²𝐸, (14)(15) respectively. The mappings 𝜎,𝑇𝜎, and 𝑇²𝜎 are defined by the sections of fibered manifolds 𝜋,𝑇𝜋, and 𝑇²𝜋.

Let us consider local coordinates on the following manifolds in the form on𝑀,𝑇𝑀,𝑇²𝑀∶𝑥^𝑖,𝑥^𝑖,𝑥^𝑖₁,𝑥^𝑖,𝑥^𝑖₁,𝑥^𝑖₂,𝑥^𝑖₁₂,on𝐸,𝑇𝐸,𝑇²𝐸∶𝑦^𝑝,𝑦^𝑝,𝑦^𝑝₁,𝑦^𝑝,𝑦^𝑝₁,𝑦^𝑝₂,𝑦^𝑝₁₂. (16) Let us also consider a function 𝑓 defined on a manifold 𝑀, whose local coordinate form is derived by means of differentials to fit the coordinates on 𝑇²𝑀: 𝑓₁≐𝑓_𝑖𝑥^𝑖₁,𝑓₂≐𝑓_𝑖𝑥^𝑖₂,𝑓₁₂≐𝑓_𝑖𝑗𝑥^𝑖₁𝑥^𝑗₂+𝑓_𝑖𝑥^𝑖₁₂,where𝑓_𝑖=𝜕𝑓𝜕𝑥^𝑖,𝑓_𝑖𝑗=𝜕²𝑓𝜕𝑥^𝑖𝜕𝑥^𝑗. (17) Furthermore, 𝑓₁=d𝑓∘𝜌₁,𝑓₂=d𝑓∘𝜌₂,𝑓₁₂=d²𝑓. We use this notation in the following formulae.

If the section 𝜎 is defined by local functions Γ^𝑝, then the sections 𝑇𝜎 and 𝑇²𝜎 are defined by its differentials Γ^𝑝₁, Γ^𝑝₂, and Γ^𝑝₁₂: 𝜎∶𝑥^𝑖⇝𝑦^𝑝=Γ^𝑝,𝑇𝜎∶𝑥^𝑖,𝑥^𝑖₁⇝𝑦^𝑝,𝑦^𝑝₁=Γ^𝑝,Γ^𝑝₁,𝑇²𝜎∶𝑥^𝑖,𝑥^𝑖₁,𝑥^𝑖₂,𝑥^𝑖₁₂⇝𝑦^𝑝,𝑦^𝑝₁,𝑦^𝑝₂,𝑦^𝑝₁₂=Γ^𝑝,Γ^𝑝₁,Γ^𝑝₂,Γ^𝑝₁₂,whereΓ^𝑝₁=Γ^𝑝_𝑖𝑥^𝑖₁,Γ^𝑝₂=Γ^𝑝_𝑖𝑥^𝑖₂,Γ^𝑝₁₂=Γ^𝑝_𝑖𝑗𝑥^𝑖₁𝑥^𝑗₂+Γ^𝑝_𝑖𝑥^𝑖₁₂. (18)

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

The case when Γ^𝑝_𝑖𝑗=𝜕Γ^𝑝_𝑖/𝜕𝑥^𝑗, where Γ^𝑝_𝑖 are arbitrary functions corresponds to a semiholonomic connection on the fibered manifold 𝜋.

The case when Γ^𝑝₁=dΓ^𝑝∘𝜌₁,Γ^𝑝₂=dΓ^𝑝∘𝜌₂,Γ^𝑝₁₂=d²Γ^𝑝 corresponds to a holonomic connection on the fibered manifold 𝜋.

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

5. Material Connection

Following the books [1, 2], the material body ℬ is a trivial manifold without boundary. A coordinate chart 𝜅₀∶ℬ→ℝ³ is identified as a reference configuration, a configuration of a material body ℬ is an embedding 𝜅∶ℬ→𝔼³. (19) Choosing a frame in 𝔼³, we can identify 𝔼³ with ℝ³. Now, one can associate with any given configuration 𝜅 the deformation 𝜒 defined as the composition: 𝜒∶𝜅∘𝜅₀⁻¹. (20) In coordinates, 𝑥^𝑖=𝜒^𝑖𝑋¹,𝑋²,𝑋³,𝐹^𝑖_𝐼=𝜕𝑥^𝑖𝜕𝑋^𝐼. (21)

For a simple hyperelastic body, the constitutive equation is of the form: 𝜓=𝜓(𝐹,𝑋). Two points 𝑋₁,𝑋₂∈ℬ are materially isomorphic if there exists a non-singular linear map 𝑃₁₂, between their tangent spaces: 𝑃₁₂∶𝑇_𝑋1ℬ→𝑇_𝑋2ℬ, (22) such that 𝜓𝐹𝑃₁₂,𝑋₁=𝜓𝐹,𝑋₂ (23) identically for all deformation gradients 𝐹. A body ℬ is materially uniform if, and only if, there exists a material isomorphism 𝑃(𝑋) from a fixed point 𝑋₀∈ℬ to each point 𝑋∈ℬ.

We shall call the point 𝑋₀ an archetypal material point, and the material isomorphisms 𝑃(𝑋) 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 𝑋₀. We will say that two vectors at two different points 𝑋₁ and 𝑋₂ of an open set 𝑈⊂ℬ 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 𝑋₀∈ℬ is material automorphism. A material symmetry 𝐺 at a point 𝑋₀ can be seen as a transformation such that 𝜓𝐹𝐺,𝑋₀=𝜓𝐹,𝑋₀. (24) The collection 𝒢₀ of all material symmetries in 𝑋₀ constitutes group called material symmetry group.

In coordinates, let 𝐸_𝛼 be the natural basis of ℝ³. By means of the uniformity maps 𝑃(𝑋) this basis induces a smooth field of bases in 𝑈, which we will denote by 𝑝_𝛼(𝑋). We now adopt a coordinate system in 𝑈, which we call 𝑋^𝐼, with natural basis 𝑒_𝑖; then 𝑝_𝛼=𝑃^𝐼_𝛼𝑒_𝐼. (25) The vector field 𝑤 can be expressed in terms of components in either basis, namely: 𝑤=𝜔^𝛼𝑝_𝛼=𝜔^𝐼𝑒_𝐼. (26) Defining the Christoffel symbols of the local material parallelism as Γ^𝐾_𝐼𝐽=𝑃^𝐾_𝛼𝜕𝑃_𝐼^−𝛼𝜕𝑋^𝐽, (27) we can write material covariant derivative of the field 𝑤 wit respect to the given material parallelism ∇^𝐾_𝐼𝐽=𝜕𝜔^𝐾𝜕𝑋^𝐽+Γ^𝐾_𝐼𝐽𝜔^𝐼. (28)

If the symmetry group 𝒢₀ is trivial identity group, the material implants are unique. The local material connection is unique if the symmetry group 𝒢₀ 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 Γ and Γ on the same base manifold 𝑀 are by coordinate formula (11), then Γ∗Γ is given by (12). Should the connections Γ and Γ be linear, the result would be obtained by substitution 𝐹^𝑝_𝑖=Γ^𝑝_𝑖𝑞𝑦^𝑞,𝐺^𝑝_𝑖=Γ^𝑝_𝑖𝑞𝑦^𝑞 (29) in (11), where Γ^𝑝_𝑖𝑞 and Γ^𝑝_𝑖𝑞 are functions of the base manifold coordinates 𝑥_𝑖. The equations of Γ∗Γ would therefore look like 𝑦^𝑝_𝑖=Γ^𝑝_𝑖𝑞𝑦^𝑞,𝑦^𝑝_0𝑖=Γ^𝑝_𝑖𝑞𝑦^𝑞,𝑦^𝑝_𝑖𝑗=𝜕Γ^𝑝_𝑖𝑞𝜕𝑥^𝑗𝑦^𝑞+Γ^𝑝_𝑖𝑞Γ^𝑞_𝑗𝑟𝑦^𝑟+Γ^𝑝_𝑖𝑞𝑦^𝑞_𝑗. (30)

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

Proof. The class [𝜃] of nonholonomic connections was introduced in Definition 7. If the element (Γ₁,Γ₂,𝐺) belongs to [𝜃], then from Theorem  6 the semiholonomity is equivalent to the property Γ₁=Γ₂. 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.