# Semiholonomic Second Order Connections Associated with Material Bodies

Authors: Jaroslav Hrdina; Petr Vašík
Journal: Journal of Applied Mathematics
 (2013)

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## 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.


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## Body

## 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 bundleπ:M1→M is defined by the structure Δh⊕Δv on a manifold M1 where Δv=kerTπ is vertical distribution tangent to the fibers and Δh is horizontal distribution complementary to the distribution Δv. The transport of the fibers along the path γ⊂M is realized by the horizontal lifts given by the distribution Δh 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 manifoldM is of dimension n and the fibers are of dimension r. Then
(1)dimΔh=n,dimΔv=r.On the neighborhoodU⊂M1, let us consider local base and fiber coordinates:
(2)(ui,uα),i=1,2,…,n;α=n+1,…,n+r.Base coordinates (ui) are determined by the projection π and the coordinates (u-i) on a neighborhood U-=π(U),ui=u-i∘π.Definition 2.
On a neighborhoodU⊂M1 we define a local (adapted) basis of the structure Δh⊕Δv:
(3)(XiXα)=(∂∂uj∂∂uβ)·(δij0Γiβδαβ),(ωiωα)=(δji0-Γjαδβα)·(dujduβ).The horizontal distributionΔhis the linear span of the vector fields (Xi) and the annihilator of the forms (ωα),
(4)Xi=∂i+Γiβ∂β,ωα=duα-Γiαdui.Definition 3.
A classical affine connection on manifoldM is seen as a linear connection on the bundle π1:TM→M. On the tangent bundle TM→M one can define the structure Δh⊕Δv. The indices in the formulas are denoted by Latin letters all of them ranging from 1 to n. The functions Γiα, Xi, and ωα are of the form (in Γiα the sign is changed to comply with the classical theory):
(5)Γiα⇝-Γjkiu1k,Xi=∂i+Γiα∂α⇝Xi=∂i-Γijku1i∂k1,ωα=duα-Γiαdui⇝U12i=u12i+Γjkiu1ku2j.Definition 4.
Higher order connections are defined as follows: on tangent bundleTM the structure Δ⊕Δ1 is defined where kerTρ1=Δ1, on T(TM) the structure Δ⊕Δ1⊕Δ2⊕Δ12 is defined where kerTρs=Δs⊕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 JrY of Y is a space of r-jets of local sections M→Y and nonholonomic prolongation J~rY of Y is defined by the following iteration: (1)
J~1Y=J1Y;   that is,  J~1Y is a space of 1-jets of sections M→Y over the target space Y;(2)
J~rY=J1(J~r-1Y→M). Clearly, we have an inclusion JrY⊂J~rY given by jxrγ↦jx1(jr-1γ). Further, rth semiholonomic prolongation J¯rY⊂J~rY is defined by the following induction. First, by β1=βY we denote the projection J1Y→Y and by βr=βJ~r-1Y the projection J~rY=J1J~r-1Y→J~r-1Y, r=2,3,…. If we set J¯1Y=J1Y and assume we have J¯r-1Y⊂J~r-1Y such that the restriction of the projection βr-1:J~r-1Y→J~r-2Y maps J¯r-1Y into J¯r-2Y, we can construct J1βr-1:J1J¯r-1Y→J1J¯r-2Y and define
(6)J¯rY={A∈J1J¯r-1Y;βr(A)=J1βr-1(A)∈J¯r-1Y}.
Obviously, Jr,J¯r, and J~r are bundle functors on the category ℱℳm,n of fibered manifolds with m-dimensional bases and n-dimensional fibres and locally invertible fiber-preserving mappings.Alternatively, one can define therth order semiholonomic prolongation J¯rY by means of natural target projections of nonholonomic jets; see [4]. For r≥q≥0 let us denote by πqr the target surjection πqr:J~rY→J~qY with πrr being the identity on J~rY. 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 Jk we have also the surjections Jkπq-kr-k:J~rY→J~qY, and, consequently, the element X∈J~rY is semiholonomic if and only if
(7)(Jkπq-kr-k)(X)=πqr(X)foranyintegers1≤k≤q≤r.
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 functorJ¯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
(8)J¯rY=J¯2(J¯r-2Y)∩J1(J¯r-1Y).To define a higher order connection we start with the definition of general connection; see [3].Definition 5.
A general connection on the fibered manifoldY→M is a section Γ:Y→J1Y of the first jet prolongation J1Y→Y.By the substitution of the target space byJrY,J¯rY, and J~rY, 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 projectionsTρs 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→J~2Y 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),(Γ-1,Γ-2,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=Γ-1 and Γ2=Γ-2.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 connectionsΓ:Y→J~rY and Γ¯:Y→J~sY, the product of Γ and Γ¯ is the (r+s)th order connection Γ*Γ¯:Y→J~r+sY defined by
(9)Γ*Γ¯=J~sΓ∘Γ¯.Concerning the holonomity, according to [4] if both Γ and Γ¯ are of the first order, then Γ*Γ¯:Y→J~2Y 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 ofΔ:Y→J~2Y is
(10)yip=Fip(x,y),y0ip=Gip(x,y),yijp=Hijp(x,y),
where F, G, and H are arbitrary smooth functions. Further, if the coordinate expressions of two first order connections Γ,Γ¯:Y→J1Y are
(11)Γ:yip=Fip(x,y),Γ¯:yip=Gip(x,y),
then the second order connection Γ*Γ¯:Y→J~2Y has equations
(12)yip=Fip,y0ip=Gip,yijp=∂Fip∂xj+∂Fip∂yqGjq.
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:J¯2Y→J¯2Y is obtained from the natural exchange map eΛ:J1J1Y→J1J1Y as a restriction to the subbundle J¯2Y⊂J1J1Y. Note that while eΛ depends on the linear connection Λ on M, its restriction e is independent of any auxiliary connections. We remark that originally the map eΛ was introduced by M. Modugno. We also recall that J. Pradines introduced a natural map J¯2Y→J¯2Y 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 connectionΓ:Y→J1Y into second order semiholonomic connection Y→J¯2Y form a one-parameter family:
(13)Γ↦k·(Γ*Γ)+(1-k)·e(Γ*Γ),k∈ℝ.To meet the classical theory mentioned in Section2 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
(14)Tπ:TE→TM,T2π:T2E→T2M,(15)Tσ:TM→TE,T2σ:T2M→T2E,
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(16)onM,TM,T2M:(xi),(xi,x1i),(xi,x1i,x2i,x12i),onE,TE,T2E:(yp),(yp,y1p),(yp,y1p,y2p,y12p).
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:
(17)f1≐fix1i,f2≐fix2i,f12≐fijx1ix2j+fix12i,wherefi=∂f∂xi,fij=∂2f∂xi∂xj.
Furthermore, f1=df∘ρ1,f2=df∘ρ2,f12=d2f. We use this notation in the following formulae.If the sectionσ is defined by local functions Γp, then the sections Tσ and T2σ are defined by its differentials Γ1p, Γ2p,  and  Γ12p:
(18)σ:xi⇝yp=Γp,Tσ:(xi,x1i)⇝(yp,y1p)=(Γp,Γ1p),T2σ:(xi,x1i,x2i,x12i)⇝(yp,y1p,y2p,y12p)=(Γp,Γ1p,Γ2p,Γ12p),whereΓ1p=Γipx1i,Γ2p=Γipx2i,Γ12p=Γijpx1ix2j+Γipx12i.The case when the coefficientsΓip and Γijp in (18) are arbitrary functions corresponds to a nonholonomic connection on the fibered manifold π.The case whenΓijp=∂Γip/∂xj, where  Γip are arbitrary functions corresponds to a semiholonomic connection on the fibered manifold π.The case whenΓ1p=dΓp∘ρ1,Γ2p=dΓp∘ρ2,Γ12p=d2Γp corresponds to a holonomic connection on the fibered manifold π.The functionsΓip and Γijp 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
(19)κ:ℬ→𝔼3.
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:
(20)χ:κ∘κ0-1.
In coordinates,
(21)xi=χi(X1,X2,X3),FIi=∂xi∂XI.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:
(22)P12:TX1ℬ→TX2ℬ,
such that
(23)ψ(FP12,X1)=ψ(F,X2)
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 pointX0 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 atX0. 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 pointX0∈ℬ is material automorphism. A material symmetry G at a point X0 can be seen as a transformation such that
(24)ψ(FG,X0)=ψ(F,X0).
The collection 𝒢0 of all material symmetries in X0 constitutes group called material symmetry group.In coordinates, letEα 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 pα(X). We now adopt a coordinate system in U, which we call XI, with natural basis ei; then
(25)pα=PαIeI.
The vector field w can be expressed in terms of components in either basis, namely:
(26)w=ωαpα=ωIeI.
Defining the Christoffel symbols of the local material parallelism as
(27)ΓIJK=PαK∂PI-α∂XJ,
we can write material covariant derivative of the field w wit respect to the given material parallelism
(28)∇IJK=∂ωK∂XJ+ΓIJKωI.If the symmetry group𝒢0 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
(29)Fip=Γiqpyq,Gip=Γ¯iqpyq
in (11), where Γiqp and Γ¯iqp are functions of the base manifold coordinates xi. The equations of Γ¯*Γ would therefore look like
(30)yip=Γiqpyq,y0ip=Γ¯iqpyq,yijp=∂Γ¯iqp∂xjyq+Γ¯iqpΓjrqyr+Γ¯iqpyjq.Theorem 10.
Letℬ 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.



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