JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 840936 10.1155/2013/840936 840936 Research Article Semiholonomic Second Order Connections Associated with Material Bodies Hrdina Jaroslav Vašík Petr Wang Baolin Institute of Mathematics, Faculty of Mechanical Engineering, Brno University of Technology Technická 2 616 69 Brno Czech Republic vutbr.cz 2013 5 2 2013 2013 24 08 2012 12 11 2012 13 11 2012 2013 Copyright © 2013 Jaroslav Hrdina and Petr Vašík. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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, . 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  for general concepts. The following section is based on .

Definition 1.

A connection on bundle π:M1M is defined by the structure ΔhΔv 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 γM 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 manifold M is of dimension n and the fibers are of dimension r. Then (1)dimΔh=n,dimΔv=r.

On the neighborhood UM1, 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-iπ.

Definition 2.

On a neighborhood UM1 we define a local (adapted) basis of the structure ΔhΔv: (3)(Xi  Xα)=(ujuβ  )·(δij0Γiβδαβ),(ωiωα)=(δji0-Γjαδβα)·(dujduβ).

The horizontal distribution Δh  is the linear span of the vector fields (Xi) and the annihilator of the forms (ωα), (4)Xi=i+Γiββ,ωα=duα-Γiαdui.

Definition 3.

A classical affine connection on manifold M is seen as a linear connection on the bundle π1:TMM. On the tangent bundle TMM one can define the structure ΔhΔv. 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+ΓiααXi=i-Γijku1ik1,ωα=duα-ΓiαduiU12i=u12i+Γjkiu1ku2j.

Definition 4.

Higher order connections are defined as follows: on tangent bundle TM the structure ΔΔ1 is defined where kerTρ1=Δ1, on T(TM) the structure ΔΔ1Δ2Δ12 is defined where kerTρs=ΔsΔ12, 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,  reads that rth order holonomic prolongation JrY of Y is a space of r-jets of local sections MY and nonholonomic prolongation J~rY of Y is defined by the following iteration:

J~1Y=J1Y;   that is,    J~1Y is a space of 1-jets of sections MY over the target space Y;

J~rY=J1(J~r-1YM).

Clearly, we have an inclusion JrYJ~rY given by jxrγjx1(jr-1γ). Further, rth semiholonomic prolongation J¯rYJ~rY is defined by the following induction. First, by β1=βY we denote the projection J1YY and by βr=βJ~r-1Y the projection J~rY=J1J~r-1YJ~r-1Y, r=2,3,. If we set J¯1Y=J1Y and assume we have J¯r-1YJ~r-1Y such that the restriction of the projection βr-1:J~r-1YJ~r-2Y maps J¯r-1Y into J¯r-2Y, we can construct J1βr-1:J1J¯r-1YJ1J¯r-2Y and define (6)J¯rY={AJ1J¯r-1Y;  βr(A)=J1βr-1(A)J¯r-1Y}. Obviously, Jr,J¯r, and J~r are bundle functors on the category 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 . For rq0 let us denote by πqr the target surjection πqr:J~rYJ~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~rYJ~qY, and, consequently, the element XJ~rY is semiholonomic if and only if (7)(Jkπq-kr-k)(X)=πqr(X)foranyintegers  1kqr. In , 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-1J¯r-2 given by the restriction of jet target projections. Then there are two canonical transformations J1J¯r-1J1J¯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 .

Definition 5.

A general connection on the fibered manifold YM is a section Γ:YJ1Y of the first jet prolongation J1YY.

By the substitution of the target space by JrY,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 projections Tρs 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  for the proof.

Theorem 6.

( 1 ) Second order nonholonomic connections θ:YJ~2Y on YM are in bijection with triples (Γ1,Γ2,G), where Γ1,Γ2:YJ1Y are first order connections on YM and G:YVY2T*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:YVYS2T*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:YVYS2T*M in addition.

4. Ehresmann Prolongation

Given two higher order connections Γ:YJ~rY and Γ¯:YJ~sY, the product of Γ and Γ¯ is the (r+s)th order connection Γ*Γ¯:YJ~r+sY defined by (9)Γ*Γ¯=J~sΓΓ¯.

Concerning the holonomity, according to  if both Γ and Γ¯ are of the first order, then Γ*Γ¯:YJ~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 of Δ:YJ~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¯:YJ1Y are (11)Γ:yip=Fip(x,y),Γ¯:yip=Gip(x,y), then the second order connection Γ*Γ¯:YJ~2Y has equations (12)yip=Fip,y0ip=Gip,yijp=Fipxj+FipyqGjq. For order three see . 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 , where the map e:J¯2YJ¯2Y is obtained from the natural exchange map eΛ:J1J1YJ1J1Y as a restriction to the subbundle J¯2YJ1J1Y. 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¯2YJ¯2Y with the same coordinate expression.

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

Proposition 9.

All natural operators transforming first order connection Γ:YJ1Y into second order semiholonomic connection YJ¯2Y form a one-parameter family: (13)Γk·(Γ*Γ)+(1-k)·e(Γ*Γ),k.

To meet the classical theory mentioned in Section 2 let us note that the corresponding operation is the following; see also . If we apply the tangent functor T two times on a projection π:EM and a section σ:ME we obtain (14)Tπ:TETM,T2π:T2ET2M,(15)Tσ:TMTE,T2σ:T2MT2E, 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)on    M,TM,T2M:    (xi),(xi,x1i),(xi,x1i,x2i,x12i),on    E,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)f1fix1i,f2fix2i,f12fijx1ix2j+fix12i,wherefi=fxi,fij=2fxixj. Furthermore, f1=dfρ1,f2=dfρ2,f12=d2f. We use this notation in the following formulae.

If the section σ is defined by local functions Γp, then the sections Tσ and T2σ are defined by its differentials Γ1p, Γ2p, and Γ12p: (18)σ:xiyp=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 coefficients Γip and Γijp in (18) are arbitrary functions corresponds to a nonholonomic connection on the fibered manifold π.

The case when Γijp=Γip/xj, where Γip are arbitrary functions corresponds to a semiholonomic connection on the fibered manifold π.

The case when Γ1p=dΓpρ1,Γ2p=dΓpρ2,Γ12p=d2Γp corresponds to a holonomic connection on the fibered manifold π.

The functions Γip and Γijp 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 κ0:3 is identified as a reference configuration, a configuration of a material body is an embedding (19)κ:𝔼3. Choosing a frame in 𝔼3, we can identify 𝔼3 with 3. Now, one can associate with any given configuration κ the deformation χ defined as the composition: (20)χ:κκ0-1. In coordinates, (21)xi=χi(X1,X2,X3),FIi=xiXI.

For a simple hyperelastic body, the constitutive equation is of the form: ψ=ψ(F,X). Two points X1,X2 are materially isomorphic if there exists a non-singular linear map P12, between their tangent spaces: (22)P12:TX1TX2, such that (23)ψ(FP12,X1)=ψ(F,X2) identically for all deformation gradients F. A body is materially uniform if, and only if, there exists a material isomorphism P(X) from a fixed point X0 to each point X.

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 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 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, let Eα be the natural basis of   3. 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=PαKPI-αXJ, we can write material covariant derivative of the field w wit respect to the given material parallelism (28)IJK=ωKXJ+ΓIJKωI.

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 (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¯iqpxjyq+Γ¯iqpΓjrqyr+Γ¯iqpyjq.

Theorem 10.

Let 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 ). 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 .

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.

Epstein M. Elżanowski M. Material Inhomogeneities and Their Evolution 2007 Berlin, Germany Springer xiv+274 Interaction of Mechanics and Mathematics 2367499 Epstein M. The Geometrical Language of Continuum Mechanics 2010 Cambridge, UK Cambridge University Press xii+312 10.1017/CBO9780511762673 2605800 Kolář I. Michor P. W. Slovák J. Natural Operations in Differential Geometry 1993 Berlin, Germany Springer vi+434 1202431 Virsik J. On the holonomity of higher order connections Cahiers de Topologie et Geometrie Differentielle Categoriques 1971 12 197 212 0305294 ZBL0223.53026 Vašík P. Transformations of semiholonomic 2- and 3-jets and semiholonomic prolongation of connections Proceedings of the Estonian Academy of Sciences 2010 59 4 375 380 10.3176/proc.2010.4.18 2752982 ZBL1213.53019 Atanasiu G. Balan V. Brînzei N. Rahula M. Differential Geometric Structures: Tangent Bundles, Connections in Bundles, Exponential Law in the Jet Space 2010 Moscow, Russia Librokom ZBL1247.43005 Rahula M. Vašík P. Voicu N. Tangent structures: sector-forms, jets and connections Journal of Physics: Conference Series 2012 346 012023 Doupovec M. Mikulski W. M. Reduction theorems for principal and classical connections Acta Mathematica Sinica, English Series 2010 26 1 169 184 2-s2.0-74849134725 10.1007/s10114-010-7333-2 ZBL1186.53036 Vašík P. On the Ehresmann prolongation Annales Universitatis Mariae Curie-Skłodowska A 2007 61 145 153 2368929 ZBL1136.53024 Rahula M. New Problems in Differential Geometry 1993 8 River Edge, NJ, USA World Scientific Publishing xx+171 1353657 Mikulski W. M. Natural transformations transforming functions and vector fields to functions on some natural bundles Mathematica Bohemica 1992 117 2 217 223 1165899 ZBL0810.58004