### Abstract

The subject of the present paper is a material connection that describes the sources of incompatibility in growing solids. There are several possibilities to introduce such a connection on the body manifold, which provides formal description of a body as a continuous collection of material particles. Two of them are discussed in detail. The first sets the geometry of Riemannian manifold, while the second sets Weitzenböck geometry. To derive particular connection functions, related with given evolutionary problem for growing solid, one has to use some intermediate configurations, whose choice is also uncertain. The purpose of this study is to find out how the ambiguity affects on the stress-strain state modelling. The main results are the following. It is proven that the geometrical invariants of considered material connections, namely the invariants of torsion and curvature, are independent on particular choice of intermediate configuration. It is shown that Weitzenböck connection contains all metric information that completely defines Riemannian ones, but, except it, provides additional description for contorsion, which characterizes inhomogeneity by specific term in balance of momentum. Thus, the two connections do not contradict each other. To describe the body’s response to deformation it is sufficient to construct more simpler Riemannian connection, while to completely describe balance laws it is advisable to obtain more complete Weitzenböck connection.

This is a preview of subscription content, access via your institution.

## Notes

Here \(S\) designates the underlying set of the structure \(\mathcal{S}\), while vertical bar signs stand for restriction of the corresponding fields.

Note, that the image \(\varkappa(\mathfrak{B})\) of a configuration \(\varkappa:\mathfrak{B}\rightarrow\mathcal{E}\) may not coincide with the whole physical space \(\mathcal{E}\). In this regard, here and in the whole paper we use the special designation. If \(f:X\rightarrow Y\) is a mapping, then \(\widehat{f}\) denotes a new mapping obtained by restricting the codomain \(Y\) to the image of \(f\), i.e., to \(f(X)\). That is,

$$\widehat{f}:X\rightarrow f(X),\quad\widehat{f}:x\mapsto f(x).$$In the paper symbols \(\delta_{ij}\), \(\delta^{i}_{j}\), and \(\delta^{ij}\) stand for the Kronecker delta.

Euclidean vectors and tensors are denoted by Latin boldface letters.

Or, more generally, the collection of charts \(\{(U_{\alpha},\sigma_{\alpha})\}_{\alpha\in A}\), where \(U_{\alpha}\) are open subsets of \(\mathcal{E}\), that cover \(\mathcal{E}\), and \(\sigma_{\alpha}:U_{\alpha}\rightarrow\mathbb{R}^{3}\) are diffeomorphisms.

Here and in what follows we use the reduced form of dependence like \(\mathcal{L}=\mathcal{L}(X,t,\gamma(X,t),\dot{\gamma}(X,t),D\gamma(X,t))\). Formally, the notation \(\mathcal{L}(X,t,\gamma,\dot{\gamma},D\gamma)\) shows that one may treat \(\mathcal{L}\) as functional of \(\gamma\), but such the interpretation is not used in the paper.

We denote the set of all linear mappings from one vector space \(\mathcal{U}\) to another, \(\mathcal{V}\), as \({\textrm{Lin}}(\mathcal{U};\mathcal{V})\).

Thus, point \(\mathcal{X}\) is a point from \(\mathcal{S}_{R}\), but considered without any surrounding geometry.

The symbol \(\iota_{M_{R}}\) stands for inclusion map \(\iota_{M_{R}}:M_{R}\hookrightarrow\mathcal{E}\). The symbol \(\mathcal{X}\) designates a point from \(M_{R}\), while the symbol \(X\) represents the similar element, but considered in space \(\mathcal{E}\).

Hereafter the symbol \({\textrm{Sec}}(E)\) designates the \(C^{\infty}(M)\)-module of all sections (tensor fields) vector bundle \(E\rightarrow M\) [18].

The symbol \({\mathfrak{X}}(M)\) stands for algebra of vector fields on \(M\).

If \(f\) is a scalar function on manifold, then \([{u},{v}]f:={u}({v}f)-{v}({u}f)\).

We mention the paper [15], in which closed time intervals are considered. In this case the final body is a manifold with boundary.

The operation \(\bigcirc\!\!\!\!\!\!\wedge\) is referred to as

*Kulkarni–Nomizu product*[21].Thus, \(\mathfrak{R}^{\flat}\in{\textrm{Sec}}(T^{\ast}M_{R}\otimes T^{\ast}M_{R}\otimes T^{\ast}M_{R}\otimes T^{\ast}M_{R})\); and in components \(\mathfrak{R}_{ijkl}={G}_{lm}\mathfrak{R}_{ijk}{}^{m}\).

Here \(e_{tab}\) and \(e^{tsl}\) are alternators.

In this regard, arguments \({u}\), \({v}\) of \(\mathfrak{K}_{{u}}{v}\) are equitable and one may write \(\mathfrak{K}({u},{v})\). Meanwhile, for further reasonings it is convenient to put the first argument in the lower index.

This formula can be obtained from the following system of \(54\) relations:

$$\mathfrak{T}^{k}{}_{ij}=\Gamma^{k}{}_{ij}-\Gamma^{k}{}_{ji},\quad i,j,k=1,2,3,$$$${}-\mathfrak{Q}_{ijk}=\partial_{i}{G}_{jk}-\Gamma^{m}{}_{ij}{G}_{mk}-\Gamma^{m}{}_{ik}{G}_{mj},\quad i,j,k=1,2,3,$$for torsion (4) and nonmetricity (6) components, and cyclic permutation of indices \((i,j,k)\) applied to the latter expression.

In dyadic representation, \((\mathbf{a}\otimes\mathbf{b}):(\mathbf{c}\otimes\mathbf{d})=(\mathbf{a}\cdot\mathbf{c})(\mathbf{b}\cdot\mathbf{d})\).

That is, if there exists a neighborhood \(U\) of \(\mathfrak{x}\), such that \({P}_{\mathfrak{y}}=T_{\mathfrak{y}}\varkappa\) for all \(\mathfrak{y}\in U\).

## REFERENCES

M. Epstein and M. Elzanowski,

*Material Inhomogeneities and Their Evolution: A Geometric Approach*(Springer Science, New York, 2007).E. Kanso, M. Arroyo, Y. Tong, A. Yavari, J. Marsden, and M. Desbrun, ‘‘On the geometric character of stress in continuum mechanics,’’ Z. Angew. Math. Phys.

**58**, 843–856 (2007). https://doi.org/10.1007/s00033-007-6141-8R. Kupferman, E. Olami, and R. Segev, ‘‘Continuum dynamics on manifolds: Application to elasticity of residually-stressed bodies,’’ J. Elast.

**128**, 61–84 (2017). https://doi.org/10.1007/s10659-016-9617-yF. Sozio and A. Yavari, ‘‘Riemannian and Euclidean material structures in anelasticity,’’ Math. Mech. Solids

**25**, 1267–1293 (2020). https://doi.org/10.1177/1081286519884719K. Kondo, ‘‘Geometry of elastic deformation and incompatibility,’’ in

*Memoirs of the Unifying Study of the Basic Problems in Engineering Science by Means of Geometry*(Gakujutsu Bunken Fukyo-Kai, Tokyo, 1955), vol. 1, pp. 5–17.K. Kondo, ‘‘Non-Riemannian geometry of imperfect crystals from a macroscopic viewpoint,’’ in

*Memoirs of the Unifying Study of the Basic Problems in Engineering Science by Means of Geometry*(Gakujutsu Bunken Fukyo-Kai, Tokyo, 1955), vol. 1, pp. 6–17.B. Bilby, R. Bullough, and E. Smith, ‘‘Continuous distributions of dislocations: A new application of the methods of non-Riemannian geometry,’’ Proc. R. Soc. London, Ser. A

**231**, 263–273 (1955).E. Kröner, ‘‘Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen,’’ Arch. Ration. Mech. Anal.

**4**, 18–334 (1959).W. Noll, ‘‘Materially uniform simple bodies with inhomogeneities,’’ Arch. Ration. Mech. Anal.

**27**, 1–32 (1967). https://doi.org/10.1007/BF00276433M. Miri and N. Rivier, ‘‘Continuum elasticity with topological defects, including dislocations and extra-matter,’’ J. Phys. A: Math. Gen.

**35**, 1727–1739 (2002). https://doi.org/10.1088/0305-4470/35/7/317A. Yavari and A. Goriely, ‘‘Riemann–Cartan geometry of nonlinear dislocation mechanics,’’ Arch. Ration. Mech. Anal.

**205**, 59–118 (2012). https://doi.org/10.1007/s00205-012-0500-0A. Yavari and A. Goriely, ‘‘Weyl geometry and the nonlinear mechanics of distributed point defects,’’ Proc. R. Soc. London, Ser. A

**468**, 3902–3922 (2012). https://doi.org/10.1098/rspa.2012.0342A. Yavari, ‘‘A geometric theory of growth mechanics,’’ J. Nonlin. Sci.

**20**, 781–830 (2010). https://doi.org/10.1007/s00332-010-9073-yF. Sozio and A. Yavari, ‘‘Nonlinear mechanics of surface growth for cylindrical and spherical elastic bodies,’’ J. Mech. Phys. Solids

**98**, 12–48 (2017). https://doi.org/10.1016/j.jmps.2016.08.012F. Sozio and A. Yavari, ‘‘Nonlinear mechanics of accretion,’’ J. Nonlin. Sci.

**29**, 1813–1863 (2019). https://doi.org/10.1007/s00332-019-09531-wG. Rudolph and M. Schmidt,

*Differential Geometry and Mathematical Physics. Part I. Manifolds, Lie Groups and Hamiltonian Systems*(Springer Science, Dordrecht, 2013). https://doi.org/10.1007/978-94-007-5345-7G. Rudolph and M. Schmidt,

*Differential Geometry and Mathematical Physics. Part II. Fibre Bundles, Topology and Gauge Fields*(Springer Science, Dordrecht, 2017). https://doi.org/10.1007/978-94-024-0959-8J. M. Lee,

*Introduction to Smooth Manifolds*(Springer, New York, 2012). https://doi.org/10.1007/978-1-4419-9982-5S. A. Lychev and K. G. Koifman, ‘‘Material affine connections for growing solids,’’ Lobachevskii J. Math.

**41**, 2034–2052 (2020). https://doi.org/10.1134/s1995080220100121S. Lychev and K. Koifman,

*Geometry of Incompatible Deformations: Differential Geometry in Continuum Mechanics*(De Gruyter, Berlin, 2018).J. M. Lee,

*Introduction to Riemannian Manifolds*(Springer, Cham, 2018). https://doi.org/10.1007/978-3-319-91755-9M. M. Postnikov,

*Lectures in Geometry: Smooth Manifolds, Semester 3*(URSS, Moscow, 1994) [in Russian].M. W. Hirsch,

*Differential Topology*(Springer Science, New York, 2012).J. Nash, ‘‘\(C^{1}\) isometric imbeddings,’’ Ann. Math.

**60**, 383–396 (1954).S. Lychev and K. Koifman, ‘‘Nonlinear evolutionary problem for a laminated inhomogeneous spherical shell,’’ Acta Mech.

**230**, 3989–4020 (2019). https://doi.org/10.1007/s00707-019-02399-7W. H. Yang and W. W. Feng, ‘‘On axisymmetrical deformations of nonlinear membranes,’’ J. Appl. Mech.

**37**, 1002–1011 (1970). https://doi.org/10.1115/1.3408651H. Weyl,

*Space, Time, Matter*(Dover, New York, 1952).L. D. Landau and E. M. Lifshitz,

*Course of Theoretical Physics,*Vol. 1:*Mechanics*(Butterworth-Heinemann, New York, 1976).R. T. Schield, ‘‘Inverse deformation results in finite elasticity,’’ Zeitschr. Angew. Math. Phys.

**18**, 490–500 (1967). https://doi.org/10.1007/bf01601719S. Chern, W. Chen, and K. Lam,

*Lectures on Differential Geometry*(World Scientific, Singapore, 1999).G. A. Maugin,

*Material Inhomogeneities in Elasticity*(CRC, Boca Raton, FL, 1993).S. Mac Lane,

*Categories for the Working Mathematician*(Springer, New York, 1978).S. A. Lychev and A. V. Manzhirov, ‘‘The mathematical theory of growing bodies. Finite deformations,’’ J. Appl. Math. Mech.

**77**, 421–432 (2013). https://doi.org/10.1016/j.jappmathmech.2013.11.011J. M. Lee,

*Introduction to Topological Manifolds*(Springer, New York, 2011). https://doi.org/10.1007/978-1-4419-7940-7T. Levi-Civita, ‘‘Nozione di parallelismo in una varietà qualunque e conseguente specificazione geometrica della curvatura riemanniana,’’ Rend. Circ. Mat. Palermo

**42**, 173–204 (1916).E. J. Cartan, ‘‘Sur les variétés á connexion affine et la théorie de la relativité généralisée,’’ Ann. Sci. Ecole Norm. Super.

**40**, 325–412 (1923).O. E. Fernandez and A. M. Bloch, ‘‘The Weitzenböck connection and time reparameterization in nonholonomic mechanics,’’ J. Math. Phys.

**52**, 012901 (2011). https://doi.org/10.1063/1.3525798C. Truesdell and W. Noll,

*The Non-Linear Field Theories of Mechanics*(Springer Science, New York, 2004). https://doi.org/10.1007/978-3-662-10388-3J. E. Marsden and T. J. Hughes,

*Mathematical Foundations of Elasticity*(Courier, North Chelmsford, MA, 1994).E. H. Lee, ‘‘Elastic-plastic deformation at finite strain,’’ J. Appl. Mech.

**36**, 1–6 (1969). https://doi.org/10.1115/1.3564580C. Goodbrake, A. Goriely, and A. Yavari, ‘‘The mathematical foundations of anelasticity: Existence of smooth global intermediate configurations,’’ Proc. R. Soc. London, Ser. A

**477**, 20200462 (2021). https://doi.org/10.1098/rspa.2020.0462F. J. Belinfante, ‘‘On the current and the density of the electric charge, the energy, the linear momentum and the angular momentum of arbitrary fields,’’ Physica (Amsterdam, Neth.)

**7**, 449–474 (1940).L. Rosenfeld, ‘‘Sur le tenseur D’Impulsion-Energie,’’ Acad. R. Belg. Cl. Sci.

**18**, 1–30 (1940).

## Funding

The study was partially supported by the Government program (contract no. AAAA-A20-120011690132-4) and partially supported by RFBR (grant no. 18-29-03228).

## Author information

### Authors and Affiliations

### Corresponding authors

## Additional information

(Submitted by A. M. Elizarov)

## Rights and permissions

## About this article

### Cite this article

Lychev, S.A., Koifman, K.G. Contorsion of Material Connection in Growing Solids.
*Lobachevskii J Math* **42**, 1852–1875 (2021). https://doi.org/10.1134/S1995080221080187

Received:

Revised:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1134/S1995080221080187