Skip to main content
SpringerLink
Log in

Contorsion of Material Connection in Growing Solids

Lobachevskii Journal of Mathematics volume 42, pages 1852–1875 (2021)Cite this article

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.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Fig. 1

Notes

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

  2. 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,

    $$\widehat{f}:X\rightarrow f(X),\quad\widehat{f}:x\mapsto f(x).$$

  3. In the paper symbols \(\delta_{ij}\), \(\delta^{i}_{j}\), and \(\delta^{ij}\) stand for the Kronecker delta.

  4. Euclidean vectors and tensors are denoted by Latin boldface letters.

  5. 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.

  6. 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.

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

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

  9. 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 \(\mathcal{E}\).

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

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

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

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

  14. The operation \(\bigcirc\!\!\!\!\!\!\wedge\) is referred to as Kulkarni–Nomizu product [21].

  15. 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 \(\mathfrak{R}_{ijkl}={G}_{lm}\mathfrak{R}_{ijk}{}^{m}\).

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

  17. 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.

  18. 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.

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

  20. 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

  1. M. Epstein and M. Elzanowski, Material Inhomogeneities and Their Evolution: A Geometric Approach (Springer Science, New York, 2007).

    MATH  Google Scholar 

  2. 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/s00033-007-6141-8

    Article  MathSciNet  MATH  Google Scholar 

  3. R. 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-y

    Article  MathSciNet  MATH  Google Scholar 

  4. F. Sozio and A. Yavari, ‘‘Riemannian and Euclidean material structures in anelasticity,’’ Math. Mech. Solids 25, 1267–1293 (2020). https://doi.org/10.1177/1081286519884719

    Article  MathSciNet  MATH  Google Scholar 

  5. K. 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.

    MATH  Google Scholar 

  6. 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.

    MATH  Google Scholar 

  7. 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).

    Article  MathSciNet  Google Scholar 

  8. E. Kröner, ‘‘Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen,’’ Arch. Ration. Mech. Anal. 4, 18–334 (1959).

    Article  MathSciNet  Google Scholar 

  9. W. Noll, ‘‘Materially uniform simple bodies with inhomogeneities,’’ Arch. Ration. Mech. Anal. 27, 1–32 (1967). https://doi.org/10.1007/BF00276433

    Article  MathSciNet  MATH  Google Scholar 

  10. M. 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/10.1088/0305-4470/35/7/317

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Yavari and A. Goriely, ‘‘Riemann–Cartan geometry of nonlinear dislocation mechanics,’’ Arch. Ration. Mech. Anal. 205, 59–118 (2012). https://doi.org/10.1007/s00205-012-0500-0

    Article  MathSciNet  MATH  Google Scholar 

  12. A. 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.2012.0342

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Yavari, ‘‘A geometric theory of growth mechanics,’’ J. Nonlin. Sci. 20, 781–830 (2010). https://doi.org/10.1007/s00332-010-9073-y

    Article  MathSciNet  MATH  Google Scholar 

  14. F. 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.2016.08.012

    Article  MathSciNet  Google Scholar 

  15. F. Sozio and A. Yavari, ‘‘Nonlinear mechanics of accretion,’’ J. Nonlin. Sci. 29, 1813–1863 (2019). https://doi.org/10.1007/s00332-019-09531-w

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  18. J. M. Lee, Introduction to Smooth Manifolds (Springer, New York, 2012). https://doi.org/10.1007/978-1-4419-9982-5

    Book  Google Scholar 

  19. S. A. Lychev and K. G. Koifman, ‘‘Material affine connections for growing solids,’’ Lobachevskii J. Math. 41, 2034–2052 (2020). https://doi.org/10.1134/s1995080220100121

    Article  MathSciNet  MATH  Google Scholar 

  20. S. Lychev and K. Koifman, Geometry of Incompatible Deformations: Differential Geometry in Continuum Mechanics (De Gruyter, Berlin, 2018).

    Book  Google Scholar 

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

    Book  MATH  Google Scholar 

  22. M. M. Postnikov, Lectures in Geometry: Smooth Manifolds, Semester 3 (URSS, Moscow, 1994) [in Russian].

    MATH  Google Scholar 

  23. M. W. Hirsch, Differential Topology (Springer Science, New York, 2012).

    Google Scholar 

  24. J. Nash, ‘‘\(C^{1}\) isometric imbeddings,’’ Ann. Math. 60, 383–396 (1954).

    Article  MathSciNet  Google Scholar 

  25. 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/s00707-019-02399-7

    Article  MathSciNet  MATH  Google Scholar 

  26. W. H. Yang and W. W. Feng, ‘‘On axisymmetrical deformations of nonlinear membranes,’’ J. Appl. Mech. 37, 1002–1011 (1970). https://doi.org/10.1115/1.3408651

    Article  MATH  Google Scholar 

  27. H. Weyl, Space, Time, Matter (Dover, New York, 1952).

    Google Scholar 

  28. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 1: Mechanics (Butterworth-Heinemann, New York, 1976).

  29. R. T. Schield, ‘‘Inverse deformation results in finite elasticity,’’ Zeitschr. Angew. Math. Phys. 18, 490–500 (1967). https://doi.org/10.1007/bf01601719

    Article  Google Scholar 

  30. S. Chern, W. Chen, and K. Lam, Lectures on Differential Geometry (World Scientific, Singapore, 1999).

    Book  Google Scholar 

  31. G. A. Maugin, Material Inhomogeneities in Elasticity (CRC, Boca Raton, FL, 1993).

    Book  Google Scholar 

  32. S. Mac Lane, Categories for the Working Mathematician (Springer, New York, 1978).

    Book  Google Scholar 

  33. 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.2013.11.011

    Article  MathSciNet  MATH  Google Scholar 

  34. J. M. Lee, Introduction to Topological Manifolds (Springer, New York, 2011). https://doi.org/10.1007/978-1-4419-7940-7

    Book  MATH  Google Scholar 

  35. T. Levi-Civita, ‘‘Nozione di parallelismo in una varietà qualunque e conseguente specificazione geometrica della curvatura riemanniana,’’ Rend. Circ. Mat. Palermo 42, 173–204 (1916).

    Article  Google Scholar 

  36. 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).

    Article  Google Scholar 

  37. 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/10.1063/1.3525798

  38. C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics (Springer Science, New York, 2004). https://doi.org/10.1007/978-3-662-10388-3

    Book  MATH  Google Scholar 

  39. J. E. Marsden and T. J. Hughes, Mathematical Foundations of Elasticity (Courier, North Chelmsford, MA, 1994).

    MATH  Google Scholar 

  40. E. H. Lee, ‘‘Elastic-plastic deformation at finite strain,’’ J. Appl. Mech. 36, 1–6 (1969). https://doi.org/10.1115/1.3564580

    Article  MATH  Google Scholar 

  41. C. 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.2020.0462

  42. F. 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).

  43. L. Rosenfeld, ‘‘Sur le tenseur D’Impulsion-Energie,’’ Acad. R. Belg. Cl. Sci. 18, 1–30 (1940).

    MATH  Google Scholar 

Download references

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

  1. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, Russia

    S. A. Lychev

  2. Bauman Moscow State Technical University, Moscow, Russia

    K. G. Koifman

Authors
  1. S. A. Lychev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. G. Koifman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to S. A. Lychev or K. G. Koifman.

Additional information

(Submitted by A. M. Elizarov)

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

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

Keywords:

search

Navigation