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Contorsion of Material Connection in Growing Solids


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.

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Fig. 1


  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 \(\mathfrak{y}\in U\).


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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).

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

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(Submitted by A. M. Elizarov)

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Lychev, S.A., Koifman, K.G. Contorsion of Material Connection in Growing Solids. Lobachevskii J Math 42, 1852–1875 (2021).

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