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Infinitesimal bending influence on the volume change. (English) Zbl 1335.53017
Summary: Applying tensor calculus we discuss change of the volume under infinitesimal bending of the surface. We obtain that the variation of the volume bounded by the surface $$s$$ and the cone joining the origin to the boundary of the surface, under infinitesimal bending of $$s$$ with the vector field of translation $$\mathbf s$$, equals one third of the flux of the field $$\mathbf s$$ through the given surface $$s$$. An example is analysed and graphically presented. The paper points to the application of the obtained result in the calculation of the volume of a solid model.
##### MSC:
 53A45 Differential geometric aspects in vector and tensor analysis 52B11 $$n$$-dimensional polytopes 74G60 Bifurcation and buckling
##### Keywords:
infinitesimal bending; variation; volume; tensor calculus
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##### References:
 [1] Abdel-Malek, K.; Blackmore, D.; Joy, K., Swept volumes: foundations, perspectives and applications, Int. J. Shape Model., 12, 1, 87-127, (2006) · Zbl 1096.68753 [2] Abdel-Malek, K.; Othman, S., Multiple sweeping using the david-hartenberg representation method, Comput.-Aid. Des., 31, 567-583, (1999) · Zbl 1041.68532 [3] Alexandrov, V. A., Remarks on sabitov’s conjecture that volume is stationary under an infinitesimal bending of a surface, Sib. Mat. Zh., 5, 16-24, (1989) [4] Alexandrov, V. A., An example of a flexible polyhedron with nonconstant volume in the spherical space, Beitr. Algebra Geom., 38, 1, 11-18, (1997) · Zbl 0881.52007 [5] Alexandrov, V. A., New manifestations of the darboux’s rotation and translation fields of a surface, New Zealand J. Math., 40, 59-65, (2010) · Zbl 1222.53006 [6] Blackmore, D.; Leu, M. C.; Shif, F., Analysis and modeling of deformed swept volume, Comput.-Aid. Des., 26, 4, 315-326, (1994) · Zbl 0805.68128 [7] R. Connelly, Assumptions and unsolved questions in the theory of bending, in: Studies in the Metric Theory of Surfaces (Russian translation), Mir, Moskva, 1980, pp. 228-238. [8] Efimov, N., Kachestvennye voprosy teorii deformacii poverhnostei, UMN, 3, 2, 47-158, (1948) [9] Gray, A., Modern differential geometry of curves and surfaces with Mathematica, (1998), CRC Press · Zbl 0942.53001 [10] Hinterleitner, I.; Mikeš, J.; Stránská, J., Infinitesimal F-planar transformations, Russ. Math., 4, 13-18, (2008) · Zbl 1162.53008 [11] Hinterleitner, I., On global geodesic mappings of ellipsoids, AIP Conf. Proc., 1460, 180-184, (2012) [12] Mikeš, J., On existence of nontrivial global geodesic mappings on n-dimensional compact surfaces of revolution, (Diff. Geom. Appl. Int. Conf., Brno, Czechoslovakia, (1990), Word Sci.), 129-137 · Zbl 0790.53003 [13] Pavlović, M. N., Symbolic computation in structural engineering, Comput. Struct., 81, 2121-2136, (2003) [14] M. Peternell, H. Pottmann, T. Steiner, H. Zhao, Swept volume, [15] Slutskiy, D. A., An infinitesimally nonrigid polyhedron with nonstationary volume in the lobachevskii 3-space, Sib. Math. J., 52, 1, 131-138, (2011) · Zbl 1215.52010 [16] Vekua, I., Obobschennye analiticheskie funkcii, Nauka, Moskva, 1959. [17] Velimirović, Lj., On variation of the volume under infinitesimal bending of a closed rotational surface, Novi Sad J. Math., 29, 3, 377-386, (1999) · Zbl 1075.53501 [18] Velimirović, Lj., Change of geometric magnitudes under infinitesimal bending, Facta Univ., 3, 11, 135-148, (2001) · Zbl 1012.53003 [19] Velimirović, Lj.; Minčić, S.; Stanković, M., Infinitesimal rigidity and flexibility of a non-symmetric affine connection space, Eur. J. Comb., 31, 4, 1148-1159, (2010) · Zbl 1194.53016 [20] Velimirović, Lj. S.; Ćirić, M. S.; Velimirović, N. M., On the Willmore energy of shells under infinitesimal deformations, Comput. Math. Appl., 61, 11, 3181-3190, (2011) · Zbl 1222.74036 [21] Velimirović, Lj. S.; Cvetković, M. D.; Ćirić, M. S.; Velimirović, N., Analysis of gaudi surfaces at small deformations, Appl. Math. Comput., 218, 6999-7004, (2012) · Zbl 1246.65035 [22] Williams, C. J.K., Use of structural analogy in generation of smooth surfaces for engineering purposes, Comput.-Aid. Des., 19, 6, 310-322, (1987) · Zbl 0655.65036 [23] Zlatanović, M. Lj., New projective tensors for equitorsion geodesic mappings, Appl. Math. Lett., 25, 5, 890-897, (2012) · Zbl 1254.53023
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