On a micropolar theory of growing solids

E. V. Murashkin ; Yu. N. Radayev

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 24 (2020) no. 3, pp. 424-444.

Résumé

The present paper is devoted to the problem of boundary conditions formulation in the growing micropolar solid mechanics. The static equations of the micropolar continuum in terms of relative tensors (pseudotensors) are derived due to virtual work principle for a solid of constant staff. The constitutive quadratic form of the elastic potential (treated as an absolute scalar) for a linear hemitropic micropolar solid is presented and discussed. The constitutive equations for symmetric and antisymmetric parts of force and couple stress tensors are given. The final forms of the static equations for the hemitropic micropolar continuum in terms of displacements and microrotations rates are obtained including the case of growing processes. A transformation of the equilibrium equations is proposed to obtain boundary conditions on the propagating growing surface in terms of relative tensors in the form of differential constraints. Those are valid for a wide range of materials and metamaterials. The algebra of rational relative invariants is intensively used for deriving the constitutive relations on the growing surface. Systems of joint algebraic rational relative invariants for force, couple stress tensors and also unit normal and tangent vectors to propagating growing surface are obtained, including systems of invariants sensitive to mirror reflections and 3D-space inversions.

Détail

Keywords: micropolar hemitropic continuum, relative tensor, 3D printing, propagating growing surface, stress, differential constraint, complete system.
Mots-clés : microrotation, pseudoscalar, constitutive equation, rational relative invariant

Bibliographie

[1] Berman B., “3-D printing: The new industrial revolution”, Business Horizons, 55:2 (2012), 155–162 | DOI

[2] Southwell R. V., An introduction to the theory of elasticity. For engineers and physicists, Oxford Engineering Science Series, Oxford Univ. Press, London, 1936 | MR | Zbl

[3] Epstein M., Maugin G. A., “Thermomechanics of volumetric growth in uniform bodies”, Int. J. Plast., 16:7–8 (2000), 951–978 | DOI | Zbl

[4] Maugin G. A., “On inhomogeneity, growth, ageing and the dynamics of materials”, J. Mech. Mater. Struct., 4:4 (2009), 731–741 | DOI

[5] Ciarletta P., Preziosi L., Maugin G. A., “Mechanobiology of interfacial growth”, J. Mech. Phys. Solids, 61:3 (2013), 852–872 | DOI | MR | Zbl

[6] Ciarletta P., Ambrosi D., Maugin G. A., Preziosi L., “Mechano-transduction in tumour growth modelling Physical constraints of morphogenesis and evolution”, Eur. Phys. J. E, 36 (2013), 23 | DOI | MR

[7] Ciarletta P., Ambrosi D., Maugin G. A., “Configurational forces for growth and shape regulations in morphogenesis”, Bull. Pol. Acad. Sci., Tech. Sci., 60:2 (2012), 253–257 | DOI

[8] Porubov A. V., Maugin G. A., “Application of non-linear strain waves to the study of the growth of long bones”, Int. J. Non-Linear Mech., 46:2 (2011), 387–394 | DOI

[9] Maugin G. A., “From the propagation of phase-transition fronts to the evolution of the growth plate in long bones”, Proc. Est. Acad. Sci., 59:2 (2010), 72–78 | DOI | MR | Zbl

[10] Ciarletta P., Preziosi L., Maugin G. A., “Thermo-mechanics of growth and mass transfer: morphogenesis of seashells”, Computer Methods in Biomechanics and Biomedical Engineering, 15:Suppl. 1 (2012), 110–112 | DOI

[11] Ciarletta P., Maugin G. A., “Elements of a finite strain-gradient thermomechanical theory for material growth and remodeling”, Int. J. Non-Linear Mech., 46:10 (2011), 1341–1346 | DOI

[12] Goriely A., The Mathematics and Mechanics of Biological Growth, Interdisciplinary Applied Mathematics, 45, Springer, New York, 2017, xxii+646 pp | DOI | MR | Zbl

[13] Veblen O., Thomas T. Y., “Extensions of Relative Tensors”, Trans. Amer. Math. Soc., 26:3 (1924), 373–377 | DOI | MR | Zbl

[14] Veblen O., Invariants of Quadratic Differential Forms, Cambridge Tracts in Mathematics and Mathematical Physics, 24, Cambridge University Press, Cambridge, 1927, viii+102 pp. | Zbl

[15] Levi–Cività T., The Absolute Differential Calculus (Calculus of Tensors), Blackie Son, London, Glasgow, 1927, xvi+450 pp. | MR | Zbl

[16] Hawking S. W., Israel W., General Relativity. An Einstein Centenary Survey, Cambridge University Press, Cambridge, 1979, xviii+919 pp. | MR | Zbl

[17] Schouten J. A., Tensor Analysis for Physicist, Oxford University Press, New York, 1951, 276 pp. | MR

[18] Sokolnikoff I. S., Tensor Analysis: Theory and Applications to Geometry and Mechanics of Continua, Applied Mathematics Series, John Wiley Sons, New York, 1964, xii+361 pp. | MR | Zbl

[19] Gurevich G. B., Foundations of the Theory of Algebraic Invariants, P. Noordhoff, Gröningen, 1964, viii+429 pp. | MR | Zbl

[20] Synge J. L., Schild A., Tensor Calculus, v. 5, Courier Corporation, New York, 1978, 334 pp. | MR

[21] Truesdell C., Toupin R., “The Classical Field Theories”, Principles of Classical Mechanics and Field Theory. Encyclopedia of Physics, Encyclopedia of Physics, v. 2/3/1, eds. S. Flügge, Springer, Berlin, Heidelberg, 1960, 226–902 | DOI

[22] Das A. J., Tensors: The mathematics of relativity theory and continuum mechanics, Springer, New York, 2007, xii+290 pp | DOI | MR | Zbl

[23] Rashba E. I., “Stresses computation in massive construction under their own weight taking into account the construction sequence”, Proc. Inst. Struct. Mech. Acad. Sci. Ukrainian SSR, 1953, no. 18, 23–27 (In Russian)

[24] Kharlab V. D., “Linear creep theory of the build-up body. Mechanics of rod systems and solid mediums”, The Proceedings of the Leningrad Civil Engineering Institute, v. 49, Leningrad Civil Engineering Institute, Leningrad, 1966, 93–119 (In Russian)

[25] Arutyunyan N. Kh., Naumov V. E., Radayev Yu. N., “Dynamic expansion of an elastic layer. Part 1. Motion of a flow of precipitated ppapers at a variable rate”, Izv. Akad. Nauk. Mekh. Tverd. Tela, 1992, no. 5, 6–24 (In Russian)

[26] Arutyunyan N. Kh., Naumov V. E., Radayev Yu. N., “Dynamical expansion of an elastic layer. Part 2. The case of drop of accreted ppapers at a constant rate”, Izv. Akad. Nauk. Mekh. Tverd. Tela, 1992, no. 6, 99–112 (In Russian)

[27] Naumov V. E., Radayev Yu. N., Thermomechanical model of an growing solids: Variational formulation, Preprint no. 527, IPMech RAS, Moscow, 1993, 39 pp. (In Russian)

[28] Dmitrieva A. M., Naumov V. E., Radayev Yu. N., Growth of thermoelastic spherical layer: Application of the variational approach, Preprint no. 528, IPMech RAS, Moscow, 1993, 64 pp. (In Russian)

[29] Kovalev V. A., Radayev Yu. N., “On a form of the first variation of the action integral over a varied domain”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 14:2 (2014), 199–209 (In Russian) | DOI | Zbl

[30] Arutyunyan N. Kh., Naumov V. E., “The boundary value problem of the theory of viscoelastic plasticity of a growing body subject to aging”, J. Appl. Math. Mech., 48:1 (1984), 1–10 | DOI | MR

[31] Trincher V. K., “On the formulation of the problem of stresses calculation in the gravitational state of a growing solid”, Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, 1984, no. 2, 119–124 (In Russian)

[32] Bykovtsev G. I., Izbrannye problemnye voprosy mekhaniki deformiruemykh sred [Selected Problems from Solid Mechanics. Collection of papers], Dal'nauka, Vladivostok, 2002, 566 pp. (In Russian)

[33] Radayev Yu. N., “The Lagrange multipliers method in covariant formulations of micropolar continuum mechanics theories”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 22:3 (2018), 504–517 (In Russian) | DOI | Zbl

[34] Courant R., Gilbert D., Metody matematicheskoi fiziki [Methods of Mathematical Physics], Gostekhteoretizdat, Moscow, Leningrad, 1933, 528 pp. (In Russian) | MR

[35] Gelfand I. M., Fomin S. V., Variatsionnoe ischislenie [Calculus of Variations], Fizmatlit, Moscow, 1961, 228 pp. (In Russian) | MR

[36] Gunter N. M., Kurs variatsionnogo ischisleniia [A Course of the Calculus of Variations], Gostekhteoretizdat, Moscow, Leningrad, 1941, 308 pp. (In Russian)

[37] Kovalev V. A., Radayev Yu. N., “On wave solutions of dynamic equations of hemitropic micropolar thermoelasticity”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 19:4 (2019), 454–463 (In Russian) | DOI | MR

[38] Radayev Yu. N., Kovalev V. A., “On plane thermoelastic waves in hemitropic micropolar continua”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 23:3 (2019), 464–474 | DOI | Zbl

[39] Rosenfeld B. A., Mnogomernye prostranstva [Multidimensional Spaces], Nauka, Moscow, 1966, 648 pp. (In Russian) | MR

[40] Rosenfeld B. A., A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space, Studies in the History of Mathematics and Physical Sciences, 12, Springer, New York, 1988, ix+471 pp | DOI | MR | Zbl

[41] Murashkin E. V., Radayev Yu. N., “On a differential constraint in asymmetric theories of the mechanics of growing solids”, Mech. Solids, 54:8 (2019), 1157–1164 | DOI

[42] Murashkin E. V., Radayev Yu. N., “On a differential constraint in the continuum theory of growing solids”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 23:4 (2019), 646–656 | DOI | MR | Zbl

[43] Murashkin E. V., Radayev Yu. N., “On a Class of Constitutive Equations on Propagating Growing Surface”, Bulletin of the Yakovlev Chuvash State Pedagogical University. Series: Mechanics of Limit State, 2019, no. 3(41), 11–29 (In Russian) | DOI

[44] Hamilton W. R., Lectures on Quaternions, Cambridge University Press, Cambridge, 1866, lx+762 pp | DOI

[45] Cayley A., “A memoir on the theory of matrices”, Philos. Trans. R. Soc. Lond., 1858, no. 148, 17–37 | DOI