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Numerical Modeling the Stresses in Incompressible and Rigid Bodies

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Progress in Continuum Mechanics

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 196))

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Abstract

Issues related to the calculation of stresses in an incompressible medium are considered. The problems of modeling the flow of a classical Newtonian viscous liquid and micropolar liquid are discussed. The problem of calculating stresses in incompressible elastomers (rubber-like) bodies is considered. The problem of determining stresses in a fully incompressible (rigid) body is considered also. Examples are given.

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References

  • Aero EL, Bulygin AN, Kuvshinskii EV (1965) Asymmetric hydromechanics. Journal of Applied Mathematics and Mechanics 29(2):333–346

    Google Scholar 

  • Bessonov NM, Schultz WW (1997) Numerical simulation of 3D rubber-like solids with application to lubricated lip seals. Mechanical Engineering and Applied Mechanics, University of Michigan, Ford Motor Co, Detroit, USA

    Google Scholar 

  • Bessonov NM, Song DJ (2001) Application of vector calculus to numerical simulation of continuum mechanics problems. Journal of Computational Physics 167(1):22–38

    Google Scholar 

  • Bessonov N, Pojman JA, Volpert V (2004) Modelling of diffuse interfaces with temperature gradients. Journal of Engineering Mathematics 49(4):321–338

    Google Scholar 

  • Bessonov, N, Pojman, J, Viner, G, Volpert, V, Zoltowski, B (2008) Instabilities of diffuse interfaces. Mathematical Modelling of Natural Phenomena 3(1):108–125

    Google Scholar 

  • Bessonov NM, Frolova KP (2022) Application of micropolar theory to the description of ultrathin liquid layers. In: Polyanskiy VA, Belyaev AK (eds) Mechanics and Control of Solids and Structures, Springer Nature, Cham,Advanced Structured Materials, vol 164, pp 49–73

    Google Scholar 

  • Bessonov NM, Koleshko SB (1988) Numerical simulation of stationary flow in artificial hearts valve on base of solution of Navier-Stokes equations on nonorthogonal grids (in Russ.). Simulation in Mechanics 22(19)

    Google Scholar 

  • Bessonov N, Aero E, Veretennikova T (1991) Numerical simulation of moment (micropolar) liquid flow in channels of complex shape (in Russ.). Preprint 56, IPME RAN

    Google Scholar 

  • Ciarlet PG (1993) Mathematical Elasticity, vol 1 - Three Dimensional Elasticity. North-Holland, Amsterdam

    Google Scholar 

  • Douglas J, Gunn J (1964) A general formulation of alternating direction methods.Part I. Parabolic and hyperbolic problems. Numerische Mathematik 6(1):428–453

    Google Scholar 

  • Eringen AC (1966) Theory of micropolar fluids. Journal of Mathematics and Mechanics 16(1):1–18

    Google Scholar 

  • Fletcher CAJ (2006) Computational Techniques for Fluid Dynamics, Scientific Computation, vol 2 - Specific Techniques for Different Flow Categories. Springer, Berlin Heidelberg

    Google Scholar 

  • Fomicheva M, Müller W, Vilchevskaya E, Bessonov N (2019) Funnel flow of a Navier-Stokes-fluid with potential applications to micropolar media. Facta Universitatis, Series: Mechanical Engineering 17(2):255–267

    Google Scholar 

  • Fomicheva M, Vilchevskaya EN, Müller WH, Bessonov N (2019) Milling matter in a crusher: modeling based on extended micropolar theory. Continuum Mechanics and Thermodynamics 31(5):1559–1570

    Google Scholar 

  • Fosdick R, Royer-Carfagni G (2004) Stress as a constraint reaction in rigid bodies. Journal of Elasticity 74(3):265–276

    Google Scholar 

  • Grioli G (1983) On the stress in rigid bodies. Meccanica 18(1):3–7

    Google Scholar 

  • Hirsch C (2007) Numerical Computation of Internal and External Flows. John Wiley & Sons, New York

    Google Scholar 

  • Kovenya VM, Yanenko NN (1981) The Method of Splitting in Gas Dynamics Problems (in Russ.). Izdatel’stvo Nauka, Novosibirsk

    Google Scholar 

  • Mooney M (2004) A Theory of Large Elastic Deformation. Journal of Applied Physics 11(9):582–592

    Google Scholar 

  • Morman Jr K (1994) A simple non-isothermal constitutive model for finitedeformation rubber viscoelasticity. In: Voyiadjis GZ, Bank LC, Jacob LJ (eds) Mechanics of Materials and Structures, Studies in Applied Mechanics, vol 35, Elsevier, pp 297–318

    Google Scholar 

  • Pojman JA, Chekanov Y, Wyatt V, Bessonov N, Volpert V (2009) Numerical simulations of convection induced by Korteweg stresses in a miscible polymermonomer system: Effects of variable transport coefficients, polymerization rate and volume changes. Microgravity Science and Technology 21(3):225–237

    Google Scholar 

  • Rivlin RS (1956) Large elastic deformations. In: Eirich FR (ed) Rheology, Academic Press, New York

    Google Scholar 

  • Roache PJ (1998) Fundamentals of Computational Fluid Dynamics. Hermosa Publishers

    Google Scholar 

  • Truesdell C, NollW(1965) The Non-Linear Field Theories of Mechanics / Die Nicht-Linearen Feldtheorien der Mechanik. In: Flügge S (ed) Handbuch der Physik, Prinzipien der theoretischen Physik / Principles of Theoretical Physics, vol III/3, Springer, Berlin Heidelberg, pp 1–541

    Google Scholar 

  • Yanenko NN (1971) Method of Fractional Steps - The Solution of Problems of Mathematical Physics in Several Variables. Hermosa Publishers

    Google Scholar 

Download references

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Correspondence to Yaroslava I. Litvinova .

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Bessonov, N.M., Litvinova, Y.I. (2023). Numerical Modeling the Stresses in Incompressible and Rigid Bodies. In: Altenbach, H., Irschik, H., Porubov, A.V. (eds) Progress in Continuum Mechanics. Advanced Structured Materials, vol 196. Springer, Cham. https://doi.org/10.1007/978-3-031-43736-6_7

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