Skip to main content
Top
Published in: Mechanics of Composite Materials 6/2022

26-01-2022

The Impliсit Finite Difference Method in the Deformation Mechanics of Homogeneous and Piecewise Homogeneous Bodies

Author: V. M. Akhundov

Published in: Mechanics of Composite Materials | Issue 6/2022

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

The implicit finite difference method for solving deformation problems of mechanics of piecewise-homogeneous bodies is presented. The method is based on approximating the sought-for quantities by polynomials with indeterminate coefficients. It allows one to approximate the derivatives in resolving equations based on a grid with an irregular, in general, arrangement of nodal points. Relations were given for one-, two-, and three-dimensional approximations of the second-order of accuracy. This method was applied to studying the deformation of an elastic rotating cylinder whose matrix is reinforced in the circumferential directions with one layer of round fibers. The material configuration of the cylinder at large displacements and deformations are presented together with the stresses of contact interaction between the matrix and fibers. Its deformation characteristic, which reflects the continuation of the solution of the problem in terms of rotation speed, is determined. The results obtained are compared with the solution of the problem for a cylinder with square fibers at the same filling found by the methods of implicit finite differences and the traditional method of finite differences. Boundary-value problems for a thick-walled cylinder made of an isotropic material are also solved in nonlinear and linear formulations with uniform and nonuniform distributions of nodal points across the cylinder thickness, and the results obtained are compared with the exact solution of the corresponding linear problem.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Literature
1.
go back to reference A. K. Malmeister, V. P. Tamuzh, and G. A. Teters, Resistance of Polymer and Composite Materials [in Russian], Riga: Zinatne (1980). A. K. Malmeister, V. P. Tamuzh, and G. A. Teters, Resistance of Polymer and Composite Materials [in Russian], Riga: Zinatne (1980).
2.
go back to reference E. I. Grigolyuk and G. M. Kulikov, Multilayer Reinforced Shells. Calculation of Pneumatic Tires [in Russian], M.: Mashinostroenie (1988). E. I. Grigolyuk and G. M. Kulikov, Multilayer Reinforced Shells. Calculation of Pneumatic Tires [in Russian], M.: Mashinostroenie (1988).
3.
go back to reference V. V. Kirichevsky, Method of Finite Elements in Mechanics of Elastomers [in Russian], K.: Nauk. Dumka (2002). V. V. Kirichevsky, Method of Finite Elements in Mechanics of Elastomers [in Russian], K.: Nauk. Dumka (2002).
4.
go back to reference K. F. Chernykh, Nonlinear Elasticity Theory in Machine-Building Calculations [in Russian], L.: Mashinostroenie, (1986). K. F. Chernykh, Nonlinear Elasticity Theory in Machine-Building Calculations [in Russian], L.: Mashinostroenie, (1986).
5.
go back to reference G. A. Holzapfel, T. C. Gasser, and R. W. Ogden, “A new constitutive framework for arterial wall mechanics and a comparative study of material models,” J. Elasticity, 61, 1-48 (2000).CrossRef G. A. Holzapfel, T. C. Gasser, and R. W. Ogden, “A new constitutive framework for arterial wall mechanics and a comparative study of material models,” J. Elasticity, 61, 1-48 (2000).CrossRef
6.
go back to reference G. A. Holzapfel, T. C. Gasser, and M. Stadler, “A structural model for the viscoelastic behavior of arterial walls: Continuum formulation and finite element analysis,” Eur. J. Mech. A / Solids, 21, 441-463 (2002).CrossRef G. A. Holzapfel, T. C. Gasser, and M. Stadler, “A structural model for the viscoelastic behavior of arterial walls: Continuum formulation and finite element analysis,” Eur. J. Mech. A / Solids, 21, 441-463 (2002).CrossRef
7.
go back to reference W. J. Poole, J. D. Embury, S. MacEwen, and U. Kocks, “Large strain deformation of a copper-tungsten composite system. 1. Strain distribution,” Philosoph. Mag. A, 69, No. 4, 645-665 (1994).CrossRef W. J. Poole, J. D. Embury, S. MacEwen, and U. Kocks, “Large strain deformation of a copper-tungsten composite system. 1. Strain distribution,” Philosoph. Mag. A, 69, No. 4, 645-665 (1994).CrossRef
8.
go back to reference P. D. Nicolaou, H. R. Piehler, and S. Saigal, “Process parameter selection for the consolidation of continuous fiber reinforced composites using finite element simulations,” Int. J. Mech. Sci., 37, No. 7, 669-690 (1995).CrossRef P. D. Nicolaou, H. R. Piehler, and S. Saigal, “Process parameter selection for the consolidation of continuous fiber reinforced composites using finite element simulations,” Int. J. Mech. Sci., 37, No. 7, 669-690 (1995).CrossRef
9.
go back to reference M. Gotoh and A. B. M. Idris, “Finite-element simulation of deformation of fiber-reinforced materials in the plastic range. Model proposition and tensile behaviors,” JSME Int. J. Ser. A. Mech. Mater. Eng., 40, No. 2, 149-157 (1997). M. Gotoh and A. B. M. Idris, “Finite-element simulation of deformation of fiber-reinforced materials in the plastic range. Model proposition and tensile behaviors,” JSME Int. J. Ser. A. Mech. Mater. Eng., 40, No. 2, 149-157 (1997).
10.
go back to reference L. Banks-Sills and V. Leiderman, “Macro-mechanical material model for fiber reinforced metal matrix composites,” Composites: Part B., 30, Jan., 443-452 (1999). L. Banks-Sills and V. Leiderman, “Macro-mechanical material model for fiber reinforced metal matrix composites,” Composites: Part B., 30, Jan., 443-452 (1999).
11.
go back to reference Yu. V. Kokhanenko, “Numerical study of edge effects in layered composites under uniaxial loading,” Prikl. Mekh., 46, No. 5, 29-45 (2010). Yu. V. Kokhanenko, “Numerical study of edge effects in layered composites under uniaxial loading,” Prikl. Mekh., 46, No. 5, 29-45 (2010).
12.
go back to reference S. Sockalingam, J. W. Gillespie, and M. Keefe, “On the transverse compression response of Kevlar KM2 using fiberlevel finite element model,” Int. J. Solids Struct., 51, Jun., 2504-2517 (2014). S. Sockalingam, J. W. Gillespie, and M. Keefe, “On the transverse compression response of Kevlar KM2 using fiberlevel finite element model,” Int. J. Solids Struct., 51, Jun., 2504-2517 (2014).
13.
go back to reference V. M. Akhundov, “Forming of a toroidal body with a cross-ply arrangement of fibers on the basis of a two-level carcas theory,” Mech. Compos. Mater., 53, No. 2, 253-266 (2017).CrossRef V. M. Akhundov, “Forming of a toroidal body with a cross-ply arrangement of fibers on the basis of a two-level carcas theory,” Mech. Compos. Mater., 53, No. 2, 253-266 (2017).CrossRef
14.
go back to reference V. M. Akhundov, “A method for calculating the near-surface effect in piecewise-homogeneous bodies at large deformations on the basis of a two-level approach. materials,” Mech. Compos. Mater., 56, No. 2, 169-184 (2020).CrossRef V. M. Akhundov, “A method for calculating the near-surface effect in piecewise-homogeneous bodies at large deformations on the basis of a two-level approach. materials,” Mech. Compos. Mater., 56, No. 2, 169-184 (2020).CrossRef
15.
go back to reference F. L. Chernousko and V. P. Banichuk, Variational Problems of Mechanics and Control [in Russian], M.: Nauka (1973). F. L. Chernousko and V. P. Banichuk, Variational Problems of Mechanics and Control [in Russian], M.: Nauka (1973).
16.
go back to reference O. K. Zenkevich and K. Morgan, Finite Elements and Approximation [Russian translation], M.: Mir (1986). O. K. Zenkevich and K. Morgan, Finite Elements and Approximation [Russian translation], M.: Mir (1986).
17.
go back to reference V. M. Akhundov and M. M. Kostrova, “Nonlinear deformation of a piecewise-homogeneous cylinder under the influence of rotation,” Mech. Compos. Mater., 54, No. 2, 231-242 (2018).CrossRef V. M. Akhundov and M. M. Kostrova, “Nonlinear deformation of a piecewise-homogeneous cylinder under the influence of rotation,” Mech. Compos. Mater., 54, No. 2, 231-242 (2018).CrossRef
18.
go back to reference V. M. Akhundov, I. Yu. Naumova, and A. A. Zabrodskaya, “Elastoreinforced pipe from three layers with ring fibers under the influence of internal pressure,” Visnik Zaporiz. Nats. Univ., No. 1, 4-13 (2019). V. M. Akhundov, I. Yu. Naumova, and A. A. Zabrodskaya, “Elastoreinforced pipe from three layers with ring fibers under the influence of internal pressure,” Visnik Zaporiz. Nats. Univ., No. 1, 4-13 (2019).
19.
go back to reference T. Shup, Solving Engineering Problems on a Computer [Russian translation], M.: Mir, (1982). T. Shup, Solving Engineering Problems on a Computer [Russian translation], M.: Mir, (1982).
20.
go back to reference N. S. Bakhvalov, N. P. Zhidkov, and G. M. Kobelkov, Numerical Methods [in Russian], M.: Nauka (1987). N. S. Bakhvalov, N. P. Zhidkov, and G. M. Kobelkov, Numerical Methods [in Russian], M.: Nauka (1987).
21.
go back to reference J. Ortega and V. Reinboldt, Iterative Methods for Solving Nonlinear Systems of Equations with Many Unknowns [Russian translation], M.: Mir (1975). J. Ortega and V. Reinboldt, Iterative Methods for Solving Nonlinear Systems of Equations with Many Unknowns [Russian translation], M.: Mir (1975).
22.
go back to reference F. P. Vasiliev, Optimization Methods [in Russian], M.: Izd “Faktorial Press,” (2002). F. P. Vasiliev, Optimization Methods [in Russian], M.: Izd “Faktorial Press,” (2002).
23.
go back to reference A. I. Lurie, Nonlinear Elasticity Theory [in Russian], M.: Nauka (1980). A. I. Lurie, Nonlinear Elasticity Theory [in Russian], M.: Nauka (1980).
24.
go back to reference V. M. Akhundov, “Analysis of elastomeric composites based on fiber systems. 1. Development of a method for calculating composite materials,” Mech. Compos. Mater., 34, No. 6, 515-524 (1998).CrossRef V. M. Akhundov, “Analysis of elastomeric composites based on fiber systems. 1. Development of a method for calculating composite materials,” Mech. Compos. Mater., 34, No. 6, 515-524 (1998).CrossRef
25.
go back to reference V. Z. Parton and P. I. Perlin, Methods of mathematical elasticity theory [in Russian], M.: Nauka (1981). V. Z. Parton and P. I. Perlin, Methods of mathematical elasticity theory [in Russian], M.: Nauka (1981).
26.
go back to reference S. D. Ponomarev, Strength Calculations in Mechanical Engineering,” S. D. Ponomarev et al., 3, 32-41, Mashgiz, Moscow (1959). S. D. Ponomarev, Strength Calculations in Mechanical Engineering,” S. D. Ponomarev et al., 3, 32-41, Mashgiz, Moscow (1959).
Metadata
Title
The Impliсit Finite Difference Method in the Deformation Mechanics of Homogeneous and Piecewise Homogeneous Bodies
Author
V. M. Akhundov
Publication date
26-01-2022
Publisher
Springer US
Published in
Mechanics of Composite Materials / Issue 6/2022
Print ISSN: 0191-5665
Electronic ISSN: 1573-8922
DOI
https://doi.org/10.1007/s11029-022-10000-x

Other articles of this Issue 6/2022

Mechanics of Composite Materials 6/2022 Go to the issue

Premium Partners