Skip to main content
Log in

Numerical Solution of Three-Dimensional Stability Problems for Elastic Bodies

  • Published:
International Applied Mechanics Aims and scope

Abstract

A solution in Cartesian coordinates to plane and spatial stability problems for composites is obtained within the framework of the second variant of the three-dimensional linearized theory of stability of deformable bodies. Two mechanical models are used: a homogeneous anisotropic medium with averaged mechanical characteristics and a piecewise-homogeneous medium with orthotropic linearly elastic components. To solve the problems, a mesh approach is applied. Discrete models are constructed using the concept of a base scheme. The calculated results are analyzed

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

REFERENCES

  1. D. S. Akopyan, A. N. Guz, and A. V. Navoyan, “Constructing a theory of stability of mine openings,” Prikl. Mekh., 18, No. 5, 3–20 (1982).

    Google Scholar 

  2. N. S. Bakhvalov, N. P. Zhidkov, and S. M. Kobel'kov, Numerical Methods [in Russian], Nauka, Moscow (1987).

    Google Scholar 

  3. I. Yu. Babich, “Stability of the equilibrium of three-dimensional orthotropic bodies under small strains,” Prikl. Mekh., 8, No. 2, 16–24 (1972).

    Google Scholar 

  4. I. N. Bronshtein and K. A. Semendyaev, Handbook of Mathematics [in Russian], Nauka, Moscow (1986).

    Google Scholar 

  5. S. A. Batugin and R. K. Nirenburg, “Approximate relationship between the elastic constants of anisotropic rock and anisotropy parameters,” Fiz.-Technol. Probl. Razrab. Polezn. Iskop., No. 1, 7–11 (1972).

    Google Scholar 

  6. H. S. Katz and J. V. Milewski (eds.), Handbook of Fillers and Reinforcements for Plastics, Van Nastrand Reinhold Company, New York (1978).

    Google Scholar 

  7. A. N. Guz, “Stability analysis of elastic systems using three-dimensional linearized elastic equations,” Prikl. Mekh., 3, No. 2, 20–29 (1967).

    Google Scholar 

  8. A. N. Guz, “Stability of orthotropic bodies,” Prikl. Mekh., 3, No. 5, 40–51 (1967).

    Google Scholar 

  9. A. N. Guz, “General solutions to three-dimensional linearized equations of the theory of elastic stability,” Dop. AN URSR, Ser. A, No. 1, 56–60 (1967).

    Google Scholar 

  10. A. N. Guz, “Stability of three-dimensional elastic bodies,” Prikl. Mat. Mekh., 32, No. 5, 930–935 (1968).

    Google Scholar 

  11. A. N. Guz, “General solutions to three-dimensional linearized stability equations for elastoplastic bodies,” Dop. AN URSR, Ser. A, No. 4, 34–37 (1968).

    Google Scholar 

  12. A. N. Guz, “Stability of elastoplastic media,” Prikl. Mekh., 5, No. 8, 11–19 (1969).

    Google Scholar 

  13. A. N. Guz, “Bifurcation of the equilibrium state of a three-dimensional elastic isotropic body under large subcritical strains,” Prikl. Mat. Mekh., 34, No. 6, 1113–1125 (1970).

    Google Scholar 

  14. A. N. Guz, “The three-dimensional theory of deformation stability of materials with rheological properties,” Izv. AN SSSR, Mekh. Tverd. Tela, No. 6, 104–107 (1970).

    Google Scholar 

  15. A. N. Guz, “Some issues of stability of a nonlinear elastic body under finite and small subcritical strains,” Prikl. Mekh., 6, No. 4, 52–58 (1970).

    Google Scholar 

  16. A. N. Guz, Stability of Three-Dimensional Deformable Bodies [in Russian], Naukova Dumka, Kiev (1971).

    Google Scholar 

  17. A. N. Guz, “Spatial equilibrium bifurcation problems for a linearly elastic incompressible body under finite homogeneous strain,” Izv. AN SSSR, Mekh. Tverd. Tela, No. 6, 72–80 (1971).

    Google Scholar 

  18. A. N. Guz, “Variational principles in the three-dimensional theory of elastic stability,” Dop. AN URSR, Ser. A, No. 11, 1013–1016 (1971).

    Google Scholar 

  19. A. N. Guz, “The three-dimensional theory of elasticity under finite subcritical strains,” Prikl. Mekh., 8, No. 12, 15–44 (1972).

    Google Scholar 

  20. A. N. Guz, “Variational principles in three-dimensional linearized problems of the theory of elasticity under large initial strains,” in: Continuum Mechanics and Related Problems [in Russian], Nauka, Moscow (1972), pp. 169–174.

    Google Scholar 

  21. A. N. Guz, “Stability problems in the mechanics of rock,” in: Problems of the Mechanics of Rock [in Russian], Nauka, Alma-Ata (1972), pp. 27–35.

    Google Scholar 

  22. A. N. Guz, Stability of Elastic Bodies under Finite Strains [in Russian], Naukova Dumka, Kiev (1973).

    Google Scholar 

  23. A. N. Guz, “Variational principles of three-dimensional stability problems for inelastic bodies,” Dop. AN URSR, Ser. A, No. 11, 1008–1012 (1973).

    Google Scholar 

  24. A. N. Guz, Fundamentals of the Theory of Stability of Mine Openings [in Russian], Naukova Dumka, Kiev (1977).

    Google Scholar 

  25. A. N. Guz, Stability of Elastic Bodies under Triaxial Compression [in Russian], Naukova Dumka, Kiev (1979).

    Google Scholar 

  26. A. N. Guz, “Variational principles of the three-dimensional theory of stability of deformable bodies under follower loads,” Dokl. AN SSSR, 246, No. 6, 1314–1316 (1979).

    Google Scholar 

  27. A. N. Guz, “Stability problems for mine openings,” Dokl. AN SSSR, 253, No. 3, 1314–1316 (1980).

    Google Scholar 

  28. A. N. Guz, Mechanics of Brittle Fracture of Initially Stressed Materials [in Russian], Naukova Dumka, Kiev (1983).

    Google Scholar 

  29. A. N. Guz, Fundamentals of the Three-Dimensional Theory of Stability of Deformable Bodies [in Russian], Vyshcha Shkola, Kiev (1986).

    Google Scholar 

  30. A. N. Guz, Fracture Mechanics of Compressed Composites [in Russian], Naukova Dumka, Kiev (1990).

    Google Scholar 

  31. A. N. Guz, “The exact solution of a plane problem on fracture of a material compressed along coplanar cracks,” Dokl. AN SSSR, 310, No. 3, 563–566 (1990).

    Google Scholar 

  32. A. N. Guz, “Brittle fracture of initially stressed materials,” Vol. 2 of the four-volume five-book series A. N. Guz (ed.), Nonclassical Problems of Fracture Mechanics [in Russian], Naukova Dumka, Kiev (1991).

    Google Scholar 

  33. A. N. Guz, “Constructing a theory of stability of mine openings,” in: Collection, Issued on the Occasion of the 70th Anniversary of L. V. Ershov [in Russian], Moscow (2000), pp. 7–12.

  34. A. N. Guz and E. Yu. Gladun, “Plane problem of three-dimensional stability of a cracked plate,” Int. Appl. Mech., 37, No. 10, 1281–1298 (2001).

    Google Scholar 

  35. A. N. Guz and I. Yu. Babich, Three-Dimensional Theory of Stability of Rods, Plates, and Shells [in Russian], Vyshcha Shkola, Kiev (1980).

    Google Scholar 

  36. A. N. Guz and I. Yu. Babich, Three-Dimensional Theory of Stability of Deformable Bodies, Vol. 4 of the six-volume series Spatial Elastic and Plastic Problems [in Russian], Naukova Dumka, Kiev (1985).

    Google Scholar 

  37. A. N. Guz, S. Yu. Babich, and V. B. Rudnitskii, Contact Interaction of Elastic Bodies with Initial Stresses [in Ukrainian], Vyshcha Shkola, Kiev (1995).

    Google Scholar 

  38. A. N. Guz and L. V. Deriglazov, “Stability of the anisotropic rock mass near two parallel mine tunnels,” Dokl. AN USSR, 325, No. 3, 450–454 (1992).

    Google Scholar 

  39. A. N. Guz, M. Sh. Dyshel', and V. M. Nazarenko, Fracture and Stability of Cracked Materials, Vol. 4, Book 1 of the four-volume five-book series A. N. Guz (ed.), Nonclassical Problems of Fracture Mechanics [in Russian], Naukova Dumka, Kiev (1992).

    Google Scholar 

  40. A. N. Guz, V. S. Zelenskii, and Yu. V. Kokhanenko, “Solution of three-dimensional problems of elastic stability of plates and rods in an inhomogeneous subcritical state,” Mekh. Komp. Mater., No. 1, 49–52 (1980).

    Google Scholar 

  41. A. N. Guz and Yu. V. Kokhanenko, “Solution of plane problems of three-dimensional elastic stability of plates in an inhomogeneous subcritical state,” Prikl. Mekh., 13, No. 12, 63–72 (1977).

    Google Scholar 

  42. A. N. Guz and Yu. V. Kokhanenko, “Brittle fracture of composites with crashed ends,” Dokl. AN SSSR, 296, No. 4, 805–808 (1987).

    Google Scholar 

  43. I. A. Guz and Yu. V. Kokhanenko, “Stability of a laminated composite compressed along microcracks,” Prikl. Mekh., 29, No. 9, 30–38 (1993).

    Google Scholar 

  44. A. N. Guz and G. G. Kuliev, “On the theory of stability of boreholes,” Prikl. Mekh., 19, No. 3, 10–17 (1983).

    Google Scholar 

  45. Yu. V. Kokhanenko, Three-Dimensional Problem on Stability of a Plate in Inhomogeneous Subcritical State [in Ukrainian], Manuscript No. 4529.76, deposited at VINITI (1976).

  46. Yu. V. Kokhanenko, “Applying the finite-difference method to an elastic-stability problem,” Dokl. AN USSR, Ser. A, No. 7, 537–539 (1973).

    Google Scholar 

  47. Yu. V. Kokhanenko, “Solution of a three-dimensional stability problem for plates in an inhomogeneous subcritical state,” Prikl. Mekh., 12, No. 2, 117–119 (1976).

    Google Scholar 

  48. Yu. V. Kokhanenko, “Numerical solution of elasticity and three-dimensional stability problems for piecewise-inhomogeneous media,” Prikl. Mekh., 22, No. 11, 46–54 (1986).

    Google Scholar 

  49. Yu. V. Kokhanenko, “One method of solving problems of three-dimensional stability of ribbon composites,” Dokl. AN USSR, Ser. A, No. 2, 31–33 (1989).

    Google Scholar 

  50. Yu. V. Kokhanenko, “Three-dimensional stability of skewed composite rods,” Dokl. AN USSR, Ser. A, No. 8, 36–39 (1989).

    Google Scholar 

  51. Yu. V. Kokhanenko, “Three-dimensional stability of stiffened composite plates,” Prikl. Mekh., 26, No. 1, 127–129 (1990).

    Google Scholar 

  52. Yu. V. Kokhanenko, “Numerical solution of a three-dimensional stability problem for a mine tunnel of rectangular cross section (piecewise-homogeneous model),” Dokl. AN USSR, Ser. A, No. 2, 38–41 (1990).

    Google Scholar 

  53. Yu. V. Kokhanenko, “Mesh analysis of composites reinforced with rectangular fibers,” Dokl. AN USSR, Ser. A, No. 9, 60–64 (1993).

    Google Scholar 

  54. Yu. V. Kokhanenko, “On one method of solving three-dimensional stability problems for laminated composites,” Prikl. Mekh., 34, No. 3, 45–56 (1998).

    Google Scholar 

  55. Yu. V. Kokhanenko, “Solution of one class of problems arising in stability analysis of laminated composites,” Prikl. Mekh., 34, No. 5, 48–56 (1998).

    Google Scholar 

  56. Yu. V. Kokhanenko, “Numerical study of three-dimensional stability problems for laminated and ribbon-reinforced composites,” Int. Appl. Mech., 37, No. 3, 317–345 (2001).

    Google Scholar 

  57. Yu. V. Kokhanenko and V. S. Zelenskii, “Numerical solution of a linearized three-dimensional stability problem for connected rectangular plates,” Dop. NAN Ukrainy, No. 9, 62–65 (1998).

  58. G. G. Kuliev, Fracture and Stability of Three-Dimensional Cracked Bodies and Some Related Problems of Mining an Oil Mechanical Engineering [in Russian], Élm, Baku (1983).

    Google Scholar 

  59. L. P. Khoroshun (ed.), Statistical Mechanics and Effective Properties of Materials, Vol. 3 of the 12-volume series A. N. Guz (ed.), Mechanics of Composites [in Russian], Naukova Dumka, Kiev (1993).

  60. J. M. Ortega and W. G. Poole, Jr., An Introduction to Numerical Methods for Differential Equations, Pitman Publishing Inc., Marshfield, Massachusetts (1981).

    Google Scholar 

  61. B. N. Parlett, The Symmetric Eigenvalue Problem, Englewood Cliffs, New Jersey (1980).

    Google Scholar 

  62. A. A. Samarskii and E. S. Nikolaev, Methods for Solution of Mesh Equations [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  63. V. M. Bystrov, “Analysis of the decay of edge effects in laminated materials on the basis of a representative element,” Int. Appl. Mech., 36, No. 6, 826–835 (2000).

    Google Scholar 

  64. E. Yu. Gladun, “Dependence of the load on the geometric characteristics of a hinged plate with a crack,” Int. Appl. Mech., 36, No. 9, 1225–1234 (2000).

    Google Scholar 

  65. A. N. Guz, “On numerical methods in three-dimensional deformable bodies stability,” in: Computational Mechanics, Tokyo (1986), pp. 169–176.

  66. A. N. Guz, “On the theory of stability of underground mine working,” in: Mechanics of Joined and Faulted Rock, A. A. Balkema, Rotterdam (1990), pp. 803–807.

    Google Scholar 

  67. A. N. Guz, Fundamentals of the Three-Dimensional Theory of Stability of Deformable Bodies, Springer-Verlag, Berlin (1999).

    Google Scholar 

  68. A. N. Guz, “Constructing the three-dimensional theory of stability of deformable bodies,” Int. Appl. Mech., 37, No. 1, 1–37 (2001).

    Google Scholar 

  69. A. N. Guz and I. A. Guz, “On the theory of stability of laminated composites,” Int. Appl. Mech., 35, No. 4, 323–329 (1999).

    Google Scholar 

  70. A. N. Guz, V. A. Dekret, and Yu. V. Kokhanenko, “Plane problems of stability of composite materials with finite-size filler,” Mech. Comp. Mater., 36, No. 1, 49–54 (2000).

    Google Scholar 

  71. A. N. Guz, Yu. N. Lapusta, and Yu. A. Mamzenko, “Stability of two fibers in an elastoplastic matrix under compression,” Int. Appl. Mech., 34, No. 5, 405–414 (1998).

    Google Scholar 

  72. Yu. V. Kokhanenko, “Finite-element solution of plane problems of the linear elasticity theory of composites,” Int. Appl.Mech., 34, No. 10, 987–997 (1998).

    Google Scholar 

  73. Yu. V. Kokhanenko, “Discrete models of problems in the elastic theory of composites in circular cylindrical coordinates. 1. Three-dimensional problems,” Int. Appl. Mech., 36, No. 8, 1067–1076 (2000).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Guz, A.N., Kokhanenko, Y.V. Numerical Solution of Three-Dimensional Stability Problems for Elastic Bodies. International Applied Mechanics 37, 1369–1399 (2001). https://doi.org/10.1023/A:1014261430281

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1014261430281

Keywords

Navigation