Skip to main content
SpringerLink
Log in

The Art of Modeling in Solid Mechanics

  • Chapter
  • First Online:
The Art of Modeling Mechanical Systems

Part of the book series: CISM International Centre for Mechanical Sciences ((CISM,volume 570))

Abstract

Modeling is one of the main tasks in engineering to predict the behavior and response of assemblies, structures, or vehicles. This contribution is aimed at modeling in solid mechanics. Due to the necessity to use numerical methods for the solution of most theoretical models it will focus on theoretical models as well as on numerical simulation models associated with engineering applications in solid mechanics.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aboudi, J. (1991). Mechanics of composite materials - a unified micromechanical approach. Amsterdam: Elsevier.

    MATH  Google Scholar 

  2. Argyris, J. H. (1960). Energy theorems and structural analysis. London: Butterworth.

    Book  Google Scholar 

  3. Argyris, J. H., & Scharpf, D. W. (1968). The SHEBA family of shell elements for the matrix displacement method. The Aeronautical Journal of the Royal Aeronautical Society, 72, 873–883.

    Google Scholar 

  4. Atluri, S. N. (1984). On constitutive relations at finite strain: Hypo-elasticity and elasto-plasticity with isotropic and kinematic hardening. Computer Methods in Applied Mechanics and Engineering, 43, 137–171.

    Article  MATH  Google Scholar 

  5. Basar, Y., & Ding, Y. (1990). Finite-rotation elements for nonlinear analysis of thin shell structures. International Journal of Solids and Structures, 26, 83–97.

    Article  MATH  Google Scholar 

  6. Basar, Y., & Ding, Y. (1996). Shear deformation models for large strain shell analysis. International Journal of Solids and Structures, 34, 1687–1708.

    Article  MATH  Google Scholar 

  7. Bathe, K. J. (1982). Finite element procedures in engineering analysis. Englewood Cliffs: Prentice-Hall.

    Google Scholar 

  8. Bathe, K. J. (1996). Finite element procedures. Englewood Cliffs: Prentice-Hall.

    MATH  Google Scholar 

  9. Bathe, K. J., & Bolourchi, S. (1979). Large displacement analysis of three-dimensional beam structures. International Journal for Numerical Methods in Engineering, 14, 961–986.

    Article  MATH  Google Scholar 

  10. Bazant, Z. P., & Cedolin, L. (2003). Stability of structures: Elastic, inelastic, fracture, and damage theories. New York: Dover Publications.

    MATH  Google Scholar 

  11. Becker, A. (1985). Berechnung ebener Stabtragwerke nach der Flies̈gelenktheorie II. Ordnung unter Berücksichtigung der Normal- und Querkraftinteraktion mit Hilfe der Methode der Finiten Elemente. Technical report, Diplomarbeit am Institut für Baumechanik und Numerische Mechanik der Universität Hannover.

    Google Scholar 

  12. Belytschko, T., Ong, J. S. J., Liu, W. K., & Kennedy, J. M. (1984). Hourglass control in linear and nonlinear problems. Computer Methods in Applied Mechanics and Engineering, 43, 251–276.

    Article  MATH  Google Scholar 

  13. Benson, D. J., Bazilevs, Y., Hsu, M. C., & Hughes, T. J. R. (2011). A large deformation, rotation-free, isogeometric shell. Computer Methods in Applied Mechanics and Engineering, 200, 1367–1378.

    Article  MathSciNet  MATH  Google Scholar 

  14. Bischoff, M., & Ramm, E. (1997). Shear deformable shell elements for large strains and rotations. International Journal for Numerical Methods in Engineering, 40, 4427–4449.

    Article  MATH  Google Scholar 

  15. Bischoff, M., Wall, W. A., Bletzinger, K. U., & Ramm, E. (2004). Models and finite elements for thin-walled structures. In E. Stein, R. de Borst, & T. J. R. Hughes (Eds.), Encyclopedia of computational mechanics (pp. 59–137). Chichester: Wiley.

    Google Scholar 

  16. Bouclier, R., Elguedj, T., & Combescure, A. (2012). Locking free isogeometric formulations of curved thick beams. Computer Methods in Applied Mechanics and Engineering, 245, 144–162.

    Article  MathSciNet  Google Scholar 

  17. Braess, D. (2007). Finite elements: Theory, fast solvers, and applications in solid mechanics. Cambridge: Cambridge University Press.

    Book  MATH  Google Scholar 

  18. Brenner, S. C., & Scott, L. R. (2002). The mathematical theory of finite element methods. New York: Springer.

    Book  MATH  Google Scholar 

  19. Büchter, N., & Ramm, E. (1992). Shell theory versus degeneration - a comparison in large rotation finite element analysis. International Journal for Numerical Methods in Engineering, 34, 39–59.

    Article  MATH  Google Scholar 

  20. Büchter, N., Ramm, E., & Roehl, D. (1994). Three-dimensional extension of non-linear shell formulation based on the enhanced assumed strain concept. International Journal for Numerical Methods in Engineering, 37, 2551–2568.

    Article  MATH  Google Scholar 

  21. Campello, E. M. B., Pimenta, P. M., & Wriggers, P. (2003). A triangular finite shell element based on a fully nonlinear shell formulation. Computational Mechanics, 31, 505–518.

    Article  MATH  Google Scholar 

  22. Cazzani, A., Malagù, M., & Turco, E. (2014). Isogeometric analysis of plane-curved beams. Mathematics and Mechanics of Solids. doi:10.1177/1081286514531265.

  23. Chadwick, P. (1999). Continuum mechanics, concise theory and problems. Mineola: Dover Publications.

    Google Scholar 

  24. Chavan, K., & Wriggers, P. (2004). Consistent coupling of beam and shell models for thermo-elastic analysis. International Journal for Numerical Methods in Engineering, 59, 1861–1878.

    Article  MATH  Google Scholar 

  25. Chinesta, F., Ammar, A., & Cueto, E. (2010). Recent advances and new challenges in the use of the proper generalized decomposition for solving multidimensional models. Archives of Computational methods in Engineering, 17, 327–350.

    Article  MathSciNet  MATH  Google Scholar 

  26. Christensen, R. M. (1980). A nonlinear theory of viscoelastocity for application to elastomers. Journal of Applied Mechanics, 47, 762–768.

    Article  MATH  Google Scholar 

  27. Cirak, F., Ortiz, M., & Schröder, P. (2000). Subdivision surfaces: A new paradigm for thin shell finite-element analysis. International Journal for Numerical Methods in Engineering, 47, 2039–2072.

    Article  MATH  Google Scholar 

  28. Cottrell, J. A., Hughes, T. J. R., & Bazilevs, Y. (2009). Isogeometric analysis: Toward integration of CAD and FEA. New York: Wiley.

    Book  Google Scholar 

  29. Courant, R. (1943). Variational methods for the solution of problems of equilibrium and vibrations. Bulletin of the American Mathematical Society, 49, 1–23.

    Article  MathSciNet  MATH  Google Scholar 

  30. Crisfield, M. A. (1991). Non-linear finite element analysis of solids and structures (Vol. 1). Chichester: Wiley.

    MATH  Google Scholar 

  31. Crisfield, M. A. (1997). Non-linear finite element analysis of solids and structures (Vol. 2). Chichester: Wiley.

    MATH  Google Scholar 

  32. de Souza Neto, E. A., Peric, D., & Owen, D. R. J. (2008). Computational methods for plasticity, theory and applications. Chichester: Wiley.

    Book  Google Scholar 

  33. Desai, C. S., & Siriwardane, H. J. (1984). Constitutive laws for engineering materials. Englewood Cliffs: Prentice-Hall.

    MATH  Google Scholar 

  34. Donnell, L. H. (1976). Beams, plates and shells. New York: McGraw-Hill Companies.

    MATH  Google Scholar 

  35. Düster, A., Bröker, H., & Rank, E. (2001). The p-version of the finite element method for three-dimensional curved thin walled structures. International Journal for Numerical Methods in Engineering, 52, 673–703.

    Article  MATH  Google Scholar 

  36. Dvorkin, E. N., Onate, E., & Oliver, J. (1988). On a nonlinear formulation for curved Timoshenko beam elements considering large displacement/rotation increments. International Journal for Numerical Methods in Engineering, 26, 1597–1613.

    Article  MATH  Google Scholar 

  37. Eberlein, R., & Wriggers, P. (1999). Finite element concepts for finite elastoplastic strains and isotropic stress response in shells: Theoretical and computational analysis. Computer Methods in Applied Mechanics and Engineering, 171, 243–279.

    Article  MATH  Google Scholar 

  38. Ehrlich, D., & Armero, F. (2005). Finite element methods for the analysis of softening plastic hinges in beams and frames. Computational Mechanics, 35(4), 237–264.

    Article  MATH  Google Scholar 

  39. Ekh, M., Johansson, A., Thorberntsson, H., & Josefson, B. (2000). Models for cyclic ratchetting plasticity integration and calibration. Journal of Engineering Materials and Technology, 122, 49.

    Article  Google Scholar 

  40. Fish, J., & Shek, K. (2000). Multiscale analysis for composite materials and structures. Composites Science and Technology, 60, 2547–2556.

    Article  Google Scholar 

  41. Flügge, W. (2013). Stresses in shells. New York: Springer Science.

    MATH  Google Scholar 

  42. Gruttmann, F., & Wagner, W. (2005). A linear quadrilateral shell element with fast stiffness computation. Computer Methods in Applied Mechanics and Engineering, 194(39–41), 4279–4300.

    Article  MATH  Google Scholar 

  43. Gruttmann, F., Sauer, R., & Wagner, W. (1998). A geometrical non-linear eccentric 3D-beam element with arbitrary cross-sections. Computer Methods in Applied Mechanics and Engineering, 160, 383–400.

    Article  MathSciNet  MATH  Google Scholar 

  44. Gruttmann, F., Sauer, R., & Wagner, W. (2000). Theory and numerics of three-dimensional beams with elastoplastic material behaviour. International Journal for Numerical Methods in Engineering, 48, 1675–1702.

    Article  MATH  Google Scholar 

  45. Gurson, A. L. (1977). Continuum theory of ductile rupture by void nucleation and growth, part i. Journal Engineering Material Technology, 99, 2–15.

    Article  Google Scholar 

  46. Hain, M., & Wriggers, P. (2008a). Computational homogenization of micro-structural damage due to frost in hardened cement paste. Finite Elements in Analysis and Design, 44, 223–244.

    Google Scholar 

  47. Hain, M., & Wriggers, P. (2008b). On the numerical homogenization of hardened cement paste. Computational Mechanics, 42, 197–212.

    Google Scholar 

  48. Hashin, D. (1983). Analysis of composite materials, a survey. Journal of Applied Mechanics, 50, 481–505.

    Article  MATH  Google Scholar 

  49. Hauptmann, R., & Schweizerhof, K. (1998). A systematic development of ‘solid-shell’ element formulation for linear and nonlinear analyses employing only displacement degree of freedom. International Journal for Numerical Methods in Engineering, 42, 49–69.

    Article  MATH  Google Scholar 

  50. Heisserer, U., Hartmann, S., Yosibash, Z., & Düster, A. (2007). On volumetric locking-free behavior of p-version finite elements under finite deformations. Communications in Numerical Methods in Engineering. Accepted for publication.

    Google Scholar 

  51. Henning, A. (1975). Traglastberechnung ebener Rahmen – Theorie II. Ordnungschweizerhof und Interaktion. Technical report 75–12, Institut für Statik, Braunschweig.

    Google Scholar 

  52. Hofstetter, G., & Mang, H. A. (1995). Computational mechanics of reinforced concrete structures. Berlin: Vieweg.

    MATH  Google Scholar 

  53. Holzapfel, G. A. (2000). Nonlinear solid mechanics. Chichester: Wiley.

    MATH  Google Scholar 

  54. Hughes, T. J. R., Cottrell, J. A., & Bazilevs, Y. (2005). Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 194(39–41), 4135–4195.

    Article  MathSciNet  MATH  Google Scholar 

  55. Hughes, T. R. J. (1987). The finite element method. Englewood Cliffs: Prentice Hall.

    MATH  Google Scholar 

  56. Hull, D., & Clyne, T. (1996). An introduction to composite materials. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  57. Ibrahimbegovic, A. (1995). A finite element implementation of geometrically nonlinear Reissner’s beam theory: Three-dimensional curved beam elements. Computer Methods in Applied Mechanics and Engineering, 122, 11–26.

    Article  MATH  Google Scholar 

  58. Idelsohn, S. R., & Cardona, A. (1984). Reduction methods and explicit time integration. Advances in Engineering Software, 6, 36–44.

    Article  Google Scholar 

  59. Ivannikov, V., Tiago, C., & Pimenta, P. (2014). Tuba finite elements: Application to the solution of a nonlinear kirchhoff-love shell theory. Shell Structures: Theory and Applications, 3, 81–84.

    Google Scholar 

  60. Jelenic, G., & Saje, M. (1995). A kinematically exact space finite strain beam model - finite element formulations by generalised virtual work principle. Computer Methods in Applied Mechanics and Engineering, 120, 131–161.

    Article  MathSciNet  MATH  Google Scholar 

  61. Johansson, G., Ekh, M., & Runesson, K. (2005). Computational modeling of inelastic large ratcheting strains. International Journal of Plasticity, 21(5), 955–980.

    Article  MATH  Google Scholar 

  62. Johnson, K. L. (1985). Contact mechanics. Cambridge: Cambridge University Press.

    Book  MATH  Google Scholar 

  63. Jones, R. M. (1975). Mechanics of composite materials (Vol. 1). New York: McGraw-Hill.

    Google Scholar 

  64. Kahn, R. (1987). Finite-Element-Berechnungen ebener Stabwerke mit Fliessgelenken und grossen Verschiebungen. Technical report F 87/1, Forschungs- und Seminarberichte aus dem Bereich der Mechanik der Universität Hannover.

    Google Scholar 

  65. Khan, A. S., & Huang, S. (1995). Continuum theory of plasticity. Chichester: Wiley.

    MATH  Google Scholar 

  66. Kiendl, J., Bletzinger, K. U., Linhard, J., & Wüchner, R. (2009). Isogeometric shell analysis with Kirchhoff-Love elements. Computer Methods in Applied Mechanics and Engineering, 198, 3902–3914.

    Article  MathSciNet  MATH  Google Scholar 

  67. Kirchhoff G. R. (1850) Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. Journal für die Reine und Angewandte Mathematik.

    Google Scholar 

  68. Kirsch, U., Bogomolni, M., & Sheinman, I. (2005). Nonlinear dynamic reanalysis for structural optimization. In D. H. van Campen, M. D. Lazurko, & W. P. J. M. van den Oever (Eds.), Proceedings of the fifth EUROMECH nonlinear dynamics conference (ENOC). Eindhoven.

    Google Scholar 

  69. Koiter, W. T. (1960). A consistent first approximation in the general theory of thin elastic shells. In W. T. Koiter (Ed.), The theory of thin elastic shells (pp. 12–33). Amsterdam: North-Holland.

    Google Scholar 

  70. Kolymbas, D. (1991). An outline of hypoplasticity. Archive of Applied Mechanics, 61, 143–151.

    MATH  Google Scholar 

  71. Konyukhov, A., & Schweizerhof, K. (2012). Geometrically exact theory for contact interactions of 1d manifolds. Algorithmic implementation with various finite element models. Computer Methods in Applied Mechanics and Engineering, 205, 130–138.

    Article  MathSciNet  MATH  Google Scholar 

  72. Korelc, J., & Wriggers, P. (1996). Consistent gradient formulation for a stable enhanced strain method for large deformations. Engineering Computations, 13, 103–123.

    Article  Google Scholar 

  73. Korelc, J., Solinc, U., & Wriggers, P. (2010). An improved EAS brick element for finite deformation. Computational Mechanics, 46, 641–659.

    Article  MATH  Google Scholar 

  74. Krysl, P., Lall, S., & Marsden, J. E. (2001). Dimensional model reduction in non-linear finite element dynamics of solids and structures. International Journal for Numerical Methods in Engineering, 51, 479–504.

    Article  MathSciNet  MATH  Google Scholar 

  75. Ladeveze, P., Passieux, J., & Neron, D. (2010). The latin multiscale computational method and the proper generalized decomposition. Computer Methods in Applied Mechanics and Engineering, 199, 1287–1296.

    Article  MathSciNet  MATH  Google Scholar 

  76. Larsson, R., Runesson, K., & Ottosen, N. (1993). Discontinuous displacement approximation for capturing plastic localization. International Journal for Numerical Methods in Engineering, 36, 2087–2105.

    Article  MATH  Google Scholar 

  77. Laursen, T. A. (2002). Computational contact and impact mechanics. Berlin: Springer.

    MATH  Google Scholar 

  78. Leger, P. (1993). Mode superposition methods. In M. Papadrakakis (Ed.), Solving large-scale problems in mechanics (pp. 225–257). New York: Wiley.

    Google Scholar 

  79. Lehmann, E., Löhnert, S., & Wriggers, P. (2012). About the microstructural effects of polycrystalline materials and their macroscopic representation at finite deformation. PAMM, 12, 275–276.

    Article  Google Scholar 

  80. Lehmann, E., Faßmann, D., Loehnert, S., Schaper, M., & Wriggers, P. (2013). Texture development and formability prediction for pre-textured cold rolled body-centred cubic steel. International Journal of Engineering Sciences, 68, 24–37.

    Article  MathSciNet  Google Scholar 

  81. Lemaitre, J. (1996). A course on damage mechanics. Berlin: Springer.

    Book  MATH  Google Scholar 

  82. Leppin, C., & Wriggers, P. (1997). Numerical simulations of the behaviour of cohesionless soil. In D. R. J. Owen, E. Hinton, & E. Onate (Eds.), Proceedings of COMPLAS 5. Barcelona: CIMNE.

    Google Scholar 

  83. Liu, W., Karpov, E., & Park, H. (2005). Nano mechanics and materials: Theory multiscale analysis and applications. Chichester: Wiley.

    Google Scholar 

  84. Love, A. E. H. (1927). A treatise on the mathematical theory of elasticity (Vol. Fourth). Cambrige: University Press.

    MATH  Google Scholar 

  85. Lubliner, J. (1985). A model of rubber viscoelasticity. Mechanics Research Communications, 12, 93–99.

    Article  Google Scholar 

  86. Lubliner, J. (1990). Plasticity theory. London: MacMillan.

    MATH  Google Scholar 

  87. Lumpe, G. (1982). Geometrisch nichtlineare berechnung von räumlichen stabwerken. Technical report 28, Institut für Statik, Universität Hannover.

    Google Scholar 

  88. MacNeal, R., & Harder, R. (1985). A proposed standard set of problems to test finite element accuracy. Finite Elements in Analysis and Design, 1, 3–20.

    Article  Google Scholar 

  89. Mäkinen, J. (2007). Total Lagrangian Reissner’s geometrically exact beam element without singularities. International Journal for Numerical Methods in Engineering, 70, 1009–1048.

    Article  MathSciNet  MATH  Google Scholar 

  90. Meyer, M., & Matthies, H. (2003). Efficient model reduction in non-linear dynamics using the Karhunen-Loeve expansion and dual-weighted-residual methods. Computational Mechanics, 31, 179–191.

    Article  MATH  Google Scholar 

  91. Miehe, C. (1998). A theoretical and computational model for isotropic elastoplastic stress analysis in shells at large strains. Computer Methods in Applied Mechanics and Engineering, 155, 193–233.

    Article  MathSciNet  MATH  Google Scholar 

  92. Miehe, C., & Schröder, J. (1994). Post-critical discontinous localization analysis of small-strain softening elastoplastic solids. Archive of Applied Mechanics, 64, 267–285.

    MATH  Google Scholar 

  93. Miehe, C., Schröder, J., & Schotte, J. (1999). Computational homogenization analysis in finite plasticity simulation of texture development in polycrystalline materials. Computer Methods in Applied Mechanics and Engineering, 171, 387–418.

    Article  MathSciNet  MATH  Google Scholar 

  94. Naghdi, P. M. (1972). The theory of shells. Handbuch der Physik, mechanics of solids II (Vol. VIa/2). Berlin: Springer.

    Google Scholar 

  95. Néron, D., & Ladevèze, P. (2010). Proper generalized decomposition for multiscale and multiphysics problems. Archives of Computational Methods in Engineering, 17, 351–372.

    Article  MathSciNet  MATH  Google Scholar 

  96. Nickell, R. E. (1976). Nonlinear dynamics by mode superposition. Computer Methods in Applied Mechanics and Engineering, 7, 107–129.

    Article  MathSciNet  MATH  Google Scholar 

  97. Niemunis, A., Wichtmann, T., & Triantafyllidis, T. (2005). A high-cycle accumulation model for sand. Computers and Geotechnics, 32, 245–263.

    Article  MATH  Google Scholar 

  98. Oden, J. T., Prudhomme, S., Hammerand, D. C., & Kuczma, M. S. (2001). Modeling error and adaptivity in nonlinear continuum mechanics. Computer Methods in Applied Mechanics and Engineering, 190, 8883–6684.

    Article  MathSciNet  MATH  Google Scholar 

  99. Ogden, R. W. (1984). Non-linear elastic deformations. Chichester: Ellis Horwood and Wiley.

    MATH  Google Scholar 

  100. Oliver, J. (1995). Continuum modeling of strong discontinuities in solid mechanics using damage models. Computational Mechanics, 17, 49–61.

    Article  MATH  Google Scholar 

  101. Onate, E., & Cervera, M. (1993). Derivation of thin plate bending elements with one degree of freedom per node: A simple three node triangle. Engineering Computations, 10(6), 543–561.

    Article  Google Scholar 

  102. Oran, C., & Kassimali, A. (1976). Large deformations of framed structures under static and dynamic loads. Computers and Structures, 6, 539–547.

    Article  MATH  Google Scholar 

  103. Perzyna, P. (1966). Fundamental problems in viscoplasticity. Advances in Applied Mechanics, 9, 243–377.

    Article  Google Scholar 

  104. Pflüger, A. (1981). Elementare Schalenstatik. New York: Springer.

    Book  MATH  Google Scholar 

  105. Pian, T. H. H. (1964). Derivation of element stiffness matrices by assumed stress distributions. AIAA Journal, 2(7), 1333–1336.

    Article  Google Scholar 

  106. Pian, T. H. H., & Sumihara, K. (1984). Rational approach for assumed stress finite elements. International Journal for Numerical Methods in Engineering, 20, 1685–1695.

    Article  MATH  Google Scholar 

  107. Pietraszkiewicz, W. (1978). Geometrically nonlinear theories of thin elastic shells. Technical report 55, Mitteilungen des Instituts für Mechanik der Ruhr-Universität Bochum.

    Google Scholar 

  108. Pimenta, P., & Yojo, T. (1993). Geometrically exact analysis of spatial frames. Applied Mechanics Review, 46, 118–128.

    Article  Google Scholar 

  109. Pimenta, P. M., Campello, E. M. B., & Wriggers, P. (2004). A fully nonlinear multi-parameter shell model with thickness variation and a triangular shell finite element formulation. Computational Mechanics, 34, 181–193.

    Article  MATH  Google Scholar 

  110. Pimenta, P. M., Campello, E. M. B., & Wriggers, P. (2008). An exact conserving algorithm for nonlinear dynamics with rotational DOFs and general hyperelasticity. Part 1: Rods. Computational Mechanics, 42, 715–732.

    Article  MathSciNet  MATH  Google Scholar 

  111. Radermacher, A., & Reese, S. (2014). Model reduction in elastoplasticity: Proper orthogonal decomposition combined with adaptive sub-structuring. Computational Mechanics, 54, 677–687.

    Article  MathSciNet  MATH  Google Scholar 

  112. Ramm, E. (1976). Geometrisch nichtlineare Elastostatik und Finite Elemente. Technical report Nr. 76-2, Institut für Baustatik der Universität Stuttgart.

    Google Scholar 

  113. Rankin, C. C., & Brogan, F. A. (1984). An element independent corotational procedure for the treatment of large rotations. In L. H. Sobel & K. Thomas (Eds.), Collapse Analysis of Structures (pp. 85–100). New York: ASME.

    Google Scholar 

  114. Reali, A., & Gomez, H. (2015). An isogeometric collocation approach for Bernoulli-Euler beams and Kirchhoff plates. Computer Methods in Applied Mechanics and Engineering, 284, 623–636.

    Article  MathSciNet  Google Scholar 

  115. Reddy, J. N. (1984). A simple higher-order theory for laminated composite plates. Journal of Applied Mechanics, 51, 745–752.

    Article  MATH  Google Scholar 

  116. Reddy, J. N. (2004). Mechanics of laminated composite plates and shells: Theory and analysis. Boca Raton: CRC Press.

    MATH  Google Scholar 

  117. Reese, S. (2005). On a physically stabilized one point finite element formulation for three-dimensional finite elasto-plasticity. Computer Methods in Applied Mechanics and Engineering, 194, 4685–4715.

    Article  MATH  Google Scholar 

  118. Reese, S., & Govindjee, S. (1998). A theory of finite viscoelasticity and numerical aspects. International Journal of Solids and Structures, 35, 3455–3482.

    Article  MATH  Google Scholar 

  119. Reissner, E. (1972). On one-dimensional finite strain beam theory, the plane problem. Journal of Applied Mathematics and Physics, 23, 795–804.

    Article  MATH  Google Scholar 

  120. Rice, J. (2010). Solid mechanics. http://esag.harvard.edu/rice/e0_Solid_Mechanics_94_10.pdf.

  121. Romero, I., & Armero, F. (2002). An objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy-momentum conserving scheme in dynamics. International Journal for Numerical Methods in Engineering, 54, 1683–1716.

    Article  MathSciNet  MATH  Google Scholar 

  122. Sansour, C. (1995). A theory and finite element formulation of shells at finite deformations involving thickness change: Circumventing the use of a rotation tensor. Ingenieur-Archiv, 65, 194–216.

    MATH  Google Scholar 

  123. Schröder, J. (2009). Anisotropic polyconvex energies. In J. Schröder (Ed.), Polyconvex analysis (pp. 1–53). CISM. Wien: Springer.

    Google Scholar 

  124. Seifert, B. (1996). Zur Theorie und Numerik Finiter Elastoplastischer Deformationen von Schalenstrukturen. Dissertation, Institut für Baumechanik und Numerische Mechanik der Universität Hannover. Bericht Nr. F 96/2.

    Google Scholar 

  125. Simo, J., Oliver, J., & Armero, F. (1993). Analysis of strong dicontinuities in rate independent softening materials. Computational Mechanics, 12, 277–296.

    Article  MathSciNet  MATH  Google Scholar 

  126. Simo, J. C. (1985). A finite strain beam formulation. The three-dimensional dynamic problem. Part I. Computer Methods in Applied Mechanics and Engineering, 49, 55–70.

    Article  MATH  Google Scholar 

  127. Simo, J. C. (1998). Numerical analysis and simulation of plasticity. In P. G. Ciarlet & J. L. Lions (Eds.), Handbook of numerical analysis (Vol. 6, pp. 179–499). Amsterdam: North-Holland.

    Google Scholar 

  128. Simo, J. C., & Armero, F. (1992). Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. International Journal for Numerical Methods in Engineering, 33, 1413–1449.

    Article  MathSciNet  MATH  Google Scholar 

  129. Simo, J. C., & Hughes, T. J. R. (1998). Computational inelasticity. New York: Springer.

    MATH  Google Scholar 

  130. Simo, J. C., & Pister, K. S. (1984). Remarks on rate constitutive equations for finite deformation problems. Computer Methods in Applied Mechanics and Engineering, 46, 201–215.

    Article  MATH  Google Scholar 

  131. Simo, J. C., & Rifai, M. S. (1990). A class of assumed strain methods and the method of incompatible modes. International Journal for Numerical Methods in Engineering, 29, 1595–1638.

    Article  MathSciNet  MATH  Google Scholar 

  132. Simo, J. C., & Vu-Quoc, L. (1986). Three dimensional finite strain rod model. Part II: Computational aspects. Computer Methods in Applied Mechanics and Engineering, 58, 79–116.

    Article  MATH  Google Scholar 

  133. Simo, J. C., Hjelmstad, K. D., & Taylor, R. L. (1984). Numerical formulations for finite deformation problems of beams accounting for the effect of transverse shear. Computer Methods in Applied Mechanics and Engineering, 42, 301–330.

    Article  MATH  Google Scholar 

  134. Simo, J. C., Fox, D. D., & Rifai, M. S. (1989). On a stress resultant geometrical exact shell model. Part I: Formulation and optimal parametrization. Computer Methods in Applied Mechanics and Engineering, 72, 267–304.

    Article  MathSciNet  MATH  Google Scholar 

  135. Simo, J. C., Fox, D. D., & Rifai, M. S. (1990). On a stress resultant geometrical exact shell model. Part III: Computational aspects of the nonlinear theory. Computer Methods in Applied Mechanics and Engineering, 79, 21–70.

    Article  MathSciNet  MATH  Google Scholar 

  136. Simo, J. C., Rifai, M. S., & Fox, D. D. (1990). On a stress resultant geometrically exact shell model. Part IV. Variable thickness shells with through-the-thickness stretching. Computer Methods in Applied Mechanics and Engineering, 81, 91–126.

    Article  MathSciNet  MATH  Google Scholar 

  137. Simo, J. C., Armero, F., & Taylor, R. L. (1993). Improved versions of assumed enhanced strain tri-linear elements for 3D finite deformation problems. Computer Methods in Applied Mechanics and Engineering, 110, 359–386.

    Google Scholar 

  138. Spiess, H. (2006). Reduction methods in finite element analysis of nonlinear structural dynamics. Technical report F06/2, Forschungs- und Seminarberichte aus dem Bereich der Mechanik der Universität Hannover.

    Google Scholar 

  139. Stein, E., & Ohnimus, S. (1996). Dimensional adaptivity in linear elasticity with hierarchical test-spaces for h- and p-refinement processes. Engineering Computations, 12, 107–119.

    Article  Google Scholar 

  140. Szabó, I. (1987). Geschichte der mechanischen Prinzipien und ihrer wichtigsten Anwendungen. Basel: Birkhäuser.

    Book  MATH  Google Scholar 

  141. Sze, K., & Yao, L. (2000). A hybrid stress ans solid-shell element and its generalization for smart structure modelling. Part isolid-shell element formulation. International Journal for Numerical Methods in Engineering, 48, 545–564.

    Article  MATH  Google Scholar 

  142. Taylor, R. L., Beresford, P. J., & Wilson, E. L. (1976). A non-conforming element for stress analysis. International Journal for Numerical Methods in Engineering, 10, 1211–1219.

    Article  MATH  Google Scholar 

  143. Temizer, I., & Wriggers, P. (2011). An adaptive multiscale resolution strategy for the finite deformation analysis of microheterogeneous structures. Computer Methods in Applied Mechanics and Engineering, 200(37), 2639–2661.

    Article  MathSciNet  MATH  Google Scholar 

  144. Timoshenko, S. (1983). History of strength of materials: With a brief account of the history of theory of elasticity and theory of structures. Courier Corporation.

    Google Scholar 

  145. Timoshenko, S., & Woinowsky-Krieger, S. (1959). Theory of plates and shells (Vol. 2). New York: McGraw-Hill.

    MATH  Google Scholar 

  146. Truesdell, C. (1985). The elements of continuum mechanics. New York: Springer.

    MATH  Google Scholar 

  147. Truesdell, C., & Noll, W. (1965). The nonlinear field theories of mechanics. In S. Flügge (Ed.), Handbuch der Physik III/3. Berlin: Springer.

    Google Scholar 

  148. Turner, M. J., Clough, R. W., Martin, H. C., & Topp, L. J. (1956). Stiffness and deflection analysis of complex structures. Journal of the Aeronautical Sciences, 23, 805–823.

    Article  MATH  Google Scholar 

  149. Tvergaard, V., & Needleman, A. (1984). Analysis of the cup-cone fracture in a round tensile bar. Archives of Mechanics, 32, 157–169.

    Google Scholar 

  150. Vlasov, V. Z. (1984). Thin-walled elastic beams. National Technical Information Service.

    Google Scholar 

  151. Vogel, U. (1985). Calibrating frames. Vergleichsrechnungen an verschiedenen Rahmen. Der Stahlbau, 10, 295–301.

    Google Scholar 

  152. Vukazich, M., Mish, K., & Romstad, K. (1996). Nonlinear dynamic response of frames using Lanczos modal analysis. Journal of Structural Engineering, 122, 1418–1426.

    Article  Google Scholar 

  153. Wagner, W., & Gruttmann, F. (1994). A simple finite rotation formulation for composite shell elements. Engineering Computations, 11, 145–176.

    Article  MathSciNet  Google Scholar 

  154. Wagner, W., & Gruttmann, F. (2002). Modeling of shell-beam transitions in the presence of finite rotations. Computer Assisted Mechanics and Engineering Sciences, 9, 405–418.

    MATH  Google Scholar 

  155. Washizu, K. (1975). Variational methods in elasticity and plasticity (2nd ed.). Oxford: Pergamon Press.

    MATH  Google Scholar 

  156. Wellmann, C., & Wriggers, P. (2012). A two-scale model of granular materials. Computer Methods in Applied Mechanics and Engineering, 205–208, 46–58.

    Article  MathSciNet  MATH  Google Scholar 

  157. Wempner, G. (1969). Finite elements, finite rotations and small strains of flexible shells. International Journal of Solids and Structures, 5, 117–153.

    Article  MATH  Google Scholar 

  158. Wilson, E. L., Taylor, R. L., Doherty, W. P., & Ghaboussi, J. (1973). Incompatible displacements models. In S. J. Fenves, N. Perrone, A. R. Robinson, & W. C. Schnobrich (Eds.), Numerical and computer models in structural mechanics (pp. 43–57). New York: Academic Press.

    Chapter  Google Scholar 

  159. Wilson, E. L., Yuan, M., & Dickens, J. M. (1982). Dynamic analysis by direct superposition of Ritz vectors. Earthquake Engineering and Structural Dynamics, 10, 813–821.

    Article  Google Scholar 

  160. Windels, R. (1970). Traglasten von Balkenquerschnitten beim Angriff von Biegemoment, Längs- und Querkraft. Der Stahlbau, 39, 10–16.

    Google Scholar 

  161. Wriggers, P. (2006). Computational contact mechanics (2nd ed.). Berlin: Springer.

    Book  MATH  Google Scholar 

  162. Wriggers, P. (2008). Nonlinear finite elements. Berlin: Springer.

    MATH  Google Scholar 

  163. Wriggers, P., & Gruttmann, F. (1989). Large deformations of thin shells: Theory and finite-element- discretization, analytical and computational models of shells. In A. K. Noor, T. Belytschko, & J. C. Simo (Eds.), ASME, CED (Vol. 3, pp. 135–159).

    Google Scholar 

  164. Wriggers, P., & Gruttmann, F. (1993). Thin shells with finite rotations formulated in biot stresses theory and finite-element-formulation. International Journal for Numerical Methods in Engineering, 36, 2049–2071.

    Article  MATH  Google Scholar 

  165. Wriggers, P., & Reese, S. (1996). A note on enhanced strain methods for large deformations. Computer Methods in Applied Mechanics and Engineering, 135, 201–209.

    Article  MATH  Google Scholar 

  166. Zienkiewicz, O. C., & Taylor, R. L. (1989). The finite element method (4th ed., Vol. 1). London: McGraw Hill.

    MATH  Google Scholar 

  167. Zienkiewicz, O. C., & Taylor, R. L. (2000). The finite element method (5th ed., Vol. 2). Oxford: Butterworth-Heinemann.

    MATH  Google Scholar 

  168. Zienkiewicz, O. C., Taylor, R. L., & Too, J. M. (1971). Reduced integration technique in general analysis of plates and shells. International Journal for Numerical Methods in Engineering, 3, 275–290.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Continuum Mechanics, Gottfried Wilhelm Leibniz Universität, Hannover, Germany

    Peter Wriggers

Authors
  1. Peter Wriggers
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter Wriggers .

Editor information

Editors and Affiliations

  1. Institute for Applied Mechanics, Technical University of Munich Institute for Applied Mechanics, Garching, Bayern, Germany

    Friedrich Pfeiffer

  2. Institute of Robotics, Johannes-Kepler-University Linz Institute of Robotics, Linz, Austria

    Hartmut Bremer

Rights and permissions

Reprints and permissions

Copyright information

© 2017 CISM International Centre for Mechanical Sciences

About this chapter

Cite this chapter

Wriggers, P. (2017). The Art of Modeling in Solid Mechanics. In: Pfeiffer, F., Bremer, H. (eds) The Art of Modeling Mechanical Systems. CISM International Centre for Mechanical Sciences, vol 570. Springer, Cham. https://doi.org/10.1007/978-3-319-40256-7_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-40256-7_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-40255-0

  • Online ISBN: 978-3-319-40256-7

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation