Abstract
This chapter gives in the first part a summary of some important elements in continuum mechanics, i.e. the decomposition of the stress tensor in its spherical and the deviatoric part and the use of stress invariants to describe the physical content of the stress tensor. In the next part, the elastic behaviour of isotropic materials based on generalised Hooke’s law is summarised and a notation appropriate for computer implementation is introduced. The constitutive description is then extended to plastic material behaviour and the description based on a yield condition, flow rule and hardening law is introduced. The concept of invariants is consistently applied and explained for the characterisation of yield conditions. A classical simple cubic cell model based on beams (Gibson/Ashby model) is investigated in the next chapter in order to highlight the assumptions and the derivation of the macroscopic material properties (elastic constants and yield stress). In the following, a strategy to determine the influence of the hydrostatic stress on the yield behaviour is proposed and conceptionally realised by a state of plane strain and a state of uniaxial strain. In addition, alternative ways to determine the complete set of elastic constants are shown. The last part covers the implementation of yield conditions into finite element codes. The understanding of the predictor-corrector concept is required to provide new constitutive equations in commercial computational codes.
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Bibliography
H. Altenbach, J. Altenbach, and A. Zolochevsky. Erweiterte Deformationsmodelle und Versagenskriterien der Werkstoffmechanik. Deutscher Verlag für Grundstoffindustrie, 1995.
J. Altenbach and H. Altenbach. Einführung in die Kontinuumsmechanik. B.G. Teubner, 1994.
H. Armen. Assumptions, models, and computational methods for plasticity. Computers and Structures, 10:161–174, 1979.
M.F. Ashby, A. Evans, N.A. Fleck, L.J. Gibson, J.W. Hutchinson, and H.N.G. Wadley. Metal foams: a design guide. Butterworth-Heinemann, 2000.
G. Backhaus. Deformationsgesetze. Akademie-Verlag, 1983.
T. Belytschko, W.K. Liu, and B. Moran. Nonlinear finite elements for continua and structures. John Wiley & Sons, 2000.
J. Betten. Kontinuumsmechanik: ein Lehrund Arbeitsbuch. Springer-Verlag, 2001.
J. Betten. Creep Mechanics. Springer-Verlag, 2005.
I.N. Bronstein and K.A. Semendjajew. Taschenbuch der Mathematik (Erg. Kap.). Verlag Harri Deutsch, 1988.
W.F. Chen and D.J. Han. Plasticity for Structural Engineers. Springer-Verlag, 1988.
W.F. Chen and A.F. Saleeb. Constitutive Equations for Engineering Materials. Volume 1: Elasticity and Modeling. John Wiley & Sons, 1982.
L.J. Cohen and O. Ishai. The elastic properties of three-phase composites. Journal of Composite Materials, 1:390–403, 1967.
M.A. Crisfield. Non-linear finite element analysis of solids and structures. Vol. 2: Advanced topics. John Wiley & Sons, 2000.
M.A. Crisfield. Non-linear finite element analysis of solids and structures. Vol. 1: Essentials. John Wiley & Sons, 2001.
E.A. de Souza Neto, D. Peric, and D.R.J. Owen. Computational Methods for Plasticity: Theory and Applications. John Wiley & Sons, 2008.
V.S. Deshpande and N.A. Fleck. Isotropic const it uitve models for metallic foams. Journal of the Mechanics and Physics of Solids, 48:1253–1283, 2000.
V.S. Deshpande and N.A. Fleck. Multi-axial yield behaviour of polymer foams. Acta Materialia, 49:1859–1866, 2001.
L.A. Feldkamp, S.A. Goldstein, A.M. Parfitt, G. Jesion, and M. Kleerekoper. The direct examination of three-dimensional bone architecture in vitro by computed tomography. Journal of Bone and Mineral Research, 4:3–10, 1989.
T. Fiedler, A. Öchsner, and J. Gracio. The uniaxial strain test — a simple method for the characterization of porous materials. Structural Engineering and Mechanics, 22:17–32, 2006.
W. Flügge. Handbook of Engineering Mechanics. McGraw-Hill Book Company, 1962.
A.H. Gent and A.G. Thomas. The deformation of foamed elastic materials. Journal of Applied Polymer Science, 1:107–113, 1959.
A.N. Gent and A.G. Thomas. Mechanics of foamed elastic materials. Rubber Chemistry and Technology, 36:597–610, 1963.
L.J. Gibson. The mechanical behaviour of cancellous bone. Journal of Biomechanics, 18:317–328, 1985.
L.J. Gibson and M.F. Ashby. The mechanics of three-dimensional cellular materials. Proceedings of the Royal Society of London Series A — Mathematical and Physical Sciences, 382:43–59, 1982.
L.J. Gibson and M.F. Ashby. Cellular Solids: Structures and Properties. Cambridge University Press, 1997.
H.G. Hahn. Elastizittslehre. B.G. Teubner, 1985.
Z. Hashin. The elastic moduli of heterogeneous materials. Journal of Applied Mechanics — Transactions of the ASME, 29:143–150, 1962.
M. Jirasek and Z.P. Bazant. Inelastic Analysis of Structures. John Wiley & Sons, 2002.
S.V. Kanakkanatt. Mechanical anisotropy of open-cell foams. Journal of Cellular Plastics, 9:50–53, 1973.
J.H. Keyak, J.M. Meagher, H.B. Skinner, and CD. Mote. Automated three-dimensional finite element modelling of bone: A new method. Journal of Biomedical Engineering, 12:389–397, 1990.
V. Kolupaev. Dreidimensionales Kriechverhalten von Bauteilen aus unverstärkten Thermoplasten. Papierflieger, 2006.
G. Lebon. Extended thermodynamics.In W. Muschik, editor, Non-Equilibrium Thermodynamics with Application to Solids. Springer-Verlag, 1992.
J.M. Lederman. The prediction of the tensile properties of flexible foams. Journal of Applied Polymer Science, 15:693–703, 1971.
J. Lemaitre. A Course on Damage Mechanics. Springer-Verlag, 1996.
J. Lubliner. Plasticity Theory. Macmillan Publishing Company, 1990.
O. Mahrenholtz and H. Ismar. Ein modell des elastisch-plastischen Über gangsverhalten metallischer Werkstoffe. Abhandlungen der Braunschweigischen Wissenschaftlichen Gesellschaft, 30:138–144, 1979.
O. Mahrenholtz and H. Ismar. Zum elastisch-plastischen Uber gangsverhalten metallischer Werkstoffe. Ingenieur-Archiv, 50:217–224, 1981.
H. Mang and G. Hofstetter. Festigkeitslehre. Springer Verlag, 2000.
I.W. Marks and T.N. Gardner. The use of strain energy as a convergence criterion in the finite element modelling of bone and the effect of model geometry on stress convergence. Journal of Biomedical Engineering, 14: 474–476, 1993.
V.A. Matonis. Elastic behavior of low density rigid foams in structural applications. SPE Journal, 20:1024–1030, 1964.
B. Moran, M. Ortiz, and C.F. Shih. A unified approach to finite deformation elastoplasticity based on the use of hyper elastic constitutive equations. International Journal for Numerical Methods in Engineering, 29:483–514, 1990.
E.P. Müller, P. Rüegsegger, and P. Seitz. Optimal ct settings for bone evaluations. Physics in Medicine and Biololgy, 30:401–409, 1985.
R. Müller and P. Rüegsegger. Three-dimensional finite element modelling of non-invasively assessed trabecular bone structures. Medical Engineering & Physics, 17:126–133, 1995.
G.C. Nayak and O.C. Zienkiewicz. Convenient form of stress invariants for plasticity. Journal of the Structural Division-ASCE, 98:1949–954, 1972.
A. Öchsner. Experimentelle und numerische Untersuchung des elastoplastischen Verhaltens zellularer Modellwerkstoffe [Experimental and Numerical Investigations of the Elastic-Plastic Properties of Model Cellular Materials]. VDI Verlag, 2003.
A. Öchsner, T. Fiedler, J. Grácio, and G. Kuhn. Experimental techniques for the investigation of the elasto-plastic transition zone of foamed materials. Advanced Engineering Materials, 8:884–889, 2006.
J.C. Simo and T. J. R. Hughes. Computational Inelasticity. Springer-Verlag, 1998.
J.C. Simo and M. Ortiz. A unified approach to finite deformation elastoplasticity based on the use of hyper elastic constitutive equations. Computational Method Appl M, 49:221–245, 1985.
B. van Rietbergen, H. Weinans, R. Huiskes, and A. Odgaard. A new method to determine trabecular bone elastic properties and loading using micromechanical finite-element models. Journal of Biomechanics, 28:69–81, 1995.
J.K. Weaver and J. Chalmers. Cancellous bone: Its strength and changes with aging and an evaluation of some methods for measuring its mineral content. Journal of Bone and Joint Surgery — American Volume, 48: 289–298, 1966.
P. Wriggers. Nichtlineare Finite-Element-Methoden. Springer-Verlag, 2001.
M. Zyczkowski. Combined Loadings in the Theory of Plasticity. PWN — Polish Scientific Publishers, 1981.
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Öchsner, A. (2010). Plasticity of Three-Dimensional Foams. In: Altenbach, H., Öchsner, A. (eds) Cellular and Porous Materials in Structures and Processes. CISM International Centre for Mechanical Sciences, vol 521. Springer, Vienna. https://doi.org/10.1007/978-3-7091-0297-8_3
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DOI: https://doi.org/10.1007/978-3-7091-0297-8_3
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