Abstract
Multi-mechanism models (MM models) have become an important tool for modeling complex material behavior. In particular, two-mechanism models are used. They are applied to model ratcheting in metal plasticity as well as steel behavior during phase transformations. We consider a small-deformation setting. The characteristic trait of multi-mechanism models is the additive decomposition of the inelastic (e.g., plastic or viscoplastic) strain into several parts. These parts are sometimes called mechanisms. In comparison with rheological models, the mechanisms can interact with each other. This leads to new properties and allows to describe important observable effects. Up to now, each mechanism has one kinematic internal variable. As a new feature, we develop multi-mechanism models (in series) with several kinematic variables for each mechanism as well as with several isotropic variables for each flow criterion. We describe this complex situation by three structural matrices which express the mutual relations between mechanisms, flow criteria, kinematic, and isotropic variables. The well-known Chaboche model with a unique inelastic strain and several kinematic variables represents a special case of these general multi-mechanism models. In this work, we also present a matrix-based approach for these new complex MM models. The presented models can form the basis for developing numerical algorithms for simulation and parameter identification.
Similar content being viewed by others
References
Abdel-Karim M.: Modified kinematic hardening rule for simulations of ratcheting. Int. J. Plast. 25, 1560–1587 (2009). doi:10.1016/j.ijplas.2008.10.004
Abdel-Karim M.: An evaluation for several kinematic hardening rules on prediction of multi-axial stress-controlled ratcheting. Int. J. Plast. 26(5), 711–730 (2010). doi:10.1016/j.ijplas.2009.10.002
Aeby-Gautier, E., Cailletaud, G.: N-phase modeling applied to phase transformations in steels: a coupled kinetics-mechanics approach. In: International Conference on Heterogeneous Material Mechanics ICHMM-2004. Chongqing, China (2004)
Bari S., Hassan T.: An advancement in cyclic plasticity modelling for multiaxial ratcheting simulation. Int. J. Plast. 18, 873–894 (2002). doi:10.1016/S0749-6419(01)00012-2
Bertram A.: Elasticity and Plasticity of Large Deformations: An Introduction. Springer, Berlin (2012)
Bertram, A., Glüge, R.: Festkörpermechanik. Otto-von-Guericke-Universitt Magdeburg (2013)
Besson J.: Damage of ductile materials deforming under multiple plastic or viscoplastic mechanisms. Int. J. Plast. 25, 2204–2221 (2009). doi:10.1016/j.ijplas.2009.03.001
Besson J., Cailletaud G., Chaboche J.L., Forest S.: Mécanique Non Linéaire des Matériaux. Hermes Science Europe Ltd, Paris (2001)
Burlet H., Cailletaud G.: Modeling of cyclic plasticity in finite element codes. In: Desai, C. (ed.) 2nd International Conference on Constitutive Laws for Engineering Materials: Theory and Applications, pp. 1157–1164. Elsevier, Tucson (1987)
Cailletaud G., Saï K.: Study of plastic/viscoplastic models with various inelastic mechanisms. Int. J. Plast. 11, 991–1005 (1995). doi:10.1016/S0749-6419(95)00040-2
Chaboche J.: A review of some plasticity and viscoplasticity constitutive theories. Int. J. Plast. 24, 1642–1693 (2008). doi:10.1016/j.ijplas.2008.03.009
Coleman B.D., Gurtin M.E.: Thermodynamics with internal state variables. J. Chem. Phys. 47(2), 597–613 (1967). doi:10.1063/1.1711937
Contesti E., Cailletaud G.: Description of creep-plasticity interaction with non-unified constitutive equations: application to an austenitic steel. Nucl. Eng. Des. 116, 265–280 (1989). doi:10.1016/0029-5493(89)90087-3
Desmorat R.: Non-saturating nonlinear kinematic hardening laws. C. R. Méc. 338(3), 146–151 (2010). doi:10.1016/j.crme.2010.02.007
Hassan T., Taleb L., Krishna S.: Influence of non-proportional loading on ratcheting responses and simulations by two recent cyclic plasticity models. Int. J. Plast. 24, 1863–1889 (2008). doi:10.1016/j.ijplas.2008.04.008
Haupt P.: Continuum Mechanics and Theory of Materials. Springer, Berlin (2002)
Helm D., Haupt P.: Shape memory behavior: modeling within continuum thermomechanics. Int. J. Solids Struct. 40, 827 (2003). doi:10.1016/S0020-7683(02)00621-2
Krawietz A.: Materialtheorie: Mathematische Beschreibung des phänomenologischen thermomechanischen Verhaltens. Springer, Berlin (1986)
Kröger, N., Böhm, M., Wolff, M.: Viskoelastisches Zwei-Mechanismen-Modell für einen eindimensionalen Stab. Tech. Rep. 12-02, Berichte aus der Technomathematik, FB 3, Universität Bremen (2012)
Kröger, N.H.: Multi-mechanism models: theory and applications. Ph.D. thesis, Universität Bremen, Germany (2013)
Kröger N.H., Böhm M., Wolff M.: Viscoelasticity within the framework of isothermal two-mechanism models. Comput. Mater. Sci. 64, 30–33 (2012). doi:10.1016/j.commatsci.2012.04.017
Lemaitre J., Chaboche J.L.: Mechanics of Solid Materials. Cambridge University Press, Cambridge (1990)
Mahnken R., Schneidt A., Antretter T.: Macro modelling and homogenization for transformation induced plasticity of a low-alloy steel. Int. J. Plast. 25, 183–204 (2009). doi:10.1016/j.ijplas.2008.03.005
Mahnken R., Schneidt A., Antretter T., Ehlenbröker U., Wolff M.: Multi-scale modeling of bainitic phase transformation in multi-variant polycrystalline low alloy steels. Int. J. Solids Struct. 54, 156–171 (2015)
Mahnken R., Wolff M., Cheng C.: A multi-mechanism model for cutting simulations combining visco-plastic asymmetry and phase transformation. Int. J. Solids Struct. 50, 3045–3066 (2013)
Mahnken R., Wolff M., Schneidt A., Böhm M.: Multi-phase transformations at large strains - thermodynamic framework and simulation. Int. J. Plast. 39, 1–26 (2012)
Maugin G.: The Thermodynamics of Plasticity and Fracture. Cambridge University Press, Cambridge (1992)
Nouailhas D., Cailletaud G., Policella H., Marquis D., Dufailly J., Lieurade H., Ribes A., Bollinger E.: On the description of cyclic hardening and initial cold working. Eng. Fract. Mech. 21(4), 887–895 (1985)
Palmov V.A.: Vibrations of elasto-plastic bodies. Springer, Berlin (1998)
Palmov V.A.: Theory of Defining Equations in Nonlinear Thermodynamics of Deformable Bodies (in Russian). St. Petersburg State Polytechnical University, St. Petersburg (2008)
Regrain C., Laiarinandrasana L., Toillon S., Saï K.: Multi-mechanism models for semi-crystalline polymer: constitutive relations and finite element implementation. Int. J. Plast. 25, 1253–1279 (2009). doi:10.1016/j.ijplas.2008.09.010
Saï, K.: Modèles à grand nombre de variables internes et méthodes numériques associées. Ph.D. thesis, National Ecole des Mines de Paris (1993)
Saï K.: Multi-mechanism models: present state and future trends. Int. J. Plast. 27, 250–281 (2011). doi:10.1016/j.ijplas.2010.05.003
Saï K., Cailletaud G.: Multi-mechanism models for the description of ratcheting: effect of the scale transition rule and of the coupling between hardening variables. Int. J. Plast. 23, 1589–1617 (2007). doi:10.1016/j.ijplas.2007.01.011
Saï K., Laiarinandrasana L., Naceur I.B., Besson J., Jeridi M., Cailletaud G.: Multi-mechanism damage-plasticity model for semi-crystalline polymer: creep damage of notched specimen of pa6. Mater. Sci. Eng. A 528, 1087–1093 (2011). doi:10.1016/j.msea.2010.09.071
Saï K., Taleb L., Cailletaud G.: Numerical simulation of the anisotropic behavior of 2017 aluminum alloy. Comput. Mater. Sci. 65, 48–57 (2012)
Shen X., Xia Z., Ellyin F.: Cyclic deformation behaviour of an epoxy polymer. part i: experimental investigation. Polym. Eng. Sci. 44(12), 2240–2246 (2004)
Taleb L.: About the cyclic accumulation of the inelastic strain observed in metals subjected to cyclic stress control. Int. J. Plast. 43, 1–19 (2013)
Taleb L., Cailletaud G.: An updated version of the multimechanism model for cyclic plasticity. Int. J. Plast. 26(6), 859–874 (2010). doi:10.1016/j.ijplas.2009.11.002
Taleb, L., Cailletaud, G.: Cyclic accumulation of the inelastic strain in the 304L SS under stress control at room temperature: ratcheting or creep? Int. J. Plast. 27, 1936–1958 (2011). doi:10.1016/j.ijplas.2011.02.001
Taleb L., Cailletaud G., Blaj L.: Numerical simulation of complex ratcheting tests with a multi-mechanism model type. Int. J. Plast. 22, 724–753 (2006). doi:10.1016/j.ijplas.2005.05.003
Taleb L., Hauet A.: Multiscale experimental investigations about the cyclic behavior of the 304L SS. Int. J. Plast. 25, 1359–1385 (2009). doi:10.1016/j.ijplas.2008.09.004
Tao G., Xia Z.: Material behaviour: ratcheting behavior of an epoxy polymer and its effect on fatigue life. Polym. Test. 26, 451–460 (2007). doi:10.1016/j.polymertesting.2006.12.010
Turki M., Ben Naceur I., Makni M., Rouis J., Saï K.: Mechanical and damage behaviour of mortar-rubber aggregates mixtures: experiments and simulations. Mater. Struct. 42, 1313–1324 (2009). doi:10.1617/s11527-008-9451-1
Videau, J.C., Cailletaud, G., Pineau, A.: Modélisation des effets mécaniques des transformations de phases pour le calcul de structures. J. de Physique IV, Colloque C3, supplément au J. de Physique III, 4 p. 227 (1994). http://hal.archives-ouvertes.fr/docs/00/25/25/28/PDF/ajp-jp4199404C331.pdf
Wolff, M., Böhm, M.: Two-mechanism models and modelling of creep. Proceedings of the 3rd International Conference on Nonlinear Dynamics, Charkiw, Ukraine (2010)
Wolff M., Böhm M., Bökenheide S., Kröger N.: Two-mechanism approach in thermo-viscoelasticity with internal variables. Tech. Mech. 32(2–5), 608–621 (2012)
Wolff M., Böhm M., Bökenheide S., Lammers D., Linke T.: An implicit algorithm to verify creep and trip behavior of steel using uniaxial experiments. Z. Angew. Math. Mech. 92, 355–379 (2012)
Wolff M., Böhm M., Helm D.: Material behavior of steel–modeling of complex phenomena and thermodynamic consistency. Int. J. Plast. 24, 746–774 (2008). doi:10.1016/j.ijplas.2007.07.005
Wolff M., Böhm M., Mahnken R., Suhr B.: Implementation of an algorithm for general material behaviour of steel taking interaction of plasticity and transformation-induced plasticity into account. Int. J. Numer. Methods Eng. 87, 1183–1206 (2011)
Wolff, M., Böhm, M., Taleb, L.: Two-mechanism models with plastic mechanisms—modelling in continuum-mechanical framework. Tech. Rep. 10-05, Berichte aus der Technomathematik, FB 3, Universität Bremen (2010)
Wolff M., Böhm M., Taleb L.: Thermodynamic consistency of two-mechanism models in the non-isothermal case. Tech. Mech. 31, 58–80 (2011)
Wolff, M., Bökenheide, S., Schlasche, J., Büsing, D., Karsch, T., Böhm, M., Zoch, H.W.: An extended approach to multi-mechanism models in plasticity - theory and parameter identification. Tech. Rep. 13-06, Berichte aus der Technomathematik, FB 3, Universität Bremen (2013)
Wolff M., Taleb L.: Consistency for two multi-mechanism models in isothermal plasticity. Int. J. Plast. 24, 2059–2083 (2008). doi:10.1016/j.ijplas.2008.03.001
Xia Z., Shen X., Ellyin F.: An assessment of nonlinearly viscoelastic constitutive models for cyclic loading: the effect of a general loading/unloading rule. Mech. Time Depend. Mater. 9, 281–300 (2006). doi:10.1007/s11043-006-9004-3
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Andreas Öchsner.
Rights and permissions
About this article
Cite this article
Wolff, M., Bökenheide, S. & Böhm, M. Some new extensions to multi-mechanism models for plastic and viscoplastic material behavior under small strains. Continuum Mech. Thermodyn. 28, 821–852 (2016). https://doi.org/10.1007/s00161-015-0418-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00161-015-0418-5