Abstract
An analytical solution of the problem of the propagation of a Lüders band in an isotropic strain gradient plasticity medium is provided based on a softening–hardening constitutive law. A detailed description is given of the plastic strain distribution in the finite size band front. The solution is shown to be harmonic in the band front and exponential in the band tail. Particular attention is paid to the conditions to be applied at the interface between both regions. This solution is then used to validate finite element simulations of the Lüders band formation and propagation in a plate in tension. The approach is shown to suppress the spurious mesh dependence exhibited by conventional finite element simulations of the Lüders behavior and to provide a finite width band front in agreement with the experimental observations from strain field measurements.
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Aifantis E.: On the microstructural origin of certain inelastic models. J. Eng. Mater. Technol. 106, 326–330 (1984)
Aifantis E.C.: The physics of plastic deformation. Int. J. Plasticity 3, 211–248 (1987)
Anand L., Aslan O., Chester S.: A large-deformation gradient theory for elastic–plastic materials: strain softening and regularization of shear bands. Int. J. Plasticity 30–31, 116–143 (2012)
Ballarin V., Perlade A., Lemoine X., Bouaziz O., Forest S.: Mechanisms and modeling of bake-hardening steels: Part II. Complex loading paths. Metall. Mater. Trans. A 40, 1367–1374 (2009)
Ballarin V., Soler M., Perlade A., Lemoine X., Forest S.: Mechanisms and modelling of bake-hardening steels: Part I. Uniaxial tension. Metall. Mater. Trans. 40, 1367–1374 (2009)
Belotteau J., Berdin C., Forest S., Parrot A., Prioul C.: Mechanical behavior and crack tip plasticity of a strain aging sensitive steel. Mater. Sci. Eng. A 526(1–2), 156–165 (2009)
Besson J., Cailletaud G., Chaboche J.L., Forest S.: Non Linear Mechanics of Materials. Springer, Berlin (2009)
Besson J., Foerch R.: Large scale object-oriented finite element code design. Comput. Meth. Appl. Mech. Eng. 142, 165–187 (1997)
de Borst R., Sluys L., Mühlhaus H., Pamin J.: Fundamental issues in finite element analyses of localization of deformation. Eng. Comput. 10, 99–121 (1993)
Butler J.F.: Lüders front propagation in low carbon steels. J. Mech. Phys. Solids 10, 313–334 (1962)
Cordero N.M., Forest S., Busso E.P.: Generalised continuum modelling of grain size effects in polycrystals. Comptes Rendus Mécanique 340, 261–274 (2012)
Cottrell A.H., Bilby B.A.: Dislocation theory of yielding and strain ageing of iron. Proc. Phys. Soc. A 62(1), 49–62 (1949)
Dell’Isola F., Seppecher P.: The relationship between edge contact forces, double forces and intersticial working allowed by the principle of virtual power. C.R. Acad. Sci. Paris IIb 321, 303–308 (1995)
Dell’Isola F., Seppecher P., Madeo A.: How contact interactions may depend on the shape of Cauchy cuts in N-th gradient continua: approach “la D’Alembert”. Zeitschrift fr Angewandte Mathematik und Physik 63, 1119–1141 (2012)
Dimitrijevic B., Hackl K.: A regularization framework for damage plasticity models via gradient enhancement of the free energy. Int. J. Numer. Meth. Biomed. Eng. 27, 1199–1210 (2011)
Dingley D.J., McLean D.: Components of the flow stress of iron. Acta Metall. 15, 885–901 (1967)
Enakoutsa K., Leblond J.: Numerical implementation and assessment of the GLPD micromorphic model of ductile rupture. Eur. J. Mech. A/solids 28, 445–460 (2009)
Engelen R., Geers M., Baaijens F.: Nonlocal implicit gradient-enhanced elasto-plasticity for the modelling of softening behaviour. Int. J. Plasticity 19, 403–433 (2003)
Ferretti, M., Madeo, A., Dell’Isola, F., Seppecher, P., Boisse, P.: Modeling the onset of shear boundary layers in fibrous composite reinforcements by second-gradient theory. Zeitschrift für Angewandte Mathematik und Physik (2013)
Forest, S.: Strain localization phenomena in generalized crystal plasticity. In: Second Euroconference and International Symposium on Material Instabiblities in Deformation and Fracture, Journal of the Mechanical Behavior of Materials, vol. 11, pp. 45–50. organized by E.C. Aifantis, Aristotle Technical University, Thessaloniki, Greece (1997)
Forest S.: Micromorphic approach for gradient elasticity, viscoplasticity, and damage. J. Eng. Mech. 135(3), 117–131 (2009)
Forest S., Aifantis E.C.: Some links between recent gradient thermo-elasto-plasticity theories and the thermomechanics of generalized continua. Int. J. Solids Struct. 47, 3367–3376 (2010)
Forest S., Bertram A.: Formulations of strain gradient plasticity. In: Altenbach, H., Maugin, G.A., Erofeev, V. (eds) Mechanics of Generalized Continua, Advanced Structured Materials, Vol. 7, pp. 137–150. Springer, Berlin (2011)
Forest S., Blazy J., Chastel Y., Moussy F.: Continuum modelling of strain localization phenomena in metallic foams. J. Mater. Sci. 40, 5903–5910 (2005)
Forest S., Sievert R.: Elastoviscoplastic constitutive frameworks for generalized continua. Acta Mech. 160, 71–111 (2003)
Fressengeas C., Beaudoin A., Lebyodkin M., Kubin L., Estrin Y.: Dynamic strain aging: a coupled dislocation-solute dynamic model. Mater. Sci. Eng. 51, 226–230 (2005)
Germain P.: La méthode des puissances virtuelles en mécanique des milieux continus, première partie: théorie du second gradient. J. de Mécanique 12, 235–274 (1973)
Germain P.: The method of virtual power in continuum mechanics. part 2: Microstructure. SIAM J. Appl. Math. 25, 556–575 (1973)
Graff, S., Forest, S., Strudel, J.L., Prioul, C., Pilvin, P., Béchade, J.L.: Strain localization phenomena associated with static and dynamic strain ageing in notched specimens: experiments and finite element simulations. Mater. Sci. Eng. A 387–389:181–185 (2004)
Graff S., Forest S., Strudel J.L., Prioul C., Pilvin P., Béchade J.L.: Finite element simulations of dynamic strain ageing effects at V-notches and crack tips. Scripta Materialia 52, 1181–1186 (2005). doi:10.1016/j.scriptamat.2005.02.007
Gurtin M.: On a framework for small-deformation viscoplasticity: free energy, microforces, strain gradients. Int. J. Plasticity 19, 47–90 (2003)
Hähner P.: Modelling the spatio-temporal aspects of the Portevin–Le Chatelier effect. Mater. Sci. Eng. A 164, 23–34 (1993)
Hähner P., Kubin L.P.: Coherent propagative structures in plastic deformation: a theory of lüders bands in polycrystals. Solid State Phenomena 23–24, 385–402 (1992)
Kochmann D., Hackl K.: The evolution of laminates in finite crystal plasticity: a variational approach. Contin. Mech. Thermodyn. 23, 65–85 (2011)
Kok S., Bharathi M., Beaudoin A., Fressengeas C., Ananthakrishna G., Kubin L., Lebyodkin M.: Spatial coupling in jerky flow using polycristal plasticity. Acta Materialia 51, 3651–3662 (2003)
Kubin L., Estrin Y.: The Portevin–Le Chatelier effect in deformation with constant stress rate. Acta Mater. 33, 397–407 (1985)
Kyriakides S., Miller J.E.: On the propagation of Lüders bands in steel strips. J. Appl. Mech. 67, 645–654 (2000)
Lambrecht M., Miehe C., Dettmar J.: Energy relaxation of non-convex incremental stress potentials in a strain-softening elastic–plastic bar. Int. J. Solids Struct. 40(6), 1369–1391 (2003)
Liebe T., Menzel A., Steinmann P.: Theory and numerics of geometrically non-linear gradient plasticity. Int. J. Eng. Sci. 41, 1603–1629 (2003)
Liebe T., Steinmann P., Benallal A.: Theoretical and computational aspects of a thermodynamically consistent framework for geometrically linear gradient damage. Comp. Methods Appl. Mech. Eng. 190, 6555–6576 (2001)
Lomer W.M.: The yield phenomenon in polycrystalline mild steel. J. Mech. Phys. Solids 1, 64–73 (1952)
Louche H., Chrysochoos A.: Thermal and dissipative effects accompanying Lüders band propagation. Mater. Sci. Eng. 307, 15–22 (2001)
Lüders W.: Über die Äusserung der Elasticität an stahlartigen Eisenstäben und Stahlstäben, und über eine beim Biegen solcher Stäbe beobachtete Molecularbewegung. Dinglers Polytech J 5, 18–22 (1860)
Marais A., Mazière M., Forest S., Parrot A., Le Delliou P.: Identification of a strain-aging model accounting for lüders behavior in a c-mn steel. Philos. Mag. 92(28–30), 3589–3617 (2012)
Maxwell J.: On the dynamical evidence of the molecular constitution of bodies. Nature 11, 357–359 (1875)
Mazière M., Besson J., Forest S., Tanguy B., Chalons H., Vogel F.: Numerical aspects in the finite element simulation of the Portevin–Le Chatelier effect. Comp. Method Appl. Mech. Eng. 199, 734–754 (2010)
McCormick P.G.: Theory of flow localization due to dynamic strain ageing. Acta Metall. 36, 3061–3067 (1988)
Mesarovic S.: Dynamic strain aging and plastic instabilities. J. Mech. Phys. Solids 43(5), 671–700 (1995)
Mindlin R., Eshel N.: On first strain gradient theories in linear elasticity. Int. J. Solids Struct. 4, 109–124 (1968)
Mühlhaus H.B., Boland J.: A gradient plasticity model for Lüders band propagation. Pure Appl. Geophys. 137(4), 391–407 (1991)
Peerlings R., Poh L., Geers M.: An implicit gradient plasticity-damage theory for predicting size effects in hardening and softening. Eng. Fract. Mech. 95, 2–12 (2012)
Piobert, G.: Expérience sur la pénétration des projectiles dans le fer forgé. Mémoire de l’Artillerie, p. 505 (1842)
Poh L., Peerlings R., Geers M., Swaddiwudhipong S.: An implicit tensorial gradient plasticity model—formulation and comparison with a scalar gradient model. Int. J. Solids Struct. 48, 2595–2604 (2011)
Rice, J.: The localisation of plastic deformation. In: Koiter, W. (ed.) Proceedings of 14th International Conference Theoretical and Applied Mechanics. Delft, North–Holland, Amsterdam, pp. 207–220 (1976)
Soler, M.: Etude du vieillissement d’un acier à bake hardening: évolution des propriétés mécaniques de traction—corrélation avec la microstructure. Ph.D. thesis, INSA Lyon (1998)
Tsukahara H., Iung T.: Finite element simulation of the Piobert-lüders behavior in an uniaxial test. Mater. Sci. Eng. A 248, 304–308 (1998)
Tsukahara H., Iung T.: Piobert–Lüders and Portevin–Le Chatelier instabilities. finite element modelling with abaqus. J. Phys. IV 9, 157–164 (1999)
Wang H.D., Berdin C., Mazière M., Forest S., Prioul C., Parrot A., Le-Delliou P.: Portevin–Le Chatelier (PLC) instabilities and slant fracture in C-Mn steel round tensile specimens. Scrypta Mater. 64, 430–433 (2011)
Wenman M.R., Chard-Tuckey P.R.: Modelling and experimental characterisation of the Lüders strain in complex loaded ferritic steel compact tension specimens. Int. J. Plasticity 26, 1013–1028 (2010)
Wulfinghoff S., Böhlke T.: Equivalent plastic strain gradient enhancement of single crystal plasticity: theory and numerics. Proc. R. Soc. A 468, 2682–2703 (2012)
Zaiser M., Mill F., Konstantinidis A., Aifantis K.: Strain localization and strain propagation in collapsible solid foams. Mater. Sci. Eng. A 567, 38–45 (2013)
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Communicated by Andreas Öchsner.
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Mazière, M., Forest, S. Strain gradient plasticity modeling and finite element simulation of Lüders band formation and propagation. Continuum Mech. Thermodyn. 27, 83–104 (2015). https://doi.org/10.1007/s00161-013-0331-8
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DOI: https://doi.org/10.1007/s00161-013-0331-8