doi: 10.3934/dcdss.2020188

Well-posedness of a one-dimensional nonlinear kinematic hardening model

1. 

Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

2. 

Istituto di Matematica Applicata e Tecnologie Informatiche E. Magenes, v. Ferrata 1, 27100 Pavia, Italy

Received  March 2018 Revised  October 2018 Published  November 2019

We investigate the quasistatic evolution of a one-dimensional elastoplastic body at small strains. The model includes general nonlinear kinematic hardening but no nonlocal compactifying term. Correspondingly, the free energy of the medium is local but nonquadratic. We prove that the quasistatic evolution problem admits a unique strong solution.

Citation: David Melching, Ulisse Stefanelli. Well-posedness of a one-dimensional nonlinear kinematic hardening model. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020188
References:
[1]

P. J. Armstrong and C. O. Frederick, A mathematical representation of the multiaxial Bauschinger effect, Materials at High Temperatures, 24 (2007), 11-26.   Google Scholar

[2]

J. Barré de Saint Venant, Mémoire sur l'établissement des équations différentielles des mouvements intérieurs opérés dans les corps solides ductiles au delà des limites où l'élasticité pourrait les ramener à leur premier état, J. Math. Pures Appl., 16 (1871), 308-316.   Google Scholar

[3]

H. Brezis, Oprateurs Maximaux Monotones, North Holland, 1973. Google Scholar

[4]

M. Brokate and P. Krejčí, On the wellposedness of the Chaboche model, in Control and Estimation of Distributed Parameter Systems, Internat. Ser. Numer. Math., 126, Birkhäuser, Basel, 1998, 67–79. doi: 10.1007/978-3-0348-8849-3_5.  Google Scholar

[5]

M. Brokate and P. Krejčí, Wellposedness of kinematic hardening models in elastoplasticity, RAIRO Math. Modél. Numer. Anal., 32 (1998), 177-209.  doi: 10.1051/m2an/1998320201771.  Google Scholar

[6]

M. Brokate and P. Krejčí, Maximum norm wellposedness of nonlinear kinematic hardening models, Contin. Mech. Thermodyn., 9 (1997), 365-380.  doi: 10.1007/s001610050077.  Google Scholar

[7]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Applied Mathematical Sciences, 121, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[8]

K. Chelminski, Mathematical analysis of the Armstrong-Frederick model from the theory of inelastic deformations of metals. First results and open problems, Contin. Mech. Thermodyn., 15 (2003), 221-245.  doi: 10.1007/s00161-002-0112-2.  Google Scholar

[9]

S. Conti and F. Theil, Single-slip elastoplastic microstructures, Arch. Ration. Mech. Anal., 178 (2005), 125-148.  doi: 10.1007/s00205-005-0371-8.  Google Scholar

[10]

G. Dal Maso and G. Lazzaroni, Quasistatic crack growth in finite elasticity with noninterpenetration, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 257-290.  doi: 10.1016/j.anihpc.2009.09.006.  Google Scholar

[11]

G. Dal MasoG. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal., 176 (2005), 165-225.  doi: 10.1007/s00205-004-0351-4.  Google Scholar

[12]

G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin-New York, 1976. doi: 10.1007/978-3-642-66165-5.  Google Scholar

[13]

M. A. Eisenberg and A. Phillips, On nonlinear kinematic hardening, Acta Mech., 5 (1968), 1-13.  doi: 10.1007/BF01624439.  Google Scholar

[14]

G. A. Francfort and U. Stefanelli, Quasi-static evolution for the Armstrong-Frederick hardening-plasticity model, Appl. Math. Res. Express. AMRX, 2013 (2013), 297-344.  doi: 10.1093/amrx/abt001.  Google Scholar

[15]

W. Han and B. D. Reddy, Plasticity: Mathematical Theory and Numerical Analysis, Interdisciplinary Applied Mathematics, 9, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4614-5940-8.  Google Scholar

[16]

S. Heinz and A. Mielke, Existence, numerical convergence and evolutionary relaxation for a rate-independent phase-transformation model, Philos. Trans. Roy. Soc. A, 374 (2016), 23pp. doi: 10.1098/rsta.2015.0171.  Google Scholar

[17]

C. Johnson, Existence theorems for plasticity problems, J. Math. Pures Appl., 55 (1976), 431-444.   Google Scholar

[18]

C. Johnson, On finite element methods for plasticity problems, Numer. Math., 26 (1976), 79-84.  doi: 10.1007/BF01396567.  Google Scholar

[19]

C. Johnson, A mixed finite element method for plasticity problems with hardening, SIAM J. Numer. Anal., 14 (1977), 575-583.  doi: 10.1137/0714037.  Google Scholar

[20]

C. Johnson, On plasticity with hardening, J. Math. Anal. Appl., 62 (1978), 325-336.  doi: 10.1016/0022-247X(78)90129-4.  Google Scholar

[21]

I. Kadashevich and V. V. Novozhilov, The theory of plasticity which takes into account residual microstresses, J. Appl. Math. Mech., 22 (1958), 104-118.  doi: 10.1016/0021-8928(58)90086-8.  Google Scholar

[22]

P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, GAKUTO International Series, Mathematical Sciences and Applications, 8, Gakkotosho Col., Ltd., Tokyo, 1996.  Google Scholar

[23] J. Lemaitre and J.-L. Chaboche, Mechanics of Solid Materials, Cambridge University Press, Cambridge, 1990.  doi: 10.1017/CBO9781139167970.  Google Scholar
[24]

M. Lévy, Extrait du mémoire sur les équations générales des mouvements intérieurs des corps solids ductiles au delà des limites où lélasticité pourrait les ramener à leur premier état, J. Math. Pures Appl., 16 (1871), 369-372.   Google Scholar

[25]

J. Lubliner, Plasticity Theory, Macmillan Publishing Company, New York, 1990. Google Scholar

[26]

A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), 73-99.  doi: 10.1007/s00526-004-0267-8.  Google Scholar

[27] G. A. Maugin, The Thermomechanics of Plasticity and Fracture, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9781139172400.  Google Scholar
[28]

E. Melan, Zur Plastizität des räumlichen Kontinuums, Ingenieur Archiv, 9 (1938), 116-126.  doi: 10.1007/BF02084409.  Google Scholar

[29]

A. Mielke and S. Müller, Lower semicontinuity and existence of minimizers in incremental finite-strain elastoplasticity, ZAMM Z. Angew. Math. Mech., 86 (2006), 233-250.  doi: 10.1002/zamm.200510245.  Google Scholar

[30]

R. von Mises, Mechanik der festen Körper im plastisch deformablen Zustand, Nachr. Akad. Wiss. Göttingen Math. Phys. Kl., (1913), 582–592. Google Scholar

[31]

R. von Mises, Mechanik der plastischen Formänderung von Kristallen, ZAMM Z. Angew. Math. Mech., 8 (1928), 161-185.  doi: 10.1002/zamm.19280080302.  Google Scholar

[32]

W. Prager, Recent developments in the mathematical theory of plasticity, J. Appl. Phys., 20 (1949), 235-241.  doi: 10.1063/1.1698348.  Google Scholar

[33]

L. T. Prandtl, Ein Gedankenmodell zur kinetischen Theorie der festen Körper, ZAMM Z. Angew. Math. Mech., 8 (1928), 85-106.  doi: 10.1002/zamm.19280080202.  Google Scholar

[34]

M. Röger and B. Schweizer, Strain gradient visco-plasticity with dislocation densities contributing to the energy, Math. Models Methods Appl. Sci., 27 (2017), 2595-2629.  doi: 10.1142/S0218202517500531.  Google Scholar

[35]

J. C. Simo and T. J. R. Hughes, Computational Inelasticity, Interdisciplinary Applied Mathematics, 7, Springer-Verlag, New York, 1998. doi: 10.1007/b98904.  Google Scholar

[36]

U. Stefanelli, Existence for dislocation-free finite plasticity, ESAIM Control Optim. Calc. Var., 25 (2018), 20pp. doi: 10.1051/cocv/2018014.  Google Scholar

[37]

H. E. Tresca, Mémoire sur l'écoulement des corps solides, Mémoire Présentés par Divers Savants, Acad. Sci. Paris, 20 (1872), 75-135.   Google Scholar

show all references

References:
[1]

P. J. Armstrong and C. O. Frederick, A mathematical representation of the multiaxial Bauschinger effect, Materials at High Temperatures, 24 (2007), 11-26.   Google Scholar

[2]

J. Barré de Saint Venant, Mémoire sur l'établissement des équations différentielles des mouvements intérieurs opérés dans les corps solides ductiles au delà des limites où l'élasticité pourrait les ramener à leur premier état, J. Math. Pures Appl., 16 (1871), 308-316.   Google Scholar

[3]

H. Brezis, Oprateurs Maximaux Monotones, North Holland, 1973. Google Scholar

[4]

M. Brokate and P. Krejčí, On the wellposedness of the Chaboche model, in Control and Estimation of Distributed Parameter Systems, Internat. Ser. Numer. Math., 126, Birkhäuser, Basel, 1998, 67–79. doi: 10.1007/978-3-0348-8849-3_5.  Google Scholar

[5]

M. Brokate and P. Krejčí, Wellposedness of kinematic hardening models in elastoplasticity, RAIRO Math. Modél. Numer. Anal., 32 (1998), 177-209.  doi: 10.1051/m2an/1998320201771.  Google Scholar

[6]

M. Brokate and P. Krejčí, Maximum norm wellposedness of nonlinear kinematic hardening models, Contin. Mech. Thermodyn., 9 (1997), 365-380.  doi: 10.1007/s001610050077.  Google Scholar

[7]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Applied Mathematical Sciences, 121, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[8]

K. Chelminski, Mathematical analysis of the Armstrong-Frederick model from the theory of inelastic deformations of metals. First results and open problems, Contin. Mech. Thermodyn., 15 (2003), 221-245.  doi: 10.1007/s00161-002-0112-2.  Google Scholar

[9]

S. Conti and F. Theil, Single-slip elastoplastic microstructures, Arch. Ration. Mech. Anal., 178 (2005), 125-148.  doi: 10.1007/s00205-005-0371-8.  Google Scholar

[10]

G. Dal Maso and G. Lazzaroni, Quasistatic crack growth in finite elasticity with noninterpenetration, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 257-290.  doi: 10.1016/j.anihpc.2009.09.006.  Google Scholar

[11]

G. Dal MasoG. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal., 176 (2005), 165-225.  doi: 10.1007/s00205-004-0351-4.  Google Scholar

[12]

G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin-New York, 1976. doi: 10.1007/978-3-642-66165-5.  Google Scholar

[13]

M. A. Eisenberg and A. Phillips, On nonlinear kinematic hardening, Acta Mech., 5 (1968), 1-13.  doi: 10.1007/BF01624439.  Google Scholar

[14]

G. A. Francfort and U. Stefanelli, Quasi-static evolution for the Armstrong-Frederick hardening-plasticity model, Appl. Math. Res. Express. AMRX, 2013 (2013), 297-344.  doi: 10.1093/amrx/abt001.  Google Scholar

[15]

W. Han and B. D. Reddy, Plasticity: Mathematical Theory and Numerical Analysis, Interdisciplinary Applied Mathematics, 9, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4614-5940-8.  Google Scholar

[16]

S. Heinz and A. Mielke, Existence, numerical convergence and evolutionary relaxation for a rate-independent phase-transformation model, Philos. Trans. Roy. Soc. A, 374 (2016), 23pp. doi: 10.1098/rsta.2015.0171.  Google Scholar

[17]

C. Johnson, Existence theorems for plasticity problems, J. Math. Pures Appl., 55 (1976), 431-444.   Google Scholar

[18]

C. Johnson, On finite element methods for plasticity problems, Numer. Math., 26 (1976), 79-84.  doi: 10.1007/BF01396567.  Google Scholar

[19]

C. Johnson, A mixed finite element method for plasticity problems with hardening, SIAM J. Numer. Anal., 14 (1977), 575-583.  doi: 10.1137/0714037.  Google Scholar

[20]

C. Johnson, On plasticity with hardening, J. Math. Anal. Appl., 62 (1978), 325-336.  doi: 10.1016/0022-247X(78)90129-4.  Google Scholar

[21]

I. Kadashevich and V. V. Novozhilov, The theory of plasticity which takes into account residual microstresses, J. Appl. Math. Mech., 22 (1958), 104-118.  doi: 10.1016/0021-8928(58)90086-8.  Google Scholar

[22]

P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, GAKUTO International Series, Mathematical Sciences and Applications, 8, Gakkotosho Col., Ltd., Tokyo, 1996.  Google Scholar

[23] J. Lemaitre and J.-L. Chaboche, Mechanics of Solid Materials, Cambridge University Press, Cambridge, 1990.  doi: 10.1017/CBO9781139167970.  Google Scholar
[24]

M. Lévy, Extrait du mémoire sur les équations générales des mouvements intérieurs des corps solids ductiles au delà des limites où lélasticité pourrait les ramener à leur premier état, J. Math. Pures Appl., 16 (1871), 369-372.   Google Scholar

[25]

J. Lubliner, Plasticity Theory, Macmillan Publishing Company, New York, 1990. Google Scholar

[26]

A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), 73-99.  doi: 10.1007/s00526-004-0267-8.  Google Scholar

[27] G. A. Maugin, The Thermomechanics of Plasticity and Fracture, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9781139172400.  Google Scholar
[28]

E. Melan, Zur Plastizität des räumlichen Kontinuums, Ingenieur Archiv, 9 (1938), 116-126.  doi: 10.1007/BF02084409.  Google Scholar

[29]

A. Mielke and S. Müller, Lower semicontinuity and existence of minimizers in incremental finite-strain elastoplasticity, ZAMM Z. Angew. Math. Mech., 86 (2006), 233-250.  doi: 10.1002/zamm.200510245.  Google Scholar

[30]

R. von Mises, Mechanik der festen Körper im plastisch deformablen Zustand, Nachr. Akad. Wiss. Göttingen Math. Phys. Kl., (1913), 582–592. Google Scholar

[31]

R. von Mises, Mechanik der plastischen Formänderung von Kristallen, ZAMM Z. Angew. Math. Mech., 8 (1928), 161-185.  doi: 10.1002/zamm.19280080302.  Google Scholar

[32]

W. Prager, Recent developments in the mathematical theory of plasticity, J. Appl. Phys., 20 (1949), 235-241.  doi: 10.1063/1.1698348.  Google Scholar

[33]

L. T. Prandtl, Ein Gedankenmodell zur kinetischen Theorie der festen Körper, ZAMM Z. Angew. Math. Mech., 8 (1928), 85-106.  doi: 10.1002/zamm.19280080202.  Google Scholar

[34]

M. Röger and B. Schweizer, Strain gradient visco-plasticity with dislocation densities contributing to the energy, Math. Models Methods Appl. Sci., 27 (2017), 2595-2629.  doi: 10.1142/S0218202517500531.  Google Scholar

[35]

J. C. Simo and T. J. R. Hughes, Computational Inelasticity, Interdisciplinary Applied Mathematics, 7, Springer-Verlag, New York, 1998. doi: 10.1007/b98904.  Google Scholar

[36]

U. Stefanelli, Existence for dislocation-free finite plasticity, ESAIM Control Optim. Calc. Var., 25 (2018), 20pp. doi: 10.1051/cocv/2018014.  Google Scholar

[37]

H. E. Tresca, Mémoire sur l'écoulement des corps solides, Mémoire Présentés par Divers Savants, Acad. Sci. Paris, 20 (1872), 75-135.   Google Scholar

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