August  2020, 13(8): 2271-2284. 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  August 2020 Early access  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 and Continuous Dynamical Systems - S, 2020, 13 (8) : 2271-2284. 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. 

[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. 

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[17]

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

[18]

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

[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.

[20]

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

[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.

[22]

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

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

[25]

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

[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.

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

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

[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.

[30]

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

[31]

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

[32]

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

[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.

[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.

[35]

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

[36]

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

[37]

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

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. 

[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. 

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[17]

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

[18]

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

[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.

[20]

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

[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.

[22]

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

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

[25]

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

[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.

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

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

[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.

[30]

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

[31]

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

[32]

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

[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.

[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.

[35]

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

[36]

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

[37]

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

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