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October  2018, 23(8): 3361-3386. doi: 10.3934/dcdsb.2018246

On geometric conditions for reduction of the Moreau sweeping process to the Prandtl-Ishlinskii operator

Dmitrii Rachinskii ,

Department of Mathematical Sciences, The University of Texas at Dallas, USA

* Corresponding author: Dmitrii Rachinskii

Received  December 2017 Revised  March 2018 Published  August 2018

The sweeping process was proposed by J. J. Moreau as a general mathematical formalism for quasistatic processes in elastoplastic bodies. This formalism deals with connected Prandtl's elastic-ideal plastic springs, which can form a system with an arbitrarily complex topology. The model describes the complex relationship between stresses and elongations of the springs as a multi-dimensional differential inclusion (variational inequality). On the other hand, the Prandtl-Ishlinskii model assumes a very simple connection of springs. This model results in an input-output operator, which has many good mathematical properties and admits an explicit solution for an arbitrary input. It turns out that the sweeping processes can be reducible to the Prandtl-Ishlinskii operator even if the topology of the system of springs is complex. In this work, we analyze the conditions for such reducibility.

Keywords: Differential inclusion, elastoplasticity, quasistatic process, model reduction, explicit solution, hysteresis loop.
Mathematics Subject Classification: Primary: 74C15, 78M34; Secondary: 47J22.
Citation: Dmitrii Rachinskii. On geometric conditions for reduction of the Moreau sweeping process to the Prandtl-Ishlinskii operator. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3361-3386. doi: 10.3934/dcdsb.2018246
References:
[1]

S. Adly, T. Haddad and L. Thibault, Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities, Math. Program Ser. B, 148 (2014), 5-47. doi: 10.1007/s10107-014-0754-4. Google Scholar

[2]

F. Al-Bender, W. Symens, J. Swevers and H. Van Brussel, Theoretical analysis of the dynamic behavior of hysteresis elements in mechanical systems, Int. J. Non-Linear Mechanics, 39 (2004), 1721-1735. doi: 10.1016/j.ijnonlinmec.2004.04.005. Google Scholar

[3]

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

[4]

M. Brokate, P. Krejci and D. Rachinskii, Some analytical properties of the multidimensional continuous Mroz model of plasticity, Control Cybernetics, 27 (1998), 199-215. Google Scholar

[5]

G. Colombo, R. Henrion, N. D. Hoang and B. S. Mordukhovich, Optimal control of the sweeping process: The polyhedral case, J. Differ. Equ., 260 (2016), 3397-3447. doi: 10.1016/j.jde.2015.10.039. Google Scholar

[6]

G. Colombo and M. D. P. Monteiro Marques, Sweeping by a continuous prox-regular set, J. Diff. Equ., 187 (2003), 46-62. doi: 10.1016/S0022-0396(02)00021-9. Google Scholar

[7]

G. Colombo and M. Palladino, The minimum time function for the controlled Moreau's sweeping process, SIAM J. Control Optimization, 54 (2016), 2036-2062. doi: 10.1137/15M1043364. Google Scholar

[8]

D. Davino, P. Krejci and C. Visone, Fully coupled modeling of magnetomechanical hysteresis through thermodynamic compatibility, Smart Mater. Struct. , 22 (2013), 095009.Google Scholar

[9]

D. Davino, P. Krejci, A. Pimenov, D. Rachinskii and C. Visone, Analysis of an operator-differential model for magnetostrictive energy harvesting, Communications Nonlinear Science and Numerical Simulation, 39 (2016), 504-519. doi: 10.1016/j.cnsns.2016.04.004. Google Scholar

[10]

W. Desch and J. Turi, The Stop operator related to a convex polyhedron, J. Diff. Equ., 157 (1999), 329-347. doi: 10.1006/jdeq.1998.3601. Google Scholar

[11]

S. Di Marino, B. Maury and F. Santambrogio, Measure sweeping processes, J. Convex Anal., 23 (2016), 567-601. Google Scholar

[12]

M. Eleuteri, J. Kopfova and P. Krejčı, A new phase field model for material fatigue in an oscillating elastoplastic beam, Discrete Continuous Dynam. Systems - A, 35 (2015), 2465-2495. doi: 10.3934/dcds.2015.35.2465. Google Scholar

[13]

G.-Y. Gu, L.-M. Zhu and C.-Y. Su, Modeling and compensation of asymmetric hysteresis nonlinearity for piezoceramic actuators with a modified Prandtl-Ishlinskii model, IEEE Trans Industrial Electronics, 61 (2014), 1583-1595. doi: 10.1109/TIE.2013.2257153. Google Scholar

[14]

A. Yu. Ishlinskii, Some applications of statistical methods to describing deformations of bodies, Izv. AN SSSR Techn. Ser., 9 (1944), 583-590. Google Scholar

[15]

M. A. Janaideh and P. Krejci, Inverse rate-dependent Prandtl-Ishlinskii model for feedforward compensation of hysteresis in a piezomicropositioning actuator, IEEE/ASME Trans. Mechatronics, 18 (2013), 1498-1507. doi: 10.1109/TMECH.2012.2205265. Google Scholar

[16]

B. Kaltenbacher and P. Krejci, A thermodynamically consistent phenomenological model for ferroelectric and ferroelastic hysteresis, ZAMM - Z. Angew. Math. Mech., 96 (2016), 874-891. doi: 10.1002/zamm.201400292. Google Scholar

[17]

M. Kamenskii and O. Makarenkov, On the response of autonomous sweeping processes to periodic perturbations, Set-Valued and Variational Analysis, 24 (2016), 551-563. doi: 10.1007/s11228-015-0348-1. Google Scholar

[18]

J. Kopfova and V. Recupero, BV-norm continuity of sweeping processes driven by a set with constant shape, J. Differ. Equ., 261 (2016), 5875-5899. doi: 10.1016/j.jde.2016.08.025. Google Scholar

[19]

M. Krasnosel'skii and A. Pokrovskii, Systems with Hysteresis, Springer, 1989. doi: 10.1007/978-3-642-61302-9. Google Scholar

[20]

P. Krejci, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakkotosho Co. Ltd, Tokyo, 1996. Google Scholar

[21]

P. Krejci and J. Sprekels, Elastic-ideally plastic beams and Prandtl-Ishlinskii hysteresis operators, Math. Meth. Appl. Sci., 30 (2007), 2371-2393. doi: 10.1002/mma.892. Google Scholar

[22]

P. Krejci, H. Lamba, S. Melnik and D. Rachinskii, Analytical solution for a class of network dynamics with mechanical and financial applications, Phys. Rev. E, 90 (2014), 032822.Google Scholar

[23]

P. Krejci, H. Lamba, S. Melnik and D. Rachinskii, Kurzweil integral representation of interacting Prandtl-Ishlinskii operators, Discrete Continuous Dynam. Systems - B, 20 (2015), 2949-2965. doi: 10.3934/dcdsb.2015.20.2949. Google Scholar

[24]

P. Krejci, H. Lamba, G. A. Monteiro and D. Rachinskii, The Kurzweil integral in financial market modeling, Mathematica Bohemica, 141 (2016), 261-286. doi: 10.21136/MB.2016.18. Google Scholar

[25]

P. Krejci and V. Recupero, Comparing BV solutions of rate independent processes, J. Convex Analysis, 21 (2014), 121-146. Google Scholar

[26]

P. Krejci and T. Roche, Lipschitz continuous data dependence of sweeping processes in BV spaces, Discrete Continuous Dynam. Systems - B, 15 (2011), 637-650. doi: 10.3934/dcdsb.2011.15.637. Google Scholar

[27]

P. Krejci and A. Vladimirov, Polyhedral sweeping processes with oblique reflection in the space of regulated functions, Set-Valued Analysis, 11 (2003), 91-110. doi: 10.1023/A:1021980201718. Google Scholar

[28]

K. Kuhnen, Modeling, identification and compensation of complex hysteretic nonlinearities: A modified Prandtl-Ishlinskii approach, Eur. J. Control, 9 (2003), 407-418. doi: 10.3166/ejc.9.407-418. Google Scholar

[29]

M. Kunze and M. D. P. Monteiro Marques, An introduction to Moreau's sweeping process, in Impacts in Mechanical Systems (ed. B. Brogliato), Lecture Notes in Physics, 551, Springer, (2000), 1-60. doi: 10.1007/3-540-45501-9_1. Google Scholar

[30]

B. Maury, A. Roudneff-Chupin and F. Santambrogio, Congestion-driven dendritic growth, Discrete Continuous Dynam. Systems, 34 (2014), 1575-1604. doi: 10.3934/dcds.2014.34.1575. Google Scholar

[31]

I. Mayergoyz, Mathematical Models of Hysteresis and Their Applications, Elsevier, 2003.Google Scholar

[32]

A. Mielke, Generalized Prandtl-Ishlinskii operators arising from homogenization and dimension reduction, Physica B, 407 (2012), 1330-1335. doi: 10.1016/j.physb.2011.10.013. Google Scholar

[33]

A. Mielke and T. Roubiček, Rate Independent Systems, Theory and Applications, Springer, 2015. doi: 10.1007/978-1-4939-2706-7. Google Scholar

[34]

J. J. Moreau, On unilateral constraints, friction and plasticity, in New Variational Techniques in Mathematical Physics (eds. G. Capriz and G. Stampacchia), Springer, (1974), 171-322. Google Scholar

[35]

J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differ. Equ., 26 (1977), 347-374. doi: 10.1016/0022-0396(77)90085-7. Google Scholar

[36]

J. J. Moreau, Application of convex analysis to the treatment of elastoplastic systems, in Applications of Methods of Functional Analysis to Problems in Mechanics (eds. P. Germain and B. Nayroles), Springer, (1976), 56-89. doi: 10.1007/BFb0088746. Google Scholar

[37]

L. Prandtl, Ein Gedankenmodell zur kinetischen Theorie der festen Körper, Z. Ang. Math. Mech., 8 (1928), 85-106. Google Scholar

[38]

D. I. Rachinskii, Equivalent combinations of stops, Automat. Remote Control, 59 (1998), 1370-1378. Google Scholar

[39]

V. Recupero, The play operator on the rectifiable curves in a Hilbert space, Math. Methods Appl. Sci., 31 (2008), 1283-1295. doi: 10.1002/mma.968. Google Scholar

[40]

V. Recupero, BV continuous sweeping processes, J. Differ. Equ., 259 (2015), 4253-4272. doi: 10.1016/j.jde.2015.05.019. Google Scholar

[41]

D. D. Rizos and S. D. Fassois, Friction identification based upon the LuGre and Maxwell slip models, IEEE Trans Control Systems Technology, 17 (2009), 153-160. Google Scholar

[42]

M. Ruderman, Presliding hysteresis damping of LuGre and Maxwell-slip friction models, Mechatronics, 30 (2015), 225-230. doi: 10.1016/j.mechatronics.2015.07.007. Google Scholar

[43]

M. Ruderman, F. Hoffmann and T. Bertram, Modeling and identification of elastic robot joints with hysteresis and backlash, IEEE Trans. Industrial Electronics, 56 (2009), 3840-3847. doi: 10.1109/TIE.2009.2015752. Google Scholar

[44]

I. Rychlik, A new definition of the rainflow cycle counting method, Int. J. Fatigue, 9 (1987), 119-121. doi: 10.1016/0142-1123(87)90054-5. Google Scholar

[45]

H. Sayyaadi and M. R. Zakerzadeh, Position control of shape memory alloy actuator based on the generalized Prandtl-Ishlinskii inverse model, Mechatronics, 22 (2012), 945-957. doi: 10.1016/j.mechatronics.2012.06.003. Google Scholar

[46]

J. P. Sethna, K. Dahmen, S. Kartha, J. A. Krumhansl, B. W. Roberts and J. D. Shore, Hysteresis and hierarchies: Dynamics of disorder-driven first-order phase transformations, Phys. Rev. Lett., 70 (1993), 3347-3350. doi: 10.1103/PhysRevLett.70.3347. Google Scholar

[47]

A. Visintin, Differential Models of Hysteresis, Springer, 1994. doi: 10.1007/978-3-662-11557-2. Google Scholar

show all references

References:
[1]

S. Adly, T. Haddad and L. Thibault, Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities, Math. Program Ser. B, 148 (2014), 5-47. doi: 10.1007/s10107-014-0754-4. Google Scholar

[2]

F. Al-Bender, W. Symens, J. Swevers and H. Van Brussel, Theoretical analysis of the dynamic behavior of hysteresis elements in mechanical systems, Int. J. Non-Linear Mechanics, 39 (2004), 1721-1735. doi: 10.1016/j.ijnonlinmec.2004.04.005. Google Scholar

[3]

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

[4]

M. Brokate, P. Krejci and D. Rachinskii, Some analytical properties of the multidimensional continuous Mroz model of plasticity, Control Cybernetics, 27 (1998), 199-215. Google Scholar

[5]

G. Colombo, R. Henrion, N. D. Hoang and B. S. Mordukhovich, Optimal control of the sweeping process: The polyhedral case, J. Differ. Equ., 260 (2016), 3397-3447. doi: 10.1016/j.jde.2015.10.039. Google Scholar

[6]

G. Colombo and M. D. P. Monteiro Marques, Sweeping by a continuous prox-regular set, J. Diff. Equ., 187 (2003), 46-62. doi: 10.1016/S0022-0396(02)00021-9. Google Scholar

[7]

G. Colombo and M. Palladino, The minimum time function for the controlled Moreau's sweeping process, SIAM J. Control Optimization, 54 (2016), 2036-2062. doi: 10.1137/15M1043364. Google Scholar

[8]

D. Davino, P. Krejci and C. Visone, Fully coupled modeling of magnetomechanical hysteresis through thermodynamic compatibility, Smart Mater. Struct. , 22 (2013), 095009.Google Scholar

[9]

D. Davino, P. Krejci, A. Pimenov, D. Rachinskii and C. Visone, Analysis of an operator-differential model for magnetostrictive energy harvesting, Communications Nonlinear Science and Numerical Simulation, 39 (2016), 504-519. doi: 10.1016/j.cnsns.2016.04.004. Google Scholar

[10]

W. Desch and J. Turi, The Stop operator related to a convex polyhedron, J. Diff. Equ., 157 (1999), 329-347. doi: 10.1006/jdeq.1998.3601. Google Scholar

[11]

S. Di Marino, B. Maury and F. Santambrogio, Measure sweeping processes, J. Convex Anal., 23 (2016), 567-601. Google Scholar

[12]

M. Eleuteri, J. Kopfova and P. Krejčı, A new phase field model for material fatigue in an oscillating elastoplastic beam, Discrete Continuous Dynam. Systems - A, 35 (2015), 2465-2495. doi: 10.3934/dcds.2015.35.2465. Google Scholar

[13]

G.-Y. Gu, L.-M. Zhu and C.-Y. Su, Modeling and compensation of asymmetric hysteresis nonlinearity for piezoceramic actuators with a modified Prandtl-Ishlinskii model, IEEE Trans Industrial Electronics, 61 (2014), 1583-1595. doi: 10.1109/TIE.2013.2257153. Google Scholar

[14]

A. Yu. Ishlinskii, Some applications of statistical methods to describing deformations of bodies, Izv. AN SSSR Techn. Ser., 9 (1944), 583-590. Google Scholar

[15]

M. A. Janaideh and P. Krejci, Inverse rate-dependent Prandtl-Ishlinskii model for feedforward compensation of hysteresis in a piezomicropositioning actuator, IEEE/ASME Trans. Mechatronics, 18 (2013), 1498-1507. doi: 10.1109/TMECH.2012.2205265. Google Scholar

[16]

B. Kaltenbacher and P. Krejci, A thermodynamically consistent phenomenological model for ferroelectric and ferroelastic hysteresis, ZAMM - Z. Angew. Math. Mech., 96 (2016), 874-891. doi: 10.1002/zamm.201400292. Google Scholar

[17]

M. Kamenskii and O. Makarenkov, On the response of autonomous sweeping processes to periodic perturbations, Set-Valued and Variational Analysis, 24 (2016), 551-563. doi: 10.1007/s11228-015-0348-1. Google Scholar

[18]

J. Kopfova and V. Recupero, BV-norm continuity of sweeping processes driven by a set with constant shape, J. Differ. Equ., 261 (2016), 5875-5899. doi: 10.1016/j.jde.2016.08.025. Google Scholar

[19]

M. Krasnosel'skii and A. Pokrovskii, Systems with Hysteresis, Springer, 1989. doi: 10.1007/978-3-642-61302-9. Google Scholar

[20]

P. Krejci, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakkotosho Co. Ltd, Tokyo, 1996. Google Scholar

[21]

P. Krejci and J. Sprekels, Elastic-ideally plastic beams and Prandtl-Ishlinskii hysteresis operators, Math. Meth. Appl. Sci., 30 (2007), 2371-2393. doi: 10.1002/mma.892. Google Scholar

[22]

P. Krejci, H. Lamba, S. Melnik and D. Rachinskii, Analytical solution for a class of network dynamics with mechanical and financial applications, Phys. Rev. E, 90 (2014), 032822.Google Scholar

[23]

P. Krejci, H. Lamba, S. Melnik and D. Rachinskii, Kurzweil integral representation of interacting Prandtl-Ishlinskii operators, Discrete Continuous Dynam. Systems - B, 20 (2015), 2949-2965. doi: 10.3934/dcdsb.2015.20.2949. Google Scholar

[24]

P. Krejci, H. Lamba, G. A. Monteiro and D. Rachinskii, The Kurzweil integral in financial market modeling, Mathematica Bohemica, 141 (2016), 261-286. doi: 10.21136/MB.2016.18. Google Scholar

[25]

P. Krejci and V. Recupero, Comparing BV solutions of rate independent processes, J. Convex Analysis, 21 (2014), 121-146. Google Scholar

[26]

P. Krejci and T. Roche, Lipschitz continuous data dependence of sweeping processes in BV spaces, Discrete Continuous Dynam. Systems - B, 15 (2011), 637-650. doi: 10.3934/dcdsb.2011.15.637. Google Scholar

[27]

P. Krejci and A. Vladimirov, Polyhedral sweeping processes with oblique reflection in the space of regulated functions, Set-Valued Analysis, 11 (2003), 91-110. doi: 10.1023/A:1021980201718. Google Scholar

[28]

K. Kuhnen, Modeling, identification and compensation of complex hysteretic nonlinearities: A modified Prandtl-Ishlinskii approach, Eur. J. Control, 9 (2003), 407-418. doi: 10.3166/ejc.9.407-418. Google Scholar

[29]

M. Kunze and M. D. P. Monteiro Marques, An introduction to Moreau's sweeping process, in Impacts in Mechanical Systems (ed. B. Brogliato), Lecture Notes in Physics, 551, Springer, (2000), 1-60. doi: 10.1007/3-540-45501-9_1. Google Scholar

[30]

B. Maury, A. Roudneff-Chupin and F. Santambrogio, Congestion-driven dendritic growth, Discrete Continuous Dynam. Systems, 34 (2014), 1575-1604. doi: 10.3934/dcds.2014.34.1575. Google Scholar

[31]

I. Mayergoyz, Mathematical Models of Hysteresis and Their Applications, Elsevier, 2003.Google Scholar

[32]

A. Mielke, Generalized Prandtl-Ishlinskii operators arising from homogenization and dimension reduction, Physica B, 407 (2012), 1330-1335. doi: 10.1016/j.physb.2011.10.013. Google Scholar

[33]

A. Mielke and T. Roubiček, Rate Independent Systems, Theory and Applications, Springer, 2015. doi: 10.1007/978-1-4939-2706-7. Google Scholar

[34]

J. J. Moreau, On unilateral constraints, friction and plasticity, in New Variational Techniques in Mathematical Physics (eds. G. Capriz and G. Stampacchia), Springer, (1974), 171-322. Google Scholar

[35]

J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differ. Equ., 26 (1977), 347-374. doi: 10.1016/0022-0396(77)90085-7. Google Scholar

[36]

J. J. Moreau, Application of convex analysis to the treatment of elastoplastic systems, in Applications of Methods of Functional Analysis to Problems in Mechanics (eds. P. Germain and B. Nayroles), Springer, (1976), 56-89. doi: 10.1007/BFb0088746. Google Scholar

[37]

L. Prandtl, Ein Gedankenmodell zur kinetischen Theorie der festen Körper, Z. Ang. Math. Mech., 8 (1928), 85-106. Google Scholar

[38]

D. I. Rachinskii, Equivalent combinations of stops, Automat. Remote Control, 59 (1998), 1370-1378. Google Scholar

[39]

V. Recupero, The play operator on the rectifiable curves in a Hilbert space, Math. Methods Appl. Sci., 31 (2008), 1283-1295. doi: 10.1002/mma.968. Google Scholar

[40]

V. Recupero, BV continuous sweeping processes, J. Differ. Equ., 259 (2015), 4253-4272. doi: 10.1016/j.jde.2015.05.019. Google Scholar

[41]

D. D. Rizos and S. D. Fassois, Friction identification based upon the LuGre and Maxwell slip models, IEEE Trans Control Systems Technology, 17 (2009), 153-160. Google Scholar

[42]

M. Ruderman, Presliding hysteresis damping of LuGre and Maxwell-slip friction models, Mechatronics, 30 (2015), 225-230. doi: 10.1016/j.mechatronics.2015.07.007. Google Scholar

[43]

M. Ruderman, F. Hoffmann and T. Bertram, Modeling and identification of elastic robot joints with hysteresis and backlash, IEEE Trans. Industrial Electronics, 56 (2009), 3840-3847. doi: 10.1109/TIE.2009.2015752. Google Scholar

[44]

I. Rychlik, A new definition of the rainflow cycle counting method, Int. J. Fatigue, 9 (1987), 119-121. doi: 10.1016/0142-1123(87)90054-5. Google Scholar

[45]

H. Sayyaadi and M. R. Zakerzadeh, Position control of shape memory alloy actuator based on the generalized Prandtl-Ishlinskii inverse model, Mechatronics, 22 (2012), 945-957. doi: 10.1016/j.mechatronics.2012.06.003. Google Scholar

[46]

J. P. Sethna, K. Dahmen, S. Kartha, J. A. Krumhansl, B. W. Roberts and J. D. Shore, Hysteresis and hierarchies: Dynamics of disorder-driven first-order phase transformations, Phys. Rev. Lett., 70 (1993), 3347-3350. doi: 10.1103/PhysRevLett.70.3347. Google Scholar

[47]

A. Visintin, Differential Models of Hysteresis, Springer, 1994. doi: 10.1007/978-3-662-11557-2. Google Scholar

Figure 1.  Hysteresis loops of a Prandtl-Ishlinsii operator with 3 springs. $g$ is the deformation and $R$ is the stress (cf. (25)). Thick (thin) lines correspond to the increasing (decreasing) deformation. The direction of motion along the loops is clockwise. Both the outer hysteresis loop and the inner loop are centrally symmetric (with respect to the corresponding loop center). The ascending branch of the inner loop is congruent to the part $AB$ of the ascending branch of the outer loop. Dashed line $OC$ is the loading curve. The ascending branch $ABC$ of the outer loop is obtained by dilation of the loading curve with the scaling factor 2 and translation which maps the initial point $O$ to $A$
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Figure 2.  Connection of 5 nodes with 9 elastic-ideal plastic springs
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Figure 3.  A nonlinear spring connecting two nodes in the model of Moreau can be represented as a combination of a dry friction element and an ideally elastic spring obeying the Hooke's law (3). The deformation $e_{ij}$ of the ideal spring is called elastic deformation; $\varepsilon_{ij}$ is the elongation of the distance between the nodes $i$ and $j$ (with respect to the distance $\Delta_{ij} = x_j^*-x_i^*$ at zero configuration). Dry friction between the box $B$ and the surface $S$ produces the friction force, which is opposite to the ideal spring force at all times (the quasistatic model). The magnitude of the friction force is limited by the maximal value $a_{ij}\rho_{ij}$. Therefore, the ideal spring deforms but the box does not move with respect to the surface $S$ as long as $|\sigma_{ij}| < a_{ij}\rho_{ij}$. When $|\sigma_{ij}| = a_{ij}\rho_{ij}$, the deformation $e_{ij}$ of the ideal spring and the force remain constant, while the box moves with respect to the surface in the direction of the spring force
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Figure 4.  An example of an input-output trajectory $c_0\, c_1\, \ldots\, c_{15}$ of a nonlinear spring with $c_n = (\varepsilon_{ij}(t_n), e_{ij}(t_n))$ and $t_0 < t_1 < \cdots$. The time series of the deformation $\varepsilon_{ij}$ and the elastic deformation $e_{ij}$ are related by the stop operator $e_{ij} = {\mathcal S}_{\rho_{ij}}[\varepsilon_{ij}]$. Slanted lines have the slope $1$; horizontal lines are $e_{ij} = \pm \rho_{ij}$
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Figure 5.  (a) Parallel connection of springs in the Prandtl-Ishlinskii model. (b) A reducible connection of springs
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Figure 6.  A "linear" connection of springs, which is equivalent to the Prandtl-Ishlinskii model
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Figure 7.  Violations of conditions of Theorem 1. The shaded area represents the polytope $\Pi\cap V$. The vector $f_0$ used in the definition (24) of the Moreau process points in the direction of the vector $B_0B_1$. (a) The prism $\Omega$ does not belong to $\Pi\cap V$. The polyline $B_0B_1B_2C_1C_2C_3$ that contains the polyline $\gamma = B_0B_1B_2$ represents the trajectory of a solution $u$ to (24) for the input $g(t)f_0$ where $g$ increases from zero to a maximum value $g_*$ and then decreases to the minimum value $-g_*$. Formula (36) defines a different polyline $B_0B_1B_2DC_3$ for the same input. (b) Conditions (31), (32) are violated because ${\rm dim}\, F_{1} = {\rm dim}\, F_2$. The polylines $B_0B_1B_2B_3C_1C_2C_3$ and $B_0B_1B_2B_3D_1D_2C_3$ are the trajectory of the inclusion (24) and the curve prescribed by formula (36) in response to the same input as in panel (a).
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Figure 8.  A 'minimal' system, which may be not reducible to the PI operator (depending on the parameters of springs)
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Figure 9.  A graph $G$ of a general linear connection of $m$ springs (thick lines). Thin dashed lines represent possible additional edges as introduced in Theorem 2
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[1]

Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569
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