##
**Stabilization of the response of cyclically loaded lattice spring models with plasticity.**
*(English)*
Zbl 07369299

Summary: This paper develops an analytic framework to design both stress-controlled and displacement-controlled \(T\)-periodic loadings which make the quasistatic evolution of a one-dimensional network of elastoplastic springs converging to a unique periodic regime. The solution of such an evolution problem is a function \(t \mapsto (e(t), p(t))\), where \(e_i(t)\) is the elastic elongation and \(p_i(t)\) is the relaxed length of spring \(i\), defined on \([t_0, \infty )\) by the initial condition \((e(t_0), p(t_0))\). After we rigorously convert the problem into a Moreau sweeping process with a moving polyhedron \(C(t)\) in a vector space \(E\) of dimension \(d\), it becomes natural to expect (based on a result by P. Krejčí [Hysteresis, convexity and dissipation in hyperbolic equations. Tokyo: Gakkotosho (1996; Zbl 1187.35003)]) that the elastic component \(t \mapsto e(t)\) always converges to a \(T\)-periodic function as \(t \rightarrow \infty \). The achievement of this paper is in spotting a class of loadings where the Krejci’s limit does not depend on the initial condition \((e(t_0), p(t_0))\) and so all the trajectories approach the same \(T\)-periodic regime. The proposed class of sweeping processes is the one for which the normals of any \(d\) different facets of the moving polyhedron \(C(t)\) are linearly independent. We further link this geometric condition to mechanical properties of the given network of springs. We discover that the normals of any \(d\) different facets of the moving polyhedron \(C(t)\) are linearly independent, if the number of displacement-controlled loadings is two less the number of nodes of the given network of springs and when the magnitude of the stress-controlled loading is sufficiently large (but admissible). The result can be viewed as an analogue of the high-gain control method for elastoplastic systems. In continuum theory of plasticity, the respective result is known as Frederick-Armstrong theorem.

### MSC:

74H55 | Stability of dynamical problems in solid mechanics |

74M05 | Control, switches and devices (“smart materials”) in solid mechanics |

74C05 | Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials) |

93C15 | Control/observation systems governed by ordinary differential equations |

### Keywords:

elastoplasticity; Moreau sweeping process; quasistatic evolution; periodic loading; stress control; displacement control; stabilization### Citations:

Zbl 1187.35003
PDF
BibTeX
XML
Cite

\textit{I. Gudoshnikov} and \textit{O. Makarenkov}, ESAIM, Control Optim. Calc. Var. 27, Paper No. S8, 43 p. (2021; Zbl 07369299)

### References:

[1] | S. Adly, M. Ait Mansour and L. Scrimali, Sensitivity analysis of solutions to a class of quasi-variational inequalities. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 8 (2005) 767-771. · Zbl 1150.49010 |

[2] | S. Adly, F. Nacry and L. Thibault, Discontinuous sweeping process with prox-regular sets. ESAIM: COCV 23 (2017) 1293-1329. · Zbl 1379.49023 |

[3] | R.B. Bapat, Graphs and matrices. Universitext. Springer, London (2010) 171. |

[4] | J. Bastien, F. Bernardin and C.-H. Lamarque, Non-smooth deterministic or stochastic discrete dynamical systems. Applications to models with friction or impact. Mechanical Engineering and Solid Mechanics Series. John Wiley & Sons, Inc., Hoboken, NJ (2013) 496. · Zbl 1280.70001 |

[5] | T.R. Bieler, N.T. Wright, F. Pourboghrat, C. Compton, K.T. Hartwig, D. Baars, A. Zamiri, S. Chandrasekaran, P. Darbandi, H. Jiang, E. Skoug, S. Balachandran, G.E. Ice and W. Liu, Physical and mechanical metallurgy of high purity Nb for accelerator cavities. Phys. Rev. Spec. Top. 13 (2010) 031002. |

[6] | I. Blechman, Paradox of fatigue of perfect soft metals in terms of micro plasticity and damage. Int. J. Fatigue 120 (2019) 353-375. |

[7] | D. Bremner, K. Fukuda and A. Marzetta, Primal-dual methods for vertex and facet enumeration. ACM Symposium on Computational Geometry (Nice, 1997). Discr. Comput. Geom. 20 (1998) 333-357. · Zbl 0910.68217 |

[8] | B. Brogliato, Absolute stability and the Lagrange-Dirichlet theorem with monotone multivalued mappings. Syst. Control Lett. 51 (2004) 343-353. · Zbl 1157.93455 |

[9] | B. Brogliato, Nonsmooth mechanics. Models, dynamics and control. 3rd edn. Communications and Control Engineering Series. Springer, Berlin (2016). · Zbl 1333.74002 |

[10] | B. Brogliato and W.P.M.H. Heemels, Observer Design for Lur’e Systems With Multivalued Mappings: A Passivity Approach. IEEE Trans. Auto. Control 54 (2009) 1996-2001. · Zbl 1367.93086 |

[11] | B. Brogliato and L. Thibault, Existence and uniqueness of solutions for non-autonomous complementarity dynamical systems. J. Convex Anal. 17 (2010) 961-990. · Zbl 1217.34026 |

[12] | M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, Berlin (1996). · Zbl 0951.74002 |

[13] | G.A. Buxton, C.M. Care and D.J. Cleaver, A lattice spring model of heterogeneous materials with plasticity. Modell. Simul. Mater. Sci. Eng. 9 (2001) 485-497. |

[14] | R. Cang, Y. Xu, S. Chen, Y. Liu, Y. Jiao and M.Y. Ren, Microstructure representation and reconstruction of heterogeneous materials via deep belief network for computational material design. J. Mech. Des. 139 (2017) 071404. |

[15] | H. Chen, E. Lin and Y. Liu, A novel Volume-Compensated Particle method for 2D elasticity and plasticity analysis. Int. J. Solids Struct. 51 (2014) 1819-1833. |

[16] | G. Colombo, R. Henrion, N.D. Hoang and B.S. Mordukhovich, Optimal control of the sweeping process over polyhedral controlled sets. J. Differ. Equ. 260 (2016) 3397-3447. · Zbl 1334.49070 |

[17] | J.B. Conway, A Course in Functional Analysis. 2nd edn. Springer, Berlin (1997). |

[18] | V. Crismale, Globally stable quasistatic evolution for a coupled elastoplastic-damage model. ESAIM: COCV 22 (2016) 883-912. · Zbl 1342.74026 |

[19] | C.O. Frederick and P.J. Armstrong, Convergent internal stresses and steady cyclic states of stress. J. Strain Anal. 1 (1966) 154-159. |

[20] | S.H. Friedberg, A.J. Insel and L.E. Spence, Linear Algebra, 4th edn. Prentice-Hall of India, New Delhi (2004). |

[21] | G. Garcea and L. Leonetti, A unified mathematical programming formulation of strain driven and interior point algorithms for shakedown and limit analysis. Internat. J. Numer. Methods Eng. 88 (2011) 1085-1111. · Zbl 1242.74115 |

[22] | W. Grzesikiewicz, A. Wakulicz and A. Zbiciak, Mathematical modelling of rate-independent pseudoelastic SMA material. Int. J. Non-Linear Mech. 46 (2011) 870-876. |

[23] | W. Han and B.D. Reddy, Plasticity. Mathematical theory and numerical analysis. 2nd edn. Interdisciplinary Applied Mathematics, 9. Springer, New York (2013). |

[24] | M. Heitzer, G. Pop and M. Staat, Basis reduction for the shakedown problem for bounded kinematic hardening material. J. Global Optim. 17 (2000) 185-200. · Zbl 1011.74009 |

[25] | H.R. Henriquez, M. Pierri and P. Taboas, On S-asymptotically ω-periodic functions on Banach spaces and applications. J. Math. Anal. Appl. 343 (2008) 1119-1130. · Zbl 1146.43004 |

[26] | J.-B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of convex analysis. Grundlehren Text Editions. Springer-Verlag, Berlin (2001) x+259. |

[27] | D.W. Holmes, J.G. Loughran and H. Suehrcke, Constitutive model for large strain deformation of semicrystalline polymers. Mech. Time-Depend Mater. 10 (2006) 281-313. |

[28] | H. Hubel, Simplified Theory of Plastic Zones. Springer, Berlin (2015). |

[29] | A. Isidori, Nonlinear control systems. An introduction. 2nd edn. Communications and Control Engineering Series. Springer-Verlag, Berlin (1989) xii+479. |

[30] | L. Jakabcin, A visco-elasto-plastic evolution model with regularized fracture. ESAIM: COCV 22 (2016) 148-168. · Zbl 1337.49017 |

[31] | M. Jirasek and Z.P. Bazant, Inelastic Analysis of Structures. Jhon Wiley & Sons, London (2002). |

[32] | P. Jordan, A.E. Kerdok, R.D. Howe and S. Socrate, Identifying a Minimal Rheological Configuration: A Tool for Effective and Efficient Constitutive Modeling of Soft Tissues. J. Biomech. Eng. 133 (2011) 041006. |

[33] | M. Kamenskii, O. Makarenkov, L. Niwanthi Wadippuli and P. Raynaud de Fitte, Global stability of almost periodic solutions to monotone sweeping processes and their response to non-monotone perturbations. Nonlinear Anal. Hybrid Syst. 30 (2018) 213-224. · Zbl 1412.34075 |

[34] | M.A. Krasnosel’skii, The operator of translation along the trajectories of differential equations. Translations of Mathematical Monographs, Vol. 19. Translated from the Russian by Scripta Technica. American Mathematical Society, Providence, R.I. (1968). |

[35] | M. Krasnosel’skii and A. Pokrovskii, Systems with Hysteresis. Springer, Berlin (1989). |

[36] | P. Krejci, Hysteresis, Convexity and Dissipation in Hyperbolic Equations. Gattotoscho, Tokyo (1996). · Zbl 1187.35003 |

[37] | P. Krejci and A. Vladimirov, Polyhedral sweeping processes with oblique reflection in the space of regulated functions. Set-Valued Anal. 11 (2003) 91-110. · Zbl 1035.34010 |

[38] | M. Kunze and M.D.P. Monteiro Marques, An introduction to Moreau’s sweeping process. Impacts in mechanical systems (Grenoble, 1999), Vol. 551 of Lecture Notes in Physics. Springer, Berlin (2000) 1-60. · Zbl 1047.34012 |

[39] | R.I. Leine and N. van de Wouw, Stability and convergence of mechanical systems with unilateral constraints, Lecture Notes in Applied and Computational Mechanics, 36. Springer-Verlag, Berlin (2008). · Zbl 1143.70001 |

[40] | R.I. Leine and N. van de Wouw, Uniform convergence of monotone measure differential inclusions: with application to the control of mechanical systems with unilateral constraints. Internat. J. Bifur. Chaos Appl. Sci. Eng. 18 (2008) 1435-1457. · Zbl 1147.34310 |

[41] | H.X. Li, Kinematic shakedown analysis under a general yield condition with non-associated plastic flow. Int. J. Mech. Sci. 52 (2010) 1-12. |

[42] | C.W. Li, X. Tang, J.A. Munoz, J.B. Keith, S.J. Tracy, D.L. Abernathy and B. Fultz, Structural Relationship between Negative Thermal Expansion and Quartic Anharmonicity of Cubic ScF_3 . Phys. Rev. Lett. 107 (2011) 195504. |

[43] | J.A.C. Martins, M.D.P Monteiro Marques and A. Petrov, On the stability of quasi-static paths for finite dimensional elastic-plastic systems with hardening. ZAMM Z. Angew. Math. Mech. 87 (2007) 303-313. · Zbl 1116.74026 |

[44] | J.L. Massera, The existence of periodic solutions of systems of differential equations. Duke Math. J. 17 (1950) 457-475. · Zbl 0038.25002 |

[45] | J.-J. Moreau, On unilateral constraints, friction and plasticity. New variational techniques in mathematical physics (Centro Internaz. Mat. Estivo (C.I.M.E.), II Ciclo, Bressanone, 1973). Edizioni Cremonese, Rome (1974) 171-32. |

[46] | L. Narici and E. Beckenstein, Topological vector spaces. 2nd edn. Vol. 296 of Pure and Applied Mathematics. CRC Press, Boca Raton, FL (2011). · Zbl 1219.46001 |

[47] | C. Polizzotto, Variational methods for the steady state response of elastic-plastic solids subjected to cyclic loads. Int. J. Solids Struct. 40 (2003) 2673-2697. · Zbl 1051.74542 |

[48] | A.R.S. Ponter and H. Chen, A minimum theorem for cyclic load in excess of shakedown, with application to the evaluation of a ratchet limit. Eur. J. Mech. A/Solids 20 (2001) 539-553. · Zbl 1002.74018 |

[49] | V. Recupero, BV continuous sweeping processes. J. Differ. Equ. 259 (2015) 4253-4272. · Zbl 1322.49014 |

[50] | R.T. Rockafellar, Convex analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, N.J. (1970). · Zbl 0193.18401 |

[51] | J. Schwiedrzik, R. Raghavan, A. Burki, V. LeNader, U. Wolfram, J. Michler and P. Zysset, In situ micropillar compression reveals superior strength and ductility but an absence of damage in lamellar bone. Nat. Mater. 13 (2014) 740-747. |

[52] | E. Svanidze, T. Besara, M.F. Ozaydin, C.S. Tiwary, J.K. Wang, S. Radhakrishnan, S. Mani, Y. Xin, K. Han, H. Liang, T. Siegrist, P. M. Ajayan and E. Morosan, High hardness in the biocompatible intermetallic compound β − Ti_3 Au. Sci. Adv.2 (2016) e1600319. |

[53] | A. Visintin, Differential Models of Hysteresis. Springer, Berlin (1994). · Zbl 0820.35004 |

[54] | D. Weichert and G. Maier, Inelastic behavior of structures under variable repeated loads. Springer, New York (2002). · Zbl 1030.00039 |

[55] | J. Zhang, B. Koo, Y. Liu, J. Zou, A. Chattopadhyay and L. Dai, A novel statistical spring-bead based network model for self-sensing smart polymer materials. Smart Mater. Struct. 24 (2015) 085022. |

[56] | N. Zouain and R. SantAnna, Computational formulation for the asymptotic response of elastoplastic solids under cyclic loads. Eur. J. Mech. A/Solids 61 (2017) 267-278. · Zbl 1406.74126 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.