Elastoplasticity with softening as a state-dependent sweeping process: non-uniqueness of solutions and emergence of shear bands in lattices of springs

Ivan Gudoshnikov; Ivan Gudoshnikov

doi:10.3934/mine.2026008

Mathematics in Engineering

2026, Volume 8, Issue 2: 209-268. doi: 10.3934/mine.2026008

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Elastoplasticity with softening as a state-dependent sweeping process: non-uniqueness of solutions and emergence of shear bands in lattices of springs

Ivan Gudoshnikov ^,

Institute of Mathematics of the Czech Academy of Sciences, Žitná 25,110 00, Praha 1, Czech Republic

Received: 04 September 2025 Revised: 07 March 2026 Accepted: 10 March 2026 Published: 25 March 2026

Plasticity with softening and fracture mechanics lead to ill-posed mathematical problems due to the loss of monotonicity. Multiple co-existing solutions are possible when softening elements are coupled together, and solutions cannot be continued beyond the point of complete degradation of the set of admissible stresses. We present a state-dependent sweeping process which solves the evolution of elasto-plastic Lattice Spring Models with arbitrary placement of softening, hardening and perfectly plastic springs. Using numerical simulations of regular grid lattices with softening, we demonstrate the emergence of non-symmetric shear bands with strain localization. At the same time, in toy examples it is easy to analytically derive multiple co-existing solutions. These solutions correspond to fixed points in the implicit catch-up algorithm and we observe a discontinuous bifurcation with the exchange of stability of those fixed points.
- state-dependent sweeping process,
- quasi-variational inequalities,
- plasticity with softening,
- non-uniqueness of solutions,
- shear band,
- strain localization
Citation: Ivan Gudoshnikov. Elastoplasticity with softening as a state-dependent sweeping process: non-uniqueness of solutions and emergence of shear bands in lattices of springs[J]. Mathematics in Engineering, 2026, 8(2): 209-268. doi: 10.3934/mine.2026008

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Abstract

Plasticity with softening and fracture mechanics lead to ill-posed mathematical problems due to the loss of monotonicity. Multiple co-existing solutions are possible when softening elements are coupled together, and solutions cannot be continued beyond the point of complete degradation of the set of admissible stresses. We present a state-dependent sweeping process which solves the evolution of elasto-plastic Lattice Spring Models with arbitrary placement of softening, hardening and perfectly plastic springs. Using numerical simulations of regular grid lattices with softening, we demonstrate the emergence of non-symmetric shear bands with strain localization. At the same time, in toy examples it is easy to analytically derive multiple co-existing solutions. These solutions correspond to fixed points in the implicit catch-up algorithm and we observe a discontinuous bifurcation with the exchange of stability of those fixed points.

References

[1]	N. C. Admal, E. B. Tadmor, The non-uniqueness of the atomistic stress tensor and its relationship to the generalized Beltrami representation, J. Mech. Phys. Solids, 93 (2016), 72–92. https://doi.org/10.1016/j.jmps.2016.03.016 doi: 10.1016/j.jmps.2016.03.016
[2]	A. Y. Alfakih, Euclidean distance matrices and their applications in rigidity theory, Springer, 2018. https://doi.org/10.1007/978-3-319-97846-8
[3]	H. Antil, R. Arndt, B. S. Mordukhovich, D. Nguyen, C. N. Rautenberg, Optimal control of a quasi-variational sweeping process, Math. Control Relat. Fields, 16 (2025), 132–156. https://doi.org/10.3934/mcrf.2025010 doi: 10.3934/mcrf.2025010
[4]	C. Baiocchi, A. Capelo, Variational and quasivariational inequalities: applications to free boundary problems, New York: Wiley, 1984.
[5]	R. B. Bapat, Graphs and matrices, Springer London, 2010. https://doi.org/10.1007/978-1-84882-981-7
[6]	E. Bauer, V. A. Kovtunenko, P. Krejčí, G. A. Monteiro, L. Paoli, A. Petrov, Non-convex sweeping processes in contact mechanics, Nonlinear Anal., 87 (2026), 104456. https://doi.org/10.1016/j.nonrwa.2025.104456 doi: 10.1016/j.nonrwa.2025.104456
[7]	H. H. Bauschke, P. L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, CMS Books in Mathematics, Springer New York, 2011.
[8]	Z. P. Bazant, Mechanics of distributed cracking, Appl. Mech. Rev., 39 (1986), 675–705. https://doi.org/10.1115/1.3143724 doi: 10.1115/1.3143724
[9]	R. de Borst, Numerical modelling of bifurcation and localisation in cohesive-frictional materials, Pure Appl. Geophys., 137 (1991), 367–390. https://doi.org/10.1007/BF00879040 doi: 10.1007/BF00879040
[10]	R. de Borst, L. J. Sluys, H. B. Muhlhaus, J. Pamin, Fundamental issues in finite element analyses of localization of deformation, Eng. Comput., 10 (1993), 99–121. https://doi.org/10.1108/eb023897 doi: 10.1108/eb023897
[11]	A. Braides, G. Dal Maso, A. Garroni, Variational formulation of softening phenomena in fracture mechanics: the one-dimensional case, Arch. Rational Mech. Anal., 146 (1999), 23–58. https://doi.org/10.1007/s002050050135 doi: 10.1007/s002050050135
[12]	H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Springer, 2011. https://doi.org/10.1007/978-0-387-70914-7
[13]	M. Brokate, P. Krejcí, H. Schnabel, On uniqueness in evolution quasivariational inequalities, J. Convex. Anal., 11 (2004), 111–130.
[14]	F. C. Campbell, Elements of metallurgy and engineering alloys, ASM International, 2008.
[15]	S. L. Campbell, C. D. Meyer, Generalized inverses of linear transformations, Society for Industrial and Applied Mathematics, 2009.
[16]	C. Castaing, A. G. Ibrahim, M. Yarou, Some contributions to nonconvex sweeping process, J. Nonlinear Convex Anal., 10 (2009), 1–20.
[17]	G. Chen, G. Baker, Energy profile and bifurcation analysis in softening plasticity, Adv. Struct. Eng., 7 (2004), 515–523. https://doi.org/10.1260/1369433042863224 doi: 10.1260/1369433042863224
[18]	H. Chen, E. Lin, Y. Jiao, Y. Liu, A generalized 2D non-local lattice spring model for fracture simulation, Comput. Mech., 54 (2014), 1541–1558. https://doi.org/10.1007/s00466-014-1075-4 doi: 10.1007/s00466-014-1075-4
[19]	J. Clark, D. A. Holton, A first look at graph theory, World Scientific, 1991.
[20]	M. A. Crisfield, Snap-through and snap-back response in concrete structures and the dangers of under-integration, Int. J. Numer. Methods Eng., 22 (1986), 751–767. https://doi.org/10.1002/nme.1620220314 doi: 10.1002/nme.1620220314
[21]	G. Dal Maso, A. DeSimone, M. G. Mora, Quasistatic evolution problems for linearly elastic–perfectly plastic materials, Arch. Rational Mech. Anal., 180 (2006), 237–291. https://doi.org/10.1007/s00205-005-0407-0 doi: 10.1007/s00205-005-0407-0
[22]	G. Dal Maso, A. DeSimone, M. G. Mora, M. Morini, A vanishing viscosity approach to quasistatic evolution in plasticity with softening, Arch. Rational Mech. Anal., 189 (2008), 469–544. https://doi.org/10.1007/s00205-008-0117-5 doi: 10.1007/s00205-008-0117-5
[23]	G. Dal Maso, A. DeSimone, M. G. Mora, M. Morini, Globally stable quasistatic evolution in plasticity with softening, Netw. Heterog. Media, 3 (2008), 567–614. https://doi.org/10.3934/nhm.2008.3.567 doi: 10.3934/nhm.2008.3.567
[24]	G. Dal Maso, R. Toader, A model for the quasi-static growth of brittle fractures: existence and approximation results, Arch. Rational Mech. Anal., 162 (2002), 101–135. https://doi.org/10.1007/s002050100187 doi: 10.1007/s002050100187
[25]	A. Demyanov, Regularity of stresses in Prandtl-Reuss perfect plasticity, Calc. Var., 34 (2009), 23–72. https://doi.org/10.1007/s00526-008-0174-5 doi: 10.1007/s00526-008-0174-5
[26]	G. Duvant, J. L. Lions, Inequalities in mechanics and physics, Grundlehren der mathematischen Wissenschaften, Vol. 219, Springer Berlin, Heidelberg, 1976. https://doi.org/10.1007/978-3-642-66165-5
[27]	F. Ebobisse, B. D. Reddy, Some mathematical problems in perfect plasticity, Comput. Methods Appl. Mech. Eng., 193 (2004), 5071–5094. https://doi.org/10.1016/j.cma.2004.07.002 doi: 10.1016/j.cma.2004.07.002
[28]	I. Ekeland, R. Temam, Convex analysis and variational problems, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics, 1999. https://doi.org/10.1137/1.9781611971088
[29]	F. Facchinei, J. S. Pang, Finite-dimensional variational inequalities and complementarity problems, Springer Series in Operations Research and Financial Engineering, Springer New York, 2003. https://doi.org/10.1007/b97543
[30]	P. E. Farrell, M. Croci, T. M. Surowiec, Deflation for semismooth equations, Optim. Method. Softw., 35 (2020), 1248–1271. https://doi.org/10.1080/10556788.2019.1613655 doi: 10.1080/10556788.2019.1613655
[31]	G. A. Francfort, A. Giacomini, J. J. Marigo, The elasto-plastic exquisite corpse: a Suquet legacy, J. Mech. Phys. Solids, 97 (2016), 125–139. https://doi.org/10.1016/j.jmps.2016.02.002 doi: 10.1016/j.jmps.2016.02.002
[32]	G. A. Francfort, J. J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids, 46 (1998), 1319–1342. https://doi.org/10.1016/S0022-5096(98)00034-9 doi: 10.1016/S0022-5096(98)00034-9
[33]	P. A. Fuhrmann, U. Helmke, The mathematics of networks of linear systems, Universitext, Springer Cham, 2015. https://doi.org/10.1007/978-3-319-16646-9
[34]	G. N. Gatica, A simple introduction to the mixed finite element method: theory and applications, SpringerBriefs in Mathematics, Springer Cham, 2014. https://doi.org/10.1007/978-3-319-03695-3
[35]	H. Goldstein, C. Poole, J. Safko, Classical mechanics, 3 Eds., Addison Wesley, 2002.
[36]	G. Gruben, D. Morin, M. Langseth, O. S. Hopperstad, Strain localization and ductile fracture in advanced high-strength steel sheets, Eur. J. Mech. - A/Solids, 61 (2017), 315–329. https://doi.org/10.1016/j.euromechsol.2016.09.014 doi: 10.1016/j.euromechsol.2016.09.014
[37]	I. Gudoshnikov, Regularity lost: the fundamental limitations and constraint qualifications in the problems of elastoplasticity, Discrete Cont. Dyn. Syst., 51 (2026), 370–438. https://doi.org/10.3934/dcds.2026034 doi: 10.3934/dcds.2026034
[38]	I. Gudoshnikov, Y. Jiao, O. Makarenkov, D. Chen, Sweeping process approach to stress analysis in elastoplastic lattice spring models with applications to network materials, Phys. Rev. E, 112 (2025), 065501. https://doi.org/10.1103/2jdr-ck1m doi: 10.1103/2jdr-ck1m
[39]	I. Gudoshnikov, O. Makarenkov, Stabilization of the response of cyclically loaded lattice spring models with plasticity, ESAIM: Control Optim. Calc. Var., 27 (2021), S8. https://doi.org/10.1051/cocv/2020043 doi: 10.1051/cocv/2020043
[40]	A. L. Gurson, Continuum theory of ductile rupture by void nucleation and growth: Part Ⅰ—Yield criteria and flow rules for porous ductile media, J. Eng. Mater. Technol., 99 (1977), 2–15. https://doi.org/10.1115/1.3443401 doi: 10.1115/1.3443401
[41]	T. Haddad, Differential inclusion governed by a state dependent sweeping process, Int. J. Difference Equ., 8 (2013), 63–70.
[42]	W. Han, B. D. Reddy, Plasticity: mathematical theory and numerical analysis, 2 Eds., Vol. 9, Interdisciplinary Applied Mathematics, Springer New York, 2013. https://doi.org/10.1007/978-1-4614-5940-8
[43]	J. Haslinger, S. Sysala, S. Repin, Inf–sup conditions on convex cones and applications to limit load analysis, Math. Mech. Solids, 24 (2019), 3331–3353. https://doi.org/10.1177/1081286519843969 doi: 10.1177/1081286519843969
[44]	M. Jirásek, Mathematical analysis of strain localization, Rev. Eur. Génie Civ., 11 (2007), 977–991. https://doi.org/10.1080/17747120.2007.9692973 doi: 10.1080/17747120.2007.9692973
[45]	M. Jirasek, S. Rolshoven, Regularized formulations of strain-softening plasticity, In: D. Kolymbas, Advanced mathematical and computational geomechanics, Lecture Notes in Applied and Computational Mechanics, Springer, Berlin, Heidelberg, 13 (2003), 269–299. https://doi.org/10.1007/978-3-540-45079-5_11
[46]	M. Kamenskii, O. Makarenkov, L. N. Wadippuli, A continuation principle for periodic BV-continuous state-dependent sweeping processes, SIAM J. Math. Anal., 52 (2020), 5598–5626. https://doi.org/10.1137/19M1248613 doi: 10.1137/19M1248613
[47]	A. M. Khludnev, V. A. Kovtunenko, Analysis of cracks in solids, Advances in Fracture Mechanics Series, WIT Press, 2000.
[48]	P. Krejčí, G. A. Monteiro, V. Recupero, Explicit and implicit non-convex sweeping processes in the space of absolutely continuous functions, Appl. Math. Optim., 84 (2021), 1477–1504. https://doi.org/10.1007/s00245-021-09801-8 doi: 10.1007/s00245-021-09801-8
[49]	P. Krejčí, G. A. Monteiro, V. Recupero, Viscous approximations of non-convex sweeping processes in the space of regulated functions, Set-Valued Var. Anal., 31 (2023), 34. https://doi.org/10.1007/s11228-023-00695-y doi: 10.1007/s11228-023-00695-y
[50]	P. Krejčí, J. Sprekels, Elastic–ideally plastic beams and Prandtl–Ishlinskii hysteresis operators, Math. Methods Appl. Sci., 30 (2007), 2371–2393. https://doi.org/10.1002/mma.892 doi: 10.1002/mma.892
[51]	M. Křížek, P. Neittaanmäki, Mathematical and numerical modelling in electrical engineering theory and applications, Mathematical Modelling: Theory and Applications, Vol. 1, Springer Dordrecht, 1996. https://doi.org/10.1007/978-94-015-8672-6
[52]	M. Kunze, M. D. P. Monteiro Marques, An Introduction to Moreau's sweeping process, In: B. Brogliato, Impacts in mechanical systems, Lecture Notes in Physics, Springer, Berlin, Heidelberg, 551 (2000), 1–60. https://doi.org/10.1007/3-540-45501-9_1
[53]	M. Kunze, M. D. P. Monteiro Marques, On parabolic quasi-variational inequalities and statedependent sweeping processes, Topol. Methods Nonlinear Anal., 12 (1998), 179–191.
[54]	A. Lemaître, Inherent stress correlations in a quiescent two-dimensional liquid: static analysis including finite-size effects, Phys. Rev. E, 96 (2017), 052101. https://doi.org/10.1103/PhysRevE.96.052101 doi: 10.1103/PhysRevE.96.052101
[55]	T. C. Lubensky, C. L. Kane, X. Mao, A. Souslov, K. Sun, Phonons and elasticity in critically coordinated lattices, Rep. Prog. Phys., 78 (2015), 073901. https://doi.org/10.1088/0034-4885/78/7/073901 doi: 10.1088/0034-4885/78/7/073901
[56]	M. Lambrecht, C. Miehe, J. Dettmar, Energy relaxation of non-convex incremental stress potentials in a strain-softening elastic–plastic bar, Int. J. Solids Struct., 40 (2003), 1369–1391. https://doi.org/10.1016/S0020-7683(02)00658-3 doi: 10.1016/S0020-7683(02)00658-3
[57]	J. A. C. Martins, M. D. P. Monteiro Marques, A. Petrov, On the stability of quasi-static paths for finite dimensional elastic-plastic systems with hardening, Z. Angew. Math. Mech., 87 (2007), 303–313. https://doi.org/10.1002/zamm.200510315 doi: 10.1002/zamm.200510315
[58]	C. Meng, Y. Liu, Damage-augmented nonlocal lattice particle method for fracture simulation of solids, Int. J. Solids Struct., 243 (2022), 111561. https://doi.org/10.1016/j.ijsolstr.2022.111561 doi: 10.1016/j.ijsolstr.2022.111561
[59]	A. Mielke, T. Roubíček, Rate-independent systems: theory and application, Applied Mathematical Sciences, Springer New York, 2015. https://doi.org/10.1007/978-1-4939-2706-7
[60]	M. G. Mora, Relaxation of the Hencky model in perfect plasticity, J. Math. Pures Appl., 106 (2016), 725–743. https://doi.org/10.1016/j.matpur.2016.03.009 doi: 10.1016/j.matpur.2016.03.009
[61]	J. J. Moreau, Application of convex analysis to the treatment of elastoplastic systems, In: P. Germain, B. Nayroles, Applications of methods of functional analysis to problems in mechanics, Lecture Notes in Mathematics, Springer, Berlin, Heidelberg 503 (1976), 56–89. https://doi.org/10.1007/BFb0088746
[62]	J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differ. Equations, 26 (1977), 347–374. https://doi.org/10.1016/0022-0396(77)90085-7 doi: 10.1016/0022-0396(77)90085-7
[63]	J. J. Moreau, On unilateral constraints, friction and plasticity, In: G. Capriz, G. Stampacchia, New variational techniques in mathematical physics, C.I.M.E. Summer Schools, Springer, Berlin, Heidelberg, 63 (1974), 171–322. https://doi.org/10.1007/978-3-642-10960-7_7
[64]	J. Müller, C. Kuttler, Methods and models in mathematical biology: deterministic and stochastic approaches, Lecture Notes on Mathematical Modelling in the Life Sciences, Springer Berlin, Heidelberg, 2015. https://doi.org/10.1007/978-3-642-27251-6
[65]	J. Nečas, I. Hlaváček, Mathematical theory of elastic and elasto-plastic bodies: an introduction, Studies in Applied Mechanics, Elsevier, 1981.
[66]	A. Needleman, V. Tvergaard, An analysis of ductile rupture in notched bars, J. Mech. Phys. Solids, 32 (1984), 461–490. https://doi.org/10.1016/0022-5096(84)90031-0 doi: 10.1016/0022-5096(84)90031-0
[67]	P. J. Olver, C. Shakiban, Applied linear algebra, 2 Eds., Undergraduate Texts in Mathematics, Springer Cham, 2018. https://doi.org/10.1007/978-3-319-91041-3
[68]	J. Outrata, M. Kocvara, J. Zowe, Nonsmooth approach to optimization problems with equilibrium constraints: theory, applications and numerical results, Nonconvex Optimization and Its Applications, Vol. 28, Springer, 2013.
[69]	G. Pijaudier-Cabot, Z. P. Bažant, M. Tabbara, Comparison of various models for strainsoftening, Eng. Comput., 5 (1988), 141–150. https://doi.org/10.1108/eb023732 doi: 10.1108/eb023732
[70]	J. Rigelesaiyin, A. Diaz, W. Li, L. Xiong, Y. Chen, Asymmetry of the atomic-level stress tensor in homogeneous and inhomogeneous materials, Proc. A, 474 (2018), 20180155. https://doi.org/10.1098/rspa.2018.0155 doi: 10.1098/rspa.2018.0155
[71]	R. T. Rockafellar, Convex Analysis, Princeton: Princeton University Press, 1970. https://doi.org/10.1515/9781400873173
[72]	J. G. Rots, R. de Borst, Analysis of concrete fracture in "direct" tension, Int. J. Solids Struct., 25 (1989), 1381–1394. https://doi.org/10.1016/0020-7683(89)90107-8 doi: 10.1016/0020-7683(89)90107-8
[73]	S. A. Sabet, R. de Borst, Structural softening, mesh dependence, and regularisation in non-associated plastic flow, Int. J. Numer. Anal. Methods Geomech., 43 (2019), 2170–2183. https://doi.org/10.1002/nag.2973 doi: 10.1002/nag.2973
[74]	B. Satco, G. Smyrlis, State-dependent sweeping processes with Stieltjes derivative, Appl. Math. Optim., 90 (2024), 24. https://doi.org/10.1007/s00245-024-10169-8 doi: 10.1007/s00245-024-10169-8
[75]	H. C. Schnabel, Zur Wohlgestelltheit des Gurson-Modells, Ph.D. Thesis, Technische Universitat Munchen, 2006.
[76]	J. C. Simo, T. J. R. Hughes, Computational inelasticity, Interdisciplinary Applied Mathematics, Vol. 7, Springer New York, 1998. https://doi.org/10.1007/b98904
[77]	G. Strang, H. Matthies, R. Temam, Mathematical and computational methods in plasticity, In: S. Nemat-Nasser, Variational methods in the mechanics of solids, Proceedings of the IUTAM Symposium on Variational Methods in the Mechanics of Solids Held at Northwestern University, Evanston, Illinois, 1980, 20–28. https://doi.org/10.1016/B978-0-08-024728-1.50008-2
[78]	S. H. Strogatz, Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, 3 Eds., CRC Press, 2024.
[79]	Supplemental material: videos and computer programs of simulations can be found at https://github.com/Ivan-Gudoshnikov/Elastoplasticity-softening-public.
[80]	P. M. Suquet, Discontinuities and plasticity, In: J. J. Moreau, P. D. Panagiotopoulos, Nonsmooth mechanics and applications, International Centre for Mechanical Sciences, 302 (1988), 279–340. https://doi.org/10.1007/978-3-7091-2624-0_5
[81]	P. M. Suquet, Existence and regularity of solutions for plasticity problems, In: S. Nemat-Nasser, Variational methods in the mechanics of solids, Proceedings of the IUTAM Symposium on Variational Methods in the Mechanics of Solids Held at Northwestern University, Evanston, Illinois, 1980,304–309. https://doi.org/10.1016/B978-0-08-024728-1.50053-7
[82]	P. M. Suquet, Sur les équations de la plasticité: existence et régularité des solutions, J. Mécanique, 20 (1981), 3–39.
[83]	P. M. Suquet, Sur un nouveau cadre fonctionnel pour les équations de la plasticité, C. R. Acad. Sci. Paris Ser., 286 (1978), 1129–1132.
[84]	R. Temam, Mathematical problems in plasticity, Modern Applied Mathematics Series, Trans-Inter-Scientia, 1985.
[85]	R. Temam, G. Strang, Functions of bounded deformation, Arch. Rational Mech. Anal., 75 (1980/81), 7–21. https://doi.org/10.1007/BF00284617 doi: 10.1007/BF00284617
[86]	V. Tvergaard, Material failure by void growth to coalescence, Adv. Appl. Mech., 27 (1989), 83–151. https://doi.org/10.1016/S0065-2156(08)70195-9 doi: 10.1016/S0065-2156(08)70195-9
[87]	H. Van Swygenhoven, P. M. Derlet, Chapter 81–Atomistic simulations of dislocations in FCC metallic nanocrystalline materials, Dislocat. Solids, 14 (2008), 1–42. https://doi.org/10.1016/S1572-4859(07)00001-0 doi: 10.1016/S1572-4859(07)00001-0

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