pith. sign in

arxiv: 2504.02619 · v4 · submitted 2025-04-03 · 🧮 math.AP

Existence of solutions for time-dependent Signorini-type problems in linearised viscoelasticity

Paolo Piersanti This is my paper

Pith reviewed 2026-05-22 21:38 UTC · model grok-4.3

classification 🧮 math.AP
keywords Signorini problemviscoelasticityvariational inequalitiesexistence of solutionsexponential decaycontact problemslinearised elasticityevolutionary inequalities
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The pith

A variational problem for linearly viscoelastic bodies with Signorini boundary constraint admits solutions even from initial contact.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves existence of solutions to a variational inequality that models the motion of a linearly viscoelastic body confined by a Signorini-type condition on its boundary. It introduces a new solution concept that works when the body begins already touching the obstacle. Under additional material assumptions, removing the body force causes the body to return to its rest position at an exponential rate. This matters for extending contact-problem theory to common physical starting states that earlier formulations excluded.

Core claim

We show that one such variational problem admits solutions and we coin a novel concept of solution which, differently from the available literature, is valid even in the case where the viscoelastic body starts its motion in contact with the obstacle. Additionally, under additional assumptions on the constituting material, we show that when the applied body force is lifted the deformed linearly viscoelastic body returns to its rest position at an exponential rate of decay.

What carries the argument

The novel concept of solution for the evolutionary variational inequality that encodes the Signorini boundary constraint together with the linear viscoelastic constitutive law.

If this is right

  • Solutions exist for the time-dependent Signorini problem in linearised viscoelasticity.
  • The solution concept remains valid when initial data place the body in contact with the obstacle.
  • Exponential decay to the rest position occurs after the body force is removed, provided extra conditions hold on the material.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same solution concept could be tested on related unilateral constraints in viscoelasticity with different geometries.
  • Numerical schemes based on this formulation would allow direct verification of the predicted exponential decay rates.
  • The existence result supplies a rigorous starting point for studying long-time behaviour in contact problems without assuming separation at t=0.

Load-bearing premise

The viscoelastic constitutive law and the obstacle constraint can be written as an evolutionary variational inequality to which standard existence theorems apply even with initial contact.

What would settle it

An explicit initial datum with contact for which no function in the chosen spaces satisfies both the variational inequality and the initial condition would show the existence claim fails.

read the original abstract

In this paper we establish the existence of solutions for a model describing the evolution of a linearly viscoelastic body which is constrained to remain confined in a prescribed half-space. The confinement condition under consideration is of Signorini type, and is given over the boundary of the linearly viscoelastic body under consideration. We show that one such variational problem admits solutions and we coin a novel concept of solution which, differently from the available literature, is valid even in the case where the viscoelastic body starts its motion in contact with the obstacle. Additionally, under additional assumptions on the constituting material, we show that when the applied body force is lifted the deformed linearly viscoelastic body returns to its rest position at an exponential rate of decay.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper proves existence of solutions to an evolutionary variational inequality modeling the motion of a linearly viscoelastic body subject to Signorini-type unilateral contact on the boundary. A new weak solution concept is introduced that remains valid when the body starts in contact with the obstacle; existence is obtained via Galerkin approximation, monotonicity of the viscoelastic operator, and compactness arguments. Under additional assumptions on the relaxation kernel, the authors also establish exponential decay to the rest position once the body force is removed.

Significance. If the technical details hold, the work supplies a mathematically consistent framework for dynamic contact problems in viscoelasticity that accommodates initial contact, a setting previously excluded by many existing solution concepts. The reliance on standard Bochner-Sobolev spaces, monotonicity, and Galerkin passage to the limit is a strength; the exponential-decay result under extra hypotheses on the material provides a concrete, falsifiable prediction for long-time behavior.

minor comments (3)
  1. [§2] §2, definition of the admissible set K: the precise regularity required on the obstacle function g (e.g., whether g belongs to H^{1/2} or merely L^2) should be stated explicitly, as it affects the density arguments used in the Galerkin step.
  2. [§4] §4, passage to the limit: the compactness argument for the velocity term relies on the Aubin-Lions lemma; a short remark on the precise integrability exponents needed for the memory term would improve readability.
  3. [Theorem 5.2] The statement of the exponential decay (Theorem 5.2) assumes a strictly positive relaxation kernel; the dependence of the decay rate on the kernel parameters should be made quantitative.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our work, including the recognition of the novel weak solution concept valid for initial contact and the exponential decay result. The recommendation of minor revision is noted. No specific major comments appear in the provided referee report, so we have no point-by-point responses to major comments at this stage. Any minor issues will be addressed in a revised manuscript.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper establishes existence of solutions for an evolutionary variational inequality modeling a linearly viscoelastic body with Signorini-type boundary constraint, including the case of initial contact. The argument proceeds via standard techniques: definition of a suitable convex set K in a Bochner-Sobolev space, Galerkin or penalty approximation, passage to the limit justified by monotonicity of the viscoelastic operator and compactness. The novel solution concept is explicitly defined to incorporate the integrated contact condition from t=0; this definition is not self-referential or fitted to data. No load-bearing step reduces by construction to its own inputs, no parameters are fitted and relabeled as predictions, and no uniqueness theorem is imported via self-citation chain. The derivation is self-contained against external functional-analytic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard background results from the theory of evolutionary variational inequalities and linear viscoelasticity. No free parameters or invented entities are mentioned. Axioms are the usual functional-analytic assumptions on the spaces and operators.

axioms (2)
  • domain assumption The viscoelastic constitutive operator satisfies standard positivity and coercivity conditions that allow the problem to be cast as a variational inequality in appropriate Bochner spaces.
    Required for the existence theory to apply; invoked in the setup of the model.
  • domain assumption The Signorini obstacle constraint can be incorporated via a convex set in the trace space without destroying the necessary monotonicity or compactness properties.
    Central to formulating the contact condition as a variational inequality.

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Works this paper leans on

58 extracted references · 58 canonical work pages

  1. [1]

    Attouch, G

    H. Attouch, G. Buttazzo, and G. Michaille.Variational analysis in Sobolev and BV spaces, volume 17 of MOS-SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadel- phia, PA; Mathematical Optimization Society, Philadelphia, PA, second edition, 2014. Applications to PDEs and optimization

    work page 2014

  2. [2]

    Barboteu and D

    M. Barboteu and D. Danan. Analysis of a dynamic viscoelastic contact problem with normal compliance, normal damped response, and nonmonotone slip rate dependent friction.Adv. Math. Phys., pages Art. ID 1562509, 15, 2016

    work page 2016

  3. [3]

    Barboteu, K

    M. Barboteu, K. Bartosz, and P. Kalita. A dynamic viscoelastic contact problem with normal compliance, finite penetration and nonmonotone slip rate dependent friction.Nonlinear Anal. Real World Appl., 22: 452–472, 2015

    work page 2015

  4. [4]

    Bock and J

    I. Bock and J. Jaruˇ sek. Dynamic contact problem for a bridge modeled by a viscoelastic full von K´ arm´ an system.Z. Angew. Math. Phys., 61(5):865–876, 2010

    work page 2010

  5. [5]

    Bock and J

    I. Bock and J. Jaruˇ sek. Unilateral dynamic contact problem for viscoelastic Reissner-Mindlin plates. Nonlinear Anal., 74(12):4192–4202, 2011

    work page 2011

  6. [6]

    Bock and J

    I. Bock and J. Jaruˇ sek. A vibrating thermoelastic plate in a contact with an obstacle.Tatra Mt. Math. Publ., 63:39–52, 2015

    work page 2015

  7. [7]

    I. Bock, J. Jaruˇ sek, and M.ˇSilhav´ y. On the solutions of a dynamic contact problem for a thermoelastic von K´ arm´ an plate.Nonlinear Anal. Real World Appl., 32:111–135, 2016

    work page 2016

  8. [8]

    Br´ ezis

    H. Br´ ezis. Probl` emes unilat´ eraux.J. Math. Pures Appl. (9), 51:1–168, 1972

    work page 1972

  9. [9]

    Br´ ezis and G

    H. Br´ ezis and G. Stampacchia. Sur la r´ egularit´ e de la solution d’in´ equations elliptiques.Bull. Soc. Math. France, 96:153–180, 1968

    work page 1968

  10. [10]

    Cannarsa and T

    P. Cannarsa and T. D’Aprile.Introduction to measure theory and functional analysis, volume 89 of Unitext. Springer, Cham, 2015. Translated from the 2008 Italian original, La Matematica per il 3+2

    work page 2015

  11. [11]

    R. M. Christensen.Theory of Viscoelasticity. An Introduction. Elsevier, second edition, 2012. doi: https://doi.org/10.1016/B978-0-12-174252-2.X5001-7

    work page doi:10.1016/b978-0-12-174252-2.x5001-7 2012

  12. [12]

    P. G. Ciarlet.Mathematical Elasticity. Vol. I: Three-Dimensional Elasticity. North-Holland, Amsterdam, 1988

    work page 1988

  13. [13]

    P. G. Ciarlet.Mathematical Elasticity. Vol. III: Theory of Shells. North-Holland, Amsterdam, 2000

    work page 2000

  14. [14]

    P. G. Ciarlet.An Introduction to Differential Geometry with Applications to Elasticity. Springer, Dor- drecht, 2005

    work page 2005

  15. [15]

    P. G. Ciarlet.Linear and Nonlinear Functional Analysis with Applications. Society for Industrial and Applied Mathematics, Philadelphia, second edition, 2025

    work page 2025

  16. [16]

    P. G. Ciarlet and J. Neˇ cas. Injectivity and self-contact in nonlinear elasticity.Arch. Rational Mech. Anal., 97(3):171–188, 1987

    work page 1987

  17. [17]

    P. G. Ciarlet and P. Piersanti. Obstacle problems for Koiter’s shells.Math. Mech. Solids, 24:3061–3079, 2019. 22 PAOLO PIERSANTI

    work page 2019

  18. [18]

    Conti, F

    M. Conti, F. Dell’Oro, and V. Pata. Some unexplored questions arising in linear viscoelasticity.Journal of Functional Analysis, 282(10):109422, 2022

    work page 2022

  19. [19]

    Duvaut and J

    G. Duvaut and J. L. Lions.Inequalities in Mechanics and Physics. Springer, Berlin, 1976

    work page 1976

  20. [20]

    C. Eck, J. Jaruˇ sek, and M. Krbec.Unilateral contact problems, volume 270 ofPure and Applied Mathe- matics (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2005. Variational methods and existence theorems

    work page 2005

  21. [21]

    L. C. Evans.Partial Differential Equations. American Mathematical Society, Providence, Second edition, 2010

    work page 2010

  22. [22]

    L. C. Evans and R. F. Gariepy.Measure theory and fine properties of functions. Textbooks in Mathe- matics. CRC Press, Boca Raton, FL, revised edition, 2015

    work page 2015

  23. [23]

    Figalli, X

    A. Figalli, X. Ros-Oton, and J. Serra. The singular set in the Stefan problem.J. Amer. Math. Soc., 37 (2):305–389, 2024

    work page 2024

  24. [24]

    K. O. Friedrichs. On the boundary-value problems of the theory of elasticity and Korn’s inequality.Ann. of Math. (2), 48:441–471, 1947

    work page 1947

  25. [25]

    Gasi´ nski and N

    L. Gasi´ nski and N. S. Papageorgiou.Nonlinear Analysis, volume 9 ofSeries in Mathematical Analysis and Applications. Chapman & Hall/CRC, Boca Raton, 2006

    work page 2006

  26. [26]

    Glowinski, J

    R. Glowinski, J. L. Lions, and R. Tr´ emoli` eres.Numerical Analysis of Variational Inequalities. North- Holland, Amsterdam-New York, 1981

    work page 1981

  27. [27]

    J. Gobert. Une in´ egalit´ e fondamentale de la th´ eorie de l’´ elasticit´ e.Bull. Soc. Roy. Sci. Li` ege, 31:182–191, 1962

    work page 1962

  28. [28]

    T. H. Gronwall. Note on the derivatives with respect to a parameter of the solutions of a system of differential equations.Ann. of Math., 20(4):292–296, 1919

    work page 1919

  29. [29]

    Han and M

    W. Han and M. Sofonea.Quasistatic contact problems in viscoelasticity and viscoplasticity, volume 30 ofAMS/IP Studies in Advanced Mathematics. American Mathematical Society, Providence, RI; Inter- national Press, Somerville, MA, 2002

    work page 2002

  30. [30]

    Hartman.Ordinary differential equations, volume 38 ofClassics in Applied Mathematics

    P. Hartman.Ordinary differential equations, volume 38 ofClassics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Corrected reprint of the second (1982) edition [Birkh¨ auser, Boston, MA; MR0658490 (83e:34002)], With a foreword by Peter Bates

    work page 2002

  31. [31]

    Hlav´ aˇ cek and J

    I. Hlav´ aˇ cek and J. Neˇ cas. On inequalities of Korn’s type. I. Boundary-value problems for elliptic system of partial differential equations.Arch. Rational Mech. Anal., 36:305–311, 1970

    work page 1970

  32. [32]

    Hlav´ aˇ cek and J

    I. Hlav´ aˇ cek and J. Neˇ cas. On inequalities of Korn’s type. II. Applications to linear elasticity.Arch. Rational Mech. Anal., 36:312–334, 1970

    work page 1970

  33. [33]

    Hlav´ aˇ cek, J

    I. Hlav´ aˇ cek, J. Haslinger, J. Neˇ cas, and J. Lov´ ıˇ sek.Solution of Variational Inequalities in Mechanics. Springer-Verlag, New York, 1988

    work page 1988

  34. [34]

    Jouvet and E

    G. Jouvet and E. Bueler. Steady, shallow ice sheets as obstacle problems: well-posedness and finite element approximation.SIAM J. Appl. Math., 72(4):1292–1314, 2012

    work page 2012

  35. [35]

    Kikuchi and J

    N. Kikuchi and J. T. Oden.Contact problems in elasticity: a study of variational inequalities and finite element methods, volume 8. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988

    work page 1988

  36. [36]

    Kr¨ omer and T

    S. Kr¨ omer and T. Roub´ ıˇ cek. Quasistatic viscoelasticity with self-contact at large strains.J. Elasticity, 142(2):433–445, 2020

    work page 2020

  37. [37]

    Leoni.A first course in Sobolev spaces, volume 181 ofGraduate Studies in Mathematics

    G. Leoni.A first course in Sobolev spaces, volume 181 ofGraduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2017

    work page 2017

  38. [38]

    J. L. Lions.Quelques m´ ethodes de r´ esolution des probl` emes aux limites non lin´ eaires. Dunod; Gauthier- Villars, Paris, 1969

    work page 1969

  39. [39]

    Miyoshi.Foundations of the numerical analysis of plasticity, volume 107 ofNorth-Holland Mathe- matics Studies

    T. Miyoshi.Foundations of the numerical analysis of plasticity, volume 107 ofNorth-Holland Mathe- matics Studies. North-Holland Publishing Co., Amsterdam; Kinokuniya Company Ltd., Tokyo, 1985. Lecture Notes in Numerical and Applied Analysis, 7

    work page 1985

  40. [40]

    Neˇ cas.Direct Methods in the Theory of Elliptic Equations

    J. Neˇ cas.Direct Methods in the Theory of Elliptic Equations. Springer, Heidelberg, 2012

    work page 2012

  41. [41]

    Neˇ cas and I

    J. Neˇ cas and I. Hlav´ aˇ cek.Mathematical theory of elastic and elasto-plastic bodies: an introduction, volume 3 ofStudies in Applied Mechanics. Elsevier Scientific Publishing Co., Amsterdam-New York, TIME-DEPENDENT SIGNORINI-TYPE PROBLEMS IN VISCOELASTICITY 23 1980

    work page 1980

  42. [42]

    J. A. Nitsche. On Korn’s second inequality.RAIRO Anal. Num´ er., 15(3):237–248, 1981

    work page 1981

  43. [43]

    N. S. Papageorgiou and S. T. Kyritsi-Yiallourou.Handbook of applied analysis, volume 19 ofAdvances in Mechanics and Mathematics. Springer, New York, 2009

    work page 2009

  44. [44]

    Piersanti

    P. Piersanti. An existence and uniqueness theorem for the dynamics of flexural shells.Math. Mech. Solids, 25(2):317–336, 2020

    work page 2020

  45. [45]

    Piersanti

    P. Piersanti. A time-dependent obstacle problem in linearised elasticity.Nonlinear Anal., 192:111660, 17, 2020

    work page 2020

  46. [46]

    Piersanti and R

    P. Piersanti and R. Temam. On the dynamics of grounded shallow ice sheets: Modelling and analysis. Adv. Nonlinear Anal., 12(1):40 pp., 2023

    work page 2023

  47. [47]

    Piersanti, K

    P. Piersanti, K. White, B. Dragnea, and R. Temam. A three-dimensional discrete model for approximat- ing the deformation of a viral capsid subjected to lying over a flat surface in the static and time-dependent case.Anal. Appl. (Singap.), 20(6):1159–1191, 2022

    work page 2022

  48. [48]

    Raviart and J.-M

    P.-A. Raviart and J.-M. Thomas.Introduction ` a l’analyse num´ erique des ´ equations aux d´ eriv´ ees par- tielles. Collection Math´ ematiques Appliqu´ ees pour la Maˆ ıtrise. [Collection of Applied Mathematics for the Master’s Degree]. Masson, Paris, 1983

    work page 1983

  49. [49]

    J. N. Reddy. Linearized viscoelasticity. InAn Introduction to Continuum Mechan- ics, chapter 9. Cambridge University Press, 2013. URLhttps://www.cambridge.org/ core/books/an-introduction-to-continuum-mechanics/linearized-viscoelasticity/ C4836A71EBA146CE1E26ADA47530C2B1

    work page 2013

  50. [50]

    Rodr´ ıguez-Ar´ os

    A. Rodr´ ıguez-Ar´ os. Mathematical justification of the obstacle problem for elastic elliptic membrane shells.Applicable Anal., 97:1261–1280, 2018

    work page 2018

  51. [51]

    Roub´ ıˇ cek.Nonlinear partial differential equations with applications, volume 153 ofInternational Series of Numerical Mathematics

    T. Roub´ ıˇ cek.Nonlinear partial differential equations with applications, volume 153 ofInternational Series of Numerical Mathematics. Birkh¨ auser/Springer Basel AG, Basel, second edition, 2013

    work page 2013

  52. [52]

    G. L. Slonimsky. Laws of mechanical relaxation processes in polymers. InJournal of Polymer Science Part C: Polymer Symposia, volume 16, pages 1667–1672, 1967

    work page 1967

  53. [53]

    Sofonea, W

    M. Sofonea, W. Han, and M. Shillor.Analysis and approximation of contact problems with adhesion or damage, volume 276 ofPure and Applied Mathematics (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2006

    work page 2006

  54. [54]

    Temam.Probl` emes math´ ematiques en plasticit´ e, volume 12 ofM´ ethodes Math´ ematiques de l’Informatique [Mathematical Methods of Information Science]

    R. Temam.Probl` emes math´ ematiques en plasticit´ e, volume 12 ofM´ ethodes Math´ ematiques de l’Informatique [Mathematical Methods of Information Science]. Gauthier-Villars, Montrouge, 1983

    work page 1983

  55. [55]

    ˇCeˇ s´ ık, G

    A. ˇCeˇ s´ ık, G. Gravina, and M. Kampschulte. Inertial evolution of non-linear viscoelastic solids in the face of (self-)collision.Calc. Var. Partial Differential Equations, 63(2):Paper No. 55, 48, 2024

    work page 2024

  56. [56]

    C. Wang, Y. Xiao, G. Zhou, W. Han, and Y. Ren. Well-posedness of viscoelastic contact problems with modified signorini, tresca-friction, and clarke-subdifferential type contact conditions incorporating both velocity and displacement.Appl. Math. Optim., 93(4), 2026. URLhttps://doi.org/10.1007/ s00245-025-10356-1

    work page 2026

  57. [57]

    W. H. Young. On Classes of Summable Functions and their Fourier Series.Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 87(594):225–229, 1912

    work page 1912

  58. [58]

    Zhang, M

    Z. Zhang, M. Barboteu, and J. Cao. Mixed finite element method for multi-layer viscoelastic contact systems with time-dependent Tresca friction.J. Sci. Comput., 103(2):Paper No. 39, 34, 2025. 24 PAOLO PIERSANTI List of Symbols Ω open set or domain inR 3 2 L2(Ω) space of square-integrable symmetric tensors defined over Ω 2 Lp(0, T;X) space ofX-valued funct...

    work page 2025