Abstract
We consider the equation of motion for one-dimensional nonlinear viscoelasticity of strain-rate type under the assumption that the stored-energy function is \(\lambda \)-convex, which allows for solid phase transformations. We formulate this problem as a gradient flow, leading to existence and uniqueness of solutions. By approximating general initial data by those in which the deformation gradient takes only finitely many values, we show that under suitable hypotheses on the stored-energy function the deformation gradient is instantaneously bounded and bounded away from zero. Finally, we discuss the open problem of showing that every solution converges to an equilibrium state as time \(t \rightarrow \infty \) and prove convergence to equilibrium under a nondegeneracy condition. We show that this condition is satisfied in particular for any real analytic cubic-like stress-strain function.
Similar content being viewed by others
References
Aaronson, J. : An introduction to infinite ergodic theory, volume 50 of mathematical surveys and monographs. American Mathematical Society.(1997)
Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows : in Metric Spaces and in the Space of Probability Measures. Birkhuser, Boston (2005)
Andrews, G.: On the existence and asymptotic behaviour of solutions to a damped nonlinear wave equation. Thesis, Heriot-Watt University (1979)
Andrews, G.: On the existence of solutions to the equation \(u_{tt}=u_{xxt}+{\sigma (u_{x})}_{x}\). J. Differ. Equ. 35, 200–231 (1980)
Andrews, G., Ball, J.M.: Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity. J. Differ. Equ. 44, 306–341 (1982)
Antman, S.S., Seidman, T.I.: Quasilinear hyperbolic-parabolic equations of one-dimensional viscoelasticity. J. Differ. Equ. 124, 132–185 (1996)
Antman, S.S., Seidman, T.I.: The parabolic-hyperbolic system governing the spatial motion of nonlinearly viscoelastic rods. Arch. Ration. Mech. Anal. 175(1), 85–150 (2005)
Ball, J.M.: Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations. J. Nonlinear Sci. 7, 475–502 (1997)
Ball, J.M., Holmes, P.J., James, R.D., Pego, R.L., Swart, P.J.: On the dynamics of fine structure. J. Nonlinear Sci. 1, 17–70 (1991)
Bartle, R.: The Elements of Integration and Lebesgue Measure. Wiley, New York (1995)
Brezis, H.: Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert. North-Holland, Amsterdam (1973)
Brunovský, P.: Poláčik, : on the local structure of \(\omega \)-limit sets of maps. Z. angew Math. Phys. 48, 976–986 (1997)
Chong, K.M.: An induction principle for spectral and rearrangement inequalities. Trans. Am. Math. Soc. 196, 371–383 (1974)
Crandall, M.G., Pazy, A.: Semi-groups of nonlinear contractions and dissipative sets. J. Funct. Anal. 3, 376–418 (1969)
Dafermos, C.M.: The mixed initial-boundary value problem for the equations of nonlinear one-dimensional viscoelasticity. J. Differ. Equ. 6, 71–86 (1969)
Demoulini, S.: Weak solutions for a class of nonlinear systems of viscoelasticity. Arch. Rational Mech. Anal. 155, 299–334 (2000)
Ericksen, J.L.: Equilibrium of bars. J. Elast. 5, 191–201 (1975)
Friesecke, G., Dolzmann, G.: Implicit time discretization and global existence for a quasi-linear evolution equation with nonconvex energy. SIAM J. Math. Anal. 28(2), 363–380 (1997)
Friesecke, G., McLeod, J.B.: Dynamics as a mechanism preventing the formation of finer and finer microstructure. Arch. Rational Mech. Anal. 133, 199–247 (1996)
Friesecke, G., McLeod, J.B.: Dynamic stability of non-minimizing phase mixtures. Proc. R. Soc. Lond. A 453, 2427–2436 (1997)
Hale, J. K.: Asymptotic behavior of dissipative systems, volume 25 of mathematical surveys and monographs. American Mathematical Society, Providence, RI.(1988)
Hale, J.K., Massatt, P.: Asymptotic behaviour of gradient-like systems. In: Bednarek, A.R., Cesari, L. (eds.) Dynamical Systems, pp. 85–101. Academic Press, New York (1982)
Hale, J.K., Raugel, G.: Convergence in gradient-like systems with applications to PDE. Z. angew Math. Phys. 43, 63–124 (1992)
Halmos, P.R., von Neumann, J.: Operator methods in classical mechanics II. Ann. Math. 43(2), 332–350 (1942)
Hartman, P.: Ordinary Differential Equations. SIAM, Philadelphia (2002)
Jendoubi, M.A.: A simple unified approach to some convergence theorems of L. Simon. J. Funct. Anal. 153(1), 187–202 (1998)
Komura, Y.: Nonlinear semi-groups in Hilbert spaces. J. Math. Soc. Japan 19, 493–507 (1967)
Kuttler, K., Hicks, D.: Initial-boundary value problems for the equation \(u_{tt}=(\sigma (u_{x}))_{x}+(\alpha (u_{x})u_{xt})_{x}+f\). Q. Appl. Math. 66, 393–407 (1988)
Lojasiewicz, S. :Une propriété topologique des sous-ensembles analytiques réels. Colloques Internationaux du C.N.R.S. 117, Les Equations aux Derivées Partielles. (1963)
Mielke, A., Stefanelli, U.: Weighted energy-dissipation functionals for gradient flows. ESAIM : COCV. Springer, New York (2009)
Natanson, I.P.: Theory of functions of a real variable. Constable & Co.Ltd, New York (1955)
Norton, R.A.: Existence of stationary solutions and local minima for 2d models of fine structure dynamics. Calc. Var. 49, 729–752 (2014)
Novick-Cohen, A.: Energy methods for the Cahn–Hilliard equation. Q. Appl. Math. 46, 681–690 (1988)
Novick-Cohen, A., Pego, R.L.: Stable patterns in a viscous diffusion equation. Trans. Am. Math. Soc. 324(1), 331–351 (1991)
Pego, R.L.: Phase transitions in one-dimensional nonlinear viscoelasticity: admissibility and stability. Arch. Rational Mech. Anal. 97, 353–394 (1987)
Pego, R.L.: Stabilization in a gradient system with a conservation law. Proc. Am. Math. Soc. 114(4), 1017–1024 (1992)
Potier-Ferry, M.: The linearization principle for stability in quasilinear parabolic equations I. Arch. Rational Mech. Anal. 77, 301–320 (1981)
Potier-Ferry, M.: On the mathematical foundations of elastic stability theory I. Arch. Rational Mech. Anal. 78, 55–72 (1982)
Rockafellar, R.T.: Convexity properties of nonlinear maximal monotone operators. Bull. Am. Math. Soc. 75(1), 74–77 (1969)
Rossi, R., Savaré, G.: Gradient flows of nonconvex functionals in Hilbert spaces and applications. ESAIM COCV 12, 564–614 (2006)
Rubinstein, J., Sternberg, P.: Nonlocal reaction-diffusion equations and nucleation. IMA J. Appl. Math. 48, 249–264 (1992)
Rudolph, D.J.: Fundamentals of Measurable Dynamics. Oxford University Press, Oxford (1990)
Rybka, P.: Dynamical modelling of phase transitions by means of viscoelasticity in many dimensions. Proc. Royal Soc. Edinb. 121A, 101–138 (1992)
Ryff, J.V.: Measure preserving transformations and rearrangements. J. Math. Anal. Appl. 31, 449–458 (1970)
Şengül, Y.: Well-posedness of dynamics of microstructure in solids. PhD thesis, University of Oxford. (2010)
Serre, D.: Asymptotics of homogenous oscillations in a compressible viscous fluid. Bol. Soc. Bras. Mat. 32(3), 435–442 (2001)
Simon, L.: Asymptotics for a class of non-linear evolution equations, with applications to geometric problems. Ann. Math. 118, 525–571 (1983)
Swart, J.P., Holmes, J.P.: Energy minimization and the formation of microstructure in dynamic anti-plane shear. Arch. Rational Mech. Anal. 121, 37–85 (1992)
Tvedt, B.: Quasilinear equations for viscoelasticity of strain-rate type. Arch. Rational Mech. Anal. 189, 237–281 (2008)
Ward, M.J.: Metastable bubble solutions for the Allen–Cahn equation with mass conservation. SIAM J. Appl. Math. 56(5), 1247–1279 (1996)
Watson, S.J.: Unique global solvability for initial-boundary value problems in one-dimensional nonlinear thermoviscoelasticity. Arch. Ration. Mech. Anal. 153(1), 1–37 (2000)
Acknowledgments
We are grateful to Gero Friesecke, Bob Pego and Endre Süli for useful discussions. We also thank the referee for valuable comments. The research of both authors was partly supported by EPSRC Grant EP/D048400/1 and by the EPSRC Science and Innovation award to the Oxford Centre for Nonlinear PDE (EP/E035027/1). The research of JMB was also supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant Agreement No 291053 and by a Royal Society Wolfson Research Merit Award. The research of YŞ was also supported by TÜBİTAK fellowship 2213.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the memory of Klaus Kirchgässner.
Rights and permissions
About this article
Cite this article
Ball, J.M., Şengül, Y. Quasistatic Nonlinear Viscoelasticity and Gradient Flows. J Dyn Diff Equat 27, 405–442 (2015). https://doi.org/10.1007/s10884-014-9410-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10884-014-9410-1
Keywords
- Viscoelasticity
- Gradient flows
- Nonlinear partial differential equations
- Infinite-dimensional dynamical systems