American Institute of Mathematical Sciences

Advanced
May  2021, 20(5): 1931-1960. doi: 10.3934/cpaa.2021053

Existence of large-data global weak solutions to a model of a strain-limiting viscoelastic body

Miroslav Bulíček 1, , Victoria Patel 2,, , Yasemin Şengül 3, and  Endre Süli 2,

1. 

Faculty of Mathematics and Physics, Charles University Prague, Sokolovská 83,186 75 Prague 8, Czech Republic
2. 

Mathematical Institute, University of Oxford, Andrew Wiles Building, Woodstock Road, Oxford, OX2 6GG, UK
3. 

Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla 34956, Istanbul, Turkey

* Corresponding author: V. Patel

Received  October 2020 Published  May 2021 Early access  March 2021

Fund Project: M. Bulíček's work is supported by the project 20-11027X financed by GAČR. M. Bulíček is a member of the Nečas Center for Mathematical Modeling. V. Patel is supported by the UK Engineering and Physical Sciences Research Council [EP/L015811/1]. Y. Şengül is partially supported by the Scientific and Technological Research Council of Turkey (TÜBITAK) under the grant 116F093

We prove the existence of a unique large-data global-in-time weak solution to a class of models of the form $ \boldsymbol{u}_{tt} = \mbox{div }\mathbb{T} + \boldsymbol{f} $ for viscoelastic bodies exhibiting strain-limiting behaviour, where the constitutive equation, relating the linearised strain tensor $ \boldsymbol{\varepsilon}( \boldsymbol{u}) $ to the Cauchy stress tensor $ \mathbb{T} $, is assumed to be of the form $ \boldsymbol{\varepsilon}( \boldsymbol{u}_t) + \alpha \boldsymbol{\varepsilon}( \boldsymbol{u}) = F( \mathbb{T}) $, where we define \(F(\mathbb{T}) = (1 + | \mathbb{T}|^a)^{-\frac{1}{a}} \mathbb{T}\), for constant parameters $ \alpha \in (0,\infty) $ and $ a \in (0,\infty) $, in any number $ d $ of space dimensions, with periodic boundary conditions. The Cauchy stress $ \mathbb{T} $ is shown to belong to $ L^{1}(Q)^{d \times d} $ over the space-time domain $ Q $. In particular, in three space dimensions, if $ a \in (0,\frac{2}{7}) $, then in fact $ \mathbb{T} \in L^{1+\delta}(Q)^{d \times d} $ for a $ \delta > 0 $, the value of which depends only on $ a $.

Keywords: Nonlinear viscoelasticity, strain-limiting theory, evolutionary problem, global existence, weak solution, regularity.
Mathematics Subject Classification: 35M13, 35K99, 74D10, 74H20.
Citation: Miroslav Bulíček, Victoria Patel, Yasemin Şengül, Endre Süli. Existence of large-data global weak solutions to a model of a strain-limiting viscoelastic body. Communications on Pure & Applied Analysis, 2021, 20 (5) : 1931-1960. doi: 10.3934/cpaa.2021053
References:
[1]

L. Beck, M. Bulíček, J. Málek and E. Süli, On the existence of integrable solutions to nonlinear elliptic systems and variational problems with linear growth, Arch. Ration. Mech. Anal., 225 (2017), 717-769.  doi: 10.1007/s00205-017-1113-4.  Google Scholar

[2]

M. Bulíček, P. Kaplický and M. Steinhauer, On existence of a classical solution to a generalized Kelvin-Voigt model, Pacific J. Math., 262 (2013), 11-33.  doi: 10.2140/pjm.2013.262.11.  Google Scholar

[3]

M. Bulíček, J. Málek, K. R. Rajagopal and E. Süli, On elastic solids with limiting small strain: modelling and analysis, EMS Surv. Math. Sci., 1 (2014), 283-332.  doi: 10.4171/EMSS/7.  Google Scholar

[4]

M. Bulíček, J. Málek and K. R. Rajagopal, On Kelvin–Voigt model and its generalizations, Evol. Equ. Control Theory, 1 (2012), 17-42.  doi: 10.3934/eect.2012.1.17.  Google Scholar

[5]

M. Bulíček, J. Málek and E. Süli, Analysis and approximation of a strain-limiting nonlinear elastic model, Math. Mech. Solids, 20 (2015), 92-118.  doi: 10.1177/1081286514543601.  Google Scholar

[6]

C. Canuto and A. Quarteroni, Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comp., 38 (1982), 67-86.  doi: 10.1090/S0025-5718-1982-0637287-3.  Google Scholar

[7]

M. Chirita and C. M. Ionescu, Models of biomimetic tissues for vascular grafts, in On biomimetics, In-Tech, 2011, 43–52. doi: 10.5772/18248.  Google Scholar

[8]

J. C. Criscione and K. R. Rajagopal, On the modeling of the non-linear response of soft elastic bodies, International Journal of Non-Linear Mechanics, 56 (2013), 20-24.  doi: 10.1016/j.ijnonlinmec.2013.05.004.  Google Scholar

[9]

H. A. Erbay and Y. Şengül, Traveling waves in one-dimensional non-linear models of strain-limiting viscoelasticity, International Journal of Non-Linear Mechanics, 77 (2015), 61-68.  doi: 10.1016/j.ijnonlinmec.2015.07.005.  Google Scholar

[10]

H. A. Erbay and Y. Şengül, A thermodynamically consistent stress-rate type model of one-dimensional strain-limiting viscoelasticity, Z. Angew. Math. Phys., 71 (2020), Paper No. 94, 10 pp. doi: 10.1007/s00033-020-01315-7.  Google Scholar

[11]

H. A. Erbay, A. Erkip and Y. Şengül, Local existence of solutions to the initial-value problem for one-dimensional strain-limiting viscoelasticity, J. Differential Equations, 269 (2020), 9720-9739.  doi: 10.1016/j.jde.2020.06.052.  Google Scholar

[12]

L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 1998.  Google Scholar

[13]

A. D. Freed and K. R. Rajagopal, A viscoelastic model for describing the response of biological fibers, Acta Mech., 277 (2016), 3367-3380.  doi: 10.1007/s00707-016-1673-7.  Google Scholar

[14]

H. Itou, V. A. Kovtunenko and K. R. Rajagopal, On the states of stress and strain adjacent to a crack in a strain-limiting viscoelastic body, Math. Mech. Solids, 23 (2018), 433-444.  doi: 10.1177/1081286517709517.  Google Scholar

[15]

H. Itou, V. A. Kovtunenko and K. R. Rajagopal, Crack problem within the context of implicitly constituted quasi-linear viscoelasticity, Math. Models Methods Appl. Sci., 29 (2019), 355-372.  doi: 10.1142/S0218202519500118.  Google Scholar

[16]

K. R. Rajagopal, On implicit constitutive theories, Appl. Math., 48 (2003), 279-319.  doi: 10.1023/A:1026062615145.  Google Scholar

[17]

K. R. Rajagopal, A note on a reappraisal and generalization of the Kelvin–Voigt model, Mechanics Research Communications, 36 (2009), 232-235.  doi: 10.1016/j.mechrescom.2008.09.005.  Google Scholar

[18]

K. R. Rajagopal, On a new class of models in elasticity, Math. Comput. Appl., 15 (2010), 506-528.  doi: 10.3390/mca15040506.  Google Scholar

[19]

K. R. Rajagopal, Non-linear elastic bodies exhibiting limiting small strain, Math. Mech. Solids, 16 (2011), 122-139.  doi: 10.1177/1081286509357272.  Google Scholar

[20]

K. R. Rajagopal, On the nonlinear elastic response of bodies in the small strain range, Acta Mech., 225 (2014), 1545-1553.  doi: 10.1007/s00707-013-1015-y.  Google Scholar

[21]

K. R. Rajagopal and G. Saccomandi, Circularly polarized wave propagation in a class of bodies defined by a new class of implicit constitutive relations, Z. Angew. Math. Phys., 65 (2014), 1003-1010.  doi: 10.1007/s00033-013-0362-9.  Google Scholar

[22]

T. Roubíček, Nonlinear partial differential equations with applications, 2$^{nd}$ edition, Birkhäuser/Springer Basel AG, Basel, 2013.  Google Scholar

[23]

M. Ruzhansky and M. Sugimoto, On global inversion of homogeneous maps, Bull. Math. Sci., 5 (2015), 13-18.  doi: 10.1007/s13373-014-0059-1.  Google Scholar

[24]

T. Saito, T. Furuta, J.-H. Hwang, S. Kuramoto, K. Nishino, N. Suzuki, R. Chen, A. Yamada, K. Ito, Y. Seno, T. Nonaka, H. Ikehata, N. Nagasako, C. Iwamoto, Y. Ikuhara and and T. Sakuma, Multifunctional alloys obtained via a dislocation-free plastic deformation mechanism, Science (New York, N.Y.), 300 (2003), 464-467.  doi: 10.1126/science.1081957.  Google Scholar

[25]

Y. Şengül, Viscoelasticity with limiting strain, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 57-70.  doi: 10.3934/dcdss.2020330.  Google Scholar

[26] J. Warga, Optimal control of differential and functional equations, Academic Press, New York-London, 1972.   Google Scholar

show all references

References:
[1]

L. Beck, M. Bulíček, J. Málek and E. Süli, On the existence of integrable solutions to nonlinear elliptic systems and variational problems with linear growth, Arch. Ration. Mech. Anal., 225 (2017), 717-769.  doi: 10.1007/s00205-017-1113-4.  Google Scholar

[2]

M. Bulíček, P. Kaplický and M. Steinhauer, On existence of a classical solution to a generalized Kelvin-Voigt model, Pacific J. Math., 262 (2013), 11-33.  doi: 10.2140/pjm.2013.262.11.  Google Scholar

[3]

M. Bulíček, J. Málek, K. R. Rajagopal and E. Süli, On elastic solids with limiting small strain: modelling and analysis, EMS Surv. Math. Sci., 1 (2014), 283-332.  doi: 10.4171/EMSS/7.  Google Scholar

[4]

M. Bulíček, J. Málek and K. R. Rajagopal, On Kelvin–Voigt model and its generalizations, Evol. Equ. Control Theory, 1 (2012), 17-42.  doi: 10.3934/eect.2012.1.17.  Google Scholar

[5]

M. Bulíček, J. Málek and E. Süli, Analysis and approximation of a strain-limiting nonlinear elastic model, Math. Mech. Solids, 20 (2015), 92-118.  doi: 10.1177/1081286514543601.  Google Scholar

[6]

C. Canuto and A. Quarteroni, Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comp., 38 (1982), 67-86.  doi: 10.1090/S0025-5718-1982-0637287-3.  Google Scholar

[7]

M. Chirita and C. M. Ionescu, Models of biomimetic tissues for vascular grafts, in On biomimetics, In-Tech, 2011, 43–52. doi: 10.5772/18248.  Google Scholar

[8]

J. C. Criscione and K. R. Rajagopal, On the modeling of the non-linear response of soft elastic bodies, International Journal of Non-Linear Mechanics, 56 (2013), 20-24.  doi: 10.1016/j.ijnonlinmec.2013.05.004.  Google Scholar

[9]

H. A. Erbay and Y. Şengül, Traveling waves in one-dimensional non-linear models of strain-limiting viscoelasticity, International Journal of Non-Linear Mechanics, 77 (2015), 61-68.  doi: 10.1016/j.ijnonlinmec.2015.07.005.  Google Scholar

[10]

H. A. Erbay and Y. Şengül, A thermodynamically consistent stress-rate type model of one-dimensional strain-limiting viscoelasticity, Z. Angew. Math. Phys., 71 (2020), Paper No. 94, 10 pp. doi: 10.1007/s00033-020-01315-7.  Google Scholar

[11]

H. A. Erbay, A. Erkip and Y. Şengül, Local existence of solutions to the initial-value problem for one-dimensional strain-limiting viscoelasticity, J. Differential Equations, 269 (2020), 9720-9739.  doi: 10.1016/j.jde.2020.06.052.  Google Scholar

[12]

L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 1998.  Google Scholar

[13]

A. D. Freed and K. R. Rajagopal, A viscoelastic model for describing the response of biological fibers, Acta Mech., 277 (2016), 3367-3380.  doi: 10.1007/s00707-016-1673-7.  Google Scholar

[14]

H. Itou, V. A. Kovtunenko and K. R. Rajagopal, On the states of stress and strain adjacent to a crack in a strain-limiting viscoelastic body, Math. Mech. Solids, 23 (2018), 433-444.  doi: 10.1177/1081286517709517.  Google Scholar

[15]

H. Itou, V. A. Kovtunenko and K. R. Rajagopal, Crack problem within the context of implicitly constituted quasi-linear viscoelasticity, Math. Models Methods Appl. Sci., 29 (2019), 355-372.  doi: 10.1142/S0218202519500118.  Google Scholar

[16]

K. R. Rajagopal, On implicit constitutive theories, Appl. Math., 48 (2003), 279-319.  doi: 10.1023/A:1026062615145.  Google Scholar

[17]

K. R. Rajagopal, A note on a reappraisal and generalization of the Kelvin–Voigt model, Mechanics Research Communications, 36 (2009), 232-235.  doi: 10.1016/j.mechrescom.2008.09.005.  Google Scholar

[18]

K. R. Rajagopal, On a new class of models in elasticity, Math. Comput. Appl., 15 (2010), 506-528.  doi: 10.3390/mca15040506.  Google Scholar

[19]

K. R. Rajagopal, Non-linear elastic bodies exhibiting limiting small strain, Math. Mech. Solids, 16 (2011), 122-139.  doi: 10.1177/1081286509357272.  Google Scholar

[20]

K. R. Rajagopal, On the nonlinear elastic response of bodies in the small strain range, Acta Mech., 225 (2014), 1545-1553.  doi: 10.1007/s00707-013-1015-y.  Google Scholar

[21]

K. R. Rajagopal and G. Saccomandi, Circularly polarized wave propagation in a class of bodies defined by a new class of implicit constitutive relations, Z. Angew. Math. Phys., 65 (2014), 1003-1010.  doi: 10.1007/s00033-013-0362-9.  Google Scholar

[22]

T. Roubíček, Nonlinear partial differential equations with applications, 2$^{nd}$ edition, Birkhäuser/Springer Basel AG, Basel, 2013.  Google Scholar

[23]

M. Ruzhansky and M. Sugimoto, On global inversion of homogeneous maps, Bull. Math. Sci., 5 (2015), 13-18.  doi: 10.1007/s13373-014-0059-1.  Google Scholar

[24]

T. Saito, T. Furuta, J.-H. Hwang, S. Kuramoto, K. Nishino, N. Suzuki, R. Chen, A. Yamada, K. Ito, Y. Seno, T. Nonaka, H. Ikehata, N. Nagasako, C. Iwamoto, Y. Ikuhara and and T. Sakuma, Multifunctional alloys obtained via a dislocation-free plastic deformation mechanism, Science (New York, N.Y.), 300 (2003), 464-467.  doi: 10.1126/science.1081957.  Google Scholar

[25]

Y. Şengül, Viscoelasticity with limiting strain, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 57-70.  doi: 10.3934/dcdss.2020330.  Google Scholar

[26] J. Warga, Optimal control of differential and functional equations, Academic Press, New York-London, 1972.   Google Scholar
[1]

Yasemin Şengül. Viscoelasticity with limiting strain. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 57-70. doi: 10.3934/dcdss.2020330
[2]

Kimie Nakashima, Wei-Ming Ni, Linlin Su. An indefinite nonlinear diffusion problem in population genetics, I: Existence and limiting profiles. Discrete & Continuous Dynamical Systems, 2010, 27 (2) : 617-641. doi: 10.3934/dcds.2010.27.617
[3]

Zhenhua Guo, Zilai Li. Global existence of weak solution to the free boundary problem for compressible Navier-Stokes. Kinetic & Related Models, 2016, 9 (1) : 75-103. doi: 10.3934/krm.2016.9.75
[4]

Feng Li, Yuxiang Li. Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5409-5436. doi: 10.3934/dcdsb.2019064
[5]

Zhong Tan, Jianfeng Zhou. Higher integrability of weak solution of a nonlinear problem arising in the electrorheological fluids. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1335-1350. doi: 10.3934/cpaa.2016.15.1335
[6]

Francesca Crispo, Paolo Maremonti. A remark on the partial regularity of a suitable weak solution to the Navier-Stokes Cauchy problem. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1283-1294. doi: 10.3934/dcds.2017053
[7]

Jiří Neustupa. A note on local interior regularity of a suitable weak solution to the Navier--Stokes problem. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1391-1400. doi: 10.3934/dcdss.2013.6.1391
[8]

Út V. Lê. Regularity of the solution of a nonlinear wave equation. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1099-1115. doi: 10.3934/cpaa.2010.9.1099
[9]

Toyohiko Aiki. On the existence of a weak solution to a free boundary problem for a model of a shape memory alloy spring. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 1-13. doi: 10.3934/dcdss.2012.5.1
[10]

Jing Li, Boling Guo, Lan Zeng, Yitong Pei. Global weak solution and smooth solution of the periodic initial value problem for the generalized Landau-Lifshitz-Bloch equation in high dimensions. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1345-1360. doi: 10.3934/dcdsb.2019230
[11]

Merab Svanadze. On the theory of viscoelasticity for materials with double porosity. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2335-2352. doi: 10.3934/dcdsb.2014.19.2335
[12]

Zhen Lei. Rotation-strain decomposition for the incompressible viscoelasticity in two dimensions. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2861-2871. doi: 10.3934/dcds.2014.34.2861
[13]

Chao Yang, Yanbing Yang. Long-time behavior for fourth-order wave equations with strain term and nonlinear weak damping term. Discrete & Continuous Dynamical Systems - S, 2021, 14 (12) : 4643-4658. doi: 10.3934/dcdss.2021110
[14]

Igor Chueshov, Irena Lasiecka. Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 777-809. doi: 10.3934/dcds.2006.15.777
[15]

Cong Qin, Xinfu Chen. A new weak solution to an optimal stopping problem. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4823-4837. doi: 10.3934/dcdsb.2020128
[16]

Guillaume Warnault. Regularity of the extremal solution for a biharmonic problem with general nonlinearity. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1709-1723. doi: 10.3934/cpaa.2009.8.1709
[17]

Boyan Jonov, Paul Kessenich, Thomas C. Sideris. Global existence of small displacement solutions for Hookean incompressible viscoelasticity in 3D. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021038
[18]

Abdelaaziz Sbai, Youssef El Hadfi, Mohammed Srati, Noureddine Aboutabit. Existence of solution for Kirchhoff type problem in Orlicz-Sobolev spaces Via Leray-Schauder's nonlinear alternative. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021015
[19]

Yue Pang, Xingchang Wang, Furong Wu. Global existence and blowup in infinite time for a fourth order wave equation with damping and logarithmic strain terms. Discrete & Continuous Dynamical Systems - S, 2021, 14 (12) : 4439-4463. doi: 10.3934/dcdss.2021115
[20]

Xiaoqiang Dai, Wenke Li. Non-global solution for visco-elastic dynamical system with nonlinear source term in control problem. Electronic Research Archive, 2021, 29 (6) : 4087-4098. doi: 10.3934/era.2021073

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (100)
  • HTML views (155)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy