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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[16]

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

[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.

[18]

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

[19]

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

[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.

[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.

[22]

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

[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.

[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.

[25]

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

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[16]

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

[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.

[18]

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

[19]

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

[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.

[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.

[22]

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

[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.

[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.

[25]

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

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