# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021053

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

 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  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 $a$.

Citation: Miroslav Bulíček, Victoria Patel, Endre Süli, Yasemin Şengül. Existence of large-data global weak solutions to a model of a strain-limiting viscoelastic body. Communications on Pure & Applied Analysis, 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

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##### 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
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