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Fractional oscillon equations; solvability and connection with classical oscillon equations
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 |
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 $.
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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