January  2021, 14(1): 57-70. doi: 10.3934/dcdss.2020330

Viscoelasticity with limiting strain

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

Dedicated to Alexander Mielke on the occasion of his 60th birthday

Received  June 2019 Revised  October 2019 Published  January 2021 Early access  April 2020

A self-contained review is given for the development and current state of implicit constitutive modelling of viscoelastic response of materials in the context of strain-limiting theory.

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

S. P. Atul Narayan and K. R. Rajagopal, Unsteady flows of a class of novel generalizations of the Navier-Stokes fluid, Appl. Math. Comput., 219 (2013), 9935-9946.  doi: 10.1016/j.amc.2013.03.049.

[2]

B. BenešováM. Kružík and A. Schlömerkemper, A note on locking materials and gradient polyconvexity, Math. Mod. Methods Appl. Sci., 28 (2018), 2367-2401.  doi: 10.1142/S0218202518500513.

[3]

C. Bridges and K. R. Rajagopal, Implicit constitutive models with a thermodynamic basis: A study of stress concentration, Z. Angew. Math. Phys., 66 (2015), 191-208.  doi: 10.1007/s00033-014-0398-5.

[4]

M. BulíčekJ. MálekK. 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.

[5]

M. BulíčekJ. 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]

R. Bustamante, Some topics on a new class of elastic bodies, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2009), 1377-1392.  doi: 10.1098/rspa.2008.0427.

[7]

R. Bustamante and K. R. Rajagopal, A note on plain strain and stress problems for a new class of elastic bodies, Math. Mech. Solids, 15 (2010), 229-238.  doi: 10.1177/1081286508098178.

[8]

R. Bustamante and K. R. Rajagopal, Solutions of some simple boundary value problems within the context of a new class of elastic materials, Int. J. Nonlinear Mech., 46 (2011), 376-386.  doi: 10.1016/j.ijnonlinmec.2010.10.002.

[9]

R. Bustamante and D. Sfyris, Direct determination of stresses from the stress equations of motion and wave propagation for a new class of elastic bodies, Math. Mech. Solids, 20 (2015), 80-91.  doi: 10.1177/1081286514543600.

[10]

J. C. Criscione and K. R. Rajagopal, On the modeling of the non-linear response of soft elastic bodies, Int. J. Nonlinear Mech., 56 (2013), 20-24.  doi: 10.1016/j.ijnonlinmec.2013.05.004.

[11]

F. Demengel and P. Suquet, On locking materials, Acta Appl. Math., 6 (1986), 185-211.  doi: 10.1007/BF00046725.

[12]

V. K. DevendiranR. K. SandeepK. Kannan and K. R. Rajagopal, A thermodynamically consistent constitutive equation for describing the response exhibited by several alloys and the study of a meaningful physical problem, Int. J. Solids and Struct., 108 (2017), 1-10.  doi: 10.1016/j.ijsolstr.2016.07.036.

[13]

H. A. Erbay and Y. Şengül, Traveling waves in one-dimensional non-linear models of strain-limiting viscoelasticity, Int. J. Nonlinear Mech., 77 (2015), 61-68.  doi: 10.1016/j.ijnonlinmec.2015.07.005.

[14]

H. A. Erbay and Y. Şengül, A thermodynamically consistent stress-rate type model of one-dimensional strain-limiting viscoelasticity, Z. Angew. Math. Phys., accepted.

[15]

H. A. Erbay, A. Erkip and Y. Şengül, Local existence of solutions to the initial-value problem for one-dimensional strain-limiting viscoelasticity, submitted.

[16]

N. Gelmetti and E. Süli, Spectral approximation of a strain-limiting nonlinear elastic model, Mat. Vesnik, 71 (2019), 63-89. 

[17]

F. Golay and P. Seppecher, Locking materials and the topology of optimal shapes, Eur. J. Mech. A Solids, 20 (2001), 631-644.  doi: 10.1016/S0997-7538(01)01146-9.

[18]

K. GouM. MallikarjunaK. R. Rajagopal and J. R. Walton, Modeling fracture in the context of a strain-limiting theory of elasticity: A single plane-strain crack, Int. J. Eng. Sci., 88 (2015), 73-82.  doi: 10.1016/j.ijengsci.2014.04.018.

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Y. L. Hao, S. J. Li, S. Y. Sun, C. Y. Zheng, Q. M. Hu and R. Yang, Super-elastic titanium alloy with unstable plastic deformation, Appl. Physc. Lett., 87 (2005), 091906. doi: 10.1063/1.2037192.

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F. Q. HouS. J. LiY. L. Hao and R. Yang, Nonlinear elastic deformation behaviour of Ti-30Nb-12Zr alloys, Scr. Mater., 63 (2010), 54-57. 

[22]

H. ItouV. A. Kovtunenko and K. R. Rajagopal, Contacting crack faces within the context of bodies exhibiting limiting strains, JSIAM Letters, 9 (2017), 61-64.  doi: 10.14495/jsiaml.9.61.

[23]

H. ItouV. A. Kovtunenko and K. R. Rajagopal, Nonlinear elasticity with limiting small strain for cracks subject to non-penetration, Math. Mech. Solids, 22 (2017), 1334-1346.  doi: 10.1177/1081286516632380.

[24]

H. ItouV. 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.

[25]

H. ItouV. A. Kovtunenko and K. R. Rajagopal, Crack problem within the context of implicitly constituted quasi-linear viscoelasticity, Math. Mod. Meth. Appl. Sci., 29 (2019), 355-372.  doi: 10.1142/S0218202519500118.

[26]

K. KannanK. R. Rajagopal and G. Saccomandi, Unsteady motions of a new class of elastic solids, Wave Motion, 51 (2014), 833-843.  doi: 10.1016/j.wavemoti.2014.02.004.

[27]

V. KulvaitJ. Málek and K. R. Rajagopal, Anti-plane stress state of a plate with a V-notch for a new class of elastic solids, Int. J. Fract., 179 (2013), 59-73. 

[28]

V. KulvaitJ. Málek and K. R. Rajagopal, Modeling gum metal and other newly developed titanium alloys within a new class of constitutive relations for elastic bodies, Arch. Mech., 69 (2017), 223-241. 

[29]

V. KulvaitJ. Málek and K. R. Rajagopal, The state of stress and strain adjacent to notches in a new class of nonlinear elastic bodies, J. Elast., 135 (2019), 375-397.  doi: 10.1007/s10659-019-09724-0.

[30]

A. B. MaganD. P. Mason and C. Harley, Two-dimensional nonlinear stress and displacement waves for a new class of constitutive equations, Wave Motion, 77 (2018), 156-185.  doi: 10.1016/j.wavemoti.2017.12.003.

[31]

T. Mai and J. R. Walton, On monotonicity for strain-limiting theories of elasticity, Math. Mech. Solids, 20 (2014), 121-139. 

[32]

R. MenesesO. Orellana and R. Bustamante, A note on the wave equation for a class of constitutive relations for nonlinear elastic bodies that are not Green elastic, Math. Mech. Solids, 23 (2018), 148-158.  doi: 10.1177/1081286516673234.

[33]

J. Merodio and K. R. Rajagopal, On constitutive equations for anisotropic nonlinearly viscoelastic solids, Math. Mech. Solids, 12 (2007), 131-147.  doi: 10.1177/1081286505055472.

[34]

A. MielkeC. Ortner and Y. Şengül, An approach to nonlinear viscoelasticity via metric gradient flows, SIAM J. Math. Anal., 46 (2014), 1317-1347.  doi: 10.1137/130927632.

[35]

A. MielkeF. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Rational Mech. Anal., 162 (2002), 137-177.  doi: 10.1007/s002050200194.

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A. Mielke and M. Thomas, Damage of nonlinearly elastic materials at small strain-existence and regularity results, Z. Angew. Math. Mech., 90 (2010), 88-112.  doi: 10.1002/zamm.200900243.

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Y. Şengül, On one-dimensional strain-limiting viscoelasticity with an arctangent type nonlinearity, submitted.

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M. ZappalortoF. Berto and K. R. Rajagopal, On the anti-plane state of stress near pointed or sharply radiused notches in strain limiting elastic materials: Closed form solution and implications for fracture assessements, Int. J. Fract., 199 (2016), 169-184.  doi: 10.1007/s10704-016-0102-1.

show all references

References:
[1]

S. P. Atul Narayan and K. R. Rajagopal, Unsteady flows of a class of novel generalizations of the Navier-Stokes fluid, Appl. Math. Comput., 219 (2013), 9935-9946.  doi: 10.1016/j.amc.2013.03.049.

[2]

B. BenešováM. Kružík and A. Schlömerkemper, A note on locking materials and gradient polyconvexity, Math. Mod. Methods Appl. Sci., 28 (2018), 2367-2401.  doi: 10.1142/S0218202518500513.

[3]

C. Bridges and K. R. Rajagopal, Implicit constitutive models with a thermodynamic basis: A study of stress concentration, Z. Angew. Math. Phys., 66 (2015), 191-208.  doi: 10.1007/s00033-014-0398-5.

[4]

M. BulíčekJ. MálekK. 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.

[5]

M. BulíčekJ. 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]

R. Bustamante, Some topics on a new class of elastic bodies, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2009), 1377-1392.  doi: 10.1098/rspa.2008.0427.

[7]

R. Bustamante and K. R. Rajagopal, A note on plain strain and stress problems for a new class of elastic bodies, Math. Mech. Solids, 15 (2010), 229-238.  doi: 10.1177/1081286508098178.

[8]

R. Bustamante and K. R. Rajagopal, Solutions of some simple boundary value problems within the context of a new class of elastic materials, Int. J. Nonlinear Mech., 46 (2011), 376-386.  doi: 10.1016/j.ijnonlinmec.2010.10.002.

[9]

R. Bustamante and D. Sfyris, Direct determination of stresses from the stress equations of motion and wave propagation for a new class of elastic bodies, Math. Mech. Solids, 20 (2015), 80-91.  doi: 10.1177/1081286514543600.

[10]

J. C. Criscione and K. R. Rajagopal, On the modeling of the non-linear response of soft elastic bodies, Int. J. Nonlinear Mech., 56 (2013), 20-24.  doi: 10.1016/j.ijnonlinmec.2013.05.004.

[11]

F. Demengel and P. Suquet, On locking materials, Acta Appl. Math., 6 (1986), 185-211.  doi: 10.1007/BF00046725.

[12]

V. K. DevendiranR. K. SandeepK. Kannan and K. R. Rajagopal, A thermodynamically consistent constitutive equation for describing the response exhibited by several alloys and the study of a meaningful physical problem, Int. J. Solids and Struct., 108 (2017), 1-10.  doi: 10.1016/j.ijsolstr.2016.07.036.

[13]

H. A. Erbay and Y. Şengül, Traveling waves in one-dimensional non-linear models of strain-limiting viscoelasticity, Int. J. Nonlinear Mech., 77 (2015), 61-68.  doi: 10.1016/j.ijnonlinmec.2015.07.005.

[14]

H. A. Erbay and Y. Şengül, A thermodynamically consistent stress-rate type model of one-dimensional strain-limiting viscoelasticity, Z. Angew. Math. Phys., accepted.

[15]

H. A. Erbay, A. Erkip and Y. Şengül, Local existence of solutions to the initial-value problem for one-dimensional strain-limiting viscoelasticity, submitted.

[16]

N. Gelmetti and E. Süli, Spectral approximation of a strain-limiting nonlinear elastic model, Mat. Vesnik, 71 (2019), 63-89. 

[17]

F. Golay and P. Seppecher, Locking materials and the topology of optimal shapes, Eur. J. Mech. A Solids, 20 (2001), 631-644.  doi: 10.1016/S0997-7538(01)01146-9.

[18]

K. GouM. MallikarjunaK. R. Rajagopal and J. R. Walton, Modeling fracture in the context of a strain-limiting theory of elasticity: A single plane-strain crack, Int. J. Eng. Sci., 88 (2015), 73-82.  doi: 10.1016/j.ijengsci.2014.04.018.

[19] M. E. Gurtin, An Introduction to Continuum Mechanics, Mathematics in Science and Engineering, 158. Academic Press, Inc., New York-London, 1981. 
[20]

Y. L. Hao, S. J. Li, S. Y. Sun, C. Y. Zheng, Q. M. Hu and R. Yang, Super-elastic titanium alloy with unstable plastic deformation, Appl. Physc. Lett., 87 (2005), 091906. doi: 10.1063/1.2037192.

[21]

F. Q. HouS. J. LiY. L. Hao and R. Yang, Nonlinear elastic deformation behaviour of Ti-30Nb-12Zr alloys, Scr. Mater., 63 (2010), 54-57. 

[22]

H. ItouV. A. Kovtunenko and K. R. Rajagopal, Contacting crack faces within the context of bodies exhibiting limiting strains, JSIAM Letters, 9 (2017), 61-64.  doi: 10.14495/jsiaml.9.61.

[23]

H. ItouV. A. Kovtunenko and K. R. Rajagopal, Nonlinear elasticity with limiting small strain for cracks subject to non-penetration, Math. Mech. Solids, 22 (2017), 1334-1346.  doi: 10.1177/1081286516632380.

[24]

H. ItouV. 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.

[25]

H. ItouV. A. Kovtunenko and K. R. Rajagopal, Crack problem within the context of implicitly constituted quasi-linear viscoelasticity, Math. Mod. Meth. Appl. Sci., 29 (2019), 355-372.  doi: 10.1142/S0218202519500118.

[26]

K. KannanK. R. Rajagopal and G. Saccomandi, Unsteady motions of a new class of elastic solids, Wave Motion, 51 (2014), 833-843.  doi: 10.1016/j.wavemoti.2014.02.004.

[27]

V. KulvaitJ. Málek and K. R. Rajagopal, Anti-plane stress state of a plate with a V-notch for a new class of elastic solids, Int. J. Fract., 179 (2013), 59-73. 

[28]

V. KulvaitJ. Málek and K. R. Rajagopal, Modeling gum metal and other newly developed titanium alloys within a new class of constitutive relations for elastic bodies, Arch. Mech., 69 (2017), 223-241. 

[29]

V. KulvaitJ. Málek and K. R. Rajagopal, The state of stress and strain adjacent to notches in a new class of nonlinear elastic bodies, J. Elast., 135 (2019), 375-397.  doi: 10.1007/s10659-019-09724-0.

[30]

A. B. MaganD. P. Mason and C. Harley, Two-dimensional nonlinear stress and displacement waves for a new class of constitutive equations, Wave Motion, 77 (2018), 156-185.  doi: 10.1016/j.wavemoti.2017.12.003.

[31]

T. Mai and J. R. Walton, On monotonicity for strain-limiting theories of elasticity, Math. Mech. Solids, 20 (2014), 121-139. 

[32]

R. MenesesO. Orellana and R. Bustamante, A note on the wave equation for a class of constitutive relations for nonlinear elastic bodies that are not Green elastic, Math. Mech. Solids, 23 (2018), 148-158.  doi: 10.1177/1081286516673234.

[33]

J. Merodio and K. R. Rajagopal, On constitutive equations for anisotropic nonlinearly viscoelastic solids, Math. Mech. Solids, 12 (2007), 131-147.  doi: 10.1177/1081286505055472.

[34]

A. MielkeC. Ortner and Y. Şengül, An approach to nonlinear viscoelasticity via metric gradient flows, SIAM J. Math. Anal., 46 (2014), 1317-1347.  doi: 10.1137/130927632.

[35]

A. MielkeF. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Rational Mech. Anal., 162 (2002), 137-177.  doi: 10.1007/s002050200194.

[36]

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Figure 1.  Limiting strain behaviour
Figure 2.  Experimental data for the stress-strain relationship for porcine carotid and thoracic artery tissues (cf. [43])
Figure 3.  Left. Model A: $ g(T) = \beta T + \alpha \left(1 + \frac{\gamma}{2} T^{2}\right)^{n} T $; Model B: $ g(T) = \frac{T}{(1 + |T|^{r})^{1/r}} $; Model C: $ g(T) = \alpha \left\{\left[1 - \exp\left(- \frac{\beta T}{1 + \delta |T|}\right)\right] + \frac{\gamma T}{1 + |T|} \right\} $; Model D: $ g(T) = \alpha \left(1-\frac{1}{1 +\frac{ T}{1 + \delta |T|}}\right) + \beta \left(1 + \frac{1}{1 + \gamma T^{2}}\right)^{n} T $, where $ \alpha, \beta, \gamma, \delta, n $ and $ r > 0 $ are constants. Right. General linear, quadratic and cubic nonlinearities
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