American Institute of Mathematical Sciences

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

Viscoelasticity with limiting strain

Yasemin Şengül

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.

Keywords: viscoelasticity, implicit constitutive relations, strain-limiting theory.
Mathematics Subject Classification: Primary: 74D99, 74A20, 35Q74; Secondary: 74A05, 74A10.
Citation: Yasemin Şengül. Viscoelasticity with limiting strain. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 57-70. doi: 10.3934/dcdss.2020330
References:
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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.  Google Scholar

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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.  Google Scholar

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M. Bulíček, J. Málek, K. 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

[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]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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[11]

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

[12]

V. K. Devendiran, R. K. Sandeep, K. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

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

[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.  Google Scholar

[18]

K. Gou, M. Mallikarjuna, K. 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.  Google Scholar

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[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.  Google Scholar

[21]

F. Q. Hou, S. J. Li, Y. L. Hao and R. Yang, Nonlinear elastic deformation behaviour of Ti-30Nb-12Zr alloys, Scr. Mater., 63 (2010), 54-57.   Google Scholar

[22]

H. Itou, V. 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.  Google Scholar

[23]

H. Itou, V. 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.  Google Scholar

[24]

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

[25]

H. Itou, V. 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.  Google Scholar

[26]

K. Kannan, K. 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.  Google Scholar

[27]

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Figure 1.  Limiting strain behaviour
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43])">Figure 2.  Experimental data for the stress-strain relationship for porcine carotid and thoracic artery tissues (cf. [43])
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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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