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Limiting strain behaviour
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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 |
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.
Mathematics Subject Classification:
Primary: 74D99, 74A20, 35Q74; Secondary: 74A05, 74A10.
Citation:
Yasemin Şengül. Viscoelasticity with limiting strain. Discrete and Continuous Dynamical Systems - S,
2021, 14
(1)
: 57-70.
doi: 10.3934/dcdss.2020330
![]()
References:
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show all references
References:
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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. Benešová, M. 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. 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. |
| [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] |
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. 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. |
| [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. 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. |
| [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. 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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [27] |
V. Kulvait, J. 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. Kulvait, J. 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] |
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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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