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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:
| [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. 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] |
V. Kulvait, J. 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. Magan, D. 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. Meneses, O. 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. Mielke, C. 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. Mielke, F. 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] |
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. |
| [37] |
A. Mielke and L. Truskinovsky,
From discrete visco-elasticity to continuum rate-independent plasticity: Rigorous results, Arch. Rational Mech. Anal., 203 (2012), 577-619.
doi: 10.1007/s00205-011-0460-9. |
| [38] |
S. Montero, R. Bustamante and A. Ortiz-Bernardin,
A finite element analysis of some boundary value problems for a new type of constitutive relation for elastic bodies, Acta Mech., 227 (2016), 601-615.
doi: 10.1007/s00707-015-1480-6. |
| [39] |
A. Muliana, K. R. Rajagopal and A. S. Wineman,
A new class of quasi-linear models for describing the nonlinear viscoelastic response of materials, Acta Mech., 224 (2013), 2169-2183.
doi: 10.1007/s00707-013-0848-8. |
| [40] |
N. Nagasako, R. Asahi and J. Hafner,
First-principles study of Gum-Metal alloys: Mechanism of ideal strength, R & D Review of Toyota CRDL, 44 (2013), 61-68.
|
| [41] |
A. Ortiz, R. Bustamante and K. R. Rajagopal,
A numerical study of a plate with a hole for a new class of elastic bodies, Acta Mech., 223 (2012), 1971-1981.
doi: 10.1007/s00707-012-0690-4. |
| [42] |
A. Ortiz-Bernardin, R. Bustamante and K. R. Rajagopal,
A numerical study of elastic bodies that are described by constitutive equations that exhibit limited strains, Int. J. Solids and Struct., 51 (2014), 875-885.
doi: 10.1016/j.ijsolstr.2013.11.014. |
| [43] |
N. Payne and K. Pochiraju,
Methodologies for constitutive model parameter identification for strain locking materials, Mech. Mater., 134 (2019), 30-37.
doi: 10.1016/j.mechmat.2019.04.004. |
| [44] |
A. Phillips,
The theory of locking materials, Trans. Soc. Rheol., 3 (1959), 13-26.
doi: 10.1122/1.548840. |
| [45] |
W. Prager,
On ideal locking materials, Trans. Soc. Rheol., 1 (1957), 169-175.
doi: 10.1122/1.548818. |
| [46] |
K. R. Rajagopal,
On implicit constitutive theories, Appl. Math., 48 (2003), 279-319.
doi: 10.1023/A:1026062615145. |
| [47] |
K. R. Rajagopal,
On implicit constitutive theories for fluids, J. Fluid Mech., 550 (2006), 243-249.
doi: 10.1017/S0022112005008025. |
| [48] |
K. R. Rajagopal,
The elasticity of elasticity, Z. Angew. Math. Phys., 58 (2007), 309-317.
doi: 10.1007/s00033-006-6084-5. |
| [49] |
K. R. Rajagopal,
On a new class of models in elasticity, J. Math. Comp. Appl., 15 (2010), 506-528.
doi: 10.3390/mca15040506. |
| [50] |
K. R. Rajagopal,
Non-linear elastic bodies exhibiting limiting small strain, Math. Mech. Solids, 16 (2011), 122-139.
doi: 10.1177/1081286509357272. |
| [51] |
K. R. Rajagopal,
Conspectus of concepts of elasticity, Math. Mech. Solids, 16 (2011), 536-562.
doi: 10.1177/1081286510387856. |
| [52] |
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. |
| [53] |
K. R. Rajagopal,
A note on the classification of anisotropy of bodies defined by implicit constitutive relations, Mech. Res. Comm., 64 (2015), 38-41.
doi: 10.1016/j.mechrescom.2014.11.005. |
| [54] |
K. R. Rajagopal,
A note on the linearization of the constitutive relations of non-linear elastic bodies, Mech. Res. Comm., 93 (2018), 132-137.
doi: 10.1016/j.mechrescom.2017.08.002. |
| [55] |
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. |
| [56] |
K. R. Rajagopal and G. Saccomandi,
Shear waves in a class of nonlinear viscoelastic solids, Quart. J. Mech. Appl. Math., 56 (2003), 311-326.
doi: 10.1093/qjmam/56.2.311. |
| [57] |
K. R. Rajagopal and A. R. Srinivasa,
A thermodynamic frame work for rate type fluid models, J. Non-Newton. Fluid Mech., 88 (2000), 207-227.
doi: 10.1016/S0377-0257(99)00023-3. |
| [58] |
K. R. Rajagopal and A. R. Srinivasa,
On thermomechanical restrictions of continua, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 460 (2004), 631-651.
doi: 10.1098/rspa.2002.1111. |
| [59] |
K. R. Rajagopal and A. R. Srinivasa,
On the nature of constraints for continua undergoing dissipative processes, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 2785-2795.
doi: 10.1098/rspa.2004.1385. |
| [60] |
K. R. Rajagopal and A. R. Srinivasa,
On the response of non-dissipative solids, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 357-367.
doi: 10.1098/rspa.2006.1760. |
| [61] |
K. R. Rajagopal and A. R. Srinivasa,
On the development of fluid models of the differential type within a new thermodynamic framework, Mech. Res. Commun., 35 (2008), 483-489.
doi: 10.1016/j.mechrescom.2008.02.004. |
| [62] |
K. R. Rajagopal and A. R. Srinivasa,
A Gibbs-potential-based formulation for obtaining the response functions for a class of viscoelastic materials, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467 (2011), 39-58.
doi: 10.1098/rspa.2010.0136. |
| [63] |
K. R. Rajagopal and A. R. Srinivasa,
An implicit thermomechanical theory based on a Gibbs potential formulation for describing the response of thermoviscoelastic solids, Int. J. Eng. Sci., 70 (2013), 15-28.
doi: 10.1016/j.ijengsci.2013.03.005. |
| [64] |
K. R. Rajagopal and J. R. Walton,
Modeling fracture in the context of a strain-limiting theory of elasticity: A single anti-plane shear crack, Int. J. Fract., 169 (2011), 39-48.
doi: 10.1007/s10704-010-9581-7. |
| [65] |
K. R. Rajagopal and A. S. Wineman,
A quasi-correspondence principle for quasi-linear viscoelastic solids, Mech. Time-Depend. Mater., 12 (2008), 1-14.
|
| [66] |
R. S. Rivlin,
Further remarks on the stress-deformation relations for isotropic materials, Indiana Univ. Math. J., 4 (1955), 681-702.
doi: 10.1512/iumj.1955.4.54025. |
| [67] |
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 T. Sakuma,
Multifunctional alloys obtained via a dislocation-free plastic deformation mechanism, Science, 300 (2003), 464-467.
doi: 10.1126/science.1081957. |
| [68] |
N. Sakaguch, M. Niinomi and T. Akahori,
Tensile deformation behavior of Ti-Nb-Ta-Zr biomedical alloys, Mater. Trans., 45 (2004), 1113-1119.
|
| [69] |
U. Saravanan, Advanced Solid Mechanics, McGraw-Hill Inc., Indian Institute of Technology Madras, 2013. |
| [70] |
A. J. M. Spencer, Theory of Invariants, Continuum Physics, Academic Press, New York, 1971.
|
| [71] |
Y. Şengül, On one-dimensional strain-limiting viscoelasticity with an arctangent type nonlinearity, submitted. |
| [72] |
R. J. Talling, R. J. Dashwood, M. Jackson and D. Dye,
On the mechanism of superelasticity in Gum metal, Acta Mater., 57 (2009), 1188-1198.
doi: 10.1016/j.actamat.2008.11.013. |
| [73] |
C. Truesdell, The Elements of Continuum Mechanics, Springer-Verlag New York, Inc., New York, 1966. |
| [74] |
A. S. Wineman and K. R. Rajagopal, Mechanical Response of Polymers: An Introduction, Cambridge University Press, Cambridge, 2000.
|
| [75] |
E. Withey, M. Jin, A. Minor, S. Kuramoto, D. C. Chrzan and J. W. Morris,
The deformation of "Gum Metal" in nanoindentation, Mater. Sci. Eng. A, 493 (2008), 26-32.
doi: 10.1016/j.msea.2007.07.097. |
| [76] |
S. Q. Zhang, S. J. Li, M. T. Jia, Y. L. Hao and R. Yang,
Fatigue properties of a multifunctional titanium alloy exhibiting nonlinear elastic deformation behavior, Scr. Mater., 60 (2009), 733-736.
doi: 10.1016/j.scriptamat.2009.01.007. |
| [77] |
M. Zappalorto, F. 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. 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] |
V. Kulvait, J. 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. Magan, D. 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. Meneses, O. 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. Mielke, C. 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. Mielke, F. 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] |
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. |
| [37] |
A. Mielke and L. Truskinovsky,
From discrete visco-elasticity to continuum rate-independent plasticity: Rigorous results, Arch. Rational Mech. Anal., 203 (2012), 577-619.
doi: 10.1007/s00205-011-0460-9. |
| [38] |
S. Montero, R. Bustamante and A. Ortiz-Bernardin,
A finite element analysis of some boundary value problems for a new type of constitutive relation for elastic bodies, Acta Mech., 227 (2016), 601-615.
doi: 10.1007/s00707-015-1480-6. |
| [39] |
A. Muliana, K. R. Rajagopal and A. S. Wineman,
A new class of quasi-linear models for describing the nonlinear viscoelastic response of materials, Acta Mech., 224 (2013), 2169-2183.
doi: 10.1007/s00707-013-0848-8. |
| [40] |
N. Nagasako, R. Asahi and J. Hafner,
First-principles study of Gum-Metal alloys: Mechanism of ideal strength, R & D Review of Toyota CRDL, 44 (2013), 61-68.
|
| [41] |
A. Ortiz, R. Bustamante and K. R. Rajagopal,
A numerical study of a plate with a hole for a new class of elastic bodies, Acta Mech., 223 (2012), 1971-1981.
doi: 10.1007/s00707-012-0690-4. |
| [42] |
A. Ortiz-Bernardin, R. Bustamante and K. R. Rajagopal,
A numerical study of elastic bodies that are described by constitutive equations that exhibit limited strains, Int. J. Solids and Struct., 51 (2014), 875-885.
doi: 10.1016/j.ijsolstr.2013.11.014. |
| [43] |
N. Payne and K. Pochiraju,
Methodologies for constitutive model parameter identification for strain locking materials, Mech. Mater., 134 (2019), 30-37.
doi: 10.1016/j.mechmat.2019.04.004. |
| [44] |
A. Phillips,
The theory of locking materials, Trans. Soc. Rheol., 3 (1959), 13-26.
doi: 10.1122/1.548840. |
| [45] |
W. Prager,
On ideal locking materials, Trans. Soc. Rheol., 1 (1957), 169-175.
doi: 10.1122/1.548818. |
| [46] |
K. R. Rajagopal,
On implicit constitutive theories, Appl. Math., 48 (2003), 279-319.
doi: 10.1023/A:1026062615145. |
| [47] |
K. R. Rajagopal,
On implicit constitutive theories for fluids, J. Fluid Mech., 550 (2006), 243-249.
doi: 10.1017/S0022112005008025. |
| [48] |
K. R. Rajagopal,
The elasticity of elasticity, Z. Angew. Math. Phys., 58 (2007), 309-317.
doi: 10.1007/s00033-006-6084-5. |
| [49] |
K. R. Rajagopal,
On a new class of models in elasticity, J. Math. Comp. Appl., 15 (2010), 506-528.
doi: 10.3390/mca15040506. |
| [50] |
K. R. Rajagopal,
Non-linear elastic bodies exhibiting limiting small strain, Math. Mech. Solids, 16 (2011), 122-139.
doi: 10.1177/1081286509357272. |
| [51] |
K. R. Rajagopal,
Conspectus of concepts of elasticity, Math. Mech. Solids, 16 (2011), 536-562.
doi: 10.1177/1081286510387856. |
| [52] |
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. |
| [53] |
K. R. Rajagopal,
A note on the classification of anisotropy of bodies defined by implicit constitutive relations, Mech. Res. Comm., 64 (2015), 38-41.
doi: 10.1016/j.mechrescom.2014.11.005. |
| [54] |
K. R. Rajagopal,
A note on the linearization of the constitutive relations of non-linear elastic bodies, Mech. Res. Comm., 93 (2018), 132-137.
doi: 10.1016/j.mechrescom.2017.08.002. |
| [55] |
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. |
| [56] |
K. R. Rajagopal and G. Saccomandi,
Shear waves in a class of nonlinear viscoelastic solids, Quart. J. Mech. Appl. Math., 56 (2003), 311-326.
doi: 10.1093/qjmam/56.2.311. |
| [57] |
K. R. Rajagopal and A. R. Srinivasa,
A thermodynamic frame work for rate type fluid models, J. Non-Newton. Fluid Mech., 88 (2000), 207-227.
doi: 10.1016/S0377-0257(99)00023-3. |
| [58] |
K. R. Rajagopal and A. R. Srinivasa,
On thermomechanical restrictions of continua, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 460 (2004), 631-651.
doi: 10.1098/rspa.2002.1111. |
| [59] |
K. R. Rajagopal and A. R. Srinivasa,
On the nature of constraints for continua undergoing dissipative processes, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 2785-2795.
doi: 10.1098/rspa.2004.1385. |
| [60] |
K. R. Rajagopal and A. R. Srinivasa,
On the response of non-dissipative solids, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 357-367.
doi: 10.1098/rspa.2006.1760. |
| [61] |
K. R. Rajagopal and A. R. Srinivasa,
On the development of fluid models of the differential type within a new thermodynamic framework, Mech. Res. Commun., 35 (2008), 483-489.
doi: 10.1016/j.mechrescom.2008.02.004. |
| [62] |
K. R. Rajagopal and A. R. Srinivasa,
A Gibbs-potential-based formulation for obtaining the response functions for a class of viscoelastic materials, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467 (2011), 39-58.
doi: 10.1098/rspa.2010.0136. |
| [63] |
K. R. Rajagopal and A. R. Srinivasa,
An implicit thermomechanical theory based on a Gibbs potential formulation for describing the response of thermoviscoelastic solids, Int. J. Eng. Sci., 70 (2013), 15-28.
doi: 10.1016/j.ijengsci.2013.03.005. |
| [64] |
K. R. Rajagopal and J. R. Walton,
Modeling fracture in the context of a strain-limiting theory of elasticity: A single anti-plane shear crack, Int. J. Fract., 169 (2011), 39-48.
doi: 10.1007/s10704-010-9581-7. |
| [65] |
K. R. Rajagopal and A. S. Wineman,
A quasi-correspondence principle for quasi-linear viscoelastic solids, Mech. Time-Depend. Mater., 12 (2008), 1-14.
|
| [66] |
R. S. Rivlin,
Further remarks on the stress-deformation relations for isotropic materials, Indiana Univ. Math. J., 4 (1955), 681-702.
doi: 10.1512/iumj.1955.4.54025. |
| [67] |
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 T. Sakuma,
Multifunctional alloys obtained via a dislocation-free plastic deformation mechanism, Science, 300 (2003), 464-467.
doi: 10.1126/science.1081957. |
| [68] |
N. Sakaguch, M. Niinomi and T. Akahori,
Tensile deformation behavior of Ti-Nb-Ta-Zr biomedical alloys, Mater. Trans., 45 (2004), 1113-1119.
|
| [69] |
U. Saravanan, Advanced Solid Mechanics, McGraw-Hill Inc., Indian Institute of Technology Madras, 2013. |
| [70] |
A. J. M. Spencer, Theory of Invariants, Continuum Physics, Academic Press, New York, 1971.
|
| [71] |
Y. Şengül, On one-dimensional strain-limiting viscoelasticity with an arctangent type nonlinearity, submitted. |
| [72] |
R. J. Talling, R. J. Dashwood, M. Jackson and D. Dye,
On the mechanism of superelasticity in Gum metal, Acta Mater., 57 (2009), 1188-1198.
doi: 10.1016/j.actamat.2008.11.013. |
| [73] |
C. Truesdell, The Elements of Continuum Mechanics, Springer-Verlag New York, Inc., New York, 1966. |
| [74] |
A. S. Wineman and K. R. Rajagopal, Mechanical Response of Polymers: An Introduction, Cambridge University Press, Cambridge, 2000.
|
| [75] |
E. Withey, M. Jin, A. Minor, S. Kuramoto, D. C. Chrzan and J. W. Morris,
The deformation of "Gum Metal" in nanoindentation, Mater. Sci. Eng. A, 493 (2008), 26-32.
doi: 10.1016/j.msea.2007.07.097. |
| [76] |
S. Q. Zhang, S. J. Li, M. T. Jia, Y. L. Hao and R. Yang,
Fatigue properties of a multifunctional titanium alloy exhibiting nonlinear elastic deformation behavior, Scr. Mater., 60 (2009), 733-736.
doi: 10.1016/j.scriptamat.2009.01.007. |
| [77] |
M. Zappalorto, F. 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. |
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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