
-
Previous Article
Adaptive time stepping in elastoplasticity
- DCDS-S Home
- This Issue
-
Next Article
Cahn-Hilliard equation with capillarity in actual deforming configurations
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.
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. 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.
|
[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. 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. |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[69] |
U. Saravanan, Advanced Solid Mechanics, McGraw-Hill Inc., Indian Institute of Technology Madras, 2013. Google Scholar |
[70] | A. J. M. Spencer, Theory of Invariants, Continuum Physics, Academic Press, New York, 1971. Google Scholar |
[71] |
Y. Şengül, On one-dimensional strain-limiting viscoelasticity with an arctangent type nonlinearity, submitted. Google Scholar |
[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. Google Scholar |
[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. 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.
|
[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. 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. |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[69] |
U. Saravanan, Advanced Solid Mechanics, McGraw-Hill Inc., Indian Institute of Technology Madras, 2013. Google Scholar |
[70] | A. J. M. Spencer, Theory of Invariants, Continuum Physics, Academic Press, New York, 1971. Google Scholar |
[71] |
Y. Şengül, On one-dimensional strain-limiting viscoelasticity with an arctangent type nonlinearity, submitted. Google Scholar |
[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. Google Scholar |
[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. |


[1] |
Stephen Doty and Anthony Giaquinto. Generators and relations for Schur algebras. Electronic Research Announcements, 2001, 7: 54-62. |
[2] |
Ziteng Wang, Shu-Cherng Fang, Wenxun Xing. On constraint qualifications: Motivation, design and inter-relations. Journal of Industrial & Management Optimization, 2013, 9 (4) : 983-1001. doi: 10.3934/jimo.2013.9.983 |
[3] |
Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409 |
[4] |
Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311 |
[5] |
Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1717-1746. doi: 10.3934/dcdss.2020451 |
[6] |
W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349 |
[7] |
John Leventides, Costas Poulios, Georgios Alkis Tsiatsios, Maria Livada, Stavros Tsipras, Konstantinos Lefcaditis, Panagiota Sargenti, Aleka Sargenti. Systems theory and analysis of the implementation of non pharmaceutical policies for the mitigation of the COVID-19 pandemic. Journal of Dynamics & Games, 2021 doi: 10.3934/jdg.2021004 |
2019 Impact Factor: 1.233
Tools
Metrics
Other articles
by authors
[Back to Top]