Analysis of Creep Curves General Properties under Step Loading Generated by the Rabotnov Nonlinear Relation for Viscoelastic Plastic Materials

Authors: Khokhlov A.V. Published: 24.05.2017
Published in issue: #3(72)/2017  
DOI: 10.18698/1812-3368-2017-3-93-123

Category: Mechanics | Chapter: Mechanics of Deformable Solid Body  
Keywords: elastoviscoplasticity, tension compression asymmetry, piecewise-constant loading, creep curves, asymptotics, recovery, fading memory, plastic strain accumulation, regular and singular models

We analyze basic properties of the theoretic creep curves under arbitrary piecewise-constant uniaxial stress histories generated by the Rabotnov constitutive relation with two material functions for elastoviscoplastic materials which exhibit a pronounced nonlinear heredity, rate sensitivity and multi-modulus behavior. Under minimal primary restrictions on the material functions of the relation, we study analytically the creep curves properties dependence on creep compliance function and loading program parameters, their asymptotic behavior at infinity, conditions of memory fading, formula for plastic strain after complete unloading (after recovery), influence of stress steps permutation, relations for strain and strain rate jumps produced by given stress jumps, etc. We compare the qualitative features of theoretic creep curves to typical test creep curves properties of rheonomous materials under multistep uniaxial loadings in order to examine the Rabotnov relation abilities to provide an adequate description of basic rheological phenomena related to creep and recovery, to find the zones of material functions influence and necessary phenomenological restrictions on material functions, to indicate the field of applicability or non-applicability of the model and to develop techniques for its identification and tuning. We compare the arsenal of capabilities of the Rabotnov nonlinear constitutive relation and its applicability scope to capabilities of the Boltzmann - Volterra linear viscoelasticity theory which was generalized to state the Rabotnov relation. We elucidate the inherited properties and the acquired properties due to the introduction of the second material function providing a sort of physical non-linearity.


[1] Rabotnov Yu.N. Creep problems in structural members. Amsterdam, London, North-Holland Publ. Co., 1969. 822 p.

[2] Bugakov I.I. Polzuchest’ polimernykh materialov [Polymer materials creep]. Moscow, Nauka Publ., 1973. 287 p.

[3] Findley W.N., Lai J.S., Onaran K. Creep and relaxation of nonlinear viscoelastic materials. Amsterdam, North Holland, 1976. 368 p.

[4] Malinin N.N. Raschety na polzuchest’ elementov mashinostroitel’nykh konstruktsiy [Creep design of machine-building constructions elements]. Moscow, Mashinostroenie Publ., 1981. 221 p.

[5] Moskvitin V.V. Tsiklicheskoe nagruzhenie elementov konstruktsiy [Cyclic loading constructive of parts]. Moscow, Nauka Publ., 1981. 344 p.

[6] Tschoegl N.W. The phenomenological theory of linear viscoelastic behavior. Berlin, Springer, 1989. 769 p.

[7] Betten J. Creep mechanics. Berlin-Heidelberg, Springer-Verlag, 2008. 367 p.

[8] Radchenko V.P., Kichaev P.E. Energeticheskaya kontseptsiya polzuchesti i vibropolzuchesti metallov [Power conception of materials creep and vibrocreep]. Samara, SamSTU Publ., 2011. 157 p.

[9] Christensen R.M. Mechanics of composite materials. New York, Dover Publications, 2012. 384 p.

[10] Bergstrom J.S. Mechanics of solid polymers. Theory and computational modeling. Elsevier, William Andrew, 2015. 520 p.

[11] Lokoshchenko A.M. Polzuchest’ i dlitel’naya prochnost’ metallov [Creep and long-term strength of metals]. Moscow, Fizmatlit Publ., 2016. 504 p.

[12] Rabotnov Yu.N. Some issues of creep theory. Vestnik MGU, 1948, no. 10, pp. 81-91 (in Russ.).

[13] Namestnikov V.S., Rabotnov Yu.N. On hereditary creep theories. Prikladnaya mekhanika i tekhnicheskaya fizika, 1961, vol. 2, no. 4, pp. 148-150 (in Russ.).

[14] Rabotnov Yu.N., Papernik L.Kh., Stepanychev E.I. Application of nonlinear heredity theory to description theory in polymers. Mekhanika polimerov, 1971, no. 1, pp. 74-87 (in Russ.).

[15] Dergunov N.N., Papernik L.Kh., Rabotnov Yu.N. Analysis of behavior of graphite on the basis of nonlinear heredity theory. Journal of Applied Mechanics and Technical Physics, 1971, vol. 12, no. 2, pp. 235-240. DOI: 10.1007/BF00850695 Available at: http://link.springer.com/article/10.1007/BF00850695

[16] Rabotnov Yu.N., Papernik L.Kh., Stepanychev E.I. Nonlinear creep of TS8/3-250 fiberglass. Mekhanika polimerov, 1971, no. 3, pp. 391-397 (in Russ.).

[17] Rabotnov Yu.N., Papernik L.Kh., Stepanychev E.I. On connection between fiberglass creep behavior and momentary deforming curve. Mekhanika polimerov, 1971, no. 4, pp. 624-628 (in Russ.).

[18] Rabotnov Yu.N., Suvorova Yu.V. On metal deformation law under uniaxial loading. Izvestiya AN SSSR. Mekhanika tverdogo tela, 1972, no. 4, pp. 41-54 (in Russ.).

[19] Rabotnov Yu.N. Elementy nasledstvennoy mekhaniki tverdykh tel [Elements of solid mechanics hereditary theory]. Moscow, Nauka Publ., 1977. 384 p.

[20] Mel’shanov A.F., Suvorova Yu.V., Khazanov S.Yu. Experimental verification of determining equation for metals under loading and unloading. Izvestiya AN SSSR. Mekhanika tverdogo tela, 1974, no. 6, pp. 166-170 (in Russ.).

[21] Suvorova Yu.V. Nonlinear effects in case of deformation of hereditary medium. Mekhanika polimerov, 1977, no. 6, pp. 976-980 (in Russ.).

[22] Osokin A.E., Suvorova Yu.V. Nonlinear governing equation of a hereditary medium and methodology of determining its parameters. Journal of Applied Mathematics and Mechanics, 1978, vol. 42, no. 6, pp. 1214-1222. DOI: 10.1016/0021-8928(78)90072-2 Available at: http://www.sciencedirect.com/science/article/pii/0021892878900722

[23] Suvorova Yu.V., Alekseeva S.I. Ninlinear model of isotropic hereditary medium under combined stress. Mekhanika kompozitnykh materialov, 1993, no. 5, pp. 602-607 (in Russ.).

[24] Suvorova Yu.V., Alekseeva S.I. Engineering application of hereditary model to description of the polymer and polymer matrix composite behavior. Zavodskaya laboratoriya. Diagnostika materialov, 2000, vol. 66, no. 5, pp. 47-51 (in Russ.).

[25] Alekseeva S.I. Nonlinear hereditary medium taking into account temperature and humidity. Doklady akademii nauk, 2001, vol. 376, no. 4, pp. 471-473 (in Russ.).

[26] Suvorova Yu.V. Yu.N. Rabotnov’s nonlinear hereditary-type equation and its applications. Izvestiya AN SSSR. Mekhanika tverdogo tela, 2004, no. 1, pp. 174-181 (in Russ.).

[27] Viktorova I., Dandurand B., Alekseeva S., Fronya M. Creep simulation of polymer nanocomposites based on alternate method of nonlinear optimization. Mekhanika kompozitnykh materialov, 2012, vol. 48, no. 6, pp. 997-1010 (in Russ.).

[28] Fung Y.C. Stress-strain-history relations of soft tissues in simple elongation, biomechanics: Its foundations and objectives. New Jersey, Prentice-Hall, 1972. Р. 181-208.

[29] Fung Y.C. Mathematical models of strain-deformation dependence for soft living tissues. Mekhanika polimerov, 1975, no. 5, pp. 850-867 (in Russ.).

[30] Woo S.L.-Y. Mechanical properties of tendons and ligaments - I. Quasi-static and nonlinear viscoelastic properties. Biorheology, 1982, vol. 19, pp. 385-396.

[31] Sauren A.A., Rousseau E.P. A concise sensitivity analysis of the quasi-linear viscoelastic model proposed by Fung. J. Biomech. Eng., 1983, vol. 105, no. 1, pp. 92-95. DOI: 10.1115/1.3138391 Available at: http://biomechanical.asmedigitalcollection.asme.org/article.aspx?articleid=1396152&resultClick=3

[32] Fung Y.C. Biomechanics. Mechanical properties of living tissues. New York, Springer-Verlag, 1993. 568 p.

[33] Funk J.R., Hall G.W., Crandall J.R., Pilkey W.D. Linear and quasi-linear viscoelastic characterization of ankle ligaments. J. Biomech. Eng., 2000, vol. 122, no. 1, pp. 15-22. DOI: 10.1115/1.429623 Available at: http://biomechanical.asmedigitalcollection.asme.org/article.aspx?articleid=1399078&resultClick=3

[34] Sarver J.J., Robinson P.S., Elliott D.M. Methods for quasi-linear viscoelastic modeling of soft tissue: Application to incremental stress-relaxation experiments. J. Biomech. Eng., 2003, vol. 125, no. 5, pp. 754-758. DOI: 10.1115/1.1615247 Available at: http://biomechanical.asmedigitalcollection.asme.org/article.aspx?articleid=1410862&resultClick=3

[35] Abramowitch S.D., Woo S.L.-Y. An improved method to analyze the stress relaxation of ligaments following a finite ramp time based on the quasi-linear viscoelastic theory. J. Biomech. Eng., 2004, vol. 126, no. 1, pp. 92-97. DOI: 10.1115/1.1645528 Available at: http://biomechanical.asmedigitalcollection.asme.org/article.aspx?articleid=1411316&resultClick=3

[36] Nekouzadeh A., Pryse K.M., Elson E.L., Genin G.M. A simplified approach to quasi-linear viscoelastic modeling. J. of Biomechanics, 2007, vol. 40, no. 14, pp. 3070-3078. DOI: 10.1016/j.jbiomech.2007.03.019 Available at: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2085233

[37] De Frate L.E., Li G. The prediction of stress-relaxation of ligaments and tendons using the quasi-linear viscoelastic model. Biomechanics and Modeling in Mechanobiology, 2007, vol. 6, no. 4, pp. 245-251. DOI: 10.1007/s10237-006-0056-8 Available at: http://link.springer.com/article/10.1007/s10237-006-0056-8

[38] Duenwald S.E., Vanderby R., Lakes R.S. Constitutive equations for ligament and other soft tissue: evaluation by experiment. Acta Mechanica, 2009, vol. 205, no. 1, pp. 23-33. DOI: 10.1007/s00707-009-0161-8 Available at: http://link.springer.com/article/10.1007/s00707-009-0161-8

[39] Lakes R.S. Viscoelastic materials. Cambridge, Cambridge Univ. Press, 2009. 461 p.

[40] Duenwald S.E., Vanderby R., Lakes R.S. Stress relaxation and recovery in tendon and ligament: Experiment and modeling. Biorheology, 2010, vol. 47, pp. 1-14. DOI: 10.3233/BIR-2010-0559 Available at: http://content.iospress.com/articles/biorheology/bir559

[41] De Pascalis R., Abrahams I.D., Parnell W.J. On nonlinear viscoelastic deformations: a reappraisal of Fung’s quasi-linear viscoelastic model. Proc. R. Soc. A., 2014, vol. 470. DOI: 10.1098/rspa.2014.0058 Available at: http://rspa.royalsocietypublishing.org/content/470/2166/20140058

[42] Babaei B., Abramowitch S.D., Elson E.L., Thomopoulos S., Genin G.M. A discrete spectral analysis for determining quasi-linear viscoelastic properties of biological materials. J. Royal. Soc. Interface, 2015, vol. 12, no. 113, pp. 20150707. DOI: 10.1098/rsif.2015.0707 Available at: http://rsif.royalsocietypublishing.org/content/12/113/20150707

[43] Khokhlov A.V. Creep and relaxation curves produced by the Rabotnov nonlinear constitutive relation for viscoelastoplastic materials. Problemy prochnosti i plastichnosti [Problems of Strength and Plasticity], 2016, vol. 78, no. 4, pp. 452-466 (in Russ.).

[44] Khokhlov A.V. Long-term strength curves generated by the nonlinear Maxwell-type model for viscoelastoplastic materials and the linear damage rule under step loading. Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. & Math. Sci.], 2016, vol. 20, no. 3, pp. 524-543 (in Russ.). DOI: 10.14498/vsgtu1512

[45] Khokhlov A.V. Constitutive relation for rheological processes: Properties of theoretic creep curves and simulation of memory decay. Mechanics of Solids, 2007, vol. 42, no. 2, pp. 291-306. DOI: 10.3103/S0025654407020148 Available at: http://link.springer.com/article/10.3103%2FS0025654407020148

[46] Khokhlov A.V. Constitutive relation for rheological processes with known loading history. Creep and long-term strength curves. Mechanics of Solids, 2008, vol. 43, no. 2, pp. 283-299. DOI: 10.3103/S0025654408020155 Available at: http://link.springer.com/article/10.3103%2FS0025654408020155

[47] Khokhlov A.V. Properties of creep curves at piecewise-constant stress generated by the linear viscoelasticity theory. Problemy prochnosti i plastichnosti [Problems of Strength and Plasticity], 2015, vol. 77, no. 4, pp. 344-359 (in Russ.). Available at: http://www.unn.ru/e-library/ppp.html?anum=322

[48] Khokhlov A.V. General properties of stress-strain curves at constant strain rate yielding from linear theory of viscoelasticity. Problemy prochnosti i plastichnosti [Problems of Strength and Plasticity], 2015, vol. 77, no. 1, pp. 60-74 (in Russ.). Available at: http://www.unn.ru/e-library/ppp.html?anum=296

[49] Khokhlov A.V. Creep recovery curves yielding from linear viscoelasticity theory and the necessary restrictions on a creep function. Problemy prochnosti i plastichnosti [Problems of Strength and Plasticity], 2013, vol. 75, no. 4, pp. 257-267 (in Russ.). Available at: http://www.unn.ru/e-library/ppp.html?anum=245

[50] Khokhlov A.V. The qualitative analysis of theoretic curves generated by linear viscoelasticity constitutive equation. Nauka i obrazovanie: nauchnoe izdanie MGTU im. N.E. Baumana [Science and Education: Scientific Publication of BMSTU], 2016, no. 5, pp. 187-245 (in Russ.). DOI: 10.7463/0516.0840650 Available at: http://technomag.bmstu.ru/en/doc/840650.html

[51] Shesterikov S.A., Yumasheva M.A. Specification of equation of state in creep theory. Izves-tiya AN SSSR. Mekhanika tverdogo tela, 1984, no. 1, pp. 86-91 (in Russ.).

[52] Dandrea J., Lakes R.S. Creep and creep recovery of cast aluminum alloys. Mechanics of Time-Dependent Materials, 2009, vol. 13, no. 4, pp. 303-315. DOI: 10.1007/s11043-009-9089-6 Available at: http://link.springer.com/article/10.1007%2Fs11043-009-9089-6

[53] Khan F., Yeakle C. Experimental investigation and modeling of non-monotonic creep behavior in polymers. Int. J. Plasticity, 2011, vol. 27, no. 4, pp. 512-521.

[54] Drozdov A.D., Dusunceli N. Unusual mechanical response of carbon black-filled thermoplastic elastomers. Mechanics of Materials, 2014, vol. 69, no. 1, pp. 116-131.

[55] Khokhlov A.V. Properties of the nonlinear Maxwell-type model with two material functions for viscoelastoplastic materials. Moscow University Mechanics Bulletin. Ser.1. Matema-tica, mekhanica, 2016, no 6, pp. 36-41 (in Russ.).

[56] Khokhlov A.V. The nonlinear Maxwell-type viscoelastoplastic model: Properties of creep curves at piecewise-constant stress and criterion for plastic strain accumulation. Mashinostroenie i inzhenernoe obrazovanie, 2016, no. 3, pp. 35-48 (in Russ.).