Nonlinear Maxwell-Type Elastoviscoplastic Model: General Properties of Stress Relaxation Curves and Restrictions on the Material Functions
Authors: Khokhlov A.V. | Published: 22.11.2017 |
Published in issue: #6(75)/2017 | |
DOI: 10.18698/1812-3368-2017-6-31-55 | |
Category: Mechanics | Chapter: Mechanics of Deformable Solid Body | |
Keywords: viscoelasticity, viscoplasticity, restrictions on the material functions, tension compression asymmetry, stress relaxation curves, equilibrium stress value, rate sensitivity, superplasticity, polymers |
The study analytically examines a nonlinear Maxwell-type constitutive relation with two arbitrary material functions in order to find out qualitative properties of the basic quasistatic curves generated by the model and to reveal its capabilities and applicability scope. The constitutive relation is targeted at adequate modeling of the main rheological phenomena set which is typical for non-ageing rheonomic materials exhibiting non-linear hereditary properties, strong positive strain rate sensitivity, secondary creep, yielding at constant stress and tension compression asymmetry. It is applicable for simulation of mechanical behavior of various polymers, their solutions and melts, solid propellants, sandasphalt concretes, composite materials, ices, titanium and aluminum alloys, ceramics at high temperature, etc. General qualitative properties of the stress relaxation curves generated by the model in uniaxial isothermal case are studied analytically under minimal primary restrictions on both material functions. We examined the relaxation rate evolution, conditions for monotonicity and convexity of relaxation curves, their asymptotics and stress limit value at infinity, their dependences on the material functions, a given strain level and initial loading stage characteristics. As a result, we reveаled the additional necessary restrictions which should be imposed on the material functions to provide an adequate description of basic rheological phenomena related to stress relaxation and typical test curves properties of a wide class of elastoviscoplastic materials. Finally, we discovered two different cases in the model behavior depending on qualitative properties of the material functions, namely, in the first case the equilibrium (limit) stress value is nonzero and the model simulates solid behavior, and in the second case the equilibrium value of stress is zero and the model simulates liquid behavior
References
[1] Khokhlov A.V. Properties of a nonlinear viscoelastoplastic model of Maxwell type with two material functions. Moscow University Mechanics Bulletin, 2016, vol. 71, iss. 6, pp. 132–136. DOI: 10.3103/S0027133016060029
[2] Khokhlov A.V. Properties of stressstrain curves generated by the nonlinear Maxwelltype viscoelastoplastic model at constant stress rates. Mashinostroenie i inzhenernoe obrazovanie, 2017, no. 1, pp. 57–71 (in Russ.).
[3] Khokhlov A.V. The nonlinear Maxwelltype model for viscoelastoplastic materials: Simulation of temperature influence on creep, relaxation and strainstress curves. Vestnik SamGTU. Ser. Fiz.Mat. Nauki, 2017, vol. 21, no. 1, pp. 160–179 (in Russ.). DOI: 10.14498/vsgtu1524
[4] Khokhlov A.V. Nonlinear Maxwelltype viscoelastoplastic model: Rate of plastic strain accumulation under cyclic loadings. Deformatsiya i razrushenie materialov, 2017, no. 7, pp. 7–19 (in Russ.).
[5] Khokhlov A.V. The nonlinear Maxwelltype viscoelastoplastic model identification techniques using creep curves under ramp loadings. P. 1. Mathematical base. Deformatsiya i razrushenie materialov, 2017, no. 9, pp. 2–9 (in Russ.).
[6] Nikitenko A.F., Sosnin O.V., Torshenov N.G., Shokalo I.K. Creep of hardening materials with different properties in tension and compression. Journal of Applied Mechanics and Technical Physics, 1971, vol. 12, no. 2, pp. 277–281. DOI: 10.1007/BF00850702
[7] Lokoshchenko A.M. Polzuchest i dlitelnaya prochnost metallov [Materials creep and longterm strength]. Moscow, Fizmatlit Publ., 2016. 504 p.
[8] Khokhlov A.V. Constitutive relation for rheological processes with known loading history. Creep and longterm strength curves. Mechanics of Solids, 2008, vol. 43, iss. 2, pp. 283–299. DOI: 10.3103/S0025654408020155
[9] Khokhlov A.V. Analysis of creep curves general properties under step loading generated by the Rabotnov nonlinear relation for viscoelastic plastic materials. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 2017, no. 3, pp. 93–123 (in Russ.). DOI: 10.18698/181233682017393123
[10] Khokhlov A.V. The qualitative analysis of theoretic curves generated by linear viscoelasticity constitutive equation. Nauka i obrazovanie: nauchnoe izdanie [Science and Education: Scientific Publication], 2016, no. 5, pp. 187–245 (in Russ.). DOI: 10.7463/0516.0840650 Available at: http://technomag.bmstu.ru/doc/840650.html
[11] Gorodtsov V.A., Leonov A.I. On kinematics, nonequilibrium thermodynamics and rheologic relation in viscoelasticity nonlinear theory. PMM, 1968, vol. 32, no. 1, pp. 70–94 (in Russ.).
[12] Leonov A.I., Lipkina E.Ch., Paskhin E.D., Prokunin A.N. Theoretical and experimental investigations of shearing in elastic polymer liquids. Rheol. Acta, 1976, vol. 15, no. 7/8, pp. 411–426. DOI: 10.1007/BF01574496
[13] Palmov V.A. Rheologic models in nonlinear mechanics of deformable bodies. Uspekhi mekhaniki, 1980, vol. 3, no. 3, pp. 75–115 (in Russ.).
[14] Prokunin A.N. On the nonlinear Maxwelltype defining equations for describing the motions of polymer liquids. Journal of Applied Mathematics and Mechanics, 1984, vol. 48, iss. 6, pp. 699–706. DOI: 10.1016/00218928(84)900376
[15] Leonov A.I. Analysis of simple constitutive equations for viscoelastic liquids. Journal of NonNewtonian Fluid Mechanics, 1992, vol. 42, iss. 3, pp. 323–350. DOI: 10.1016/03770257(92)870176
[16] Leonov A.I., Prokunin A.N. Nonlinear phenomena in flows of viscoelastic polymer fluids. London, Chapman and Hall, 1994. 475 p.
[17] Leonov A.I. Constitutive equations for viscoelastic liquids: Formulation, analysis and comparison with data. Rheology Series, 1999, vol. 8, pp. 519–575. DOI: 10.1016/S01693107(99)800409
[18] Kremple E., Ho K. Inelastic compressible and incompressible, isotropic, small strain viscoplasticity theory based on overstress (VBO). In: Handbook of Materials Behavior Models. Vol. 1. New York, Academic Press, 2001, pp. 336–348.
[19] Rabotnov Yu.N. Polzuchest elementov konstruktsiy [Construction elements creep]. Moscow, Nauka Publ., 1966. 752 p.
[20] Malinin N.N. Raschety na polzuchest elementov mashinostroitelnykh konstruktsiy [Creep design of mechanicalengineering construction parts]. Moscow, Mashinostroenie Publ., 1981. 221 p.
[21] Betten J. Creep mechanics. Berlin, Heidelberg, SpringerVerlag, 2008. 367 p.
[22] Takagi H., Dao M., Fujiwara M. Prediction of the constitutive equation for uniaxial creep of a powerlaw material through instrumented microindentation testing and modeling. Materials Transactions, 2014, vol. 55, no. 2, pp. 275–284. DOI: 10.2320/matertrans.M2013370 Available at: https://www.jstage.jst.go.jp/article/matertrans/55/2/55_M2013370/_article
[23] Astarita G., Marrucci G. Principles of nonNewtonian fluid mechanics. McGrawHill, 1974. 304 p.
[24] Vasin R.A., Enikeev F.U. Vvedenie v mekhaniku sverkhplastichnosti [Introduction to superplasticity mechanics]. Ufa, Gilem Publ., 1998. 280 p.
[25] Nieh T.G., Wadsworth J., Sherby O.D. Superplasticity in metals and ceramics. Cambridge University Press, 1997. 287 p.
[26] Segal V.M., Beyerlein I.J., Tome C.N., Chuvildeev V.N., Kopylov V.I. Fundamentals and engineering of severe plastic deformation. New York, Nova Science Publ., 2010. 542 p.
[27] Naumenko K., Altenbach H., Gorash Y. Creep analysis with a stress range dependent constitutive model. Arch. Appl. Mech., 2009, vol. 79, no. 67, pp. 619–630. DOI: 10.1007/s0041900802875
[28] Cao Y. Determination of the creep exponent of a powerlaw creep solid using indentation tests. Mech. TimeDepend. Mater., 2007, vol. 11, no. 2, pp. 159–172. DOI: 10.1007/s1104300790336
[29] Radchenko V.P., Shapievskiy D.V. Analysis of a nonlinear generalized Maxwell model. Vestnik SamGTU. Ser. Fiz.Mat. Nauki, 2005, no. 38, pp. 55–64 (in Russ.). DOI: 10.14498/vsgtu372
[30] Lu L.Y., Lin G.L., Shih M.H. An experimental study on a generalized Maxwell model for nonlinear viscoelastic dampers used in seismic isolation. Engineering Structures, 2012, vol. 34, pp. 111–123. DOI: 10.1016/j.engstruct.2011.09.012
[31] Vinogradov G.V., Malkin A.Ya. Reologiya polimerov [Polymer rheology]. Moscow, Khimiya Publ., 1977. 440 p.
[32] Rohn C.L. Analytical polymer rheology. Munich, Hanser Publishers, 1995. 314 p.
[33] Brinson H.F., Brinson L.C. Polymer engineering science and viscoelasticity. Springer Science & Business Media, 2008. 446 p.
[34] Christensen R.M. Mechanics of composite materials. New York, Dover Publications, 2012. 384 p.
[35] Bergstrom J.S. Mechanics of solid polymers. Theory and computational modeling. William Andrew, 2015. 520 p.
[36] Truesdell C. A first course in rational continuum mechanics. Academic Press, 1972. 304 p.