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IssuesArchive of Issues2018-3pp.307-328

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A.V. Khokhlov, "Two-Sided Estimates for the Relaxation Function of the Linear Theory of Heredity via the Relaxation Curves during the Ramp-Deformation and the Methodology of Identification," Mech. Solids. 53 (3), 307-328 (2018)
Year 2018 Volume 53 Number 3 Pages 307-328
DOI 10.3103/S0025654418070105
Title Two-Sided Estimates for the Relaxation Function of the Linear Theory of Heredity via the Relaxation Curves during the Ramp-Deformation and the Methodology of Identification
Author(s) A.V. Khokhlov (Institute of Mechanics, Lomonosov Moscow State University, Michurinskii pr. 1, Moscow, 119192, Russia, andrey-khokhlov@ya.ru)
Abstract The general properties of the relaxation curves with the initial stage of deformation and their dependence on the duration of the initial stage of deformation and the properties of the relaxation function are analytically studied. These curves are induced by the constitutive equation of viscoelasticity with an arbitrary relaxation. New accurate two-sided estimates for the relaxation curves and their absolute and relative deviations from the relaxation curves for instantaneous deformation are derived. The uniform convergence of the set of relaxation curves when the duration of the initial stage tends to zero is proved.

Effective universal two-sided estimates for the relaxation function (at any time instant) are obtained via the relaxation curves of the material during ramp-deformation. On their basis, simple and effective formulas for determining the relaxation function from the relaxation curves for ramp deformation obtained in the material tests have been proposed. The error estimations of these approximations are given. The uniform convergence of the set of approximations to the relaxation function when the duration of the initial stage tends to zero is proved. The higher accuracy of the estimates found and the proposed approximation in comparison with the known related approaches to the determination of the RF (relaxation function) is established.

The results of the analysis are useful for clarifying the many of the possibilities of the linear theory, the domain and indicators of its (non)applicability (and also of a number of its generalizing nonlinear constitutive equations of viscoelasticity, for example, proposed by Rabotnov, Ilyushin, Pobedrya, etc.) to improve the methods of selection, identification and adjustment of linear models. In particular, they are useful for obtaining reliable estimates of the lower bound for the observation window of the relaxation function from the experimental relaxation curve of the material in terms of the initial stage duration to clarify the "ten-times rule" and expand the observation window to the region of small time values.
Keywords viscoelasticity, relaxation curves, influence of the initial stage, estimates for the relaxation function, identification, convergence of the set of curves, memory attenuation
References
1.  M. A. Koltunov, "On Allocation of the Main Part of Hereditary Functions of Influence for the Description of Relaxation Processes in the Initial Period," Mekh. Polim. 4, 625-635 (1967).
2.  M. A. Koltunov, "Determination of the Characteristics for Elastoviscous Media from Quasi-static Experiments," Mekh. Polim. 5, 803-811 (1967).
3.  L. J. Zapas and J. C. Phillips, "Simple Shearing Flows in Polyisobutylene Solutions," J. Res. Nat. Bur. Stds. 75A (1), 33-41 (1971).
4.  W. N. Findley, J. S. Lai, and K. Onaran, Creep and Relaxation of Nonlinear Viscoelastic Materials (North Holland, Amsterdam, 1976).
5.  M. A. Koltunov, Creep and Relaxation (Vysshaya shkola, Moscow, 1976) [in Russian].
6.  A. Ya. Malkin, A. A. Askadsky, and V. V. Kovriga, Methods for Measuring the Mechanical Properties of Polymers (Khimiya, Moscow, 1978) [in Russian].
7.  J. Meissner, "Combined Constant Strain Rate and Stress Relaxation Test for Linear Viscoelastic Studies," J. Polym. Sci. Polym. Phys. Ed. 16, 915-919 (1978).
8.  T. L. Smith, "Evaluation of the Stress Relaxation Modulus from the Response to a Constant Rate of Strain Followed by a Constant Strain," J. Polym. Sci. Polym. Phys. Ed. 17, 2181-2188 (1979).
9.  M. A. Koltunov, V. P. Mayboroda, and V. G. Zubchaninov, Strength Calculations of Products Made of Polymer Materials (Mashinostroyeniye, Moscow, 1983) [in Russian].
10.  Yu. S. Urzhumtsev, and V. P. Mayboroda, Technical Means and Methods for Determining the Strength Characteristics of Structures Made of Polymers (Mashinostroyeniye, Moscow, 1984) [in Russian].
11.  N. W. Tschoegl, The Phenomenological Theory of Linear Viscoelastic Behavior (Springer, Heidelberg, 1989).
12.  R. W. Kolkka, D. S. Malkus, and T. R. Rose, "Finite Rise Time Step Strain Modeling of Nearly Monodisperse Polymer Melts and Solutions," Rheol. Acta. 30 (5), 430-446 (1991).
13.  A. D. Drozdov, Mechanics of Viscoelastic Solids (Wiley & Sons, New York, 1998).
14.  S. Lee and W. G. Knauss, "A Note on the Determination of Relaxation and Creep Data from Ramp Tests," Mech. Time-Dep. Mater. 4 (1), 1-7 (2000).
15.  A. A.  Adamov, V. P.  Matveenko, N. A.  Trufanov, and I. N. Shardakov, Methods of Applied Viscoelasticity (UrO RAN, Ekaterinburg, 2003) [in Russian].
16.  A. Flory and G. B. McKenna, "Finite Step Rate Corrections in Stress Relaxation Experiments: a Comparison of Two Methods," Mech. Time-Dep. Mater. 8 (1), 17-37 (2004).
17.  S. D. Abramowitch and S. L.-Y. Woo, "An Improved Method to Analyze the Stress Relaxation of Ligaments Following a Finite Ramp Time Based on the Quasi-linear Viscoelastic Theory," J. Biomech. Engng 126, 92-97 (2004).
18.  J. A.`Jimbel, J. J. Sarver, and L. J. Soslowski, "The Effect of Overshooting the Target Strain on Estimating Viscoelastic Properties from Stress Relaxation Experiments," J. Biomech. Engng 126 (6), 844-848 (2004).
19.  M. L. Oyen, "Spherical Indentation Creep Following Ramp Loading," J. Mater. Res. 20 (8), 2094-2100 (2005).
20.  F. Khan, "Loading History Effects on the Creep and Relaxation Behavior of Thermoplastics," Trans. ASME J. Engng Mater. Technol. 128, 564-571 (2006).
21.  J. Sorvari and M. Malinen, "Determination of the Relaxation Modulus of a Linearly Viscoelastic Material," Mech. Time-Dep. Mater. 10 (2), 125-133 (2006).
22.  J. Sorvari, M. Malinen, and J. Hämäläinen, "Finite Ramp Time Correction Method for Non-linear Viscoelastic Material Model," Int. J. Non-Lin. Mech. 41, 1050-1056 (2006).
23.  J. Sorvari and M. Malinen, "On the Direct Estimation of Creep and Relaxation Functions," Mech. Time-Dep. Mater. 11 (2), 143-157 (2007).
24.  A. V. Khokhlov, "Constitutive Relation for Rheological Processes: Properties of Theoretic Creep Curves and Simulation of Memory Decay," Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, No. 2, 147-146 (2007) [Mech. Solids (Engl. Transl.) 42 (2), 291-306 (2007)].
25.  W. G. Knauss and J. Zhao J., "Improved Relaxation Time Coverage in Ramp-strain Histories," Mech. Time-Dep. Mater. 11 (3), 199-216 (2007).
26.  W. G. Knauss, I. Emri, and H. Lu, "Mechanics of Polymers: Viscoelasticity," in Springer Handbook of Experimental Solid Mechanics, ed. by W. N. Sharpe (Springer, New York, 2008), pp. 49-96.
27.  D. Craiem, F. J. Rojo, J. M. Atienza, et al. "Fractional-order Viscoelasticity Applied to Describe Uniaxial Stress Relaxation of Human Arteries," Phys. Med. Biol. 53, 4543-4554 (2008).
28.  A. V. Khokhlov, "Constitutive Relation for Rheological Processes with Known Loading History. Creep and LongTerm Strength Curves," Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, No. 2, 140-160 (2008) [Mech. Solids (Engl. Transl.) 43 (2), 283-299 (2008)].
29.  A. V. Khokhlov, "Fracture Criteria Under Creep with Strain History Taken into Account, and Long-Term Strength Modelling," Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, No. 4, 121-135 (2009) [Mech. Solids (Engl. Transl.) 44 (4), 596-607 (2009)].
30.  A. McClung and M. Ruggles-Wrenn, "Strain Rate Dependence and Short-term Relaxation Behavior of a Thermoset Polymer at Elevated Temperature: Experiment and Modeling," J. Press. Vessel Technol. 131 (3), 31405-31413 (2009).
31.  S. E. Duenwald, R. Vanderby, and R. S. Lakes, "Viscoelastic Relaxation and Recovery of Tendon," Annals Biomed. Engng 37 (6) 1131-1140 (2009).
32.  S. E. Duenwald, R. Vanderby, and R. S. Lakes, "Constitutive Equations for Ligament and Other Soft Tissue: Evaluation by Experiment," Acta Mech. 205 23-33 (2009).
33.  R. S. Lakes, Viscoelastic Materials (Cambridge Univ. Press, Cambridge, 2009).
34.  S. E. Duenwald, R. Vanderby, and R. S. Lakes, "Stress Relaxation and Recovery in Tendon and Ligament: Experiment and Modeling," Biorheology 47, 1-14 (2010).
35.  D. Tscharnuter, M. Jerabek, Z. Major, and R .W. Lang, "On the Determination of the Relaxation Modulus of PP Compounds from Arbitrary Strain Histories," Mech. Time-Dep. Mater. 15 (1), 1-14 (2011).
36.  A. Stankiewiez, "Identification of Relaxation Modulus of Viscoelastic Materials from Non-ideal Ramp-test Histories - Problem and Method," TEKA Commission of Motorization and Energetics in Agriculture 13 (1), 169-176 (2013).
37.  F. Stan, and C. Fetecau C., "Study of Stress Relaxation in Polytetrafluoroethylene Composites by Cylindrical Macroindentation," Composites, Part B: Engineering 47, 298-307 (2013).
38.  R. M. Guedes and J. L. Morais, "A Simple and Effective Scheme for Data Reduction of Stress Relaxation Incorporating Physical-aging effects: An Analytical and Numerical Analysis," Polymer Testing 32 (5), 961-971 (2013).
39.  M. Di Paola, V. Fiore, F. Pinnola, and A. Valenza, "On the Influence of the Initial Ramp for a Correct Definition of the Parameters of Fractional Viscoelastic Materials," Mech. Mater. 69 (1), 63-70 (2014).
40.  V. A. Fernandes and D. S. De Focatiis, "The Role of Deformation History on Stress Relaxation and Stress Memory of Filled Rubber," Polymer Testing 40, 124-132 (2014).
41.  D. Mathiesen, D. Vogtmann, and R. Dupaix, "Characterization and Constitutive Modeling of Stress-relaxation Behavior of Polymethyl Methacrylate (PMMA) Across the Glass Transition Temperature," Mech. Mater. 71, 74-84 (2014).
42.  J. Sweeneya, M. Bonnerb, and I. Ward, "Modelling of Loading, Stress Relaxation and Stress Recovery in a Shape Memory Polymer," J. Mech. Behav. Biomed. Mater. 37, 12-23 (2014).
43.  H. Zhang, K. Lamnawar, A. Maazouz, and J. M. Maia, "Experimental Considerations on the Step Shear Strain in Polymer Melts: Sources of Error and Windows of Confidence," Rheologica Acta 54 (2), 121-138 (2015).
44.  Scientific Report 5302 "Properties of the Relaxation Curves with the Initial Stage of Deformation under a Constant Rate Arising by Linear Integral Ratio of Viscoelasticity, and Methods of its Identification" (Institute of Mechanics, Lomonosov Moscow State University, 2016) [in Russian].
45.  A. V. Khokhlov, "General Properties of Relaxation Curves in the Case of the Initial Stage of Strain with a Constant Rate in the Linear Heredity Theory," Vestnik Moskov. Univ. Ser. I Mat. Mekh., 72 (3), 44-47 (2017) [Moscow Univ. Mech. Bull. (Engl. Transl.) 72 (3), 55-58 (2017)].
46.  A.  V. Khokhlov, "Identification of a Nonlinear Viscoelastoplastic Model of Maxwell Type by the Creep Curves with Initial Loading Stage. Part 2," Deform. Razrush. Mater., No 10, 2-9 (2017).
47.  A. A. Il'yushin, and B. E. Pobedrya, Foundations of Mathematical Theory of Thermoviscoelasticity (Nauka, Moscow, 1970) [in Russian].
48.  Yu. N.  Rabotnov, Elements of Hereditary Solid Mechanics (Nauka, Moscow, 1977) [in Russian].
49.  G. V. Vinogradov and A. Ya. Malkin, Rheology of Polymers (Khimiya, Moscow, 1977) [in Russian].
50.  D. V. Georgievskii, D. M. Klimov, and B. E. Pobedrya, "Specific Features of the Behavior of Viscoelastic Models," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 1, 119-157 (2004) [Mech. Solids (Engl. Transl.) 39 (1), 88-120 (2004)].
51.  J. S. Bergstrom, Mechanics of Solid Polymers. Theory and Computational Modeling (Elsevier, William Andrew, 2015).
52.  M. Baumgaertel and H. H. Winter, "Determination of Discrete Relaxation and Retardation Time Spectra from Dynamic Mechanical Data," Rheologica Acta 28 (6), 511-519 (1989).
53.  J. Honerkamp and J. Weese, "A Nonlinear Regularization Method for the Calculation of Relaxation Spectra," Rheologica Acta 32 (1), 65-73 (1993).
54.  D. W. Mead, "Numerical Interconversion of Linear Viscoelastic Material Functions," J. Rheology 38 (6), 1769-1795 (1994).
55.  J. Janno and L. Von Wolfersdorf, "Inverse Problems for Identification of Memory Kernels in Viscoelasticity," Math. Methods Appl. Sci. 20, 291-314 (1997).
56.  S. Gerlach and A. Matzenmiller, "Comparison of Numerical Methods for Identification of Viscoelastic Line Spectra from Static Test Data," Int. J. Numer. Meth. Engng 63, 428-454 (2005).
57.  F. J. Stadler and C. Bailly , "A New Method for the Calculation of Continuous Relaxation Spectra from Dynamic-Mechanical Data," Rheologica Acta 48 (1), 33-49 (2009).
58.  J. Luk-Cyr, T. Crochon, C. Li, and M. Levesque, "Interconversion of Linearly Viscoelastic Material Functions Expressed as Prony Series: A Closure," Mech. Time-Dep. Mater. 17 (1), 53-82 (2012).
59.  D. Jalocha, A. Constantinescu, and R. Neviere R. "Revisiting the Identification of Generalized Maxwell Models from Experimental Results," Int. J. Solids Struct. 67-68, 169-181 (2015).
60.  A. V.  Khokhlov, "Analysis of the General Properties of Creep Curves for Stepwise Loading, Generated by the Nonlinear Rabotnov Relation for Viscoelastic Plastic Materials," Vestnik MGTU. Estestv. Nauki, No. 3, 93-123 (2017).
61.  A. V. Khokhlov, "Analysis of the General Properties of Creep Curves under Cyclic Stepwise Loading, Generated by the Linear Theory of Heredity," Vestnik Samarsk. Gos. Techn. Univ. Ser. Fiz.-Math. Nauki 21 (2), 326-361 (2017).
62.  A. V. Khokhlov, "General Properties of Deformation Diagrams of Linear Models of Viscoelasticity at a Constant Strain Rate," Probl. Proch. Plast. 77 (1), 60-74 (2015).
63.  A. V.  Khokhlov, "Properties of Creep Curve Sets with Stepwise Loading of the Linear Constitutive Equation of Viscoelasticity," Probl. Proch. Plast. 77 (4), 344-359 (2015).
64.  A. V.  Khokhlov, "Analysis of properties of creep curves generated by the linear viscoelasticity theory under arbitrary loading programs at initial stage," Vestnik Samarsk. Gos. Techn. Univ. Ser. Fiz.-Math. Nauki 22 (1), 65-95 (2018).
65.  A. V.  Khokhlov, "The Qualitative Analysis of Theoretic Curves Generated by Linear Viscoelasticity Constitutive Equation," Nauka Obrazov. MGTU, No. 5, 187-245 (2016).
66.  L. J. Zapas and T. Craft, "Correlation of Large Longitudinal Deformations with Different Strain Histories," J. Res. Nat. Bur. Stds., 69A (6), 541-546 (1965).
Received 29 May 2016
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