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Erschienen in: Mechanics of Composite Materials 4/2018

22.09.2018

Analysis of Properties of Ramp Stress Relaxation Curves Produced by the Rabotnov Nonlinear Hereditary Theory

verfasst von: А. V. Khokhlov

Erschienen in: Mechanics of Composite Materials | Ausgabe 4/2018

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Abstract

The possibilities and applicability limits of the Rabotnov physically nonlinear constitutive equation containing two arbitrary material functions for nonaging materials are investigated. Analytically examined are the general properties of the resulting stress relaxation curves with an initial stage of constant strain rate and their dependence on duration of the stage, strain rate, and characteristics of the two material functions. Investigated are the monotonicity and convexity intervals and asymptotics of the relaxation curves, the jump of stress derivative at the end of the initial stage, the character of convergence of the family of relaxation curves as duration of the initial stage tends to zero, and other properties. Found are estimates for the deviation of the relaxation curves with an initial stage from those obtained in an instantaneous loading, and it is proved that this deviation tends to zero if the time tends to infinity or duration of the initial stage approaches zero. Revealed are the necessary restrictions on both material functions that allow an adequate modeling of the typical properties of experimental stress relaxation curves. Indicated are the effects which principally cannot be described whatever the material functions used. The possibilities of the Rabotnov nonlinear constitutive equation are compared with those of the Boltzmann–Volterra linear viscoelasticity theory (which were generalized introducing a second material function), and the additional effects that can be described by the nonlinear theory owing to presence of the second material function in it.

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Literatur
1.
Zurück zum Zitat Yu. N. Rabotnov, “Equilibrium of elastic medium with an afteraction,” Prikl. Matem. Mekh., 12, Iss. 1, 53-62 (1948). Yu. N. Rabotnov, “Equilibrium of elastic medium with an afteraction,” Prikl. Matem. Mekh., 12, Iss. 1, 53-62 (1948).
2.
Zurück zum Zitat Yu. N. Rabotnov, “Some questions of the theory of creep,” Vest. MGU, No. 10, 81-91 (1948). Yu. N. Rabotnov, “Some questions of the theory of creep,” Vest. MGU, No. 10, 81-91 (1948).
3.
Zurück zum Zitat V. S. Namestnikov and Yu. N. Rabotnov, “On the hereditary theories of creep,” Zhurn. Priklad. Mekh. Tekhn. Fiz., 2, No. 4, 148-150 (1961). V. S. Namestnikov and Yu. N. Rabotnov, “On the hereditary theories of creep,” Zhurn. Priklad. Mekh. Tekhn. Fiz., 2, No. 4, 148-150 (1961).
4.
Zurück zum Zitat Yu. N. Rabotnov, Creep of Structural Elements [in Russian], M., Nauka (1966). Yu. N. Rabotnov, Creep of Structural Elements [in Russian], M., Nauka (1966).
5.
Zurück zum Zitat Yu. N. Rabotnov, L. Kh. Papernik, and E. I. Stepanichev, “Application of the nonlinear theory of a heredity to the description of time effects in polymeric materials,” Polym. Mekh., 7, No. 1, 63-73 (1971). Yu. N. Rabotnov, L. Kh. Papernik, and E. I. Stepanichev, “Application of the nonlinear theory of a heredity to the description of time effects in polymeric materials,” Polym. Mekh., 7, No. 1, 63-73 (1971).
6.
Zurück zum Zitat Yu. N. Rabotnov, L. Kh. Papernik, and E. I. Stepanichev, “Relation between the creep characteristics of glass-fiber-reinforced plastics and the instantaneous stress-strain curve,” Polym. Mekh., 7, No. 4, 555-558 (1971). Yu. N. Rabotnov, L. Kh. Papernik, and E. I. Stepanichev, “Relation between the creep characteristics of glass-fiber-reinforced plastics and the instantaneous stress-strain curve,” Polym. Mekh., 7, No. 4, 555-558 (1971).
7.
Zurück zum Zitat N. N. Dergunov, L. Kh. Papernik, and Yu. N. Rabotnov, “Analysis of the behavior of graphite on the basis of the nonlinear hereditary theory,” Zhurn. Priklad. Mekh. Tekhn. Fiz., No. 2, 76-82 (1971). N. N. Dergunov, L. Kh. Papernik, and Yu. N. Rabotnov, “Analysis of the behavior of graphite on the basis of the nonlinear hereditary theory,” Zhurn. Priklad. Mekh. Tekhn. Fiz., No. 2, 76-82 (1971).
8.
Zurück zum Zitat Yu. N. Rabotnov and Yu. V. Suvorova, “On the deformation law for metals in uniaxial loading,” News АH the USSR. Mechanics of a firm body, No. 4, 41-54 (1972). Yu. N. Rabotnov and Yu. V. Suvorova, “On the deformation law for metals in uniaxial loading,” News АH the USSR. Mechanics of a firm body, No. 4, 41-54 (1972).
9.
Zurück zum Zitat Yu. N. Rabotnov, L. Kh. Papernik, and E. I. Stepanichev, “Description of creep of composition materials under tension compression,” Polym. Mekh., 9, No. 5, 690-695 (1973). Yu. N. Rabotnov, L. Kh. Papernik, and E. I. Stepanichev, “Description of creep of composition materials under tension compression,” Polym. Mekh., 9, No. 5, 690-695 (1973).
10.
Zurück zum Zitat Yu. N. Rabotnov, Elements of the Hereditary Mechanics of Solids [in Russian], M., Nauka (1977). Yu. N. Rabotnov, Elements of the Hereditary Mechanics of Solids [in Russian], M., Nauka (1977).
11.
Zurück zum Zitat Yu. V. Suvorova and S. I. Alekseeva, “Nonlinear model of an isotropic hereditary medium in complex stress state,” Mech. Compos. Mater., 29, No. 5, 602-607 (1993). Yu. V. Suvorova and S. I. Alekseeva, “Nonlinear model of an isotropic hereditary medium in complex stress state,” Mech. Compos. Mater., 29, No. 5, 602-607 (1993).
12.
Zurück zum Zitat Yu. V. Suvorova, “On the Yu. N. Rabotnov nonlinear hereditary equation and its applications,” Izv. AN SSSR, Mekh. Tverd. Tela, No. 1, 174-181 (2004). Yu. V. Suvorova, “On the Yu. N. Rabotnov nonlinear hereditary equation and its applications,” Izv. AN SSSR, Mekh. Tverd. Tela, No. 1, 174-181 (2004).
13.
Zurück zum Zitat S. I. Alekseeva, M. A. Fronya, and I. V. Viktorova, “Analysis of the viscoelastic properties of polymer composites with carbon fillers,” Kompoz. Nanostrukt., No. 2, 28-39 (2011). S. I. Alekseeva, M. A. Fronya, and I. V. Viktorova, “Analysis of the viscoelastic properties of polymer composites with carbon fillers,” Kompoz. Nanostrukt., No. 2, 28-39 (2011).
14.
Zurück zum Zitat Y. C. Fung, “Stress-strain history relations of soft tissues in simple elongation,” Biomechanics, its Foundations and Objectives, ed. by Fung Y. C. et al., New Jersey: Prentice-Hall, 181-208 (1972). Y. C. Fung, “Stress-strain history relations of soft tissues in simple elongation,” Biomechanics, its Foundations and Objectives, ed. by Fung Y. C. et al., New Jersey: Prentice-Hall, 181-208 (1972).
15.
Zurück zum Zitat Y. C. Fang, “Mathematical stress-strain models for live soft tissue,” Mekh. Polym., 11, No. 5, 850-867 (1975). Y. C. Fang, “Mathematical stress-strain models for live soft tissue,” Mekh. Polym., 11, No. 5, 850-867 (1975).
16.
Zurück zum Zitat Y. C. Fung, Biomechanics. Mechanical Properties of Living Tissues, N.Y., Springer Verlag (1993). Y. C. Fung, Biomechanics. Mechanical Properties of Living Tissues, N.Y., Springer Verlag (1993).
17.
Zurück zum Zitat J. R. Funk, G. W. Hall, J. Crandall, and W. D. Pilkey, “Linear and quasilinear viscoelastic characterization of ankle ligaments,” J. Biomech. Eng., 122, 15-22 (2000).CrossRef J. R. Funk, G. W. Hall, J. Crandall, and W. D. Pilkey, “Linear and quasilinear viscoelastic characterization of ankle ligaments,” J. Biomech. Eng., 122, 15-22 (2000).CrossRef
18.
Zurück zum Zitat J. J. Sarver, P. S. Robinson, and D. M. Elliott, “Methods for quasilinear viscoelastic modeling of soft tissue: application to incremental stressrelaxation experiments,” J. Biomech. Eng., 125, No. 5, 754-758 (2003).CrossRef J. J. Sarver, P. S. Robinson, and D. M. Elliott, “Methods for quasilinear viscoelastic modeling of soft tissue: application to incremental stressrelaxation experiments,” J. Biomech. Eng., 125, No. 5, 754-758 (2003).CrossRef
19.
Zurück zum Zitat 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 quasilinear viscoelastic theory,” J. Biomech. Eng., 126, 92-97 (2004).CrossRef 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 quasilinear viscoelastic theory,” J. Biomech. Eng., 126, 92-97 (2004).CrossRef
20.
Zurück zum Zitat A. Nekouzadeh, K. M. Pryse, E. L. Elson, and G. M. Genin, “A simplified approach to quasilinear viscoelastic modeling,” J. of Biomechanics, 40, No. 14, 3070-3078 (2007).CrossRef A. Nekouzadeh, K. M. Pryse, E. L. Elson, and G. M. Genin, “A simplified approach to quasilinear viscoelastic modeling,” J. of Biomechanics, 40, No. 14, 3070-3078 (2007).CrossRef
21.
Zurück zum Zitat L. E. De Frate and G. Li, “The prediction of stress relaxation of ligaments and tendons using the quasilinear viscoelastic model,” Biomechanics and Modeling in Mechanobiology, 6, No. 4, 245-251 (2007).CrossRef L. E. De Frate and G. Li, “The prediction of stress relaxation of ligaments and tendons using the quasilinear viscoelastic model,” Biomechanics and Modeling in Mechanobiology, 6, No. 4, 245-251 (2007).CrossRef
22.
Zurück zum Zitat S. E. Duenwald, R. Vanderby, and R. S. Lakes, “Constitutive equations for ligament and other soft tissue: evaluation by experiment,” Acta Mechanica, 205, 23-33 (2009).CrossRef S. E. Duenwald, R. Vanderby, and R. S. Lakes, “Constitutive equations for ligament and other soft tissue: evaluation by experiment,” Acta Mechanica, 205, 23-33 (2009).CrossRef
23.
Zurück zum Zitat R. S. Lakes, Viscoelastic Materials, Cambridge: Cambridge Univ. Press (2009).CrossRef R. S. Lakes, Viscoelastic Materials, Cambridge: Cambridge Univ. Press (2009).CrossRef
24.
Zurück zum Zitat 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). 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).
25.
Zurück zum Zitat A. Nekouzadeh and G. M. Genin, Adaptive Quasi-Linear Viscoelastic Modeling, Studies in Mechanobiology, Tissue Engineering and Biomaterials, 10, Berlin Heidelberg: Springer, 47-83 (2013).CrossRef A. Nekouzadeh and G. M. Genin, Adaptive Quasi-Linear Viscoelastic Modeling, Studies in Mechanobiology, Tissue Engineering and Biomaterials, 10, Berlin Heidelberg: Springer, 47-83 (2013).CrossRef
28.
Zurück zum Zitat A. V. Khokhlov, “Creep and relaxation curves produced by the Rabotnov nonlinear constitutive equation for viscoelastoplastic materials,” Probl. Prochn. Plast., 78, No. 4, 452-466 (2016). A. V. Khokhlov, “Creep and relaxation curves produced by the Rabotnov nonlinear constitutive equation for viscoelastoplastic materials,” Probl. Prochn. Plast., 78, No. 4, 452-466 (2016).
29.
Zurück zum Zitat A. V. Khokhlov, “Asyimptotics of creep curves generated by the Yu. N. Rabotnov nonlinear heredity theory in piecewise constant loadings and a condition of fading memory,” Vest. Moscow Univ., Ser. 1: Matem. Mekh., No. 5 26-31 (2017). A. V. Khokhlov, “Asyimptotics of creep curves generated by the Yu. N. Rabotnov nonlinear heredity theory in piecewise constant loadings and a condition of fading memory,” Vest. Moscow Univ., Ser. 1: Matem. Mekh., No. 5 26-31 (2017).
30.
Zurück zum Zitat A. V. Khokhlov, “Analysis of general properties of creep curves at step loadings generated by the Rabotnov nonlinear relation for viscoelastoplastic materials,” Vest. N. E. Baumam MGTU, ser. Estestv. Nauki, No. 3, 93-123 (2017). A. V. Khokhlov, “Analysis of general properties of creep curves at step loadings generated by the Rabotnov nonlinear relation for viscoelastoplastic materials,” Vest. N. E. Baumam MGTU, ser. Estestv. Nauki, No. 3, 93-123 (2017).
31.
Zurück zum Zitat A. V. Khokhlov, “Properties of the family of deformation diagrams generated by the Rabotnov nonlinear relation for viscoelastoplastic materials,” Izv. RAN Mekh. Tverd. Tela (in press) (2018). A. V. Khokhlov, “Properties of the family of deformation diagrams generated by the Rabotnov nonlinear relation for viscoelastoplastic materials,” Izv. RAN Mekh. Tverd. Tela (in press) (2018).
32.
Zurück zum Zitat N. W. Tschoegl, The Phenomenological Theory of Linear Viscoelastic Behavior, Heidelberg: Springer (1989).CrossRef N. W. Tschoegl, The Phenomenological Theory of Linear Viscoelastic Behavior, Heidelberg: Springer (1989).CrossRef
33.
Zurück zum Zitat A. A. Adamov, V. P. Matveenko, N. A. Trufanov, and I. N. Shardakov, “Methods of Applied Viscoelasticity [in Russian], Ekaterinburg: izd. UrO RAN (2003). A. A. Adamov, V. P. Matveenko, N. A. Trufanov, and I. N. Shardakov, “Methods of Applied Viscoelasticity [in Russian], Ekaterinburg: izd. UrO RAN (2003).
34.
Zurück zum Zitat F. Khan, “Loading history effects on the creep and relaxation behavior of thermoplastics,” Trans. ASME J. Eng. Mater. Technol., 128, 564-571 (2006).CrossRef F. Khan, “Loading history effects on the creep and relaxation behavior of thermoplastics,” Trans. ASME J. Eng. Mater. Technol., 128, 564-571 (2006).CrossRef
35.
Zurück zum Zitat J. Sorvari, M. Malinen, and J. Hämäläinen, “Finite ramp time correction method for nonlinear viscoelastic material model,” Int. J. Non-Linear Mech., 41, 1050-1056 (2006).CrossRef J. Sorvari, M. Malinen, and J. Hämäläinen, “Finite ramp time correction method for nonlinear viscoelastic material model,” Int. J. Non-Linear Mech., 41, 1050-1056 (2006).CrossRef
36.
Zurück zum Zitat W. G. Knauss and J. Zhao, “Improved relaxation time coverage in rampstrain histories,” Mechanics of Time-Dependent Materials, 11, No. 3, 199-216 (2007).CrossRef W. G. Knauss and J. Zhao, “Improved relaxation time coverage in rampstrain histories,” Mechanics of Time-Dependent Materials, 11, No. 3, 199-216 (2007).CrossRef
37.
Zurück zum Zitat S. Choi, S. W. Cha, and B. H. Oh, “Identification of viscoelastic behavior for earlyage concrete based on measured strain and stress histories,” Mater. and Struct., 43, 1161-1175 (2010).CrossRef S. Choi, S. W. Cha, and B. H. Oh, “Identification of viscoelastic behavior for earlyage concrete based on measured strain and stress histories,” Mater. and Struct., 43, 1161-1175 (2010).CrossRef
38.
Zurück zum Zitat 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, No. 1, 63-70 (2014).CrossRef 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, No. 1, 63-70 (2014).CrossRef
39.
Zurück zum Zitat 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, 24-132, (2014).CrossRef 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, 24-132, (2014).CrossRef
40.
Zurück zum Zitat 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, No. 2, 121-138 (2015).CrossRef 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, No. 2, 121-138 (2015).CrossRef
41.
Zurück zum Zitat J. S. Bergstrom, Mechanics of Solid Polymers. Theory and Computational Modeling, Elsevier, William Andrew (2015).CrossRef J. S. Bergstrom, Mechanics of Solid Polymers. Theory and Computational Modeling, Elsevier, William Andrew (2015).CrossRef
42.
Zurück zum Zitat A. V. Khokhlov, “Identification of a Maxwell-type nonlinear model of viscoelastoplastic by using creep curves with an initial ramp loading. Part 1. Mathematical foundations,” Deform. Razruch. Mater., No. 9, 2-9 (2017). A. V. Khokhlov, “Identification of a Maxwell-type nonlinear model of viscoelastoplastic by using creep curves with an initial ramp loading. Part 1. Mathematical foundations,” Deform. Razruch. Mater., No. 9, 2-9 (2017).
43.
Zurück zum Zitat A. V. Khokhlov, “A qualitative analysis of the general properties of theoretical curves of the linear constitutive equation of viscoelasticity,” Nauka i Obrazov.: Nauch Izd. N. E. Bauman MGTU, Elektron. Zhurn., No. 5, 187-245 (2016). A. V. Khokhlov, “A qualitative analysis of the general properties of theoretical curves of the linear constitutive equation of viscoelasticity,” Nauka i Obrazov.: Nauch Izd. N. E. Bauman MGTU, Elektron. Zhurn., No. 5, 187-245 (2016).
44.
Zurück zum Zitat A. V. Khokhlov, “Two-sided bonds for the relaxation modulus of the linear viscoelasticity via relaxations curves at ramp-strain histories and identification techniques,” Izv, RAN, Mekh. Tverd. Tela, No. 3, 81-104 (2018). A. V. Khokhlov, “Two-sided bonds for the relaxation modulus of the linear viscoelasticity via relaxations curves at ramp-strain histories and identification techniques,” Izv, RAN, Mekh. Tverd. Tela, No. 3, 81-104 (2018).
45.
Zurück zum Zitat A. V. Khokhlov, “Analysis of the properties of creep curves generated by the linear hereditary theory under arbitrary loading programs at the initial stage,” Vest. Samar. Gos. Univ., Ser. Fiz. Mat. Nauk, 22, No. 1 (2018). A. V. Khokhlov, “Analysis of the properties of creep curves generated by the linear hereditary theory under arbitrary loading programs at the initial stage,” Vest. Samar. Gos. Univ., Ser. Fiz. Mat. Nauk, 22, No. 1 (2018).
46.
Zurück zum Zitat A. V. Khokhlov, “Characteristic features of the families of deformation curves of linear models of viscoelasticity,” Probl. Prochn. Plast., 77, No. 2, 139-154 (2015). A. V. Khokhlov, “Characteristic features of the families of deformation curves of linear models of viscoelasticity,” Probl. Prochn. Plast., 77, No. 2, 139-154 (2015).
47.
Zurück zum Zitat S. A. Shesterikov and M. A. Yumasheva, “Specification of the constitutive equation of creep,” Izv. AN SSSR, Mekh. Tverd. Tela,. No 1, 86-91 (1984). S. A. Shesterikov and M. A. Yumasheva, “Specification of the constitutive equation of creep,” Izv. AN SSSR, Mekh. Tverd. Tela,. No 1, 86-91 (1984).
48.
Zurück zum Zitat A. D. Drozdov, “Time-dependent response of polypropylene after strain reversal,” Int. J. Solids and Structures, 47, 3221-3233 (2010).CrossRef A. D. Drozdov, “Time-dependent response of polypropylene after strain reversal,” Int. J. Solids and Structures, 47, 3221-3233 (2010).CrossRef
49.
Zurück zum Zitat F. Khan and C. Yeakle, “Experimental investigation and modeling of nonmonotonic creep behavior in polymers,” Int. J. Plasticity, 27, 512-521 (2011).CrossRef F. Khan and C. Yeakle, “Experimental investigation and modeling of nonmonotonic creep behavior in polymers,” Int. J. Plasticity, 27, 512-521 (2011).CrossRef
50.
Zurück zum Zitat A. D. Drozdov and N. Dusunceli, “Unusual mechanical response of carbon blackfilled thermoplastic elastomers,” Mechanics of Materials, 69, 116-131 (2014).CrossRef A. D. Drozdov and N. Dusunceli, “Unusual mechanical response of carbon blackfilled thermoplastic elastomers,” Mechanics of Materials, 69, 116-131 (2014).CrossRef
Metadaten
Titel
Analysis of Properties of Ramp Stress Relaxation Curves Produced by the Rabotnov Nonlinear Hereditary Theory
verfasst von
А. V. Khokhlov
Publikationsdatum
22.09.2018
Verlag
Springer US
Erschienen in
Mechanics of Composite Materials / Ausgabe 4/2018
Print ISSN: 0191-5665
Elektronische ISSN: 1573-8922
DOI
https://doi.org/10.1007/s11029-018-9757-1

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