Nonlinear reptation in molecular based hysteresis models for polymers
Authors:
H. T. Banks, Negash G. Medhin and Gabriella A. Pinter
Journal:
Quart. Appl. Math. 62 (2004), 767-779
MSC:
Primary 74D10; Secondary 35R10, 35R30, 74E35, 74H45
DOI:
https://doi.org/10.1090/qam/2104273
MathSciNet review:
MR2104273
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Abstract: We extend the linear “stick-slip” models of Doi-Edwards and Johnson-Stacer to nonlinear tube reptation models. We then show that such models, when combined with probabilistic formulations allowing distributions of relaxation times, provide a good description of dynamic experiments with highly filled rubber in tensile deformations. A connection to other applications including dielectric polarization and reptation in other viscoelastic materials (e.g., living tissue) is noted.
- Azmy S. Ackleh, H. T. Banks, and Gabriella A. Pinter, Well-posedness results for models of elastomers, J. Math. Anal. Appl. 268 (2002), no. 2, 440–456. MR 1896208, DOI https://doi.org/10.1006/jmaa.2000.7281
R.D. Andrews, Correlation of dynamic and static measurements on rubberlike materials, Ind. Engr. Chem., 44 (1952), 707-715.
- H. T. Banks, J. H. Barnes, A. Eberhardt, H. Tran, and S. Wynne, Modeling and computation of propagating waves from coronary stenoses, Comput. Appl. Math. 21 (2002), no. 3, 767–788. MR 1992370
- H. T. Banks and Kathleen L. Bihari, Modelling and estimating uncertainty in parameter estimation, Inverse Problems 17 (2001), no. 1, 95–111. MR 1818494, DOI https://doi.org/10.1088/0266-5611/17/1/308
- H. T. Banks, David Bortz, Gabriella Pinter, and Laura Potter, Modeling and imaging techniques with potential for application in bioterrorism, Bioterrorism, Frontiers Appl. Math., vol. 28, SIAM, Philadelphia, PA, 2003, pp. 129–154. MR 2036544
- H. T. Banks and N. G. Medhin, A molecular based dynamic model for viscoelastic responses of rubber in tensile deformations, Comm. Appl. Nonlinear Anal. 8 (2001), no. 3, 1–18. MR 1856067
- H. T. Banks, Negash G. Medhin, and Gabriella A. Pinter, Multiscale considerations in modeling of nonlinear elastomers, Int. J. Comput. Methods Eng. Sci. Mech. 8 (2007), no. 2, 53–62. MR 2300863, DOI https://doi.org/10.1080/15502280601149346
H.T. Banks, H.T. Tran and S. Wynne, A well-posedness result for a shear wave propagation model, Intl. Series Num. Math., Vol.143, Birkhauser Verlag, Basel, 2002, 25-40.
- H. T. Banks and G. A. Pinter, Approximation results for parameter estimation in a class of abstract nonlinear hyperbolic systems, Appl. Math. Lett. 12 (1999), no. 6, 129–133. MR 1751420, DOI https://doi.org/10.1016/S0893-9659%2899%2900091-9
H.T. Banks, G.A. Pintér, L.K. Potter, M.J. Gaitens and L.C. Yanyo, Modeling of nonlinear hysteresis in elastomers under uniaxial tension, J. Intelligent Material Systems and Structures 10 (1999), 116-134.
G. Bishko, T.C.B. McLeish, O.G. Harlen and R.G. Larson, Theoretical molecular rheology of branched polymers in simple and complex flows: The pom-pom model, Phys. Rev. Lett., 79 (1997), 2352-2355.
R. Blackwell, O.G. Harlen and T.C.B. McLeish, Theoretical linear and nonlinear rheology of symmetric treelike polymer melts, Macromolecules 34 (2001), 2579-2596.
R. Blackwell, T.C.B. McLeish and O.G. Harlen, Molecular drag-strain coupling in branched polymer melts, J. Rheology 44 (2000), 121-136.
C.J.F. Böttcher and P. Bordewijk, Theory of Electric Polarization, Vol.II: Dielectrics in Time Dependent Fields, Elsevier, Amsterdam, 1978.
M. Doi and M. Edwards, The Theory of Polymer Dynamics, Oxford, New York, 1986.
J.D. Ferry, E.R. Fitzgerald, L.D. Grandine and M.L. Williams, Temperature dependence of dynamic properties of elastomers: relaxation distributions, Ind. Engr. Chem., 44 (1952), 703-706.
Y.C. Fung, Biomechanics: Mechanical Properties of Living Tissues, Springer Verlag, New York, 1993.
R.S. Graham, T.C.B. McLeish and O.G. Harlen, Using the pom-pom equations to analyze polymer melts in exponential shear, J. Rheology 45 (2001), 275-290.
A.R. Johnson, C.J. Quigley and J.L. Mead, Large strain viscoelastic constitutive models for rubber, part I: Formulations, Rubber Chemistry Technology 67 (1994), 904-917.
A.R. Johnson and R.G. Stacer, Rubber viscoelasticity using the physically constrained systems’ stretches as internal variables, Rubber Chemistry Technology 66 (1993), 567-577.
A.R. Johnson, C.J. Quigley, D.G. Young and J.A. Danik, Viscohyperelastic modeling of rubber vulcanization, Tire Sci. Tech., 21 (1993), 179-199.
T.C.B. McLeish and R.G. Larson, Molecular constitutive equations for a class of branched polymers: The pom-pom polymer, J. Rheology 42 (1998), 81-110.
F. Schwarzl and A.J. Staverman, Higher approximation methods for the relaxation spectrum from static and dynamic measurements of viscoelastic materials, Appl. Sci. Res., A4 (1953), 127-141.
- D. ter Haar, A phenomenological theory of visco-elastic behaviour. I, II, III, Physica 16 (1950), 719–737, 738–752, 839–850. MR 44345
E.R. Von Schweidler, Studien uber die anomalien im verhalten der dielektrika, Ann. Physik 24 (1907), 711-770.
K.W. Wagner, Zur theorie der unvollkommenen dielektrika, Ann. Physik 40 (1913), 817-855.
M.L. Williams and J.D. Ferry, Second approximation calculations of mechanical and electrical relaxation and retardation distributions, J. Poly. Sci., 11 (1953), 169-175.
A.C. Ackleh, H.T. Banks and G.A. Pinter, Well-posedness results for models of elastomers, J. Math. Analysis and Applications 268 (2002), 440-456.
R.D. Andrews, Correlation of dynamic and static measurements on rubberlike materials, Ind. Engr. Chem., 44 (1952), 707-715.
H.T. Banks, J.H. Barnes, A. Eberhardt, H.T. Tran and S. Wynne, Modeling and computation of propagating waves from coronary stenoses, Comp. and Applied Math., 21 (2002), 1-22.
H.T. Banks and K. Bihari, Modeling and estimating uncertainty in parameter estimation, CRSCTR99-40, NCSU, December 1999; Inverse Problems 17 (2001), 1-17.
H.T. Banks, D. Bortz, G.A. Pinter and L.K. Potter, Modeling and imaging techniques with potential application in bioterrorism, CRSC-TR03-02, NCSU, January 2003; Chapter 6 in Bioterrorism: Mathematical Modeling Applications in Homeland Security, (H.T. Banks and C. Castillo-Chavez, eds.), Frontiers in Applied Mathematics, Vol.28, SIAM, Philadelphia, 2003, 129-154.
H.T. Banks and N.G. Medhin, A molecular based dynamic model for viscoelastic responses of rubber in tensile deformations, CRSC-TR00-27, NCSU, October, 2000; Communications on Applied Nonlinear Analysis 8 (2001), 1-18.
H.T. Banks, N.G. Medhin and G.A. Pinter, Multiscale considerations in modeling of nonlinear elastomers, CRSC-TR03-42, NCSU, October, 2003; J. Comp. Meth. Sci. Engr., to appear.
H.T. Banks, H.T. Tran and S. Wynne, A well-posedness result for a shear wave propagation model, Intl. Series Num. Math., Vol.143, Birkhauser Verlag, Basel, 2002, 25-40.
H.T. Banks and G.A. Pinter, Approximation results for parameter estimation in a class of abstract nonlinear hyperbolic systems, Appl. Math. Letters 12 (1999), 129-133.
H.T. Banks, G.A. Pintér, L.K. Potter, M.J. Gaitens and L.C. Yanyo, Modeling of nonlinear hysteresis in elastomers under uniaxial tension, J. Intelligent Material Systems and Structures 10 (1999), 116-134.
G. Bishko, T.C.B. McLeish, O.G. Harlen and R.G. Larson, Theoretical molecular rheology of branched polymers in simple and complex flows: The pom-pom model, Phys. Rev. Lett., 79 (1997), 2352-2355.
R. Blackwell, O.G. Harlen and T.C.B. McLeish, Theoretical linear and nonlinear rheology of symmetric treelike polymer melts, Macromolecules 34 (2001), 2579-2596.
R. Blackwell, T.C.B. McLeish and O.G. Harlen, Molecular drag-strain coupling in branched polymer melts, J. Rheology 44 (2000), 121-136.
C.J.F. Böttcher and P. Bordewijk, Theory of Electric Polarization, Vol.II: Dielectrics in Time Dependent Fields, Elsevier, Amsterdam, 1978.
M. Doi and M. Edwards, The Theory of Polymer Dynamics, Oxford, New York, 1986.
J.D. Ferry, E.R. Fitzgerald, L.D. Grandine and M.L. Williams, Temperature dependence of dynamic properties of elastomers: relaxation distributions, Ind. Engr. Chem., 44 (1952), 703-706.
Y.C. Fung, Biomechanics: Mechanical Properties of Living Tissues, Springer Verlag, New York, 1993.
R.S. Graham, T.C.B. McLeish and O.G. Harlen, Using the pom-pom equations to analyze polymer melts in exponential shear, J. Rheology 45 (2001), 275-290.
A.R. Johnson, C.J. Quigley and J.L. Mead, Large strain viscoelastic constitutive models for rubber, part I: Formulations, Rubber Chemistry Technology 67 (1994), 904-917.
A.R. Johnson and R.G. Stacer, Rubber viscoelasticity using the physically constrained systems’ stretches as internal variables, Rubber Chemistry Technology 66 (1993), 567-577.
A.R. Johnson, C.J. Quigley, D.G. Young and J.A. Danik, Viscohyperelastic modeling of rubber vulcanization, Tire Sci. Tech., 21 (1993), 179-199.
T.C.B. McLeish and R.G. Larson, Molecular constitutive equations for a class of branched polymers: The pom-pom polymer, J. Rheology 42 (1998), 81-110.
F. Schwarzl and A.J. Staverman, Higher approximation methods for the relaxation spectrum from static and dynamic measurements of viscoelastic materials, Appl. Sci. Res., A4 (1953), 127-141.
D. Ter Haar, A phenomenological theory of viscoelastic behavior, Physica 16 (1950), 839-850.
E.R. Von Schweidler, Studien uber die anomalien im verhalten der dielektrika, Ann. Physik 24 (1907), 711-770.
K.W. Wagner, Zur theorie der unvollkommenen dielektrika, Ann. Physik 40 (1913), 817-855.
M.L. Williams and J.D. Ferry, Second approximation calculations of mechanical and electrical relaxation and retardation distributions, J. Poly. Sci., 11 (1953), 169-175.
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