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Journal of Thermal Stresses Volume 42, 2019 - Issue 1: Special Issue to commemorate the 90th birthday of Richard B. Hetnarski and 40 years of the Journal of Thermal Stresses: Part II
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Special Issue to commemorate the 90th birthday of Richard B. Hetnarski and 40 years of the Journal of Thermal Stresses

Coupled 1-D stress and thermal waves in temperature dependent nonlinear elastic and viscoelastic media*

Harry H. HiltonAerospace Engineering Department, College of Engineering, University of Illinois at Urbana–Champaign (UIUC), Urbana, Illinois, USA; ;Computing and Data Sciences Division, National Center for Supercomputing Applications, University of Illinois at Urbana–Champaign (UIUC), Urbana, Illinois, USA; Correspondence[email protected]
,
Mohamed H. ElalfyAerospace Engineering Department, College of Engineering, University of Illinois at Urbana–Champaign (UIUC), Urbana, Illinois, USA; ;Aerospace Engineering Department, University of Science and Technology (UST), Zewail City, Egypt
,
Hazem S. EldahshanAerospace Engineering Department, College of Engineering, University of Illinois at Urbana–Champaign (UIUC), Urbana, Illinois, USA; ;Aerospace Engineering Department, University of Science and Technology (UST), Zewail City, Egypt
&
Mohamed KhairyAerospace Engineering Department, College of Engineering, University of Illinois at Urbana–Champaign (UIUC), Urbana, Illinois, USA; ;Aerospace Engineering Department, University of Science and Technology (UST), Zewail City, Egypt
Pages 122-151 | Received 18 Oct 2018, Accepted 19 Oct 2018, Published online: 26 Feb 2019

References

  • H. H. Hilton, “Coupled 1-D thermal and stress waves in temperature dependent nonlinear elastic and viscoelastic media,” Proceedings 11th International Congress on Thermal Stresses, Salerno, Italy, 2016.
  • A. D. Drozdov, Finite Elasticity and Viscoelasticity – A Course in Nonlinear Mechanics of Solids. Singapore: World Scientific, 1996.
  • R. A. Schapery, “On the characterization of nonlinear viscoelastic materials,” Polym. Eng. Sci., vol. 9, pp. 295–319, 1969. doi:10.1002/pen.760090410
  • R. M. Christensen, “A nonlinear theory of viscoelasticity for application to elastomers, nASME J. Appl. Mech., vol. 47, pp. 762–768, 1980. doi:10.1115/1.3153787
  • S. Yi, H. H. Hilton, and M. F. Ahmad, “Nonlinear thermo-viscoelastic analysis of interlaminar stresses in laminated composites,” ASME J. Appl. Mech., vol. 63, pp. 218–224, 1996. doi:10.1115/1.2787201
  • D. R. Bland, The Theory of Linear Viscoelasticity. New York: Pergamon Press, 1960.
  • S. C. Hunter, (1960) “Viscoelastic waves,” Prog. Solid Mech., vol. 1, pp. 357.
  • H. H. Hilton, “Coupled longitudinal 1-D thermal and viscoelastic waves in media with temperature dependent material properties,” Eng. Mech., vol. 21, pp. 219–238, 2014.
  • H. H. Hilton, “Nonlinear elastic and viscoelastic 1-D wave propagation modeling and analysis,” J. Math. Eng. Sci. Aerosp., vol. 8, pp. 301–332, 2016.
  • T. Alfrey, Jr, “Non-homogeneous stress in viscoelastic media,” Quarter. Appl. Math., vol. 2, pp. 113–119, 1944. doi:10.1090/qam/10499
  • T. Alfrey, Jr, Mechanical Behavior of High Polymers. New York: Interscience Publishers, Inc., 1948.
  • J. J. Aklonis, W. J. MacKnight, and M. C. Shen, Introduction to Polymer Viscoelasticity. New York: Wiley, 1972.
  • J. J. Aklonis, and W. J. MacKnight, Introduction to Polymer Viscoelasticity. New York: Wiley, 1983.
  • H. A. Barnes, J. F. Hutton, and K. Walters, An Introduction to Rheology. Amsterdam: Elsevier, 1989.
  • G. M. Bartenev, and Y. S. Zuyev, Strength and Failure of Viscoelastic Materials. Oxford: Pergamon Press, 1968.
  • Z. P. Bazant, (Ed.) Mathematical Modeling of Creep and Shrinkage of Concrete. New York: John Wiley and Sons, 1988.
  • Z. P. Bazant, and L. Cedolin, Stability of Structures - Elastic, Inelastic, Fracture and Damage Theories. New York: Oxford University Press, 1991.
  • Z. P. Bazant, Scaling of Structural Strength, 2nd ed. New York: Elsevier, 2005.
  • J. T. Bergen, Viscoelasticity; Phenomenological Aspects. New York: Academic Press, 1960.
  • J. Betten, Creep Mechanics, 2nd ed. New York: Springer, 2005.
  • M. A. Biot, Mechanics of Incremental Deformations. New York: John Wiley & Sons, 1965.
  • D. R. Bland, Nonlinear Dynamic Elasticity. Waltham, MA: Blaisdell, 1969.
  • H. F. Brinson, and L. Catherine Brinson, Polymer Engineering Science and Viscoelasticity: An Introduction. New York: Springer, 2008.
  • R. M. Christensen, Theory of Viscoelasticity – An Introduction, 2nd ed. New York: Academic Press, 1982.
  • A. D. Drozdov, and V. B. Kolmanovski, Stability in Viscoelasticity. Amsterdam: North-Holland, 1994.
  • A. D. Drozdov, Mechanics of Viscoelastic Solids. New York: John Wiley & Sons, (1998)
  • A. D. Drozdov, Viscoelastic Structures Mechanics of Growth and Aging. San Diego, CA: Academic Press, 1998.
  • A. C. Eringen, Nonlinear Theory of Continuous Media. New York: McGraw-Hill, 1962.
  • M. Fabrizio, and A. Morro, Mathematical Problems in Linear Viscoelasticity. Philadelphia, PA: SIAM, 1992.
  • J. D. Ferry, Viscoelastic Properties of Polymers. New York: John Wiley & Sons, 1980.
  • W. N. Findley, J. S. Lai, and K. Onaran, Cree and Relaxation of Nonlinear Materials. Amsterdam: North-Holland Publ. Co, 1976.
  • W. Flügge, Viscoelasticity. Waltham, MA: Blaisdell Pub. Co, 1967.
  • A. M. Freudenthal, The Inelastic Behavior of Engineering Materials and Structures. New York: John Wiley & Sons, 1950.
  • Y. C. Fung, Foundations of Solid Mechanics. Englewood Cliffs, NJ: Prentice-Hall, 1965.
  • L. Garibaldi, and H. N. Onah, Viscoelastic Material Damping Technology. Turin, Italy: Becchis Osiride, 1996.
  • J. M. Golden, and C. A. C. Graham, Boundary Value Problems in Linear Viscoelasticity. Berlin: Springer Verlag, 1988.
  • B. Gross, Mathematical Structure of the Theories of Viscoelasticity. Paris: Hermann & Cie, 1953.
  • Y. M. Haddad, Viscoelasticity of Engineering Materials. London: Chapman & Hall, 1995.
  • P. Haupt, Viskoelastizität Und Plastizität: Thermomechanisch Konsistente Materialgleichungen. Berlin: Springer, 1977.
  • H. H. Hilton, “An introduction to viscoelastic analysis,” in Engineering Design for Plastics, E. Baer, ed. New York: Reinhold Publishing Corp., 1964, 199–276.
  • H. H. Hilton, “Equivalences and contrasts between thermo-elasticity and thermo-viscoelasticity: a comprehensive critique,” J. Therm. Stress., vol. 34, pp. 488–535, 2011. doi:10.1080/01495739.2011.564010
  • R. S. Lakes, Viscoelastic Solids. Boca Rotan, FL: CRC Press, 1998.
  • R. S. Lakes, Viscoelastic Materials. New York: Cambridge University Press, 2009.
  • B. J. Lazan, Damping of Materials and Members in Structural Mechanics. Oxford: Pergamon Press, 1968.
  • P. Le Tallec, Numerical Analysis of Viscoelastic Problems. Berlin: Springer-Verlag, 1990.
  • F. Levi, and G. Pizzetti, Fluage, Plasticité, Précontrainte. Paris: Dunod, 1951.
  • F. J. Lockett, Nonlinear Viscoelastic Solids. London: Academic Press, 1972.
  • A. S. Lodge, Body Tensor Fields in Continuum Mechanics with Applications to Polymer Rheology. New York: Academic Press, 1974.
  • A. S. Lodge, M. Renardy and J. A. Nohel (Eds.), Viscoelasticity and Rheology. New York: Academic Press, New York. (1985)
  • N. G. McCrum, B. E. Read, and G. Williams, Anelastic and Dielectric Effects in Polymeric Solids. New York: John Wiley & Sons, 1967.
  • A. D. Nashif, D. I. G. Jones, and J. P. Henderson, Vibration Damping. New York: John Wiley & Sons, 1985.
  • A. C. Pipkin, Lectures on Viscoelasticity Theory. Berlin: Springer-Verlag, 1972.
  • W. T. Read, “Stress analysis for compressible viscoelastic materials,” J. Appl. Phys., vol. 21, pp. 671–674, 1950. doi:10.1063/1.1699729
  • M. Reiner, Deformation, Strain and Flow: An Elementary Introduction to Rheology. London: H. K. Lewis, 1969.
  • M. Reiner, Advanced Rheology. London: H. K. Lewis, 1971.
  • M. Renardy, W. J. Hrusa, and A. N. John, Mathematical Problems in Viscoelasticity. Burnt Mill: Longmans Scientific and Technical Press, 1987.
  • C. M. Roland, Viscoelastic Behavior of Rubbery Materials. New York: Oxford University Press, 2011.
  • J. Salençon, Viscoélasticité. Paris: Presses ENPC, 1983.
  • N. W. Tschoegl, The Phenomenological Theory of Linear Viscoelastic Behavior: An Introduction. New York: Springer, 1989.
  • G. V. Vinogradov, and A. Ya. Malkin, Rheology of Polymers: Viscoelasticity and Flow of Polymers. Moscow: Mir Publishers, 1980.
  • A. S. Wineman, and K. R. Rajakopal, Mechanical Response of Polymers – An Introduction. New York: Cambridge University Press, 2000.
  • C. Zener, Elasticity and An Elasticity of Metals. Chicago: University of Chicago Press, 1948.
  • J. C. Maxwell, “On reciprocal figures and diagrams of forces,” Philos. Magazine, vol. 26, pp. 250–261, 1864.
  • J. D. Achenbach, Wave Propagation in Elastic Solids. Amsterdam: North-Holland Publishing Co., 1973.
  • D. R. Bland, Vibrating Strings; an Introduction to the Wave Equation. London: Routledge and Paul, 1960.
  • D. R. Bland, Wave Theory and Applications. New York: Clarendon Press, 1988.
  • J. L. Davis, Mathematics of Wave Propagation. Princeton: Princeton University Press, 2000.
  • J. F. Doyle, Wave Propagation in Structures. New York: Springer, 1997.
  • Doyle, Ja F. “Viscoelastic wave propagation,” in Encyclopedia of Thermal Stresses, Richard B. Hetnarski (Ed.) vol. 11. New York: Springer, 2014, pp. 6462–6467
  • K. F. Graff, Wave Motion in Elastic Solids. New York: Dover Publications, 1975.
  • J. G. Harris, Linear Elastic Waves. New York: Cambridge University Press, 2001.
  • P. Hagedorn, Non-Linear Oscillations. Oxford: Clarendon Press, 1981.
  • H. Kolsky, Stress Waves in Solids. London: Clarendon Press, 1953.
  • A. H. Nayfeh, Wave Propagation in Layered Anisotropic Media with Applications to Composites. Amsterdam: North-Holland, 1995.
  • J. Ignaczak, and M. Ostoja-Starzewski, Thermoelasticity with Finite Wave Speeds. Oxford: Oxford Press, 2010.
  • M. Ostoja-Starzewski, “Viscothermoelasticity with finite wave speeds: thermomechanical laws,” Acta Mech., vol. 225, pp. 1277–1285, 2014. doi:10.1007/s00707-013-1075-z
  • B. A. Boley, and J. A. Weiner, Theory of Thermal Stresses. New York, NY: John Wiley & Sons, 1960.
  • J. M. C. Duhamel, “Second mémoire sur les phénomenes thermomecaniques,” Journal de L’École Polytechnique de Paris, vol. 43, pp. 356–399, 1837.
  • J. B. Joseph Baron de Fourier, Théorie de la chaleur. Paris: Didot, 1822.
  • G. E. Gatewood, Thermal Stresses. New York: McGraw-Hill, 1957.
  • D. E. Carlson, “Linear thermoelasticity,” Handbuch Der Physik, vol. 2, Clifford Truesdell, Ed. New York: Springer, 1972, pp. 297–346.
  • R. B. Hetnarski, and J. Ignaczak, The Mathematical Theory of Elasticity, 2nd ed. Philadelphia, PA: Taylor & Francis, 2011.
  • R. B. Hetnarski, and M. Reza Eslami, Thermal Stresses – Advanced Theory and Applications (Solid Mechanics and Its Applications. New York: Springer, 2010.
  • R. B. Hetnarski, Thermal Stresses I (Mechanics and Mathematical Methods – Series of Handbooks), vol. 1. Amsterdam: North Holland, 1986.
  • R. B. Hetnarski, Ed. Encyclopedia of Thermal Stresses. Berlin: Springer, 2013.
  • S. Jiang, and R. Racke, Evolution Equations in Thermoelasticity. Boca Raton: Chapman & Hall/CRC, 2000.
  • D. J. Johns, Thermal Stress Analysis. New York: Pergamon Press, 1965.
  • A. D. Kovalenko, Thermoelasticity. Basic Theory and Applications. Amsterdam: Groningen, Wolters-Noordhoff, 1969.
  • E. Melan, and H. Parkus, Wärmespannungen Infolge Stationärer Temperaturfelder. Wien: Springer, 1953.
  • W. Nowacki, Thermoelasticity. Reading, MA: Addison-Wesley, 1962.
  • W. Nowacki, Dynamic Problems of Thermoelasticity. New York: Springer, 1975.
  • H. Parkus, Wärmespannungen. Wien: Springer, 1959.
  • T. R. Tauchert, and R. B. Hetnarski, Bibliography on Thermal Stresses. Washington, DC: Hemisphere, 1986.
  • H. Ziegler, Introduction to Thermomechanics, 2nd ed., Amsterdam: North Holland, 1983.
  • D. S. Berry, “Stress propagation in viscoelastic bodies,” J. Mech. Phys. Solid., vol. 6, pp. 177–185, 1958. doi:10.1016/0022-5096(58)90024-3
  • B. D. Coleman, M. E. Gurtin, and I. Herrera, “Waves in materials with memory I, the velocity of one dimensional shock and acceleration waves,v Archives for Rational Mechanics and Analysis, vol. 1, pp. 1–19, 1965.
  • B. D. Coleman, and M. E. Gurtin, “Waves in materials with memory III, thermodynamic influences on the growth and decay of acceleration waves,v Archive for Rational Mechanics and Analysis, vol. 19, pp. 266–298, 1965. doi:10.1007/BF00250214
  • Mainardi, Francesco (Ed.) (1982) Wave Propagation in Viscoelastic Media. Boston, MA: Pitman Publishing Co.
  • F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. London: Imperial College Press, 2010.
  • W. G. Knauss, “Uniaxial wave propagation in a viscoelastic material using measured material properties,” ASME J. Appl. Mech., vol. 35, pp. 449–453, 1968. doi:10.1115/1.3601234
  • M. I. Mustafa, and S. A. Messaoud, “General stability result for viscoelastic wave equations,” J. Math. Phys., vol. 53, pp. 1–8, 2012.
  • N. H. Ricker, Transient Waves in Visco-Elastic Media. Amsterdam: Elsevier, 1977.
  • M. Schanz, Wave Propagation in Viscoelastic and Poroelastic Continua: A Boundary Element Approach. New York: Springer-Verlag, 2001.
  • H. H. Hilton, (1951) “Thermal stresses and strains in circular thin plates and thick-walled cylinders exhibiting temperature dependent elastic and viscoelastic properties,” PhD Thesis in Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign.
  • A. M. Freudenthal, “Effect of rheological behavior on thermal stresses,” J. Appl. Phys., vol. 25, pp. 1110–1117, 1954. doi:10.1063/1.1721824
  • H. H. Hilton, “Thermal stresses in thick walled cylinders exhibiting temperature dependent viscoelastic properties of the Kelvin type,” in Second United States National Congress for Applied Mechanics, pp. 547–553, 1954.
  • G. U. Losi, and W. G. Knauss, “Thermal stresses in nonlinearly viscoelastic solids,” ASME J. Appl. Mech., vol. 59, pp. S43–S49, 1992. doi:10.1115/1.2899506
  • J. L. Sackman, “A remark on transient stresses in nonhomogeneous viscoelastic materials,” J. Aerosp. Sci., vol. 29, pp. 1015–1016, 1962.
  • R. Muki, and E. Sternberg, “On transient thermal stresses in viscoelastic materials with temperature dependent properties,” ASME J. Appl. Mech., vol. 28, pp. 193–207, 1961. doi:10.1115/1.3641651
  • H. H. Hilton, “Thermal distributions without thermal stresses in nonhomogeneous media,” J. Appl. Mech., vol. 26, pp. 137–138, 1959.
  • H. H. Hilton, and H. G. Russell, “An extension of Alfrey’s analogy to thermal stress problems in temperature dependent linear viscoelastic media,” J. Phys. Solid., vol. 9, pp. 152–164, 1961. doi:10.1016/0022-5096(61)90014-X
  • H. H. Hilton, and J. R. Clements, “Formulation and evaluation of approximate analogies for temperature dependent linear viscoelastic media,” Therm. Load. Creep., vol. 6, pp. 17–24, 1964.
  • S. A. Arrhenius, “Über die dissociationswärme und den einfluß der temperatur auf den dissociationsgrad der elektrolyte,” Zeitschrift Für Phys. Chem., vol. 4, pp. 96–116, 1889.
  • S. A. Arrhenius, “Über die Reaktionsgeschwindigkeit bei der Inversion von Rohrzucker durch Säuren,” Zeitschrift Für Phys. Chem., vol. 4, 226–248, 1889.
  • K. J. Laidler, The World of Physical Chemistry. New York: Oxford University Press, 1993.
  • M. L. Williams, R. F. Landel, and J. D. Ferry, “The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids,” J. Am. Chem. Soc., vol. 77, pp. 3701–3707, 1955. doi:10.1021/ja01619a008
  • R. F. Landel, “A two-part tale: the WLF equation and beyond linear viscoelasticity,” Rubber Chem. Technol., vol. 79, no. 3, pp. 381–401, 2006. doi:10.5254/1.3547943
  • F. Schwarzl, and A. J. Staverman, “Time-temperature dependence of linear viscoelastic behavior,” J. Appl. Phys., vol. 23, no. 8, pp. 838–843, 1952. doi:10.1063/1.1702316
  • P. Haupt, “Thermorheologically simple materials: Thermodynamic restrictions upon material functions in finite linear viscoelasticity,” Mech. Res. Commun., vol. 1, no. 4, pp. 203–208, 1974. doi:10.1016/0093-6413(74)90065-2
  • G. Prony and C. F. M. R. de Baron, “Essai experimental et analytique,” J. de L’École Polytechnique de Paris, vol. 1, pp. 24–76, 1795.
  • L. E. Boltzmann, “Zur theorie der elastischen nachwirkung,” Sitzungsberichte Kaiserliche Akademie Wissenhaft Wien Mathematische-Naturwissenhaft, vol. 70, pp. 275–306, 1874.
  • M. Michaeli, A. Shtark, H. Grosbein, E. Altus, and H. H. Harry, (2014) “Uncertainty in characterizations of isotropic linear viscoelastic moduli/compliances from 1-D tensile experiments,” Proceedings AIAA SciTech SDM 2014 Conference, AIAA Paper 2014-0506, National Harbor, MD.
  • M. Abramowitz, and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover Publications, 1970.
  • H. H. Hilton, et al., Probabilistic Characterizations of Linear Isotropic Viscoelastic Properties from 1-D Statistical Quasi-Static and Dynamic Tensile and Compressive Experiments. Berlin: Springer, 2018.
  • I. N. Sneddon, Special Functions of Mathematical Physics and Chemistry. London: Oliver and Boyd, 1961.
  • A. E. Green, and J. E. Adkins, Large Elastic Deformations and Non-Linear Continuum Mechanics. London: Oxford University Press, 1960.
  • É. Picard, (1925) Traité D’analyse, 3rd ed., vol. 2, Paris: Gauthier-Villars, 95–106.
  • J. H. Poincaré, “ 67s Henrillarsy Pressons and Non-Li NASA TT F-450, 451, 452, 1967.
  • J. H. Poincaré, Les méthods nouvelles de la mécanique celeste, vol. I, II, III, Paris: Gauthier-Villars, 1892, 1893, 1899.
  • S. M. J. Lighthill, “A technique for rendering approximate solutions to physical problems uniformly valid,” Philosop. Magazine, vol. 40, pp. 1179–1120, 1949.
  • Y. H. Kuo, “On the flow of an incompressible viscous fluid past a flat plate at moderate reynolds numbers,” Journal of Mathematics and and Physics, vol. 32, no. 1-4, pp. 83–51, 1953. doi:10.1002/sapm195332183
  • Y. H. Kuo, “Viscous flow along a flat plate moving at high supersonic speeds,” J. Aeronaut. Sci., vol. 23, pp. 125–136, 1956. doi:10.2514/8.3520
  • B. E. Pobedrya, “Method of successive approximations in the nonlinear theory of viscoelasticity,” Polym. Mech., vol. 5, pp. 191–196, 1969.
  • H. S. Tsien, “The Poincaré-Lighthill-Kuo method,” J. Adv. Appl. Math., vol. 4, pp. 281–349, 1956.
  • R. R. Bracewell, The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 2000.
  • R. N. Bracewell, Fourier Analysis and Imaging. Boston, MA: Springer, 2003.
  • S. M. J. Lighthill, Introduction to Fourier Analysis and Generalized Functions. Cambridge: University Press, 1958.
  • D. Voelker, and G. Doetsch, Die Zweidimensionale Laplace-Transformation: Eine Einführung in Ihre Anwendung Zur Lösung Von Randwertproblemen Nebst Tabellen Von Korrespondenzen Der Exakten Wissenschaften. Basel, Switzerland: Birkhäuser, 1950.
  • S. Okui, “Tables of one- and two-dimensional inverse Laplace transforms of complete elliptic integrals,” J. Res B. Math. Sci., vol. 81, pp. 1–35, 1977.
  • P. P. Valkó, and J. Abate, “Numerical inversion of 2-D Laplace transforms applied to fractional diffusion equations,” Applied Numerical Mathematics, vol. 53, no. 1, pp. 73–88, 2005. doi:10.1016/j.apnum.2004.10.002
  • V. A. Ditkin, and A. P. Prudnikov, Operational Calculus in Two Variables and Its Applications. New York: Pergamon Press, 1962.
  • G. D. Bergland, “A guided tour of the fast Fourier transform,” IEEE Spectrum, vol. 6, no. 7, pp. 41–52, 1969. doi:10.1109/MSPEC.1969.5213896
  • E. O. Brigham, The Fast Fourier Transform. Englewood Cliffs, NJ: Prentice Hall Inc, 1974.
  • H. Dubner, and J. Abate, “Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform,” J. ACM, vol. 15, no. 1, pp. 115–123, 1968. doi:10.1145/321439.321446
  • F. Durbin, “Numerical inversion of Laplace transforms: an efficient improvement to Dubner and Abate’s method,” Comput. J., vol. 17, no. 4, pp. 371–376, 1974. doi:10.1093/comjnl/17.4.371
  • C. F. V. Loan, Computational Frameworks for the Fast Fourier Transform. Philadelphia, PA: SIAM, 1962.
  • E. L. Post, “Generalized differentiation,” Trans. Am. Math. Soc., vol. 32, no. 4, pp. 723–781, 1930. doi:10.1090/S0002-9947-1930-1501560-X
  • S. Yi, and H. H. Hilton, “Dynamic finite element analysis of viscoelastic composite plates in the time domain,” Int. J. Num. Method. Eng., vol. 37, no. 23, pp. 4081–4096, 1994. doi:10.1002/nme.1620372309
  • A. R. Zak, “Structural analysis of realistic solid propellant materials,” AIAA J. Spacecraft., vol. 5, no. 3, pp. 270–275, 1968. doi:10.2514/3.29237
  • R. L. Taylor, K. L. Pister, and G. L. Goudreau, “Thermomechanical analysis of viscoelastic solids,” Int. J. Num. Method. Eng., vol. 2, no. 1, pp. 45–59, 1970. doi:10.1002/nme.1620020106
  • B. Hartman, and J. Jarzynski, “Polymer sound speeds and elastic constants,” NOLTR. Silver Springs, MD: Naval Ordnance Laboratory, 1972, pp. 72–269.
  • S. G. Mikhlin, Variational Methods in Mathematical Physics. New York: Pergamon Press, (1964)
  • M. W. Kutta, “Beitrag zur näherungsweisen integration totaler differential gleichungen,” Zeitschrift Der Math. Phys., vol. 46, pp. 435–453, 1901.
  • C. Runge, “Über die numerische auflösung von differentialgleichungen,” Math. Ann., vol. 46, no. 2, pp. 167–178, 1895. doi:10.1007/BF01446807
  • H. H. Hilton, and G. Sossou, “Computational protocols for viscoelastic Prony series hereditary integrals and for variable coefficient integral-differential equations,” submitted to J. Aerosp. Inform. Syst., 2017.
  • R. L. Bisplinghoff, H. Ashley, and R. L. Halfman, Aeroelasticity. Cambridge, MA: Addison-Wesley Publishing Company, 1980 (New York: Dover Publications, 1955).
  • J. R. Wright, and J. E. Cooper, Introduction to Aircraft Aeroelasticity and Loads, 2nd ed. Hoboken, NJ: John Wiley, 2015.
  • H. H. Hilton, “The divergence of supersonic, linear viscoelastic lifting surfaces, including chord-wise bending,” J. Aero/Space Sci., vol. 27, pp. 926–934, 1960. doi:10.2514/8.8813
  • C. G. Merrett, and H. H. Hilton, “Fractional order derivative aero-servo-viscoelasticity,” Int. J. Dynam. Control., vol. 5, pp. 229–251, 2017.
  • C. E. Beldica, H. H. Hilton, and C. Greffe, (2001) “The relation of experimentally generated wave shapes to viscoelastic material characterizations – analytical and computational simulations,” Proceedings of the Sixteenth Annual Technical Conference of the American Society for Composites CD-ROM, Vol. 1–11.
  • E. Grüneisen, “Theorie des festen zustandes einatomiger elemente,” Annalen Der Physik, vol. 344, no. 12, pp. 257–306, 1912. doi:10.1002/andp.19123441202
  • J. P. Frankel, Principles of the Properties of Materials. New York: McGraw-Hill, 1957.
  • P. T. Landsberg, Thermodynamics and Statistical Mechanics. New York: Dover Publications, Inc., 1990.
  • H. Ziegler, and C. Wehrli, “The derivation of constitutive relations from the free energy and the dissipation functions,” Adv. Appl. Mech., vol. 25, pp. 183–238, 1987.

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