Abstract
An existing description of composite materials modeled as interpenetrating solid continua is extended to include constitutive response functionals over the histories of the constitutive variables of the n-consistent composite. As in the earlier work, the relative displacement of the individual constituents is required to be infinitesimal in order that the composite not rupture even though the motion of the combined composite is finite. Conditions on the response functionals under the assumption of fading memory are obtained from the Clausius-Duhem inequality for the combined composite. Relaxation and cyclical processes are discussed and it is shown that the equilibrium response of the composite considered is the same as that of the elastic composites treated in the earlier work. Some detailed results are obtained for the case of two-constituents under the additional assumption of finite linear viscoelastic response.
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References
H. T. Hahn & S. W. Tsai, “Nonlinear Elastic Behavior of Unidirectional Composite Laminae,” J. Comp. Mat., 7, 102 (1973).
C. T. Sun, W. H. Feng & S. L. Koh, “A Theory for Physically Nonlinear Elastic Fiber-Reinforced Composites,” Int. J. Engng. Sci., 12, 919 (1974).
R. W. Ogden, “On the Overall Moduli of Nonlinear Elastic Composite Materials,” J. Mech. Phys. Solids, 22, 541 (1974).
C. T. Sun, J. D. Achenbach & G. Herrmann, “Continuum Theory for a Laminated Medium,” J. Appl. Mech., 35, 467 (1968).
R. A. Grot & J. D. Achenbach, “Linear Anisothermal Theory for a Viscoelastic Laminated Composite,” Acta Mechanica, 9, 245 (1970).
R. A. Grot & J. D. Achenbach, “Large Deformations of a Laminated Composite,” Int. J. Solids Structures, 6, 641 (1970).
J. Aboudi, “A Mixture Theory for a Thermoelastic Laminated Medium, with Application to a Laminated Plate under Impulsive Loads,” J. Sound Vib., 33, 187 (1974).
Y. Benveniste & J. Aboudi, “A Nonlinear Mixture Theory for the Dynamic Response of a Laminated Composite under Large Deformations,” J. Appl. Math. Mech. (Z.A.M.M.), 28, 1067 (1977).
G. A. Hegemier, G. A. Gurtman & A. H. Nayfeh, “A Continuum Mixture Theory of Wave Propagation in Laminated and Fiber-Reinforced Composites,” Int. J. Solids and Structures, 9, 395 (1973).
G. A. Hegemier & G. A. Gurtman, “Finite Amplitude Elastic-Plastic Wave Propagation in Fiber-Reinforced Composites,” J. Appl. Phys., 45, 4254 (1974).
A. Bedford & M. Stern, “A Multicontinuum Theory for Composite Elastic Materials,” Acta Mechanica, 14, 85 (1972).
M. Stern & A. Bedford, “Wave Propagation in Elastic Laminates Using a Multicontinuum Theory,” Acta Mechanica, 15, 21 (1972).
H. F. Tiersten & M. Jahanmir, “A Theory of Composites Modeled as Interpenetrating Solid Continua,” Arch. Rational Mech. Anal., 65, 153 (1977).
L. P. Khoroshun, “Theory of Interpenetrating Elastic Mixtures,” Sov. Appl. Mech., 13, 10, 124 (1978).
P. Marinov, “A Three Continuum Theory for Composite Thermomechanical Materials,” Bull. Soc. Roy. Sci de Liege, No. 1–2, 106–118 (1974).
P. Marinov, “Toward a Thermoviscoelastic Theory of Two Component Materials,” Int. J. Engng. Sci., 16, 533 (1978).
A. Bedford & M. Stern, “Toward a Diffusing Continuum Theory of Composite Materials,” J. Appl. Mech., 38, 8 (1971).
J. Meixner, “The Fundamental Inequality in Thermodynamics,” Physica, 59, 305 (1972).
B. D. Coleman & W. Noll, “The Thermodynamics of Elastic Materials with Heat Conduction and Viscosity,” Arch. Rational Mech. Anal., 13, 167 (1963).
B. D. Coleman & W. Noll, “Foundations of Linear Viscoelasticity,” Rev. Mod. Phys., 33, 239 (1961).
B. D. Coleman, “Thermodynamics of Materials with Memory,” Arch. Rational Mech. Anal., 17, 1 (1964).
V. L. Kolpashchikov & A. I. Shnipp, “Thermodynamics and Properties of Relaxation Functions of Materials with Memory,” Int. J. Engng. Sci., 16, 503 (1978).
C. Truesdell & W. Noll, “The Nonlinear Field Theories of Mechanics,” in Encyclopedia of Physics, edited by S. Flügge (Springer-Verlag, Berlin, 1965), Vol. III/2.
In order to derive most of the results of this paper it suffices to follow Coleman [21] and assume that the response functionals are C 1. However, for the purposes of Sections 8 and 9 it is necessary to assume that \(\hat \psi \)(·) is C 2.
A. S. Wineman & A. C. Pipkin, “Material Symmetry Restrictions on Constitutive Equations,” Arch. Rational Mech. Anal., 17, 184 (1964).
A. E. Green & R. S. Rivlin, “The Mechanics of Nonlinear Materials with Memory,” Arch. Rational Mech. Anal., 1, 1 (1957).
E. H. Dill, “Simple Materials with Fading Memory,” in Continuum Physics, editor, A. C. Eringen (Academic Press, New York, 1975), Vol. 2.
R. M. Christensen, Theory of Viscoelasticity (Academic Press, New York, 1971).
M. M. Carroll, “Finite Deformations of Incompressible Simple Solids I. Isotropic Solids,” Quart. Jour. Mech. Appl. Math., 21, 147 (1968).
B. D. Coleman & V. J. Mizel, “A General Theory of Dissipation in Materials with Memory,” Arch. Rational Mech. Anal., 27, 255 (1967).
W. A. Day, The Thermodynamics of Simple Materials with Fading Memory (Springer-Verlag, Berlin, 1972).
An inequality of this type was derived earlier by Osborn [33].
R. B. Osborn, “Material Property Inequalities in Viscoelasticity,” Int. J. Engng. Sci., 9, 713 (1971).
H. F. Tiersten, “On the Mechanics of Interpenetrating Solid Continua,” S. M. Study No. 12, Continuum Models of Discrete Systems, University of Waterloo Press, 719 (1977).
M. E. Gurtin & I. R. Herrera, “On Dissipation Inequalities and Linear Viscoelasticity,” Quart. Appl. Math., 23, 235 (1965).
N. S. Wilkes, “Thermodynamic Restrictions on Viscoelastic Materials,” Quart. Jour. Mech. Appl. Math., 30, 209 (1977).
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McCarthy, M.F., Tiersten, H.F. A theory of viscoelastic composites modeled as interpenetrating solid continua with memory. Arch. Rational Mech. Anal. 81, 21–51 (1983). https://doi.org/10.1007/BF00283166
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DOI: https://doi.org/10.1007/BF00283166