Equivalences and Contrasts of Thermo-Elasticity and Thermo-Viscoelasticity: A Comprehensive Critique

REFERENCES

• W. T. Kelvin , On the Elasticity and Viscosity of Metals , Proceedings of the Royal Society , vol. 14 , pp. 289 – 297 , 1865 .
   Google Scholar
• J. C. Maxwell , On the Dynamical Theory of Gases , Philosophical Transactions of the Royal Society of London , vol. A157 , pp. 49 – 88 , 1867 .
   Google Scholar
• L. Boltzmann , Zur Theorie der Elastischen Nachwirkung, Sitzungsberichte Kaiserliche Akademie Wissenhaft Wien Mathematische-Naturwissenhaft , vol. 70, pp. 275–306, 1874.
   Google Scholar
• W. Voigt , Theoretischen Studien über das Elasticitatsverhaltnisse der Kristalle , Abhandlung Geselschaft der Wissenschaft zu Göttingen , vol. 34 , pp. 52 – 79 , 1887 .
   Google Scholar
• W. Voigt , Über innere Reibung der Festen Körper, Insbesondere der Kristalle , Abhandlung der Königlichen Geselschaft der Wissenschaft zu Göttingen , vol. 36 , no. 1 , 1889 .
   Google Scholar
• W. Voigt , Über innere Reibung der Festen Körper , Insbesondere der Metalle , Annallen der Physik , vol. 47 , pp. 671 – 639 , 1892 .
   Google Scholar
• V. Volterra , Sulle Equazioni Integrodifferenzialli dela Teoria Dell'Elastica , Atti Reale Accademia Nazionale dei Lincei , vol. 18 , pp. 295 – 301 , 1909 .
   Google Scholar
• V. Volterra , M. Tomassetti , and F. S. Zarlatti , Leçons sur les Équations Integral et les Équations Integro-Differentielles . Guthier-Villars , Paris , 1913 .
   Google Scholar
• T. Alfrey , Jr., Non-Homogeneous Stress in Viscoelastic Media , Quarterly of Applied Mathematics , vol. 2 , pp. 113 – 119 , 1944 .
   Google Scholar
• T. Alfrey Jr. , and P. Doty , The Methods of Specifying Properties of Viscoelastic Materials , Journal of Applied Physics , vol. 16 , pp. 700 – 713 , 1945 .
   Google Scholar
• T. Alfrey Jr. , Mechanical Behavior of High Polymers . Interscience Publishers, Inc. , New York , 1948 .
   Google Scholar
• W. T. Read , Stress Analysis for Compressible Viscoelastic Materials , Journal of Applied Physics , vol. 21 , pp. 671 – 674 , 1950 .
   Web of Science ®Google Scholar
• A. M. Freudenthal , The Inelastic Behavior of Engineering Materials and Structures . John Wiley & Sons , New York , 1950 .
   Google Scholar
• B. Gross , Mathematical Structure of the Theories of Viscoelasticity . Hermann & cie. , Paris , 1953 .
   Google Scholar
• F. Schwarzl and A. J. Staverman , Time-Temperature Dependence of Linear Viscoelastic Behavior , Journal of Applied Physics , vol. 23 , pp. 838 – 843 , 1952 .
   Web of Science ®Google Scholar
• D. R. Bland , The Theory of Linear Viscoelasticity . Pergamon Press , New York , 1960 .
   Google Scholar
• M. E. Gurtin and E. Sternberg , On the Linear Theory of Viscoelasticity , Archive for Rational Mechanics and Analysis , vol. 11 , pp. 291 – 356 , 1962 .
   Web of Science ®Google Scholar
• A. C. Pipkin , Convolutions as Products in Viscoelastic Theory , in K. R. Rajagopal ( ed. ), Recent Advances in Elasticity, Viscoelasticity and Inelasticity , pp. 82 – 107 . World Scientific , Singapore , 1995 .
   Google Scholar
• M. E. Gurtin , Variational Principles in the Linear Theory of Viscoelasticity , Archive for Rational Mechanics and Analysis , vol. 13 , pp. 179 – 185 , 1963 .
   Google Scholar
• M. E. Gurtin and E. Sternberg , A Reciprocal Theory of Anisotropic Viscoelastic Solids , Journal of the Society for Industrial Applied Mathematics , vol. 11 , pp. 607 – 613 , 1963 .
   Google Scholar
• E. Sternberg , and M. E. Gurtin , Uniqueness in the Theory of Thermorheologically Simple Ablating Materials , Progress in Applied Mechanics – Prager Anniversary , vol. 373–384. Macmillan , New York , 1963 .
   Google Scholar
• C. B. Navarro , Asymptotic Stability in Linear Thermoviscoelasticity , Journal of Mathematical Analysis and Applications , vol. 65 , pp. 399 – 431 , 1978 .
   Web of Science ®Google Scholar
• H. H. Hilton , and S. B. Dong , An Analogy for Anisotropic, Nonhomogeneous, Linear Viscoelasticity Including Thermal Stresses , Development in Mechanics , pp. 58 – 73 , Pergamon Press , New York , 1964 .
   Google Scholar
• H. H. Hilton , An Introduction to Viscoelastic Analysis , in E. Baer ( ed. ), Engineering Design for Plastics , pp. 199 – 276 , Reinhold Publishing Corp. , 1964 .
   Google Scholar
• A. C. Pipkin , Lectures on Viscoelasticity Theory . Springer-Verlag , Berlin , 1972 .
   Google Scholar
• R. M. Christensen , Theory of Viscoelasticity – An Introduction, , 2nd ed. , Academic Press , New York , 1982 .
   Google Scholar
• J. Salençon , Viscoélasticité , Presses ENPC , Paris , 1983 .
   Google Scholar
• M. Fabrizio and A. Morro , Mathematical Problems in Linear Viscoelasticity . SIAM , Philadelphia , PA , 1992.
   Google Scholar
• A. D. Drozdov , Viscoelastic Structures , Academic Press , New York , 1998 .
   Google Scholar
• H. F. Brinson and L. C. Brinson , Polymer Engineering Science and Viscoelasticity: An Introduction , Springer , New York , 2008 .
   Google Scholar
• H. Kolsky , Stress Waves in Solids , Clarendon Press , London , 1953 .
   Google Scholar
• J. D. Achenbach , Wave Propagation in Elastic Solids , North-Holland Publishing Co. , Amsterdam , 1973 .
   Google Scholar
• B. A. Boley and J. A. Weiner , Theory of Thermal Stresses , John Wiley & Sons , New York , NY , 1960 .
   Google Scholar
• W. Nowacki , Thermo-Elasticity , Addison-Wesley , Reading , MA , 1962 .
   Google Scholar
• W. Nowacki , Dynamic Problems of Thermoelasticity , Springer , New York , 1975 .
   Google Scholar
• A. D. Kovalenko , Thermoelasticity. Basic Theory and Applications , Groningen , Wolters-Noordhoff , Amsterdam , 1969 .
   Google Scholar
• D. S. Berry , Stress Propagation in Viscoelastic Bodies , Journal of Mechanics and Physics of Solids , vol. 6 , pp. 177 – 185 , 1958 .
   Google Scholar
• S. C. Hunter , Viscoelastic Waves , Progress in Solid Mechanics , vol. 1 , pp. 3 – 57 , Pergamon Press , New York , 1960 .
   Google Scholar
• C. E. Beldica and H. H. Hilton , The Influence of Viscoelastic Wave Generation on Experimental Material Property Characterization – Analytical and Computational Simulations, Proceedings of the IUTAM Symposium on Mechanical Waves for Composite Structures Characterization, pp. 27–32, 2000.
   Google Scholar
• F. Mainardi , ( Ed. ) Wave Propagation in Viscoelastic Media , Pitman Publishing Co. , Boston , 1982 .
   Google Scholar
• C. E. Beldica , H. H. Hilton , and C. Greffe , 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, 2001.
   Google Scholar
• C. F. M. R. Prony , Essai Experimental et Analytique , Journal de l’École Polytechnique de Paris , vol. 1 , pp. 24 – 76 , 1795 .
   Google Scholar
• S.-D. Poisson , Mémoire sur l’Équilibre et le Mouvement des Corps Élastiques , Mémoires de l'Académie Royal des Sciences de l'Institut de France , vol. 8 , pp. 357 – 570 , 623–627 , 1829 .
   Google Scholar
• K. A. Khan and H. H. Hilton , On Inconstant Poisson's Ratios in Non-Homogeneous Elastic Media , Journal of Thermal Stresses , vol. 33 , pp. 1 – 8 , 2010 .
   Google Scholar
• H. H. Hilton , On the Inadmissibility of Separation of Variable Solutions in Linear Anisotropic Viscoelasticity , International Journal of Mechanics of Composite Materials and Structures , vol. 3 , pp. 97 – 100 , 1996 .
   Google Scholar
• H. H. Hilton and S. Yi , The Significance of Anisotropic Viscoelastic Poisson Ratio Stress and Time Dependencies , International Journal of Solids and Structures , vol. 35 , pp. 3081 – 3095 , 1998 .
   Web of Science ®Google Scholar
• H. H. Hilton , Implications and Constraints of Time Independent Poisson Ratios in Linear Isotropic and Anisotropic Viscoelasticity , Journal of Elasticity , vol. 63 , pp. 221 – 251 , 2001 .
   Web of Science ®Google Scholar
• H. H. Hilton , The Elusive and Fickle Viscoelastic Poisson's Ratio and its Relation to the Elastic–Viscoelastic Correspondence Principle , Journal of Mechanics of Materials and Structures , vol. 4 , pp. 1341 – 1364 , 2009 .
   Google Scholar
• H. H. Hilton , Clarifications of Certain Ambiguities and Failings of Poisson's Ratios in Linear Viscoelasticity, accepted for publication , Journal of Elasticity , 2011 .
   Google Scholar
• N. W. Tschoegl , Time Dependence in Material Properties: An Overview , Mechanics of Time-Dependent Materials , vol. 1 , pp. 3 – 31 , 1997 .
   Google Scholar
• N. W. Tschoegl , W. G. Knauss , and I. Emri , Poisson's Ratio in Linear Viscoelasticity – A Critical Review, Mechanics of Time-Dependent Materials , vol. 6, pp. 3–51, 2002.
   Web of Science ®Google Scholar
• R. S. Lakes and A. Wineman , On Poisson's Ratio in Linearly Viscoelastic Solids , Journal of Elasticity , vol. 85 , pp. 45 – 63 , 2006 .
   Web of Science ®Google Scholar
• R. S. Lakes , Foam Structures with a Negative Poisson's Ratio , Science , vol. 235 , pp. 1038 – 1040 , 1987 .
   PubMed Web of Science ®Google Scholar
• R. S. Lakes , The Time-Dependent Poisson's Ratio of Viscoelastic Materials can Increase or Decrease , Cellular Composites , vol. 11 , pp. 466 – 469 , 1992 .
   Google Scholar
• C. P. Chen and R. S. Lakes , Viscoelastic Behaviour of Composite Materials with Conventional- or Negative- Poisson's Ratio Foam as One Phase , Journal of Materials Science , vol. 28 , pp. 4288 – 4299 , 1993 .
   Google Scholar
• R. S. Lakes , http://silver.neep.wisc.edu/ ~lakes/Poisson.html, 2005.
   Google Scholar
• T. M. Jaglinski and R. S. Lakes , Negative Stiffness and Negative Poisson's Ratio in Materials which Undergo a Phase Transformation , in D. Wagg , I. Bond , P. Weaver and M. Friswell ( eds. ), Adaptive Structures – Engineering Applications , pp. 231 – 246 , John Wiley & Sons , New York , 2007 .
   Google Scholar
• K. Ravi-Chandar , Simultaneous Measurement of Nonlinear Bulk and Shear Relaxation Behavior , Proceedings of the 2nd International Conference on Mechanics of Time-Dependent Materials SEM , pp. 30 – 31 , Ljubljana , Slovenia , 1998 .
   Google Scholar
• K. Ravi-Chandar , Inelastic Deformation in Polymers Under Multiaxial Compression , Mechanics of Time-Dependent Materials , vol. 4 , pp. 333 – 357 , 2000 .
   Web of Science ®Google Scholar
• A. Shtark , H. Grosbein , G. Sameach , and H. H. Hilton , An Alternate Protocol for Determining Viscoelastic Material Properties Based on Tensile Tests Without Use of Poisson Ratios , Proceedings of the 2007 International Mechanical Engineering Congress and Exposition. ASME Paper IMECE 2007-41068 , Seattle , WA , 2007 .
   Google Scholar
• A. Shtark , H. Grosbein , and H. H. Hilton , Analytical Determination Without Use of Poisson Ratios of Temperature Dependent Viscoelastic Material Properties Based on Uniaxial Tensile Experiments , Proceedings of the 2009 International Mechanical Engineering Congress and Exposition. ASME Paper IMECE 2009-10332 . Buena Vista , FL , 2009 .
   Google Scholar
• A. Shtark , H. Grosbein , G. Sameach , and H. H. Hilton , An Alternate Protocol for Determining Viscoelastic Material Properties Based on Tensile Tests Without Use of Poisson Ratios, accepted for publication ASME Journal of Applied Mechanics. JAM08-1361, 2010.
   Google Scholar
• H. H. Hilton and A. R. A. El Fouly , Designer Auxetic Viscoelastic Sandwich Column Materials Tailored to Minimize Creep Buckling Failure Probabilities and Prolong Survival Times , Proceedings 48th AIAA/ASME/ASCE/AHS/ASC SDM Conference, AIAA Paper 2007-2400 , 2007 .
   Google Scholar
• W. C. Butlar , E. V. Dave , H. H. Hilton , and G. H. Paulino , A Review of Elastic and Viscoelastic Functionally Graded Materials – Characterization and Analysis, to appear Applied Mechanics Reviews, 2011.
   Google Scholar
• H. H. Hilton , 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 , 1951 .
   Google Scholar
• H. H. Hilton , Thermal Stresses in Thick Walled Cylinders Exhibiting Temperature-Dependent Viscoelastic Properties of the Kelvin Type , Second United States National Congress for Applied Mechanics , pp. 547 – 553 , 1954 .
   Google Scholar
• J. L. Sackman , A Remark on Transient Stresses in Nonhomogeneous Viscoelastic Materials , Journal of the Aerospace Sciences , vol. 29 , pp. 1015 – 1016 , 1962 .
   Google Scholar
• R. Muki and E. Sternberg , On Transient Thermal Stresses in Viscoelastic Materials with Temperature Dependent Properties, ASME Journal of Applied Mechanics , vol. 28, pp. 193–207, 1961.
   Google Scholar
• H. H. Hilton , Designer Linear Viscoelastic Material Properties Tailored to Minimize Probabilistic Failures or Thermal Stress Induced Dynamic Column Creep Buckling , Journal of Thermal Stresses , vol. 29 , pp. 403 – 421 , 2006 .
   Web of Science ®Google Scholar
• H. H. Hilton and H. G. Russell , An Extension of Alfrey's Analogy to Thermal Stress Problems in Temperature Dependent Linear Viscoelastic Media , Journal of Physics and Solids , vol. 9 , pp. 152 – 164 , 1961 .
   Web of Science ®Google Scholar
• H. H. Hilton , and J. R. Clements , Formulation, and Evaluation of Approximate Analogies for Temperature Dependent Linear Viscoelastic Media , Thermal Loading and Creep , vol. 6 , pp. 6 – 17 , 6–24 , Institution of Mechanical Engineers , London , 1964 .
   Google Scholar
• H. H. Hilton , Sensitivity Analysis of Linear and Nonlinear Elastic and Viscoelastic Panel Responses to Aerodynamic Noise , International Journal of Applied Mathematical Sciences , vol. 4 , pp. 3841 – 3867 , 2010 .
   Google Scholar
• H. H. Hilton , and J. J. Piechocki , Shear Center Motion in Beams with Temperature Dependent Linear Elastic or Viscoelastic Properties , Proceedings Fourth United States National Congress of Applied Mechanics , vol. 2 , pp. 1279 – 1289 , ASME , 1962 .
   Google Scholar
• S. P. Timoshenko , On the Correction for Shear of the Differential Equation Transverse Vibration of Prismatic Bars , Philosophical Magazine, Series 6 , vol. 41 , pp. 744 – 746 , 1921 .
   Google Scholar
• S. P. Timoshenko , On the Transverse Vibrations of Beams of Uniform Cross-Section , Philosophical Magazine, Series 6 , vol. 43 , pp. 125 – 131 , 1922 .
   Google Scholar
• S. P. Timoshenko , Vibration Problems in Engineering , pp. 329 – 331 , Van Nostrand , New York , 1928 .
   Google Scholar
• C. W. Bert , Simplified Analysis Static Shear Correction Factors Beams with Non-Homogeneous Cross Section , Journal of Composite Materials , vol. 7 , pp. 525 – 529 , 1973 .
   Web of Science ®Google Scholar
• G. R. Cowper , The Shear Coefficient in Timoshenko's Beam Theory , ASME Journal of Applied Mechanics , vol. 33 , pp. 335 – 340 , 1966 .
   Web of Science ®Google Scholar
• H. H. Hilton , Viscoelastic Timoshenko Beam Theory , Mechanics of Time-Dependent Materials , vol. 13 , pp. 1 – 10 , 2009 .
   Google Scholar
• H. H. Hilton , Creep Collapse of Viscoelastic Columns with Initial Curvatures , Journal of the Aeronautical Sciences , vol. 19 , pp. 844 – 846 , 1952 .
   Google Scholar
• C. Libove , Creep Buckling of Columns , Journal of the Aeronautical Sciences , vol. 19 , pp. 459 – 467 , 1952 .
   Google Scholar
• H. H. Hilton , Designer Linear Viscoelastic Material Properties Tailored to Minimize Probabilistic Failures or Thermal Stress Induced Dynamic Column Creep Buckling , Journal of Thermal Stresses , vol. 29 , pp. 403 – 421 , 2006 .
   Web of Science ®Google Scholar
• H. H. Hilton , Dynamic creep buckling of linear viscoelastic columns including failure probabilities and survival times, in press Mechanics of Time-Dependent Materials, 2011.
   Google Scholar
• H. H. Hilton , The Divergence of Supersonic, Linear Viscoelastic Lifting Surfaces, Including Chordwise Bending , Journal of the Aero/Space Sciences , vol. 27 , pp. 926 – 934 , 1960 .
   Google Scholar
• H. H. Hilton , Pitching Instability of Rigid Lifting Surfaces on Viscoelastic Supports in Subsonic or Supersonic Potential Flow , Advances in Solid Mechanics , pp. 1 – 19 , Edwards Bros. , Ann Arbor , 1957 .
   Google Scholar
• H. H. Hilton and C. F. Vail , Bending-Torsion Flutter of Linear Viscoelastic Wings Including Structural Damping , Proceedings AIAA/ASME/ASCE/AHS/ASC 33rd Structures, Structural Dynamics and Materials Conference, AIAA Paper 93-1475 , vol. 3 , pp. 1461 – 1481 , 1993 .
   Google Scholar
• B. Etkin , Dynamics of Atmospheric Flight , John Wiley & Sons , New York , 1972 .
   Google Scholar
• B. Etkin and L. D. Reid , Dynamics of Flight: Stability and Control , John Wiley & Sons , New York , 1996.
   Google Scholar
• C. E. Beldica and H. H. Hilton , Piezoelectric and Viscoelastic Control of Lifting Surface Aerodynamic Derivatives and Aileron Reversal, Proceedings of the 40th AIAA/ASME/AHS/ASC Structures, Structural Dynamics and Material Conference, 3:1869-1879, AIAA Paper No. 99-1423, 1999.
   Google Scholar
• J. M. C. Duhamel , Second Mémoire sur les Phénomenes Thermomecaniques , Journal de l’École Polytechnique , 1837 .
   Google Scholar
• J. B. J. Fourier , Théorie de la Chaleur . Didot , Paris , 1822 .
   Google Scholar
• C. E. Beldica and H. H. Hilton , Analytical Simulations of Optimum Anisotropic Linear Viscoelastic Damping Properties , Journal of Reinforced Plastics and Composites , vol. 18 , pp. 1658 – 1676 , 1999 .
   Google Scholar
• C. E. Beldica and H. H. Hilton , Analytical and Computational Simulations of Experimental Determinations of Deterministic and Random Linear Viscoelastic Constitutive Relations, accepted for publication , Journal of Sandwich Structures and Materials , 2011 .
   Google Scholar
• N. J. Hoff , The Analysis of Structures , John Wiley & Sons , New York , 1956 .
   Google Scholar
• S. Yi , H. H. Hilton , and M. F. Ahmad , Cure Cycle Simulations of Composites with Temperature and Cure Dependent Anisotropic Viscoelastic Properties and Stochastic Delaminations , Mechanics of Composite Materials and Structures , vol. 5 , pp. 81 – 101 , 1998 .
   Google Scholar
• S. Yi and H. H. Hilton , Effects of Thermo-Mechanical Properties of Composites on Viscosity, Temperature and Degree of Cure in Thick Thermosetting Composite Laminates During Cure Processes , Journal of Composite Materials , vol. 32 , pp. 600 – 622 , 1998 .
   Google Scholar
• S. Yi , H. H. Hilton , and M. F. Ahmad , A Finite Element Approach for Cure Simulation of Thermosetting Matrix Composites , International Journal of Computers and Structures , vol. 64 , pp. 383 – 388 , 1997 .
   Google Scholar
• H. H. Hilton , Optimum Viscoelastic Designer Materials for Minimizing Failure Probabilities During Composite Cure , Journal of Thermal Stresses , vol. 26 , pp. 547 – 557 , 2003 .
   Web of Science ®Google Scholar
• F. Vohwinkel , Cold Curing Epoxy Adhesives with Enhanced Resistance to Heat and Moisture , Adhesion , vol. 34 , pp. 25 – 29 , 1990 .
   Google Scholar
• F. Zhao and F. R. Jones , Thermal Loading of Short Fibre Composites and the Induction of Residual Shear Stresses , Composites: Part A , vol. 38 , pp. 2374 – 2381 , 2007 .
   Google Scholar
• H. H. Hilton , Creep Collapse of Viscoelastic Columns with Initial Curvatures , Journal of the Aeronautical Sciences , vol. 19 , pp. 844 – 846 , 1952 .
   Google Scholar
• G. Gerard , Introduction to Structural Stability Theory . McGraw-Hill , New York , 1962 .
   Google Scholar
• F. R. Steinbacker and G. Gerard , Aircraft Structural Mechanics , Pitman Publishing Co. , New York , 1952 .
   Google Scholar
• F. R. Shanley , and E. I. Ryder , Stress Ratios: The Answer to the Combined Loading Problem , Aviation , vol. 36, pp. 28–29, 43, 66, 69–70 , 1937 .
   Google Scholar
• D. A. Dillard and H. F. Brinson , A Numerical Procedure for Predicting Creep and Delayed Failures in Laminated Composites , in T. K. O'Brien ( ed. ), Long-Term Behavior of Composites, ASTM STP 813 , pp. 23 – 37 , ASTM , Philadelphia , 1983 .
   Google Scholar
• C. Hiel , M. Sumich , and D. P. Chappell . A Curved Beam Test Specimen for Determining the Interlaminar Tensile Strength of a Laminated Composite , Journal of Composite Materials , vol. 25 , pp. 854 s– 868 , 1991 .
   Google Scholar
• J. M. Lifshitz and A. Rotem , Time-Dependent Longitudinal Strength of Unidirectional Fibrous Composites , Fibre Science and Technology , vol. 3 , pp. 1 – 20 , 1970 .
   Google Scholar
• S. L. Phoenix , Statistical Aspects of Failure of Fibrous Materials, in S. W. Tsai ( ed.), Composite Materials: Testing and Design, ASTM STP 674 , pp. 455–483, 1979.
   Google Scholar
• S. L. Phoenix and L.-J. Tierney , A Statistical Model for the Time Dependent Failure of Unidirectional Composite Materials Under Local Elastic Load-Sharing Among Fibers , Engineering Fracture Mechanics , vol. 18 , pp. 193 – 215 , 1982 .
   Google Scholar
• A. S. Watson and R. L. Smith , An Examination of Statistical Theories for Fibrous Materials in the Light of Experimental Data , Journal of Materials Science , vol. 20 , pp. 3260 – 3270 , 1985 .
   Web of Science ®Google Scholar
• H. H. Hilton and S. Yi , Stochastic Viscoelastic Delamination Onset Failure Analysis of Composites , Journal of Composite Materials , vol. 27 , pp. 1097 – 1113 , 1993 .
   Web of Science ®Google Scholar
• H. H. Hilton and S. T. Ariaratnam , Invariant Anisotropic Large Deformation Deterministic and Stochastic Combined Load Failure Criteria , International Journal of Solids and Structures , vol. 31 , pp. 3285 – 3293 , 1994 .
   Google Scholar
• W. Weibull , A Statistical Distribution Function of Wide Applicability , ASME Journal of Applied Mechanics , vol. 18 , pp. 293 – 297 , 1951 .
   Web of Science ®Google Scholar
• E. S. Lindquist , Strength of Materials and the Weibull Distribution , Journal of Probabilistic Engineering Mechanics , vol. 9 , pp. 191 – 194 , 1994 .
   Google Scholar
• D. N. Murthy , M. X. Prabhakar , and R. Jiang , Weibull Models , John Wiley & Sons , Hoboken , 2004 .
   Google Scholar
• V. H. Schmidt , D. Brandt , F. Holloway ., and A. M. Vinogradov , Piezoelectric Polymer Actuator and Material Properties , Proceedings of the Tenth IEEE International Symposium on Applications of Ferroelectrics. , vol. 1 , pp. 377 – 380 , 1996 .
   Google Scholar
• A. M. Vinogradov and F. Holloway , Mechanical Testing and Characterization of PVDF, a Thin Film Piezoelectric Polymer , Journal of Advanced Materials , vol. 29 , pp. 11 – 17 , 1997 .
   Web of Science ®Google Scholar
• A. M. Vinogradov and F. Holloway , Cyclic Creep of Piezoelectric Polymer Polyvinylidene Fluoride , AIAA Journal , vol. 39 , pp. 2227 – 2229 , 1999 .
   Google Scholar
• A. M. Vinogradov and F. Holloway , Electro-Mechanical Properties of the Piezoelectric Polymer PVDF , Ferroelectrics , vol. 226 , pp. 169 – 181 , 1999 .
   Web of Science ®Google Scholar
• A. M. Vinogradov , Nonlinear Characteristics of Piezoelectric Polymers, Proceedings of 2001 ASME International Mechanical Congress and Exposition, IMECE 2001/AD-23736. 2001.
   Google Scholar
• C. M. Merrett and H. H. Hilton , Generalized Linear Aero-Servo-Viscoelasticity: Theory and Applications , Proceedings 49th AIAA/ASME/ASCE/ AHS Structures, Structural Dynamics and Materials Conference, AIAA Paper AIAA-2008-1997 , Schaumburg , IL , 2008 .
   Google Scholar
• H. H. Hilton , J. R. Vinson , and S. Yi , Anisotropic Piezo-Electro-Thermo-Viscoelastic Theory with Applications to Composites , Proceedings of the 11th International Conference on Composite Materials (ICCM-11) , vol. VI , pp. 4881 – 4890 , Australian Composites Society , Gold Coast , Australia , 1997 .
   Google Scholar
• C. E. Beldica , H. H. Hilton , and S. Yi , A Sensitivity Study of Viscoelastic, Structural and Piezoelectric Damping for Flutter Control , Proceedings 39th AIAA/ASME/ ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, AIAA Paper 98-1848 ,, vol. 2 , pp. 1304 – 1314 , 1998 .
   Google Scholar
• S. Yi , S. F. Ling , M. Ying , H. H. Hilton , and J. R. Vinson , Finite Element Formulation for Anisotropic Coupled Piezo-Hygro-Thermo-Viscoelasto-Dynamic Problems , International Journal of Numerical Methods in Engineering , vol. 45 , pp. 1531 – 1546 , 1999 .
   Web of Science ®Google Scholar
• H. H. Hilton , On the Inadmissibility of Separation of Variables Solutions in Linear Anisotropic Viscoelasticity , International Journal of Mechanics of Composite Materials and Structures , vol. 3 , pp. 97 – 100 , 1996 .
   Google Scholar
• C. V. Loan , Computational Frameworks for the Fast Fourier Transform , SIAM , Philadelphia , 1992.
   Google Scholar
• E. Post , Generalized Differentiation , Transactions American Mathematical Society , vol. 32 , pp. 723 – 781 , 1930 .
   Google Scholar
• A. R. Zak , Structural Analysis of Realistic Solid Propellant Materials , Journal of Spacecraft , vol. 5 , pp. 270 – 275 , 1968 .
   Google Scholar
• R. L. Taylor , K. S. Pister , and G. S. Goudreau , Thermomechanical Analysis of Viscoelastic Solids , International Journal of Numerical Methods in Engineering , vol. 2 , pp. 45 – 59 , 1970 .
   Google Scholar
• S. Nagarajan and A. R. Zak , Finite Element Model for Orthotropic Beams , Computers & Structures , vol. 20 , pp. 443 – 449 , 1985 .
   Google Scholar
• S. Yi , Finite Element Analysis of Anisotropic Viscoelastic Composite Structures, and Analytical Determination of Optimum Viscoelastic Material Properties , Ph.D. Thesis , Aerospace Engineering Department, University of Illinois at Urbana-Champaign , USA , 1991 .
   Google Scholar
• S. Yi and H. H. Hilton , Dynamic Finite Element Analysis of Viscoelastic Composite Plates in the Time Domain , International Journal of Numerical Methods in Engineering , vol. 37 , pp. 4081 – 4096 , 1994 .
   Web of Science ®Google Scholar
• M. Kaliske and H. Rothert , Formulation and Implementation of Three-Dimensional Viscoelasticity at Small and Finite Strains , Computational Mechanics , vol. 19 , pp. 228 – 239 , 1997 .
   Web of Science ®Google Scholar
• M. A. Zocher , S. E. Groves , and D. H. Allen , A Three-Dimensional Finite Element Formulation for Thermoviscoelastic Orthotropic Media , International Journal of Numerical Methods in Engineering , vol. 40 , pp. 2267 – 2288 , 1997 .
   Web of Science ®Google Scholar
• S. Yi , M. F. Ahmad , and H. H. Hilton , Finite Element Algorithms for Dynamic Simulations of Viscoelastic Composite Shell Structures Using Conjugate Gradient Method on Coarse Grained Machines , International Journal of Numerical Methods in Engineering , vol. 40 , pp. 1857 – 1875 , 1997 .
   Google Scholar
• J. C. Simo and T. J. R. Hughes , Computational Inelasticity , Springer , New York , 1998 .
   Google Scholar
• A. C. Pipkin , A Course in Integral Equations . Springer-Verlag , Berlin , 1991 .
   Google Scholar
• P. Harrison and M. F. Ahmad , A Material Point Time Integration Procedure for Anisotropic, Thermo-Rheologically Simple, Viscoelastic Solids , Computational Mechanics , vol. 21 , pp. 236 – 242 , 1998 .
   Google Scholar
• A. H. Muliana and K. A. Khan , A Time-Integration Algorithm for Thermo-Rheologically Complex Polymers , Computational Materials Science , vol. 41 , pp. 576 – 588 , 2008 .
   Google Scholar
• H. H. Hilton and S. R. Rishipathak , Computational Issues in Linear and Nonlinear Viscoelastic Integral-Differential Relations for Composite Materials , in A. Johnston , J. Laliberté , L. Petrescue , P. V. Straznicky ( eds. ), Proceedings Fourth Canadian International Composite Conference , CD-ROM 32 , pp. 1 – 16 , Ottawa , Canada , 2003 .
   Google Scholar
• S. Yi and H. H. Hilton , Hygro-Thermal Effects on Viscoelastic Responses of Laminated Composites , Composites Engineering , vol. 5 , pp. 183 – 193 , 1995 .
   Google Scholar
• R. W. Sullivan , M. Razzaghi , and J. Simsiriwong , Numerical Iterative Method for Volterra Equations of the Convolution Type , Mathematical Methods in Applied Sciences , DOI: 10.1002/mma.1340 , 2010 .
   Google Scholar
• H. H. Hilton , An Efficient Protocol for Evaluating Viscoelastic Prony Series Hereditary Integrals , Submitted to Journal of Numerical Methods in Engineering , 2011 .
   Google Scholar
• H. H. Hilton , J. Hsu , and J. S. Kirby , Linear Viscoelastic Analysis with Random Material Properties , Journal of Probabilistic Engineering Mechanics , vol. 6 , pp. 57 – 69 , 1991 .
   Google Scholar
• H. H. Hilton , Dynamic Thermal Stress Creep Buckling, Probabilistic Failures and Survival Times of Viscoelastic Columns with Follower Loads, Proceedings 46th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics and Materials Conference, AIAA Paper 2005-2144 , 2005.
   Google Scholar
• M. Bouleau , Interpretation Probabiliste de la Viscoélasticité Linéaire , Mechanics Research Communications , vol. 19 , pp. 15 – 20 , 1992 .
   Google Scholar
• H. Suemasu , An Analytical Study of Probabilistic Aspects of Strength of Unidirectional Fiber Reinforced Composites Under Tensile Loads , Journal of the Society of Composite Materials , vol. 8 , pp. 29 – 36 , 1982 .
   Google Scholar
• R. W. Sullivan , On the Use of a Spectrum-Based Model for Linear Viscoelastic Materials , Mechanics of Time-Dependent Materials , vol. 10 , pp. 215 – 228 , 2008 .
   Google Scholar
• I. Babu[sbreve]ka , Solution of Interface Problems by Homogenization , SIAM Journal of Mathematical Analysis , vol. 7 , pp. 603 – 634 , 1976 .
   Google Scholar
• I. Babu[sbreve]ka , Homogenization Approach in Engineering , in R. Glowinski and J.-L. Lions ( eds. ), Lecture Notes in Economics and Mathematical Systems, Computing Methods in Applied Sciences and Engineering , Vol. 134 , pp. 137 – 153 , Springer-Verlag , Berlin , 1976 .
   Google Scholar
• G. Bouchité , G. Butazzo , and P. Suquet ( Eds. ), Calculus of Variations, Homogenization and Continuum Mechanics , World Scientific , Singapore , 1994 .
   Google Scholar
• M. Bouzidi and P. Duhamel , Thermal Diffusion in Cyclic Laminated Composites: Spectral Properties and Application to Homogenization , International Journal of Heat and Mass Transfer , vol. 36 , pp. 2715 – 2723 , 1993 .
   Google Scholar
• S. Brahim-Otsmane , G. A. Francfort , and F. Murat , Homogenization in Thermoelasticity , in R. V. Kohn and G. W. Milton ( eds. ), Random Media and Composites , pp. 13 – 45 , SIAM , Philadelphia , 1989 .
   Google Scholar
• P. de Buhan , Homogénéisation en Calcul à la Rupture: le cas du Matériau Composite Multicouche , Comptes Rendus des Séances de l'Académie des Sciences, Série II, Paris , vol. 296 , pp. 933 – 936 , 1983 .
   Google Scholar
• C. S. Chang and J. W. Ju ( Eds. ), Homogenization and Constitutive Modeling for Heterogeneous Materials, AMD-166. ASME , 1993 .
   Google Scholar
• G. A. Francfort , D. Leguillon , and P. M. Suquet , Homogénéisation de Millieux Viscoelastic Linéaires de Kelvin-Voigt , Comptes Rendus des Séances de l'Acadé- mie des Sciences, Série I, Mathematique, Paris , vol. 296 , pp. 287 – 290 , 1983 .
   Google Scholar
• G. A. Francfort , Q. S. Nguyen , and P. M. Suquet , Thermodynamique et les Lois de Compartment Thermomécanique Homogénéisées , Comptes Rendus des Séances de l'Académie des Sciences, Série II, Paris , vol. 296 , pp. 1007 – 1010 , 1983 .
   Google Scholar
• G. A. Francfort , Homogenization and Linear Thermoelasticity , SIAM Journal of Mathematical Analysis , vol. 14 , pp. 696 – 708 , 1984 .
   Google Scholar
• G. A. Francfort and P. M. Suquet , Homogenization and Mechanical Dissipation in Thermoviscoelasticity , Archives for Rational Mechanics and Analysis , vol. 96 , pp. 265 – 296 , 1986 .
   Google Scholar
• G. A. Francfort and F. Murat , Homogenization and Optimal Bounds in Linear Elasticity , Archives for Rational Mechanics and Analysis , vol. 94 , pp. 307 – 334 , 1986 .
   Web of Science ®Google Scholar
• E. Ya. Khruslov , Homogenized Models of Composite Media , in G. dal Maso and G. F. dell’ Antonio ( eds. ), Composite Media and Homogenization Theory , pp. 159 – 182 , Birkhäuser , Boston , 1991 .
   Google Scholar
• M.-L. Macarenhus , A Linear Homogenization Problem with Time Dependent Coefficients , Transactions American Mathematical Society , vol. 281 , pp. 179 – 195 , 1984 .
   Google Scholar
• M.-L. Macarenhus , Homogenization of a Viscoelastic Equation with Non-Periodic Coefficients , Proceedings of the Royal Society of Edinburgh , vol. 106A , pp. 143 – 160 , 1987 .
   Google Scholar
• G. dal Maso and G. F. dell’ Antonio ( Eds.), Composite Media and Homogenization Theory , Birkhäuser , Boston , 1991.
   Google Scholar
• O. A. Oleinik , A. S. Shamaev , and G. A. Yosifian , Mathematical Problems in Elasticity and Homogenization , North-Holland , Amsterdam , 1992 .
   Google Scholar
• E. Sanchez-Palencia and A. Zaoui (Eds .), Homogenization Techniques for Composite Media , Springer-Verlag , Berlin , 1987 .
   Google Scholar
• P. M. Suquet , Elements of Homogenization for Inelastic Solid Mechanics , CISM No. 272 Udine , Springer-Verlag , Berlin , 1985 .
   Google Scholar
• P. M. Suquet , G. Bouchitte , and G. Butttazzo ( Eds. ), Calculus of Variations, Homogenization and Continuum Mechanics , World Scientific , Singapore , 1994 .
   Google Scholar
• L. Tartar , Quelques Remarques sur l'Homogénéisation , in H. Fujita ( ed. ), Functional Analysis and Numerical Analysis , Japan Society for the Promotion of Science , 1976 .
   Google Scholar
• L. Tartar , Estimations de Coefficients Homogénéisés , Computing Methods in Applied Sciences and Engineering, I, Lecture Notes in Mathematics , Vol. 704 , 364 – 373 , Springer-Verlag , Berlin , 1979 .
   Google Scholar
• L. Tartar , Estimations Fines de Coefficients Homogénéisés , in P. Krée ( ed. ), Ennio de Giorgi Colloquium, Research Notes in Mathematics , vol. 125 , Pitman , New York , 1985 .
   Google Scholar
• L. Tartar , Remarks on Homogenization , in J. L. Ericksen , D. Kinderlehrer , R. Kohn and J.-L. Lions ( eds. ), Homogenization and Effective Moduli of Materials and Media , pp. 228 – 246 , Springer-Verlag , New York , 1986 .
   Google Scholar
• L. Tartar , H-Measures, and Small Amplitude Homogenization , in R. V. Kohn , and G. W. Milton ( eds. ), Random Media and Composites , pp. 89 – 99 , SIAM , Philadelphia , 1988 .
   Google Scholar
• A. Toledano and H. Murakami , High-Order Mixture Homogenization of Fiber-Reinforced Composites , Journal of Energy Resources Technology , vol. 113 , pp. 254 – 263 , 1991 .
   Google Scholar
• Q. Yu and J. Fish , Multiscale Asymptotic Homogenization for Multiphysics Problems with Multiple Spatial and Temporal Scales: A Coupled Thermo-Viscoelastic Example Problem , International Journal of Solid and Structures , vol. 39 , pp. 6429 – 6452 , 2002 .
   Web of Science ®Google Scholar
• V. V. Zhikov , S. M. Kozlov , and O. A. Oleinik , Homogenization of Differential Operators and Integral Functionals , Springer-Verlag , Berlin , 1994 .
   Google Scholar
• G. U. Losi and W. G. Knauss , Thermal Stresses in Nonlinearly Viscoelastic Solids , ASME Journal of Applied Mechanics , vol. 59 , pp. S43 – S49 , 1992 .
   Google Scholar
• Y. C. Lou and R. A. Schapery , Viscoelastic Characterization of a Nonlinear Fiber-Reinforced Plastic , Journal of Composite Materials , vol. 5 , pp. 208 – 234 , 1971 .
   Web of Science ®Google Scholar
• S. P. C. Marques and G. J. Creus , Nonlinear Finite Element Analysis of Viscoelastic Composites Materials Under Mechanical and Hygrothermal Loads , in W. R. Blain , W. P. De Wilde and C. A. Brebbia ( eds. ), Computer Aided Design in Composite Material Technology IV , Computational Mechanics Publications , Southampton , 1995 .
   Google Scholar
• D. F. Payne , B. A. Biskup , A. Letton , and T. W. Strganac , On Nonlinear Viscoelastic Analysis Using a Dynamic Mechanical Approach , Proceedings of the 34th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, AIAA Paper 93-1578 , Washington , DC , 1993 .
   Google Scholar
• M. Poitier-Ferry , An Existence and Stability Theorem in Nonlinear Viscoelasticity , in S. Nemat-Nasser ( ed. ), Variational Methods in the Mechanics of Solids , pp. 327 – 331 , Pergamon Press , Elmsford , NY , 1980 .
   Google Scholar
• R. A. Schapery , A Theory of Nonlinear Thermoviscoelasticity Based on Irreversible Thermodynamics , Proceedings of the Fifth U. S. National Congress of Applied Mechanics , pp. 511 – 530 . ASME , NY , 1966 .
   Google Scholar
• R. A. Schapery , On the Characterization of Nonlinear Viscoelastic Materials , Polymer Engineering and Science , vol. 9 , pp. 295 – 310 , 1969 .
   Web of Science ®Google Scholar
• R. A. Schapery , On Nonlinear Viscoelastic Constitutive Equations for Composite Materials, Proceedings VII International Congress of Applied Mechanics , vol. I, pp. 9–21, Society for Experimental Mechanics , 1992.
   Google Scholar
• R. A. Schapery , Nonlinear Viscoelastic Effects in the Compressive Behavior of Fiber Composites , in Y. D. S. Rajapakse ( ed. ), Mechanics of Thick Composites , pp. 81 – 90 , AMD-162, ASME , 1993 .
   Google Scholar
• J. Smart and J. G. Williams , A Comparison of Single Integral Nonlinear Viscoelasticity Theories , Journal of Mechanics and Physics of Solids , vol. 20 , pp. 313 – 324 , 1972 .
   Web of Science ®Google Scholar
• T. W. Strganac , A. Letton , D. F. Payne , and B. A. Biskup , Characterization of Nonlinear Viscoelastic Behavior Using a Dynamic Mechanical Approach , AIAA Journal , vol. 33 , pp. 904 – 910 , 1995 .
   Google Scholar
• S. Yi , H. H. Hilton , and M. F. Ahmad , Nonlinear Thermoviscoelastic Analysis of Interlaminar Stresses in Laminated Composites , Proceedings 35th AIAA/ASME/ ASCE/AHS/ASCT SDM Conference, AIAA Paper 94-1637 , 1994 .
   Google Scholar
• S. Yi , M. F. Ahmad , and H. H. Hilton , Nonlinear Thermoviscoelastic Analysis of Interlaminar Stresses in Laminated Composites , ASME Journal of Applied Mechanics , vol. 63 , pp. 218 – 224 , 1996 .
   Google Scholar
• C. M. Dafermos and J. A. Nohel , A Nonlinear Hyperbolic Volterra Equation in Viscoelasticity , vol. MRC TR 2095 , Mathematics Research Center, University of Wisconsin , Madison , 1980 .
   Google Scholar
• A. C. Pipkin and T. G. Rogers , A Nonlinear Integral Representation for Viscoelastic Behavior , Journal of Mechanics and Physics of Solids , vol. 16 , pp. 59 – 74 , 1968 .
   Web of Science ®Google Scholar