Summary
Non-linear constitutive equations are developed for highly filled polymeric materials. These materials typically exhibit an irreversible stress softening called the “Mullins’ Effect.” The development stems from attempting to mathematically model the failing microstructure of these composite materials in terms of a linear cumulative damage model. It is demonstrated that pth order Lebesgue norms of the deformation history can be used to describe the state of damage in these materials and can also be used in the constitutive equations to characterize their time dependent response to strain distrubances. This method of analysis produces time dependent constitutive equations, yet they need not contain any internal viscosity contributions. This theory is applied to experimental data and shown to yield accurate stress predictions for a variety of strain inputs. Included in the development are analysis methods for proportional stress boundary valued problems for special cases of the non-linear constitutive equation.
The research reported herein was supported in part by the Air Force Office of Scientific Research THEMIS Contract F-44620-68-C0022. The work was performed at the University of Utah, Department of Civil Engineering, and is part of the author’s Ph.D. dissertation.
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References
Farris, R. J., “Homogeneous Constitutive Equations for Materials with Permanent Memory,” Ph.D Thesis, University of Utah, Department of Civil Engineering (June 1970).
Farris, R. J., “Applications of Viscoelasticity to Filled Materials,” Master’s Thesis, University of Utah, Department of Civil Engineering (June 1969).
Mullins, L. J., “Effect of Stretching on the Properties of Rubber,” J. Rubber Res., 16, 275–289, (1947).
Mullins, L. J., “Permanent Set in Vulcanized Rubber,” Ind. Rubber World, 63–69 (1949).
Mullins, L. J., “Studies in the Absorption of Energy by Rubber,” J. Rubber Res., 16, 180–185 (1947).
Oberth, A. E., “Principle of Strength Reinforcement in Filled Polymers,” Rubber Chem. Tech., 40, 1337–1362 (1967).
Bueche, F., “Molecular Basis for the Mullins Effect,” J. Appl. Polymer Sci., 4, 107–114 (1960).
Bueche, F., “Mullins Effect and Rubber-Filler Interaction,” J. Appl. Polymer Sci., 5, 271–281 (1961).
Farris, R. J., “Dilatation of Granular Filled Elastomers Under High Rates of Strain,” J. Appl. Polymer Sci., 8, 25–25 (1964).
Farris, R. J., “The Character of the Stress-Strain Function for Solid Propellants,” Trans. Soc. Rheol., 12, 281–301 (1968).
Farris, R. J., “The Influence of Vacuole Formation on the Response and Failure of Highly Filled Polymers,” Trans. Soc. Rheo., 12, 315–334 (1968).
Williams, M. L., Blatz, P. J., and Schapery, R. A., “Fundamental Studies Relations to Systems Analysis of Solid Propellants,” GALCIT SM 61-S, California Institute of Technology, Pasadena, California (Feb 1961). A.so published in Interagency Chemical Rocket Propulsion Group, Solid Propellant Mechanical Behavior Manual, Chemical Propulsion Information Agency Pub. No. 21, Section 2. 3 (1963).
Royden, H. L., Real Analysis, III, 2nd Edition, The Macmillan Company, New York (1968).
Coleman, B. D., and Noll, W., “An Approximation Theorem for Functionals with Applications in Continuum Mechanics,” Arch. Rat. Mech. Anal., 6, 355–370 (1960).
Timoshenko, S., and Goodier, J. N., Theory of Elasticity, McGraw-Hill Book Co., New York (1951).
Fung, Y. C., Foundations of Solid Mechanics, Prentice-Hall Inc., Englewood Cliffs, New Jersey (1965).
Sokolnikoff, I. S., Mathematical Theory of Elasticity, McGraw-Hill Book Co., New York (1956).
Eringen, A. C., Mechanics of Continua, John Wiley and Sons, Inc., New York (1967).
Eringen, A. C., Non-Linear Theory of Continuous Media, McGraw-Hill Book Co., New York (1962).
Malvern, L. E., Introduction to the Mechanics of a Continuous Media, Prentice-Hall, Inc., New Jersey (1969).
Volterra, V., Theory of Functionals and of Integral and Integro-Differential Equations, Dover Publications, Inc., New York (1959).
Green, A. E., and Rivlin, R. S., “The Mechanics of Non-Linear Materials with Memory, Part One,” Arch. Rat. Mech. Anal., 1, 1–21 (1959).
Green, A. E., Rivlin, R. S., and Spencer, A. J. M., “The Mechanics of Non-Linear Materials with Memory, Part Two,” Arch. Rat. Mech. Anal., 3, 82–90 (1959).
Green, A. E., and Rivlin, R. S., “The Mechanics of Non-Linear Materials with Memory, Part Three,” Arch. Rat. Mech. Anal., 4, 387–404 (1959).
Rivlin, R. S., “Non-Linear Viscoelastic Solids,” SIAM Review, 7, 323–340 (1965).
Pipkin, A. C., and Rivlin, R. S., “Small Deformations Superposed on Large Deformations in Materials with Fading Memory,” Arch. Rat. Mech. Anal., 8, 297–308 (1961).
Freda, E.,,“Il Teorema di Eulers per le Funzioni di Linea Omogenee,” R. Acc. du LincEi, Rend., XXIV, 5, (1915).
Wylie, C. R., Jr., Advanced Engineering Mathematics, McGraw-Hill Book Co., New York (1960).
Love, A. E. H., The Mathematical Theory of Elasticity, 4th Ed., Dover Publications, Inc., New York (1944).
Swanson, S. R., “Development of Constiutive Equations for Rocks,” Ph.D. Dissertation, Dept. of Mechanical Eng., Univ. of Utah (1969).
Noll, W., “Mathematical Theory of the Mechanical Behavior of Continuous Media,” Arch. Rat. Mech. Anal., 2, 197–226, (1958).
Pipkin, A. C., “Small Finite Deformations of Viscoelastic Solids,” Rev. Mod. Phys., 36, 1034–1041 (1964).
Pipkin, A. C., and Rogers, T. G., “A Non-Linear Integral Representation for Viscoelastic Behavior,” J. Mech. Phys. Solids, 16, 59–72, (1968).
Fréchet, M., “Sur Les Fonctionnelles Continues,” Ann. de L’Ecole Normale Sup., 27, 3rd Series (1910).
Herrmann, L. R., “On a General Theory of Viscoelasticity,” J. Franklin Inst., 280, 244–255 (1965).
Onaran, K., and Findley, W. N., “Combined Stress-Creep Experiments on a Non-Linear Viscoelastic Material to Determine the Kernel Functions for a Multiple Integral Representation of Creep,” Trans. Soc. Rheol. 1, 299–327, (1965).
Williams, M. L., “Structural Analysis of Viscoelastic Materials,” AIAA J., 2, 785–808 (1964).
Tobolsky, A. V., Properties and Structure of Polymers, 160, John Wiley and Sons, Inc., New York (1960).
Schapery, R. A., “A Theory of Non-Linear Thermoviscoelasticity Based on Irreversible Thermodynamics,” Proc. Fifth U.S. Nat. Cong. of Appt. Mech., ASME, 511–530 (1966).
Schapery, R. A., “On the Characterization of Non-Linear Viscoelastic Materials,” Polymer Eng. Sci., 9, 295–310 (1969).
McGuirt, C. W., and Lianis, G., “Experimental Investigation of Non-Linear Non-Isothermal Viscoelasticity,” Int. J. Eng. Sci., 7, 579–599 (1969).
Frudenthal, A. M., “Strain Sensitive Response of Filled Elastomers,” Technical Report No. 24, Department of Civil Engineering and Engineering Mechanics, Columbia Univ., etc.
Flory, P. J., Principles of Polymer Chemistry, Cornell University Press, Ithaca, New York (1953).
Treloar, L. R. G., The Physics of Rubber Elasticity, Oxford, Clarendon Press, (1949).
Graham, P. H., and Robinson, C. N., “Analysis of Cumulative Damage in Solid Rocket Propellants by Application of Reaction Rate Methods to the Binder-Filler Separation Process,” Minutes of the Fourth Cumulative Damage Technical Coordination Meeting, North American Rockwell Corporation, McGregor, Texas (May 1968).
Bills, K. W., Jr., Svob, G. J., Planck, R. W., and Erickson, T. L., “A Cumulative-Damage Concept for Propellant-Liner Bonds in Solid Rocket Motors,” J. of Spacecraft, 3, 408–412, (1966).
Fitzgerald, J. E., “Thermomechanical Coupling in Viscoelastic Materials,” Presentation at the International Conference on Structure, Solid Mechanics and Engineering Design in Civil Engineering Materials, Southampton, England (1969), Discussion to be published by John Wiley and Sons, Inc., New York.
Coleman, B. D., and Mizel, V. J., “Norms and Semi-Groups in the Field of Fading Memory,” Arch. Rat. Mech. Anal., 23, 87–123 (1966).
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Farris, R.J. (1971). The Stress-Strain Behavior of Mechanically Degradable Polymers. In: Chompff, A.J., Newman, S. (eds) Polymer Networks. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-6210-5_17
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DOI: https://doi.org/10.1007/978-1-4757-6210-5_17
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