Abstract
Initial applications of elastic–plastic and other inelastic constitutive relations in predicting overall response of heterogeneous materials had focused on polycrystalline metals, modeled as a multiphase system of randomly orientated single crystal grains which were assigned certain yield conditions and slip mechanisms. Early work includes the slip theory of Batdorf and Budiansky (1949), the rigid-plastic single crystal system of Bishop and Hill (1951), the elastic–plastic K.B.W. model of Kröner (1961) and the self-consistent approximation by Hershey (1954) and by Budiansky and Wu (1962). Further developed by Hill (1965c, 1967) and implemented by Hutchinson (1970), the SCM approximation extended the elasticity form of the method to polycrystals and two-phase composites. That and numerous other extensions of elastic micromechanical methods to inelastic systems provide an interface with the latter. However, they often assume uniform elastic and inelastic deformation in each grain, or in the entire matrix of a particulate or fibrous composite, according to a specified constitutive relation. Since local deformation is not uniform, the overall response predicted by such theories is not supported by experiments, as shown in Sect. 12.2.2. Nonuniform local deformation was examined on composite cylinders under axisymmetric and thermal loads, and in shakedown state, by Dvorak and Rao (1976a, b), Tarn, et al. (1975). General loading effects were investigated with models which constrained only longitudinal deformation by elastic fibers (Dvorak and Bahei-El-Din 1979, 1980, 1982). More recent work, supported by numerical methods, has focused on realistic aspects of deformation mechanisms of polycrystals and composites, as reviewed by Dawson, Hutchinson, Torquato and others in a report on research trends in solid mechanics (Dvorak 1999).
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References
Aboudi, J. (1991). Mechanics of composite materials – A unified micromechanical approach. Amsterdam: Elsevier.
Accorsi, M. L., & Nemat-Nasser, S. (1986). Bounds on the overall elastic and instantaneous elastoplastic moduli of periodic composites. Mechanics of Materials, 5, 209–220.
Bahei-El-Din, Y. A. (1992). Uniform fields, yielding and thermal hardening in fibrous composite laminates. International Journal of Plasticity, 8, 867–892.
Bahei-El-Din, Y. A. (1996). Finite element analysis of viscoplastic composite materials and structures. Mechanics of Composite Materials and Structures, 3, 1–28.
Bahei-El-Din, Y. A. (2009). Modeling electromechanical coupling in woven composites exhibiting damage. Journal of Aerospace Engineering, Proceedings of the Institution of Mechanical Engineering, 223(Part G), 485–495.
Bahei-El-Din, Y. A., & Dvorak, G. J. (2000). Micromechanics of inelastic composite materials. In A. Kelly & C. Zweben (Eds.), Comprehensive composite materials. In T.-W. Chou (Eds.), I: Fiber Reinforcements and General Theory of Composites, Ch. 1.14. Amsterdam: Elsevier Science B. V., pp. 403–430.
Bahei-El-Din, Y. A., Dvorak, G. J., & Wu, J. F. (1989). Fracture of fibrous metal matrix composites – II. Modeling and numerical analysis. Engineering Fracture Mechanics, 34, 105–123.
Bahei-El-Din, Y. A., Khire, R., & Hajela, P. (2010). Multiscale transformation field analysis of progressive damage in fibrous laminates. International Journal of Multiscale Computational Engineering, 8, 69–80.
Batdorf, S. B., & Budiansky, B. (1949). A matrhmatical theory of plasticity based on the concept of slip (Techical Note 1871). Washington, DC: National Advisory Committee for Aeronautics.
Baweja, S., Dvorak, G. J., & Bazant, Z. P. (1998). Composite model for basic creep of concrete. Journal of Engineering Mechanics, 124, 959–965.
Benveniste, Y. (1987a). A new approach to the application of Mori-Tanaka theory in composite materials. Mechanics of Materials, 6, 147–157.
Benveniste, Y. (1987b). A differential effective medium theory with a composite sphere embedding. ASME Journal of Applied Mechanics, 54, 466–468.
Bishop, J. F. W., & Hill, R. (1951). A theory of the plastic distortion of a polycrystalline aggregate under combined stress. Philosophical Magazine, 42, 414–427.
Brockenbrough, J. R., Suresh, S., & Wienecke, H. A. (1991). Deformation of fiber-reinforced metal-matrix composites: Geometrical effects of fiber shape and distribution. Acta Metallurgica et Materialia, 39, 735–752.
Budiansky, B., & Wu, T. T. (1962). Theoretical prediction of plastic strains of polycrystals. In Proceedings of the Fourth U. S. National Congress of Applied Mechanics (pp. 1175–1185). New York: ASME.
Chaboche, J.-L. (1989). Constitutive equations for cyclic plasticity and visco-plasticity. International Journal of Plasticity, 5, 274–302.
Chaboche, J. L., Kruch, S., & Pottier, T. (1998). Micromechanics versus macromechanics: A combined approach for the metal matrix composites constitutive modelling. European Journal of Mechanics – A/Solids, 17, 885–908.
Chaboche, J. L., Kruch, S., Maire, J. F., & Pottier, T. (2001). Towards a micromechanics based inelastic and damage modeling of composites. International Journal of Plasticity, 17, 411–439.
Chaboche, J. L., Kanoute, P., & Roos, A. (2005). On the capabilities of mean-field approaches for the description of plasticity in metal matrix composites. International Journal of Plasticity, 21, 1409–1434.
Christensen, R. M. (1969). Viscoelastic properties of heterogeneous media. Journal of the Mechanics and Physics of Solids, 17, 23.
Christensen, R. M. (1998). Two theoretical elasticity micromechanics models. Journal of Elasticity, 50, 15–25.
Christensen, R. M. (2003). Mechanics of cellular and other low density materials. International Journal of Solids and Structures, 37, 93–104.
deBotton, G., & Ponte Castañeda, P. (1993). Elastoplastic constitutive relations for fiber-reinforced solids. International Journal of Solids and Structures, 30, 1865–1890.
Dvorak, G. J. (1990). On uniform fields in heterogeneous media. Proceedings of the Royal Society of London A, 431, 89–110.
Dvorak, G. J. (1992). Transformation field analysis of inelastic composite materials. Proceedings of the Royal Society London, A437, 311–327.
Dvorak, G. J. (Ed.) (1999) Research trends in solid mechanics, a Report from the U.S. National Committee on Theoretical and Applied Mechanics. Oxford: Elsevier Science Ltd. Also in International Journal of Solids and Structures 37(1&2) (2000).
Dvorak, G. J., & Bahei-El-Din, Y. A. (1987). A bimodal plasticity of theory of fibrous composite materials. Acta Mechanica, 69, 219–241.
Dvorak, G. J., & Teply, J. (1985). Periodic hexagonal array models for plasticity analysis of composite materials. In A. Sawczuk & V. Bianchi (Eds.), Plasticity today: Modeling, methods and applications (W. Olszak memorial volume, pp. 623–642). Amsterdam: Elsevier Scientific Publishing Company.
Dvorak, G. J., Rao, M. S. M., & Tarn, J. Q. (1974). Generalized yield surfaces for unidirectional composites. Journal of Applied Mechanics, 41, 249–253.
Dvorak, G. J., Bahei-El-Din, Y. A., Macheret, Y., & Liu, C. H. (1988). An experimental study of elastic-plastic behavior of fibrous boron-aluminum composites. Journal of the Mechanics and Physics of Solids, 36, 665–687.
Dvorak, G. J., Bahei-El-Din, Y. A., Shah, R., & Nigam, H. (1991). Experiments and modeling in plasticity of fibrous composites. In G. J. Dvorak (Ed.), Inelastic deformation of composite materials (pp. 270–293). New York: Springer.
Dvorak, G. J., Bahei-El-Din, Y. A., & Wafa, A. M. (1994). Implementation of the transformation field analysis for inelastic composite materials. Computational Mechanics, 14, 201–228.
Farez, N., & Dvorak, G. J. (1989). Large elastic-plastic deformations of fibrous metal matrix composites. Journal of the Mechanics and Physics of Solids, 39, 725–744.
Farez, N., & Dvorak, G. J. (1993). Finite deformation constitutive relations for elastic-plastic fibrous metal matrix composites. Journal of Applied Mechanics, 60, 619–625.
Findley, W. N., Lai, J. S., & Onaran, K. (1976). Creep and relaxation of nonlinear viscoelastic materials. Amsterdam: North Holland Publishing Co.
Fish, J., & Shek, K. L. (1999). Finite deformation plasticity of composite structures: Computational models and adaptive strategies. Computer Methods in Applied Mechanics and Engineering, 172, 145–174.
Fish, J., Shek, K. L., Shephard, M. S., & Pandheeradi, M. (1997). Computational plasticity for composite structures based on mathematical homogenization: Theory and practice. Computer Methods in Applied Mechanics and Engineering, 157, 69–94.
Fish, J., Yu, Q., & Shek, K. L. (1999). Computational damage mechanics for composite materials based on mathematical homogenization. International Journal for Numerical Methods in Engineering, 45, 1657–1679.
Freed, A. D., & Walker, K. P. (1993). Viscoplasticity with creep and plasticity bounds. International Journal of Plasticity, 9, 213–242.
Gavazzi, A. C., & Lagoudas, D. C. (1990). On the numerical evaluation of Eshelby’s tensor and its application to elastoplastic fibrous composites. Computational Mechanics, 7, 13–19.
Ghahremani, F. (1977). Numerical evaluation of the stresses and strains in ellipsoidalinclusions in an anisotropic elastic material. Mechanics Research Communications, 4, 89–91.
Hashin, Z. (1970). Complex moduli of viscoelastic composites I. General theory and applications to particulate composites. International Journal of Solids and Structures, 6, 539–552.
Hershey, A. V. (1954). The elasticity of an isotropic aggregate of anisotropic cubic crystals. ASME Journal of Applied Mechanics, 21, 236–240.
Hill, R. (1948). A theory of the yielding and plastic flow of anisotropic metals. Proceedings of the Royal Society London, A193, 281–297.
Hill, R. (1965c). A self-consistent mechanics of composite materials. Journal of the Mechanics and Physics of Solids, 13, 213–222.
Hill, R. (1967). The essential structure of constitutive laws for metal composites and polycrystals. Journal of the Mechanics and Physics of Solids, 15, 79–95.
Hutchinson, J. W. (1970). Elastic-plastic behaviour of polyrystalline metals and composites. Proceedings of the Royal Society London, A319, 247–272.
Kanoute, P., Boso, D. P., Chaboche, J. L., & Schrefler, B. A. (2009). Multiscale methods for composites: A review. Archives of Computational Methods in Engineering, 16, 31–75.
Kattan, P., & Voyiadjis, G. (1993). Overall damage and elastoplastic deformation in fibrous metal matrix composites. International Journal of Plasticity, 9, 931–949.
Knauss, W. G., & Emri, I. J. (1981). Non-linear viscoelasticity based on free volume consideration. Computers and Structures, 13, 123–128.
Krempl, E. (2000). Visoplastic models for high temperature applications. International Journal of Solids and Structures, 37, 279–291.
Lissenden, C. J. (2010). Experimental investigation of initial and subsequent yield surfaces for laminated metal matrix composites. International Journal of Plasticity, 26, 1606–1628.
Michel, J. C., & Suquet, P. (2003). Nonuniform transformation field analysis. International Journal of Solids and Structures, 40, 6937–6955.
Miller, M. F., Christian, J. L., & Wennhold, W. F. (1973). Design, manufacture, development, test and evaluation of boron/aluminum structural components for Space Shuttle. General Dynamics/Convair Aerospace (Contract No. NAS 8-27738).
Moulinec, H., & Suquet, P. (1994). A fast numerical method for computing the linear and nonlinear mechanical properties of composites. Comptes Rendus de I’Academie des Sciences Paris, 318(Ser. II), 1417–1423.
Mulhern, J. F., Rogers, T. G., & Spencer, A. J. M. (1967). A continuum model for fibre-reinforced plastic materials. Proceedings of the Royal Society London, A301, 473–492.
Mulhern, J. F., Rogers, T. G., & Spencer, A. J. M. (1969). A continuum theory of a plastic-elastic fibre-reinforced material. International Journal of Engineering Science, 7, 129–152.
Nemat-Nasser, S. (1992). Phenomenological theories of elastoplasticity and strain localization at high strain rates. Applied Mechanics Reviews, 45, 519–545.
Nemat-Nasser, S. (2004). Plasticity: A treatise on finite deformation of heterogeneous inelastic materials. Cambridge: Cambridge University Press.
Nemat-Nasser, S., & Hori, M. (1999). Micromechanics: Overall properties of hetero-geneous materials (2nd ed.). Amsterdam: Elsevier.
Nigam, H., Dvorak, G. J., & Bahei-El-Din, Y. A. (1994). An experimental investigation of elastic-plastic behavior of a fibrous boron-aluminum composite: I. Matrix-dominated mode. II. Fiber dominated mode. International Journal of Plasticity, 10, 23–62.
Oskay, C., & Fish, J. (2007). Eigendeformation-based reduced order homogenization for failure analysis of heterogeneous materials. Computer Methods in Applied Mechanics and Engineering, 196, 1216–1243.
Phillips, A., & Weng, G. J. (1975). An analytical study of an experimentally verified hardening law. Journal of Applied Mechanics, 42, 375–378.
Ponte Castaneda, P. (1991). The effective mechanical properties of nonlinear isotropic composites. Journal of the Mechanics and Physics of Solids, 39, 45–71.
Ponte Castaneda, P. (1996). A second-order theory for nonlinear composite materials. Computes Rendus de I’Academie des Sciences Paris, 322(Série II b), 3–10.
Ponte Castaneda, P., & Suquet, P. (1998). Nonlinear composites. Advances in Applied Mechanics, 34, 171–302.
Ponte Castaneda, P., & Willis, J. R. (1995). The effect of spatial distribution on the effective behavior of composite materials and cracked media. Journal of the Mechanics and Physics of Solids, 43, 1919–1951.
Sacco, E. (2009). A nonlinear homogenization procedure for periodic masonry. European Journal of Mechanics A/Solids, 28, 2090–2222.
Schapery, R. A. (1997). Nonlinear viscoelastic and viscoplastic constitutive equations based on thermodynamics. Mechanics of Time-Dependent Materials, 1, 209–240.
Smith, G. E., & Spencer, A. J. M. (1970). A continuum theory of a plastic-rigid solid reinforced by two families of inextensible fibres. Quarterly Journal of Mechanics and Applied Mathematics, 23, 489–504.
Spencer, A. J. M. (1972). Deformation of fibre-reinforced materials. London: Oxford University Press.
Spencer, A. J. M. (1987). Kinematic constraints, constitutive equations and failure rules for anisotropic materials. In J. P. Boehler (Ed.), Chapter 10 of Applications of tensor functions in continuum mechanics, CISM Courses and Lectures (No. 292, pp. 187–201). Wien: Springer.
Spencer, A. J. M. (1992). Plasticity theory for fibre-reinforced composites. Journal of Engineering Mathematics, 26, 107–118.
Suquet, P. (1987). Elements of homogenization for inelastic solid mechanics. In E. Sanchez-Palencia & A. Zaoui (Eds.), Homogenization techniques for composite media. New York: Springer.
Suquet, P. (1997). Effective properties of nonlinear composites. In P. Suquet (Ed.), Continuum micromechanics (Vol. 337 of CISM Lecture Notes, pp. 197–264). New York: Springer.
Talbot, D. R. S., & Willis, J. R. (1985). Variational principles for inhomogeneous nonlinear media. Journal of Applied Mathematics, 35, 39–54.
Talbot, D. R. S., & Willis, J. R. (1992). Some simple explicit bounds for the overall behaviour of nonlinear composites. International Journal of Solids and Structures, 29, 1981–1987.
Talbot, D. R. S., & Willis, J. R. (1997). Bounds of third order for the overall response of nonlinear composites. Journal of the Mechanics and Physics of Solids, 45, 87–111.
Teply, J. L., & Dvorak, G. J. (1987). Dual estimates of instantaneous properties of elastic-plastic composites. In A. J. M. Spencer (Ed.), Continuum models of discrete systems (pp. 205–2l6). Rotterdam: A. A. Balkema Press.
Teply, J. L., & Dvorak, G. J. (1988). Bounds on overall instantaneous properties of elastic-plastic composites. Journal of the Mechanics and Physics of Solids, 36, 29–58.
Teply, J. L., & Reddy, J. N. (1990). A unified formulation of micromechanics models of fiber-reinforced composites. In G. J. Dvorak (Ed.), Inelastic deformation of composite materials (pp. 341–372). New York: Springer.
Teply, J. L., Reddy, J. N., & Brockenbrough, J. R. (1992). A unified formulation of micromechanics models of fiber-reinforced composites. In J. N. Reddy & A. V. Krishna Murty (Eds.), Composite structures (pp. 294–325). New Delhi: Narosa Publication House.
Walker, K. P., Jordan, E. H., & Freed, A. D. (1990). Equivalence of Green’s function and the Fourier series representation of composites with periodic structure. In G. J. Weng, M. Taya, & H. Abé (Eds.), Micromechanics and inhomogeneity, The T. Mura 65-th anniversary volume (pp. 535–558). New York: Springer.
Willis, J. R. (1991). On methods for bounding the overall properties of nonlinear composites. Journal of the Mechanics and Physics of Solids, 39, 73–86. Errata ibid. 40, (1992) 441–445.
Green, A. E., & Atkins, J. E. (1960). Large elastic deformations and non-linear continuum mechanics. Oxford: Clarendon Press.
Franciosi, P., & Berberinni, S. (2007). Heterogeneous crystal and poly-crystal plasticity modeling from a transformation field analysis with a regularized Schmid law. Journal of the Mechanics and Physics of Solids, 55, 2265–2299.
Michel, J. C., & Suquet, P.(2004). Computational analysis of nonlinear composite structures using the nonuniform transformation field analysis. Computer Methods in Applied Mechanics and Engineering, 193, 5477–5502.
Prochazka, P. (1997). Slope optimization using transformation field analysis. Engineering Analysis with Boundary Elements, 20, 179–184.
Wu, J. F., Shephard, M. S., Dvorak, G. J., & Bahei-El-Din, Y. A. (1989). A material Model for the finite element analysis of metal matrix composites. Composites Science and Technology, 35, 347–366.
Dvorak, G. J., & Benveniste, Y. (1997). On micromechanics of inelastic and piezoelectric composites. In T. Tatsumi, E. Watanabe, & T. Kambe (Eds), Theoretical and Applied Mechanics 1996 (pp. 217 – 237 ). Elsevier Science B.V.
Dvorak, G. J (2001). Damage evolution and prevention in composite materials. In H. Aref & J. W. Phillips (Eds), Mechanics for the New Millenium Proceedings of ICTAM 2000, the 20th International Congress of Theoretical and Applied Mechanics (pp. 197–210). Kluver Academic Publishers.
Michel, J. C., Moulinec, H., & Suquet, P. (1999). Effective properties of composite materials with periodic microstructure:a computational approach. Computer Methods in Applied Mechanics and Engineering, 172, 109–143.
Dvorak, G. J., Rao, M.S.M., & Tarn, J. Q. (1973). Yielding in unidirectional composites under external loads and temperature changes. Journal of Composite Materials, 7, l94–216.
Dvorak, G. J. (1997). Thermomechanics of heterogeneous media. Journal of Thermal Stresses, 20, 799–817.
Dvorak, G. J., & Johnson, W. S. (1980). Fatigue of metal matrix composites. International Journal of Fracture, 16, 585–607.
Tarn, J. Q., Dvorak, G, J., & Rao, M.S.M. (1975). Shakedown of unidirectional composites, Intl. J. Solids Structures, 6, 75l–764.
Dvorak, G. J., Lagoudas. D. C., & Huang, C-M. (2000). Shakedown and fatigue damage in metal matrix composites. In D. Weichert & G. Maier (Eds), Inelastic Analysis of Structures under Variable Repeated Loads (pp. 193 – 196). Kluver Academic Publishers.
Brinson, H. F., & Brinson, L. C. (2008). Polymer Engineering Science and Viscoelasticity: An Introduction. Springer Science, New York.
Krempl, E. (1985). Inelastic work and thermomechanical coupling in viscoplasticity. In A. Sawczuk and V. Bianchi (Eds), Plasticity Today: Modeling Methods and Applications. Elsevier Scientific Publishing Company, Amsterdam, The Netherlands.
Dvorak, G, J., & Rao, M.S.M. (1976a). Axisymmetric plasticity theory of fibrous composites. International Journal of Engineering Science, 14, 36l–373.
Dvorak, G, J., & Rao, M.S.M. (1976b). Thermal stresses in heat-treated fibrous composites. ASME Journal of Applied Mechanics. 43, 6l9–624.
Maier, G. (1969) Shakedown theory in perfect elastoplasticity with associated and nonassociated flow laws: A finite element linear programing approach. Meccanica. 4, 250–260.
Maier, G. (1973) A shakedown matrix theory allowing for workhardening and second order geometric effects. In Foundations of Plasticity, Vol. 1, Sawczuck A. (ed). Noordoff: Leyden, 417–433.
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Dvorak, G.J. (2013). Inelastic Composite Materials. In: Micromechanics of Composite Materials. Solid Mechanics and Its Applications, vol 186. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4101-0_12
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