On the effective viscoelastic moduli of two-phase media. III. Rigorous bounds on the complex shear modulus in two dimensions

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 455, Issue 1986, 1999, Pages 2117-2149

  Gibiansky, Leonid V ^a,d   Milton, Graeme W ^b   Berryman, James G ^c  

^a Princeton University   (United States)

^b UNIVERSITY OF UTAH   (United States)

^c LAWRENCE LIVERMORE NATIONAL LABORATORY   (United States)

^d NONE   (United States)

Author keywords

Hashin shtrikman bounds; Homogenization; Shear modulus; Translation bounds; Two component composite material; Viscoelastic constants

Indexed keywords

EID: 13544264960     PISSN: 13645021     EISSN: None     Source Type: Journal    
DOI: 10.1098/rspa.1999.0395     Document Type: Article

References (28)

1. Brinson, L. C. & Lin, W. S. 1998 Comparison of micromechanics methods for effective properties of multiphase viscoelastic composites. Composite Struct. 41, 353-367.
2. Cherkaev, A. V. &: Gibiansky, L. V. 1992 The exact coupled bounds for effective tensors of electrical and magnetic properties of two-component two-dimensional composites. Proc. R. Soc. Edinb. A122, 93-125.
3. Cherkaev, A. V. & Gibiansky, L. V. 1993 Coupled estimates for the bulk and shear moduli of a two-dimensional isotropic elastic composite. J. Mech. Phys. Solids 41, 937-980.
4. Cherkaev, A. V. & Gibiansky, L. V. 1994 Variational principles for complex conductivity, viscoelasticity, and similar problems in media with complex moduli. J. Math. Phys. 35, 1-19.
5. Cherkaev, A. V. & Kohn, R. V. (eds) 1997 Topics in the mathematical modelling of composite materials. Progress in Nonlinear Differential Equations and their Applications, vol. 31. Berlin: Birkhäuser.
6. Cherkaev, A. V., Lurie, K. A. & Milton, G. W. 1992 Invariant properties of the stress in plane elasticity and equivalence classes of composites. Proc. R. Soc. Land. A 438, 519-529.
7. Christensen, R. M. 1969 Viscoelastic properties of heterogeneous media. J. Mech. Phys. Solids 17, 23-41.
8. Francfort, G. A. & Murât, F. 1986 Homogenization and optimal bounds in linear elasticity. Arch. Ration. Mech. Analysis 94, 307-334.
9. Gibiansky, L. V. & Lakes, R. 1997 Bounds on the complex bulk and shear moduli of a twodimensional two-phase viscoelastic composite. Mech. Mater. 25, 79-95.
10. Gibiansky, L. V. & Milton, G. W. 1993 On the effective viscoelastic moduli of two-phase media. I. Rigorous bounds on the complex bulk modulus. Proc. R. Soc. Land. A 440, 163-188.
11. Gibiansky, L. V. & Torquato, S. 1998 New method to generate three-point bounds on effective properties of composites: application to viscoelasticity. J. Mech. Phys. Solids 46, 749-783.
12. Gibiansky, L. V., Lakes, R. & Milton, G. W. 1993 Viscoelastic composites with extremal prop-erties. In Structural Optimization 93, Proc. World Congress on Optimal Design of Structural Systems, Rio de Janeiro, Brazil, 2-6 August 1993, vol. I (ed. J. Herskovits), pp. 369-376.
13. Hashin, Z. 1965 On elastic behaviour of fibre reinforced materials of arbitrary transverse phase geometry. J. Mech. Phys. Solids 13, 119-134.
14. Hashin, Z. & Shtrikman, S. 1963 A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids 11, 127-140.
15. Li, J. & Weng, G. J. 1994 Strain-rate sensitivity, relaxation behavior, and complex moduli of a class of isotropic viscoelastic composites. ASME J. Engng Mater. Technol. 116, 495-504.
16. Li, J. & Weng, G. J. 1997 Influence of inclusion microgeometry on some thermomechanical properties of isotropic polymer-matrix composites. ASME J. Engng Mater. Technol. 119, 242-250.
17. Lurie, K. A. & Cherkaev, A. V. 1984a G-closure of some particular sets of admissible material characteristics for the problem of bending of thin plates. J. Opt. Theor. Appl. 42, 305-316.
18. Lurie, K. A. & Cherkaev, A. V. 19846 The problem of formation of optimal isotropic multiphase composite. Preprint 895, loffe Physico-Technical Institute (Leningrad).
19. Milton, G. W. 1986o Modelling the properties of composites by laminates. In Homogenization and effective moduli of materials and media (ed. J. Ericksen, D. Kinderlehrer, R. Kohn & J. L. Lions), pp. 150-174. New York: Springer.
20. Milton, G. W. 1986& A proof that laminates generate all possible effective conductivity functions of two-dimensional, two-phase media. In Advances in multiphase flow and related problems (ed. G. Papanicolaou), pp. 136-146. Philadelphia: SIAM.
21. Milton, G. W. 1991 The field equation recursion method. In Composite media and homogenization theory, progress in nonlinear differential equations and their applications, vol. 5 (ed. G. Dal Maso & G. F. Dell'Antonio), pp. 223-245. Berlin: Birkhäuser.
22. Milton, G. W. & Berryman, J. G. 1997 On thé effective viscoelastic moduli of two-phase media. II. Rigorous bounds on the complex shear modulus in three dimensions. Proc. R. Soc. Land. A 453, 1849-1880.
23. Murât, F. 1987 A survey on compensated compactness. In Contributions to modem calculus of variations (ed. L. Cesari), pp. 145-183. Pitman Research Notes in Mathematics, vol. 148. Harlow: Longman.
24. Norris, A. N. 1985 A differential scheme for the effective moduli of composites. Mech. Maier. 4, 1-16.
25. Schulgasser, K. 1977 Concerning the effective transverse conductivity of a two-dimensional twophase material. Int. J. Heat Transfer 20, 1273-1280.
26. Tartar, L. 1979 Compensated compactness and applications to partial differential equations. In Nonlinear Analysis and Mechanics: Heriot-lVatt Symp., vol. IV (ed. R. Knops), pp. 136-212. Pitman Research Notes in Mathematics, vol. 39. Harlow: Longman.
27. Walpole, L. J. 1966 On bounds for the overall elastic moduli of inhomogeneous systems. I. J. Mech. Phys. Solids 14, 151-162.
28. Wang, Y. M. & Weng, G. J. 1992 The influence of inclusion shape on the overall viscoelastic behavior of composites. ASME J. Appl. Mech. 59, 510-518.