Bounds and estimates for linear composites with strain gradient effects

Abstract Overall mechanical properties are studied for linear composites demonstrating a size effect. Variational principles of Hashin-Shtrikman type are formulated for incompressible composites involving the gradient of strain in their constitutive description. These variational principles are applied to linear, statistically homogeneous and isotropic two-phase composites. Upper and lower bounds of Hashin-Shtrikman type for the effective shear modulus and related self-consistent estimates are derived in terms of volume fraction and a two-point correlation function accounting for the scale of microstructure. An alternative selfconsistent scheme for matrix-inclusion strain-gradient composites is also proposed by a development of the approach laid down by Budiansky and Hill. Some numerical results are given to demonstrate the size effect.

[1]  Pedro Ponte Castañeda.  The effective mechanical properties of nonlinear isotropic composites , 1991 .

[2]  J. D. Eshelby.  The determination of the elastic field of an ellipsoidal inclusion, and related problems , 1957, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[3]  S. Shtrikman,et al.  A variational approach to the theory of the elastic behaviour of multiphase materials , 1963 .

[4]  J. Willis,et al.  Polarization approach to the scattering of elastic waves—I. Scattering by a single inclusion , 1980 .

[5]  E. Aifantis.  On the Microstructural Origin of Certain Inelastic Models , 1984 .

[6]  Hans Muhlhaus,et al.  A variational principle for gradient plasticity , 1991 .

[7]  J. Willis,et al.  The overall elastic response of composite materials , 1983 .

[8]  J. Willis,et al.  On methods for bounding the overall properties of nonlinear composites , 1991 .

[9]  Elias C. Aifantis,et al.  The physics of plastic deformation , 1987 .

[10]  H. Zbib,et al.  On the gradient-dependent theory of plasticity and shear banding , 1992 .

[11]  J. Willis,et al.  A simple self-consistent analysis of wave propagation in particulate composites , 1988 .

[12]  J. Willis.  The nonlocal influence of density variations in a composite , 1985 .

[13]  R. D. Mindlin.  Second gradient of strain and surface-tension in linear elasticity , 1965 .

[14]  E. Aifantis.  On the role of gradients in the localization of deformation and fracture , 1992 .

[15]  S. Shtrikman,et al.  On some variational principles in anisotropic and nonhomogeneous elasticity , 1962 .

[16]  B. Budiansky.  On the elastic moduli of some heterogeneous materials , 1965 .

[17]  J. Willis,et al.  Self-consistent analysis of waves in a matrix-inclusion composite—I. Aligned spheroidal inclusions , 1993 .

[18]  J. Willis.  Bounds and self-consistent estimates for the overall properties of anisotropic composites , 1977 .

[19]  Graeme W. Milton,et al.  Modelling the Properties of Composites by Laminates , 1986 .

[20]  Norman A. Fleck,et al.  A phenomenological theory for strain gradient effects in plasticity , 1993 .

[21]  M. Ashby,et al.  Strain gradient plasticity: Theory and experiment , 1994 .

[22]  R. Hill.  A self-consistent mechanics of composite materials , 1965 .

[23]  J. Willis,et al.  Self-consistent analysis of waves in a matrix-inclusion composite. III: A matrix containing cracks , 1993 .

[24]  J. Willis,et al.  Self-consistent analysis of waves in a matrix-inclusion composite—II. Randomly oriented spheroidal inclusions , 1993 .

[25]  J. Willis,et al.  Variational Principles for Inhomogeneous Non-linear Media , 1985 .

[26]  W. T. Koiter.  Couple-stresses in the theory of elasticity , 1963 .

[27]  J. Willis,et al.  Variational and Related Methods for the Overall Properties of Composites , 1981 .