Minimum energy combined and separated bounds on elastic constants of transversely-isotropic composites

The considered linearly-elastic transversely-isotropic composite (TIC) is composed of $n$ isotropic, or more generally, transversely-isotropic components sharing the materials’ common symmetry axis with that of the macroscopic material. Using the basic minimum energy and complementary energy principles with certain free-parameter-dependent mixed-longitudinal-transverse-mode strain and stress trial fields, various combination bounds involving some sets of the macroscopic (effective) mixed-mode elastic constants of the composite, which are inter-connected via the constitutive relations, have been established. Choosing the appropriate parameter values of/or optimizing over the free parameters in those inequalities, the separated bounds on the major effective mixed-transverse-longitudinal-mode elastic constants, including the transverse bulk modulus $K^{eff}$, the longitudinal Young modulus $E^{eff}$, and the longitudinal Poisson’s ratio ${\nu }^{eff}$, are derived, beside the classical arithmetic and harmonic average bounds on the pure-mode ones — the transverse shear (${\mu }^{eff}$) and longitudinal shear (${\bar{\mu }}^{eff}$) moduli. The separated bounds on 4 remaining effective mixed-mode elastic constants are also obtained. The illustrative numerical comparisons of the bounds, in the two component case, with those for the special subclass of unidirectional transversely-isotropic composites (UTIC), having the unidirectional cylindrical boundaries between the component materials parallel to their symmetry axis, and the exact coated-cylinder assemblage and laminate models are presented. The extreme models cover substantial parts between the bounds for TIC; however the laminate models lie outside the bounds for the subclass UTIC.