Minimum energy and iteration estimates for longitudinal Young’s modulus and Poisson’s ratio of transversely-isotropic unidirectional composites

Highlights

• Bounds on longitudinal elastic constants of transversely-isotropic unidirectional composites are constructed.

• Mixed longitudinal–transverse deformations modes are explored to derive specific original (starting) and iteration bounds.

• Illustrating numerical examples are presented for comparisons with previous results.

Abstract

Some formal inequalities restricting the macroscopic (effective) longitudinal elastic constants of the $n$-component transversely-isotropic unidirectional composites, based on the minimum energy and complementary energy principles with the help of appropriate constant strain and piece-wise constant stress trial fields and optimization techniques, have been established. Exploring further the possible longitudinal stress–strain modes, additional starting original bounds are derived leading to the new set of iteration bounds on both effective longitudinal constants, which broaden and improve upon the ones of the first set constructed earlier (Pham, 2020) for the composites with positive longitudinal Poisson’s ratio. Another new set of iteration bounds is constructed for the composites having negative longitudinal Poisson’s ratio. Illustrating numerical examples are presented for comparisons.

Introduction

Exact determination of the macroscopic (effective) elastic moduli of multicomponent materials, including the unidirectional ones, is often not an easy task because of the complex microstructure of the composites. Microscopic models have been developed to approximate the elastic micro-structured materials (Hill, 1963, Hashin, 1965, Mori and Tanaka, 1973, Christensen, 1979, Norris, 1985, Norris, 1989, Qiu and Weng, 1991, McCartney, 2010, Nava-Gómez et al., 2012, Tran and Pham, 2015, Gua and He, 2017, Pham et al., 2017, Hervé-Luanco, 2020). Alternatively, rigorous variational approaches have been developed to estimate the effective moduli from above and below, starting with Voigt–Reuss–Hill–Paul bounds involving only the elastic moduli and volume proportions of the constituent materials: the bounds have been obtained by minimum energy principles, with constant strain and stress trial fields (Hill, 1952, Paul, 1960). Tighter bounds could be constructed for specific classes of composites and if more statistical informations about the microstructure of the composites are available and can be incorporated into the bounds (Hashin and Shtrikman, 1963, Hashin and Rosen, 1964, Hill, 1964, Beran, 1968, Miller, 1969, Yeh, 1973, Willis, 1977, Milton and Phan-Thien, 1982, Berdichevski, 1983, Phan-Thien and Milton, 1983, Milton and Kohn, 1988, Pham, 1993, Pham, 1994, Pham, 1997, Pham, 1998, Pham, 1999, Pham, 2011, Pham, 2012, Pham and Phan-Thien, 1998, Milton, 2001, Torquato, 2002, Brito-Santana et al., 2009). Composites having exotic negative Poisson’s ratio also attract attentions. Reentrant polymer foam materials with negative Poisson’s ratios have been observed by Lakes (1987). Milton (1992) showed that one can construct elastically isotropic two- and three-dimensional composites with Poisson’s ratio arbitrarily close to −1, while Ting and Chen (2005) indicated that Poisson’s ratio for anisotropic elastic materials can have no bounds.

In distinction from those concerning the longitudinal shear modulus, and the transverse bulk and shear ones, direct construction of the bounds on the 2 remaining longitudinal elastic constants of $n$-component transversely-isotropic unidirectional composites encounters technical difficulties resulted from non-separation of the longitudinal deformation modes, except in the specific two-component case when one has Hill (1964) formulae relating the longitudinal constants to the transverse bulk one. In (Pham, 2020) some formal inequalities restricting the macroscopic longitudinal and transverse-bulk elastic constants of the $n$-component transversely-isotropic unidirectional composites have been constructed, using minimum energy principles, with appropriate constant strain and piece-wise constant stress trial fields. Some specific original bounds and a set of iteration ones on the effective longitudinal constants of the composites have been derived. In this paper new additional original bounds will be derived leading to additional new sets of iteration bounds to improve the estimates and complete the approach.

In Section 2 some general formal inequalities and bounds on the longitudinal and transverse bulk elastic constants constructed from the minimum energy principles and those corresponding to specified stress–strain modes are summarized. In Section 3 new original upper and lower bounds on the effective longitudinal Poisson’s ratio are derived leading to the new sets of iteration bounds on both effective longitudinal constants. Then another new set of iteration bounds is constructed for certain composites with negative longitudinal Poisson’s ratio. Illustration examples are provided in Section 4. Conclusion Section 5 completes the paper.