Evaluation of elastic moduli of bilaminate filament-wound composites

This paper investigates the elastic behavior of bilaminate-composite coupons segmented from a filament-wound rocket case and of a laboratory prepared flat panel for direct tension-compression and flexural loading under static and dynamic conditions. New methods of testing have been developed which are primarily applicable to composite constructions. The dynamic test consists of exciting the primary free-free resonant mode of a specimen. Flexure tests utilizing a unique pure-bend system are employed for the static evaluations. The composite moduli determined from the static and dynamic test are compared with analytic values. The analytic values for the composite are derived from tension-compression and flexure analytical models using the material properties of the constituents. The measured elastic moduli compared favorably with analytical prediction and are indicative of the history of loading effects as well as the crazed condition of the composite constructions. The moduli determined by the dynamic test showed the closest agreement with analytic values, with a difference of 0 to 16 percent.     

Canonical elastic moduli

Every linear planar anisotropic elastic material is equivalent, under a linear change of coordinates, to an orthotropic material. Consequently, up to linear changes of variables, there are just two canonical planar elastic moduli which determine the properties of any linearly elastic material. Extensions to three-dimensional elasticity and applications are indicated.     

Symmetry conditions for third order elastic moduli and implications in nonlinear wave theory

Several results are presented concerning symmetry properties of the tensor of third order elastic moduli. It is proven that a set of conditions upon the components of the modulus tensor are both necessary and sufficient for a given direction to be normal to a plane of material symmetry. This leads to a systematic procedure by which the underlying symmetry of a material can be calculated from the 56 third order moduli. One implication of the symmetry conditions is that the nonlinearity parameter governing the evolution of acceleration waves and nonlinear wave phenomena is identically zero for all transverse waves associated with a plane of material symmetry.     

Evaluation of the elastic moduli of a transversely isotropic aggregate of cubic crystals

The crystals and the aggregate have the same bulk modulus. The other three overall elastic moduli in the simplified stress-strain relations of Walpole (1985) are placed between upper and lower, Voigt and Reuss, bounds and some exact calculations are given for particular fibre textures.     

Wave propagation in elastic laminates using a second order microstructure theory

A generalization of the simpler microstructure theory developed earlier for elastic laminates by Sun, Achenbach and Herrmann is used to analyze steady state plane wave propagation. This new version incorporates higher-order thickness variations in the displacement functions and includes restrictions on both displacement and stress at the laminate interfaces.To assess the potential of a second-order microstructure theory for accurate modeling of mechanical processes in laminates, dispersion results and especially mode shape data for both displacements and stresses are obtained and compared to corresponding solutions obtained by the theory of elasticity. The comparisons indicate that while dispersion results may be nearly identical, extremely significant differences may be observed in the mode shapes.     

Effective elastic moduli of granular media

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 91–96, January–February, 1990.     

The elastic moduli estimation of the solid-water mixture

Conceptually, the undrained elastic constants estimated by the poroelasticity theory should be identical to the effective moduli of the two-phase composite of a porous material saturated with pore water. Here we show numerically that the undrained elastic constants determined by an effective moduli estimate are almost identical with those calculated by poroelasticity theory, and if pore shapes are not exactly known and the porosity is around 50%, estimating the elastic constant as the average value of its Voigt and Reuss bounds is reasonably accurate. This is the situation in bone and dentin, the materials that are our primary intended application. This result will hold for situations in which the totally enclosed water phase is constrained to small deformations by virtue of its confinement. Importantly, in this work we assume that water is an isotropic elastic solid with a shear modulus that is 10^?4 times the bulk modulus of the water. Note that it is compressible, but almost incompressible with a Poisson’s ratio of 0.4999.     

Connexions between the moduli for anistropic elastic materials

Summary For homogeneous isotropic elastic materials there are simple interrelations connecting Young's modulus, Poisson's ratio, the rigidity modulus and the modulus of compression. However for anisotropic materials the situation is quite different. Young's modulus is a function of direction and Poisson's ratio and the rigidity modulus are functions of pairs of orthogonal directions. Here some simple universal connexions between the moduli for various directions are simply derived for general anisotropic materials. No particular symmetry is assumed in the material.     

Effective elastic moduli of third-order composite materials

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 118–123, November–December, 1990.     

Theory of the elastic moduli of a heterogeneous medium

It is shown that the theory of random functions permits the expansion of the effective tensor X~jkl for the elastic moduli with respect to correlation functions and that it leads in the second approximation in the Voigt-Reuss scheme to values that lie to one side of the Xijkl, while in the third approximation it brackets the latter. The analysis is used to refine the Hashin limits to the elastic moduli for a mechanical mixture of isotrcpic components and polycrystalline aggregates of cubic structure.There are two methods for calculating the effective elastic moduli of heterogeneous solids: virial expansion [2] (as a power series in the concentration of one of the components) and the method of correlation functions [2] (expansion with respect to relative fluctuation of the elastic moduli). Identical results should be obtained in the two cases if all terms are incorporated, but great mathematical difficulties restrict one to the lowest approximations. The first approximation in the virial method gives better results when the concentration of one component is low, while the method of correlation functions gives better results when the fluctuations in the elastic moduli are small and the concentrations are similar.Methods have been developed for determining the upper and lower bounds in both approaches, and various schemes of averaging are used for this purpose in the correlation-function method. The upper bound is established by renormalizing the equation of equilibrium, while the lower one is found by renormalizing the equation of incompatibility. The range of the bracketing can be reduced by means of higher approximations. The range can be reduced in the limit to zero, which implies passing from an approximate effective tensor to the true one, which relates the means in stress and strain over the material. Here we show that the two methods of renormalization give identical results when all terms of the series are summed.If the tensor has a Gaussian distribution, the moment functions of odd order are zero, while the even ones are expressed via combinations of the binary functions [3]. However, a mechanical mixture of several components is not Gaussian, and the odd moments are not zero. Splitting of the higher-order correlation functions is possible also for mechanical mixtures having determinate phase interfaces, but this involves various simplifying assumptions. A derivation is given for a moment of arbitrary order, which allows one to formulate the conditions under which such splitting is possible. The results are used in calculating the exact value of the effective bulk modulus for a medium with a homogeneous shear modulus.We are indebted to V. V. Bolotin for a discussion.     

On the overall moduli of non-linear elastic composite materials

From the work of R. Hill on constitutive macro-variables it is known that for an inhomogeneous elastic solid under finite strain an overall elastic constitutive law may be defined. In particular, the volume average of the strain energy of the solid is a function only of the volume-averaged deformation gradient. In view of the importance of this result it is re-derived in this paper as a prelude to a discussion of composite materials. A composite material consisting of a dilute suspension of initially spherical inclusions embedded in a matrix of different material is considered. For second-order isotropic elasticity theory an expression for the overall bulk modulus of the composite material is obtained in terms of the moduli of the constituents. When the inclusions are vacuous a known result for the bulk modulus of porous materials is recovered. In certain situations the strengthening/ weakening effects of the inclusions are less pronounced in the second-order theory than in the linear theory.     

Improved bounds for the elastic moduli of perfectly-random composites

New upper and lower bounds are constructed for the elastic moduli of a class of isotropic composites with perfectly-random microgeometries ([1–3]), which improve upon the bounds on the elastic shear modulus given in [1].     

Asymptotics of the homogenized moduli for the elastic chess-board composite

We find the asymptotic behavior of the homogenized coefficients of elasticity for the chess-board structure. In the chess board white and black cells are isotropic and have Lamé constants (, ,) and (, ) respectively. We assume that the black cells are soft, so 0. It turns out that the Poisson ratio for this composite tends to zero with .     

On the quasistatic effective elastic moduli for elastic waves in three-dimensional phononic crystals

Effective elastic moduli for 3D solid–solid phononic crystals of arbitrary anisotropy and oblique lattice structure are formulated analytically using the plane-wave expansion (PWE) method and the recently proposed monodromy-matrix (MM) method. The latter approach employs Fourier series in two dimensions with direct numerical integration along the third direction. As a result, the MM method converges much quicker to the exact moduli in comparison with the PWE as the number of Fourier coefficients increases. The MM method yields a more explicit formula than previous results, enabling a closed-form upper bound on the effective Christoffel tensor. The MM approach significantly improves the efficiency and accuracy of evaluating effective wave speeds for high-contrast composites and for configurations of closely spaced inclusions, as demonstrated by three-dimensional examples.