Rigorous bounds on the effective moduli of composites and inhomogeneous bodies with negative-stiffness phases

We review the theoretical bounds on the effective properties of linear elastic inhomogeneous solids (including composite materials) in the presence of constituents having non-positive-definite elastic moduli (so-called negative-stiffness phases). We show that for statically stable bodies the classical displacement-based variational principles for Dirichlet and Neumann boundary problems hold but that the dual variational principle for traction boundary problems does not apply. We illustrate our findings by the example of a coated spherical inclusion whose stability conditions are obtained from the variational principles. We further show that the classical Voigt upper bound on the linear elastic moduli in multi-phase inhomogeneous bodies and composites applies and that it imposes a stability condition: overall stability requires that the effective moduli do not surpass the Voigt upper bound. This particularly implies that, while the geometric constraints among constituents in a composite can stabilize negative-stiffness phases, the stabilization is insufficient to allow for extreme overall static elastic moduli (exceeding those of the constituents). Stronger bounds on the effective elastic moduli of isotropic composites can be obtained from the Hashin-Shtrikman variational inequalities, which are also shown to hold in the presence of negative stiffness

Microstructure-based modeling of elastic functionally graded materials: One dimensional case

Functionally graded materials (FGMs) are two-phase composites with continuously changing microstructure adapted to performance requirements. Traditionally, the overall behavior of FGMs has been determined using local averaging techniques or a given smooth variation of material properties. Although these models are computationally efficient, their validity and accuracy remain questionable, since a link with the underlying microstructure (including its randomness) is not clear. In this paper, we propose a modeling strategy for the linear elastic analysis of FGMs systematically based on a realistic microstructural model. The overall response of FGMs is addressed in the framework of stochastic Hashin-Shtrikman variational principles. To allow for the analysis of finite bodies, recently introduced discretization schemes based on the Finite Element Method and the Boundary Element Method are employed to obtain statistics of local fields. Representative numerical examples are presented to compare the performance and accuracy of both schemes. To gain insight into similarities and differences between these methods and to minimize technicalities, the analysis is performed in the one-dimensional setting.Comment: 33 pages, 14 figure

Effective elastic properties of 3D stochastic bicontinuous composites

We study effective elastic properties of 3D bicontinuous random composites (such as, e.g., nanoporous gold filled with polymer) considering linear and infinitesimal elasticity and using asymptotic homogenization along with the finite element method. For the generation of the microstructures, a leveled-wave model based on the works of J. W. Cahn [J.W. Cahn. Phase separation by spinodal decomposition in isotropic systems. The Journal of Chemical Physics, 42(1):93-99, 1965.] and Soyarslan et al. [C. Soyarslan, S. Bargmann, M. Pradas, and J. Weissmüller. 3D stochastic bicontinuous microstructures: generation, topology and elasticity. Acta Materialia, 149: 326-340, 2018.] is used. The influences of volume element size, phase contrast, relative volume fraction of phases and applied boundary conditions on computed apparent elastic moduli are investigated. The nanocomposite behaves distinctly different than its nanoporous counterpart without any filling as determined by scrutinized macroscopic responses of gold-epoxy nanocomposites of various phase volume fractions. This is due to the fact that, in the space-filling nanocomposite the force transmission is possible in all directions whereas in the nanoporous gold the load is transmitted along ligaments, which hinges upon the phase topology through network connectivity. As a consequence, we observe a distinct elastic scaling law for bicontinuous metal-polymer composites. A comparison of our findings with the Hashin-Shtrikman, the three-point Beran-Molyneux and the Milton-Phan-Tien analytical bounds for the same composites show that computational homogenization using periodic boundary conditions is justified to be the only tool in accurate and efficient determination of the effective properties of 3D bicontinuous random composites with high contrast and volume fraction bias towards the weaker phase

On the dependence of orientation averaging mean field homogenization on planar fourth-order fiber orientation tensors

A comprehensive study on the influence of planar fourth-order fiber orientation tensors on effective linear elastic stiffnesses predicted by orientation averaging mean field homogenization is given. Fiber orientation states of sheet molding compound (SMC) are identified to be in most cases approximately planar. In the planar case, all possible fourth-order fiber orientation tensors are given by a minimal invariant set of structurally differing planar fourth-order fiber orientation tensors. This set defines a three-dimensional body and forms the basis for a comprehensive study on the influence of a fiber orientation distribution in terms of a fourth-order tensor on homogenized stiffnesses. The methodology of this study is the main contribution of this work and can be adopted to analyze the orientation dependence of any quantity which is a function of a planar fourth-order fiber orientation tensor. At specific points inside the set of planar fiber orientation tensors, effective stiffnesses are calculated with selected mean field homogenization schemes. These schemes are based on orientation averaging of transversely isotropic elasticity tensors following Advani and Tucker (1987), which is explicitly recast as linear invariant composition in the fiber orientation tensors of second and fourth order of Kanatani third kind. A maximum entropy reconstruction of a fiber orientation distribution function based on leading fiber orientation tensors, enables a new numerical formulation of the Advani and Tucker average for the special planar case. Polar plots of Young’s modulus and generalized bulk modulus obtained by selected homogenization schemes are arranged on two-dimensional slices within the body of admissible fiber orientation tensors, visualizing the influence of the orientation tensor on the stiffness tensor. The orientation-dependence of the generalized bulk modulus differs significantly between selected homogenizations. Restrictions on the effective anisotropic material response caused by orthotropy of closure approximations are discussed

Homogenization of the Linear and Non-linear Mechanical Behavior of Polycrystals

This work is dedicated to the numerically efficient simulation of the material response of polycrystalline aggregates. Therefore, crystal plasticity is combined with a new non-linear homogenization scheme, which is based on piecewise constant stress polarizations with respect to a homogeneous reference medium and corresponds to a generalization of the Hashin-Shtrikman scheme. This mean field approach accounts for the one- and two-point statistics of the microstructure

Homogenization of the Linear and Non-linear Mechanical Behavior of Polycrystals

This work is dedicated to the numerically efficient simulation of the material response of polycrystalline aggregates. Therefore, crystal plasticity is combined with a new non-linear homogenization scheme, which is based on piecewise constant stress polarizations with respect to a homogeneous reference medium and corresponds to a generalization of the Hashin-Shtrikman scheme. This mean field approach accounts for the one- and two-point statistics of the microstructure

Homogenization Relations for Elastic Properties Based on Two-Point Statistical Functions

In this research, the homogenization relations for elastic properties in isotropic and anisotropic materials are studied by applying two-point statistical functions to composite and polycrystalline materials. The validity of the results is investigated by direct comparison with experimental results. In todays technology, where advanced processing methods can provide materials with a variety of morphologies and features in different scales, a methodology to link property to microstructure is necessary to develop a framework for material design. Statistical distribution functions are commonly used for the representation of microstructures and also for homogenization of materials properties. The use of two-point statistics allows the materials designer to consider morphology and distribution in addition to properties of individual phases and components in the design space. This work is focused on studying the effect of anisotropy on the homogenization technique based on two-point statistics. The contribution of one-point and two-point statistics in the calculation of elastic properties of isotropic and anisotropic composites and textured polycrystalline materials will be investigated. For this purpose, an isotropic and anisotropic composite is simulated and an empirical form of the two-point probability functions are used which allows the construction of a composite Hull. The homogenization technique is also applied to two samples of Al-SiC composite that were fabricated through extrusion with two different particle size ratios (PSR). To validate the applied methodology, the elastic properties of the composites are measured by Ultrasonic methods. This methodology is then extended to completely random and textured polycrystalline materials with hexagonal crystal symmetry and the effect of cold rolling on the annealing texture of near- Titanium alloy are presented.Ph.D.Committee Chair: Hamid Garmestani; Committee Co-Chair: Arun Gokhale; Committee Member: David McDowell; Committee Member: Naresh Thadhani; Committee Member: W. Steven Johnso

Unified mean-field modeling of viscous short-fiber suspensions and solid short-fiber reinforced composites

Mean-field homogenization is an established and computationally efficient method estimating the effective linear elastic behavior of composites. In view of short-fiber reinforced materials, it is important to homogenize consistently during process simulation. This paper aims to comprehensively reflect and expand the basics of mean-field homogenization of anisotropic linear viscous properties and to show the parallelism to the anisotropic linear elastic properties. In particular, the Hill–Mandel condition, which is generally independent of a specific material behavior, is revisited in the context of boundary conditions for viscous suspensions. This study is limited to isothermal conditions, linear viscous and incompressible fiber suspensions and to linear elastic solid composites, both of which consisting of isotropic phases with phase-wise constant properties. In the context of homogenization of viscous properties, the fibers are considered as rigid bodies. Based on a chosen fiber orientation state, different mean-field models are compared with each other, all of which are formulated with respect to orientation averaging. Within a consistent mean-field modeling for both fluid suspensions and solid composites, all considered methods can be recommended to be applied for fiber volume fractions up to 10%. With respect to larger, industrial-relevant, fiber volume fractions up to 20%, the (two-step) Mori–Tanaka model and the lower Hashin–Shtrikman bound are well suited