Hashin Shtrikman Research Articles

Systematic study of homogenization and the utility of circular simplified representative volume element

Although both computational and analytical homogenization are well-established today, a thorough and systematic study to compare them is missing in the literature. This manuscript aims to provide an exhaustive comparison of numerical computations and analytical estimates, such as Voigt, Reuss, Hashin–Shtrikman, and composite cylinder assemblage. The numerical computations are associated with canonical boundary conditions imposed on either tetragonal, hexagonal, or circular representative volume elements using the finite-element method. The circular representative volume element is employed to capture an effective isotropic material response suitable for comparison with associated analytical estimates. The analytical results from composite cylinder assemblage are in excellent agreement with the numerical results obtained from a circular representative volume element. We observe that the circular representative volume element renders identical responses for both linear displacement and periodic boundary conditions. In addition, the behaviors of periodic and random microstructures with different inclusion distributions are examined under various boundary conditions. Strikingly, for some specific microstructures, the effective shear modulus does not lie within the Hashin–Shtrikman bounds. Finally, numerical simulations are carried out at finite deformations to compare different representative volume element types in the nonlinear regime. Unlike other canonical boundary conditions, the uniform traction boundary conditions result in nearly identical effective responses for all types of representative volume element, indicating that they are less sensitive with respect to the underlying microstructure. The numerical examples furnish adequate information to serve as benchmarks.

Sound velocities of the 23 Å phase at high pressure and implications for seismic velocities in subducted slabs

Dense hydrous magnesium silicates (DHMS) are believed to play an important role in transporting water back into the deep interior of the Earth. Recently, a new Al-bearing hydrous Mg-silicate, named the 23 Å phase [ideal composition Mg12Al2Si4O16(OH)14], was reported (Cai et al., 2015), which could be another important hydrous phase in subducting slabs. In this paper, we report new measurements of the compressional (P) and shear (S) wave velocities of the 23 Å phase under pressures up to 14 GPa at room temperature, using a bulk sample with a grain size of less than 20 μm and density of 2.947 g/cm3. The acoustic measurements were conducted in a 1000-ton uniaxial split-cylinder multi-anvil apparatus using ultrasonic interferometric techniques (Li et al., 1996). The pressures were determined in situ by using an alumina buffer rod as the pressure marker (Wang et al., 2015). A dual-mode piezoelectric transducer enabled us to measure P and S wave travel times simultaneously, which in turn allowed a precise determination of the sound velocities and elastic bulk (KS) and shear (G) moduli at high pressures. The velocities of this 23 Å phase (ambient VP = 7.53 km/s, VS = 3.72 km/s) are lower than those of phase A, olivine, pyrope, etc. (especially for S waves), yet the VP/VS ratio (from 2.02 to 1.94, decreasing with increasing pressure) is much higher compared to those mantle minerals (1.7–1.8). A fit to the acoustic data using finite strain analysis combined with the Hashin-Shtrikman (HS) averaging scheme yields: KS0 = 111.4(6) GPa, G0 = 41.0(2) GPa, and KS′ = 3.6, G′ = 1.8 for the bulk and shear moduli and their pressure derivatives. Using these data, the density and velocity profiles of the hydrous harzburgite and lherzolite were calculated along a warm slab P-T condition. The results suggest that a hydrous pyrolitic assemblage containing 23 Å phase should be distinguishable from a dry assemblage at the high pressure and temperature conditions relevant to Al-bearing subducted slabs.

Estimation of Elastic Properties of Oil Palm Kernel Shell Concrete by Semianalytical Methods of Homogenisation

Despite its importance as one of the key parameters in the design of structural elements, the modulus of elasticity (MoE) is one of the least researched areas in oil palm kernel shell (OPKS) concrete. In the present study, we determined the MoE of OPKS concrete, using micromechanical models based on the classic approaches of homogenisation of Hashin–Shtrikman (HS) and Mori–Tanaka (MT). The MoE values for OPKSC (fcu < 35 MPa and OPKS volume fraction < 42%) estimated using the HS and MT models in the literature showed a good correlation with the experimental values. An empirical linear correlation between the volume fraction of inclusions and the MoE was proposed. The results obtained can enable better control of the mix design of structural concrete based on the proportion of OPKS coarse aggregate in tropical countries producing palm oil.

An efficient reduced computational method for nonlinear homogenization problems: the Hashin–Shtrikman type Finite Element method (HSFE)

AbstractThis contribution is dedicated to a numerically efficient homogenization method for nonlinear microstructures (HSFE), which combines the small computational effort of the Hashin‐Shtrikman type approach with the accuracy of full‐field finite element solutions. The key point of the method is a model order reduction scheme based on data‐clustering, which leads to a significant reduction of computational time. The derivation of the analytical tangent operator realises the coupling between the microscopic and macroscopic scale. The presented approach is tested in the context of small strains by a numerical example, where a nonlinear RVE is attached to each integration point of a 2D macrostructure. A comparison to the FE‐simulations shows that the global‐ and local‐responses are well captured by the HSFE method.

Effective permittivity of liquid crystal nanodispersions and composites with different pore structure

ABSTRACTThe volume filling fraction dependence of the effective permittivity of the nematic liquid crystal 4-n-pentyl-4’-cyanobiphenyl embedded in different porous membranes and dispersed with aerosil nanoparticles was determined using broadband dielectric spectroscopy in the frequency range from 106 to 109 Hz. The experimental data were analyzed and compared with some existing theories based on the effective medium approximation and their modifications. The obtained effective permittivities as a function of the volume filling fraction lie between the lower limits of the Wiener and Hashin–Shtrikman bounds. The observed shift of the experimental points reflects the changes in the structure of the investigated composites.

Two-scale simulation of the hot stamping process based on a Hashin–Shtrikman type mean field model

The hot stamping process of sheet metals into a w-shape profile is considered in experiments and numerical simulation using the FEM tool ABAQUS. The influence of the processing parameters of the hot stamping process on the final shape and the residual stress state are investigated. Tensile tests of thermomechanically treated specimens serve as a starting point for the modeling of the thermomechanical behavior of the steel grade 22MnB5. Based on the thermo-micromechanical (TMM) model for small deformations suggested by Neumann and Böhlke (2016), the updated Lagrangian formulation is used to render the finite deformation process.The model captures in a two-scale framework the temperature dependent elasto-plastic effect of the different phases, the phase transformation effect, and the phase transformation accompanying transformation strain occurring during the quenching of the austenitized sheet metal. Additionally, a model for the transformation induced plasticity (TRIP) effect in the context of the two-scale approach is suggested. The homogenization and localization of the thermomechanical properties is performed with a HashinöShtrikman (HS) type homogenization scheme. In this work, a reduced formulation for isotropic elastic behavior of the different phases is found reducing the amount of numerical costs. Furthermore, a self-consistent scheme based on the HS type method is introduced.Finally, the suggested TMM model is validated by considering the time-temperature-transformation- (TTT) and continuous-cooling-time-curves (CCT), dilatation tests under external load, and the hot stamping of w-shape profiles. The TMM model is compared to both experimental results and a phenomenological reference model usually used for the simulation of hot stamping processes.It is found that the suggested TMM model fits the final shape and the residual stress state of hot stamped parts better than the phenomenological reference model.

Topology optimization of binary microstructures involving various non-volume constraints

In this paper, we use the Topology Optimization of Binary Structures (TOBS) method recently developed by Sivapuram and Picelli (2018) for microstructural optimization. This is the first work in topology optimization addressing various non-volume microstructural constraints with discrete (0/1) design variables. The objective and constraint functions are linearized at each iteration, and the obtained linear problem is solved through Integer Linear Programming (ILP) using sensitivities computed from asymptotic homogenization. A periodic filter is used to make the optimized solutions checkerboard-free and mesh-independent. Volume minimization problems subject to elastic and thermal constraints are considered. The examples consider different sets of constraints, including bulk and shear moduli, square/cubic symmetry, isotropy, thermal conductivity and a combination of them in two and three dimensions. The non-volume constraints are treated explicitly, i.e., without the use of Lagrange multiplier/penalty as used in conventional gradient-based binary topology optimization methods (Huang and Xie, 2010). The resulting microstructures are observed to be convergent in all the examples presented and in agreement with the Hashin–Shtrikman bounds.

Measurement and prediction of thermal properties of alkali-activated fly ash/slag binders at elevated temperatures

This paper presents a comprehensive experimental study of thermal properties of various alkali-activated binders at ambient and elevated temperatures. The binders were prepared using alkali-activated low calcium fly ash/ground granulated blast-furnace slag at ratios of 100/0, 90/10, 50/50 and 0/100 wt%. These binders can be considered as a composite of solid, water and air. Accordingly, a three-phase model is applied to predict thermal conductivity of the binders at ambient temperature. At elevated temperatures, the Hashin–Shtrikman model is used to estimate the bounds of thermal conductivity for alkali-activated binders containing of fly ash. To validate the above models, a transient plane source measurement technique was applied to measure the thermal conductivity and heat capacity at temperatures ranging from 23 to 600 °C. Data generated is then utilised to develop analytical expressions for estimating thermal properties as a function of temperature. The simplified relationships can be used for estimating the fire resistance of structural elements made from alkali-activated cementitious materials.

Elastic properties of a unidirectional composite reinforced with hexagonal array of fibers

It is difficult to measure the transverse shear modulus of the fibrous composites. Thus, theoretical investigations by means of analytical and numerical techniques are paramount. In particular, they are important for the regime with high-concentration of fibers. We apply general techniques to study the mechanical properties of unidirectional fibers with a circular section embedded into the matrix and organized into the hexagonal array. Our theoretical considerations are designed to include two regimes, of low and high concentrations of inclusions. The former regime is controlled by Hashin–Shtrikman lower bounds, while the latter is controlled by square-root singularity. We derived the analytical formulae for the effective shear, Young and bulk moduli in the form of the rational expressions valid up to O ( f 7 ) by the method of functional equations. The obtained formulae contains elastic constants of components in a symbolic form as well as the concentration f . The general scheme based on the asymptotically equivalent transformations is developed to extend the obtained analytical formulae to the critical concentration of touching fibers. A comparison with the numerical FEM is performed for all concentrations of inclusions. Good agreement is achieved for all available concentrations.

Isotropic Ti–6Al–4V lattice via topology optimization and electron-beam melting

Electron-beam melting (EBM) exhibits advantages over other metal-additive manufacturing techniques owing to its low residual stress, rapid fabrication speed, and high energy efficiency. However, in EBM, metal powder is preheated and sintered to stabilize the temperature gradient and powder position during melting with a high-power electron beam. When making a lattice structure by EBM, a certain size of the powder-removing hole is required to remove the sintered remaining metal powder from the lattice. However, a large powder-removing hole can reduce the lattice mechanical performance. We conducted topology optimization to derive an optimal lattice structure shape with high isotropic stiffness assuming fabrication by EBM and minimizing the performance reduction owing to fixed large powder-removing holes. The optimized structure was fabricated via the EBM of a Ti–6Al–4V alloy. The optimal lattice structure achieved 83% of the performance of the Hashin–Shtrikman upper bound in numerical simulations, but an approximate 20% stiffness reduction was observed in the experiments. The isotropy was high with an error in Young's modulus and a strength of less than 9% and 6%, respectively. These results are discussed based on numerical and experimental results.

Old and New Approaches Predicting the Diffusion in Porous Media

An accurate quantitative description of the diffusion coefficient in a porous medium is essential for predictive transport modeling. Well-established relations, such as proposed by Buckingham, Penman, and Millington–Quirk, relate the scalar diffusion coefficient to the porous medium’s porosity. To capture the porous medium’s structure in more detail, further models include fitting parameters, geometric, or shape factors. Some models additionally account for the tortuosity, e.g., via Archie’s law. A validation of such models has been carried out mainly via experiments relating the proposed description to a specific class of porous media (by means of parameter fitting). Contrary to these approaches, upscaling methods directly enable calculating the full, potentially anisotropic, effective diffusion tensor without any fitting parameters. As input only the geometric information in terms of a representative elementary volume is needed. To compute the diffusion–porosity relations, supplementary cell problems must be solved numerically and their (flux) solutions must be integrated. We apply this approach to provide easy-to-use quantitative diffusion–porosity relations that are based on representative single grain, platy, blocky, prismatic soil structures, porous networks, or random porous media. As a discretization method, we use the discontinuous Galerkin method on structured grids. To make the relations explicit, interpolation of the obtained data is used. We furthermore compare the obtained diffusion–porosity relations with the well-established relations mentioned above and also with the well-known Voigt–Reiss or Hashin–Shtrikman bounds. We discuss the ranges of validity and further provide the explicit relations between the diffusion and surface area and comment on the role of a tortuosity–porosity relation.

Representation of Hashin–Shtrikman Bounds in Terms of Texture Coefficients for Arbitrarily Anisotropic Polycrystalline Materials

The present work generalizes the results of Bohlke and Lobos (Acta Mater. 67:324–334, 2014) by giving an explicit representation of the Hashin–Shtrikman (HS) bounds of linear elastic properties in terms of tensorial Fourier texture coefficients not only for cubic materials but for arbitrarily anisotropic linear elastic polycrystalline materials. Based on the HS bounds as given by Walpole (J. Mech. Phys. Solids 14(3):151–162, 1966) and tensor functions for the representation of the crystallite orientation distribution function, it is shown that the HS bounds are represented in terms of the exact same second- and fourth-order texture coefficients which appear in the consideration of the Voigt and Reuss bounds. The derived representations in terms of tensorial texture coefficients are valid for all symmetry classes in elasticity and present expressions highly attractive for the description of physical quantities in terms of tensorial variables. In order to make these results also accessible for the community of quantitative texture analysis, transformation relations between experimentally obtained Bunge’s or Roe’s coefficients and the tensorial texture coefficients are given. The representation of the present work offers a finite and low dimensional parametrization of the fully anisotropic Hashin–Shtrikman bounds, which can be used in inverse materials design problems in order to explore the set of possible materials properties or for the determination of optimal microstructural influence with respect to prescribed material properties. Examples for orthotropic polycrystals of cubic materials and transversely isotropic polycrystals of hexagonal materials (showing the connection and applicability of the results also to fiber orientation distributions) are discussed. Finally, an implementation in Mathematica® 11 of the HS bounds for arbitrarily anisotropic materials is offered, such that readers can reproduce all the results of this work and use them for their own purposes.

A parametric class of composites with a large achievable range of effective elastic properties

In this paper, we study an instance of the G-closure problem for two-dimensional periodic metamaterials. Specifically, we consider composites with isotropic homogenized elasticity tensor, obtained as a mixture of two isotropic materials. We focus on the case when one material has zero stiffness, i.e., single-material structures with voids. This problem is important, in particular, in the context of designing small-scale structures for metamaterials that can be manufactured using additive fabrication. A range of effective metamaterial properties can be obtained this way using a single base material.We demonstrate that two closely related simple parametric families based on the structure proposed by Sigmund in [26] attain good coverage of the space of isotropic properties satisfying Hashin–Shtrikman bounds. In particular, for positive Poisson’s ratio, we demonstrate that Hashin–Shtrikman bound can be approximated arbitrarily well, within limits imposed by numerical approximation: a strong evidence that these bounds are achievable in this case. For negative Poisson’s ratios, we numerically obtain a bound which we hypothesize to be close to optimal, at least for metamaterials with rotational symmetries of a regular triangle tiling.

The Dispersion Tensor and Its Unique Minimizer in Hashin–Shtrikman Micro-structures

In this paper, we introduce a macroscopic quantity, namely the dispersion tensor or the Burnett coefficients in the class of generalized Hashin–Shtrikman micro-structures (Tartar in The general theory of homogenization, volume 7 of Lecture notes of the Unione Matematica Italiana, Springer, Berlin, p 281, 2009). In the case of two-phase materials associated with the periodic Hashin–Shtrikman structures, we settle the issue that the dispersion tensor has a unique minimizer, which is the so called Apollonian–Hashin–Shtrikman micro-structure.

Equivalent inclusion approach and approximations for conductivity of isotropic matrix composites with sphere-like, platelet, and fibrous fillers

The construction starts from certain typical effective medium approximations for conductivity of idealistic isotropic matrix composites with randomly oriented inclusions of perfect spherical, platelet, and circular fiber forms, which obey Hashin–Shtrikman bounds over all the ranges of volume proportions of the component materials. Equivalent inclusion approach is then developed to account for possible diversions, such as non-idealistic geometric forms of the inhomogeneities, imperfect matrix-inclusion contacts, filler dispersions, and when the particular values of the fillers’ properties are unspecified, using available numerical or experimental reference conductivity data for particular composites. Illustrating applications involving experimental data from the literature show the usefulness of the approach.

A parametric model for the changes in the complex valued conductivity of a lung during tidal breathing

Classical homogenization theory based on the Hashin–Shtrikman coated ellipsoids is used to model the changes in the complex valued conductivity (or admittivity) of a lung during tidal breathing. Here, the lung is modeled as a two-phase composite material where the alveolar air-filling corresponds to the inclusion phase. The theory predicts a linear relationship between the real and the imaginary parts of the change in the complex valued conductivity of a lung during tidal breathing, and where the loss cotangent of the change is approximately the same as of the effective background conductivity and hence easy to estimate. The theory is illustrated with numerical examples based on realistic parameter values and frequency ranges used with electrical impedance tomography (EIT). The theory may be potentially useful for imaging and clinical evaluations in connection with lung EIT for respiratory management and control.

An iterative analytical model for heterogeneous materials homogenization

Abstract The purpose of this study was to establish a method based on an iterative scheme to approximate the numerical solution obtained from finite elements analysis for an RVE in two and three dimensions based on the homogenization concept for the assessment of the effective properties. The bounds of Hashin–Shtrikman and Voigt–Reuss were considered in the iterative process based on an updating of the constitutive relations of these models respectively. In this study, by assumption, we took the particular case of the heterogeneous materials with several elastic isotopic phases. The output variables considered using the iterative process are the bulk, shear modulus and the thermal conductivity. We have found a fast convergence of the iterative solution to the numerical result with a suitable concordance between the two solutions at the final step.

Young’s modulus and thermal conductivity of model materials with convex or concave pores – from analytical predictions to numerical results

The effective Young’s modulus and thermal conductivity of porous materials can be rigorously bounded from above via micromechanical bounds (upper Wiener–Paul bounds and upper Hashin–Shtrikman bounds), and several model relations are commonly used as tentative approximate predictions (Maxwell-type, Coble–Kingery-type, power-law and exponential relations). Based on numerical calculations on computer-generated digital model microstructures, both periodic and random, it is shown that these model relations provide rough approximations that are more or less appropriate for microstructures with essentially convex pores, but are not suitable for microstructures with concave pores. On the other hand, the Pabst–Gregorová cross-property relation provides a very accurate (better than 0.04 relative property units) analytical prediction for the relative Young’s modulus of isotropic porous materials with isometric pores, both convex and concave, when the relative thermal conductivity is known. It is shown that this cross-property relation is the best prediction currently available for isotropic porous materials with isometric pores.

Thermophysical properties of heat-treated U-7Mo/Al dispersion fuel

In this study, the effects of interaction layer (IL) on thermophysical properties of U-7Mo/Al dispersion fuel were examined. Microstructural analyses revealed that ILs were formed uniformly on U-Mo particles during heating of U-7Mo/Al samples. The IL volume fraction was measured by applying image analysis methods. The uranium loadings of the samples were calculated based on the measured meat densities at 298 K. The density of the IL was estimated by using the measured density and IL volume fraction. Thermal diffusivity and heat capacity of the samples after the heat treatment were measured as a function of temperature and volume fractions of U-Mo and IL. The thermal conductivity of IL-formed U-7Mo/Al was derived by using the measured thermal diffusivity, heat capacity, and density. The thermal conductivity obtained in the present study was lower than that predicted by the modified Hashin–Shtrikman model due to the theoretical model's inability to consider the thermal resistance at interfaces between the meat constituents.

Electrohydrodynamic fibrillation governed enhanced thermal transport in dielectric colloids under a field stimulus.

Electrorheological (ER) fluids are known to exhibit enhanced viscous effects under an electric field stimulus. The present article reports the hitherto unreported phenomenon of greatly enhanced thermal conductivity in such electro-active colloidal dispersions in the presence of an externally applied electric field. Typical ER fluids are synthesized employing dielectric fluids and nanoparticles and experiments are performed employing an in-house designed setup. Greatly augmented thermal conductivity under a field's influence was observed. Enhanced thermal conduction along the fibril structures under the field effect is theorized as the crux of the mechanism. The formation of fibril structures has also been experimentally verified employing microscopy. Based on classical models for ER fluids, a mathematical formalism has been developed to predict the propensity of chain formation and statistically feasible chain dynamics at given Mason numbers. Further, a thermal resistance network model is employed to computationally predict the enhanced thermal conduction across the fibrillary colloid microstructure. Good agreement between the mathematical model and the experimental observations is achieved. The domineering role of thermal conductivity over relative permittivity has been shown by proposing a modified Hashin-Shtrikman (HS) formalism. The findings have implications towards better physical understanding and design of ER fluids from both 'smart' viscoelastic as well as thermally active materials points of view.