Hashin Shtrikman Research Articles

Effective field hypothesis in Hashin–Shtrikman bounds estimations for effective moduli of composites with noncanonical shape of inclusions

One consider linear elastic composite materials (CMs), which consist of a homogeneous matrix containing a statistically homogeneous random set of identical aligned inhomogeneous inclusions of non-canonical (i.e. nonellipsoidal) shape. Estimates of the Hashin-Shtrikman type are developed by extremizing of the classical variational functional involving a general integral equation (GIE) connecting the strain fields in the point being considered with the strain fields in the surrounding points. An improvement of the approach proposed is based on the use of a piece wise constant approximation of the trial effective field rather than the trial polarisation tensor that, in its turn, allows one to take into account any inhomogeneity of the strain distributions inside the inclusions. Due to a sensitivity of the proposed approach to the inhomogeneity of these strain distributions, a final Hashin-Shtrikman bounds representation demonstrates more significant dependence on the shape of noncanonical inclusions than the corresponding classical bounds (that is shown by the concrete numerical examples).

Micromechanical modeling of 3D printable interpenetrating phase composites with tailorable effective elastic properties including negative Poisson's ratio

3D printable two-phase interpenetrating phase composites (IPCs) are designed by embedding a 3D periodic re-entrant lattice structure (as the reinforcing phase) in a matrix phase. These IPCs display the cubic or tetragonal symmetry. A micromechanical model is developed to evaluate effective elastic properties of the IPCs. Effective Young's moduli, shear moduli and Poisson's ratios (PRs) of each IPC are determined from the effective stiffness and compliance matrices of the composite, which are obtained through a homogenization analysis using a unit cell-based finite element (FE) model incorporating periodic boundary conditions. The FE simulation results are also compared with those based on various analytical bounding techniques in micromechanics, including the Voigt–Reuss, Hashin–Shtrikman, and Tuchinskii bounds. The effective properties of the IPC can be tailored by adjusting five geometrical parameters, including two strut lengths, two re-entrant angles and one strut diameter, and elastic properties of the two constituent materials. The numerical results reveal that IPCs with a negative PR can be generated by using a compliant matrix material and large re-entrant angles. In addition, it is found that the two re-entrant angles can greatly affect other effective elastic properties of the IPC: the effective shear modulus can be enhanced, while the effective Young's modulus can be enhanced or compromised with the increase of the re-entrant angles. Furthermore, it is seen that by adjusting one of the two re-entrant angles or one of the two strut lengths, the material symmetry exhibited by the IPC can be changed from cubic to tetragonal.

Model order reduction of nonlinear homogenization problems using a Hashin–Shtrikman type finite element method

This work presents a computational nonlinear homogenization approach, the starting point of which is a model order reduction method based on data-clustering. To this end, the micromechanical data from numerical experiments (snapshots) is analyzed in order to identify characteristic microstructural deformation patterns. These describe how the macroscopic strain typically localizes within the microstructure. The outcome of the procedure is a subdivision of the microstructure into a set of clusters of material points. Within each cluster the strain is then approximated as being constant.The mechanical problem is formulated in terms of a three-field Hashin–Shtrikman type variational formulation which is based on the introduction of a linear-elastic reference medium. After discretization, most of the global unknowns can be eliminated via static condensation leaving the piecewise constant cluster strains as the primary unknowns. The resulting homogenization scheme includes, as special cases, the finite element method as well as Hashin–Shtrikman and Talbot–Willis type homogenization approaches with phase-wise constant trial fields (as well as related bounds). The limit case ’finite element method’ allows to transfer knowledge from finite element technology and thus provides new strategies for the choice of the stiffness of the reference material. The method is applied to several nonlinear microstructures with different inclusion volume fractions and varying degree of anisotropy. The results are shown to be in good agreement with full-field FE-simulations. Furthermore, the method is used to compute a refined upper bound of the Talbot–Willis type (compared to phase-wise constant trial fields), which converges to the finite element solution with increasingly refined discretization.

High-stiffness and strength porous maraging steel via topology optimization and selective laser melting

Recent additive manufacturing technologies can be used to fabricate porous metals with precise internal pore structures and effective performance. We use topology optimization to derive an optimal pore structure shape with high stiffness that is verified experimentally. The design maximizes the effective bulk modulus and isotropic stiffness, and the performance is compared with Hashin–Shtrikman (HS) bounds. The optimized structure is fabricated via selective laser melting of maraging steel, which is a high-strength, iron-nickel steel that cannot easily be made porous with conventional methods. The optimal porous structure achieved 85% of the performance of the HS upper bound in numerical simulations, and at least 90% of them were realized in compressive testing. Finally, the performance is discussed relative to that of other metals.

Acoustic and Petrophysical Evolution of Organic-Rich Chalk Following Maturation Induced by Unconfined Pyrolysis

Thermal maturation is known to influence the rock physics of organic-rich rocks. While most studies were performed on low-porosity organic-rich shales, here we examine the effect of thermal maturation on a high-porosity organic-rich chalk. We compare the physical properties of native state immature rock with the properties at two pyrolysis-simulated maturity levels: early-mature and over-mature. We further evaluate the applicability of results from unconfined pyrolysis experiments to naturally matured rock properties. Special attention is dedicated to the elastic properties of the organic phase and the influence of bitumen and kerogen contents. Rock physics is studied based on confined petrophysical measurements of porosity, density and permeability, and measurements of bedding-normal acoustic velocities at estimated field stresses. Geochemical parameters like total organic carbon (TOC), bitumen content and thermal maturation indicators are used to monitor variations in density and volume fraction of each phase. We find that porosity increases significantly upon pyrolysis and that P wave velocity decreases in accordance. Solids density versus TOC relationships indicate that the kerogen increases its density from 1.43 to 1.49 g/cc at the immature and early-mature stages to ~ 2.98 g/cc at the over-mature stage. This density value is unusually high, although increase in S wave velocity and backscatter SEM images of the over-mature samples verify that the over-mature kerogen is significantly denser and stiffer. Using the petrophysical and acoustic properties, the elastic moduli of the rock are estimated by two Hashin–Shtrikman (HS)-based models: “HS + BAM” and “HS kerogen.” The “HS + BAM” model is calibrated to the post-pyrolysis measurements to describe the mechanical effect of the unconfined pyrolysis on the rock. The absence of compaction in the pyrolysis process causes the post-pyrolysis samples to be extremely porous. The “HS kerogen” model, which simulates a kerogen-supported matrix, depicts a compacted version of the matrix and is believed to be more representative of a naturally matured rock. Rock physics analysis using the “HS kerogen” model indicates strong mechanical dominance of porosity and organic content, and only small maturity-associated effects.

On the thermal conductivity of CGI and SGI cast irons

The thermal conductivity of Compacted Graphite Iron (CGI) and spheroidal graphite iron (SGI) was established in the temperature range from room temperature up to 500 °C using the experimental thermal diffusivity, density and specific heat values. The influence of nodularity, graphite amount, silicon content and temperature on the thermal conductivity of fully ferritic high-silicon cast irons was investigated. It was found that the CGI materials showed higher thermal conductivity than the SGI materials. The thermal conductivity tended to increase with increasing temperature until it reached a maximum followed by a subsequent decrease as temperature was increased up to 500 °C. Conventional models were applied to estimate thermal conductivity and the predictive accuracy of each model was evaluated. The thermal conductivity could be estimated by the Helsing model. The Maxwell model, Bruggeman model and Hashin–Shtrikman model were also in fair agreement using the thermal conductivity value of graphite parallel to the basal planes in graphite.

Thermal conductivity of polymer composite pigmented with titanium dioxide

The aim of this work is to provide a numerical modeling of thermal conductivity of a polymer matrix polystyrene composite filled with titanium dioxide spheres, and to compare the obtained results with theoretical prediction models and the experimental data as a function of the quenching temperature. For this purpose, a numerical study was conducted using the finite element method to predict the effective thermal conductivity of the composite. In addition, a comparison with the results from the analytical models showed that the proposed numerical model is in good agreement with the analytical models of Hatta–Taya and Hashin–Shtrikman. Finally, the comparison of the numerical model to experimental results based on the quenching temperature shows that the best quenching temperature that agrees well with the theoretical model Hashin–Shtrikman is 20 °C.

On Hashin–Shtrikman-type bounds for nonlinear conductors

For linear composite conductors, it is known that the celebrated Hashin–Shtrikman bounds can be recovered by the translation method. We investigate whether the same conclusion extends to nonlinear composites in two dimensions. To that purpose, we consider two-phase composites with perfectly conducting inclusions. In that case, explicit expressions of the various bounds considered can be obtained. The bounds provided by the translation method are compared with the nonlinear Hashin–Shtrikman-type bounds delivered by the Talbot–Willis (1985) [2] and the Ponte Castañeda (1991) [3] procedures.

Different arrangements of simplified models to predict effective thermal conductivity of open-cell foams

It is often desirable to predict the effective thermal conductivity (ETC) of a homogenous material like open-cell foams based on its composition, particularly when variations in composition are expected. A combination of five fundamental simplified thermal conductivity bounds and models (series, parallel, Hashin–Shtrikman, effective medium theory, and reciprocity models) is proposed to predict ETC of open-cell foams. Usually, these models use a parameter as the weighted mean to account the proportion of each bound arranged in arithmetic and geometric schemes. Based on ETC data obtained on numerous virtual Kelvin-like foam samples, the dependence of this parameter has been deduced as a function of morphology and phase thermal conductivity ratio. Various effective thermal conductivity correlations are derived based on material properties and foam structure. This is valid for open-cell foams filled with any arbitrary working fluid over a solid conductivity of materials range ( $$\lambda_{s} /\lambda_{f}$$ = 10–30,000) and over a wide range of porosity (0.60 $$< \varepsilon_{o} <$$ 0.95). Arrangement of series and parallel models together using the simplest models for both, arithmetic and geometric schemes, is found to predict excellent results among all the generic combinations.

Effective interfacial conditions for the Stokes flow of a fluid on periodically rough surfaces

This work is concerned with the Stokes flow of a fluid through the channel between two parallel solid walls, of which one is smooth and homogeneous and the other is periodically rough and heterogeneous. The main purpose of this work is to homogenize the resulting periodically rough solid/fluid interface so as to replace it by an equivalent smooth homogeneous solid/fluid interface characterized by appropriate effective slip lengths. To achieve this objective, a semi-analytic approach is elaborated by carrying out the Fourier expansions of the velocity and pressure fields in the plane normal to the thickness direction of the channel and by resorting to the point collocation method. In comparison with the relevant approaches reported in the literature, the present one has the advantage of being valid for the general case where the distance between the two parallel walls is arbitrary. With the help of the elaborated approach, the effective slip lengths are determined for a variety of periodically rough surface microstructures including, for example, longitudinal grooves, transverse grooves, chessboard texture and the Hashin–Shtrikman’s nested circles. The results obtained by the proposed semi-analytical approach are compared systematically with the ones provided by the finite element method and discussed in the light of some available bounds or analytical results. It follows from these comparisons and discussions that our semi-analytical approach is particularly efficient and accurate.

Hill–Mandel condition and bounds on lower symmetry elastic crystals

Despite advances in contemporary micromechanics, there is a void in the literature on a versatile method for estimating the effective properties of polycrystals comprising of highly anisotropic single crystals belonging to lower symmetry class. Basing on variational principles in elasticity and the Hill–Mandel homogenization condition, we propose a versatile methodology to fill this void. It is demonstrated that the bounds obtained using the Hill–Mandel condition are tighter than the Voigt and Reuss [1,2] bounds, the Hashin–Shtrikman [3] bounds as well as a recently proposed self-consistent estimate by Kube and Arguelles [4] even for polycrystals with highly anisotropic single crystals.

Negligible microstructure in a class of porous nonlinear materials via variational bounds

This work is devoted to the estimation of the effective energy density of porous nonlinear materials by means of variational bounds. Because of the unavailability of an improved upper bound, attention is turned to the improved bound obtained by considering constant nonzero polarization fields in the formal lower bound, which follows from the generalization to nonlinear behavior of the Hashin–Shtrikman variational principles. A particular class of nonlinearity is considered which is relevant in different physical situations. Also, 11 different microstructures are considered. Several computational experiments performed show that the elementary upper bound and the improved lower bound are indistinguishable, suggesting that nonlinearity dominates over the microstructural effects. In other words, at least for the nonlinearity considered here, the influence of microstructure on the effective behavior is negligible. So, in this case, there is no need to search for an improved upper bound, as the elementary one provides a simple but accurate estimate of the effective energy density. Copyright © 2017 John Wiley &amp; Sons, Ltd.

Homogenization and Materials Design of Anisotropic Multiphase Linear Elastic Materials Using Central Model Functions

It is shown in the present work that for linear elastic multiphase materials the orientation average of a stiffness tensor can be given in a closed form if central model functions are taken into consideration for the description of the material orientation distribution. This holds for arbitrary numbers of material constituents and for arbitrary degree of anisotropy of any of the constituents. The first-order bounds of Voigt and Reuss, the geometric average of Aleksandrov and Aisenberg and, specifically, the Hashin–Shtrikman bounds are given in closed forms depending on the single crystal stiffnesses of the material constituents, on their volume fractions and, for elasticity, on a central orientation and two scalar parameters per central model function. The central orientation and the two scalar parameters, referred to as texture eigenvalues in the present work, reflect the complete influence of the corresponding central model function on the elastic properties. The set of all admissible texture eigenvalues is derived. The closed form expressions can be used, e.g., for efficient and low dimensional homogenization procedures taking the crystallographic texture into account or, e.g., for the determination of estimates for unknown single crystal behavior. It is further shown, that central crystallite orientation distributions exist which conserve the anisotropy of the single crystal behavior but invert the direction of the anisotropy. The closed form expressions documented in the present work are used for an exemplary materials design problem in order to determine favorable volume fractions and texture eigenvalues for the orientation distributions of considered constituents for given prescribed anisotropic properties.

Bounds for nonlinear composite conductors via the translation method

Hashin–Shtrikman type bounds are proposed for nonlinear isotropic composite conductors in two dimensions. Those bounds are obtained by combining the translation method with the idea of embedding the original two-dimensional problem in an extended problem of dimension 6. Invariance properties allow the evaluation of the bounds to be dramatically simplified. Explicit results are obtained for the problem of dielectric breakdown. Numerical results are given for two-phase composites governed by power-law energy functions. The obtained bounds are shown to improve on the linear comparison bounds of the Hashin–Shtrikman type that are delivered by the Talbot-Willis (1985) approach and the Ponte Castañeda (1991) variational method.

Towards improved Hashin–Shtrikman bounds on the effective moduli of random composites

The celebrated bounds of Hashin and Shtrikman on the effective properties of composites are valid for a very wide class of materials. However, they incorporate only a very limited amount of information on the microstructure (volume fraction of each phase in the case of isotropic microstructures). As a result, they are generally not tight. In this work, we present an attempt at improving these bounds by incorporating explicitly the local volume fraction to the set of local descriptors of the microstructure. We show that, quite unexpectedly, the process fails in the sense that the classical bounds are retrieved. We further show that this negative result applies to so-called weakly isotropic local descriptors of the microstructure (to be defined in this paper). This suggests that improved bounds may be obtained with anisotropic descriptors.

Damage assessment of random multiwalled carbon nanotube-reinforced polymer nanocomposites

Abstract The paper presents a numerical procedure to evaluate the mechanical properties and predict the damage initiation of random multiwalled carbon nanotube-reinforced polymer nanocomposites (MWCNT-RPNC). The Hashin-Shtrikman (H-S) random prediction model is used to compute the properties of the reinforced polymer matrix, whereas the Chamis model is used to compute the lamina properties and the Hashin progressive damage model within the ABAQUS environment is used as a finite element analysis (FEA) tool to predict the damage initiation in the reinforced composite material. Experimental testing is employed to validate the numerical results and to adjust the H-S prediction model for MWCNT-RPNC.

Porous metal produced by selective laser melting with effective isotropic thermal conductivity close to the Hashin–Shtrikman bound

Additive manufacturing may be a novel method for fabricating porous materials. These materials can achieve effective performance because of their internal geometries. Metal-additive manufacturing is expected to utilize thermal conduction materials and devices. We have developed a porous metal with effective isotropic thermal conductivity by using metal-selective laser melting additive manufacturing. The internal pore structure was designed by topology optimization, which is the most effective structural optimization technique to maximize effective thermal conductivity. The designed structure was converted to a three-dimensional STL model, which is a native digital format of additive manufacturing, and assembled as a test piece. Effective thermal conductivity was measured by a steady-state method in which the effective thermal conductivity was calculated from a one-dimensional temperature gradient and the heat flux of the test pieces. The test pieces showed an effective thermal conductivity close to the Hashin–Shtrikman or Maxwell–Eucken bound, which is the theoretical limit of effective performance with an error less than 10%.

Mean field homogenization and experimental investigation of short and long fiber reinforced polymers

AbstractThe effective elastic material properties of short fiber reinforced polypropylen are determined by means of the self‐consistent (SC) method and the interaction direct derivative (IDD) method. In order to account for thermoelastic effective material properties, a Hashin‐Shtrikman (HS) based two‐step homogenization method with variable reference stiffness is used. The influence of the reference stiffness, dependent on a scalar parameter is investigated. Information on the microstructure are derived by computed tomography scans (µCT) and considered within the homogenization schemes. Thermomechanical properties of a long fiber reinforced polymer (LFRP) and a short fiber reinforced polymer (SFRP) are obtained by means of dynamic mechanical analysis (DMA). Simulation results for SFRP are compared to experimental results. (© 2016 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Evaluation of effective elastic properties of 3D printable interpenetrating phase composites using the meshfree radial point interpolation method

ABSTRACTInterpenetrating phase composites (IPCs) have recently been fabricated using three-dimensional (3D) printing methods. In a two-phase IPC, the two phases are topologically interconnected and mutually reinforced in the three dimensions. As a result, such IPCs exhibit higher stiffness, strength, and toughness than particle- or fiber-reinforced composites. In the current study, three unit cell models for the IPCs with the simple cubic (SC), face-centered cubic (FCC), and body-centered cubic (BCC) microstructures are developed using the meshfree radial point interpolation method. Radial basis functions with polynomial reproduction are applied to construct shape functions, and the Galerkin method is employed to formulate discretized equations. These unit cell-based meshfree models are used to evaluate effective elastic properties of 3D printable IPCs. The simulation results are compared with those based on the finite element (FE) method and various analytical bounding techniques in micromechanics, including the Voigt–Reuss, Hashin–Shtrikman, and Tuchinskii bounds. It is found that all of the simulation results for the effective Young's modulus and shear modulus fall between the Voigt–Reuss upper and lower bounds for each IPC considered, with the FE models predicting higher values than the meshfree models. In addition, it is seen that the SC microstructure leads to higher effective Young's modulus than the BCC and FCC microstructures. Furthermore, the numerical results reveal that the IPCs with the SC, BCC, and FCC microstructures can be approximated as isotropic materials (with the Zener anisotropic ratio varying between 0.9 and 1.0), with the BCC IPC being the most isotropic one, and the SC IPC being the least isotropic one among the three types of IPCs.

Sensitivity analysis of effective fluid and rock bulk modulus due to changes in pore pressure, temperature and saturation

Fluid substitution plays a vital role in time-lapse seismic modeling and interpretation. It is, therefore, very important to quantify as exactly as possible the changes in fluid bulk modulus due to changes in reservoir parameters. In this paper, we analyze the sensitivities in effective fluid bulk modulus due to changes in reservoir parameters like saturation, pore-pressure and temperature. The sensitivities are analyzed for two extreme bounds, i.e. the Voigt average and the Reuss average, for various fluid combinations (i.e. oil–water, gas–water and gas–oil). We quantify that the effects of pore-pressure and saturation changes are highest in the case of gas–water combination, while the effect of temperature is highest for oil–gas combination. Our results show that sensitivities vary with the bounds, even for same amount of changes in any reservoir parameter. In 4D rock physics studies, we often neglect the effects of pore-pressure or temperature changes assuming that those effects are negligible compare to the effect due to saturation change. Our analysis shows that pore-pressure and temperature changes can be vital and sometimes higher than the effect of saturation change. We investigate these effects on saturated rock bulk modulus. We first compute frame bulk modulus using the Modified Hashin Shtrikman (MHS) model for carbonate rocks and then perform fluid substitution using the Gassmann equation. We consider upper bound of the MHS as elastic behavior for stiffer rocks and lower bound of the MHS as elastic behavior for softer rocks. We then investigate four various combinations: stiff rock with upper bound (the Voigt bound) as effective fluid modulus, stiff rock with lower bound (Reuss bound) as effective fluid modulus, soft rock with upper bound as effective fluid modulus and soft rock with lower bound as effective fluid modulus. Our results show that the effect of any reservoir parameter change is highest for soft rock and lower bound combination and lowest for stiff rock and upper bound combination.