Three-phase Composite Model Research Articles

Analytical Solution for Predicting the Elastic Modulus of a Cement Slurry System with the Effect of Calcium Dissolution.

The dissolution of calcium ions in concrete in a low-alkalinity environment is an important factor causing a significant increase in the porosity of internal concrete, leading to a deterioration in its mechanical properties and affecting the durability of the concrete structure. In order to improve the reliability of concrete durability design and significantly increase the service life of concrete structures located in soft water environments, it is crucial to establish an analytical method to predict the elastic modulus (Edc) of cement slurry systems suffering from calcium dissolution. Firstly, the hydrated cement particles are regarded as a three-phase composite sphere composed of unhydrated cement particles (UC), a high-density hydrated layer (H-HL), and a low-density hydrated layer (L-HL). By introducing the equivalent inclusion phase (EQ) composed of UC and H-HL, the three-phase composite sphere model can be simplified into an equivalent hydrated cement particle model composed of EQ and L-HL. Finally, the Edc of the two-phase composite sphere composed of the equivalent hydrated cement particles and the porosity of the dissolved cement slurry system are solved by using elasticity theory. The effectiveness of the developed analytical method is verified by comparing it with third-party numerical results. Based on this method, the effects of hydration degree, volume ratio of calcium hydroxide (CH) to hydrated calcium silicate (C-S-H), and volume ratio of inner C-S-H to outer C-S-H on the Edc of the dissolved cement slurry system are analyzed. The parameter analysis indicates that among the three influencing parameters, the hydration degree has the greatest effect on the Edc of the dissolved cement slurry system. This study provides an analytical method for predicting Edc, which can provide some references for the durability design of concrete after calcium dissolution.

Structural Reconstruction of Hydrothermally Grown BaTiO3 Nanoparticles Based on High-Resolution Synchrotron Powder X-ray Diffraction

Barium titanate (BaTiO3) is a versatile material for industrial and research purposes owing to its various and excellent properties, including its ferroelectricity. The demand for ultrafine BaTiO3 nanoparticles via a robust synthesis route capable of mass production, such as hydrothermal reactions, has increased owing to the rapid advancement of highly integrated electronic components. In this study, BaTiO3 nanopowders smaller than 150 nm are synthesized using a facile hydrothermal method to obtain ultrafine and highly uniform nanoparticles. The synthesized nanopowders are analyzed via high-resolution synchrotron powder X-ray diffraction experiments. Based on Rietveld refinement studies, a three-phase composite model is shown to be reasonable for investigating the nanopowders. The analytical results are summarized in detail for a set of BaTiO3 nanopowders. Nanoparticles are reconstructed based on the thicknesses obtained from the calculated phase weight fractions for each case.

Electrical response, elastic property, and pressure sensing under bending of hybrid graphene/CNT/elastomer nanocomposites

The hybrid graphene/carbon nanotube (CNT)/elastomer nanocomposite is studied as a pressure sensor under bending. The main idea is that, due to the uneven elastic deformation between the softer elastomer matrix and the stiffer nanofillers, the volume fractions of graphene and CNTs both increase as a result of bending, and this in turn triggers higher electrical conductivity and superb sensing performance of the nanocomposite. This effect is analyzed through a three-phase composite model that contains graphene nanoplatelets (GNPs) and CNTs in a fluorinated elastomer matrix. We also show how the elastic and electrical properties of the hybrid nanocomposite depend on the graphene and CNT volume fractions. It is demonstrated that, with the increase of bending pressure from 0 to 1 KPa, the sensor resistance reduced by several orders of magnitude, which makes the hybrid sensor conductive from an insulator and an ideal pressure sensor.

Experimental and numerical analysis of crack propagation in reinforced concrete structures using a three-phase concrete model

A new multi-scale numerical method is proposed to simulate crack propagation in reinforced concrete structures. First of all, a three-phase composite material model is used to build the concrete structures at meso-level. And a modified Weibull distribution model is introduced to describe further the microscopic characteristics of concrete. Then different constitutive relations are adopted for components of concrete and reinforcing bars. Next, the numerical program written in ANSYS parametric design language and tool control language is executed to investigate the fracture behavior of reinforced concrete structures. The program, which can check the development of multiple cracks, is greatly improved by contrast with traditional methods. Finally, relevant experiments are conducted for comparing with the numerical results. The analysis results indicate that the program has strong robustness. Moreover, a good agreement is achieved between simulation results and experimental observation.

Towards Sophisticated 3D Interphase Modelling of Advanced Bionanocomposites via Atomic Force Microscopy

Nanomechanical properties and interphase dimensions of PVA bionanocomposites reinforced with halloysite nanotubes (HNTs) and Cloisite 30B montmorillonite (MMT) were evaluated by means of peak force quantitative nanomechanical mapping (PFQNM). A three-phase theoretical composite model was established based on hard-core–soft-shell structures consisting of hard mono-/polydispersed anisotropic particles and soft interphase and matrices. Halpin-Tsai model and Mori-Tanaka model were employed to predict experimentally determined tensile moduli of PVA bionanocomposites where effective volume fraction of randomly oriented nanoparticles resulted from the inclusion of interphase properties and volume fractions. Overall, it was suggested that the estimation of elastic modulus according to effective volume fraction of nanoparticles revealed better agreement with experimental data as opposed to that based upon their nominal volume fraction. In particular, the use of polydispersed HNTs and Cloisite 30B MMT clays with Fuller particulate gradation was proven to yield the best prediction when compared with experimental data among all proposed theoretical models. This study overcomes the neglected real interphase characteristics in modelling nanocomposite materials with much more accurate estimation of their mechanical properties.

Insight into the inherent randomness of concrete properties using the stochastic micromechanics

The concrete properties inherently fluctuate even using the same manufacturing process. A Legendre polynomial based micromechanical framework is proposed to investigate the inherent randomness of the concrete’s elastic behavior. The three-phase composite model is employed to represent the concrete material at microscale level. The volume fractions of different constituents are reached using a simplified analytical approach. Predictions with and without particle interactions for the concrete properties are obtained through the two-level homogenization process. By describing the microstructures with high-dimensional random vector, the stochastic micromechanical framework is obtained based on the presented three-phase model to incorporate the concrete’s inherent randomness. The unbiased probability density functions (PDFs) of the properties are attained using efficient simulation framework consisting of the Legendre polynomial approximations, Monte Carlo simulations and the linear transformations. Numerical examples indicate that the proposed framework is able to describe the probabilistic characters of concrete properties from microscale level and obtain the material’s unbiased PDFs with high efficiency. Finally, the statistical effects of ITZ are discussed on the concrete’ s macroscopic probabilistic behaviors.

Meso-scale modelling of stress effect on chloride diffusion in concrete using three-phase composite sphere model

At mesoscopic scale of modelling, concrete can be conceived as a three-phase composite material which is composed of coarse aggregate, mortar matrix and interfacial transition zone (ITZ) between the coarse aggregate and the mortar. This paper proposes a numerical model of stress effect on chloride diffusion in concrete from the meso-scale perspective. The concrete is idealised by using the three-phase composite sphere model. Based on the theory of elasticity, the constitutive relationships between the porosity of mortar, porosity of ITZ and the stress of concrete are established. Subsequently, the relationship among apparent chloride diffusion coefficient of concrete, chloride diffusion coefficients of mortar and ITZ, and volume fractions of three phases is established, and an apparent chloride diffusion coefficient model with respect to the elastic deformation stage of concrete is obtained. Finally, to extend the applicability of the model to a wide range of concrete stress levels under tension and compression, the model is modified to cater for the inelastic deformation stage of concrete. The modified model is applied to evaluate the apparent chloride diffusion coefficient of experimental specimens in the literature, and good agreement between experimental data and computed results is achieved. For concrete under tensile stress, the maximum prediction error of the modified model in the elastic stage and inelastic stage of concrete is 8.1% and − 11.6%, respectively. For concrete under compressive stress, the maximum prediction error of the modified model in the elastic stage and inelastic stage of concrete is − 8.2% and − 9.3%, respectively. Hence, the modified model is of high accuracy and can serve as a useful tool for durability design and service life prediction of concrete structures, especially those susceptible to chloride attack.

Pore structure and chloride diffusivity of recycled aggregate concrete with nano-SiO2 and nano-TiO2

Nano-SiO2 and nano-TiO2 particles were mixed into recycled aggregate concrete (RAC) to enhance its properties. Mercury intrusion porosimetry was used to test the pore structure of the mortar in RAC, and the rapid chloride migration instrument was selected to investigate the diffusivity of RAC. It is found that the addition of nanoparticle refines the pore structure of RAC and enhances the resistance to chloride diffusivity of RAC. Furthermore, a three-phase composite sphere model for the matrix in RAC was proposed by considering the new mortar, old attached mortar and original aggregate as continuous phases. The three-phase model considering the microstructure of RAC was verified by experiment results, which can be applied to predict the chloride diffusivity of RAC with nanoparticles.

Asymptotic Analysis of Fiber-Reinforced Composites of Hexagonal Structure

The fiber-reinforced composite materials with periodic cylindrical inclusions of a circular cross-section arranged in a hexagonal array are analyzed. The governing analytical relations of the thermal conductivity problem for such composites are obtained using the asymptotic homogenization method. The lubrication theory is applied for the asymptotic solution of the unit cell problems in the cases of inclusions of large and close to limit diameters, and for inclusions with high conductivity. The lubrication method is further generalized to the cases of finite values of the physical properties of inclusions, as well as for the cases of medium-sized inclusions. The analytical formulas for the effective coefficient of thermal conductivity of the fiber-reinforced composite materials of a hexagonal structure are derived in the cases of small conductivity of inclusions, as well as in the cases of extremely low conductivity of inclusions. The three-phase composite model (TPhM) is applied for solving the unit cell problems in the cases of the inclusions with small diameters, and the asymptotic analysis of the obtained solutions is performed for inclusions of small sizes. The obtained results are analyzed and illustrated graphically, and the limits of their applicability are evaluated. They are compared with the known numerical and asymptotic data in some particular cases, and very good agreement is demonstrated.

Three-phase model for a composite material with cylindrical circular inclusions. Part I: Application of the boundary shape perturbation method

The problem of determining the effective thermal conductivity of a composite material with periodic cylindrical inclusions of a circular cross-section arranged in a square grid is analyzed. Defining mathematical relationships are derived on the basis of a three-phase composite model, asymptotic homogenization technique and application of the boundary shape perturbation method to solve the derived unit cell problems. The analytical expression for the effective coefficient of thermal conductivity is obtained in the zero-order approximation and the correction to this expression is derived in the first-order approximation. This correction allows taking into account the geometry of inclusion, not just its volume fraction. It is shown that a small ε1-order perturbation of the unit cell contour yields the ε12-order contribution to the homogenized relations. The obtained solution is analyzed and compared with known results in some particular cases, and the limits of its applicability are evaluated.

Three-phase model for a composite material with cylindrical circular inclusions. Part II: Application of Padé approximants

The solution of the Three-phase composite model (coinciding with the Maxwell Garnett formula) is transformed using the Padé approximants. The obtained Padé approximants fundamentally expand the applicability limits of the Maxwell Garnett solution. The explicit analytical formulae for the effective coefficient of thermal conductivity of the composite materials with periodic cylindrical inclusions of a circular cross-section are derived. The developed Padé approximants provide the accurate quantitative results as well as a qualitative picture of the asymptotic behavior for the effective coefficient of thermal conductivity for composites with inclusion of any conductivity, including the limiting cases of non-conducting and absolutely conducting inclusions; and with inclusions of small and large sizes, including very large, close to the limiting sizes.

Application of an improved three-phase model to calculate effective characteristics for a composite with cylindrical inclusions

A modified three-phase composite model yielding reliable effective characteristics of composite structures has been proposed. In particular, the problem of effective heat transfer coefficient of the composite structure with periodically located inclusions of circular cross-sections located on a square net is solved. Advantages of the proposed model in comparison to the classical three-phase model are illustrated and discussed.

Three-phase model for a fiber-reinforced composite material

The fiber-reinforced composite materials with cylindrical inclusions of a square cross-section are analyzed using three-phase composite model combined with the asymptotic homogenization. The problem of thermal conductivity of such composite structure is solved in the zero- and first-order approximations using the boundary shape perturbation method. And the explicit analytical expression for the effective conductivity is derived. Analysis of the effective thermal conductivity is carried out and the formulae for the effective conductivity are obtained over the entire range of geometric and physical properties of the composite material. The obtained asymptotic solution is compared with the known numerical and analytical results. It is shown that three-phase model for a composite structure with cylindrical inclusions of a square cross-section provides accurate solution for the effective thermal conductivity in the entire range of sizes of the inclusions and their thermal conductivities, including the limiting cases.

Effect of Aggregate Shape on the Chloride Diffusivity of Concrete

The effect of aggregate shape on the chloride diffusivity of concrete is studied. To represent the heterogeneity of concrete, a three-phase composite ellipse model is constructed and the relative dimensions are determined by calculating the area fraction of interfacial transition zone in a numerical manner. The height of each bar in the lattice mesh is derived analytically. The lattice method is then used to estimate the chloride diffusivity of concrete. After the validity of the numerical method is verified with experimental results, the effect of aggregate shape on the chloride diffusivity of concrete is evaluated in a quantitative manner.

Effect of Aggregate Shape on the Chloride Diffusivity of Concrete

The effect of aggregate shape on the chloride diffusivity of concrete is studied. To represent the heterogeneity of concrete, a three-phase composite ellipse model is constructed and the relative dimensions are determined by calculating the area fraction of interfacial transition zone in a numerical manner. The height of each bar in the lattice mesh is derived analytically. The lattice method is then used to estimate the chloride diffusivity of concrete. After the validity of the numerical method is verified with experimental results, the effect of aggregate shape on the chloride diffusivity of concrete is evaluated in a quantitative manner.

A Prediction Method of Tensile Young's Modulus of Concrete at Early Age

Knowledge of the tensile Young's modulus of concrete at early ages is important for estimating the risk of cracking due to restrained shrinkage and thermal contraction. However, most often, the tensile modulus is considered equal to the compressive modulus and is estimated empirically based on the measurements of compressive strength. To evaluate the validity of this approach, the tensile Young's moduli of 6 concrete and mortar mixtures are measured using a direct tension test. The results show that the tensile moduli are approximately 1.0–1.3-times larger than the compressive moduli within the material's first week of age. To enable a direct estimation of the tensile modulus of concrete, a simple three-phase composite model is developed based on random distributions of coarse aggregate, mortar, and air void phases. The model predictions show good agreement with experimental measurements of tensile modulus at early age.

ITZ volume fraction in concrete with spheroidal aggregate particles and application: Part II. Prediction of the chloride diffusivity of concrete

Owing to its importance in the durability assessment of reinforced concrete structures located in a marine or de-icing salt environment, it is essential to determine the chloride diffusivity of concrete. The intention of the present paper is to present an analytical solution for the chloride diffusivity of concrete with aggregate shape effect. A three-phase composite spheroid model for the concrete matrix is proposed to represent the heterogeneous nature of concrete and to take into account the aggregate shape effect. By reducing the three-phase composite spheroid to two two-phase composite spheroids, an analytical solution for the chloride diffusivity of concrete is derived. After the validity of the derived analytical solution is verified with two sets of experimental results, the effects of main factors that affect the chloride diffusivity of concrete are evaluated in a quantitative manner. It is found that the chloride diffusivity of concrete increases with the increase of the thickness and chloride diffusivity of the interfacial transition zone (ITZ), but decreases with the increase of the aspect ratio of spheroidal particles and the maximum aggregate diameter. It is also found that the aggregate gradation has a significant influence on the chloride diffusivity of concrete. The paper concludes that the analytical solution derived here can predict the chloride diffusivity of concrete with reasonable accuracy.

Micro-scale modelling of bovine cortical bone fracture: Analysis of crack propagation and microstructure using X-FEM

Bone-fracture susceptibility is increased by factors such as bone loss, microstructure changes, and variations in material properties. Therefore, investigation of the effect of microstructure and material properties of bone on crack propagation in it and of its global response at macro-scale is important. A non-uniform distribution of osteons in a cortical bone tissue results in a localization of deformation processes. Such localization can affect bone’s performance under external loads and initiate fracture or assist its propagation. Once the fracture initiates, that distribution can play an important role in the crack propagation process at micro-scale; subsequently, the global response at macro-scale could also be affected. In this study, a two-dimensional numerical (finite-element) fracture model for osteonal bovine cortical bone was developed with account for its microstructure using extended finite element method (X-FEM). The topology of a transverse-radial cross section of a bovine cortical bone was captured using optical microscopy. Mechanical properties for the bone’s micro-structural features in the cross section were obtained with a use of the nanoindentation technique. Both the topology and nanoindentation data were used as input to the model formulated with the Abaqus finite-element software. The area, directly reflecting micro-scale information, was embedded into the region with homogenised properties of the cortical bone. Numerical simulations provide the macro-scale global response, crack propagation paths and distribution of maximum principal stress at the microscopic level for three different topologies – homogeneous, three-phase composite and four-phase composite model under tensile loading conditions.The calculated stress fields for various cases of topologies demonstrate different patterns due to implementation of micro-structural features in the finite-element models, confirming an important role of the microstructure in the crack propagation scenarios. The suggested approach emphasizes the importance of micro-structural features, especially cement lines, in development of bone failure.

Assessing the influence of ITZ on the steady-state chloride diffusivity of concrete using a numerical model

In this study, the influence of the aggregate-cement paste interfacial transition zone (ITZ) on the steady-state chloride diffusivity of mortars and concretes was examined using a semi-empirical, three-phase composite sphere model. Mortars and concretes were modelled as three-phase composites consisting of the aggregate, bulk cement paste and an inhomogeneous ITZ. The latter was divided into a series of homogenous concentric shell elements of equal thickness. The initial porosity and cement gradients at the ITZ were first estimated from the overall water/cement ratio ( w 0/ c). The evolution of the porosity, solid hydration products and remnants of unreacted cement were then calculated from the hydration degree and local water/cement ratio ( w/ c) using Powers' empirical model. Based on the Laplacian equation, an element transfer matrix was derived analytically to predict the steady-state chloride diffusivity. The model was calibrated using the available experimental data and then applied to perform a sensitivity analysis to evaluate the effects of aggregate content, water/cement ratio, curing period, ITZ width, maximum aggregate size and aggregate gradation on diffusivity. Some of these variables are impractical to quantify by laboratory experimentation. Implications of the findings with regard to the role of ITZ on mass transport properties are discussed.

A numerical algorithm for the ITZ area fraction in concrete with elliptical aggregate particles

In view of the importance of the interfacial transition zone (ITZ) to the macro-properties of concrete, it is essential to determine the ITZ area fraction in concrete. Once the ITZ area fraction is known, the physico-mechanical properties of concrete can be predicted based on the three-phase composite model. The intention of the present paper is to develop a numerical algorithm for the ITZ area fraction in concrete with elliptical aggregate particles. By introducing a contact function for two ellipses, the elliptical aggregate particles with various sizes are distributed within a square element with periodic boundary conditions. The Monte Carlo method is then adopted to evaluate the ITZ area fraction. After verifying the developed numerical algorithm with the analytical solution for the ITZ area fraction in concrete with circular aggregate particles, the effects of key factors that affect the ITZ area fraction are examined in a quantitative manner. It is found in the paper that the ITZ area fraction increases with the increase of the ITZ thickness, but decreases with the increase of the maximum aggregate diameter and the ratio between the major and minor axes of elliptical aggregate particles. It is also found that the aggregate gradation has a significant influence on the ITZ area fraction. The paper concludes that the numerical algorithm presented in the paper can predict the ITZ area fraction with reasonable accuracy.