Equivalent Inclusion Research Articles

Thermoelastic behavior analysis of finite composites embedded in ellipsoidal inhomogeneities with inclusion-based boundary element method

The thermoelastic properties of composites play a pivotal role in various practical engineering challenges. This paper employs the inclusion-based boundary element method (iBEM) to analyze the thermoelastic field and predicts the influence of key parameters in the particle phase on composite properties. Utilizing the equivalent inclusion theory, material mismatch effects are characterized by eigenstrain and eigen-temperature gradient. Incorporating the boundary integral equation, the boundary effects are addressed, which facilitates the computation of internal field outcomes via stress and heat flux equivalent conditions. Validation against finite element and experimental data confirms the efficacy of iBEM in accurately simulating thermoelastic behavior and predicting effective properties of particle composites. Noteworthy findings indicate that optimal composite properties align with the long axis orientation of ellipsoidal particles, while the particle distribution significantly impacts the effective Poisson’s ratio.

On the thermo-visco-elastic behaviour of neat and aged PPS composites

Reinforced PPS thermoplastics exposed to variations in temperature and humidity are subject to aging and interface degradation. These phenomena can lead to ruptures due to the interphase weakening. In this study, a micromechanical model based on the equivalent inclusion model is implemented using the viscoelastic behaviours of the neat and aged grades identified by spectrometric analyses. The presented coated inclusion model allows to extract the viscoelastic behavior of the interphase in the dry-as-molded composite and in the aged composite by inverse method conducted for viscoelastic behavior. The presence of the interphase testifies to the degradation of the thermomechanical properties in the vicinity of the reinforcements. In the aged composite, the interphase also undergoes an aging phenomenon. Thus, the model compares the behavior of the interphase in the dry-as-molded and aged grades of the composite and separates the effects linked to the degradation of adhesion of the fibers from the effects linked to specific hydroscopic aging.

Effective toughness based on Eshelby transformation theory for heterogeneous composites

Predicting fracture toughness of heterogeneous composites is an important and challenging problem in physics and mechanics. The dependence of effective toughness on elastic properties of phases remains unclear. Considering that energy plays an essential role in crack propagation, an energy approach is proposed to obtain effective toughness in this study. We built the relationship between effective toughness and the homogenized local surface energy. The energy is constructed by generalizing Eshelby’s equivalent inclusion formulation to heterogeneous case, which couples physical features with elastic properties. An analytical formula of effective toughness can be derived for heterogeneous composites. Based on this formula, effects of toughness and elastic properties of the phases are discussed in depth, which reveals that how elastic heterogeneity can influence the effective toughness fundamentally. It is demonstrated that the predictions of concretes and metal toughening glasses agree well with experimental evidences.

Interfacial performance evolution of ceramics-in-polymer composite electrolyte in solid-state lithium metal batteries

The incorporation of ceramics into polymers, forming solid composite electrolytes (SCEs) leads to enhanced electrical performance of all-solid-state lithium metal batteries. This is because the dispersed ceramics particles increase the ionic conductivity, while the polymer matrix leads to better contact performance between the electrolyte and the electrode. In this study, we present a model, based on Hybrid Elements Methods, for the time-dependent Li metal and SCE rough interface mechanics, taking into account for the oxide (ceramics) inclusions (using the Equivalent Inclusion method), and the viscoelasticity of the matrix. We study the effect of LLTO particle size, weight concentration, and spatial distribution on the interface mechanical and electrical response. Moreover, considering the viscoelastic spectrum of a real PEO matrix, under a given stack pressure, we investigate the evolution over time of the mechanical and electrical performance of the interface. The presented theoretical/numerical model might be pivotal in tailoring the development of advanced solid state batteries with superior performance; indeed, we found that conditions in the SCE mixture which optimize both the contact resistivity and the interface stability in time.

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.

Micromechanical analysis for effective elastic moduli and thermal expansion coefficient of composite materials containing ellipsoidal fillers oriented randomly

In this study, we examine to derive the solutions of effective elastic moduli and thermal expansion coefficient for composite materials containing ellipsoidal fillers oriented randomly in the material using homogenization theories, which are the self-consistent method and the Mori–Tanaka method. This analysis is carried out by micromechanics combining Eshelby’s equivalent inclusion method for each theory. The solutions for effective elastic moduli and thermal expansion coefficient obtained on each theory are expressed by common coefficients composed of both the physical properties of the constituents of the composite material and geometrical factors depending upon the shape of the fillers. Moreover, these solutions enable us to calculate effective elastic moduli and thermal expansion coefficient for composite materials that contain randomly oriented fillers of various shapes and physical properties. By taking the limit of eliminating the existence of the matrix for these solutions, we can derive effective physical properties of polycrystalline materials. Using the obtained solutions, we investigate the effects of the shape of the fillers on the effective elastic moduli and thermal expansion coefficient. As a result, we confirm that these effective properties fall within the lower and upper bounds, and find that a characteristic result appears when the shape of the fillers is flake or oblate. Through comparisons between the analytical and experimental results, we confirm the practical usability of the solutions obtained in this analysis. Furthermore, we determine originally the shape factor for the filler and can show that this factor has the potential to provide guidelines for the optimal design of filler shape to improve the effective elastic properties of materials.

A novel two-scale nonlinear damage accumulation model for vibration fatigue life prediction

Vibration is the main factor causing high cycle fatigue. The macro-response stress and strain of the structure under the excitation of vibration load are generally in the elastic stage. However, fatigue failure still occurs due to long-term cyclic load, which is unexplainable by the macro-plastic accumulation method. Therefore, a two-scale nonlinear damage accumulation model is proposed for vibration fatigue based on the physical characteristics of multi-scale fatigue damage evolution. In this model, two-scale elastic-plastic constitutive equations based on the equivalent inclusion theory are built, and a two-stage damage accumulation strategy with two critical damages is designed. A resonance fatigue life prediction algorithm is established based on the proposed model, and a validation test is carried out using the cantilever beam made of GH4169 alloy. The results show that the deviation between the vibration fatigue life prediction and test results is within a scatter band of factor 2, indicating that the two-scale nonlinear damage accumulation model is effective in the vibration fatigue region.

Analytical calculation model for elastic modulus and strength of foamed concrete

Foamed concrete has gained increasing attention for its excellent engineering performance. Endeavors have been made to experimentally and numerically elucidate the mechanism of foamed concrete, while few theoretical methods are used to analytically calculate mechanical properties. This paper proposes a theoretical framework to describe the elastic behavior of foamed concrete by combining Eshelby’s equivalent inclusion theory with the Mori–Tanaka method. In the analytical model, foamed concrete is regarded as a two-phase composite comprising a homogeneous cement matrix and randomly oriented ellipsoidal air void. During derivation, the elastic modulus, strength, bulk modulus, shear modulus, and Poisson ratio are measured. A good agreement is established between the proposed method and experimental data.

Fracture criteria and initiation mechanism of concrete based on material configurational mechanics

AbstractIn this study, the interaction force between an in‐plane elliptical inclusion and a crack is evaluated using the Eshelby equivalent inclusion and material configurational force method. The toughening effect of coarse aggregates on concrete is investigated. The fracture criteria for mixed‐mode cracks based on the characteristics of the crack‐tip plastic zone assessed by the principal configurational stress difference are proposed and verified by experimental results in the literature. The fracture probability is then used to improve the fracture criteria, and the dimensionless factor associated with stress intensity factors is introduced to determine the fracture probability. Additionally, the volume fraction of coarse aggregates, the age and fatigue life of concrete are considered in the present fracture criteria. Theoretical studies show that the interaction between a crack and a hard or soft inclusion exhibits mutual repulsion and attraction. The size and shape of the crack‐tip plastic zones assessed by the Mises stress and the principal configurational stress difference are essentially the same. The fracture load envelopes increase with the increase in the volume fraction of coarse aggregates and the age of concrete and shrink with the increase in the life ratio.

A macro-mesoscopic constitutive model for porous and cracked rock under true triaxial conditions

The complex mechanical and damage mechanisms of rocks are intricately tied to their diverse mineral compositions and the formation of pores and cracks under external loads. Numerous rock tests reveal a complex interplay between the closure of porous defects and the propagation of induced cracks, presenting challenges in accurately representing their mechanical properties, especially under true triaxial stress conditions. This paper proposes a conceptualization of rock at the mesoscopic level as a two-phase composite, consisting of a bonded medium matrix and frictional medium inclusions. The bonded medium is characterized as a mesoscopic elastic material, encompassing various minerals surrounding porous defects. Its mechanical properties are determined using the mixed multi-inclusion method. Transformation of the bonded medium into the frictional medium occurs through crack extension, with its elastoplastic properties defined by the Drucker–Prager yield criterion, accounting for hardening, softening, and extension. Mori–Tanaka and Eshelby’s equivalent inclusion methods are applied to the bonded and frictional media, respectively. The macroscopic mechanical properties of the rock are derived from these mesoscopic media. Consequently, a True Triaxial Macro-Mesoscopic (TTMM) constitutive model is developed. This model effectively captures the competitive effect and accurately describes the stress-deformation characteristics of granite. Utilizing the TTMM model, the strains resulting from porous defect closure and induced crack extension are differentiated, enabling quantitative determination of the associated damage evolution.

A data-driven model based on the numerical solution of the equivalent inclusion problem for the analysis of nonlinear short-fibre composites

The paper introduces a novel data-driven homogenization model developed for the computation of stress-strain curves for composites featuring arbitrary orientation distributions, aspect ratios, and volume fractions of fibres. The objective of the proposed model is to deliver both time-efficient and accurate results while taking into account a relatively small database. The database is generated through numerical solutions of the equivalent inclusion problem, encompassing results for predefined orientations and aspect ratios of the single inclusion (fibre). The database created for a specific matrix-fibre system can be employed by the surrogate model to predict outcomes for various cases involving different fibre configurations. The surrogate model comprises four key modules: aspect ratio interpolation, a discrete orientation averaging procedure involving a genetic algorithm, a spatial transformation procedure, and an iterative Mori-Tanaka scheme. The validity of the proposed data-driven model has been confirmed by comparing the obtained results with those acquired through analysis of the complex representative volume element (RVE) using Fast Fourier Transform (FFT)-based homogenization, as well as through comparisons with experimental results from the existing literature.

Performance Study of Black Shale Modified Soil for Road Use Based on Eshelby–Mori–Tanaka Theory

Black shale, as a type of soft rock, exhibits high strength when freshly exposed. However, it easily disintegrates upon contact with water, making it unsuitable for direct use in roadbed construction. Using it as discarded material not only increases construction costs but also pollutes the environment. Therefore, the reuse and modification of black shale have become particularly important. Based on the theory of composite material equivalent inclusions, this study investigates the strength and water stability characteristics of black shale gravel after being mixed with cement and compacted with clay. The results show that the strength of cemented soil increases linearly with the cement content. The water absorption properties of the modified soils with different amounts of black shale added are similar, with an average water absorption rate of about 2.53%. The strength of black shale modified soil is generally positively correlated with the cement content, although the linear correlation is not significant. The modified black shale soil used in the experiment is suitable for the subgrade of medium- and light-grade secondary roads and below. The recommended mass ratio is Mshale:Mclay:Mcement = 70:21:9. The unconfined compressive strength of the material under 7-day curing is 1.36 MPa. The relationship between the strength of modified soil, clay strength, cement content, and gravel addition has been established, clarifying the physical significance of each parameter. The “drying and soaking” cycle can accelerate the strength degradation of modified soil. It is recommended to strengthen the construction of roadbed drainage facilities during construction to maintain a stable and dry environment for the modified soil as a roadbed filling material. The research results not only provide clear technical indicators for the reuse of discarded black shale in engineering but also serve as a basis for proportion of crushed stone discarded material as roadbed fill.

Analytical modeling of the electrical conductivity of CNT-filled polymer nanocomposites

Electrical conductivity of most polymeric insulators can be drastically enhanced by introducing a small volume fraction [Formula: see text] of conductive nanofillers. These nanocomposites find wide-ranging engineering applications from cellular metamaterials to strain sensors. In this work, we present a mathematical model to predict the effective electrical conductivity of carbon nanotubes (CNTs)/polymer nanocomposites accounting for the conductivity, dimensions, volume fraction, and alignment of the CNTs. Eshelby’s classical equivalent inclusion method (EIM) is generalized to account for electron-hopping—a key mechanism of electron transport across CNTs, and is validated with experimental data. Two measurements, namely, the limit angle of filler orientation and the probability distribution function, are used to control the alignment of CNTs within the composites. Our simulations show that decreasing the angle from a uniformly random distribution to a fully aligned state significantly reduces the transverse electrical conductivity, while the longitudinal conductivity shows less sensitivity to angle variation. Moreover, it is observed that distributing CNTs with non-uniform probability distribution functions results in an increase in longitudinal conductivity and a decrease in transverse conductivity, with these differences becoming more pronounced as the volume fraction of CNTs is increased. A reduction in CNT length decreases the effective electrical conductivity due to the reduced number of available conductive pathways while reducing CNT diameter increases the conductivity.

Stress discontinuity and singularity around the vertex of a triangular inhomogeneity

The stress field around a vertex of angular inhomogeneity has been investigated by Eshelby’s equivalent inclusion method (EIM). Different from Eshelby’s problem for ellipsoidal inhomogeneity with a uniform or polynomial eigenstrain, a singular eigenstrain field is derived by Airy’s stress function and asymptotic analysis, in which the singular elastic fields can be expressed in terms of the distance to the vertex. The discontinuity of Eshelby’s tensor along the boundary has been analytically derived and are used in the stress equivalence condition at the vertex, which can be converted into an eigenvalue problem with the integral of the Green’s function and singular eigenstrain over the inhomogeneity. To verify the solution, when the opening angle 2β at the vertex approaches zero, the triangular void reduces to a slit-like crack, and the paper reproduces the classic solution of 1/r singularity. When β increases from 0 to π/2, the singularity parameter λ around the vertex of the triangular void reduces at different pace under the symmetric and antisymmetric loading conditions. When a triangular inhomogeneity exhibits nonzero stiffness and different angles, λ changes with the stiffness ratio, β, and loading conditions, and the dominant stress singularity around the vertex and stress discontinuity across the boundary are analytically provided. Particularly, the stress singularity for an adhesive interface with varying stiffness provides insight for structure repair and integration.

Analysis of adhesive contact of heterogeneous elastic materials

This paper reports a three-dimensional numerical model to solve adhesive elastic contact problem of heterogeneous materials with an engineered rough surface. This model integrates the Maugis-Dugdale (MD) adhesive contact model, the Boussinesq-Cerruti integral equations, the equivalent inclusion method (EIM), and the discrete convolution and fast Fourier transform (DC-FFT) algorithm, aiming at accurate determination of contact results for heterogeneous materials. An adhesive displacement-driven conjugate gradient method (CGM) was also developed to support rapid solutions of contact pressure and contact area. Four cases were analyzed to verify this model with satisfaction. This model was used to analyze the effects of inhomogeneity (location, size, and material properties) and adhesion on micro-adhesive-elastic contact, and the results lead to a curve-fitting equation for quick evaluation of the effects of inhomogeneity properties and adhesion characteristics on normal traction. It was also utilized to study the influence of the roughness level of a type of rough surface on the surface adhesion behavior.

Viscoelastic behavior of composite materials with multi-coated ellipsoidal reinforcements and imperfect interfaces modeled by an equivalent inclusion

Viscoelastic behavior of composite materials with multi-coated ellipsoidal reinforcements and imperfect interfaces modeled by an equivalent inclusion

Elasto-plastic partial slip contact modeling of graded layers

A calculation model is developed to solve elasto-plastic partial slip contact problems of graded layer-substrate systems. With the high dependence of contact stress on the local region near the contact spot, the graded layer-substrate system of interest is modeled as a homogeneous half-plane with an embedded, finitely long inclusion. Unlike the conventional finite element method, the elasto-plastic responses are solved within a considerably small region rather than an oversized truncated domain. The computational domain allows arbitrary distributions of elasto-plastic properties and is directly discretized without tedious separation and assembling skills. The resulting disturbance fields are calculated by the numerical equivalent inclusion method. Additionally, the plastic strain is determined via the return-mapping algorithm, and the generated residual responses are directly obtained based on the explicit theoretical solutions. The conjugate gradient method is utilized to solve surface tractions, and the induced elasto-plastic fields are calculated by the technique of fast Fourier transform. Some parametric studies are subsequently conducted based on the developed model to explore the effects of property inhomogeneity and friction coefficient on the elasto-plastic response.

A semi-analytical solution for inhomogeneous material in the quarter space

A semi-analytical model is first presented that thoroughly examines stress distribution patterns in inhomogeneous materials within a quarter space: a scenario of great significance and complexity observed in components such as rail-wheels, roller bearings, and gears. The model utilizes the Hetényi Image Method and Eshelby’s Equivalent Inclusion Method, effectively handling the impact of inhomogeneities and satisfying the boundary conditions of the quarter space. To ensure efficient and accurate analysis, numerical techniques are incorporated, notably the Conjugate Gradient Method and the Discrete Convolution-Fast Fourier Transform algorithm. The model’s validity and robustness are demonstrated through comparisons with the Finite Element Method results, displaying excellent agreement. Parametric analyses reveal that distance, depth to the free surfaces, and shape of the inhomogeneities significantly influence the results. The introduction of an edge decay coefficient, Keg, serves as a practical tool for rapid estimation of edge effects in inhomogeneous materials, significantly contributing to precise predictions of material fatigue and failure.

Thermoelastic analysis of a bi-layered system with the single domain inclusion-based boundary element method

When a bi-layered composite system is subjected to prescribed thermal and elastic boundary conditions, material mismatch of layers and inhomogeneities causes intensive stress concentration, particularly with an initial crack, which causes delamination of the system. This paper combines the boundary element method and our recent work on the dual equivalent inclusion method (DEIM), namely the inclusion-based boundary element method (iBEM), to simulate the thermoelastic behavior without the whole domain integral. Using Galerkin’s stress vector technique, the bimaterial thermoelastic Green’s function mathematically transforms the entire domain integral to boundary integral equations. Because the inhomogeneities and matrix generally exhibit thermal and mechanical material mismatch, the DEIM simulates the inhomogeneity problem by inclusion with continuously distributed polynomial-form eigentemperature and eigenstrain. Therefore, iBEM avoids conventional multi-region solution scheme and convert the entire domain into boundary integral equations plus domain integrals within the inhomogeneity domain. The energy release rate (J-integral) can be utilized to reveal and judge the stability of the crack. For elastic problems, when one of the layers becomes stiffer or is embedded with the stiffer inhomogeneity, the J-integral decreases, and the system is more stable. However, this conclusion is not established for thermoelastic problems. Moreover, locations, distribution, and material mismatches are essential in (modified) J-integrals. Numerical case studies validate the single-domain algorithm, and parametric studies illustrate the effects of inhomogeneities on the (modified) J-integral for a bi-layered system.

Modeling layered composite rock with excavated tunnels subjected to ground loads

Tunnels excavation in layered composite rock subjected to ground loads to make the best of underground space is an important engineering problem in more and more mega-cities. Such a three-dimensional case, layered composite rock with excavated tunnels, is usually simplified as a two-dimensional plane strain problem in mechanics. The present study reports a Semi-Analytical Method (SAM) to predict elastic field of an excavated layered composite rock with the assistance of the Equivalent Inclusion Method (EIM) that can handle elastic field of the layers and tunnels simultaneously. In the EIM, the layer and tunnel are considered as inhomogeneities of different elastic moduli from the base rock, and the inhomogeneities are further equivalent to inclusions of identical elastic modulus to the base rock but including eigenstrains. Effectiveness of the SAM is examined by the Finite Element Method (FEM) through comparing their results. Moreover, a yield approximate index (YAI) is introduced to evaluate safety of excavated layered composite rock and acoustic emission (AE) event is used to count the number of plastic yield elements. Finally influences of depth of and distance between the tunnels, thickness, elastic modulus, number of the layer, range of ground load on principal stress, YAI, plastic zones are discussed, which may have potentials to provide guidance for practical engineering applications of tunnel excavation.