Volume Fractions Of Constituents Research Articles

Mechanical behavior of porous functionally graded plates

In this study, the effect of porosity distribution pattern on the bending and free vibration analysis of porous FG plates is studied by using sinusoidal shear deformation theory. This theory account for sinusoidal variation of transverse shear strain through the depth of the plate and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. The material properties of the plate and the porosities within the plate are considered to vary continuously through the thickness direction according to the volume fraction of constituents defined by the modified rule of the mixture, this includes porosity volume fraction with four different types of porosity distribution over the cross-section. The governing partial differential equation of motion for the bending and free vibration analysis is obtained using sinusoidal shear deformation theory. An analytical solution is presented for the governing equation. Results of the presented solution are compared and validated by the available results in the literature. Moreover, the effects of material and porosity distribution and geometrical parameters on bending and free vibrational properties are investigated.

A micromechanical approach for homogenization of elastic multiphase periodic composites

This work presents a micromechanical model for the linear elastic homogenization of composites with periodic microstructures. The model is constructed for composites with an arbitrary number of phases and geometric shapes of the inclusions, differently from the most homogenization approaches available in the literature. In addition, no restriction is made with respect to the mismatch between the properties of the constituent phases and volume fractions of the inclusions. The model formulation is based on the Eshelby equivalent inclusion method and corresponds to an extension of a procedure originally derived to evaluate the effective elastic moduli of periodic composites with only two constituent phases. The fluctuating elastic fields within of the repeating unit cell (RUC) are represented by Fourier series, resulting in Lippmann-Schwinger integral equations involving the unknown eigenstrain fields of the inclusions. These integral equations are solved by using a straightforward approach that employs a scheme of partition of the domain of each inclusion. Applications to composites with different arrays of coated fibers and constituent materials are presented to show the efficiency of the model.

A boundary element solution for free vibration of FGM beams

The main purpose in this article is to derive a new solution based on Boundary Element Method (BEM) for dynamic analysis of Euler-Bernoulli beams made of composites such as Functionally Graded Materials (FGM) that have properties that vary continuously through the thickness direction according to the volume fraction of constituents. In contrast, laminated composite materials are oriented plies bonded together, where each ply may be made of a different material. A boundary element solution for any problem is obtained using integral equations and fundamental solutions defined on continuum, and algebraic equations for the discretized problem. In this paper, original discussions on deriving both integral equations and fundamental solutions for dynamic composite beam problems are properly made. Algebraic representation is also obtained, incorporating rigid and elastic end supports into the BEM solution. Numerical results of different cases of mechanical properties and boundary conditions are considered and compared to other published works.

Antiplane effective properties of two‐phase micropolar elastic fiber‐reinforced composites with parallelogram‐like unit cells

AbstractIn this contribution, heterogeneous micropolar elastic fiber‐reinforced composites (FRCs) with a periodic structure are analyzed using the two‐scale asymptotic homogenization method (AHM). We focus on predicting the antiplane effective properties of micropolar two‐phase FRCs with parallelogram‐like unit cells. The periodic structure is defined by unidirectional, infinitely long, and concentric cylindrical fibers embedded in a homogeneous matrix. Constituent materials are assumed centro‐symmetric isotropic materials, and perfect interface conditions are considered. The AHM allows us to address the local problems on the periodic cell and determine the corresponding effective properties. This is achieved by employing two‐scale asymptotic expansions for the displacement and microrotation fields, which depend on both macro‐ and micro‐scales. The complex variable theory, combined with the complex‐potential method and doubly periodic Weierstrass elliptic functions, is applied to determine the solution of the antiplane local problems. Simple closed‐form formulas are provided for the antiplane stiffness and torque effective properties of two‐phase micropolar elastic FRCs, which depend on the physical properties and volume fractions of constituents. Finally, numerical examples are reported and discussed. Comparisons with other theoretical models are also presented, and good agreements are obtained.

Impact of porosity distribution and grading parameters on free vibration of functionally graded beams

This study focuses on investigating the free vibration of porous functionally graded (FG) beams based on neutral surface position and the impact of porosity distribution on the properties and performance of materials. To achieve this, the study assumes that the beam’s material characteristics vary continuously along its thickness direction. The volume fraction of constituents is specified using the modified rule of the mixture, which includes porosity volume fraction with changed porosity patterns across the cross-section. The study uses the parabolic shear deformation theory without shear correction factors to derive the equations of motion by applying Hamilton’s principle. The study employs Navier’s method to obtain analytical results for the free vibration of porous FG beams. To verify the proposed formulation, the study compares the results with relevant findings from the literature. The study also examines the relationship between frequency of vibration and grading parameter in homogeneous porosity distributions, as well as the frequency of vibration in beam structures with different porosity patterns. Additionally, numerical examples are provided to investigate the impact of parameters like power-law index, span to depth ratio, porosity distribution pattern, and porosity volume fraction on the natural frequencies of FG beams.

Nonlinear vibrations of temperature-dependent FGM cylindrical panels subjected to instantaneous surface heating

An analysis is performed in this research to deal with the large amplitude vibrations caused by rapid surface heating in an FGM cylindrical panel. It is assumed that the properties of the panel are distributed through the thickness in terms of volume fraction of constituents. Following the first-order shear deformation theory, von Karman type of kinematics, and Donnell kinematic assumptions, definition of kinetic and strain energies of the shell are established. These obtained equations are discretized via the method of Ritz, where the shape functions are estimated by the Chebyshev polynomials. Temperature distribution within the body is also obtained using the one-dimensional heat conduction equation. This equation is discretized using the finite different method and solved iteratively following the Picard and Crank-Nicolson methods. Thermally induced force and moment resultants are evaluated and inserted into the matrix representation of equations of motion. The temporal evolution of displacement is obtained using the iterative Picard method and Newmark time marching method. Results of this study are compared with the available data in the open literature and after that novel results are given to explore the effects of geometrical, mechanical, and thermal parameters. It is highlighted that power law index, shallowness, thickness, edge supports, temperature dependency, geometrical nonlinearity and thermal boundary conditions are all important factors on the response of the structure under thermal shock. It is highlighted that thermally induced vibrations indeed exists especially in thin class of shells. Besides, thermally induced deflections may be controlled through using a proper power law index.

Effect of Microstructural Constituents on Hydrogen Embrittlement Resistance of API X60, X70, and X80 Pipeline Steels

This study describes how microstructural constituents affected the hydrogen embrittlement resistance of high-strength pipeline steels. The American Petroleum Institute (API) X60, X70, and X80 pipeline steels demonstrated complicated microstructure comprising polygonal ferrite (PF), acicular ferrite, granular bainite (GB), bainitic ferrite (BF), and secondary phases, e.g., the martensite-austenite (MA) constituent, and the volume fraction of the microstructures was dependent on alloying elements and processing conditions. To evaluate the hydrogen embrittlement resistance, a slow strain rate test (SSRT) was performed after electrochemical hydrogen charging. The SSRT results indicated that the X80 steel with the highest volume fraction of the MA constituent demonstrated relatively high yield strength but exhibited the lowest hydrogen embrittlement resistance because the MA constituent acted as a reversible hydrogen trap site.

A refined shell theory considering the effects of transverse normal strain for the analysis of functionally graded porous shallow shells with double curvature

In this paper, a static and free vibration analysis of functionally graded porous shallow shells with double curvature containing an even distribution of porosity is investigated using a trigonometric shear and normal deformation theory. Both the effects of transverse shear and normal strains are included in the present theory to investigate their effects on the responses of FGM porous shells. The theory satisfies zero transverse shear stress conditions at the top and bottom surfaces of the shell. Hamilton’s principle is used for obtaining governing equations and boundary conditions of the present theory for functionally graded porous shallow shells. The FGM porous shell is assumed isotropic at any point within the shell domain, with its elastic properties varying across its thickness by a power law in terms of the volume fractions of the shell constituents. The simply-supported FG porous shell is analyzed in the present study using the Navier method. The numerical results of FGM porous plates are obtained and compared with other higher-order theories available in the literature to verify the present theory. Further, the numerical results are presented for the FGM porous shells with double curvature. The effects of a/h ratios, porosity distribution factor, and radii of curvature on deflections, stresses, and fundamental frequencies are presented. The solution of hyperbolic and elliptical FGM porous shells is presented for the first time in this paper.

Vibration analysis of perforated functionally graded circular plates

Functionally graded materials have garnered significant attention due to their vast array of industrial applications. These substances possess a composition that is non-uniform, resulting in material characteristics that fluctuate across the depth, transitioning between different surfaces. In this research paper, our exploration delves into the analysis of unconstrained vibrations occurring in circular plates that are fully clamped, formed from materials with functional gradation. The influence of diverse elements like the thickness of the plate and the count of perforations on the intrinsic frequencies of the plate is examined. Additionally, a comparison between the inherent frequencies of functionally graded circular plates and their isotropic counterparts, aiming to distinguish disparities in performance characteristics is undertaken. In the present study, power law distribution is utilized to represent the modulation in material attributes, grounded in the volume fraction of constituents. Through this comprehensive investigation, we aim to gain insights into the intricate behavior of functionally graded materials and provide valuable information regarding the performance of the plate in practical applications.

Effect of Microstructural Constituents on Hydrogen Embrittlement Resistance of API X60, X70, and X80 Pipeline Steels

This study describes how microstructural constituents affected the hydrogen embrittlement resistance of high-strength pipeline steels. The American Petroleum Institute (API) X60, X70, and X80 pipeline steels demonstrated complicated microstructure comprising polygonal ferrite (PF), acicular ferrite, granular bainite (GB), bainitic ferrite (BF), and secondary phases, e.g., the martensite-austenite (MA) constituent, and the volume fraction of the microstructures was dependent on alloying elements and processing conditions. To evaluate the hydrogen embrittlement resistance, a slow strain rate test (SSRT) was performed after electrochemical hydrogen charging. The SSRT results indicated that the X80 steel with the highest volume fraction of the MA constituent demonstrated relatively high yield strength but exhibited the lowest hydrogen embrittlement resistance because the MA constituent acted as a reversible hydrogen trap site.

Stochastic reconstruction and performance prediction of cathode microstructures based on deep learning

The effective properties of lithium-ion battery (LIB) cathode are determined by both the volume fractions of constituents and the morphological features of microstructure. However, it is difficult to establish an accurate quantitative relationship between the macroscopic effective properties and microstructural features. Deep learning techniques, due to their exceptional nonlinear fitting capabilities, have been widely applied in various complex fields. Our study presents a generation scheme of numerous three-dimensional (3D) digital microstructures of cathode, using a deep convolutional neural network (CNN)-based stochastic reconstruction algorithm combining with the scanning electron microscope (SEM) images. The reconstructed samples are substituted with the corresponding finite element (FE) models, and the effective mechanical and electrochemical properties are assessed through the FE-based homogenization theory. Finally, the generated cathode samples and their effective properties are used to train the 3D CNN for performance prediction. This study demonstrates that the deep learning approaches can accurately and rapidly reconstruct the microstructure of cathode and predict their effective properties. Furthermore, the established framework can be extended to other heterogeneous materials.

Damped Vibration and Optimization of the Geometrical Parameters of FG Stiffened Cylindrical Shells Resting on Elastic Foundation Using GA

Functionally graded materials with their structural characteristics, such as the type of distribution, the size of phases, gradually change from one surface to another and provide a wide range of applications. In this research, the optimization of geometric parameters of FG structure rested on the elastic foundation, including damping effects is of interest. The cylindrical shell’s material properties are assumed to vary smoothly and continuously across the thickness according to the power law distribution of the volume fraction of constituents. The governing equations of stiffened functionally graded cylindrical shell resting on an elastic foundation are obtained using Hamilton’s principle, the first-order shear deformation theory and finite element method. The numerical results are obtained for studying the effect of various factors, such as distribution of volume fraction, length and thickness, different boundary conditions, stiffness and damping coefficients, and also the natural frequency of the dynamic responses of the cylindrical shell. The damping effects are considered which were not presented in previous studies. The frequency responses of the cylindrical FGM shell resting on elastic foundation were obtained by adding the damping term into the governing equations. Besides, analytical formulation results were considered as an objective function (input) for optimization in Genetic Algorithms. Finally, the compatibility of the analytical modeling results with the commercial finite element software was checked which showed good agreement.

An investigation on the shear deformation under bending conditions of cantilever FGM beams using a new polynomial shear function

In this paper, a static analysis to establish a mathematical model using high order bending theories considering shear strains in displacement fields which have not been taken into account by other theories for functionally graded material (FGM) cantilever beams under bending. A new polynomial shear function is used in this investigation, when satisfied the stress-free boundary conditions. These theories do not require a shear correction factor and consider a hyperbolic shape function. Material properties are assumed to vary in the thickness direction, a simple power-law distribution in terms of volume fractions of constituents is considered. Illustrative cases, a cantilever FGM beam subjected to a concentrated shear force at the free end, and also as a cantilever FGM beam with uniformly distributed load are presented the originality of this research work, in this investigation. The mathematical model is established by differential equations which are derived by the principle of virtual work. Equilibrium equations and boundary conditions are introduced. The solution model is based on a variation approach (integrals) to predict the field component of displacements and the basic constitutive laws. The solution of the analytical model is presented. The results in terms of displacement fields including rotation of the section, and shear stresses, predicted from the proposed model, are presented.

Effect of collagen fibril orientation on the anisotropic properties of peri-implant bone.

In orthopedic and dental surgery, the implantation of biomaterials within the bone to restore the integrity of the treated organ has become a standard procedure. Their long-term stability relies on the osseointegration phenomena, where bone grows onto and around metallic implants, creating a bone-implant interface. Bone is a highly hierarchical material that evolves spatially and temporally during this healing phase. A deeper understanding of its biomechanical characteristics is needed, as they are determinants for surgical success. In this context, we propose a multiscale homogenization model to evaluate the effective elastic properties of bone as a function of the distance from the implant, based on the tissue's structure and composition at lower scales. The model considers three scales: hydroxyapatite foam (nanoscale), ultrastructure (microscale), and tissue (mesoscale). The elastic properties and the volume fraction of the elementary constituents of bone matrix (mineral, collagen, and water), the orientation of the collagen fibril relative to the implant surface, and the mesoscale porosity constitute the input data of the model. The effect of a spatiotemporal variation in the collagen fibrils' orientation on the bone anisotropic properties in the proximity of the implant was investigated. The findings revealed a strong variation of the components of the effective elasticity tensor of the bone as a function of the distance from the implant. The effective elasticity appears to be primarily sensitive to the porosity (mesoscale) rather than to the collagen fibrils' orientation (sub-micro scale). However, the orientation of the fibrils has a significant influence on the isotropy of the bone. When analyzing the symmetry properties of the effective elasticity tensor, the ratio between the isotropic and hexagonal components is determined by a combination of the porosity and the fibrils' orientation. A decrease in porosity leads to a decrease in bone isotropy and, in turn, an increase in the impact of the fibrils' orientation. These results demonstrate that the collagen fibril orientation should be taken into account to properly describe the effective elastic anisotropy of bone at the organ scale.

Free Vibration Response of Functionally Graded Porous Metallic Plates Embedded with Piezoelectric Layers

The objective of this study is to determine the natural frequencies of Functionally Graded (FG) metallic plates comprising piezoelectric layers on both the top and bottom surfaces. The material characteristics of the FG plates are expected to exhibit a gradual variation along the thickness direction in accordance with a power-law model along with porosity. The governing equations are derived using the principle of virtual displacements, taking into account the first-order shear deformation plate theory. A commercial finite element programme in ANSYS Parametric Design Language (APDL) is developed to compute the natural frequency and mode shapes of functionally graded porous plates embedded with piezoelectric layers. The obtained natural frequencies results are used for different boundary condition to show their variations with respect to the constituent volume fractions, boundary condition, and piezoelectric thickness for the parameteric study. The present paper highlights some important characteristics of Functionally Graded Materials (FGM) plate embedded with piezoelectric layers that can be advantageous in the design of smart structures.

Static and Fatigue Mechanical Properties of Polymer Matrix Composite Rods

Tendons (cables or rods) are commonly employed as tension members in civil infrastructure, as well as buildings and offshore engineering structures. This study focuses on reliability evaluation of composite rods consisting of polymer matrix (carbon fiber/glass fiber hybrid thermoplastic composite rods (hybrid composite rods) and basalt fiber/polypropylene composite rod (BF/PP composite rod)). Optical and gravimetric methods were used to characterize the morphologies, including constituent volume fractions, of the composite rods. The hybrid composite rods are braided structures with varying diameters and braid angles. The BF/PP composite rod exhibits slight twisting. The volume fractions of the constituent elements (carbon fiber, glass fiber, basalt fiber, matrix, and void) were evaluated. Tensile and flexural tests were conducted under static and fatigue loadings. During the static tensile test, the stress applied to the composite rods was almost linearly proportional to the strain. The fiber-dominant behaviors of the composite rods were observed. During the static flexural test, the stress-strain relationship was initially linear, but as the stress approached its maximum, deformation became non-linear, and finally, the fibers fractured rapidly. During the fatigue tensile and flexural tests, the regression lines of the full-logarithmic curves showed good agreement with the fatigue test data. In addition, data was collected and statistical analyses were performed to assess the effects of environmental factors, such as temperature, on the static properties of the composite rods.

Microstructural Characteristics and Mechanical Properties of Nanostructured Bainite Processed through High and Low Temperature Ausforming and Isothermal Holding near Ms in a Medium Carbon Steel

Laboratory rolling simulations, comprising of thermomechanically controlled rolling followed by ausforming and isothermal holding close to Ms temperature, were conducted on a medium-carbon (C) steel in order to understand the combined effect of prior straining and low temperature holding on the bainite transformation characteristics and resultant properties. Field emission scanning electron microscopy (FE-SEM) was employed to examine the morphology, size and volume fraction of different phase constituents and the observed microstructural features were correlated with the mechanical properties of the steels. The morphology of bainite and/or retained austenite (RA) after low temperature ausforming was found to be extremely fine compared to the sample ausformed at high temperature. An excellent combination of high yield (~1200 MPa) and tensile (~1800 MPa) strengths, good ductility (~18 %) and reasonable ambient temperature impact toughness (~15 J/cm2) were achieved in the low temperature ausformed steel that was attributed to significant refinement of bainite sheaves and presence of high fractions of film-like RA. The dislocations introduced by ausforming hindered the growth of bainite and promoted enhanced carbon diffusion, resulting in high fractions of finely divided film-like RA with high stability. Recent results obtained on a nanostructured /ultrafine medium-carbon bainitic steel elucidating the ausforming conditions and corresponding microstructures and properties are illustrated here.

Effect of cooling rate on the final microstructure and tensile property in an Fe–Mn–Si–C-based multiphase TRIP steel

The effect of cooling rate on the formation of the final microstructure and the tensile property was studied in Fe–Mn–Si–C-based multiphase steel. Three different cooling rates of 5 °C/s, 10 °C/s, and 30 °C/s (CR5, CR10, and CR30) were applied from 800 °C to 400 °C and then the specimens held at 400 °C for 30 min. EPMA, EBSD, and TEM analyses clarified the constituent phases' volume fractions and the mechanical stability of retained γ. Appling a faster cooling led to a remarkable increase of tensile properties without a loss of elongation. This increase was accompanied by a higher fraction of bainite as the cooling rate increased All the specimens showed a similar fraction and mechanical stability of γ. In-situ tensile EBSD experiment revealed the accelerated deformation-induced martensite transformation (DIMT) in the faster cooling rate condition. The increased flow stress from a larger bainite fraction brought faster DIMT kinetics and improves tensile properties. This study provides a detailed understanding of the influence of cooling rate on microstructural evolution and tensile properties.

Impact response analysis of stiffened sandwich functionally graded porous materials doubly-curved shell with re-entrant honeycomb auxetic core

Owing to unique deformation behavior and high capacity of energy absorption, auxetic structures have garnered significant research attention. To reveal the energy-absorbing properties of the auxetic structure, the impact response of stiffened sandwich functionally graded porous materials (FGPM) doubly-curved shell with re-entrant honeycomb auxetic core is studied by the established analytical model in the present work. The material properties in the thickness direction of FGPM face sheets are adjusted according to the volume fraction of constituents by power law and exponential law, while the core layer is made of the re-entrant honeycomb auxetic core with negative Poisson's ratio. On the basis of the Hertz theory for elastic impacts and the first-order shear deformation theory (FSDT), an analytical model is developed to study its impact phenomenon. Equations of motion are derived using Hamilton's principle, which are further solved analytically using Navier's technique of assuming unknown variables in the double trigonometric series. The time-dependent contact force is obtained by applying the spring-mass (SM) model consisting two-degrees-of freedom. Afterward, the transverse central displacements of the stiffened sandwich FGPM doubly-curved shell are calculated by adopting the Duhamel integration. The impact force and transverse displacements are compared with the previously published results, indicating that the theoretical results agree with the previously published results.Parametric analyses have been carried out to investigate the effect of variation of various parameters such as volume fraction exponents, radii of principal curvatures, porosity volume fraction, core layer inclined angle on the impact response of the sandwich shell. Additionally, the results show that compared with the traditional hexagonal honeycomb core type, the stiffened sandwich FGPM doubly-curved shell with re-entrant honeycomb core type has significant advantages in impact energy absorption. This work may provide an extensive reference and valuable guidance in the impact design and application of auxetic sandwich shell structures.

Application of Chebyshev collocation element method to Green-Naghdi thermoelasticity of a bi-layered media consists of FGM and viscoelastic domains

In this study, the generalized coupled thermoelastic wave propagation and temporal evolution in a bi-layered system are analyzed. The main layer is made of functionally graded materials (FGM), while the holder layer is viscoelastic. The constituent volume fractions in the FGM layer are calculated by a power law function. Moreover, the Voigt rule of mixtures is employed to obtain the homogenized material properties. On the other hand, the constitutive law for the viscoelastic environment is considered based on the Kelvin-Voigt model. The heat conduction equation for each layer is represented based on the general form of the Green-Naghdi theory. First, for each layer, two motion and heat conduction coupled equations are derived, as well as the boundary and appropriate continuity conditions. The Chebyshev collocation element (CCE) method is used to solve the spatial-dependent parts of equations and also assemble the governing matrices into a global system. Finally, the Newmark numerical integration method is implemented to solve the ODEs and calculate the system response. The achieved response is verified with limited articles in the literature. Some parametric results are shown to evaluate the performance of FG and viscoelastic characteristics on the FGM layer response which is subjected to a thermal shock.