Effects Of Material Properties Research Articles

Effective thermal and electromagnetic properties of composite materials containing ellipsoidal-coated fillers oriented randomly

This study aims to derive explicit solutions for the effective thermal and electromagnetic properties of composite materials containing coated fillers oriented randomly. In this study, the coated fillers are modeled as double inhomogeneous inclusions with anisotropic properties and ellipsoidal shapes, without loss of generality. This model is analyzed by combining the double inclusion method with the self-consistent method or the Mori–Tanaka method. The derived solutions for the effective properties are expressed in explicit form by the physical properties of constituents and the difference in shape between the inner and outer regions of the coated fillers, and confirmed to be consistent in the limit of no coating layer. In addition, the solutions of effective properties are also derived for composite materials containing many kinds of coated fillers with different shapes and physical properties. The solutions allow the prediction of thermal conductivity, permittivity, electrical conductivity, and magnetic permeability of a wide range of composite materials by analogy with thermal and electromagnetic properties. These solutions are applied to the following two composites: diamond fillers coated with TiC/Al, and Co fillers coated with Co 3 O 4/Mg. The effect of the thickness of the coating layer on the effective thermal conductivity of these materials is investigated. As a result, our analytical results for these composite materials generally agree with the experimental results. Moreover, from the viewpoint of maximizing the thermal conductivity of diamond fillers coated with TiC/Al, the optimal combinations of the thickness of the coating layer and the shape of the diamond fillers are identified. For Co coated with Co 3 O 4/Mg, we find that the shape of Co fillers should be long fibrous from the viewpoint of giving high magnetic permeability while maintaining high thermal conductivity due to Mg. Thus, the analytical solutions are demonstrated to be powerful tools for designing composite materials containing coated fillers.

A novel dynamic stiffness matrix for the nonlocal vibration characteristics of porous functionally graded nanoplates on elastic foundation with small-scale effects

The present paper deals with the development of a dynamic stiffness matrix to evaluate the free vibration response of functionally graded nanoplate (FG-nP) resting on the Winkler-Pasternak elastic foundation. The complete mathematical modeling of the dynamic stiffness matrix for nanostructures is given for the first time. The equation of motion for rectangular FG-nP plates supported on an elastic foundation is derived using Hamilton’s principle in conjunction with nonlocal elasticity theory. The non-local theory is incorporated to account for the size effect in the small-scale plate. The effective material property of the porous FG-nP has been calculated using three recently developed models of porosity. The developed dynamic stiffness matrix is solved using the Wittrick-Williams algorithm to extract the natural frequencies of the FG-nP. The variation of natural frequencies with the change of numerical values, such as nonlocal parameter, aspect ratio, elastic foundation parameters, and porosity volume fraction is analyzed. The validity and accuracy of the results are confirmed through comparison with the available literature. The use of non-local theory in dynamic stiffness analysis is shown to be effective in predicting the natural frequency of the FG-nP on a Winkler-Pasternak elastic foundation, providing new insights into the dynamic behavior of small-scale structures.

The toughness of interlocking metasurfaces

Interlocking metasurfaces (ILMs) are arrays of autogenous latching unit cells patterned across a surface. These create structural joints similar to bio‐inspired suture joints but patterned over a 2D surface rather than a 1D seam. This enables ILMs to be an alternative to conventional joining technologies such as bolts, welds, and adhesives. However, compared to conventional joining methods, relatively little is known of the engineering considerations for designing structural ILMs. In this study, the interfacial toughness of an archetypal ILM was examined for the first time. Under the conditions studied here, the ILM was substantially tougher than the material from which it was made; in this case exhibiting up to a 50% increase in interfacial crack initiation energy over the solid base material, a photocured 3D printed polymer. Through experimental tests using in‐situ digital image correlation along with complementary computational analyses, the mechanism of toughening in the ILM structure and the origins of toughness anisotropy were revealed. The increase in toughness is associated with cross‐cell interactions, i.e. load‐sharing across unit cells, which gives rise to a finite process zone length with different effective material properties. In this way, ILM toughening is analogous to crack blunting in ductile materials or fiber bridging in composites; yet here, the ILM was composed of a single‐phase base material and so the architected toughening is geometric in nature and hence amenable to future topological optimization.This article is protected by copyright. All rights reserved.

Micro-CT image-based computation of effective thermal and mechanical properties of fibrous porous materials

Fibrous porous materials are extensively employed as heat shielding material for space vehicles and capsules. To predict the performance of these materials using computational modeling at the vehicle/capsule scale, the effective material properties are needed. In this work, the effective thermal conductivity and elastic constants of a carbon fibrous porous material called FiberForm (precursor of PICA) were calculated using a direct image-based approach. In this image-based approach, the microstructures obtained using X-ray computed tomography technique are represented as binary voxel data. Nonlocal interactions are introduced between neighboring voxels. Integro-differential equations, with respect to spatial and temporal dimensions respectively, are developed to govern material thermal and mechanical behaviors. Using this approach, energy-based computational procedures were developed to calculate the effective material properties of fibrous and porous materials irrespective of the periodicity of underlying microstructures. The size of representative volume element was determined by convergence of effective properties with respect to microstructure size.

Peridynamic Differential Operator for Stress Analysis of Imperfect Functionally Graded Porous Sandwich Beams Based on Refined Zigzag Theory

This study focuses on the stress analysis of imperfect functionally graded porous (FGP) sandwich beams using the Peridynamic Differential Operator (PDDO) and Refined Zigzag Theory (RZT). Functionally graded materials (FGMs) can be found in diverse engineering applications since they offer smooth transitions in the mechanical properties of distinct materials, unlike traditional composite materials. Micro-voids and porosities may appear inside the FGMs due to technical challenges during the manufacturing process of such materials. Therefore, understanding the stress variations of the FG sandwich beams with porosities/micro-voids, which are called imperfect FGMs, under loads is of vital importance. In order to achieve the porous configuration in the FG structures, the modified rule of mixtures is applied to calculate the effective material properties of FGMs. The RZT is very suitable for stress analysis of laminated composites, especially for thick and moderately thick beams. It contains only four kinematic variables and eliminates the use of the shear correction factors. In this study, the equilibrium equations of the RZT are solved by means of the PDDO for the stress analysis of imperfect FG sandwich beams having uniform and non-uniform porosity distributions. The PDDO transforms the differential equations into their nonlocal form, namely integral form, providing highly accurate approximations of the derivatives. A comprehensive PDDO analysis is performed for the investigation of the influence of the material variation, and even and uneven porosity distributions on the stress variations of the imperfect FG sandwich beams. It is noted that when the FG core exhibits soft material distributions, the effect of porosity is evident. The even type porosity distribution pattern is more influential on the axial displacement and stress variations in comparison with those of the even type porosity distribution pattern.

Analysis of the embedded cell method in 2D for the numerical homogenization of metal-ceramic composite materials

Abstract In this paper, we extend our analysis of the embedding cell method, an algorithm which has been developed for the numerical homogenization of metal-ceramic composite materials, from [W.-P. Düll, B. Hilder and G. Schneider, Analysis of the embedded cell method in 1D for the numerical homogenization of metal-ceramic composite materials, J. Appl. Anal. 24 2018, 1, 71–80]. We show the convergence of the iteration scheme of this algorithm and the coincidence of the material properties predicted by the limit with the effective material properties provided by the analytical homogenization theory for two-dimensional linear hyperelastic isotropic materials with constant shear modulus and slightly varying first Lamé parameter.

Experimentally calibrated FE modelling of helical buckling of anisotropic slender shafts

Helical buckling is very common in slender structures and it is characterized by the development of helical deformation patterns along their length. This phenomenon is particularly acute in very slender drill-strings as associated high stresses decrease drilling efficiency and shorten life expectancy of expensive drilling assemblies. Hence, understanding and predicting buckling behaviour is of paramount importance. In this paper, we investigated buckling phenomena of slender flexible shafts in forms of stranded wire ropes mimicking drill-strings, by experimental and numerical means. From experiments we obtained the loading curves for the flexible shafts and developed governing buckling equations using dimensional analysis. Based on these equations, we calculated the effective material properties of the stranded ropes and carried out FE modelling and analysis in Abaqus. Finally, we developed a robust method to predict the helical buckling behaviour, which matches well with experiment results.

R‐Vine Copulas for Data‐Driven Quantification of Descriptor Relationships in Porous Materials

AbstractLocal variations in the 3D microstructure can control the macroscopic behavior of heterogeneous porous materials. For example, the permittivity through porous sheets or membranes is governed by local high‐volume pathways or bottlenecks. Due to local variations, unfeasibly large amounts of microstructure data may be needed to reliably predict such material properties directly from image data. Here it is demonstrated that a vine copula approach provides parametric models for local microstructure descriptors that compactly capture the 3D microstructure including its local variations and efficiently probe it with respect to selected, measurable properties. In contrast to common methods of complexity reduction, the proposed approach creates parametric models for the multivariate probability distribution of high‐dimensional descriptor vectors that inherently contain the complex, nonlinear dependencies between these descriptors. Therein, material properties are offered in physically motivated distributions of microstructure descriptors rather than as normally distributed data. Applied to porous fiber networks (paper) before and after unidirectional compression, it is shown that the copula‐based models reveal material‐characteristic relationships between two or more microstructure descriptors. In this way, the presented modeling approach can provide deeper insight into the microscopic origin of effective macroscopic properties of heterogeneous porous materials.

Effective Elastic Properties and Micro-mechanical Damage Evolution of Composite Granular Rocks: Insights from Particulate Discrete Element Modelling

AbstractThe mechanical behavior of composite granular rocks is a multifaceted phenomenon with broad relevance in various geomechanical applications. Traditional homogenization models and continuum mechanics-based numerical methods often fall short of accurately capturing the intricacies of granular materials. Granular materials exhibit heterogeneity and arching mechanisms governing the force networks that ensure system stability. Unlike continuum-based approaches, discrete element methods (DEM) have an advantage in assessing effective material properties by considering material heterogeneity and grain-level physical interactions. This study evaluates effective elastic properties using DEM with flat-joint contact law for composite binary mixtures with a stiff inclusion embedded within a matrix material. We examine variations in the elastic properties across different structural and laminated geometrical distributions of inclusion materials. Our findings closely adhere to the Voigt-Reuss and Hashin–Strikman bounds within their specific conditions, demonstrating the promising application of DEM in the analysis of composite materials. In addition, our research provides an in-depth analysis of the distinctive stress-evolution and damage-evolution characteristics exhibited by various geometrical configurations of inclusions under unconfined compressive loading. These results offer invaluable insights into the mechanical behavior of composite granular rocks and underscore the potential applications of DEM in addressing rock physics modeling problems encountered in petroleum engineering.

Метод чисельного визначення динамічних характеристик лопаті повітряного гвинта з композиційного матеріалу

The subject matter of this article is the dynamic characteristics of a composite propeller blade. Determination of the modes and frequencies of natural vibrations is necessary to predict the dangerous resonance modes of aircraft engines and to identify the most stressed local zones of the blade surface. The goal of this study is to develop a verified method for determining the dynamic characteristics of composite rotor parts of aircraft engines based on the known properties of the structural components of the composite material. A general mathematical statement of the problem of elasticity theory for the analysis of the dynamic characteristics of composite structures is described. A complete system of equations that describes the mechanical state of the body within the framework of the continuum mechanics approach was developed. Geometric modeling of the propeller blades was performed. Modeling and numerical investigation of the natural frequencies and mode shapes of the propeller blade were performed using the finite element method using the ANSYS software package. Based on the results of numerical research of the stressed and deformed state of the propeller blade, the first five natural vibration frequencies and the distribution of the local stress field were determined. The most stressed local zones on the blade surface were determined for each form of natural vibration. The propeller blade model was verified using an experimental study of the first five forms and frequencies of natural blade vibrations. The eigenfrequencies of the blade vibration were experimentally determined using the method of free (natural) vibrations. The resonance method was used to experimentally determine the resonant frequencies and vibration modes of the blade. The distribution of the blade deformation field was investigated using the strain gauge method. The highest error in the verification of numerical and experimental research is 4.11% for the fourth vibration frequency. Conclusions. The scientific novelty of the results obtained is that the effective elastic properties of the composite material for calculations should be determined using the procedure of numerical homogenisation of composite materials of different reinforcement structures by the properties of the matrix and fibers. The method does not require experimental determination of the effective elastic constants for the layers of blade components of different weave architecture patterns.

Non-linear behavior of functionally graded elastoplastic beam under torsion

The torsional behavior of beams graded in one and two directions under large displacements and angular deformations was analyzed using the power law and sinusoidal functions. Functionally graded material is elastoplastic, consisting of ceramic and metal. A nonlinear finite element method with isoparametric hexahedral elements was used. The finite element formulation was made using the updated Lagrangian formulation based on the virtual displacement principle. An iterative solution using Newton-Raphson and updated Newton-Raphson methods was used to solve the nonlinear equation system. The propagation of the plastic zone was calculated based on the flow theory of plasticity. Elastoplastic behavior and effective material properties were determined according to the TTO model. Numerical investigations have shown that functionally graded beams behave quite differently from homogeneous beams under torsion. Yielding of the material starts at the outer boundaries of the section of the homogeneous beams, and the plastic zone propagates symmetrically. On the other hand, yielding and propagation of plastic zones tend to shift to regions with more ceramic volume with higher effective Young modulus in functionally graded beams. Unlike homogeneous beams, the largest shear stresses can occur within the section rather than at the outer boundaries of the section. In beams graded from ceramic to metal using the power law, the section moves along the transverse direction in addition to the rotation. This transverse displacement occurs in the grading direction, and its magnitude is about 3% of the thickness at 12.5° rotation angle. Also, the shear stresses are not zero in the section's midpoint. The effects of material distribution on displacements, stresses, and plastic zone propagation were examined, and essential points were reported.

Investigating Microstructure–Property Relationships of Nonwovens by Model-Based Virtual Material Testing

Quantifying the relationship between geometric descriptors of microstructure and effective properties like permeability is essential for understanding and improving the behavior of porous materials. In this paper, we employ a previously developed stochastic model to investigate microstructure–property relationships of nonwovens. First, we show the capability of the model to generate a wide variety of realistic nonwovens by varying the model parameters. By computing various geometric descriptors, we investigate the relationship between model parameters and microstructure morphology and, in this way, assess the range of structures which may be described by our model. In a second step, we perform virtual materials testing based on the simulation of a wide range of nonwovens. For these 3D structures, we compute geometric descriptors and perform numerical simulations to obtain values for permeability as an effective material property. We then examine and quantify the relationship between microstructure morphology and permeability by fitting parametric regression formulas to the obtained data set, including but not limited to formulas from the literature. We show that for structures which are captured by our model, predictive power may be improved by allowing for slightly more complex formulas.

Analysis of sandwich graphene origami composite plate sandwiched by piezoelectric/piezomagnetic layers: A higher-order electro-magneto-elastic analysis

This work applies a higher order thickness-stretched model for the electro-elastic analysis of the composite graphene origami reinforced square plate sandwiched by the piezoelectric/piezomagnetic layers subjected to the thermal, electric, magnetic and mechanical loads. The plate is manufactured of a copper matrix reinforced with graphene origami where the effective material properties are calculated based on the micromechanical models as a function of volume fraction and folding degree of graphene origami, material properties of matrix, reinforcement, and local temperature. The governing equations are derived using the virtual work principle in terms of the bending, shear and stretching functions, in-plane displacements, electric, and magnetic potentials. The numerical results including various displacement components, maximum electric, and magnetic potentials are presented with changes of volume fraction, folding degree of reinforcement, electrical, magnetic, and thermal loading. A verification investigation is presented for approve of the methodology, and the solution procedure. The main novelty of this work is simultaneous effect of the thickness stretching and the multi-field loading on the electromagnetic bending results of the sandwich plate. Another novelty of this work is usage of graphene origami nano-reinforcement as a controllable material in a sandwich structure subjected to multi-field loadings. The results show an increase in bending, shear, and stretching deflections with an increase in electromagnetic loads, and folding degree as well as a decrease in volume fraction of reinforcement.

Buckling and post-buckling of anisogrid lattice-core sandwich plates with nanocomposite skins

This research offers an in-depth examination of the buckling and post-buckling responses observed in sandwich plates distinguished by the presence of an anisogrid lattice core and face layers composed of functionally graded carbon nanotube-reinforced composites (FG-CNTRC). To model this structure, a global continuous model for the lattice core is employed, while the displacement field is approximated using the First-Order Shear Deformation Theory (FSDT). Geometrically nonlinear strain relations based on the von-Kármán assumptions are incorporated to accurately capture the structural response. The homogenization of the FG-CNTRC skins is achieved through the modified rule of mixture, enabling estimations of effective material properties. To separate pre-buckling and post-buckling phases, we implement the adjacent-equilibrium criterion. Solving the linear (pertaining to buckling) and nonlinear (related to post-buckling) equations is based on the Generalized Differential Quadrature (GDQ) technique alongside an iterative method grounded in the displacement control strategy. Our analysis explores the influence of core and skin characteristics on the stability of these sandwich plates. By examining critical buckling thresholds, post-buckling equilibrium paths, and the underlying mechanisms driving these responses, we contribute insights into the structural stability and performance of such composite structures. This study not only enhances our understanding of buckling behavior in anisogrid lattice core sandwich plates but also holds promise for diverse engineering applications where structural stability is paramount.

A unified expression for estimating Poisson’s ratio from a hinged–hinged beam with a use for abnormality detection

Cycle fatigue weakens a structure. Poisson’s ratio is a sensitive indicator of cycle fatigue. The amount of fatigue is proportional to the severity of an abnormality, such as a crack, delamination or other imperfection. Our prior work confirms that direct estimates of Poisson’s ratio by Cornu’s method must be identical for all antinodes of a mode shape of a homogeneous, thin, hinged–hinged beam. Therefore, if a direct estimate at any antinode was found to be significantly different from the other estimates at the other antinodes, then that would simultaneously indicate fatigue and identify the location of the abnormality. We use simulations of Besselograms and apply Cornu’s method to directly estimate Poisson’s ratio at all antinodes of a beam. We transform finite element simulations of the out-of-plane bending displacements of mode shapes by simulating time-average scanning digital holography. We show that amount of disagreement between the direct estimation Poisson’s ratio and the true value of Poisson’s ratio depends on the ratio of the distance between nodes of a mode shape divided by the width of the beam. We call this distance between the nodes of a mode shape the ‘Span’. We then define a unified expression for predicting the simulated direct estimate given the true value of Poisson’s ratio and the Span-to-Width ratio. Next, we invert the unified expression, which enables us to use the Span-to-Width ratio and simulated direct estimates of Poisson’s ratio as the independent variables to compute the adjusted value of Poisson’s ratio. We find that our inverse expression makes it possible to obtain an estimate of Poisson’s ratio that is within ±2% of the true value for thin hinged–hinged beams and plates that have a Span-to-Width ratio greater than 1. We also find that if the material is known to have a Poisson’s ratio greater than zero, which can be assessed by our previous statement, then we can extend our confidence to Span-to-Width ratios greater than 0.1. That was the lower limit of our testing. Finally, we show how to use direct estimates of Poisson’s ratio to statistically detect and localize an abnormality along a hinged–hinged plate or beam. We believe that our combination of Cornu’s method and time-averaging could eventually offer the ability to directly estimate Poisson’s ratio from hinged–hinged beams and plates as well as to unambiguously monitor cycle fatigue for changes in effective material properties, predict time-to-failure, and do so without contact.

Multi-factors effects analysis of nonlinear vibration of FG-GNPRC membrane using machine learning

Functionally graded graphene nanoplatelet reinforced composite (FG-GNPRC) have exhibited significant potential for the development of high-performance and multifunctional structures. In this paper, we present a machine learning (ML) assisted uncertainty analysis of nonlinear vibration of FG-GNPRC membranes under the influence of multi-factor coupling. Effective medium theory (EMT), Mori-Tanaka (MT) model and rule of mixture are utilized to evaluate the effective material properties of the composite membrane. Governing equations are derived via an energy method with the frameworks of the hyperelastic membrane theory, Neo-Hookean constitutive model and the couple dielectric theory. Randomly generated inputs after data pre-processing are fed into governing equations, which are solved by numerical methods for outputs. Three ML models, including artificial neural network (ANN), support vector regression (SVR) and AutoGluon-Tabular (AGT), are adopted to capture the complex relationship between the systematic inputs (i.e., structural dimensions, attributes of GNPs and pores, external electric field, etc.), frequency ratio and dimensionless amplitude of FG-GNPRC membranes. The results demonstrate that all three ML models demonstrate exceptional computational efficiency, and AGT presents higher prediction accuracy compared to the other two models. Based on the Shapley additive explanations (SHAP) approach, the effects of uncertainties of system parameters and multi-factor coupling on the nonlinear vibration of the FG-GNPRC membrane are analyzed. It is found that the uncertainty of structural parameters has the greatest impact on the nonlinear vibration of FG-GNPRC membranes, particularly when the membrane is subjected to a voltage of 10 V and smaller stretching ratio.

Joint micromechanical model for determination of effective elastic and electromagnetic properties of porous materials

In this paper we propose an approach for calculating the effective physical properties of porous materials (for example, sedimentary rocks) which is based on the unified structure of the pore space. This approach is based on the Generalized Differential Effective Medium (GDEM) method. This method generalizes the classical differential scheme (DEM) for the case of many types of inclusions. The physical properties of a composite calculated using the GDEM depend on how the solution is constructed.A porous medium is represented by the elastic weakly conductive matrix with embedded inclusions of two types (spheroidal and cylindrical), saturated with a conductive liquid. The cylindrical inclusions appear in the system when the porosity value exceeds the void percolation. Parameters, that characterize the inclusions (the aspect ratio of spheroidal inclusions and the relative part of cylindrical inclusions), are determined in the inverse problem solving process for the experimental data approximation of the effective conductivity as a porosity function. These parameters, obtained by solving the inverse problem, were used to calculate the effective elastic moduli, electrical conductivity, and dielectric permittivity of porous media. The results obtained describe well the available experimental data for different effective physical properties.

Multifield and higher-order analysis of sandwich smart curved beams made of graphene origami auxetic metamaterial

This article is organized to investigate effect of material and geometric characteristics of graphene origami as well as multifield loading on the electro-magneto-elastic bending results of the sandwich smart nanocomposite curved beam. The analysis is performed using the higher-order shear deformation theory accounting out of plane normal strain. The constitutive relations are extended in the cylindrical coordinate in the presence of thermal, electrical, and magnetic loads. The governing equations are derived using the principle of virtual work with accounting electric and magnetic works. The effective material properties of graphene origami reinforced composite core are evaluated through Halpin–Tsai micromechanical models and rule of mixture for modulus of elasticity, Poisson’s ratio, and density. The analytical solution is presented for simply supported boundary conditions based on trigonometric functions. A verification test is performed for validation of the formulation, solution procedure, and numerical results. The stress and deformation components are evaluated with changes of multifield loading and characteristics of graphene origami along the thickness direction. The results show a significant effect of electrical, thermal, and magnetic loading on the stress and deformation results.

State-Space Formulation for Buckling and Free Vibration of Axially Functionally Graded Graphene Reinforced Nanocomposite Microbeam under Axially Varying Loads.

This paper focuses on the size-dependent free vibration and buckling behaviors of the axially functionally graded (AFG) graphene platelets (GPLs) reinforced nanocomposite microbeams subjected to axially varying loads (AVLs). With various axial grading patterns, the GPL nano-reinforcements are distributed throughout the polymer matrix against microbeam length, and the improved Halpin-Tsai micromechanics model and the rule of mixture are adopted to evaluate the effective material properties. Eigenvalue equations of the microbeams governing the static stability and vibration are derived based on the modified couple stress Euler-Bernoulli beam theory via the state-space method, and are analytically solved with the discrete equilong segment model. The effects of axial distribution patterns, weight fraction, and geometric parameters of GPLs, as well as different types of AVLs, on the size-dependent buckling load and natural frequency are scrutinized in detail. The results show that the synchronized axial distributions of GPLs and AVLs could improve the buckling resistance and natural frequency more powerfully.

Bulk wave propagation analysis of imperfect FG bio-composite beams resting on variable elastic medium

Bio-composites are outstanding alternatives to prevalent biomaterials in medical instruments due to their supreme material properties. On the other hand, mechanical analyses of bio-systems should be carried out to optimize the designed biomedical applications such as medical implants. Therefore, bulk wave propagation analysis of functionally graded (FG) bio-composite beams could serve as design targets of biomedical structures. Some applications like medical implants have a layered structure and FG composites will be an appropriate choice for constitutive material of such structures. In the current study, the influence of imperfection on the propagation of bulk waves in FG bio-composite beams with biomedical applications lying on a variable elastic medium has been evaluated for the first time. The constitutive materials of FG bio-composite beams are hydroxyapatite and gold alloy which are common materials in the biomedical industry. To determine the effective material properties of the structure under study, the conventional porosity-dependent homogenization scheme has been implemented. Various distribution patterns have been accounted for variable elastic medium. Moreover, a refined higher-order shear deformation theory and Hamilton’s principle have been exerted to obtain the partial differential governing equations and the derived equations are analytically solved via a harmonic function. To check the accuracy of the used methodology, the obtained result has been validated firstly with former investigations. Eventually, the sensitivity of various considerable parameters has comprehensively been probed and discussed. It is found that some parameters including wave number, power law index, and elastic foundation have positive effects and some parameters including slenderness ratio, ξ parameter, and imperfection coefficient have negative effects on the propagated bulk waves. Also, the sinusoidal pattern possesses less increasing effect than other ones.