Islamic Azad University - Arak BranchJournal of Solid Mechanics2008-350512320200930Non-Linear Response of Torsional Buckling Piezoelectric Cylindrical Shell Reinforced with DWBNNTs Under Combination of Electro-Thermo-Mechanical Loadings in Elastic Foundation50552067753610.22034/jsm.2019.581546.1365ENMSarvandiDepartment of Mechanical Engineering, Arak Branch, Islamic Azad University, Arak, IranM.MNajafizadehDepartment of Mechanical Engineering, Arak Branch, Islamic Azad University, Arak, IranHSeyyedhasaniSchool of Plant and Environmental Sciences, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USAJournal Article20200329Nanocomposites provide new properties and exploit unique synergism between materials. Polyvinylidene fluoride (PVDF) is an ideal piezoelectric matrix applicable in nanocomposites in a broad range of industries from oil and gas to electronics and automotive. And boron nitride nanotubes (BNNTs) show high mechanical, electrical and chemical properties. In this paper, the critical torsional load of a composite tube made of PVDF reinforced with double-walled BNNTs is investigated, under a combination of electro-thermo-mechanical loading. First, a nanocomposite smart tube is modeled as an isotropic cylindrical shell in an elastic foundation. Next, employing the classical shell theory, strain-displacement equations are derived so loads and moments are obtained. Then, the total energy equation is determined, consisting of strain energy of shell, energy due to external work, and energy due to elastic foundation. Additionally, equilibrium equations are derived in cylindrical coordinates as triply orthogonal, utilizing Euler equations; subsequently, stability equations are developed through the equivalent method in adjacent points. The developed equations are solved using the wave technique to achieve critical torsional torque. Results indicated that critical torsional buckling load occurred in axial half-wave number m = 24 and circumferential wave number n = 1, for the investigated cylindrical shell. The results also showed that with the increase in the length-to-radius ratio and in the radius-to-shell thickness ratio, the critical torsional buckling load increased and decreased, respectively. Lastly, results are compared in various states through a numerical method. Moreover, stability equations are validated via comparison with the shell and sheet equations in the literature. http://jsm.iau-arak.ac.ir/article_677536_7d220f50472d3b4128234d01c545f4d4.pdfIslamic Azad University - Arak BranchJournal of Solid Mechanics2008-350512320200930The Effects of Initial In-Plane Loads on the Response of Composite-Sandwich Plates Subjected to Low Velocity Impact: Using a New Systematic Iterative Analytical Process52153867745710.22034/jsm.2019.573380.1320ENKMalekzadeh FardDepartment of Aerospace Engineering, Malek-e-Ashtar University of Technology, Tehran, IranAAzarniaDepartment of Aerospace Engineering, Malek-e-Ashtar University of Technology, Tehran, IranJournal Article20200330A new systematic iterative analytical procedure is presented to predict the dynamic response of composite sandwich plates subjected to low-velocity impact phenomenon with/without initial in-plane forces. In this method, the interaction between indenter and sandwich panel is modeled with considering the exponential equation similar to the Hertzian contact law and using the principle of minimum potential energy and the energy-balance model. In accordance with the mentioned procedure and considering initial in-plane forces, the unknown coefficients of the exponential equation are obtained analytically. Accordingly, the traditional Hertzian contact law is modified for use in the composite sandwich panel with the flexible core under biaxial pre-stresses. The maximum contact force using the two-degrees-of-freedom (2DOF) spring-mass model is calculated through an iterative systematic analytical process. Using the present method, in addition to reducing the runtime, the problem-solving process is carried out with appropriate convergence. The numerical results of the analysis are compared with the published experimental and theoretical results. The effects of some important geometrical and physical parameters on contact force history are examined in details.http://jsm.iau-arak.ac.ir/article_677457_af56a7c6d30dd076755494f603479552.pdfIslamic Azad University - Arak BranchJournal of Solid Mechanics2008-350512320200930On the Optimum Die Shape in Rod Drawing Process Considering Work-Hardening Effect of Material53955067430210.22034/jsm.2019.585582.1400ENM.MMahdaviMechanical Engineering Department, Razi University, Kermanshah, IranHHaghighatMechanical Engineering Department, Razi University, Kermanshah, IranJournal Article20200528The assessment of the influence of the work-hardening of material on the optimum die profile and drawing force in rod drawing process is the main objective of the present paper. The upper bound solution, based on the assumption of perfect plasticity, has been extended to consider the work-hardening of the material during the rod drawing process through curved dies. Analytical results of drawing forces for rod drawing process through four different types of streamlined die profiles are compared with the finite element simulation data using the finite element code, DEFORM 2D. It is shown that as the work-hardening exponent increases, the optimum die length increases, the required drawing force decreases and maximum possible reduction in area increases. Based on this proposed modeling technique, drawing process of real materials through various curved dies can be optimized. http://jsm.iau-arak.ac.ir/article_674302_14df6cb1089d1bfdf758f2339cf910eb.pdfIslamic Azad University - Arak BranchJournal of Solid Mechanics2008-350512320200930Fatigue Life Prediction of Rivet Joints55155867757210.22034/jsm.2019.1867882.1436ENM. MAmiriResearch Institute of Petroleum Industry, Islamic Republic of, Tehran, Iran0000-0003-1406-1069Journal Article20200530Strength reduction in structures like an aircraft could be resulted as cyclic loads over a period of time and is an important factor for structural life prediction. Service loads are emphasized at the regions of stress concentration, mostly at the connection of components. The initial flaw prompting the service life was expected by using the Equivalent Initial Flaw Size (EIFS) which has been recognized as a powerful design tool for life prediction of engineering structures. This method was introduced 30 years ago in an attempt to study the initial quality of structural details. In this paper, the prediction of life based on failure mechanics in a riveted joint has been addressed through the concept of EIFS. For estimation of initial crack length by EIFS, extrapolation method has been used. The EIFS value is estimated using the coefficient of cyclic intensity (Δ<em>K</em>) and using the cyclic integral (Δ<em>J</em>), and the results are compared with each other. The simulation results show that the if the coefficient of tension been used in EIFS estimation, which based on the Paris law, the EIFS value will be dependent on the loading domain, while the use of the <em>J</em>-Cyclic integral in the EIFS decrease its dependence on the load domain dramatically.http://jsm.iau-arak.ac.ir/article_677572_67fd91d8b76a5338f6d6eabe0ec80a7d.pdfIslamic Azad University - Arak BranchJournal of Solid Mechanics2008-350512320200930A New Approach for Stress State - Dependent Flow Localization Failure Bounded Through Ductile Damage in Dynamically Loaded Sheets55956967756210.22034/jsm.2019.1869860.1449ENFHosseini MansoubDepartment of Mechanical Engineering, University Campus, University of Guilan, Rasht, IranABastiDepartment of Mechanical Engineering, University of Guilan, Rasht, IranADarvizehDepartment of Mechanical Engineering, University of Guilan, Rasht, IranAZajkaniDepartment of Mechanical Engineering, Imam Khomeini International University, Qazvin, IranJournal Article20200605In this paper, a new approach is proposed for stress state - dependent flow localization in bifurcation failure model bounded through ductile damage in dynamically loaded sheets. Onset of localized necking is considered in phenomenological way for different strain rates to draw the forming limit diagram (FLD). Using a strain metal hardening exponent in the Vertex theory related to the strain rate helps investigate rate- dependent metal forming limits. Besides, the paper utilizes the model of ductile damage as a function of strain condition, stress states (triaxiality and Lode parameters), and the symbols of stiffness strain to predict the onset of the necking. It is worth noting that updated level of elasticity modulus in the plastic deforming is attributed as an essential index for the ductile damage measuring. According to original formulations, a UMAT subroutine is developed in the finite element simulation by ABAQUS code to analyze and connect the related constitutive models. Results reveal that the FLD levels increase for St 13 material through enhancing the strain rate.http://jsm.iau-arak.ac.ir/article_677562_86850cc6a1c2791faa18a78ac3978ee4.pdfIslamic Azad University - Arak BranchJournal of Solid Mechanics2008-350512320200930Vibration Analysis of Size-Dependent Piezoelectric Nanobeam Under Magneto-Electrical Field57058567731010.22034/jsm.2019.1864682.1446ENMGhadiriDepartment of Mechanics, Imam Khomeini International University, Qazvin, IranMKarimi AslDepartment of Mechanics, Imam Khomeini International University, Qazvin, IranMNorooziDepartment of Mechanics, Imam Khomeini International University, Qazvin, IranJournal Article20200609The damping vibration characteristics of magneto-electro-viscoelastic (MEV) nanobeam resting on viscoelastic foundation based on nonlocal strain gradient elasticity theory (NSGT) is studied in this article. For this purpose, by considering the effects of Winkler-Pasternak, the viscoelastic medium consists of linear and viscous layers. with respect to the displacement field in accordance with the refined shear deformable beam theory (RSDT) and the Kelvin-Voigt viscoelastic damping model, the governing equations of motion are obtained using Hamilton’s principle based on nonlocal strain gradient theory (NSGT). Using Fourier Series Expansion, The Galerkin’s method adopted to solving differential equations of nanobeam with both of simply supported and clamped boundary conditions. Numerical results are obtained to show the influences of nonlocal parameter, the length scale parameter, slenderness ratio and magneto-electro-mechanical loadings on the vibration behavior of nanobeam for both types of boundary conditions. It is found that by increasing the magnetic potential, the dimensionless frequency can be increased for any value of the damping coefficient and vice versa. Moreover, negative/positive magnetic potential decreases/increases the vibration frequencies of thinner nanobeam. Also, the vibrating frequency decreases and increases with increasing nonlocal parameter and length scale parameter respectively. http://jsm.iau-arak.ac.ir/article_677310_2b293bfffb298ceeba1371d8793a9e92.pdfIslamic Azad University - Arak BranchJournal of Solid Mechanics2008-350512320200930An Analytical Solution on Size Dependent Longitudinal Dynamic Response of SWCNT Under Axial Moving Harmonic Load58659967731410.22034/jsm.2019.1875642.1476ENFKhosraviDepartment of Aerospace Engineering, K.N. Toosi University of Technology, Tehran, IranMSimyariDepartment of Mechanical Engineering, University of Tehran, Tehran, IranS. AHosseiniDepartment of Industrial, Mechanical and Aerospace Engineering, Buein Zahra Technical University, Buein Zahra, Qazvin, IranMGhadiriFaculty of Engineering, Department of Mechanics, Imam Khomeini International University, Qazvin, IranJournal Article20200611The main purposes of the present work are devoted to the investigation of the free axial vibration, as well as the time-dependent and forced axial vibration of a SWCNT subjected to a moving load. The governing equation is derived through using Hamilton's principle. Eringen’s nonlocal elasticity theory has been utilized to analyze the nonlocal behaviors of SWCNT. A Galerkin method based on a closed-form solution is applied to solve the governing equation. The boundary conditions are considered as clamped-clamped (C-C) and clamped-free (C-F). Firstly, the nondimensional natural frequencies are calculated, as well as the influence of the nonlocal parameter on them are explained. The results of both boundary conditions are compared together, and both of them are compared to the results of another study to verify the accuracy and efficiency of the present results. The novelty of this work is related to the study of the dynamic forced axial vibration due to the axial moving harmonic force in the time domain. The previously forced vibration studies were devoted to the transverse vibrations. The effect of the geometrical parameters, velocity of the moving load, excitation frequency, as well as the small-scale effect, are explained and discussed in this context. According to the lack of accomplished studies in this field, the present work has the potential to be used as a benchmark for future works. http://jsm.iau-arak.ac.ir/article_677314_cd32f0cf76c778aa98dc00a2aeb59e96.pdfIslamic Azad University - Arak BranchJournal of Solid Mechanics2008-350512320200930Reliability-Based Robust Multi-Objective Optimization of Friction Stir Welding Lap Joint AA1100 Plates60060667746410.22034/jsm.2019.1877986.1493ENESarikhaniAutomotive Simulation and Optimal Design Research Laboratory, School of Automotive Engineering, Tehran, IranAKhalkhaliUniversity of Science and Technology, Tehran, IranJournal Article20200616The current paper presents a robust optimum design of friction stir welding (FSW) lap joint AA1100 aluminum alloy sheets using Monte Carlo simulation, NSGA-II and neural network. First, to find the relation between the inputs and outputs a perceptron neural network model was obtained. In this way, results of thirty friction stir welding tests are used for training and testing the neural network. Using such obtained neural network model, for the reliability robust design of the FSW, a multi-objective genetic algorithm is employed. In this way, the statistical moments of the forces, temperature, strength, elongation, micro-hardness of welded zone, grain size and welded zone thickness are considered as the conflicting objectives. The optimization process was followed by multi criteria decision making process, NIP and TOPSIS, to propose optimum points for each of the pin profiles. It is represented that some beneficial design principles are involved in FSW which were discovered by the proposed optimization process.http://jsm.iau-arak.ac.ir/article_677464_e223066a9498afa31f7c8f3573bbaa94.pdfIslamic Azad University - Arak BranchJournal of Solid Mechanics2008-350512320200930Vibration of Timoshenko Beam-Soil Foundation Interaction by Using the Spectral Element Method60761967731610.22034/jsm.2020.1879476.1503ENSHamioudDepartment of Civil Engineering, University of Jijel, Jijel, AlgeriaSKhalfallahDepartment of Mechanical Engineering, National Polytechnic School, Constantine, AlgeriaSBoudaaCivil Engineering Department, University of Constantine, Constantine, AlgeriaJournal Article20200620This article presents an analysis of free vibration of elastically supported Timoshenko beams by using the spectral element method. The governing partial differential equation is elaborated to formulate the spectral stiffness matrix. Effectively, the non classical end boundary conditions of the beam are the primordial task to calibrate the phenomenon of the Timoshenko beam-soil foundation interaction. Non-dimensional natural frequencies and shape modes are obtained by solving the partial differential equations, numerically. Upon solving the eigenvalue problem, non-dimensional frequencies are computed for the first three modes of vibration. Obtained results of this study are intended to describe multiple objects, such as: (1) the establishment of the modal analysis with and without elastic springs, (2) the quantification of the influence of the beam soil foundation interaction, (3) the influence of soil foundation stiffness’ on free vibration characteristics of Timoshenko beam. For this propose, the first three eigenvalues of Timoshenko beam are calculated and plotted for various stiffness of translational and rotational springs.http://jsm.iau-arak.ac.ir/article_677316_9f91814aa7024a7daa258ec07e8e8595.pdfIslamic Azad University - Arak BranchJournal of Solid Mechanics2008-350512320200930Numerical Analysis of the Effect of External Circumferential Cracks in Transition Thickness Zone of Pressurized Pipes Using XFEM – Elastic-Plastic Behavior62063167728310.22034/jsm.2019.1882104.1511ENHSalmiDepartment of National Higher School of Mechanics, ENSEM, Laboratory of Control and Mechanical Characterization of Materials and Structures, MoroccoKhEL HadInstitute of Maritims Studies, Laboratory of Materials and Structures Casablanca, MoroccoHEL BhilatDepartment of National Higher School of Mechanics, ENSEM, Laboratory of Control and Mechanical Characterization of Materials and Structures, MoroccoAHachimInstitute of Maritims Studies, Laboratory of Materials and Structures Casablanca, MoroccoJournal Article20200622The elastic-plastic behavior of the material is considered to analyze the effect of an external circumferential crack on a pipe with thickness transition and double slopes. Using the extended finite element method (XFEM), the <em>J</em>-integral of 3D cracks were investigated and compared between straight pipes and pipes with thickness transition and different slopes. Considering internal pressure, this work highlighted the investigation of a 3D crack problem in a thickness transition pipe with a double slope, In the extended finite element method (XFEM), the level sets and the enrichment zone were defined. A crack is easily modeled by enrichment functions. The comparison between the values of the <em>J</em>-integral showed that the pipe containing thickness transition with double slopes is more sensitive to the considered cracks, more precisely, the parameters of the first thickness transition have more influence on the variation of <em>J</em>-integral than the parameters of the second thickness transition. The decreasing of the angle of the slopes and the increase of the ratio of the thicknesses is one effective method of reducing the <em>J</em>-integral.http://jsm.iau-arak.ac.ir/article_677283_e9b047037cf262002643552ae5c27da7.pdfIslamic Azad University - Arak BranchJournal of Solid Mechanics2008-350512320200930Investigation of Strain Gradient Theory for the Analysis of Free Linear Vibration of Nano Truncated Conical Shell63264867740210.22034/jsm.2019.1882391.1513ENA.RSheykhiDepartment of Mechanical Engineering , Science and Research Branch, Islamic Azad University, Tehran, IranShHosseini HashemiSchool of Mechanical Engineering, Iran University of Science and Technology, Tehran, IranAMaghsoudpourDepartment of Mechanical Engineering , Science and Research Branch, Islamic Azad University, Tehran, IranShEtemadi HaghighiDepartment of Mechanical Engineering , Science and Research Branch, Islamic Azad University, Tehran, IranJournal Article20200626In this paper the nano conical shell model is developed based on modified strain gradient theory. The governing equations of the nano truncated conical shell are derived using the FSDT, and the size parameters through modified strain gradient theory have been taken into account. Hamilton’s principle is used to obtain the governing equations, and the shell’s equations of motion are derived with partial differentials along with the classical and non-classical boundary conditions. Galerkin’s method and the Generalized Differential Quadrature (GDQ) approach are applied to obtain the linear free vibrations of the carbon nano cone (CNC). The CNC is studied with simply supported boundary condition. The results of the new model are compared with those of the classical and couple stress theories, which point to the conclusion that the classical and couple stress models are special cases of modified strain gradient theory. Results also reveal that rigidity of the nano truncated conical shell in the strain gradient theory is greater than that in the modified couple stress and classical theories respectively, which leads to an increase in dimensionless natural frequency ratio. Moreover, the study investigates the effect of the size parameters on nano shell vibration for different lengths and vertex angles. http://jsm.iau-arak.ac.ir/article_677402_afdf1e4e863a9785c3f67384d3a86f0b.pdfIslamic Azad University - Arak BranchJournal of Solid Mechanics2008-350512320200930Study of Torsional Vibrations of Composite Poroelastic Spherical Shell-Biot’s Extension Theory64966267745510.22034/jsm.2020.1885789.1529ENRGurijalaDepartment of Mathematics, Kakatiya University, Warangal, Telangana, IndiaMReddy PeratiDepartment of Mathematics, Kakatiya University, Warangal, Telangana, IndiaJournal Article20200628Torsional vibrations of composite poroelastic dissipative spherical shell are investigated in the framework of Biot’s extension theory.Here composite poroelastic spherical shell consists of two spherical shells, one is placed on other, and both are made of different poroelastic materials. Consideration of the stress-free boundaries of outer surface and the perfect bonding between two shells leads to complex valued frequency equation. Limiting case when the ratio of thickness to inner radius is very small is investigated numerically. In this case, thick walled composite spherical shell reduces to thin composite spherical shell. For illustration purpose, four composite materials, namely, Berea sandstone saturated with water and kerosene, Shale rock saturated with water and kerosene are employed. The particular cases of a poroelastic solid spherical shell and poroelastic thick walled hollow spherical shell are discussed. If the shear viscosity of fluid is neglected, then the problem reduces to that of classical Biot’s theory. Phase velocity and attenuation are computed and the results are presented graphically. Comparison is made between the results of Biot’s extension theory and that of classical Biot’s theory. It is conclude that shear viscosity of fluid is causing the discrepancy of the numerical results.http://jsm.iau-arak.ac.ir/article_677455_8576dddb09a8d539acbe1066885ebc2d.pdfIslamic Azad University - Arak BranchJournal of Solid Mechanics2008-350512320200930Three Dimensional Thermal Shock Problem in Magneto-Thermoelastic Orthotropic Medium66368067728410.22034/jsm.2020.1885944.1530ENSBiswasDepartment of Mathematics, University of North Bengal, Darjeeling, India0000-0002-3599-3431S. MAbo-DahabDepartment of Mathematics, Faculty of Science, Taif University, Saudi Arabia----
Department of Mathematics, Faculty of Science, South Valley University, EgyptJournal Article20200628The paper is concerned with the study of magneto-thermoelastic interactions in three dimensional thermoelastic medium under the purview of three-phase-lag model of generalized thermoelasticity. The medium under consideration is assumed to be homogeneous orthotropic medium. The fundamental equations of the three-dimensional problem of generalized thermoelasticity are obtained as a vector-matrix differential equation form by employing normal mode analysis which is then solved by eigenvalue approach. Stresses and displacements are presented graphically for different thermoelastic models.http://jsm.iau-arak.ac.ir/article_677284_317712f8e20be9749b2ec31ff69cd3ed.pdfIslamic Azad University - Arak BranchJournal of Solid Mechanics2008-350512320200930Moving Three Collinear Griffith Cracks at Orthotropic Interface68169967731110.22034/jsm.2020.1894276.1554ENPMandalDepartment of Civil Engineering, Indian Institute of Technology, Hyderabad, IndiaS.CMandalDepartment of Mathematics, Jadavpur University, IndiaJournal Article20200629This work deals with the interaction of P-waves between a moving central crack and a pair of outer cracks situated at the interface of an orthotropic layer and an elastic half-space. Initially, we considered a two-dimensional elastic wave equation in orthotropic medium. The Fourier transform has been applied to convert the basic problem to solve the set of four integral equations. These set of integral equations have been solved to to get the analytical expressions for the stress intensity factor (SIF) and crack opening displacements (COD) by using the finite Hilbert transform technique and Cooke’s result. The main objective of this work is to investigate the dynamic stress intensity factors and crack opening displacement at the tips of the cracks. The aims of the study of these physical quantities (SIF, COD) is the prediction of possible arrest of the cracks within a certain range of crack velocity by monitoring applied load. SIF and COD have been depicted graphically for various types of orthotropic materials. We presented a parametric study to explore the influence of crack growing and propagation. This result is very much applicable in bridges, roads, and buildings fractures.http://jsm.iau-arak.ac.ir/article_677311_9f52ab4ae7f11ac9a318c896c7ef4b28.pdfIslamic Azad University - Arak BranchJournal of Solid Mechanics2008-350512320200930Thermoelastic Damping and Frequency Shift in Kirchhoff Plate Resonators Based on Modified Couple Stress Theory With Dual-Phase-Lag Model70071267755510.22034/jsm.2020.1896290.1569ENSDeviDepartment of Mathematics, Abhilashi University, Mandi, Himachal Pradesh, IndiaRKumarDepartment of Mathematics, Kurukshetra University, Kurukshetra, Haryana, IndiaJournal Article20200629The present investigation deals with study of thermoelastic damping and frequency shift of Kirchhoff plate resonators by using generalized thermoelasticity theory of dual-phase-lag model. The basic equations of motion and heat conduction equation are written with the help of Kirchhoff-Love plate theory and dual phase lag model. The analytical expressions for thermoelastic damping and frequency shift of modified couple stress dual-phase-lag thermoelastic plate have been obtained. A computer algorithm has been constructed to obtain the numerical results. Influences of modified couple stress dual-phase-lag thermoelastic plate, dual- phase-lag thermoelastic plate and Lord-Shulman (L-S, 1967) thermoelastic plate with few vibration modes on the thermoelastic damping and frequency shift are examined. The thermoelastic damping and frequency shift with varying values of length and thickness are shown graphically for clamped-clamped and simply-supported boundary conditions. It is observed from the results that the damping factor and frequency shift have noticed larger value in the presence of couple stress for varying values of length but opposite effect are shown for varying values of thickness in case of both vibration modes and boundary conditions.http://jsm.iau-arak.ac.ir/article_677555_24f4d0a144e19d9021fb65d19e552a01.pdfIslamic Azad University - Arak BranchJournal of Solid Mechanics2008-350512320200930Influence of Rigidity, Irregularity and Initial Stress on Shear Waves Propagation in Multilayered Media71372867731210.22034/jsm.2020.1896884.1572ENR.KPooniaDepartment of Mathematics, Chandigarh University, Mohali, Punjabp-140413, India0000-0002-5732-2006NBasatiyaDepartment of Mathematics, Chandigarh University, Mohali, Punjabp-140413, IndiaVKaliramanDepartment of Mathematics, Chaudhary Devi Lal University, Sirsa-Haryana-125055, India0000-0002-4583-4442Journal Article20200630The propagation of shear waves in an anisotropic fluid saturated porous layer over a prestressed semi-infinite homogeneous elastic half-space lying under an elastic homogeneous layer with irregularity present at the interface with rigid boundary has been studied. The rectangular irregularity has been taken in the half-space. The dispersion equation for shear waves is derived by using the perturbation technique followed by Fourier transformations. The dimensionless phase velocity is plotted against dimensionless wave number for the different size of ratios of depth of rectangular irregularity with the height of the layer and anisotropy parameters with the help of MATLAB graphical routines in presence and absence of initial stress. From the graphical results, it has been seen that the phase velocity is significantly influenced by the wave number, the depth of the irregularity, rigid boundary and initial stress. The acquired outcomes can be valuable for the investigation of geophysical prospecting and understanding the cause and evaluating of damage due to earthquakes.http://jsm.iau-arak.ac.ir/article_677312_5d0f299b9a62d6d2d60fd90806e18f81.pdf