Inhomogeneous Constants Research Articles

Pull-in behaviors of micro-beam made of bidirectional functionally grade materials based on modified couple stress theory

In this paper, pull-in behavior of cantilever micro/nano beams made of bi directional functionally graded materials (BDFGM) with small scale effects under electrostatic force is investigated. Couple stress theory is employed to study the influence of small-scale on pull-in behavior. The material properties except Poisson’s ratio obey the arbitrary function in the thickness and length direction. The approximate analytical solutions for the pull-in voltage and pull-in displacement of the microbeams are derived using the Galerkin method. Comparison between the results of present work with other article for pull-in behavior of a microbeams made of isotropic material reveals the accuracy of this study. Numerical results explored the effects of material length scale parameter, inhomogeneity constant, gap distance and dimensionless thickness.

Effect of Inhomogeneity Constant on Equivalent Stresses in Elastic Analysis of Hollow Cylinder Made from Functionally Graded Material

In this study, the determination of the equivalent stresses required for the elastic analysis of a hollow cylinder made of functional graded material (FGM) subjected to internal and external pressures was determined quickly and accurately, and its evaluation was discussed. The Poisson’s ratio is thought to be constant. The functional grading for the modulus of elasticity varies radially along the thickness of the cylinder and depending on a simple power law function. Radial, tangential (hoop) and equivalent stresses with radial displacements in the cylinder are determined rapidly by modeling both analytically and by the finite element method (FEM) numerically. Outcomes of both methods were compared and found to be in harmony. At the same time, differently from previous studies in my paper, the influence of the inhomogeneity constant of the material on equivalent stresses was investigated and the results are presented in graphical form.

Practical Jointed Approach to Thermal Performance of Functionally Graded Material Annular Fin

Using the complementary functions method, a general solution for the one-dimensional steady-state heat conduction equation for the fin together with the boundary conditions made of functionally graded material is presented. Two cases for the variation of the functionally graded material wall thermal conductivity were considered. These are 1) power and 2) exponential types of variation of the thermal conductivity varying with the radial coordinate. In the present paper, a semi-analytical iterative technique, one of the most efficient unified methods, is employed to solve the heat conduction equation. For different values of the inhomogeneity constant, distributions of temperature, heat transfer rate, fin efficiency, and fin effectiveness, as a function of radial direction, are obtained. Various material models from the literature are used and corresponding temperature distributions are computed. Verification of the proposed method is done using benchmark solutions available in the literature for some special cases, and virtually exact results are obtained.

Multiaxial ratcheting evaluation of functionally graded cylindrical shell by means of Ohno–Wang’s type models

In the present paper, the ratcheting responses of functionally graded (FG) pipe by means of nonlinear kinematic hardening rules of the Ohno–Wang (O–W), McDowell, Jiang–Sehitoglu (J–S) and Chen–Jiao–Kim (C–J–K) models are investigated. The FG pipe is considered to be subjected to a broad class of non-proportional/proportional with different loading types including tension–torsion, tension/thermal–internal pressure with different loading sequences and directions. In the current constitutive models of FG pipe, not only the physical and mechanical properties are variables but also the coefficients of the kinematic hardening rules vary as a power law through thickness. An implicit integration scheme implemented within user subroutine UMAT in ABAQUS/standard is presented for the relatively complicated constitutive models. Comparing with the novel experiments and available results in the literature, the predicted results by the proposed numerical method are demonstrated to be reliable. Results reveal the significant influences of the adopted hardening rules incorporated in the constitutive model and also FG inhomogeneity constant on the multiaxial ratcheting responses of FG pipe.

Torsional vibration of functionally graded nano-rod under magnetic field supported by a generalized torsional foundation based on nonlocal elasticity theory

This article contains the nonlocal elasticity theory to capture size effects in functionally graded (FG) nano-rod under magnetic field supported by a torsional foundation. Torque effect of an axial magnetic field on an FG nano-rod has been defined using Maxwell’s relation. The material properties were assumed to vary according to the power law in radial direction. The Navier equation and boundary conditions of the size-dependent FG nano-rod were derived by the Hamilton’s principle. These equations were solved by employing the generalized differential quadrature method (GDQM). Presented model has the ability to turn into the classical model if the material length scale parameter is taken to be zero. The effects of some parameters, such as inhomogeneity constant, magnetic field and small-scale parameter, were studied. As an important result of this study can be stated that an FG nano-rod model based on the nonlocal elasticity theory behaves softer and has smaller natural frequency.

Pull-in behavior of functionally graded micro/nano-beams for MEMS and NEMS switches

In this paper, pull-in behavior of cantilever micro/nano-beams made of functionally graded materials (FGM) with small-scale effects under electrostatic force is investigated. Consistent couple stress theory is employed to study the influence of small-scale on pull-in behavior. According to this theory, the couple tensor is skew-symmetric by adopting the skew-symmetric part of the rotation gradients. The material properties except Poisson’s ratio obey the power law distribution in the thickness direction. The approximate analytical solutions for the pull-in voltage and pull-in displacement of the microbeams are derived using the Rayleigh–Ritz method. Comparison between the results of the present work with Osterberg and Senturia’s article for pull-in behavior of microbeams made of isotropic material reveals the accuracy of this study. Numerical results explored the effects of material length scale parameter, inhomogeneity constant, gap distance and dimensionless thickness. Presented model has the ability to turn into the classical model if the material length scale parameter is taken to be zero. A comparison between classical and consistent couple stress theories is done which reveals the application of the consistent couple stress theory. As an important result of this study can be stated that a micro/nano-beams model based on the couple stress theory behaves stiffer and has larger pull-in voltages.

Consistent couple-stress theory for free vibration analysis of Euler-Bernoulli nano-beams made of arbitrary bi-directional functionally graded materials

In this paper, using consistent couple stress theory and Hamilton\'s principle, the free vibration analysis of Euler- Bernoulli nano-beams made of bi-directional functionally graded materials (BDFGMs) with small scale effects are investigated. To the best of the researchers\' knowledge, in the literature, there is no study carried out into consistent couple-stress theory for free vibration analysis of BDFGM nanostructures with arbitrary functions. In addition, in order to obtain small scale effects, the consistent couple-stress theory is also applied. These models can degenerate into the classical models if the material length scale parameter is taken to be zero. In this theory, the couple-tensor is skew-symmetric by adopting the skew-symmetric part of the rotation gradients as the curvature tensor. The material properties except Poisson\'s ratio are assumed to be graded in both axial and thickness directions, which it can vary according to an arbitrary function. The governing equations are obtained using the concept of Hamilton principle. Generalized differential quadrature method (GDQM) is used to solve the governing equations for various boundary conditions to obtain the natural frequencies of BDFG nano-beam. At the end, some numerical results are presented to study the effects of material length scale parameter, and inhomogeneity constant on natural frequency.

Exact and Numerical Elastic Analysis for the FGM Thick-Walled Cylindrical Pressure Vessels with Exponentially-Varying Properties

Abstract Assuming exponential-varying properties in the radial direction and based on the elasticity theory, an exact closed-form analytical solution is obtained to elastic analysis of FGM thick-walled cylindrical pressure vessels in the plane strain condition. Following this, radial distribution of radial displacement, radial stress, and circumferential stress are plotted for different values of material inhomogeneity constant. The displacements and stresses distributions are compared with the solutions of the finite element method (FEM).

Thermo-Elasto-Plastic Analysis of Thick-Walled Spherical Pressure Vessels Made of Functionally Graded Materials

An exact closed-form analytical solution is presented to solve the thermo-elasto-plastic problem of thick-walled spherical vessels made of functionally graded materials (FGMs). Assuming that the inner surface is exposed to a uniform heat flux, and that the outer surface is exposed to an airstream. The heat conduction equation for the one-dimensional problem in spherical coordinates is used to obtain temperature distribution in the sphere. Material properties are graded in the thickness direction according to a power law distribution, whereas the Poisson’s ratio is kept constant. The Poisson’s ratio due to slight variations in engineering materials is assumed constant. The plastic model is based on von Mises yield criterion and its associated flow rules under the assumption of perfectly plastic material behavior. For various values of inhomogeneity constant, the so-obtained solution is then used to study the distribution of limit heat flux, displacement and stresses versus the radial direction. Moreover, the effect of increasing the heat flux and pressure on the propagation of the plastic zone are investigated. Furthermore, the effect of change in Poisson’s ratio on the value of the critical material parameter is demonstrated. The present study is also validated by comparing the numerical results for thick elasto-plastic spherical shells available in the literature. To the best of the authors’ knowledge, in previous studies, exact thermo-elasto-plastic behavior of FGM thick-walled sphrical pressure vessels has not investigated.

Eringen's non-local elasticity theory for bending analysis of bi-directional functionally graded Euler–Bernoulli nano-beams

In this paper, the problem of the static bending of Euler–Bernoulli nano-beams made of bi-directional functionally graded material (BDFGM) with small scale effects is formulated. The model is based on the Eringen's nonlocal elasticity theory applied to Euler–Bernouilli nano-beams. To the best of the researchers’ knowledge, in the literature, there is no study carried out into non-local elasticity theory for bending analysis of BDFGM nanostructures with arbitrary functions. The novelty of the present study is that it seeks to investigate size effects on bending analysis of bi-directional functionally graded (BDFG) Euler–Bernoulli nano-beams based on Eringen's non-local elasticity theory. Material properties of nano-beam are assumed to change along the thickness and length direction according to arbitrary function. The governing equations are obtained, using the concept of the principle of minimum potential energy. Generalized differential quadrature method (GDQM) is used to solve the governing equations for various boundary conditions to obtain the deflection of FG nano-beam. These models can degenerate into the classical models if the material length scale parameter is taken to be zero. Finally, some numerical results are presented to study the effects of material length scale parameter and inhomogeneity constant on bending analysis of FGM Euler–Bernoulli nano-beams.

Non-local analysis of free vibration of bi-directional functionally graded Euler–Bernoulli nano-beams

In this paper, the free vibration analysis of Euler–Bernoulli nano-beams made of bi-directional functionally graded material (BDFGM) with small scale effects is investigated. To study the small scale effects on free vibration, the non-local elasticity theory is applied. To the best of the researchers’ knowledge, in the literature, there is no study carried out into non-local elasticity theory for free vibration analysis of BDFGM nanostructures with arbitrary functions. The novelty of the present work is that it seeks to investigate size effects on free vibration analysis of bi-directional functionally graded (BDFG) Euler–Bernoulli nano-beams based on Eringen's non-local elasticity theory. The material properties obey the arbitrary function in thickness and length direction. The governing equations are obtained, using the concept of Hamilton principle. Generalized differential quadrature method (GDQM) is used to solve the governing equations for various boundary conditions to obtain the natural frequencies of FG nano-beam. These models can degenerate into the classical models if the material length scale parameter is taken to be zero. Finally, some numerical results are presented to study the effects of material length scale parameter and inhomogeneity constant on free vibration analysis of FGM Euler–Bernoulli nano-beams.

Buckling analysis of arbitrary two-directional functionally graded Euler–Bernoulli nano-beams based on nonlocal elasticity theory

Based on the nonlocal elasticity theory, buckling analysis of the nano-beams made of two-directional functionally graded materials (FGM) with small scale effects is carried out. To the best of the authors’ knowledge, so far all previous solutions to the buckling analysis of arbitrary FGM Euler–Bernoulli nano-beams have addressed the case of properties varying in one direction only. The novelty of the current work is to present a solution by taking into account the variation of properties in two-directional functionally graded materials with arbitrary functions. The material properties obey the arbitrary function in thickness and length direction. The governing equations are obtained, employing the principle of minimum potential energy. Generalized differential quadrature method (GDQM) is selected in order to analyze the nonlocal beams with arbitrary boundary conditions along them to obtain the critical buckling load of FG nano-beam. These models can degenerate into the classical models if the material length scale parameter is taken to be zero. Finally, some numerical results are presented to study the effects of material length scale parameter and inhomogeneity constant on size dependent critical buckling load.

A unified method for stresses in FGM sphere with exponentially-varying properties

Using the Complementary Functions Method (CFM), a general solution for the one-dimensional steady-state thermal and mechanical stresses in a hollow thick sphere made of functionally graded material (FGM) is presented. The mechanical properties are assumed to obey the exponential variations in the radial direction, and the Poisson's ratio is assumed to be constant, with general thermal and mechanical boundary conditions on the inside and outside surfaces of the sphere. In the present paper, a semi-analytical iterative technique, one of the most efficient unified method, is employed to solve the heat conduction equation and the Navier equation. For different values of inhomogeneity constant, distributions of radial displacement, radial stress, circumferential stress, and effective stress, as a function of radial direction, are obtained. Various material models from the literature are used and corresponding temperature distributions and stress distributions are computed. Verification of the proposed method is done using benchmark solutions available in the literature for some special cases and virtually exact results are obtained.

Direct application of the Complementary Functions Method (CFM) to the static analysis of rotating disks with both parabolic-varying thickness profile and functionally graded (FG) material

In this study elastic stresses and displacements of a rotating variable-thickness disk made of a radially functionally graded (FG) material are calculated numerically in an accurate manner based on the Complementary Functions Method (CFM). The general parabolic function which determines the variation of the thickness of the disk includes concave, convex and linear disk profiles. It is assumed that the inner surface is to be all-metal and the outer surface is to be all-ceramic. Young’s modulus and density vary between inner and outer surface by obeying the volume-fraction material grading rule. The disk is fixed to a shaft at the inner surface and free expansion exists at the outer surface. For the Ti-6Al-4V - Si3N4 metal-ceramic pair, the effect of the inhomogeneity constants on the stress and displacement is investigated. Results are presented in both tabular and graphical forms. Those presented in tables may form benchmark data for purely numerical calculations.

Effect of mechanical and electrical behavior of gelatin hydrogels on drug release and cell proliferation

The present study was aimed to explore the effect of the mechanical and the electrical properties of the gelatin hydrogels on the mammalian cell proliferation and drug release properties. FTIR analysis of the hydrogels suggested that gelatin retained its secondary protein structure. A decrease in the diffusion constant of the water molecules was observed with the increase in the gelatin concentration in the hydrogels. The mechanical and the electrical stabilities of the hydrogels were enhanced with the increase in the gelatin content. Stress relaxation and creep studies were modeled using Weichert and Burger׳s models, respectively. The relaxation time (stress relaxation study) did not follow a concentration-dependent relationship and was found to affect the MG-63 cell (human osteoblast) proliferation. The impedance profile of the hydrogels was modeled using a (RQ)Q model. Release of ciprofloxacin from the hydrogels was inversely dependent on the rate of swelling. The release of the drug was not only dependent on the Fickian diffusion but also on the relaxation process of the gelatin chains. The inhomogeneous constant of the constant phase element representing the hydrogel-electrode interface indicated improved cell proliferation rate with a decrease in the inhomogeneous constant. In gist, the rate of cell proliferation could be related to the relaxation time (stress relaxation) and the inhomogeneous constant of the sample-electrode constant phase element (electrical study) properties, whereas, the drug release properties can be related to the bulk resistance of the formulations.

Elastic analysis of pressurized thick FGM cylinders with exponential variation of material properties using TSDT

In this research, the general governing set of differential equations for axisymmetric thick FG pressurized cylinders with exponential function of material properties is derived based on third order shear deformation theory. Afterwards, a general analytical solution of governing equations based on Eigen values problems is conducted for cylinders under clamped ends condition. Furthermore, a numerical modeling is done in order to compare the results of two different solution and prove the accuracy of analysis. The displacements and stresses resulted from FEM and TSDT are depicted for a case study along the radial and longitudinal direction of the cylinder. Afterwards, the effect of internal and external pressure, FGM inhomogeneity constants and higher order approximation is investigated. The results of SDT and FEM show good agreement and prove the fact that usage of FGM cylinders causes lower values of displacements and stresses.

Exact elasto-plastic analysis of rotating disks made of functionally graded materials

Presented in this paper is an exact closed-form analytical solution for elasto-plastic deformations and stresses in a rotating disk made of functionally graded materials (FGMs) in which the elasto-perfectly-plastic material model is employed. As for the yield criterion, the well-known Tresca’s model is used. The material properties are assumed to vary according to the power law whereas the Poisson’s ratio is kept constant. For various values of inhomogeneity constant, the so-obtained solution is then used to study the distribution of limit angular speed, displacement and stresses versus the radial direction. Moreover, the effect of increasing the angular speed on the propagation of the plastic zone is investigated. This analysis reveals that the plasticity can occur in different regions of the disk. To the best of authors’ knowledge, so far all previous exact solutions for the elasto-plastic analysis of FGM disks address the case of constant material density. The novelty of this work is to present an exact solution by taking into account the variation of density. The results show that the variation of density has a significant influence on the distribution of deformations and stresses. Furthermore, the effect of changing Poisson’s ratio on the value of the critical material parameter is demonstrated.

Time-dependent thermo-creep analysis of rotating FGM thick-walled cylindrical pressure vessels under heat flux

In the present study, time-dependent thermoelastic creep response for isotropic rotating thick-walled cylindrical pressure vessels made of functionally graded material (FGM) has been investigated, taking into account the creep behavior of the FGM pressure vessels, as described in Norton’s model. For the purpose of stress analysis in an FGM pressure vessel, material creep behavior and the solutions of the stresses at a time equal to zero (i.e. the initial stress state) are needed. This corresponds to the solution of materials with linear elastic behavior. Therefore, using equations of equilibrium, stress–strain and strain–displacement, a differential equation for displacement is obtained and subsequently the stresses at a time equal to zero are calculated. Using Norton’s law in the multi-axial form in conjunction with the above-mentioned equations in the rate form, the radial displacement rate is obtained and then the radial, circumferential and axial creep stress rates are calculated for the conditions of plane strain and plane stress. When the stress rates are known, the stresses at any time are calculated iteratively. Assuming that the inner surface is exposed to a uniform flux, and that the outer surface is exposed to an airstream, the heat conduction equation for the one-dimensional problem in polar coordinates is used to obtain temperature distribution in the cylinder. Assuming that material properties are a function of the radius of the cylinder and that the Poisson’s ratio is constant, creep stresses, creep strains and radial displacement are plotted against dimensionless radius and time for different values of the powers of the material properties. It has been found that in-homogeneity constants have significant influence on the distributions of the creep stresses and radial displacement.

Elastic analysis of exponential FGM disks subjected to internal and external pressure

Abstract Assuming exponential varying properties in the radial direction and constant Poisson’s ratio, a closed-form analytical solution based on the elasticity theory is obtained to elastic analysis of disks made of functionally graded materials (FGMs) subjected to internal and external pressure. Following this, radial displacement, radial stress, and circumferential stress profiles are plotted for different values of material inhomogeneity constant, as a function of radial direction. The displacements and stresses distributions are compared with the solutions of the finite element method (FEM) and comparison with the corresponding numerical solution indicates that the proposed solution has excellent convergence and accuracy.

COMPLETE CLOSED-FORM SOLUTION FOR PRESSURIZED HETEROGENEOUS THICK SPHERICAL SHELLS

On the basis of plane elasticity theory (PET), the governing equations for axisymmetric thick spherical shells made of nonhomogeneous functionally graded materials (FGMs) subjected to internal and external pressure in general case are derived. It is assumed that the modulus of elasticity varies nonlinearly in the radial direction, and the Poisson’s ratio is constant. The analytical solution of the equations for real, double and complex roots are obtained. The radial stress, meridional stress and radial displacement distributions depending on an inhomogeneity constant are compared with those of the homogeneous case as well as with the solution using finite element method (FEM) and presented in the form of graphs. The obtained result shows that the property of FGMs has a significant influence to the stress distribution along the radial direction. Results are useful for engineers to design a sphere made of FGMs. DOI: http://dx.doi.org/10.5755/j01.mech.18.5.2702