• Structural Engineering and Mechanics
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• v.65 no.4
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• pp.465-476
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• 2018

This article investigates buckling behavior of a multi-phase nanocrystalline nanobeam resting on Winkler-Pasternak foundation in the framework of nonlocal couple stress elasticity and a higher order refined beam model. In this model, the essential measures to describe the real material structure of nanocrystalline nanobeams and the size effects were incorporated. This non-classical nanobeam model contains couple stress effect to capture grains micro-rotations. Moreover, the nonlocal elasticity theory is employed to study the nonlocal and long-range interactions between the particles. The present model can degenerate into the classical model if the nonlocal parameter, and couple stress effects are omitted. Hamilton's principle is employed to derive the governing equations and the related boundary conditions which are solved applying an analytical approach. The buckling loads are compared with those of nonlocal couple stress-based beams. It is showed that buckling loads of a nanocrystalline nanobeam depend on the grain size, grain rotations, porosities, interface, elastic foundation, shear deformation, surface effect, nonlocality and boundary conditions.

• Structural Engineering and Mechanics
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• v.76 no.3
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• pp.413-420
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• 2020

Considering inverse cotangential shear strain function, the present paper studies nonlinear stability of nonlocal higher-order refined beams made of metal foams based on Chebyshev-Ritz method. Based on inverse cotangential beam model, it is feasible to incorporate shear deformations needless of shear correction factor. Metal foam is supposed to contain different distributions of pores across the beam thickness. Also, presented Chebyshev-Ritz method can provide a unified solution for considering various boundary conditions based on simply-supported and clamped edges. Nonlinear effects have been included based upon von-karman's assumption and nonlinear elastic foundation. The buckling curves are shown to be affected by pore distribution, geometric imperfection of the beam, nonlocal scale factor, foundation and geometrical factors.

• Advances in nano research
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• v.5 no.4
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• pp.281-301
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• 2017

In this disquisition, an exact solution method is developed for analyzing the vibration characteristics of magneto-electro-elastic functionally graded (MEE-FG) beams by considering porosity distribution and various boundary conditions via a four-variable shear deformation refined beam theory for the first time. Magneto-electroelastic properties of porous FG beam are supposed to vary through the thickness direction and are modeled via modified power-law rule which is formulated using the concept of even and uneven porosity distributions. Porosities possibly occurring inside functionally graded materials (FGMs) during fabrication because of technical problem that lead to creation micro-voids in FG materials. So, it is necessary to consider the effect of porosities on the vibration behavior of MEE-FG beam in the present study. The governing differential equations and related boundary conditions of porous MEE-FG beam subjected to physical field are derived by Hamilton's principle based on a four-variable tangential-exponential refined theory which avoids the use of shear correction factor. An analytical solution procedure is used to achieve the natural frequencies of porous-FG beam supposed to magneto-electrical field which satisfies various boundary conditions. A parametric study is led to carry out the effects of material graduation exponent, porosity parameter, external magnetic potential, external electric voltage, slenderness ratio and various boundary conditions on dimensionless frequencies of porous MEE-FG beam. It is concluded that these parameters play noticeable roles on the vibration behavior of MEE-FG beam with porosities. Presented numerical results can be applied as benchmarks for future design of MEE-FG structures with porosity phases.

• Earthquakes and Structures
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• v.15 no.4
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• pp.369-382
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• 2018

A free vibration analysis and wave propagation of functionally graded porous beams has been presented in this work using a high order hyperbolic shear deformation theory. Unlike other conventional shear deformation theories, a new displacement field that introduces indeterminate integral variables has been used to minimize the number of unknowns. The constituent materials of the beam are assumed gradually variable along the direction of height according to a simple power law distribution in terms of the volume fractions of the constituents. The variation of the pores in the direction of the thickness influences the mechanical properties. It is therefore necessary to predict the effect of porosity on vibratory behavior and wave velocity of FG beams in this study. A new function of the porosity factor has been developed. Hamilton's principle is used for the development of wave propagation equations in the functionally graded beam. The analytical dispersion relationship of the FG beam is obtained by solving an eigenvalue problem. Illustrative numerical examples are given to show the effects of volume fraction distributions, beam height, wave number, and porosity on free vibration and wave propagation in a functionally graded beam.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.41 no.5
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• pp.337-343
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• 2017

As a shop practice, a strength estimation method for die cast parts is suggested, in which various defects such as pores can be allowed. The equivalent porosity is evaluated by combining the stiffness data from a simple elastic test at the part level during the shop practice and the theoretical stiffness data, which are defect free. A porosity equation is derived from Eshelby's inclusion theory. Then, using the Mori-Tanaka method, the porosity value is used to draw a stress-strain curve for the porous material. In this paper, the Hollomon equation is used to capture the strain hardening effect. This stress-strain curve can be used to estimate the strength of a die cast part with porous defects. An elastoplastic theoretical solution is derived for the three-point bending of a die cast beam by using the plastic hinge method as a reference solution for a part with porous defects.

• Steel and Composite Structures
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• v.45 no.1
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• pp.67-82
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• 2022

In this paper, buckling and free vibration of imperfect, functionally graded beams, including porosities, are investigated, using a higher order shear strain theory. Due to defects during the manufacturing process, micro porosities may appear in the material, hence the appearance of this imperfection in the structure. The material properties of the beams are assumed to vary regularly, with power and sigmoid law, in the direction of thickness. A novel porosity distribution affecting the functionally graded volume fraction is presented. For the compact formulation used for cementite-based materials and already used in P-FGM, we have adapted it for the distribution of S-FGM. The equations of motion in the FG beam are derived using Hamilton's principle. The boundary conditions for beam FG are assumed to be simply supported. Navier's solution is used to obtain the closed form solutions of the FG beam. The numerical results of this work are compared with those of other published research to verify accuracy and reliability. The comparisons of different shear shape functions, the influence of porosity, thickness and inhomogeneity parameters on buckling and free vibration of the FG beam are all discussed. It is established that the present work is more precise than certain theories developed previously.

• Advances in nano research
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• v.10 no.3
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• pp.281-293
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• 2021

This paper presents a new nonlocal Hyperbolic Shear Deformation Beam Theory (HSDBT) for the free vibration of porous Functionally Graded (FG) nanobeams. A new displacement field containing integrals is proposed which involves only three variables. The present model incorporates the length scale parameter (nonlocal parameter) which can capture the small scale effect and its account for shear deformation by a hyperbolic variation of all displacements through the thickness without using the shear correction factor. It has been observed that during the manufacture of Functionally Graded Materials (FGMs), micro-voids and porosities can occur inside the material. Thus, in this work, the investigation of the free vibration analysis of FG beams taking into account the influence of these imperfections is established. Four different porosity types are considered for FG nanobeam. Material characteristics of the FG beam are supposed to vary continuously within thickness direction according to a power-law scheme which is modified to approximate material characteristics for considering the influence of porosities. Based on the nonlocal differential constitutive relations of Eringen, the equations of motion of the nanobeam are derived using Hamilton's principle. The effects of nonlocal parameter, aspect ratio, and the porosity types on the dynamic responses of the nanobeam are discussed.

• Advances in nano research
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• v.10 no.3
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• pp.235-251
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• 2021

In the current study, the free vibrational behavior of a Porous Micro (PM) beam which is integrated with Functionally Graded Piezoelectric (FGP) layers with initial curvature is considered based on the two trigonometric shear deformation theories namely SSDBT and Tan-SDBT. The structure's mechanical properties are varied through its thicknesses following the given functions. The curved microbeam is exposed to electro-mechanical preload and also is rested on a Pasternak type of elastic foundation. Hamilton's principle is used to extract the motion equations and the MCST is used to capture the size effect. Navier's solution method is selected as an analytical method to solve the motion equations for a simply supported ends case and by validating the results for a simpler state with previously published works, effects of different important parameters on the behavior of the structure are considered. It is found that although increasing the porosity reduces the natural frequency, but enhancing the volume fraction of CNTs increasing it. Also, by increasing the central angle of the curved beam the vibrations of the structure increases. Designing and manufacturing more efficient smart structures such as sensors and actuators are of the aims of this study.

• Steel and Composite Structures
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• v.50 no.1
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• pp.15-24
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• 2024

This research presents dynamical reaction investigation of pore-dependent and nano-thickness beams having functional gradation (FG) constituents exposed to a movable particle. The nano-thickness beam formulation has been appointed with the benefits of refined high orders beam paradigm and nonlocal strain gradient theory (NSGT) comprising two scale moduli entitled nonlocality and strains gradient modulus. The graded pore-dependent constituents have been designed through pore factor based power-law relations comprising pore volumes pursuant to even or uneven pore scattering. Therewith, variable scale modulus has been thought-out until process a more accurate designing of scale effects on graded nano-thickness beams. The motion equations have been appointed to be solved via Ritz method with the benefits of Chebyshev polynomials in cosine form. Also, Laplace transform techniques help Ritz-Chebyshev method to obtain the dynamical response in time domain. All factors such as particle speed, pores and variable scale modulus affect the dynamical response.

• Journal of Sensor Science and Technology
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• v.5 no.5
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• pp.21-29
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• 1996

A piezoresistive silicon acceleration sensor with 8 beams, utilized by an unique silicon micromachining technique using porous silicon etching method which was fabricated on the selectively diffused (111)-oriented $n/n^{+}/n$ silicon subtrates. The width, length, and thickness of the beam was $100\;{\mu}m$, $500\;{\mu}m$, and $7\;{\mu}m$, respectively, and the diameter of the mass paddle (the region suspended by the eight beams) was 1.4 mm. The seismic mass on the mass paddle was formed about 2 mg so as to measure accelerations of the range of 50g for automotive applications. For the formation of the mass, the solder mass was loaded on the mass paddle by dispensing Pb/Sn/Ag solder paste. After the solder paste is deposited, Heat treatment was carried out on the 3-zone reflow equipment. The decay time of the output signal to impulse excitation of the fabricated sensor was observed for approximately 30 ms. The sensitivity measured through summing circuit was 2.9 mV/g and the nonlinearity of the sensor was less than 2% of the full scale output. The output deviation of each bridge was ${\pm}4%$. The cross-axis sensitivity was within 4% and the resonant frequency was found to be 2.15 KHz from the FEM simulation results.