Theory Of Elasticity In Order Research Articles

Accurate Small Sized Load Bearing Components Displacement Analysis

Load bearing components with cross sections in scale of millimeters attracted popularity with the advance in material engineering. As standard size scale of load bearing structures are around 100 times bigger than millimeter sized components, classical stability analyzes may be insufficient to simulate all stress and strains in load bearing structures which may lead to fail in whole structure. In this paper, the displacement analysis of small cross-sectional load bearing metals is presented using finite element analyzes and higher order elasticity theory. To take the size effect into consideration surface elasticity theory is selected. It is obtained higher order analyzes results differs from F.E.A. results. This result reveals that classical methods are insufficient to give accurate success in millimeters scale.

SYMMETRY PROPERTIES AND SIMPLIFICATION OF MULTISCALE SECOND-ORDER GRADIENT ELASTICITY

The symmetry conditions of the gradient distortion models were investigated. We studied the variational significance of order-of-differentiation symmetry conditions of strain gradient elasticity. This symmetry is a necessary and sufficient conditions for the continuity of the first derivatives of displacements (i.e., conditions for the absence of defects in the medium under consideration). We believe that ignoring this symmetry can lead to ill-posedness in boundary value problems and, as a result, to erroneous results in modeling. Thus the problem under consideration is relevant to all applied models of higher order elasticity theory that take into account size effects and are used for modeling in micro and nanomechanics. The discussed symmetry properties must be account for, due to formulation of boundary value problems for applied gradient distortion theories and especially for the vector variants of such models. Otherwise, the boundary value problem may lose correctness and in the result, the energetically insignificant components of the sixth rank tensor can be appeared in the formulation of the boundary conditions. A example is given, indicating the possibility of the occurrence of ill-posedness of the boundary value problem.

The grain parameters determination based on elements of the elasticity theory

The grain of agricultural crops during the machine process undergoes loads exceeding the permissible ones, as a result of which, under the influence of the mechanisms’ units, the grain material becomes damaged. The paper is devoted to search of the solution for the problem of reducing the grain mechanical damages which appears when the seeds pass through the machining. Using buckwheat as an example, the author considers the application of methods of the theory of elasticity in order to assess the magnitudes of stresses arising in caryopses under their force loading. In this regard, it is recommended to represent the grain model of this crop in the form of a cylindrical trihedron. The text of the work contains the experimentally proven correspondence of the volume of the proposed model to the real one, depending on the humidity conditions, the value of the elastic modulus is determined. The considered method for assessing the magnitudes of stresses at a known value of the compressive force and the shape of the seed makes it possible to take into account the design parameters when designing agricultural machines.

Higher order mixture nonlocal gradient theory of wave propagation

The higher order mixture nonlocal gradient theory of elasticity is conceived via consistent unification of the higher order stress‐ and strain‐driven mixture nonlocal elasticity and the higher order strain gradient theory. The integro‐differential constitutive law is established applying an abstract variational approach and appropriately replaced with the equivalent differential condition subject to nonclassical boundary conditions. The introduced higher order elasticity theory provides, as special cases, a variety of generalized elasticity theories adopted in nanomechanics to assess size effects in continua with nanostructural features. The well‐posed higher order mixture nonlocal gradient theory is elucidated and invoked to examine the flexural wave propagation. The closed‐form wave propagation relation between the phase velocity and the wave number is analytically derived. The determined wave propagation response and ensuing results are compared and calibrated with the pertinent molecular dynamic simulations. The demonstrated results of the phase velocity of the flexural wave propagation detect new benchmarks for numerical analyses. The proposed higher order size‐dependent elasticity approach can be profitably employed in rigorous analysis of pioneering nanotechnological devices.

Two‐phase local/nonlocal gradient mechanics of elastic torsion

The higher order two‐phase local/nonlocal elasticity model and the higher order strain gradient theory are unified via an abstract variational scheme. The higher order constitutive integral convolution is established in a consistent variational framework governed by ad hoc functional space of test fields. Equivalent differential constitutive law equipped with nonclassical boundary conditions of constitutive type is determined. The proposed higher order elasticity theory provides as special cases a range of well‐known size‐dependent elasticity models such as nonlocal, two‐phase local/nonlocal, strain gradient, modified nonlocal strain gradient, and nonlocal strain‐driven gradient models. Evidences of well‐posedness of the introduced higher order two‐phase local/nonlocal gradient problems are elucidated by rigorous examination of the elastostatic torsional response of structural schemes of applicative interest in nano‐mechanics. The exact analytical solution of the torsion problem of elastic nano‐beams is derived, graphically demonstrated, and compared with analogous outcomes in the literature. The conceived higher order elasticity theory can efficiently characterize advanced nano‐materials and structural elements of modern nano‐systems.

Axial vibration analysis of a Rayleigh nanorod with deformable boundaries

In this study, the free axial vibration of Rayleigh nanorods with axial restraints is studied via Eringens’ nonlocal elasticity theory. This higher order elasticity theory takes into account the size effect into the formulation due to dealing with micro and nanostructures. The boundary conditions and equation of motion are obtained using Hamilton’s principle. Two symmetrical axial elastic springs are attached to a nanorod at both ends. The novelty of the present study is that it seeks to obtain a general eigen value algorithm for the angular frequencies subjected to the rigid or restrained boundary conditions in a nanorod for the first time. A Fourier sine series is used to work Stokes’ transformation for the Rayleigh nanorods with elastic springs at the ends. Afterward, the effect of the spring coefficient on the the eigen-frequency is investigated. Also, the effects of the nonlocal parameter and the elastic springs on the eigen-frequency is reported.

Size-Dependent Torsional Buckling of Carbon Nano-Peapods Based on the Modified Couple Stress Theory

Using the modified couple stress theory and the shell model, this paper investigates the torsional instability of carbon nano peapods (CNPs), which are the hybrid structures made by inserting C[Formula: see text] fullerenes into CNT. Based on the modified couple stress theory as a higher order elasticity theory and using the first-order shear deformation shell theory, the equilibrium equations are derived through the principle of the minimum potential energy. The critical torsional moment is determined for the (10,10) CNT and the C[Formula: see text]@(10,10) CNP, and compared with that of MD simulations in the literature. Then, the increase in critical torsional moment is represented with the increase in the length scale parameter in both (10,10) CNT and C[Formula: see text]@(10,10) CNP based on the modified couple stress theory. Besides, by comparing the critical torsional moment in the C[Formula: see text]@(10,10) CNP and the (10,10) CNT, 100 percent increase in buckling resistance due to the presence of C[Formula: see text] fullerene is exhibited as well. In addition, the CNP surrounding is modeled as Pasternak foundation, and the effects of Pasternak stiffness are shown on the critical torsional moment.

An Accurate Study on Capacitive Microphone with Circular Diaphragm Using a Higher Order Elasticity Theory

THIS STUDY HAS BEEN UNDERTAKEN TO INVESTIGATE THE MECHANICAL BEHAVIOR OF THE CAPACITIVE MICROPHONE WITH CLAMPED CIRCULAR DIAPHRAGM USING MODIFIED COUPLE STRESS THEORY IN COMPARISON TO THE CLASSICAL ONE. PRESENCE OF THE LENGTH SCALE PARAMETER IN MODI-FIED COUPLE STRESS THEORY PROVIDES THE MEANS TO EVALUATE THE SIZE EFFECT ON THE MICROPHONE MECHANICAL BEHAVIOR. INVESTIGATING PULL-IN PHENOMENON AND DYNAMIC BEHAVIOR OF THE MICROPHONE ARE THE MATTERS PROVIDED DUE TO THE APPLICATION OF A STEP DC VOLTAGE. ALSO THE EFFECTS OF DIFFERENT AIR DAMPING COEFFICIENTS ON DYNAMIC PULL-IN VOLTAGE AND PULL-IN TIME HAVE BEEN STUDIED. THE OUTPUT LEVEL OR SENSITIVITY OF THE MICROPHONE HAS BEEN STUDIED BY INVESTIGATING THE FREQUENCY RESPONSE IN TERM OF MAGNITUDE FOR DIFFERENT LENGTH SCALE PARAMETERS TO FIGURE OUT HOW THE LENGTH SCALE PARA-METER AFFECTS ON THE SENSITIVITY OF THE CAPACITIVE MICROPHONE. TO ACHIEVE THESE ENDS, THE NONLINEAR DIFFERENTIAL EQUATION OF THE CIRCULAR DIAPHRAGM HAS BEEN EXTRACTED USING KIRCHHOFF THIN PLATE THEORY. THEN, A STEP-BY-STEP LINEARIZATION METHOD (SSLM) HAS BEEN USED TO ESCAPE FROM THE NONLINEARITY OF THE DIFFERENTIAL EQUATION .AFTERWARDS, GALERKIN-BASED REDUCED-ORDER MODEL HAS BEEN APPLIED TO SOLVE THE OBTAINED EQUATION.

On the Buckling of Isotropic, Transversely Isotropic and Laminated Composite Rectangular Plates

This paper presents the uniaxial and biaxial buckling analysis of rectangular plates based on new trigonometric shear and normal deformation theory. The theory accounts for the cosine distribution of the transverse shear strain through the plate thickness and on the free boundary conditions on the plate surfaces without using the shear correction factor. Governing equations and boundary conditions of the theory are derived by the principle of virtual work. The Navier type solutions for the buckling analysis of simply supported isotropic, transversely isotropic, orthotropic and symmetric cross-ply laminated composite rectangular plates subjected to uniaxial and biaxial compressions are presented. The effects of variations in the degree of orthotropy of the individual layers, side-to-thickness ratio and aspect ratio of the plate are examined on the buckling characteristics of composite plates. The present results are compared with those of the classical plate theory (CPT), first order shear deformation theory (FSDT) and exact three-dimensional (3D) elasticity theory wherever applicable. Good agreement is achieved of the present results with those of higher order shear deformation theory (HSDT) and elasticity theory.

Acoustoelastic effects on the resonance frequencies of prestressed concrete beams—Short-term measurements

In this study resonant acoustic spectroscopy was applied during static loading and unloading of three prestressed concrete beams in the context of acoustoelasticity. At each load step multiple modes of vibration were measured using an accelerometer and a small impact source. It was found that the measured resonance frequencies increased with increasing compressive stress in a manner which can be predicted using a non-linear finite element model based on Murnaghan's third order elastic theory. Previous results from experimental studies on concrete bridges indicate that measured structural resonance frequencies increase with increasing prestress forces. These results have been difficult to explain theoretically and the results from this study can possibly provide a new theoretical basis for the observed stress dependency of the resonance frequencies of concrete structures. Furthermore, the results indicate that a change in the state of stress in a concrete structure can be detected or monitored by measuring one or several resonance frequencies.

Free vibration of nanorings/arches based on nonlocal elasticity

This paper deals with the free vibration problem of nanorings/arches. The problem is formulated on the basis of Eringen’s nonlocal theory of elasticity in order to allow for the small length scale effect. Exact vibration frequencies are derived for the nanorings/arches and the effects of small length scale, defects, and elastic boundary conditions are investigated. The small length scale effect lowers the vibration frequencies. The defects and the use of elastic boundary conditions (instead of fixed restraints) also significantly reduce the frequencies and alter the vibration mode shapes of circular rings/arches. The results presented should be useful to engineers who are designing nanorings/arches for microelectromechanical and nanoelectromechanical devices.

Solitary and compactlike shear waves in the bulk of solids

We show that a model proposed by Rubin, Rosenau, and Gottlieb [J. Appl. Phys. 77, 4054 (1995)] for dispersion caused by an inherent material characteristic length belongs to the class of simple materials. Therefore, it is possible to generalize the idea of Rubin, Rosenau, and Gottlieb to include a wide range of material models, from nonlinear elasticity to turbulence. Using this insight, we are able to fine-tune nonlinear and dispersive effects in the theory of nonlinear elasticity in order to generate pulse solitary waves and also bulk traveling waves with compact support.

Load-carrying Behavior Of Unstiffened Elements At Elevated Temperatures In Fire

Unstiffened elements affect markedly the resistance of commonly used open steel sections under fire conditions. The distinct non-linear material behavior of steel at elevated temperatures requires large strains to activate the increase of cross-sectional capacity due to plastification. The yield line theory in conjunction with temperaturedependent second order elastic theory including equivalent geometric imperfections has been used to describe the load-carrying behavior of unstiffened elements in compression and bending at elevated temperatures in fire. Using both theories, the complete loadcarrying behavior in the pre- and post-buckling range can be analyzed. In addition, the results obtained from these analytical solutions were compared to numerical results using a geometrical and physical non-linear finite element approach.

Spherically-symmetric solutions for a spherical shell in finite pseudo-elasticity

In a recent paper the authors have incorporated the notion of discontinuous internal variables into the theory of finite elasticity in order to model certain coexistence of phases phenomena encountered in the deformation of solids. For example, phase changes in polymeric solids may be associated with changes in material properties induced by damage, in which case the internal variables are interpreted as damage variables. In the present paper the inflation of a thick-walled spherical shell of incompressible isotropic elastic material with a single internal variable is examined on the basis of this theory. The development of two-phase solutions is demonstrated for the loading process, and, for unloading, an example in which a residual deformation remains after removal of the pressure is given and the corresponding residual stress is calculated.

Two-Dimensional Stress Analysis and Strength of Band Adhesive Butt Joints of Dissimilar Adherends Subjected to External Bending Moments

The stresses of band adhesive butt joints, in which the interfaces were partially bonded, were analyzed using a two-dimensional theory of elasticity in order to establish the fracture criteria when the joints of dissimilar adherends were subjected to external bending moments. In the analysis, the dissimilar adherends and the adhesive were replaced with finite strips when the interfaces were bonded by the adhesive at two regions. In the numerical calculations, the effect of the ratios of Young's moduli between adherends and the adhesive, the thickness of the adhesive, the bonding area and the position on the stress distributions at the interfaces were demonstrated. As a result, it was seen that band adhesive joints were available when the bonding area and positions were determined taking into account the external load distributions. Experiments were performed on the strains of adherends and the joint strength. Analytical results were consistent with the experimental ones.

Flexoelectric polarization and second order elasticity for nematic liquid crystals

The flexoelectric properties of nematic liquid crystals are analysed. It is shown that in the frame of the usual elasticity two coefficients characterize completely the flexoelectric properties of the liquid crystal. These coefficients, in the limit of small scalar order parameter, i.e. near the clearing point, are approximately equal. More precisely they have the same linear term in the scalar order parameter, and differ for terms quadratic in this parameter. Their difference behaves, hence, as the usual nematic liquid crystal elastic constants, whereas their sum depends on the temperature, like the mixed splay-bend elastic constant. It is shown furthermore that in the frame of a second order elastic theory in the flexoelectric polarization there are no terms from the second order spatial derivatives of the nematic director, or the nematic tensor order parameter. Consequently also in the frame of the second order elasticity the flexoelectric polarization is given by the usual expression. This conclusion is important in connection with the surface polarization recently discovered in pretilted nematic liquid crystal samples.

Two-Dimensional Stress Analysis and Strength Evaluation of Band Adhesive Butt Joints of Dissimilar Adherends Subjects to Tensile Loads.

The stress of band adhesive butt joints in which the interfaces are partially bonded, was analyzed using a two-dimensional, theory of elasticity in order to establish fracture criteria for the case in which the joints of dissimilar adherends are subjected to tensile loads. In the analysis, when the interfaces are bonded by an adhesive at two regions, the dissimilar adherends and the adhesive are replaced with finite strips. In the numerical calculations, the effects of the ratio of Young's modulus of adherends to that of adhesives, the thickness of the adhesives, the bonding area and position and load distributions on the stress distributions at the interfaces were demonstrated. In addition, with use of the stress distribution, a method for evaluating joint strength was proposed. As a result, it was observed that band adhesive joints were available when the bonding area and positions were varied taking into account external load distribution. For verification, experiments were performed on the strain of adherends and the joint strength. Analytical results were consistent with the experimental ones.

A Two-Dimensional Stress Analysis of Butt Adhesive Joints Filled with Rigid Circular Fillers in an Adhesive Subjected to Tensile Loads

This paper deals with a two-dimensional stress analysis of a butt adhesive joint with rigid fillers in an adhesive subjected to a tensile load. Assuming that adherends are rigid and replacing an adhesive with a finite strip including rigid fillers, the analysis was done using the two-dimensional theory of elasticity in order to examine the joint strength. The effects of the location and size of rigid fillers on the stress distributions around the fillers and at the interfaces were shown by numerical calculations. For verification, photoelastic experiments were performed. The analytical result was in fairly good agreement with the experimental result. It was seen that based on the amount of shift of the location and the filler diameter, the joint strength was more improved than that of the joints without fillers.

A Two-Dimensional Stress Analysis of a Butt Adhesive Joint Filled with Rigid Circular Fillers in the Adhesive Subjected to a Cleavage Load.

This paper deals with a two-dimensional stress analysis of a butt adhesive joint with rigid fillers in the adhesive subjected to a cleavage load. Similar adherends were replaced with finite strips, and an adhesive was replaced with a finite strip including rigid fillers. The analysis was done using the two-dimensional theory of elasticity in order to evaluate the joint strength. The effects of the location and size of rigid fillers on the stress distributions around the fillers and at the inter face were shown by numerical calculations. An experiment concerning the strains of adhesive was performed. The analytical result was consistent with the experimental result. It was seen that with an increase of the shift of location and size of fillers, the joint strength increased more than that of the joint without a filler. The stress singularity at the edge of the interface was discussed.

A Two-Dimensional Stress Analysis and Strength of Band Adhesive Butt Joints of Dissimilar Adherends Subjected to Externl Bending Moments.

The stresses of band adhesive butt joints, in which the interfaces were partially bonded, were analyzed using a two-dimensional theory of elasticity in order to establish the fracture criteria when the joints of dissimilar adherends were subjected to external bending moments. In the analysis, the dissimilar adherends and the adhesive were replaced with finite strips when the interfaces were bonded by an adhesive at two regions. In the numerical calculations, the effects of the ratios of Young's modulus among adherends and an adhesive, the thickness of the adhesive, the bonding area and the position on the stress distributions at the interfaces were demonstrated. As a result, it was seen that band adhesive joints were available when the bonding area and positions were determined taking into account the external load distributions. Experiments were performed on the strains of adherends and the joint strengths. Analytical results were consistent with the experimental ones.