Showing papers in "European Journal of Mechanics A-solids in 2013"
TL;DR: In this paper, the analysis of free vibration problems of functionally graded shells is performed by radial basis functions collocation, according to a higher-order shear deformation theory that accounts for through-the-thickness deformation.
Abstract: This paper deals with free vibration problems of functionally graded shells. The analysis is performed by radial basis functions collocation, according to a higher-order shear deformation theory that accounts for through-the-thickness deformation. The equations of motion and the boundary conditions are obtained by Carrera’s Unified Formulation resting upon the principle of virtual work, and further interpolated by collocation with radial basis functions. Numerical results include spherical as well as cylindrical shell panels with all edges clamped or simply supported and demonstrate the accuracy of the present approach.
143 citations
TL;DR: In this article, a strain energy function for isotropic incompressible rubbers is proposed, which satisfies all the necessary characteristic properties of an efficient hyperelastic model, and complete analysis of the Mooney plot over a wide range of stretch in extension-compression is carried out.
Abstract: Hyperelastic behavior of isotropic incompressible rubbers is studied to develop a strain energy function which satisfies all the necessary characteristic properties of an efficient hyperelastic model. The proposed strain energy function includes only three material parameters which are somehow related to the physical quantities of the material molecular network. Moreover, the model benefits from mathematical simplicity, well suitting in all ranges of stretch and possessing the property of deformation-mode-independency. This reduces the required number of experimental tests for parameter calibration of the model. Results of the proposed model are compared with results of some available models as well as experimental data. Moreover, complete analysis of the Mooney plot over a wide range of stretch in extension–compression is carried out. It is found that the proposed model gives reasonable predictions in comparison with those of experiments.
141 citations
TL;DR: In this article, the effect of matrix shear strength upon the dynamic response was explored by testing: (i) CFRP plates with both a cured and uncured matrix and (ii) UHMWPE laminates with identical fibres but with two matrices of different shear strengths.
Abstract: The ballistic performance of clamped circular carbon fibre reinforced polymer (CFRP) and Ultra High Molecular Weight Polyethylene (UHMWPE) fibre composite plates of equal areal mass and 0/90° lay-up were measured and compared with that of monolithic 304 stainless steel plates. The effect of matrix shear strength upon the dynamic response was explored by testing: (i) CFRP plates with both a cured and uncured matrix and (ii) UHMWPE laminates with identical fibres but with two matrices of different shear strength. The response of these plates when subjected to mid-span, normal impact by a steel ball was measured via a dynamic high speed shadow moire technique. Travelling hinges emanate from the impact location and travel towards the supports. The anisotropic nature of the composite plate results in the hinges travelling fastest along the fibre directions and this results in square-shaped moire fringes in the 0/90° plates. Projectile penetration of the UHMWPE and the uncured CFRP plates occurs in a progressive manner, such that the number of failed plies increases with increasing velocity. The cured CFRP plate, of high matrix shear strength, fails by cone-crack formation at low velocities, and at higher velocities by a combination of cone-crack formation and communition of plies beneath the projectile. On an equal areal mass basis, the low shear strength UHMWPE plate has the highest ballistic limit followed by the high matrix shear strength UHMWPE plate, the uncured CFRP, the steel plate and finally the cured CFRP plate. We demonstrate that the high shear strength UHMWPE plate exhibits Cunniff-type ballistic limit scaling. However, the observed Cunniff velocity is significantly lower than that estimated from the laminate properties. The data presented here reveals that the Cunniff velocity is limited in its ability to characterise the ballistic performance of fibre composite plates as this velocity is independent of the shear properties of the composites: the ballistic limit of fibre composite plates increases with decreasing matrix shear strength for both CFRP and UHMWPE plates.
134 citations
TL;DR: In this paper, a Kirchhoff micro-plate model based on the modified strain gradient elasticity theory was presented to capture size effects, in contrast with the classical plate theory, and the analysis is general and can be reduced to the modified couple stress plate model or classical plate model once two or all material length scale parameters in the theory are set zero respectively.
Abstract: A Kirchhoff micro-plate model is presented based on the modified strain gradient elasticity theory to capture size effects, in contrast with the classical plate theory. The analysis is general and can be reduced to the modified couple stress plate model or classical plate model once two or all material length scale parameters in the theory are set zero respectively. Governing equation and boundary conditions of an isotropic rectangular micro-plate are derived using minimum potential energy principle. Various boundary conditions including simply supported and clamped edges are covered by the analysis. The extended Kantorovich method (EKM) which is an accurate approximate closed-form solution is applied to solve the resulting sixth order boundary value problem. Application of EKM to the partial differential equation (PDE) yields two ordinary differential equations (ODEs) in the independent x and y coordinates. The resulted ODEs are solved in an iterative manner. Exact closed-form solutions are presented for both ODEs in all of the iteration. It is shown that the method provides accurate predictions with very fast convergence. Numerical results reveal that the differences between the deflection predicted by the modified strain gradient model, the couple stress model and the classical model are large when the plate thickness is small and comparable to the material length scale parameters. However, the differences decrease with increasing the plate thickness. Validation of the presented EKM solution shows good agreement with available literature.
122 citations
TL;DR: In this paper, a number of refined beam theories were obtained by expanding the unknown displacement variables over the beam section axes by adopting Taylor's polynomials, trigonometric series, exponential, hyperbolic and zig-zag functions.
Abstract: A number of refined beam theories are discussed in this paper. These theories were obtained by expanding the unknown displacement variables over the beam section axes by adopting Taylor's polynomials, trigonometric series, exponential, hyperbolic and zig-zag functions. The Finite Element method is used to derive governing equations in weak form. By using the Unified Formulation introduced by the first author, these equations are written in terms of a small number of fundamental nuclei, whose forms do not depend on the expansions used. The results from the different models considered are compared in terms of displacements, stress and degrees of freedom (DOFs). Mechanical tests for thick laminated beams are presented in order to evaluate the capability of the finite elements. They show that the use of various different functions can improve the performance of the higher-order theories by yielding satisfactory results with a low computational cost.
115 citations
TL;DR: In this paper, it was shown that a necessary consequence of this reduced form of the strain energy function is that the infinitesimal shear moduli are identical, an assumption that is not supported by experimental data.
Abstract: Skeletal muscles, ligaments and tendons are typically assumed to be incompressible, transversely isotropic, non-linearly hyperelastic materials. If one adopts the phenomenological approach to modelling, then the corresponding strain-energy function can be represented as an arbitrary function of two invariants of the Cauchy–Green strain tensors, representing the isotropic contribution, and two pseudo-invariants, representing the anisotropic contribution. For mathematical convenience, dependence on one of these pseudo-invariants is usually dropped. It will be shown here that a necessary consequence of this reduced form of the strain-energy function is that the infinitesimal shear moduli are identical, an assumption that is not supported by experimental data. It will also be shown that a further consequence is that two out of the three shearing modes are identical over the full range of deformation. The conclusion is that transversely isotropic biological, soft tissue must be modelled using both anisotropic invariants.
107 citations
TL;DR: In this article, a modified variational method for dynamic analysis of ring-stiffened conical-cylindrical shells subjected to different boundary conditions is presented, which involves partitioning the stiffened shell into appropriate shell segments in order to accommodate the computing requirement of high-order vibration modes and responses.
Abstract: This work presents a modified variational method for dynamic analysis of ring-stiffened conical–cylindrical shells subjected to different boundary conditions. The method involves partitioning of the stiffened shell into appropriate shell segments in order to accommodate the computing requirement of high-order vibration modes and responses. All essential continuity constraints on segment interfaces are imposed by means of a modified variational principle and least-squares weighted residual method. Reissner-Naghdi's thin shell theory combined with the discrete element stiffener theory to consider the ring-stiffening effect is employed to formulate the theoretical model. Double mixed series, i.e., the Fourier series and Chebyshev orthogonal polynomials, are adopted as admissible displacement functions for each shell segment. To test the convergence, efficiency and accuracy of the present method, both free and forced vibrations of non-stiffened and stiffened shells are examined under different combinations of edge support conditions. Two types of external excitation forces are considered for the forced vibration analysis, i.e., the axisymmetric line force and concentrated point force. The numerical results obtained from the present method show good agreement with previously published results and those from the finite element program ANSYS. Effects of structural damping on the harmonic vibration responses of the stiffened conical–cylindrical–conical shell are also presented.
106 citations
TL;DR: In this article, a transfer function method was developed to obtain closed-form and uniform solution for the vibration analysis of EulereBernoulli beams with different boundary conditions, and analytical expressions for critical viscoelastic parameters, damping parameters and limiting frequencies were obtained.
Abstract: The dynamic characteristics of damped viscoelastic nonlocal beams are studied in this paper. The KelvineVoigt and three-parameter standard viscoelastic models, velocity-dependent external damping and nonlocal EulereBernoulli beam theory are employed to establish the governing equations of motion for the bending vibration of nanobeams. A transfer function method (TFM) is developed to obtain closed-form and uniform solution for the vibration analysis of EulereBernoulli beams with different boundary conditions. New analytical expressions for critical viscoelastic parameters, damping parameters and limiting frequencies are obtained. Considering a carbon nanotube as a numerical example, the effects of the nonlocal and viscoelastic constants on the natural frequencies and damping factors are discussed. The results demonstrate the efficiency of the proposed modeling and analysis methods for free vibration analysis of viscoelastic damped nonlocal EulereBernoulli beams.
92 citations
TL;DR: In this article, a modified couple stress theory was proposed for analyzing the static bending, free vibration and buckling behaviors of size-dependent Mindlin micro-plates, which can be directly used to predict the size effect on the Mindlin nano-plates.
Abstract: This paper presents a novel Mindlin plate element based on the framework of a modified couple stress theory for analyzing the static bending, free vibration and buckling behaviors of size-dependent Mindlin micro-plates. The element proposed is a four-node rectangular element which has 15-DOF (degrees of freedom) at C0 each node with considering both bending and stretching deformations, and 9-DOF with only considering bending deformation. Unlike the classical Mindlin plate element, this element satisfies the continuity and C1 weak continuity and contains a material length scale parameter. It can be directly used to predict the size effect on the Mindlin micro-plates. Finite element formulations are derived by applying the corresponding weak form equations. To illustrate the applicability and accuracy of the developed Mindlin plate element, the static bending, free vibration and buckling problems for rectangular Mindlin micro-plates with various boundary conditions are investigated. Convergence and comparison studies are carried out to examine the reliability of the numerical solutions. It is shown that the typical numerical results are in good agreement with those available results reported in previous literature. In addition, the numerical results illustrate that the size effect on Mindlin micro-plates can be effectively predicted by using the proposed plate element. These predicted trends agree with those observed in micro-torsion test of thin copper wires and micro-bending test of epoxy polymeric beams. Some results are believed to be the first known in the open literature and can be used as benchmark for further studies.
82 citations
TL;DR: In this paper, the free vibration of a radially polarized piezoelectric cylindrical shell of finite length was analyzed and it was shown that the surface effect has a remarkable influence on the natural frequency of the shell at the nano-scale.
Abstract: Two-dimensional (2D) general equations of piezoelectric shells with nano-thickness are presented in an orthogonal curvilinear coordinate system, in which the surface effect is considered by treat the shell as a bulk core plus two surface layers. The general 2D equations can be directly degenerated into those of particular shells such as flat plates, cylindrical shells and so on by setting the Lame coefficients and the principal radii of curvature to certain values. Using the derived 2D equations of cylindrical shells, the free vibration of a radially polarized piezoelectric cylindrical shell of finite length is analyzed. Numerical results show that the surface effect has a remarkable influence on the natural frequency of the shell at the nano-scale.
79 citations
TL;DR: In this paper, a refined beam model is proposed for tubes to improve the prediction of the transverse shear deformation and study its influence on the mechanical response of tubes, which can capture the geometric features of tubes very well and, more importantly, shear stress can be vanished readily on inner and outer surfaces.
Abstract: A refined beam model is proposed for tubes to improve the prediction of the transverse shear deformation and study its influence on the mechanical response of tubes. The current beam model adopts a Laurent series expansion form for the displacement field, instead of the conventional Taylor series expansion form used by most higher-order beam models. This new form can capture the geometric features of tubes very well and, more importantly, shear stress can be vanished readily on the inner and outer surfaces. Both a generalized theory and a simplified third-order model are presented. It is found that the transverse shear stress distribution of the third-order model is very close to that predicted by the Saint-Venant solution. Moreover, the static bending, wave propagation, and free vibration problems are studied, whose results are also compared with those obtained from other beam models and three-dimensional elasticity solutions to show the advantage of the current model.
TL;DR: The constitutive laws for the three-phase lag micropolar thermoelasticity theory are given in this paper, and the uniqueness and reciprocal theorems are proved and a variational principle is established for a linear anisotropic and inhomogeneous thermo-elastic solid.
Abstract: The constitutive laws for the three-phase-lag micropolar thermoelasticity theory are given. The uniqueness and reciprocal theorems are proved and a variational principle is established for a linear micropolar anisotropic and inhomogeneous thermoelastic solid. A continuous dependence result is given for isotropic solid.
TL;DR: In this paper, the free vibration of beams made of functionally graded materials (FGMs) containing any arbitrary number of open edge cracks is studied and a parametric investigation is carried out to examine the influences of crack depth, crack location, total number of cracks, material property distribution, and boundary conditions on the natural frequencies of the damaged FGM beams.
Abstract: Free vibration of beams made of functionally graded materials (FGMs) containing any arbitrary number of open edge cracks is studied. The study is based on Euler–Bernoulli beam and massless rotational springs connecting two intact segments of the beam. It is assumed that the material gradients follow exponential distribution through beam thickness direction. Frequency equations are obtained for flawed FGM beams with fixed–fixed, fixed–hinged, fixed–free, hinged–hinged, and spring–spring end boundaries. Detailed parametric investigation is carried out to examine the influences of crack depth, crack location, total number of cracks, material property distribution, and boundary conditions on the natural frequencies of the damaged FGM beams. The frequency equation for a damaged FGM beam with any kind of two end supports and any arbitrary number of cracks are established through a third order determinant. Compared to previous studies, this decrease in the determinant order can lead to significant advantages in the computational time.
TL;DR: In this paper, the buckling problem of thin rectangular functionally graded plates subjected to proportional biaxial compressive loadings with arbitrary edge supports is investigated, and a classical plate theory based on the physical neutral plane is applied to derive the stability equations.
Abstract: In this paper, the buckling problem of thin rectangular functionally graded plates subjected to proportional biaxial compressive loadings with arbitrary edge supports is investigated Classical plate theory (CPT) based on the physical neutral plane is applied to derive the stability equations Mechanical properties of the FGM plate are assumed to vary continuously along its thickness according to a power law function The displacement function is considered to be in the form of a double Fourier series whose derivatives are determined using Stokes' transformation The advantage of this method is capability of considering any possible combination of boundary conditions with no necessity to be satisfied in the Fourier series To give generality to the problem, the plate is assumed to be elastically restrained by means of rotational and translational springs at the four edges Numerical examples are presented, and the effects of the plate aspect ratio, the FGM power index, and the loading proportionality factor on the buckling load of an FGM plate with different usual boundary conditions are studied The present results are compared with those have been previously reported by other analytical and numerical methods, and very good agreement is seen between the findings indicating validity and accuracy of the proposed approach in the buckling analysis of FGM plates
TL;DR: In this article, the bending analysis of functionally graded plates by a n th-order shear deformation theory and meshless global collocation method based on the thin plate spline radial basis function was studied.
Abstract: This paper focus on the bending analysis of functionally graded plates by a n th-order shear deformation theory and meshless global collocation method based on the thin plate spline radial basis function. Reddy's third-order theory can be considered as a special case of present n th-order theory ( n = 3). The governing equations are derived by the principle of virtual work. The displacement and stress of a simply supported functionally graded plate under sinusoidal load are calculated to verify the accuracy and efficiency of the present theory.
TL;DR: In this article, a closed-form solution is obtained for the thermal post-buckling and nonlinear free vibration analysis of SMA fiber reinforced hybrid composite beams with symmetric and asymmetric lay-up.
Abstract: In this article, large amplitude vibration and thermal post-buckling of shape memory alloy (SMA) fiber reinforced hybrid composite beams with symmetric and asymmetric lay-up are analytically investigated. To predict the behavior of the smart laminated beam, the Euler–Bernoulli beam theory and the nonlinear von-Karman strain field are employed. Also, one-dimensional Brinson SMA model is utilized to calculate the recovery stress of SMA fibers in the case of restrained strain. Nonlinear governing equations of motion are derived via the Hamilton principle. Using an analytical approach based on the Galerkin procedure together with the simple harmonic motion assumption, a closed-form solution is obtained for the thermal post-buckling and nonlinear free vibration analysis of SMA fiber reinforced hybrid composite beams. Due to lack of any results on the free vibration and thermal stability of SMA fiber reinforced composite beams, the results obtained from the present solution for laminated composite beams without SMA fiber are compared with counterpart data in the open literature, which validate the present solution. Then, a set of parametric study is carried out to show the influence of SMA volume fraction, amount of prestrain in the SMA fiber, orientation of composite fiber, SMA-reinforced layer thickness to total thickness ratio, location of SMA layer, vibration amplitude, boundary conditions and temperature on the vibration characteristic of the laminated beam reinforced with SMA in the pre- and post-buckled domains.
TL;DR: In this article, a finite axisymmetric inflation of an initially stretched flat circular hyperelastic membrane has been analyzed, where the membrane material has been assumed to be a homogeneous and isotropic Mooney-Rivlin solid.
Abstract: In this paper, finite axisymmetric inflation of an initially stretched flat circular hyperelastic membrane has been analyzed. The membrane material has been assumed to be a homogeneous and isotropic Mooney–Rivlin solid. The inflation problem has been reduced to a set of three first order ordinary differential equations using a set of appropriately defined variables. An interesting method based on the invariance of these equations to scaling has been used to solve the two point boundary value problem without much effort. This method does not require any special technique for negotiating the limit points in the pressure–stretch relations of the membrane. Several inflation results of an initially unstretched and pre-stretched circular membrane for various material parameters are obtained. The roles of pre-stretch and internal pressure on the inflation mechanics are clearly delineated. The initial stretch is observed to have some interesting counter-intuitive effects on the inflation of the membrane.
TL;DR: In this paper, a PML approach is used to simulate the dissipation of waves radiated from the anchor into the substrate and provide several guidelines for a robust application to micro-structures.
Abstract: Several barriers exist to the development and optimization of high frequency Micro-Electro-Mechanical (MEMS) resonators, primarily adequate control and understanding of dissipation phenomena. There is growing experimental evidence that anchor losses contribute significantly to damping. A reliable, large scale and native 3D numerical approach for estimating the anchor loss contribution in general is hence a much demanded tool. In this paper we discuss the implementation of a PML approach to simulate dissipation of waves radiated from the anchor into the substrate and provide several guidelines for a robust application to micro-structures. Next we employ the codes developed to perform extensive benchmarks against analytical solutions and verify the applicability of possible simplifications. In particular we show that the commonly adopted decoupling assumption between the resonator and the substrate might induce severe errors especially in 2D.
TL;DR: In this paper, the authors deal with the analysis of a spherical parallel manipulator (3RCC) to determine the error on the pose of the end effector as a function of the manufacturing errors of the different links and the presence of a clearance in the joints.
Abstract: This paper deals with the analysis of a spherical parallel manipulator (3RCC) to determine the error on the pose of the end effector as a function of the manufacturing errors of the different links and the presence of a clearance in the joints. The obtained model allowed us to identify the error on the platform in three cases, i.e., only manufacturing errors were considered, then only clearance in the joints was considered and finally the case of both sources of error were present in the system. It was shown, in particular, that the axial displacement in the C joints is quite important. The second result is the fact that the superposition principle does not work when we consider both the manufacturing errors and the clearance despite the assumption of small displacements.
TL;DR: In this paper, the authors presented a simple formulation to obtain the natural frequencies and the associated mode shapes of a multi-step beam carrying arbitrary various concentrated elements (including eccentric lumped masses with rotary inertias, linear springs, rotational springs and spring-mass systems) in various boundary conditions.
Abstract: The continuous-mass transfer matrix method (CTMM) is one of the few practical approaches to yield the “exact” solutions for free vibrations of a non-uniform beam carrying any number of concentrated elements. However, most of the existing CTMM does not consider the effects of shear deformation (SD), rotary inertia (RI), joint action term of SD and RI, and axial load. Thus, the objective of this paper is to present a simple formulation so that one can easily obtain the “exact“ natural frequencies and the associated mode shapes of a multi-step beam carrying arbitrary various concentrated elements (including eccentric lumped masses with rotary inertias, linear springs, rotational springs and spring-mass systems) in various boundary conditions with all the above-mentioned effects considered by using the modified CTMM. In addition to comparing with the existing relevant data, most of the numerical results obtained from the modified CTMM are also compared with those of the conventional finite element method (FEM) and good agreements are achieved.
TL;DR: In this article, a fractional order generalized electro-magneto-thermo-elasticity (FOGEMTE) theory is developed for anisotropic and linearly electromagnetic media by introducing the dynamic electro-MAGnetic fields.
Abstract: Built upon the fractional order generalized thermoelasticity (FOGTE), which is based on ETE (extended thermoelasticity), a fractional order generalized electro-magneto-thermo-elasticity (FOGEMTE) theory is developed for anisotropic and linearly electro-magneto-thermo-elastic media by introducing the dynamic electro-magnetic fields, with various generalized thermoelasticity considered, such as ETE, TRDTE (temperature rate dependent thermoelasticity), TEWOED (thermoelasticity without energy dissipation), TEWED (thermoelasticity with energy dissipation), DPLTE (dual-phase-lag thermoelasticity). The two temperature (thermodynamics and conductive temperature) model is also introduced. In addition, to numerically deal with the multi-physics problems expressed by a series of partial differential equations especially a fractional one, the corresponding variational principle based on the variational integral method is proposed, and various degenerated variational theorems are presented. A generalized variational theorem is obtained for the unified theory by using the semi-inverse method. Finally, two examples are numerically validated, and concluding remarks are also given.
TL;DR: In this paper, a Jeffcott rotor with a transverse breathing crack is examined, and stability of the system is investigated by Floquet theory considering the crack depth and rotating speed.
Abstract: Transverse breathing cracks have been considered a primary mode of damage in studies of rotordynamic systems In this paper, a Jeffcott rotor with a transverse breathing crack is examined, and stability of the system is investigated by Floquet theory considering the crack depth and rotating speed New breathing functions proposed in a recent publication are adopted to approximate the actual breathing mechanism of the crack Unlike previous studies wherein stability diagrams without detailed information about the stable and unstable regimes of the motion have been provided, in this work we perform a detailed study of the corresponding eigenvalues of the cracked rotor in the complex plane, and the effect of damping on the instability regions has been investigated Our study indicates that the unstable regions appear as the speed of the rotor approaches an integer fraction or an integer multiple of the critical speed of the rotor, whereas bifurcations are detected in certain unstable regimes The results also shows damping has significant influence on the structures of instability regions
TL;DR: In this article, the elasticity solutions for curved cantilever beams with n orthotropic functionally graded layers by means of the Airy stress function method are presented for a general model and boundary and continuity conditions to determine integral constants are given.
Abstract: Elasticity solutions are presented for curved beams with n orthotropic functionally graded layers by means of the Airy stress function method. The beams are subjected to a uniform load on the outer surface and may have various constraints or/and loads at ends. Firstly, the stresses and displacements are expressed in terms of three unknown functions for a general model. Secondly, the unknown functions are deduced for two slightly general forms of elastic compliance parameters and represented by the generalized hypergeometric functions. Thirdly, the boundary and continuity conditions to determine integral constants are given. As the application, a curved cantilever beam, with three different variations in the elastic compliance parameters and under two types of loads, is discussed.
TL;DR: In this article, the authors analyzed the simple shear state of an incompressible hyperelastic solid under large deformation by experimental and theoretical approaches and obtained a nonlinear stress-strain response.
Abstract: The aim of the present work is to analyze the simple shear state of an incompressible hyperelastic solid under large deformation by experimental and theoretical approaches. The experimental procedure was performed using a single lap shear test and the displacement fields were determined by the Digital Image Correlation method. The applied force and the measured angular distortion were used to evaluate the shear stress and the amount of shear. Hence, a nonlinear stress-strain response was achieved. In addition, the normal stress components were obtained from the experimental data by assuming two hypotheses: the first one was based on a plane stress condition, while the normal component of the traction on the inclined surfaces was assumed to be zero on the second. Finally, to verify the presented results, the initial shear modulus of the hyperelastic material was estimated and compared with the value obtained using the data from a planar tensile test. The Lopez-Pamies strain energy function was used in the inverse analysis in order to estimate the material property, which was similar for both experimental tests.
TL;DR: In this article, the authors proposed reasonable evolution equations for the length and orientation of the axes of the ellipsoidal voids, which are not attached to this specific model and could be used in conjunction with any similar criterion accounting for void shape effects.
Abstract: In Part I, Madou and Leblond (2012a,b)’s criterion for plastic porous materials containing arbitrary ellipsoidal voids was validated by comparing its predictions with the results of some numerical limit-analyses of elementary cells containing such voids. In the present Part II, our aim is now to complete the model by proposing reasonable evolution equations for the length and orientation of the axes of the voids. Again, however, the equations proposed are not attached to this specific model and could be used in conjunction with any similar criterion accounting for void shape effects. In the definition of the evolution equations looked for, a central role is played by “elastic” expressions for the strain and rotation rates of the voids proposed by Ponte-Castaneda and Zaidman (1994) and Kailasam and Ponte-Castaneda (1998) from homogenization theory. The importance of plastic effects however makes it necessary to modify these expressions; this is done heuristically by introducing stress-dependent correction factors determined numerically in a number of reference cases and suitably interpolated between these cases.
TL;DR: In this article, the radial vibration of nanoscale spherical shells based on the nonlocal elasticity theory is derived in terms of radial displacement, where the shell is considered elastic, homogeneous and isotropic.
Abstract: This paper presents the radial vibration of nanoscale spherical shells based on the nonlocal elasticity theory. The shell is considered elastic, homogeneous and isotropic. The nonlocal differential equation of radial motion is derived in terms of radial displacement. The relation between the nonlocal and local frequencies is also investigated. Considering the small-scale effect, the general characteristic equation for radial vibration of spherical shell is obtained by applying boundary conditions. Moreover, the characteristic equations for two special cases are presented. To demonstrate the accuracy of the present formulation, theoretical calculations of the fundamental frequency have been compared with those available in the literature and a good agreement is achieved. The variations of the frequencies with the nonlocal parameter, radius ratio and Poisson's ratio are also examined. It is observed that the frequencies are affected when the size effect is taken into consideration.
TL;DR: In this paper, the pull-in instability of fixed-fixed nano-switches subjected to electrostatic forces produced by an applied voltage, and intermolecular forces are investigated.
Abstract: In this article, pull-in instability of cantilever and fixed–fixed nano-switches subjected to electrostatic forces produced by an applied voltage, and intermolecular forces are investigated. A linear distributed load model is considered to approximately model the nonlinear intermolecular and electrostatic interactions acting on the nano-beam. The effect of small length-scale is taken into account using hybrid nonlocal Euler–Bernoulli beam model. The effects of small length-scale on the pull-in instability and freestanding behavior of the cantilever and fixed–fixed nano-beams are presented and compared with the Eringen's nonlocal and classical beam models. It is found that the Eringen's nonlocal beam model produces unreasonable pull-in voltages, minimum gaps and detachment lengths. It is shown that shortcomings of the Eringen's nonlocal beam theory can be resolved by using hybrid nonlocal beam model.
TL;DR: In this paper, a numerical limit-analysis of elementary cells of arbitrary ellipsoidal shape and containing confocal ellipseidal voids was performed by the finite element method, and the results confirmed the general validity of Madou and Leblond's proposed criterion, although slight corners not accounted for in this criterion are apparent on the numerical yield surfaces of cylindrical cells.
Abstract: This work is devoted to some numerical limit-analyses, performed by the finite element method, of elementary cells of arbitrary ellipsoidal shape and containing confocal ellipsoidal voids. The aim is essentially, in the present Part I, to validate an approximate analytic criterion for such cells recently proposed by Madou and Leblond, 2012a , Madou and Leblond, 2012b , and in Part II, to supplement the model by proposing reasonable evolution equations for the length and orientation of the axes of the voids. The results obtained are however not specifically attached to this particular model and could assist in the definition of any similar one for plastic porous materials incorporating void shape effects. The present Part I is devoted to the numerical determination of the yield surfaces corresponding to eight different void geometries (prolate and oblate spheroids, circular and elliptic cylinders, circular and elliptic cracks, two general ellipsoids). The results obtained confirm the general validity of Madou and Leblond, 2012a , Madou and Leblond, 2012b 's proposed criterion, although slight corners not accounted for in this criterion are apparent on the numerical yield surfaces of cylindrical cells.
TL;DR: In this article, an integral model of dry friction components is built under assumption of classical Coulomb friction law and fully developed sliding on the contact area of general shape and arbitrary contact pressure distribution.
Abstract: There are many examples of mechanical systems with non-point friction contacts (billiard ball, Thompson top, wobblestone, electric polishing machine, the wobblestone, the Celtic stone), where the assumption of one-dimensional dry friction model do not necessarily lead to satisfactory accuracy of the numerical simulation Moreover the rolling resistance often plays an important role in such systems The paper is devoted to the problem of developing an approximate coupled model of resulting dry friction force and moment as well as rolling resistance, suitable for fast numerical simulation of rigid bodies with friction contacts, ie allowing to avoid the space discretization An integral model of dry friction components is built under assumption of classical Coulomb friction law and fully developed sliding on the contact area of general shape and arbitrary contact pressure distribution Then the special model of stress distribution over the elliptic contact area is developed, being a kind of generalization of Hertzian normal stress distribution, with addition of special distortion related to the rolling resistance Finally some original approximate models of friction force and moment are proposed, based on Pade approximants and their generalizations as well as in the form of piecewise polynomial functions
TL;DR: In this paper, stability in parametric resonance of axially moving viscoelastic plates subjected to plane stresses is investigated, where the plate material obeys the Kelvin-Voigt model in which the material time derivative is used.
Abstract: In this paper, stability in parametric resonance of axially moving viscoelastic plates subjected to plane stresses is investigated. The plate material obeys the Kelvin–Voigt model in which the material time derivative is used. The generalized Hamilton principle is employed to obtain the governing equation. The axial speed is characterized as a simple harmonic variation about the constant mean speed. The governing equation can be regarded as a continuous gyroscopic system with small periodically parametric excitations and a damping term. The method of multiple scales is applied to the governing equation to establish the solvability conditions in principal and summation parametric resonances. The natural frequencies and modes of linear generating equation are numerically calculated based on the given boundary conditions. The necessary and sufficient condition of the stability is derived from the Routh–Hurwitz criterion. Some numerical examples are presented to demonstrate the effects of related parameters on the frequencies and the stability boundaries. The differential quadrature scheme is developed to solve numerically the linear generating system and the primitive equation model. The numerical calculations confirm the analytical results.