Showing papers in "Acta Mechanica in 2016"

A higher-order nonlocal strain gradient plate model for buckling of orthotropic nanoplates in thermal environment

Abstract: In this paper, a new size-dependent plate model is developed based on the higher-order nonlocal strain gradient theory. The influences of higher-order deformations in conjunction with the higher- and lower-order nonlocal effects are taken into account. The presence of three different kinds of scale parameters in the formulation results in a theory which is capable of capturing both reduction and increase in the stiffness of structures at nanoscale. The governing differential equations are derived for the buckling of nanoplates resting on a two-parameter elastic foundation using the principle of virtual work. The nanoplate is assumed to be orthotropic with size-dependent material properties. The influence of thermal stress caused by a temperature change is taken into consideration. An exact closed-form solution is obtained for the critical buckling loads of graphene sheets. The higher-order governing differential equation is also solved by the differential quadrature method. The results of the two solution methods are compared with each other. Excellent agreement between the exact and numerical results is observed. For numerical results, three types of graphene sheets with different aspect ratio are considered. The effects of various scale parameters together with the other parameters such as the coefficients of the elastic medium, temperature change and the length of the nanoplate on the buckling behavior of graphene sheets are investigated.

On effective properties of materials at the nano- and microscales considering surface effects

Abstract: In the last years, the rapid increase in the technical capability to control and design materials at the nanoscale has pushed toward an intensive exploitation of new possibilities concerning optical, chemical, thermoelectrical and electronic applications. As a result, new materials have been developed to obtain specific physical properties and performances. In this general picture, it was natural that the attention toward mechanical characterization of the new structures was left, in a sense, behind. Anyway, once the theoretically designed objects proceed toward concrete manufacturing and applications, an accurate and general description of their mechanical properties becomes more and more scientifically relevant. The aim of the paper is therefore to discuss new methods and techniques for modeling the behavior of nanostructured materials considering surface/interface properties, which are responsible for the main differences between nano- and macroscale, and to determine their actual material properties at the macroscale. Our approach is intended to study the mechanical properties of materials taking into account surface properties including possible complex inner microstructure of surface coatings. We use the Gurtin–Murdoch model of surface elasticity. We consider the inner regular and irregular surface thin coatings (i.e., ordered or disordered nanofibers arrays) and present few examples of averaged 2D properties of them. Since the actual 2D properties depend not only on the mechanical properties of fibers or other elements of a coating, but also on the interaction forces between them, the analysis also includes information on the geometry of the microstructure of the coating, on mechanical properties of elements and on interaction forces. Further we use the obtained 2D properties to derive the effective properties of solids and structures at the macroscale, such as the bending stiffness or Young’s modulus.

A generalized layerwise higher-order shear deformation theory for laminated composite and sandwich plates based on isogeometric analysis

Abstract: This paper presents a generalized layerwise higher-order shear deformation theory for laminated composite and sandwich plates. We exploit a higher-order shear deformation theory in each layer such that the continuity of the displacement and transverse shear stresses at the layer interfaces is ensured. Thanks for enforcing the continuity of the displacement and transverse shear stresses at an inner-laminar layer, the minimum number of variables is retained from the present theory in comparison with other layerwise theories. The method requires only five variables, the same as what obtained from the first- and higher-order shear deformation theories. In comparison with the shear deformation theories based on the equivalent single layer, the present theory is capable of producing a higher accuracy for inner-laminar layer shear stresses. The free boundary conditions of transverse shear stresses at the top and bottom surfaces of the plate are fulfilled without any shear correction factors. The discrete system equations are derived from the Galerkin weak form, and the solution is obtained by isogeometric analysis (IGA). The discrete form requires the C1 continuity of the transverse displacement, and hence NURBS basis functions in IGA naturally ensure this condition. The laminated composite and sandwich plates with various geometries, aspect ratios, stiffness ratios and boundary conditions are studied. The obtained results are compared with the 3D elasticity solution, the analytical as well as numerical solutions based on various plate theories.

Hygro-mechanical vibration analysis of a rotating viscoelastic nanobeam embedded in a visco-Pasternak elastic medium and in a nonlinear thermal environment

Abstract: The vibration behavior of the rotating viscoelastic nanobeam embedded in the visco-Pasternak foundation is studied. The governing equation is extracted by using the surface elasticity and the nonlocal elasticity theory. The influence of the humidity on the vibration frequencies of the viscoelastic nanobeam is investigated in the thermal environment. The effects of the linear and the nonlinear thermal stress cases on the vibration frequencies of the viscoelastic nanobeam are studied. The vibration frequencies are obtained based on the differential quadrature method. Also, the numerical results are compared with those which are reported in the literature, and a good correlation is obtained. The results are presented for the different boundary conditions as well as the effect of the torsion spring on the viscoelastic nanobeam ends is investigated. This study focuses on the combined effects of the angular velocity, the internal and external damping, the humidity change, the temperature change, the surface effects (surface density, surface elasticity and surface stress), the nonlocal parameter, the boundary conditions, the visco-Pasternak foundation, the cross section geometry, and the torsion spring. The results of this study could be used to design and manufacture nanosensors, biosensors, atomic force microscope and the NEMS/MEMS devices.

Vibration analysis of functionally graded carbon nanotube-reinforced composite shell structures

Abstract: This article deals with the vibration analysis of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) shell structures. The material properties of an FG-CNTRC shell are graded smoothly through the thickness direction of the shell according to uniform distribution and some other functionally graded (FG) distributions (such as FG-X, FG-V, FG-O and FG- $${\Lambda}$$ ) of the volume fraction of the carbon nanotube (CNT), and the effective material properties are estimated by employing the extended rule of mixture. An eight-noded shell element considering transverse shear effect according to Mindlin’s hypothesis has been employed for the finite element modelling and analysis of the composite shell structures. The formulation of the shell midsurface in an arbitrary curvilinear coordinate system based on the tensorial notation is also presented. The Rayleigh damping model has been implemented in order to study the effects of carbon nanotubes (CNTs) on the damping capacity of such shell structures. Different types of shell panels have been analyzed in order to study the impulse and frequency responses. The influences of CNT volume fraction, CNT distribution, geometry of the shell and material distributions on the dynamic behavior of FG-CNTRC shell structures have also been presented and discussed. Various types of FG-CNTRC shell structures (such as spherical, ellipsoidal, doubly curved and cylindrical) have been analyzed and discussed in order to compare studies in terms of settling time, first resonant frequency and absolute amplitude corresponding to first resonant frequency based on the impulse and frequency responses, and the effects of CNTs on vibration responses of such shell structures are also presented. The results show that the CNT distribution and volume fraction of CNT have a significant effect on vibration and damping characteristics of the structure.

Nonlinear free vibration of temperature-dependent sandwich beams with carbon nanotube-reinforced face sheets

Abstract: In this research, large amplitude free vibrations of a sandwich beam with stiff core and carbon nanotube (CNT)-reinforced face sheets are analysed. The distribution of CNTs across the thickness of the face sheets may be uniform or functionally graded. The equivalent single- layer theory of Timoshenko is used to construct the Hamiltonian of the beam under the von Karman type of geometrical nonlinearity assumptions. A uniform temperature field through the beam is also included in the formulation. The Ritz method with polynomial basis functions is used to discrete the equations of motion and establish the matrix representation of the governing equations. A nonlinear eigenvalue problem is obtained and solved using a standard continuation procedure. After validating the developed solution method and formulation, parametric studies are conducted to examine the influences of thermal environment, core thickness-to-face sheet thickness ratio, boundary conditions, amplitude of vibrations, CNTs volume fraction and their distribution pattern. It is concluded that an increase in the volume fraction of CNTs results in higher fundamental frequency and decreases the nonlinear-to-linear frequency ratio.

Water-hammer control in pressurized-pipe flow using an in-line polymeric short-section

Abstract: Water-hammer control strategies constitute an essential and critical task for both hydraulic designers and manufacturers to ensure the global economic efficiency and safety operations of hydraulic utilities. The primary objective of this paper is to present an alternative strategy to control water-hammer up- and down-surges, induced into a steel piping system. The proposed technique is based on replacing a short-section of the transient sensitive regions of the existing piping system by another one made of polymeric material. Two types of polymeric materials, used for the short-section and including high- or low-density polyethylene (HDPE) or (LDPE), are addressed in this study. The 1-D pressurized-pipe flow model is used to describe the hydraulic system, along with the Ramos formulation, based on two decay coefficients being used for considering the pipe-wall viscoelastic behavior and unsteady friction effects. Numerical computations were performed using the fixed-grid method of characteristics. The efficiency of the numerical model is first verified against experimental data available from the literature. Thereafter, critical flow scenarios relating to water-hammer up- and down-surges, including a cavitating flow, are revealed and discussed to point out the efficiency of the used protection technique. From the case studied, it is found that such a technique could mitigate critical water-hammer surges and, hence, might greatly enhance the reliability of the industrial hydraulic systems and urban water utilities, while safeguarding operators. Despite the available protection measures, the utilized technique can substantially soften both up- and down-surge waves induced by severe water-hammer events. It is also found that the amortization of pressure rise and pressure drop is slightly more important for the case of a short-section made of LDPE polymeric material than that using an HDPE polymeric material. It is also observed that other factors contributing to the damping rate depended upon the short-section length and diameter. In fact, the examination of the pressure peak magnitude sensitivity, with the short-section length and diameter being the controlling variables, provides optimum values of these parameters for sizing the replaced polymeric short-section.

Frequency analysis of doubly curved functionally graded carbon nanotube-reinforced composite panels

Abstract: Vibration characteristics of moderately thick doubly curved functionally graded composite panels reinforced by carbon nanotube are analyzed. Here, special cases of doubly curved shell panels such as spherical, cylindrical and hyperbolic paraboloid panels and five different distributions of carbon nanotubes through the thickness direction are considered. By utilizing the modified rule of mixture, mechanical properties are estimated. The equations of motion are derived via the first-order shear deformation theory, and non-dimensional frequencies are obtained by the use of Galerkin’s method. The suggested model is justified by a good agreement between the results given by present model and available data in the literature. The influences of volume fraction of carbon nanotubes, thickness ratio, aspect ratio, curvature ratio, and shallowness ratio on the frequencies of moderately thick doubly curved nanocomposite shell panels are also examined. Furthermore, the effect of various boundary conditions on the frequency analysis of doubly curved nanocomposite panels is studied, and the corresponding mode shapes are depicted.

Dynamic instability analysis of doubly clamped cylindrical nanowires in the presence of Casimir attraction and surface effects using modified couple stress theory

Abstract: The dynamic instability of free-standing size-dependent nanowires by considering the Casimir force and surface effects is investigated in the following research work. The study is carried out for nanosystems with circular cross section and cylinder–plate geometry for which the governing equation of motion is derived based on the Gurtin–Murdoch model and modified couple stress theory. Two methods including the proximity force approximation for small separations and Dirichlet asymptotic approximation for large separations are utilized to formulate the Casimir attraction of a free-standing cylinder–plate geometry. To solve the complex nonlinear problem faced in this work, a stepwise numerical procedure is developed and the effects of length scale parameter, surface energy and vacuum fluctuations on the dynamic instability and adhesion time of nanowires are studied. Based on the obtained results, the phase portrait of Casimir-induced nanowires shows periodic and homoclinic orbits.

Modeling and experimental validation of a new type of tuned liquid damper

Abstract: This paper introduces a new type of tuned liquid damper (TLD) having a relatively simple, easy-to-model behavior and high effectiveness in controlling structural vibrations. It consists of a traditional TLD with addition of a floating roof. Since the roof is much stiffer than water, it prevents wave breaking, hence making the response linear even at large amplitudes. The roof also facilitates the incorporation of supplemental devices with which the level of damping of the liquid vibration can be substantially augmented. This newly proposed TLD, denoted as tuned liquid damper with floating roof (TLD-FR), maintains the traditional advantages of TLDs (low cost, easy installation and tuning), but its numerical characterization is much simpler because the floating roof suppresses higher sloshing vibration modes, resulting in a system that can be represented by a single-degree-of-freedom model. An efficient numerical scheme, where the dynamic behavior of the TLD-FR is expressed as a second-order lineal system of equations, is discussed and validated by scaled experimental tests. The equations of motion of a structure equipped with a TLD-FR are then derived and manipulated to offer a unifying representation dependent upon only four model characteristics of the TLD-FR: The first three (mass, frequency and damping ratios) are common for all type of mass dampers, whereas the final one, termed efficiency index, is related to a similar parameter used to characterize liquid column dampers. Through this approach, the behavior of the proposed TLD-FR can be easily correlated with the behavior of other well-known linear mass damper devices. The relationship between these parameters and the geometrical characteristics of the TLD-FR is also examined. Finally, the identification of the optimal characteristic of the TLD-FR (natural frequency and damping) under stationary stochastic excitation is discussed.

Analysis of wave in a functionally graded magneto-electro-elastic nano-rod using nonlocal elasticity model subjected to electric and magnetic potentials

Abstract: The purpose of this research is to present the wave propagation analysis of a functionally graded nano-rod made of magneto-electro-elastic material subjected to an electric and magnetic potential. The unified nonlocal elasticity theory and Love's rod model are used in this study. All mechanical, electrical and magnetic properties are assumed to be variable along the thickness direction based on a power law distribution. Two-dimensional electric and magnetic potential distributions due to an applied potential and a magnet at the top of the rod are considered. The governing equations of motion are obtained using equilibrium and nonlocal theory of elasticity in conjunction with the Hamilton principle. The effect of important parameters of the functionally graded magneto-electro-elastic nano-rod such as nonlocal parameters, power index, wave number, applied magnetic and electric potentials on the wave propagation characteristics is studied.

A general Fourier formulation for vibration analysis of functionally graded sandwich beams with arbitrary boundary condition and resting on elastic foundations

Abstract: Free vibration analysis of functionally graded sandwich beams with general boundary conditions and resting on a Pasternak elastic foundation is presented by using strong form formulation based on modified Fourier series. Two types of common sandwich beams, namely beams with functionally graded face sheets and isotropic core and beams with isotropic face sheets and functionally graded core, are considered. The bilayered and single-layered functionally graded beams are obtained as special cases of sandwich beams. The effective material properties of functionally graded materials are assumed to vary continuously in the thickness direction according to power-law distributions in terms of volume fraction of constituents and are estimated by Voigt model and Mori–Tanaka scheme. Based on the first-order shear deformation theory, the governing equations and boundary conditions can be obtained by Hamilton’s principle and can be solved using the modified Fourier series method which consists of the standard Fourier cosine series and several supplemented functions. A variety of numerical examples are presented to demonstrate the convergence, reliability and accuracy of the present method. Numerous new vibration results for functionally graded sandwich beams with general boundary conditions and resting on elastic foundations are given. The influence of the power-law indices and foundation parameters on the frequencies of the sandwich beams is also investigated.

Surface effects on the bending, buckling and free vibration analysis of magneto-electro-elastic beams

Abstract: Surface effects are responsible for the size dependence and should be taken into account for dielectric structures at nanoscale dimensions. By incorporating the effects of surface stress, surface piezoelectricity, surface elasticity and surface piezomagneticity, this paper investigates the bending, buckling and free vibration of magneto-electro-elastic (MEE) beams based on the Euler–Bernoulli beam theory. The governing differential equation and its corresponding boundary conditions are derived by Hamilton’s principle. The analytical solutions for the magneto-electro-elastic bending deflection, buckling magnetic potentials and frequency equations of MEE beams are obtained. In contrast to the previously published works, the positive surface stress is found to stiffen the MEE beams, as evidenced by the decrease in the deflections, the increase in the buckling magnet potentials and the increase in the resonant frequencies. Numerical studies show the importance of the surface effects, the electric and magnetic potentials and boundary conditions on the static and dynamic behavior of MEE beams. This work may be of special interest in the design and application of smart composite MEE beams.

Nonlocal buckling and vibration analysis of thick rectangular nanoplates using finite strip method based on refined plate theory

Abstract: The buckling and vibration of thick rectangular nanoplates is analyzed in this article. A graphene sheet is theoretically assumed and modeled as a nanoplate in this study. The two-variable refined plate theory (RPT) is applied to obtain the differential equations of the nanoplate. The theory accounts for parabolic variation of transverse shear stress through the thickness of the plate without using a shear correction factor. Besides, the analysis is based on the nonlocal theory of elasticity to take the small-scale effects into account. For the first time, the finite strip method (FSM) based on RPT is employed to study the vibration and buckling behavior of nanoplates and graphene sheets. Hamilton’s principle is employed to obtain the differential equations of the nanoplate. The stiffness, stability and mass matrices of the nanoplate are formed using the FSM. The displacement functions of the strips are evaluated using continuous harmonic function series which satisfy the boundary conditions in one direction and a piecewise interpolation polynomial in the other direction. A matrix eigenvalue problem is solved to find the free vibration frequency and buckling load of the nanoplates subjected to different types of in-plane loadings including the uniform and nonuniform uni-axial and biaxial compression. Comparison studies are presented to verify the validity and accuracy of the proposed nonlocal refined finite strip method. Furthermore, a number of examples are presented to investigate the effects of various parameters (e.g., boundary conditions, nonlocal parameter, aspect ratio, type of loading) on the results.

Using the modified strain gradient theory to investigate the size-dependent biaxial buckling analysis of an orthotropic multi-microplate system

Abstract: On the basis of the modified strain gradient theory, this research presents an analytical approach to analyze elastic instability of an orthotropic multi-microplate system (OMMPS) embedded in a Pasternak elastic medium under biaxial compressive loads. Kirchhoff plate theory and the principle of total potential energy are applied to obtain the partial differential equations and corresponding boundary conditions. Various types of “chain” boundary conditions for the ends of the microplates system are assumed such as “Clamped-Chain,” “Free-Chain” and “Cantilever-Chain” systems. In order to analytically obtain the buckling load of the OMMPS, we use Navier’s approach which satisfies the simply supported boundary conditions and trigonometric method. In order to show the dependability of the presented formulation in this paper, several comparison studies are carried out to compare with existing data in the literature. Numerical results are presented to reveal variations of the buckling load of OMMPS corresponding to various values of the number of microplates, the length scale parameter $${\left({\frac{h}{l}}\right)}$$ , aspect ratio, Pasternak elastic medium parameters and the thickness of the microplate and the biaxial compression ratio. Some numerical results of this paper illustrate that when the number of microplates is small, especially becoming 2, there is an important difference between buckling loads obtained for “Clamped-Chain,” “Free-Chain” and “Cantilever-Chain” systems. Also, it is shown that by increasing the number of microplates in the system, the influence of the Pasternak elastic medium on the buckling load of system is reduced. It is anticipated that the results reported in this work are applied as a benchmark in future microstructure issues.

Three-dimensional micromechanical analysis of the CNT waviness influence on the mechanical properties of polymer nanocomposites

Abstract: The effects of carbon nanotube (CNT) waviness on the elastic characterizations of polymer nanocomposites are investigated using a three-dimensional unit cell-based micromechanical model. The most important advantages of this model are its accuracy, simplicity, and efficiency. Both random and regular CNT arrangements can be included in the modeling. The wavy CNTs are modeled as sinusoidal solid CNT fibers while at any location along the length of CNT, the CNT is considered as transversely isotropic material. The polymer and interphase formed due to non-bonded interaction between a CNT and the polymer are assumed to be homogeneous and isotropic as well. Results show that the effect of CNT waviness is not important for the effective coefficients $$C_{11}$$ , $$C_{12}$$ , and $$C_{13}$$ of the nanocomposites. CNT waviness plays a critical role in determining the effective coefficients $$C_{22}$$ , $$C_{23}$$ , $$C_{33}$$ , and $$C_{44}$$ of the nanocomposites. Also, it is found that the CNT waviness slightly affects the effective values of $$C_{55}$$ and $$C_{66}$$ . The effects of volume fraction of CNT and interphase on the mechanical properties of the nanocomposite are examined. Comparison of the present model results shows very good agreement with other available micromechanical analysis and experiment. As comparing with the finite element method, the present model requires much less computational time for obtaining the effective properties of the nanocomposites. Consequently, the results emphasize that all four important parameters, i.e., CNT behavior and waviness, CNT random arrangement, and interphase contributions, should be precisely included in the modeling to predict a more realistic outcome.

A promising approach for modeling biological fibers

Abstract: The final-to-initial stiffness ratio is very large (>100) for many biological fibers, and as such, these materials have been modeled as being strain limiting. We propose an unconventional structure for a stored energy function that leads to a constitutive relation capable of describing this observed strain-limiting behavior. The model can attain infinite stress at a finite strain while storing a finite amount of internal energy. Many biological fibers have a mechanical response that starts out as being compliant and nonlinear, and transitions into one that is stiff and linear. We present a biological fiber model comprised of a strain-limiting fiber (strain being attributed to molecular reconfiguration) loaded in conjunction with a Hookean fiber (strain being attributed to molecular stretch). The model’s parameters are physical, intuitive and readily extracted from a stress/strain curve. Chordae tendineae data are used to demonstrate the efficacy of the model.

3D mass-redistributed finite element method in structural–acoustic interaction problems

Abstract: A 2D mass-redistributed finite element method (MR-FEM) for pure acoustic problems was recently proposed to reduce the dispersion error. In this paper, the 3D MR-FEM is further developed to solve more complicated structural–acoustic interaction problems. The smoothed Galerkin weak form is adopted to formulate the discretized equations for the structure, and MR-FEM is applied in acoustic domain. The global equations of structural–acoustic interaction problems are then established by coupling the MR-FEM for the acoustic domain and the edge-based smoothed finite element method for the structure. The perfect balance between the mass matrix and stiffness matrix is able to improve the accuracy of the acoustic domain significantly. The gradient smoothing technique used in the structural domain can provide a proper softening effect to the “overly-stiff” FEM model. A number of numerical examples have demonstrated the effectiveness of the mass-redistributed method with smoothed strain.

Development of an efficient algorithm to analyze the elastic wave in acoustic metamaterials

Abstract: A novel modified edge-based smoothed finite element method (modified ES-FEM) is developed to compute the band gap of acoustic metamaterials. The stiffness in the modified ES-FEM is created by the edge-based smoothed finite element method (ES-FEM) which is aimed at softening the overly stiffness of standard finite element method (FEM). On the other hand, the mass matrix is constructed by mass-redistributed method to tune the balance between the smoothed stiffness and mass matrix. The present modified ES-FEM adopts linear triangular elements generated automatically, which enables automation in computation and saving computational cost in mesh generation. Two numerical examples are presented to verify the computational efficiency of the modified ES-FEM. The numerical results have clearly demonstrated that the modified ES-FEM is very effective to minimize the dispersion errors in the simulation of band gap of acoustic metamaterials.

Inverse problems in free surface flows: a review

Abstract: Free surface flows occur frequently in our daily lives and many natural or industrial settings. Our understanding of such flows has grown tremendously with progress in mathematical modelling and numerical simulations. As a consequence, the response of a free surface to an external perturbation can often be computed and quantified. The free surface response is characteristic of the imposed perturbation and can be thought of as the signature of this perturbation. In this review paper, we survey the literature which deals with the inverse problem of identifying unknown flow conditions or fluid properties from an observed response of the free surface.

Rapid heating of FGM rectangular plates

Abstract: Thermally induced vibrations of functionally graded material rectangular plates are investigated in this research. The thermomechanical properties of the plate are assumed to be temperature and position dependent.DependencyontemperatureisexpressedbasedontheTouloukianformula,andpositiondependency is written as a power-law function. The ceramic-rich surface of the plate is subjected to temperature rise or heat flux, whereas the metal rich surface is kept at reference temperature or thermally insulated. Temporal evolution of the temperature profile across the plate thickness is obtained by the solution of one-dimensional heat conduction equation. This equation is originally nonlinear since temperature dependency of thermal conductivity is taken into account. The solution of this equation is obtained by means of the generalized differential quadrature (GDQ) accompanied with the successive Runge-Kutta algorithm in time domain. The motionequationsoftheplateareobtainedbasedonthefirst-ordersheardeformationtheoryofplatesundersmall strains and small deformations assumptions. Hamilton's principle is used to establish the motion equations. Theseequationsarediscretedintheplatedomainbymeansofthetwo-dimensionalGDQmethod.Theresulting equations are linear time-dependent coupled equations which are traced in time by means of the Newmark time-marching method. Conducting comparison studies to assure the validity and accuracy of the proposed model, parametric studies are carried out to examine the influences of temperature dependency, thermal and mechanical boundary conditions, power-law index, plate geometry and boundary conditions. It is shown that thermally induced vibrations exist for thin plates.

Exact solutions for free vibrations of axially inhomogeneous Timoshenko beams with variable cross section

Abstract: A novel method is proposed to simplify the governing equations for the free vibration of Timoshenko beams with both geometrical nonuniformity and material inhomogeneity along the beam axis. For a wide class of Timoshenko beams, this method enables us to reduce the coupled governing differential equations with variable coefficients to a pair of uncoupled second-order differential equations of Sturm–Liouville type with respect to the rotation angle due to bending. The reduced equations contain two important parameters, one describing the variations of translational inertia and bending rigidity along the beam axis, and the other reflecting the comprehensive effect of rotatory inertia and shear deformation. A series of exact analytical solutions are derived from the reduced equations for the first time, and several examples are also provided as benchmarks.

A study on fiber orientation influence on the mechanical response of a short fiber composite structure

Abstract: This paper deals with the structural analysis of composite materials with non-homogenous orientation of the reinforcement. During this research, a short fiber-reinforced polymer matrix composite is studied. In this case, inhomogeneity of the reinforcement orientation caused by injection molding manufacturing process is analyzed. The main objective of the paper is the investigation of an influence of process-induced orientation of the reinforcement on mechanical properties of the material in comparison with unidirectional and random reinforcement orientation. In particular, natural frequencies and transient response of an exemplary composite component are investigated. To specify effective properties of the composite, Mori–Tanaka’s micromechanical model is assumed. Orientation distribution of the reinforcement is determined by injection molding simulation. To determine elastic material properties dependent on non-homogenous orientation of the reinforcement, an orientation averaging procedure is taken into account. Therefore, during this study, effectiveness of the orientation averaging procedure and different closure approximations influence on the results are studied. Orientation averaging results are compared with numerical results obtained by finite element-based homogenization of composites with prescribed second-order orientation tensor. Finally, the obtained material parameters were applied into a macroscale finite element model, and numerical simulation with different boundary conditions was conducted.

Receding contact problem for two-layer functionally graded media indented by a rigid punch

Abstract: The present paper examines the plane strain receding frictionless contact problem of two functionally graded layers indented by a rigid cylindrical punch and with mismatched material properties at the interface. The shear moduli of the layers are assumed to vary in exponential form along the thickness direction and the Poisson’s ratios are taken as constant. With use of the Fourier integral transform, the governing equations are reduced to a system of two singular integral equations, in which the unknowns are the contact pressure and the contact widths. These integral equations are solved numerically using Gauss–Chebychev integration formulas. The main objective of this paper is to study the effect of the material inhomogeneity parameters and interface material property mismatch on the contact pressure and the size of the contact regions.

Analytical treatment of the nonlinear free vibration of cylindrical nanoshells based on a first-order shear deformable continuum model including surface influences

Abstract: Surface stresses can significantly affect the mechanical behavior of structures when they are scaled down to deep submicron dimensions. The Gurtin–Murdoch surface elasticity theory has the capability to capture the size-dependent behavior of nanostructures due to the surface stress effect in a continuum manner. The present work is concerned with the application of Gurtin–Murdoch theory to the nonlinear free vibration analysis of circular cylindrical nanoshells with considering surface stress and shear deformation effects. The nonlinear governing equations of motion together with the corresponding boundary conditions are firstly derived using Hamilton’s principle, the first-order shear deformation shell theory and von Karman’s assumption. An analytical approach is then presented to solve the nonlinear free vibration problem. Selected numerical results are given to illustrate the effects of surface energy on the nonlinear free vibration behavior of shear deformable nanoshells with different material and geometrical parameters. It is shown that there is a large difference between the results of Gurtin–Murdoch theory and those of its classical counterpart for very thin nanoshells.

Nonlinear model of an axially moving plate in a mixed Eulerian–Lagrangian framework

Abstract: We consider finite deformations and bending of an elastic plate moving across a given domain. Velocities of the plate are kinematically prescribed at two parallel lines, which bound the region in the direction of motion. Inhomogeneity of the velocity profile at the exit from the domain results in planar deformations and out-of-plane buckling of the plate. The presented quasistatic analysis features a novel kinematic description, in which the coordinate in the direction of motion is a Eulerian one, while the displacements in transverse and the out-of-plane directions are modeled in a Lagrangian framework. The material volume is traveling across a finite element mesh, which is aligned to the boundaries of the domain. A concise mathematical formulation results in a robust numerical scheme without the need to solve the advection (transport) equation at each time step. The model is validated against solutions of a benchmark problem with a conventional Lagrangian finite element scheme. The approach is further demonstrated by modeling the time evolution of deformation of a moving plate.

Vibration analysis of a single-layered graphene sheet-based mass sensor using the Galerkin strip distributed transfer function method

Abstract: The free vibration of a single-layered graphene sheet (SLGS)-based mass sensor is analyzed using the Galerkin strip distributed transfer function method (GSDTFM) based on the nonlocal Kirchhoff plate theory. The dynamic equations of the SLGS-based mass sensor are formulated, and the semi-analytical solutions of the frequency shift are computed with the GSDTFM. The effects of the nonlocal parameter, the attached nanoparticle locations, the plate side length, as well as the boundary conditions, on the frequency shift are studied. The simulated results show that the frequency shift of the SLGS-based mass sensor becomes smaller when the nonlocal parameter increases. The SLGS-based mass sensor is more sensitive when the attached nanoparticle is closer to the SLGS center or the side length of the SLGS becomes smaller. The boundary conditions strongly affect the frequency shift. Stiffer boundary condition causes larger frequency shift.

Granular micromechanics model of anisotropic elasticity derived from Gibbs potential

Abstract: This paper presents a Gibbs potential-based granular micromechanics approach capable of modeling materialswith complete anisotropy. The deformation energy of each grain–pair interaction is taken as a function of the inter-granular forces. The overall classical Gibbs potential of a material point is then defined as the volume average of the grain–pair deformation energy. As a first-order theory, the inter-granular forces are related to the Cauchy stress tensor using a modified static constraint that incorporates directional distribution of the grain–pair interactions. Further considering the conjugate relationship of the macroscale strain tensor and the Cauchy stress, a relationship between inter-granular displacement and the strain tensor is derived. To establish the constitutive relation, the inter-granular stiffness coefficients are introduced considering the conjugate relation of inter-granular displacement and forces. Notably, the inter-granular stiffness introduced in this manner is by definition different from that of the isolated grain–pair interactive. The integral form of the constitutive relation is then obtained by defining two directional density distribution functions; one related to the average grain–scale combined mechanical–geometrical properties and the other related to purely geometrical properties. Finally, as the main contribution of this paper, the distribution density function is parameterized using spherical harmonics expansion with carefully selected terms that has the capability of modeling completely anisotropic (triclinic) materials. By systematic modification of this distribution function, different elastic symmetries ranging from isotropic to completely anisotropic (triclinic) materials are modeled. As a comparison, we discuss the results of the present method with those obtained using a kinematic assumption for the case of isotropy and transverse isotropy, wherein it is found that the velocity of surface quasi-shear waves can show different trends for the two methods.

Size-dependent dynamic modeling and vibration analysis of MEMS/NEMS-based nanomechanical beam based on the nonlocal elasticity theory

Abstract: Accurate modeling and analysis of micro-/nanoelectromechanical systems (MEMS/NEMS) has an immense contribution in identification and improvement of the performance of such systems. This article investigates a nonclassical formulation for dynamicmodeling and vibration analysis of a piezo-actuatedmicrocantilever considering the Euler–Bernoulli beam model. Regarding the size effects in micro- to nanoscales, the size-dependent nonlocal continuum theory is employed to derive dynamic equations of the nonclassical microbeam taking into account the beam discontinuities. The nonlocal formulation of the beam and piezoelectric actuator is taken into consideration. Furthermore, the size effects on the resonant vibration characteristics of the beam are studied and some results are obtained. The results illustrate the size-dependent behavior of the beam particularly at higher resonant modes of vibrations. Also, it is indicated that the nonlocality and piezoelectric characteristics have a significant influence on the resonance behavior of the beam. However, this effect is more significant at higher resonant modes of vibrations.

Improved quaternion-based integration scheme for rigid body motion

Abstract: Rotation quaternions are frequently used for describing the orientation of non-spherical rigid bodies. Their compact representation by four numbers and disappearance of numerical problems, such as gimbal lock, are reasons for using them. We describe an improvement of a predictor–corrector direct multiplication (PCDM) method for numerically integrating the rigid body equations of motion with rotation quaternions. The method only uses quaternions to describe the orientation, so no rotation matrices are needed in the implementation. A predictor–corrector approach is used to update the quaternions each time step, such that no renormalization is needed at the end of the time step. The PCDM method suggested by Zhao and Van Wachem is improved such that forces and torques are calculated at the correct time using position and orientation information at that same time. This is achieved by using a leapfrog approach in the improved version, in which the linear and angular velocities and rotation quaternions are defined at half time steps, while whole time step information of these quantities is calculated as part of the improved integration scheme. The improved PCDM scheme is compared with the original implementation for rotational kinetic energy conservation, accuracy of object orientation and angular velocity, and rate of convergence for different time steps. With the modifications that we propose, the improved method has a true second-order rate of convergence, without the need for explicit renormalization of the quaternions. Furthermore, the method is applicable to problems with position and velocity dependent torques, while still only a single force/torque evaluation is needed per time step. For objects experiencing torque, the improved PCDM method performs better than the original method, now showing a true second-order rate of convergence, and much smaller errors in the prediction of object orientation and angular velocity while still requiring only a single torque evaluation per time step.