In-plane Mechanical Loading Research Articles

A meshfree formulation for size-dependent thermal buckling and post-buckling behaviour of porous microplates on elastic foundation subjected to localized heating

The semi-analytical framework is developed for in-plane thermal stresses within the microplates under localized heating. Localized heating is modelled using Fourier Series expansions over the complete domain of the plate. These stresses within the plate are estimated after solving the in-plane thermoelasticity problem using Airy's stress approach. Utilizing these stresses, present work also studies the buckling and post-buckling behaviour of a porous metal foam microplate resting on Winkler and Pasternak elastic foundations. The plate is modelled using Reddy's third-order shear deformation theory (TSDT) and von Kármán geometric nonlinearity, and the size-dependent effects are included using the modified strain gradient theory (MSGT). Galerkin's weighted residual method converts the partial differential equations (PDEs) into algebraic equations. The post-buckling equilibrium path is obtained using the modified Newton-Raphson method. The mode shapes of the microplate for the various configurations of the plate with different boundary conditions due to localized thermal loading are plotted. Also, these mode shapes are compared with the mode shapes of the microplate due to in-plane mechanical loading. The parametric effect of porosity, elastic foundation parameters, thickness of plate, size of plate, boundary conditions and loading concentrations on the buckling and post-buckling behaviour of the plate is studied.

Nonlinear Dynamics Suppression of Bending Deflection in a Composite Laminated Plate Using a Beam Stiffener Due to Critical Buckling Load and Shear Force

In this paper, the nonlinear dynamics behavior of the bending deflection of a stiffened composite laminated plate is suppressed using beam stiffeners at different fiber volume fractions and different aspect ratios. The non-periodic motion and chaos in a stiffened composite laminated plate is detected using the largest Lyapunov exponent parameter and power density function of a fast Fourier transform (FFT). The critical buckling load is calculated at different thickness ratios, numbers of stiffeners, lamination angles and stiffener–depth ratios based on different boundary conditions. The nonlinear response of the bending deflection is analyzed analytically, numerically and experimentally. The analytic solution has been derived using Levy and Navier solutions of classical laminate plate theory at different boundary conditions (CLPT). The numerical simulation was conducted using the ANSYS program while the experiment test was carried out using a strain gauge through a strain meter device. Experimentally, a Southwell plot is used to investigate the value of the critical buckling load. The combined loading are the in-plane compression mechanical load and shear force. All the values of the largest Lyapunov exponent are positive, which gives indication to non-periodic motion and chaos. The nonlinear dynamics behavior of the bending deflection is decreased with the increasing of number of stiffeners in which the value of largest Lyapunov exponent has been decreased. The nonlinear dynamics behavior is increased with the increasing of aspect ratios and fiber volume fractions. The system with an aspect ratio (2.5) and fiber volume fraction (υf = 80%) for an un-stiffened plate is more chaotic than the other systems.

Analytical buckling response of sectorial porous plates integrated with piezoelectric layers

This paper is concerned with influence of porosity and mechanical/electrical boundary conditions on buckling of moderately thick, porous sector and annular sector plates integrated with piezoelectric layers. Presence of both porous and piezoelectric elements in a configuration provides new opportunities for manipulating the buckling load broadly. To demonstrate these possibilities, across the thickness of the core plate, varying porosity profiles are considered, yet the network of pores is assumed to be free of fluid. The governing equations of the sandwich plate under uniform in-plane mechanical loading are obtained by taking the Mindlin plate theory and von Kármán assumptions into account; and later the nonlinear nature of the problem is simplified through the use of the adjacent equilibrium criterion. Next, decoupled formulae for all the unknown kinematic variables are reached by introducing two transformation functions. The critical buckling loads are computed analytically by assuming simply supported radial edges and classical boundary conditions for the circumferential edges. Eventually, the effect of the mechanical/electrical boundary conditions, geometrical parameters and porosity is realized by comprehensive parametric studies.

In-plane stress analysis in a cracked functionally graded piezoelectric plane

The fracture behavior of a cracked functionally graded piezoelectric plane was investigated under in-plane mechanical and electrical loading. The electro-elastic properties of the plane exponentially varied along the y-axis. Employing the Fourier transform, stress analysis containing a Volterra edge dislocation in a functionally graded piezoelectric plane was carried out. These solutions were used to derive a system of Cauchy singular integral equations for analyzing the straight and curved cracks. Linear, arc, parabolic and sine wave-shaped cracks were evaluated. The integral equations were numerically solved using the Lobatto–Chebyshev integration formula for dislocation density functions on the crack face which were employed to calculate field intensity factors. The effects of the shaped crack, loading parameters, and non-homogeneity parameter on the stress and electric intensity factors were considered.

Buckling treatment of piezoelectric functionally graded graphene platelets micro plates

Micro-electro-mechanical systems (MEMS) are widely employed in sensors, biomedical devices, optic sectors, and micro-accelerometers. New reinforcement materials such as carbon nanotubes as well as graphene platelets provide stiffer structures with controllable mechanical specifications by changing the graphene platelet features. This paper deals with buckling analyses of functionally graded graphene platelets micro plates with two piezoelectric layers subjected to external applied voltage. Governing equations are based on Kirchhoff plate theory assumptions beside the modified couple stress theory to incorporate the micro scale influences. A uniform temperature change and external electric field are regarded along the micro plate thickness. Moreover, an external in-plane mechanical load is uniformly distributed along the micro plate edges. The Hamilton's principle is employed to extract the governing equations. The material properties of each composite layer reinforced with graphene platelets of the considered micro plate are evaluated by the Halpin–Tsai micromechanical model. The governing equations are solved by the Navier's approach for the case of simply-supported boundary condition. The effects of the external applied voltage, the material length scale parameter, the thickness of the piezoelectric layers, the side, the length and the weight fraction of the graphene platelets as well as the graphene platelets distribution pattern on the critical buckling temperature change and on the critical buckling in-plane load are investigated. The outcomes illustrate the reduction of the thermal buckling strength independent of the graphene platelets distribution pattern while meanwhile the mechanical buckling strength is promoted. Furthermore, a negative voltage, -50 Volt, strengthens the micro plate stability against the thermal buckling occurrence about 9% while a positive voltage, 50 Volt, decreases the critical buckling load about 9% independent of the graphene platelet distribution pattern.

Buckling analysis of composite laminated plate in different boundary conditions under thermo mechanical loading

Abstract Composite material consists of two or more components at the microscopic level, which improves its properties. In recent year, it is widely used in Thermo mechanical loading where the structural performances of these capacities are required to perform at elevated temperature. The consideration of such loading with high temperature is known as thermo mechanical loading, and this type of loading in which thermal and mechanical loading are applied together. In this paper the buckling analysis of composite rectangular plate with combined in-plane mechanical & thermal loading has been carried out. The analysis is performed using the ANSYS APDL software package, in which SHELL element is used. The effects of temperature on the plate have been investigated. It was found from the analysis that the natural frequencies of a variable thickness rectangular plate decrease with the increase of the temperature. Effect of ply angle for different condition is also analyzed and represented with 2D graphs for the effective comparison of various conditions. The load carrying capacity increases with increasing number of ply, at the maximum carrying capacity of load i.e. forty five degree orientation of the plate and shell. If the number of elemental length increases load carrying capacity also increases. Buckling analysis is widely useful in the aerospace industry, in the shipbuilding industry, manufacturing industry therefore the ANSYS FEA software is a good tool for the calculation of critical buckling load.

Influence of debonding on nonlinear deflection responses of curved composite panel structure under hygro-thermo-mechanical loading–macro-mechanical FE approach

The excess geometrical deformation due to the in-plane hygrothermal and transverse mechanical loading are computationally (through a customized MATLAB code) obtained for the weakly bonded structure using the different kinematic theories in combination with the finite element steps. To evaluate the nonlinear deflection data a macro mechanical model is prepared mathematically considering the stretching effect (through the thickness), as well as the constant functions of displacement. The model includes the variation of composite properties due to the change in environmental temperature and moisture content including the necessary continuity assumptions between the separated layers for the weak bonding. The role of delamination i.e. location, position and size on the nonlinear deflection parameters are also computed under the combined loading (Hygral/Thermal/Mechanical/Hygro-Thermo-Mechanical). The solution accuracy and the elemental sensitivity have been computed for the proposed macro mechanical finite element model. Moreover, a set of numerical examples are solved to show the environmental effect on the flexural strength of a weakly bonded composite panel including the design-dependent parameters. The final results are indicating the necessity of the various kinematic model for the analysis of laminated structure with/without hygrothermal loading as well as the debonding.

Wrinkling of soft magneto-active plates

Coupled magneto-mechanical wrinkling has appeared in many scenarios of engineering and biology. Hence, soft magneto-active (SMA) plates buckle when subject to critical uniform magnetic field normal to their wide surface. Here, we provide a systematic analysis of the wrinkling of SMA plates subject to an in-plane mechanical load and a transverse magnetic field. We consider two loading modes: plane-strain loading and uni-axial loading, and two models of magneto-sensitive plates: the neo-Hookean ideal magneto-elastic model and the neo-Hookean magnetization saturation Langevin model. Our analysis relies on the theory of nonlinear magneto-elasticity and the associated linearized theory for superimposed perturbations. We derive the Stroh formulation of the governing equations of wrinkling, and combine it with the surface impedance method to obtain explicitly the bifurcation equations identifying the onset of symmetric and antisymmetric wrinkles. We also obtain analytical expressions of instability in the thin- and thick-plate limits. For thin plates, we make the link with classical Euler buckling solutions. We also perform an exhaustive numerical analysis to elucidate the effects of loading mode, load amplitude, and saturation magnetization on the nonlinear static response and bifurcation diagrams. We find that antisymmetric wrinkling modes always occur before symmetric modes. Increasing the pre-compression or heightening the magnetic field has a destabilizing effect for SMA plates, while the saturation magnetization enhances their stability. We show that the Euler buckling solutions are a good approximation to the exact bifurcation curves for thin plates.

A comprehensive electro-magneto-elastic buckling and bending analyses of three-layered doubly curved nanoshell, based on nonlocal three-dimensional theory

The present paper studies the three-dimensional magneto-electro-elastic bending and buckling analyses of three-layered doubly curved nanoshells based on nonlocal elasticity theory. The kinematic relations are developed based on two-variable sinusoidal transverse shear and thickness deformation theory. The transverse deflection is decomposed into bending, shear and thickness deformation components by using a two-variable sinusoidal shear deformation theory. The nanoshell is subjected to in-plane mechanical, electrical and magnetic loads in which these forces are accounted as external works in constitutive relations for buckling analysis. This introduces a variable transverse displacement along the thickness direction. The principle of virtual work is used to derive the governing equations of bending and buckling. Numerical results are obtained based on Navier’s technique for simply-supported boundary conditions. A parametric analysis is performed to investigate the influence of the nonlocal parameter, the applied electric and magnetic potentials on the magneto-electro-elastic bending and buckling responses of doubly curved nanoshells.

Nonlinear electromechanical coupling in graded soft materials: Large deformation, instability, and electroactuation.

Subject to an applied electric field, soft dielectrics with intrinsic low moduli can easily achieve a large deformation through the so-called electrostatic Maxwell stress. Meanwhile, the highly nonlinear electromechanical coupling between the mechanical and electric loads in soft dielectrics gives a variety of failure modes, especially pull-in instability. These failure modes make the application of soft dielectrics highly limited. In this paper, we investigate the large deformation, pull-in instability, and electroactuation of a graded circular dielectric plate subject to the in-plane mechanical load and the applied electric load in the thickness direction. The results obtained herein cover, as special cases, the electromechanical behaviors of homogeneous dielectrics. There is a universal physical intuition that stiffer dielectrics can sustain higher electromechanical loads for pull-in instability but achieve less deformation, and vice versa. We show this physical intuition theoretically in different homogeneous dielectrics and graded dielectrics. Interestingly, we find that the ability to sustain a high electric field or a large deformation in a stiff or soft homogeneous circular dielectric plate can be achieved by just using a graded circular dielectric plate. We only have to partly change the modulus of a circular plate, with a stiff or soft outer region. The change makes the same electromechanical behavior as that of a homogeneous dielectric, even increases the maximum electroactuation stretch from 1.26 to 1.5. This sheds light on the effects of the material inhomogeneity on the design of advanced dielectric devices including actuators and energy harvestors.

Buckling behavior of porous CNT-reinforced plates integrated between active piezoelectric layers

This work presents the buckling resistance of a novel active multidisciplinary sandwich plate (AMSP) under in-plane mechanical load or temperature change. The proposed sandwich plate includes an advanced porous core reinforced with carbon nanotubes (CNTs) integrated between two active piezoelectric faces. Functional graded (FG) profiles are considered for the dispersions of CNTs and porosity along the thickness of core layer. In addition, the effect of CNT agglomeration in core layer on the buckling resistance of the proposed AMSP has been studied by employing Eshelby-Mori-Tanaka (EMT)'s approach. A third order shear deformation theory (TSDT) of plates is adopted to obtain governing Eigen value equations for the thermal and mechanical buckling analyses of AMSP. The critical buckling resistance of each analysis has been extracted from the governing equations through a developed mesh-free solution based on moving least squares (MLSs) shape functions. The impacts of porosity, CNTs and geometrical dimensions on the buckling resistance of AMSPs have been investigated. The results show that embedding porosity in core results in a slight reduction in mechanical responses and a significant improvement in thermal buckling resistance. Moreover, reinforcing core layer with CNTs leads to remarkable drop and increase in thermal and mechanical buckling resistances, respectively. However, the formation of CNT agglomerations significantly reduces such CNTs impacts.

In-Plane Analysis of an FGP Plane Weakened by Multiple Moving Cracks

In this paper, the analytical solution of an electric and Volterra edge dislocation in a functionally graded piezoelectric (FGP) medium is obtained by means of complex Fourier transform. The system is subjected to in-plane mechanical and electrical loading. The material properties of the medium vary exponentially with coordinating parallel to the crack. In this study, the rate of the gradual change of the shear moduli and mass density is assumed to be same. At first, the Volterra edge dislocation solutions are employed to derive singular integral equations in the form of Cauchy singularity for an FGP plane containing multiple horizontal moving cracks. Then, these equations are solved numerically to obtain dislocation density functions on moving crack surfaces. Finally, the effects of the crack moving velocity, material properties, electromechanical coupling factor and cracks arrangement on the normalized mode I and mode II stress intensity factors and electric displacement intensity factor are studied.

Wave propagation in FG porous GPLs-reinforced nanoplates under in-plane mechanical load and Lorentz magnetic force via a new quasi 3D plate theory

In this paper, the nonlocal strain gradient theory and a new quasi 3D plate theory are employed to investigate the wave propagation in functionally graded (FG) porous graphene platelets (GPLs)-reinforced nanoplates on elastic foundations subjected to in-plane mechanical load and magnetic field. The present theory takes into account the shear deformation as well as the thickness stretching effect. The internal porosities and the GPLs are uniformly or non-uniformly distributed into the matrix according to four different types. The properties of the nanocomposites plates are calculated by utilizing the modified Halpin-Tsai pattern. Lorentz magnetic force is derived from Maxwell’s equations for the conducting material. The motion equations are derived employing Hamilton’s principle according to a new shear and normal deformations plate theory. Detailed parametric investigations on the wave frequency and phase velocity of the porous GPLs-reinforced nanoplates are implemented considering the influences of porosity coefficient, GPLs weight fraction, magnetic parameter and foundation stiffnesses on the results. It can be found that an increment occurs in the wave frequency with increasing the GPLs weight fraction and magnetic field parameter. While, it decreases as the pore coefficient increases.

Research on non-pneumatic tire with gradient anti-tetrachiral structures

In this paper, a new type of non-pneumatic tire is designed. The pneumatic parts of standard pneumatic tire are replaced by gradient anti-tetrachiral structure. Firstly, the in-plane mechanical properties and elastic limit load of anti-tetrachiral structures with different gradient factors are studied by experiment, finite element method and theory. Based on this, the structural design of non-pneumatic tire is carried out, and its mechanical properties under compression load are studied by finite element method. Results show that the gradient anti-tetrachiral structure and the non-pneumatic tire based on this structure have certain similarities in the deformation mode. The basic research of gradient anti-tetrachiral structure is of guiding significance to the design of non-pneumatic tire. It also shows that the non-pneumatic tire with the gradient anti-tetrachiral structure has good load-carrying capacity, and has a wide engineering application potential.

Nonlinear Limit Cycle Oscillation and Flutter Analysis of Clamped Curved Plates

This Paper studies the instability (flutter) and poststability limit cycle oscillations of elastic shallow shells in a supersonic gas flow. In a previous paper by the authors, a nonlinear dynamic aeroelastic analysis of streamwise square curved plates was made to compare with the experimental results of Anderson. In the present Paper, computational results are presented for a much larger range of several parameters, which are important for flutter and limit cycle oscillations including both spanwise and streamwise curved plates with various aspect ratios, various boundary conditions, and the effects of static pressure and in-plane mechanical loadings. No previous study has included such a wide range of parameters and parameter values. Von Kármán shallow shell theory, a modal expansion along with the Galerkin method, and the piston theory aerodynamic model for supersonic/hypersonic flow are employed to obtain these results.

A dielectric breakdown model for an electrode along an interface between two piezoelectric materials

A plane strain problem for an interface electrically charged thin electrode in a bimaterial piezoelectric space is considered. Remote in-plane mechanical loading and an electrical field parallel to the electrode faces are applied. The known solution of this problem demonstrates the electromechanical field singularities at the electrode tips. To remove these singularities the dielectric breakdown model is applied. Using the presentations of electromechanical quantities via sectional analytic functions the Riemann-Hilbert problem with discontinuous right side is formulated and solved exactly. The dielectric breakdown zone lengths are found from the electric field finiteness at the end points of these zones. It is shown that this length depends on the external electric field and the electrode net charge. The variations of electromechanical characteristics along the electrode and its continuation are presented in graphical form. Particular cases of an uncharged electrode and homogeneous material are considered and compared with previously obtained results. A verification of the obtained solutions by means of the ABAQUS code is carried out and good agreement of analytical and numerical results is revealed.

New analytical solutions for modified polarization saturation models in piezoelectric materials

In this paper, we present new analytical solutions for modified polarization saturation (PS) models for arbitrary polarized and semipermeable 2-D piezoelectric media. The PS model is modified to various other non-linear fracture models by varying the normal electric displacement saturated conditions in place of a constant saturated value. These variations are hereby defined as interpolating linear, quadratic and cubic type’s polynomials into saturated value interpolated on the basis of possible electric displacement values at the centre of the crack, actual crack tip and saturated zone tip. To present the analytical study, a centre crack problem in 2-D semipermeable piezoelectric media under arbitrary poling direction and combined in-plane mechanical and electric displacement loadings is considered. Using extended Stroh formalism and complex variable approach, these models are mathematically reduced into generalized non-homogeneous Riemann–Hilbert problems. Applying Riemann–Hilbert technique, the solution of these problems is obtained in terms of generalized complex potential functions representing traction forces and electric displacement components. These lead to obtain the explicit expressions for saturated zone length, crack opening displacement, crack opening potential and local intensity factor in each case. Also, the results of saturation zones and local intensity factor are obtained in each case and compared with the established numerical results. The studies show that saturated zone length increases with increasing electric loading for each case and also depends on the polynomial varying saturation condition. For a particular electric loading applied within the range of critical value, it increases with increasing degree of polynomial type saturated condition i.e. from constant to cubic type’s normal electric displacement saturated condition.

Closed form solutions – Analysis and optimal design of supersonic composite laminated flat plates considering mechanical and thermal effects

Bolotin (1961) published the book presenting the analytical solution of the flutter problem for isotropic plates having finite dimensions (i.e. a width and a length). His results are much more broader than those discussed by Hedgepeth (1956) and Houbolt (1958) for plates having one infinite dimension (i.e. in fact reduced to the analysis of beams). The present paper is an extension of Bolotin’s analytical method to the analysis of multilayered laminated plates and includes the analysis of anisotropic plates with different wavenumbers m, n at the x and y directions, respectively. It is shown how easily that method can be used in the optimal analysis of laminated plates searching for the maximal aerodynamic pressure. In our opinion the obtained results can be easily adopted as the benchmark for numerical (finite element) investigations of flutter problems for plates. The effects of in-plane mechanical and thermal loads, boundary conditions, stacking sequences and orthotropic modulus ratio on flutter characteristics are examined in an analytical way. The analysis is conducted for the normal oncoming flow.

Effect of various characteristics of graphene nanoplatelets on thermal buckling behavior of FGRC micro plate based on MCST

The size dependent thermal buckling analysis of composite micro plate is studied in this paper based on modified couple stress theory (MCST) and sinusoidal shear deformation theory. The composite micro plate is composed of epoxy reinforced with functionally graded graphene nanoplatelets which is distributed along the thickness direction based on various distributions (parabolic, linear and uniform distributions). After calculation of effective material properties including modulus of elasticity and Poisson ratio of composite plate, Halpin-Tsai model and rule of mixture, the governing equations are derived based on principle of virtual work. The in-plane mechanical and thermal loads are included in the work of external loads. The novelty of this work is the application of a modified couple stress theory to predict the mechanical and thermal buckling loads of micro plate reinforced composites. The effect of weight fraction of graphene nanoplatelets and various distributions of reinforcement was studied on the mechanical and thermal buckling loads. It is concluded that with increase of weight fraction of graphene nanoplatelets, the mechanical buckling loads are increased for all distributions, while the thermal buckling loads are increased for parabolic distribution, are decreased for linear distribution and are insensitive for uniform distribution.

Stress fields and effective modulus of piezoelectric fiber composite with arbitrary shaped inclusion under in-plane mechanical and anti-plane electric loadings

The piezoelectric fiber composite, which has been widely used in vibration control of aeronautic and aerospace structures, contains an active layer constructed by piezoceramic fibers embedded in polymer matrix. In this paper, an analytical framework is developed to predict the stress fields and the effective elastic and piezoelectric modulus of a piezoelectric fiber composite with arbitrary shaped inclusion under in-plane mechanical and anti-plane electric loadings. Firstly, a three-phase model is presented based on the general self-consistent method. It assumes that a representative volume element, consisting of an arbitrary shaped piezoelectric fiber and non-piezoelectric matrix, is embedded in infinite equivalent medium of piezoelectric fiber composite. Also, in order to obtain the complex potentials in different phases, a new conformal mapping method is proposed to translate two arbitrary connected domains into an annular domain. Then, the general solution of each complex potential can be expressed in series form with some coefficients. The continuity conditions of the interfaces and the homogeneous conditions are used to build up a set of equations to determine the coefficients. After the complex potentials are solved, the exact stress responses and effective modulus of piezoelectric fiber composite with arbitrary shaped inclusion are obtained. It is shown that the piezoelectric fiber shape and volume fraction have significant influence on the stress distributions around the piezoelectric fiber and effective modulus of the piezoelectric fiber composite. Furthermore, the proposed analytical framework provides a powerful instrument for solving related problems about fiber-reinforced composites with imperfect interface or with coated phase.