Uniform In-plane Loading Research Articles

Circular inclusion with a refined linearized version of Steigmann–Ogden model

The plane deformation of an infinite elastic matrix enclosing a single circular inclusion incorporating stretching and bending resistance for the inclusion–matrix interface is revisited using a refined linearized version of the Steigmann–Ogden model. This refined version of the Steigmann–Ogden model differs from other linearized counterparts in the literature mainly in that the tangential force of the interface defined in this version depends not only on the stretch of the interface but also on the bending moment and initial curvature of the interface (the corresponding bending moment relies on the change in the real curvature of the interface during deformation). Closed-form results are derived for the full elastic field in inclusion–matrix structure induced by an arbitrary uniform in-plane far-field loading. It is identified that with this refined version of the Steigmann–Ogden model a uniform stress distribution could be achieved inside the inclusion for any non-hydrostatic far-field loading when R = 3 χ int / λ int (where R is the radius of the inclusion, while λ int and χ int are the stretching and bending stiffness of the interface). Explicit expressions are also obtained for the effective transverse properties of composite materials containing a large number of unidirectional circular cylindrical inclusions using, respectively, the dilute and Mori–Tanaka homogenization methods. Numerical examples are presented to illustrate the differences between the refined version and two typical counterparts of the Steigmann–Ogden model in evaluating the stress field around a circular nanosized inclusion and the effective properties of the corresponding homogenized composites.

Isogeometric technique for dynamic instability analysis of nanocomposite folded plates based on higher-order shear deformation theory

This paper proposes a dynamic instability isogeometric analysis (IGA) of functionally graded (FG) folded plates strengthened by carbon nanotubes (CNTs) under a uniform in-plane loading. The folded plates are modeled by two patches to overcome the existing discontinuity in the geometry. A recently developed logarithmic higher-order shear deformation theory (HSDT) is used to describe the kinematic relations of each patch. Then, IGA in conjunction with Hamilton’s principle is used to discretize the governing equations of the patch. Afterwards, an appropriate coordinate transformation is applied to transfer the patch stiffness, mass and geometric stiffness matrices to the global coordinates. Moreover, the continuity conditions along the joined line are enforced by means of the bending strip method (BSM). The final form of the governing equations, describing the dynamic instability behavior of folded plates, is solved by Bolotin’s method in order to estimate the boundaries of the principal instability regions with the first and second-order approximations. Various types of numerical examples are investigated to verify the precision and reliability of the present solution method. These comparative studies confirm that the proposed logarithmic HSDT-based isogeometric formulation can precisely predict natural frequencies, critical buckling loads and principal instability regions of FG carbon nanotube-reinforced composite (CNTRC) folded plates. The effects of first and second-order approximations, various configurations of CNTRCs, in-plane loading behavior and geometrical parameters on the aforementioned mechanical analyses of the plate are investigated.

Buckling behaviour and design aids of cylindrical panel subjected to uniform in-plane loading

Buckling behaviour and design aids of cylindrical panel subjected to uniform in-plane loading

Buckling behaviour and design aids of cylindrical panel subjected to uniform in-plane loading

Buckling behaviour and design aids of cylindrical panel subjected to uniform in-plane loading

Uniqueness of the shape of neutral inhomogeneities with interface effects in anisotropic plane elasticity

In the context of classical interface elasticity theory, the existence and uniqueness of elliptical neutral inhomogeneities in an isotropic plane have been verified in the literature for plane deformation when the interface effects are described by variable interface parameters and the matrix is subjected to a uniform external loading. In this paper, we examine similar existence and uniqueness of neutral inhomogeneities in an anisotropic plane under the same conditions. We show that for an anisotropic inhomogeneity-matrix system under plane deformation and uniform external in-plane loadings, (i) the inhomogeneity is necessarily elliptical to achieve neutrality even if the interface parameters are allowed to vary at the interface, and (ii) the inhomogeneity must be circular to ensure its neutrality when the interface parameters keep constant on the entire interface.

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.

Need the Uniform Stress Field Inside Multiple Interacting Inclusions Be Hydrostatic?

Since the pioneering work of Eshelby on a single ellipsoidal inclusion embedded in an infinite space, much attention has been devoted in the literature to the question of the uniformity of the stress field inside inclusions surrounded by an elastic matrix. Over the last decade or so, researchers have established the existence of multiple (interacting) inclusions enclosing uniform internal stress distributions when embedded in an infinite elastic matrix subjected to a uniform far-field loading and identified a variety of shapes of such inclusions. In the design of multiple inclusions with uniform internal stresses, it is customary to assume that the uniform stress field inside each inclusion (each with different shear modulus distinct from that of the matrix) is hydrostatic. In this paper, we examine whether this assumption is actually necessary to ensure the required existence of multiple inclusions enclosing uniform stresses. By establishing several theorems in the theory of functions of a complex variable, we prove rigorously that for any collection of multiple inclusions enclosing uniform stresses in an infinite isotropic plane subjected to uniform remote in-plane loading, the internal uniform stress field must indeed be hydrostatic if the corresponding inclusion’s shear modulus is distinct from that of the matrix.

Static and dynamic instability analysis of tapered CNTRC sandwich plates under uniform and non-uniform in-plane loadings using spline finite strip method

This paper presents the buckling analysis of rectangular sandwich plates with pure polymeric tapered cores and functionally graded carbon nanotube (FG-CNT) reinforced composite face sheets under static and harmonic dynamic loads. One uniform and four linearly-varying patterns are considered for the applied in-plane loads. The effective material properties of face sheets, in which the polymeric matrix is reinforced with aligned single-walled carbon nanotubes (SWCNT), are estimated using the extended rule of mixture. A higher-order zigzag shear deformation theory, which has the same number of dependent unknowns as the first-order theory, is employed to express the plate's displacement field. The governing equations are derived using Hamilton's principle and discretized by the spline finite strip method. The dynamic instability regions of plates with uniform and variable thickness are obtained by employing Bolotin's method. The accuracy of the present method is demonstrated by considering the problems for which solutions are available. A detailed numerical study is conducted to examine and compare the effects of different parameters, including the taper angle, type of the applied in-plane load, static and dynamic load factors, length-to-width ratios, the volume fraction of CNT, distribution pattern of CNTs along the thickness direction, temperature, and boundary conditions on the behavior of sandwich plate.

Effect of Thermal Environment and Peripheral Loading on Axisymmetric Vibrations of Non-uniform FG Circular Plates via Generalized Differential Quadrature Method

In this article, axisymmetric vibrations of a non-uniform functionally graded circular plate subjected to uniform in-plane peripheral loading and non-linear temperature rise across the thickness have been analyzed on the basis of classical theory of plates. The thickness of the plate is assumed to vary exponentially along the radial direction. The plate material is graded in thickness direction using a power-law model and its mechanical properties are temperature-dependent. Keeping the uniform thermal environment over the top and bottom surfaces, the equations for thermoelastic equilibrium and axisymmetric motion for such a plate model have been derived by Hamilton’s energy principle. Employing generalized differential quadrature method, the frequency equations for clamped and simply supported plates have been obtained and solved numerically using MATLAB. The lowest three roots have been retained as the frequencies for the first three modes of vibration. The influence of various parameters, such as power-law index, in-plane force parameter and temperature difference with varying values of thickness parameter, has been analyzed on the vibration characteristics of the plate. By allowing the frequency to approach zero, the critical buckling loads with varying values of other parameters have been computed. The benchmark results for linear as well as uniform temperature rise have also been computed. The validity of the present technique is verified by comparing the results with the published work.

Free vibration and stability of graphene platelet reinforced porous nano-composite cylindrical panel: Influence of grading, porosity and non-uniform edge loads

Buckling and vibration characteristics of functionally graded(FG) porous(P), graphene platelet (GPL) reinforced cylindrical panel are presented. A multilayer model is considered for analysis with graphene and internal pores distribution varying in an uniformly or two different non-uniformly manner along the thickness. To evaluate the effective mechanical properties, extended rule of mixture together with modified Halpin-Tsai micromechanics model and mechanical properties of open-cell metal foams is used. Considering a higher order shear deformation theory, characteristics of the FG-P-GPL reinforced cylindrical panel under different edge loads such as uniform, triangular, trapezoidal and parabolic are investigated. The Hamilton’s principle is used to formulate the governing partial differential equations and buckling and free vibration solutions are obtained by employing the Galerkins method. The influences of grading of GPL and internal pores, porosity coefficient on buckling and dynamic characteristics of functionally graded GPL reinforced porous cylindrical panel under uniform and non-uniform in-plane loads are presented.

Macro-Scale Numerical Modeling of Unreinforced Brick Masonry Squat Pier Under In-Plane Shear

Numerical modeling of brick masonry behaviour under different performance conditions has always remained a challenging task. Several modeling strategies have been developed for masonry, in general, through the course of time that have been simplified to speed up modeling and analysis duration. This ranges from a simplified strut model to a highly discontinuous micro-scale nonlinear model. With the current advent of high-speed computing and modeling tools, more realistic numerical modeling of masonry is now possible. In this paper, the strategy adopted is based on macro-scale modeling, where isotropic material properties are considered for the homogenous continuum. ABAQUS is used as a state-of-the-art finite element-based analysis and modeling tool. The Concrete Damage Plasticity (CDP) model is used for simulating inelastic material behaviour of brick and mortar, which is available in the ABAQUS library. This material model can be used in both implicit and explicit schemes of integration but the explicit procedure is highly preferred as it overcomes the convergence issues. Various parameters required for CDP modeling of brick and mortar are adapted from literature. The model is assembled in two parts, first part is modeled for masonry with both elastic and plastic properties, while the other part simulates a rigid beam at the top of the masonry part to create a uniform in-plane shear loading effect. The masonry part has been fixed at the bottom with free vertical ends, while horizontal in-plane displacement was applied to the top rigid beam. The load-displacement curves were generated from these models for monotonic push, to compare them with the envelopes of experimental results, loaded similarly. Since brick masonry is a highly disjointed material, it is a complicated procedure to develop an exact model and predict its exact behaviour. However, the overall representative load-displacement curve developed numerically was in good agreement with the ones produced experimentally.

Buckling behaviour of composite laminates of trapezoidal panel with cutout subjected to non-uniform in-plane edge loads

Abstract Critical buckling loads of composite laminates are usually calculated by using analytical solutions based on the assumptions of uniform in-plane loads, despite of the fact that real structures are often subjected to various nonuniform loads. The present work is focused on the buckling behavior of composite trapezoidal laminates panel with cutouts, subjected to various nonuniform in-plane loads. The effect of variation is size of trapezoidal plate on the buckling behavior has been studied using finite element method. The governing differential eigenvalue equations have been solved using subspace algorithm. The analytical solution for such problems is very difficult. Hence in this work, a finite element software FORTRAN is used to study the buckling behavior of thin and thick laminated panels with cutouts subjected to various kinds of non-uniform edge loads. The panel is modeled by using eight node elements with five degrees of freedom at each node. The accuracy of the model is confirmed by comparing present results with available literature. The effect of various parameters such as cutout size, boundary conditions and ply-orientations of panel are included in this work. It is observed from this study that the non-uniform edge loads and cutout size has remarkable effect on the buckling behavior of said panels.

Comprehensive effect of in-plane load and nonlinear thermal field on dynamic response of embedded bi-directional functionally graded tapered thick nanobeams

This article presents a numerical model based on shear deformation theory in conjunction with Eringen’s nonlocal theory to investigate the thermo-elastic vibration behavior of bi-directional functionally graded nonuniform thick nanobeams supported on Pasternak's foundation. These beams are subjected to nonlinear temperature field along the thickness and uniform in-plane loading. Two directional variations in material properties are described by a power-law model across the thickness and as per the exponential function along the length. Employing Hamilton's energy principle, the governing equations are derived and then solved for frequency parameter via the generalized differential quadrature method for clamped, simply-supported, and combination of both the boundary conditions. A parametric study is carried out for understanding the vibration characteristic of considered beams, which shows that the combination of in-plane and thermal loadings with the foundation parameter is the most critical combination for bi-directional functionally graded nanobeams. Results are compared with the published work and found in good accordance. The results will be of great use in designing aerospace shuttles and scanning probe microscopes where nonuniform embedded nanobeams are subjected to compression in the thermal environment.

Geometrically nonlinear analysis of doubly curved imperfect shallow shells made of functionally graded carbon nanotube reinforced composite (FG_CNTRC)

This article presents an analytical approach to investigate the geometrically nonlinear response of doubly curved imperfect shallow shells made of functionally graded carbon nanotube reinforced composite (FG_CNTRC) under in-plane and lateral loads. For all of the case studies, the edges of shells are assumed to be diaphragm supports, which are subjected to in-plane edge-loads. The governing equations are developed for thin shells based on the classical theory considering the geometric nonlinear terms using the Von Karman–Donnell sense. The Galerkin method is used to solve the extended nonlinear system of differential equilibrium and compatibility equations. The solution results in a closed-form formula representing the equilibrium path (load-deflection relationship) of the shell which is validated by numerical methods, such as finite element and mesh-free methods. The parametric studies are performed for doubly curved and cylindrical shells subjected to uniform external pressure and in-plane edge loads, respectively. The effects of various parameters such as volume fraction of FG_CNTRCs and geometrical properties on the equilibrium paths are examined in detail.

Effect of Winkler Foundation on Radially Symmetric Vibrations of Bi-Directional FGM Non-Uniform Mindlin’s Circular Plate Subjected to In-Plane Peripheral Loading

An analysis has been presented of the effect of elastic foundation and uniform in-plane peripheral loading on the natural frequencies and mode shapes of circular plates of varying thickness exhibiting bi-directional functionally graded characteristics, on the basis of first order shear deformation theory. The material properties of the plate are varying following a power-law in both the radial and transverse directions. The numerical solutions of the coupled differential equations leading the motion of simply supported and clamped plates acquired by using Hamilton’s principle, is attained by harmonic differential quadrature method. The effect of different plate parameters namely gradient index, heterogeneity parameter, density parameter, taper parameter and thickness parameter is illustrated on the vibration characteristics for the first three modes of vibration for various values of in-plane peripheral loading parameter together with foundation parameter. Critical buckling loads in compression are calculated for both the boundary conditions by putting the frequencies to zero. The reliability of the present technique is confirmed by comparing the results with exact values and results of published work.

Buckling analysis of functionally graded thin square plates with triangular cut-out subjected to uni-axial loads

In this article, the buckling behaviour of simply supported functionally graded (FG) thin square plates with triangular cut-out subjected to mechanical loading is investigated. Both uniform and non-uniform (linear triangle and trapezoidal) in-plane uniaxial compressive loadings are considered. Finite element method based on first order shear deformation theory (FSDT) is used to find out the critical buckling loads with the help of ANSYS – a commercial finite element software. Eight noded shell element with six degrees of freedom at each node is used for discretization. The material properties of the plates are assumed to vary along the thickness according to a simple power law distribution in terms of volume fraction exponent. The effects of volume fraction exponent and cut-out size for the various types of loads considered are presented. Finally, from the results it is observed that the buckling load capacity decreases with the increase in volume fraction exponent and cut-out size for the considered types of loading

A screw dislocation near a non-elliptical piezoelectric inhomogeneity with internal uniform electroelastic field

Conformal mapping and analytic continuation are employed to prove the existence of an internal uniform electroelastic field inside a non-elliptical piezoelectric inhomogeneity interacting with a screw dislocation. We focus specifically on the case when the piezoelectric matrix surrounding the inhomogeneity is subjected to uniform remote anti-plane mechanical and in-plane electrical loading and a constraint is imposed between the remote loading and the screw dislocation. The constraint can be expressed in a relatively simple decoupled form by utilizing orthogonality relationships between two corresponding eigenvectors. The internal uniform electroelastic field is found to be independent of the presence of the screw dislocation; moreover, it can be expressed in decoupled form.

A new approach for electro-elastic analysis of piezoelectric fiber composites with arbitrary shaped inclusions under anti-plane shear and in-plane electric loadings

Electro-elastic analysis of piezoelectric fiber composites with arbitrary shaped inclusions subjected to far field uniform anti-plane shear and in-plane electric loadings is performed in this study in order to increase the electromechanical performance and protect the vulnerable piezoelectric inclusion from electromechanical damage. A three-phase model with a representative volume element of piezoelectric fiber composites with arbitrary shaped inclusions is proposed based on the generalized self-consistent theory. Meanwhile, a new conformal mapping method based on Cauchy integral and series expansion is also proposed to map an arbitrary two-connected domain into an annular domain. Both of the electro-elastic responses and the effective moduli of the piezoelectric fiber composites with arbitrary shaped inclusions for the anti-plane problem are obtained using the complex potentials in series form and the homogenization strategy. Comparisons with literature show a good agreement. Additionally, the effects of the fiber shape and the fiber volume fraction on the normal shear stress and the tangential electric field distributions outside the piezoelectric fiber, and the effective moduli are investigated using the proposed model and method. It is shown that the normal shear stresses and the tangential electric fields outside piezoelectric fiber have intense fluctuations at the corner of the inclusion. Both of the electro-elastic responses and the effective moduli of piezoelectric fiber composites become larger as the shape of piezoelectric inclusion sharpens and the fiber volume fraction increases. Furthermore, the proposed model and method can be used for determining the appropriate shape and volume fraction of the piezoelectric inclusion to significantly reduce stress and electric field concentrations around the inclusion and optimize the electromechanical performance of piezoelectric fiber composites.

Stress concentration around an elliptical hole with surface tension based on the original Gurtin–Murdoch model

It is widely appreciated that surface tension plays a central role in the mechanics of soft solids such as gels. The original Gurtin–Murdoch (G–M) surface model incorporated the contribution of surface tension also via the dependence of surface stress on surface deformation. In many subsequent researches, this contribution was neglected presumably to simplify the analysis of the corresponding models of deformation. Here, we restore the original G–M surface model and re-examine the problem of an elliptical hole with a constant surface tension in an elastic infinite plane under a uniform remote in-plane loading. Semi-analytic solutions for the full stress field are derived based on series expansion techniques. Numerical examples are presented to compare the original G–M model with its simplified counterpart (which neglects the deformation-induced change in the configuration of the hole) in predicting the hoop stress around the hole. We show that in the absence of remote loading the use of the simplified G–M model generally overestimates the surface tension-induced residual hoop stress around the hole especially when the hole is slender. In the context of the original G–M model, we find that the presence of surface tension may intensify the remote loading-induced hoop stress concentration significantly at the endpoints of the hole's major axis as the hole becomes slender (for example, in the case of a remote tensile loading perpendicular to the major axis of the hole), which certainly does not happen for the simplified G–M model.

Buckling analysis of FGM plates under uniform, linear and non-linear in-plane loading

The paper investigates the buckling responses of functionally graded material (FGM) plate subjected to uniform, linear, and non-linear in-plane loads. New nonlinear in-plane load models are proposed based on trigonometric and exponential function. Non-dimensional critical buckling loads are evaluated using non-polynomial based higher order shear deformation theory. Navier’s method, which assures minimum numerical error, is employed to get an accurate explicit solution. The equilibrium conditions are determined utilizing the principle of virtual displacements and material property are graded in the thickness direction using simple Voigt model or exponential law. The present formulation is accurate and efficient in analyzing the behavior of thin, thick and moderately thick FGM plate for buckling analysis. It is found that with the help of displacement-buckling load curve, critical buckling load can be derived and maximum displacement due to the instability of inplane load can be obtained. Also, the randomness in the values of transverse displacement due to inplane load increases as the extent of uniformity of the load on the plate is disturbed. Furthermore, the parametric varying studies are performed to analyse the effect of span-to-thickness ratio, volume fraction exponent, aspect ratio, the shape parameter for non-uniform inplane load, and non-dimensional load parameter on the non-dimensional deflections, stresses, and critical buckling load for FGM plates.