Two-dimensional Elasticity Solution Research Articles

Transverse stretching vibration analysis of cross-ply composite and sandwich beams by means of an equivalent single-layer theory

Due to largely different stiffness between the face sheets and the core, the vibration modes corresponding to the transverse stretching displacement will occur in the sandwich plates. The third-order models will encounter difficulty to reasonably produce these frequencies corresponding to transverse stretching displacement modes, which will influence engineering design. Therefore, a global-local higher-order theory including transverse stretching effect has been proposed to study transverse stretching vibration, which can meet in advance the continuity conditions and the free surface conditions of transverse shear stresses. In the light of Hamilton’s principle, the equation of motion is presented, and analytical solutions of the simply-supported beams can be acquired by using Navier’s technique. In order to verify the capability of the proposed model, free vibration of composite beams is firstly investigated and the results have been compared to two-dimensional elasticity solutions. After the proposed model is assessed by two-dimensional elasticity solutions, the proposed model is extended to study dynamic behaviors corresponding to the transverse stretching displacement vibration modes, which will be verified by utilizing the three-dimensional finite element (3 D-FEM). Furthermore, variation of natural frequencies of the sandwich beams is also investigated by altering the thickness of the face sheets, which can further illustrate the capability of the proposed model.

Flexure of a rigidly clamped orthotropic sandwich plate strip – An elasticity solution using superposition method

A simple two-dimensional elasticity solution is presented here for flexure of an infinite sandwich strip with rigidly clamped ends; it is based on the superposition of a strong form solution for a simply supported strip under transverse loading and a Ritz solution for end-loading. Results useful for future comparisons are presented in tabular form. On the basis of these results, the accuracy of classical plate theory and first-order shear deformation theory is critically examined. Finally, the difference between rigidly clamped end conditions and a softer version of clamped conditions is highlighted with reference to far-field response.

Upper and Lower Bounds for the Parameters of One-Dimensional Theories for Sandwich Fracture Specimens

AbstractUpper and lower bounds for the parameters of one-dimensional theories used in the analysis of sandwich fracture specimens are derived by matching the energy release rate with two-dimensional elasticity solutions. The theory of a beam on an elastic foundation and modified beam theory are considered. Bounds are derived analytically for foundation modulus and crack length correction in single cantilever beam (SCB) sandwich specimens and verified using accurate finite element results and experimental data from the literature. Foundation modulus and crack length correction depend on the elastic mismatch between face sheets and core and are independent of the core thickness if this is above a limit value, which also depends on the elastic mismatch. The results in this paper clarify conflicting results in the literature, explain the approximate solutions, and highlight their limitations. The bounds of the model parameters can be applied directly to specimens satisfying specific geometrical/material ratios, which are given in the paper, or used to support and validate numerical calculations and define asymptotic limits.

Two-Dimensional Mechanical Behavior Analysis of Multilayered Solids Subjected to Surface Contact Loading Based on a Semi-Analytical Method

In this paper, a two-dimensional semi-analytical method is developed for the mechanical behavior analysis of multilayered solids subjected to surface contact loading, which is indispensable for realizing an optimized tribological performance from the mechanical behavior point of view. Firstly, the explicit analytical frequency response functions of the multilayered solid are derived in a recursive form by analytically solving a system of linear equations established according to the boundary conditions and the interface continuous conditions. Then, the two-dimensional elastic field solution in the subsurface of multilayered solids in the space domain is converted from its corresponding frequency response functions by employing a numerical conversion method based on the inverse fast Fourier transformation. The present method is validated by comparing with the solution given by other methods. Lastly, the stress analysis of multilayered coatings with various structure layouts and various layer number of the multilayers were performed with the present method.

Vibration analysis of functionally graded graphene oxide-reinforced composite beams using a new Ritz-solution shape function

In this study, a new Ritz-solution shape function, in the form of a combination of polynomials and general exponential functions for various boundary conditions, is constructed from two-dimensional elasticity solutions using an inverse method. In conjunction with an improved third-order shear deformation theory, a reliable and accurate model is developed for the analysis of the mechanical behavior of composite beams. Free and forced vibrations of a functionally graded (FG) polymer nanocomposite beam reinforced with a low content of graphene oxide (GO) and excited by a moving load with a constant velocity are investigated. The weight fraction of the GOs is assumed to vary continuously and smoothly in the thickness direction. The modified Halpin–Tsai micromechanics model is used to evaluate the effective Young’s modulus of the FG GO-reinforced composites. The governing equations of motion are derived using the Lagrange method. The Newmark-β method is adopted to solve the forced vibration problem of a beam subjected to a moving load. A parametric study is conducted to demonstrate the effects of GO distribution patterns, weight fraction, and size on the vibration response of the nanocomposite beam with various classical boundary conditions.

Time-dependent behavior of laminated functionally graded beams bonded by viscoelastic interlayer based on the elasticity theory

A two-dimensional (2-D) elasticity solution is developed to investigate the time-dependent response of laminated functionally graded beam with viscoelastic interlayer. The elastic modulus in each functionally graded layer varies through the thickness following an exponential function, and the mechanical property of each layer is described by the exact 2-D elasticity equations. Viscoelastic property in the interlayer is simulated by the Maxwell–Wiechert model, in which the memory effect of it is neglected. By virtue of the recursive matrix method, the solution of stress and deformation can be obtained efficiently. The comparison studies indicate that the relative error of the Euler–Bernoulli solution is slight for slender beam with small gradient factor, but it significantly increases as the beam thickness or the gradient factor grows. The finite element solution is close to the present one with large number of sub-layers in FE modeling and small gradient factor of FG layer, while the error of FE solution increases as the number of sub-layers decrease or the gradient factor increases. Besides, the influences of material parameters upon the time-dependent behavior and the optimization for the stress and deformation in the beam are discussed in detail.

Two-dimensional elasticity solution for free vibration of simple-supported beams with arbitrarily and continuously varying thickness

This paper studies the analytical solution for the vibration of simply supported beams with arbitrarily and continuously varying thickness based on the two-dimensional elasticity theory. The general expression of stress function, which exactly satisfies the governing differential equations and the boundary conditions, is derived. Frequency equation governing the free vibration of beams with variable thickness can be obtained by using the Fourier sinusoidal series expansion on the upper and lower surfaces of the beam. The present solution method ensures a rapid convergence and meets the need of high accuracy in modern precise instruments. Several examples are provided to show the application of the proposed solution method which can be used to assess the validity of various approximate solutions and numerical methods for the beams with arbitrarily and continuously varying thickness.

Shielding or anti-shielding effects of solute hydrogen near a finite length crack: A new possible mechanism of hydrogen embrittlement

A two-dimensional elastic solution for the stress field of a dilatation line near an elliptical void in an infinite matrix is obtained. The dilatation line is used to represent a row of equally spaced solute atoms. This elastic solution is employed to study the shielding or anti-shielding effect of solute atoms near a finite length crack. The stress intensity factor of stress field induced by a dilatation line is analytically obtained. It is shown that whether solute atoms have a shielding or anti-shielding effect on the crack tips depends on the position of solute atoms. An equaling zero curve is defined, on which the solute dilatation line induces zero Mode I stress intensity factor at the crack-tip. It is found that the shape of equaling zero curve is independent of any model parameters. The shielding and anti-shielding effects of a lot of solute hydrogen (hydrogen atmospheres) are studied through an integral scheme. The hydrogen concentration field is assumed to equilibrate with external or internal stress fields. We consider two cases. One case is that an externally uniaxial load in the direction vertical to crack is applied. Another case is that there exists an edge dislocation near crack. Based on these studies, a new possible mechanism of hydrogen embrittlement is proposed, which can be referred to as hydrogen induced anti-shielding crack mechanism.

On the interaction of solute atoms with circular inhomogeneity and edge dislocation

A two-dimensional elastic solution for the stress field of a line dilatation near a circular inhomogeneity is obtained. A line dilatation is used to represent a row of solute atoms, referred to as a solute rod. The image effect is investigated with an especial attention. It is shown that a soft inhomogeneity attracts solute rods and a hard inhomogeneity repels solute rods. Based on this, the interactions of solute atoms with inhomogeneity-edge dislocation are studied by inserting more solute atoms into the matrix one-by-one, with an especial focus on the distribution of solute atoms (rods) around inhomogeneity and edge dislocation. As two typical applications of the present method, two simple cases are analyzed in details, with the inhomogeneity considered as a nano-void and a hard nano-fiber, respectively. It is shown that the equilibrium concentration distribution of solutes near an inhomogeneity is non-local, which not only depends on the local hydrostatic stress but also on the hydrostatic stress in the adjacent region. Besides, the shear stress on an edge dislocation, generated by hydrogen atmospheres between a circular nano-void/fiber and the dislocation, is calculated numerically. The results show that the hydrogen atmosphere around dislocations can shield the long-range elastic interaction between dislocations and other internal stress sources, supporting the so-called hydrogen enhanced localized plasticity.

Free and Forced Vibration of Laminated Composite Beams

A new higher-order zig-zag theory is proposed for the free and forced vibration analysis of laminated composite and sandwich beams with different thickness at each ply. In-plane displacements in the proposed model nonlinearly distribute through the thickness and exhibit an abrupt discontinuity of slope at interfaces, which coincide with three-dimensional analysis. Differing from previous approaches, the transverse shear stress satisfies the continuity condition at interface by integrating the three-dimensional equilibrium equations, whereas the higher-order derivatives of displacement parameters have been eliminated using the three-field Hu-Washizu (HW) mixed variational principle. Moreover, the accurate transverse shear stresses are introduced in the equations of motion by using Hamilton’s principle, which actively impact the accurate prediction of natural frequencies of the laminated composite beams with arbitrary layouts. The performance of the proposed model is tested with different examples. It is observed that the present model gives most accurate results with respect to two-dimensional elasticity solutions for the free and forced vibration analysis compared with the existing higher-order models recently reported in the literature.

Postbuckling up to Collapse of Polar Orthotropic Linearly Elastic Rings Subjected to External Pressure

Post-buckling and collapse phenomenon of a thick closed ring subjected to external pressure on its outer surface is analyzed in this research. Ring is made of a linearly elastic polar orthotropic material. Constitutive law of linear elasticity under plane stress conditions is used. Virtual work principle is implemented to obtain the governing equations. Unlike the available noncompressible ring theories, a two-dimensional elasticity solution including the complete nonlinear Green strain field is used. The finite element method is used to solve the highly nonlinear coupled equilibrium equations. The well-known Newton–Raphson iterative technique is applied to the matrix representation of the equilibrium equations to trace the post-buckling response of the ring up to the collapse point where two antipodal points on the interior side of the ring are collided with each other. It is shown that, including the complete Green strain field is necessary in accurate estimation of the post-buckling path, especially for higher load levels and collapse load. Furthermore, orthotropic rings under external pressure load follow the bifurcation type of buckling accompanied with a stable post-buckling path.

Two-dimensional elasticity solution of elastic strips and beams made of functionally graded materials under tension and bending

This paper presents a theoretical approach to solve elastic problems of functionally graded materials (FGMs). For FGMs with exponential gradient, based on a two-dimensional theory of elasticity, a governing equation is derived by means of the Airy stress function method together with the strain compatibility equation. Simple uniaxial tension and bending are solved. For an FGM layer with transversely and/or vertically varying material properties, stress distribution and strain field under simple tension are determined according to two different assumptions. The obtained results indicate that for a thin elastic layer of thickness-wise gradient as a transition zone linking two dissimilar materials, there is a horizontal displacement difference across the transition zone due to mismatch of the material properties. In particular, when the thickness of the FGM layer reduces to zero, the horizontal displacement difference has a severe mismatch across the interface of two perfectly bonded dissimilar materials. An FGM beam subjected to a bending moment is also analyzed. The normal stress exhibits a nonlinear distribution and may arrive at its maximum tensile stress inside the beam, not at the surface. The obtained elasticity solution is useful for better understanding of the mechanical behaviors of FGMs subjected to different combined loads.

Two-dimensional elasticity solution for bending of functionally graded beams with variable thickness

This paper studies the stress and displacement distributions of functionally graded beam with continuously varying thickness, which is simply supported at two ends. The Young’s modulus is graded through the thickness following the exponential-law and the Poisson’s ratio keeps constant. On the basis of two-dimensional elasticity theory, the general expressions for the displacements and stresses of the beam under static loads, which exactly satisfy the governing differential equations and the simply supported boundary conditions at two ends, are analytically derived out. The unknown coefficients in the solutions are approximately determined by using the Fourier sinusoidal series expansions to the boundary conditions on the upper and lower surfaces of the beams. The effect of Young’s modulus varying rules on the displacements and stresses of functionally graded beams is investigated in detail. The two-dimensional elasticity solution obtained can be used to assess the validity of various approximate solutions and numerical methods for the aforementioned functionally graded beams.

Spectral finite element based on an efficient layerwise theory for wave propagation analysis of composite and sandwich beams

In this paper, we present a spectral finite element model (SFEM) using an efficient and accurate layerwise (zigzag) theory, which is applicable for wave propagation analysis of highly inhomogeneous laminated composite and sandwich beams. The theory assumes a layerwise linear variation superimposed with a global third-order variation across the thickness for the axial displacement. The conditions of zero transverse shear stress at the top and bottom and its continuity at the layer interfaces are subsequently enforced to make the number of primary unknowns independent of the number of layers, thereby making the theory as efficient as the first-order shear deformation theory (FSDT). The spectral element developed is validated by comparing the present results with those available in the literature. A comparison of the natural frequencies of simply supported composite and sandwich beams obtained by the present spectral element with the exact two-dimensional elasticity and FSDT solutions reveals that the FSDT yields highly inaccurate results for the inhomogeneous sandwich beams and thick composite beams, whereas the present element based on the zigzag theory agrees very well with the exact elasticity solution for both thick and thin, composite and sandwich beams. A significant deviation in the dispersion relations obtained using the accurate zigzag theory and the FSDT is also observed for composite beams at high frequencies. It is shown that the pure shear rotation mode remains always evanescent, contrary to what has been reported earlier. The SFEM is subsequently used to study wavenumber dispersion, free vibration and wave propagation time history in soft-core sandwich beams with composite faces for the first time in the literature. (C) 2014 Elsevier Ltd. All rights reserved.

The Two-Dimensional Elasticity Solution for the Buckling of a Thick Orthotropic Ring Under External Pressure Loading

This paper is concerned with the 2D elasticity solution for the buckling of a thick orthotropic ring under external hydrostatic pressure loading. The bifurcation buckling problem is first formulated using two methods, distinguished by the manner in which the external work done by the pressure loading during the buckling transition is treated. In doing so, the correct buckling equations and associated traction boundary conditions are derived. The resulting sets of equations and associated boundary conditions are then cast in a weak form, amenable to a numerical solution using the finite element method. The necessity of using the correct pairs of energetically conjugate stress and strain measures for the buckling problem is pointed out. Errors in using the incorrect traction boundary condition and terms that influence the buckling load and that have been omitted in popular commercial codes are pointed out and their significance in influencing the buckling load is identified. Results from the present two-dimensional analysis to predict the critical pressure are compared with previous theoretical results. The formulation and results presented here can be used as the correct benchmark solution to establish the accuracy in computing the buckling load of thick orthotropic composite structures, of contemporary interest, due to the increased use of thick-walled composite shell type structures in diverse engineering applications.

A new two-dimensional refined higher order theory for in-plane free vibrations of sandwich and composite thick curved beams with flexible cores

Closed-form formulations of two-dimensional refined higher order beam theories and higher order beam theories for the free vibration analysis of simply supported cross-ply laminated composite and sandwich curved beams are presented. The present refined higher order beam theory analysis incorporates a trapezoidal shape factor that arises due to the fact that the stresses through the thickness of the beam are integrated over a curved surface. Therefore, the present theory refines other higher order beam theories established hitherto for free vibration analysis of strongly thick circular curved beams. The stress–strain relationship is derived from an orthotropic lamina in a three-dimensional state of stress. Numerical results are presented for the natural frequencies of laminated composite and sandwich thin/thick straight beams and shallow/deep and moderately/strongly thick curved beams. The closed-form solutions presented herein are compared with the available exact two-dimensional elasticity and analytical solutions in the available literature and excellent agreement is obtained.

Two-dimensional elasticity solution for transient response of simply supported beams under moving loads

A semi-analytical analysis for the transient elastodynamic response of an arbitrarily thick simply supported beam due to the action of an arbitrary moving transverse load is presented, based on the linear theory of elasticity. The solution of the problem is derived by means of the powerful state space technique in conjunction with the Laplace transformation with respect to the time coordinate. The inversion of Laplace transform has been carried out numerically using Durbin’s approach based on Fourier series expansion. Special convergence enhancement techniques are invoked to completely eradicate spurious oscillations and obtain uniformly convergent solutions. Detailed numerical results for the transient vibratory responses of concrete beams of selected thickness parameters are obtained and compared for three types of harmonic moving concentrated loads: accelerated, decelerated and uniform. The effects of the load velocity, pulsation frequency and beam aspect ratio on the dynamic response are examined. Also, comparisons are made against solutions based on Euler–Bernoulli and Timoshenko beam models. Limiting cases are considered, and the validity of the model is established by comparison with the solutions available in the existing literature as well as with the aid of a commercial finite element package.

Limits of Application for Two-Dimensional Elasticity Solutions in Plates Loaded in Tension and Out-of-Plane Bending

Elasticity solutions of plates have been widely researched and used in practice. However, elasticity solutions of plane problems are applicable to structures only under the premise that such structures are very thin. The precise values of thickness for which elasticity solutions are valid have not been studied in detail. It is necessary to clarify the range of appropriate thickness values in order to apply two-dimensional elasticity solutions properly. In this study, we use finite element analyses to study three flat-plate models each containing a U-shaped hole of the same radius, and compared the three-dimensional finite element solutions with two-dimensional elasticity solutions. From the results, the limits of application of two-dimensional elasticity solutions are discussed.

Two-dimensional elasticity solutions for temperature-dependent in-plane vibration of FGM circular arches

Abstract Temperature-dependent in-plane vibration of functionally graded (FGM) circular arches based on the two-dimensional theory of elasticity is investigated. An analytical solution using the state space formulation and Fourier series expansion is obtained for a simply supported circular arch. For such functionally graded arches, the state equation has variable coefficients. Because a definite, continuously varying FG model through the thickness is impractical if not impossible, an approximate laminate model is constructed to derive an asymptotic solution through the thickness direction. The transfer relationship between the state vectors at the inner and outer surfaces is ultimately obtained by considering the continuity conditions at the artificial interfaces. The new formulation is validated by comparing some numerical solutions with established results in open literature, such as functionally graded straight beams, curved sandwich beams and laminated thick circular arches. Effective material properties are predicted using the Mori–Tanaka model and taken as temperature-dependent. Effects of the gradient index, temperature and geometric parameters, i.e. the curvature, length-to-thickness ratio, subtended angle, on the vibration frequency are analyzed and discussed.

An explicit elastic solution for a brittle film with periodic cracks

A two-dimensional explicit elastic solution is derived for a brittle film bonded to a ductile substrate through either a frictional interface or a fully bonded interface, in which periodically distributed discontinuities are formed within the film due to the applied tensile stress in the substrate and consideration of a “weak form stress boundary condition” at the crack surface. This solution is applied to calculate the energy release rate of three-dimensional channeling cracks. Fracture toughness and nominal tensile strength of the film are obtained through the relation between crack spacing and tensile strain in the substrate. Comparisons of this solution with finite element simulations show that the proposed model provides an accurate solution for the film/substrate system with a frictional interface; whereas for a fully bonded interface it produces a good prediction only when the substrate is not overly compliant or when the crack spacing is large compared with the thickness of the film. If the section is idealized as infinitely long, this solution in terms of the energy release rate recovers Beuth’s exact solution for a fully cracked film bonded to a semi-infinite substrate. Interfacial shear stress and the edge effect on the energy release rate of an asymmetric crack are analyzed. Fracture toughness and crack spacing are calculated and are in good agreement with available experiments.