Mesh‐free models for static analysis of smart laminated composite beams

This paper is concerned with the development of mesh‐free models for the static analysis of smart laminated composite beams. The overall smart composite beam is composed of a laminated substrate composite beam and a piezoelectric layer attached partially or fully at the top surface of the substrate beam. The piezoelectric layer acts as the distributed actuator layer of the smart beam. A layer‐wise displacement theory and an equivalent single‐layer theory have been used to derive the models. Several cross‐ply substrate beams are considered for presenting the numerical results. The responses of the smart composite beams computed by the present new mesh‐free model based on the layer‐wise displacement theory excellently match with those obtained by the exact solutions. The mesh‐free model based on the equivalent single‐layer theory cannot accurately compute the responses due to transverse actuation by the piezoelectric actuator. The models derived here suggest that the mesh‐free method can be efficiently used for the numerical analysis of smart structures. Copyright © 2016 John Wiley & Sons, Ltd.     

A new FE model based on higher order zigzag theory for the analysis of laminated sandwich beam with soft core

A new finite element (FE) model has been developed based on higher order zigzag theory (HOZT) for the static analysis of laminated sandwich beam with soft core. In this theory, the in-plane displacement variation is considered to be cubic for both the face sheets and the core. The transverse displacement is assumed to vary quadratically within the core while it remains constant in the faces beyond the core. The proposed model satisfies the condition of transverse shear stress continuity at the layer interfaces and the zero transverse shear stress condition at the top and bottom of the beam. The nodal field variables are chosen in an efficient manner to overcome the problem of continuity requirement of the derivatives of transverse displacements. A C₀ quadratic beam finite element is implemented to model the HOZT for the present analysis. Numerical examples covering different features of laminated composite and sandwich beams are presented to illustrate the accuracy of the present model. Many new results are also presented which should be useful for future research.     

Analytical solution for composite layered beam subjected to uniformly distributed load

The article presents an analytical theory for multilayered composite beams subjected to transverse uniformly distributed loads. The formulation is based on a layerwise model characterized by third-order approximation of the axial displacements and fourth-order approximation of the transverse displacements. The layerwise kinematical model is rewritten in terms of generalized variables. The beam equilibrium equations, expressed in terms of stress resultant, allow writing the boundary value governing problem. The layerwise fields are obtained by postprocessing steps. The main advantage is to ensure the accuracy level associated to the layerwise formulations preserving the computational efficiency of the equivalent-single-layer theories.     

A generalized micromechanics model with shear-coupling

Summary A unified mathematical framework for a higher-order transverse shear-normal stress coupled micromechanical model is presented. The model is developed based on the analysis of a repeating unit cell in a doubly periodic array of fibers. The behavior in subregions within the unit cell is modeled using an expansion for the displacement field. The order and form of the displacement expansions in the subregions are arbitrary. The higher-order terms in the displacement expansion result in coupling between the transverse shearing and the normal deformation responses (shear coupling). The formulation is sufficiently general to allow generic elastic, plastic, viscoelastic, viscoplastic, or damage constitutive models (within the context of infinitesimal strain theory) for history-dependent behavior to be incorporated into the micromechanical framework. The proposed approach is analytical and provides closed-form expressions for the effective macroscopic behavior of a continuous fiber composite.The model is validated by comparison with existing micromechanics models. The agreement between the predicted effective moduli obtained from the current model and other existing models indicates that the current formulation accurately predicts the effective elastic behavior of a composite. Furthermore, comparison with existing data for the local elastic stress distributions around the inclusion indicates that the current model correctly captures the trends and magnitudes in these distributions. The predictions obtained from the current theory are shown to be more accurate than the corresponding MOC predictions. The ability to more accurately capture the spatial stress distributions can be directly attributed to the incorporation of the shear-coupling phenomena.Finally, the influence of the presence of shear coupling on the local field distributions is considered for the simple macroscopic loading cases of transverse tension and transverse shearing. It is shown that signficant coupling between the local transverse shearing and normal deformation responses exists even when the composite is subjected to a macroscopically simple loading field. The existence of this coupling has potentially significant implications in the implementation of history-dependent constitutive models.     

A higher-order finite element for analysis of composite laminated structures

A new six-node higher-order triangular composite layered shell finite element with six degrees of freedom at each node is presented. With respect to the inplane variables, the in-plane and the out-of-plane displacement fields of the element are quadratic and cubic respectively. By using Utku's method (AFFDL-TR-71-160, Air Force Third Conf., Wright Paterson, Ohio, 1971), the transverse shear strain energy is computed directly from the displacement field rather than from the stress couple field. Some typical bending problems for composite laminated beams and plates with different stack sequences are analyzed. Excellent agreements are obtained when compared to the exact solutions, the first order shear deformation theory (FSDT), the higher order shear deformation theory (HSDT) and some other existing finite element models. ‘Shear locking’ is avoided when the plate is thin.     

A refined higher-order composite box beam theory

A three-dimensional theory is developed to model composite box beams with arbitrary wall thicknesses. The theory, which is based on a refined displacement field, approximates the three-dimensional elasticity solution so that the beam cross-sectional properties are not reduced to one-dimensional beam parameters. Both in-plane and out-of-plane warping are included automatically in the formulation. The model can accurately capture the transverse shear stresses through the thickness of each wall while satisfying stress-free boundary conditions on the inner and outer surfaces of the beam. Numerical results are presented for beams with varying wall thicknesses and aspect ratios. The static results are correlated with available experimental data and show excellent agreement. Results presented for thick-walled box beams show the importance of including transverse shear in the formulation and the difficulty of defining a ‘beam’ twist for the entire cross-section.     

Enhanced magnetoelectric effect in multiferroic composite beams due to flexoelectricity and transverse deformations

This paper is concerned with the exploration of the role of transverse normal and shear deformations on enhancing the magnetoelectric (ME) coefficient of multiferroic bilayer composite beams composed of a piezoelectric layer and a piezomagnetic layer. Analytical models have been derived based on the displacement field which accounts for both the transverse normal and shear deformations, Timoshenko beam theory and Euler Bernoulli beam theory. The induced flexoelectricity in the piezoelectric layer due to axial strain gradient and transverse shear strain gradient has also been taken into consideration for estimating the ME coefficient. It has been found that the contribution of transverse normal strain in the piezoelectric layer for enhancing the ME coefficient is significantly larger than that due to axial strain, transverse shear strain and flexoelectricity. For the particular values of the thicknesses of the piezoelectric layer and the piezomagnetic layer, the ME coefficient remains invariant for both thick and thin multiferroic composite beams.     

增强型Reddy层合梁理论与热应力分析

为推广Reddy理论准确分析复合材料层合/夹层结构的热变形和应力, 通过使用横法向热变形和自由表面条件, 发展了增强型Reddy层合梁理论。虽然考虑了横法向应变, 增强型Reddy理论位移变量数与Reddy理论相同。用虚功原理推导了复合材料层合梁平衡方程, 并分析了简支复合材料层合/夹层梁热膨胀问题。数值结果表明, 增强型Reddy理论能准确分析复合材料层合/夹层结构热膨胀问题, 而Reddy理论分析热膨胀问题误差较大。     

Fracture-mechanics failure criteria for RC beams strengthened with FRP strips––a simplified approach

A simplified approximation approach for the evaluation of a fracture mechanics based criterion for the edge delamination failure of reinforced concrete beams strengthened with externally bonded composite materials is presented. The proposed approach is based on evaluation of the energy release rate (ERR) through the virtual crack extension method using various analytical and numerical stress analysis models. The investigated models include the high-order model, two types of “elastic foundation” or “springs” models, a simplified beam model, and finite elements analysis. The stress and displacement fields, the governing equations and their closed form solutions, and the expressions for the release rate of the total potential energy of the various models are presented. The proposed approach sets up the basis for an energetic failure criterion, in which the ERR is compared to the specific fracture energy of the bonded system. This criterion replaces the traditional allowable stress approach in describing the initiation and stable or unstable growth of the delamination crack. The capabilities of the proposed approach and its ability to evaluate the ERR through simplified and approximated models is investigated numerically. The accuracy of the simplified approach is numerically examined through comparison with the J-integral formulation. Numerical results in terms of stresses near the edge of the bonded strip, the ERR associated with initiation and growth of the interfacial crack, and the critical loads and crack lengths are presented. The paper closes with a summary and conclusions.     

法向应力和法向应变对复合材料层板自由振动基础频率的影响

一种包含法向应力,法向应变和横向剪切影响的复合材料层板自由振动的新位移模式在本文中提出.通过应用层板自由振动的Rayleigh-Ritz分析,比较了在简支边界条件下复合材料层板自由振动的YNS理论解[1]和本文新位移模式解,从而得出在简支边界条件下法向应力和法向应变对复合材料层板自由振动基础频率的影响.     

Conference diary

A variational higher-order theory for bending and stretching of linearly elastic orthotropic beams including the deformations due to transverse shearing and stretching of the transverse normal fibre is presented. The theory assumes a linear distribution for the longitudinal displacement and a parabolic variation of the transverse displacement across the thickness. Additionally, independent expansions are introduced for the through-thickness displacement gradients with the requirement of a least-square compatibility for the transverse strains and the satisfaction of exact stress boundary conditions at the top/bottom beam surfaces. The theory is shown to be well suited for finite element development requiring simple C⁰- and C^?1- continuous displacement interpolation fields. To demonstrate the computational utility of the theory, a simple two-node stretching-bending finite element is formulated. The analytic and finite element results are obtained for a simple bending problem for which an exact elasticity solution is available. It is shown that the inclusion of the transverse normal deformation in the present theory enables improved displacement, strain and stress predictions, particularly, in the analysis of deep beams.     

A simple finite element formulation of a higher-order theory for unsymmetrically laminated composite plates

A higher-order theory which satisfies zero transverse shear stress conditions on the bounding planes of a generally laminated fibre-reinforced composite plate subjected to transverse loads is developed. The displacement model accounts for non-linear distribution of inplane displacement components through the plate thickness and the theory requires no shear correction coefficients. A C∘ continuous displacement finite element formulation is presented and the coupled membrane-flexure behaviour of laminated plates is investigated. The nodal unknowns are the three displacements, two rotations and two higher-order functions as the generalized degrees of freedom. The simple isoparametric formulation developed here is capable of evaluating transverse shears and transverse normal stress accurately by using the equilibrium equations. The accuracy of the nine-noded Lagrangian quadrilateral element is then established by comparing the present results with the closed-form, three-dimensional elasticity and other finite element available solutions.     

C<Superscript>0</Superscript>-type global–local theory with non-zero normal strain for the analysis of thick multilayer composite plates

In this paper, an accurate and efficient C⁰-type third-order global–local model incorporating effects of the transverse normal strain is proposed to study the thermal/mechanical behaviors of thick multilayer cross-ply plates. Transverse displacement is assumed to be a linear distribution through the thickness direction, for which the normal strain could be readily computed. Based on the interlaminar continuity conditions of in-plane displacement and transverse shear stresses, layer-dependent variables could be reduced. Employing shear stress free condition at the upper and the lower surfaces, derivatives of transverse displacement are eliminated from the displacement field, so that C⁰ interpolation functions are only required for the finite element implementation. As a result, the number of variables is independent of the number of layers of the laminate. To assess the proposed model, the classical quadratic eight-node isoparametric element is used for the interpolation of all the displacement parameters defined at each nodal point on the composite plate. Comparing with various existing composite plate models, it is found that simple C⁰ finite elements with non-zero normal strain could produce accurate deformations and stresses of thick multilayer composite plates subjected to thermal and mechanical loads.     

Layerwise modeling of progressive damage in fiber-reinforced composite laminates

This paper investigates the effects of discrete layer transverse shear strain and discrete layer transverse normal strain on the predicted progressive damage response and global failure of fiber-reinforced composite laminates. These effects are isolated using a hierarchical, displacement-based 2-D finite element model that includes the first-order shear deformation model (FSD), type-I layerwise models (LW1) and type-II layerwise models (LW2) as special cases. Both the LW1 layerwise model and the more familiar FSD model use a reduced constitutive matrix that is based on the assumption of zero transverse normal stress; however, the LW1 model includes discrete layer transverse shear effects via in-plane displacement components that are C ⁰ continuous with respect to the thickness coordinate. The LW2 layerwise model utilizes a full 3-D constitutive matrix and includes both discrete layer transverse shear effects and discrete layer transverse normal effects by expanding all three displacement components as C ⁰ continuous functions of the thickness coordinate. The hierarchical finite element model incorporates a 3-D continuum damage mechanics (CDM) model that predicts local orthotropic damage evolution and local stiffness reduction at the geometric scale represented by the homogenized composite material ply. In modeling laminates that exhibit either widespread or localized transverse shear deformation, the results obtained in this study clearly show that the inclusion of discrete layer kinematics significantly increases the rate of local damage accumulation and significantly reduces the predicted global failure load compared to solutions obtained from first-order shear deformable models. The source of this effect can be traced to the improved resolution of local interlaminar shear stress concentrations, which results in faster local damage evolution and earlier cascading of localized failures into widespread global failure.     

On approximate solutions for the deformation of plane anisotropic beams

Plane deformation of anisotropic beams with narrow rectangular cross sections exhibits coupling of stretching, bending and transverse shearing. For anisotropic cantilever beams with a stiff end-cap under end forces and an end couple, assessments were made for approximate solutions by comparing these with numerically exact finite element (FE) solutions. Specific attention is given to point-wise or approximate satisfaction of the end-fixity conditions. As approximate methodologies, (i) the elementary polynomial form of Airy's stress function for the plane stress problem in a rectangular region, (ii) a Timoshenko-type beam theory, and (iii) the Bernoulli-Euler beam theory were selected. Among these, only the polynomial form of Airy's stress function violates the point-wise end-fixity conditions. Both the polynomial Airy stress function and the Timoshenko-type beam theory successfully model the effects of transverse shear deformation and the coupling of stretching and transverse deflection. Analytical solutions demonstrate that the normal shear coupling effect increases linearly with the thickness-to-span ratios in axial normal stress and axial displacement, while the coupling manifests quadratically in transverse displacement. The comparison of end displacements with the numerically exact FE solutions indicates that the polynomial form of Airy's stress function is no better than the Timoshenko-type beam theory. Similar conclusions were reached for the problem of uniformly loaded cantilever beams. It has been found that the accurate prediction of the deformation of thick anisotropic beams with significant normal-shear coupling requires the use of higher order theories.     

Deflection of thick beams of multimodular materials

A transfer-matrix analysis is presented for determining the static behaviour of thick beams of ‘multimodular materials’ (i.e. materials which have different elastic behaviour in tension and compression, with nonlinear stress–strain curves approximated as piecewise linear, with four or more segments). To validate the transfer-matrix method results, a closed-form solution is also presented for cases in which the neutral-surface location is constant along the beam axis. Numerical results for axial displacement, transverse deflection, bending slope, bending moment, transverse shear, axial force and location of neutral surface are presented for multimodular and bimodular models of unidirectional aramid cordrubber. The transfer-matrix method results agree very well with the closed-form solutions.     

Some closed-form solutions in random vibration of Bernoulli-Euler beams

The only closed-form solutions for random vibration of beams are that due to Houdijk, for the tip mean-square displacement of a cantilever beam under space- and time-wise ideal white noise, and that due to Eringen for a simply-supported beam under identical excitation. In both instances, beams possessing transverse damping were treated. In the present study closed-form solutions are found for uniform, simply supported beams subjected to a stationary excitation that is white both in space and time. The beams possess either structural, Voigt or rotary damping mechanisms. Expressions are obtained for the space-time correlation functions of displacement, velocity and stress. Previously derived interesting conclusions by Crandall and Yildiz on divergence of the mean-square stress for a beam with Voigt damping, and its convergence for the beam with combined transverse and rotary damping, are confirmed. Moreover, the closed form solution is obtained for the probabilistic characteristics of a beam under a number of separate or combined dampings.     

Transient dynamic and damping analysis of laminated anisotropic plates using a refined plate theory

A refined model is presented for the linear transient dynamic and damping analysis of laminated anisotropic composite plates. Experimental measurements of specific damping capacity of unidirectional composite beams are used to predict the specific damping capacity of laminated composite plates in various modes of vibration. A finite element idealization is adopted, and the quadratic Lagrangian element is used together with selective/reduced integration. A viscous damping approximation is then employed to calculate the damped transient response of laminated plates. The effects of transverse shear deformation, symmetry condition, boundary conditions, anisotropy, aspect ratio, fibre orientation and the lamination scheme on specific damping capacity and damped transient response are investigated. Realistic examples illustrate the importance of these parameters. The present results agree very closely with experimental results available in the literature and can serve as a benchmark for future comparison by other investigators.     

Assessment of coupled 1D models for hybrid piezoelectric layered functionally graded beams

A unified formulation is presented for coupled efficient zigzag, third order, consistent third order and first order models for hybrid piezoelectric layered FGM beams under thermoelectromechanical load. The transverse as well as inplane electric fields are considered. The potential and thermal fields are discretised sublayerwise as piecewise linear which can adequately approximate the actual distributions of these fields across the thickness. The zigzag theory accounts for the non-uniform deflection across the thickness due to the potential and thermal fields and satisfies exactly the conditions on transverse shear stress at the top, bottom and layer interfaces. The governing equilibrium equations have been derived from a variational principle. For the first time, the accuracy of the 1D models is assessed in direct comparison with the exact 2D solutions for elastic as well as piezoelectric hybrid layered FGM beams for mechanical, potential and thermal loads. The effects of the inhomogeneity parameter, span-to-thickness ratio and electric boundary conditions are investigated. It is established that the zigzag theory results are superior to those of other 1D models, even though they have the same number of displacement variables.