Free vibration of orthotropic laminated plates according to partial hybrid plate element

Abstract

The free vibration analysis of orthotropic composite laminates is investigated by using the partial hybrid plate element. The Hellinger‐Reissner principle is modified by adding kinetic energy. The through thickness effect is properly predicted since the transverse shear stress fields are assumed in the hybrid stress version. The natural frequencies are therefore accurately predicted. Apparently, the present study is more accurate than the displacement‐based higher‐order plate element.     

An optimal versatile partial hybrid stress solid‐shell element for the analysis of multilayer composites

In the present contribution we propose an optimal low‐order versatile partial hybrid stress solid‐shell element that can be readily employed for a wide range of geometrically linear elastic structural analyses, that is, from shell‐like isotropic structures to multilayer anisotropic composites. This solid‐shell element has eight nodes with only displacement degrees of freedom and only a few internal parameters that provide the locking‐free behavior and accurate interlaminar shear stress resolution through the element thickness. These elements can be stacked on top of each other to model multilayer composite structures, fulfilling the interlaminar shear stress continuity at the interlayer surfaces and zero traction conditions on the top and bottom surfaces of composite laminates. The element formulation is based on the modified form of the well‐known Fraeijs de Veubeke–Hu–Washizu multifield variational principle with enhanced assumed strains formulation and assumed natural strains formulation to alleviate the different types of locking phenomena in solid‐shell elements. The distinct feature of the present formulation is its ability to accurately calculate the interlaminar shear stress field in multilayer structures, which is achieved by the introduction of the assumed interlaminar shear stress field in a standard enhanced assumed strains formulation based on the Fraeijs de Veubeke–Hu–Washizu principle. The numerical testing of the present formulation, employing a variety of popular numerical benchmark examples related to element patch test, convergence, mesh distortion, shell and laminated composite analyses, proves its accuracy for a wide range of structural analyses.Copyright © 2012 John Wiley & Sons, Ltd.     

An effective method of stress intensity factor calculation for cracks emanating from a triangular or square hole under biaxial loads

This paper is concerned with stress intensity factors for cracks emanating from a triangular or square hole under biaxial loads by means of a new boundary element method. The boundary element method consists of the constant displacement discontinuity element presented by Crouch and Starfied and the crack‐tip displacement discontinuity elements proposed by the author. In the boundary element implementation, the left or the right crack‐tip displacement discontinuity element is placed locally at the corresponding left or right crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. The method is called a Hybrid Displacement Discontinuity Method (HDDM). Numerical examples are included to show that the method is very efficient and accurate for calculating stress intensity factors for plane elastic crack problems. In addition, the present numerical results can reveal the effect of the biaxial loads on stress intensity factors.     

Hybrid‐stress six‐node prismatic elements

This paper presents three novel hybrid‐stress six‐node prismatic elements. Starting from the element displacement interpolation, the equilibrating non‐constant stress modes for the first element are identified and orthogonalized with respect to the constant stress modes for higher computational efficiency. For the second element, the non‐constant stress modes are non‐equilibrating and chosen for the sake of stabilizing the reduced‐integrated element. The first two elements are intended for three‐dimensional continuum analysis with both passing the patch test for three‐dimensional continuum elements. The third element is primarily intended for plate/shell analysis. Shear locking is alleviated by a new assumed strain scheme which preserves the element accuracy with respect to the twisting load. Furthermore, the Poisson's locking along the in‐plane and out‐of‐plane directions is overcome by using the hybrid‐stress modes of the first element. The third element passes the patch test for plate/shell elements. Unless the element assumes the right prismatic geometry, it fails the patch test for three‐dimensional continuum elements. It will be seen that all the proposed elements are markedly more accurate than the conventional fully integrated element. Copyright © 2004 John Wiley & Sons, Ltd.     

A new variable‐order singular boundary element for two‐dimensional stress analysis

A new variable‐order singular boundary element for two‐dimensional stress analysis is developed. This element is an extension of the basic three‐node quadratic boundary element with the shape functions enriched with variable‐order singular displacement and traction fields which are obtained from an asymptotic singularity analysis. Both the variable order of the singularity and the polar profile of the singular fields are incorporated into the singular element to enhance its accuracy. The enriched shape functions are also formulated such that the stress intensity factors appear as nodal unknowns at the singular node thereby enabling direct calculation instead of through indirect extrapolation or contour‐integral methods. Numerical examples involving crack, notch and corner problems in homogeneous materials and bimaterial systems show the singular element's great versatility and accuracy in solving a wide range of problems with various orders of singularities. The stress intensity factors which are obtained agree very well with those reported in the literature. Copyright © 2002 John Wiley & Sons, Ltd.     

A SUPERCONVERGENT STRESS RECOVERY TECHNIQUE WITH EQUILIBRIUM CONSTRAINT

A stress recovery technique is developed to extract more accurate nodal stress values from the raw stress values obtained directly from the finite element analysis. In the present method a stress field is assumed over a patch of elements, and a least-squares functional is formed using the discrete stress errors at the superconvergent stress points and the residual of the equilibrium equation expressed in the virtual work form. The results of numerical tests conducted on one-dimensional and two-dimensional example problems demonstrate the validity and effectiveness of the present method. The introduction of an equilibrium constraint allows a patch stress field of higher order than is possible without the equilibrium constraint and this leads to a recovered stress field of higher accuracy. Because the residual of equilibrium is expressed in the virtual work form, the proposed method can easily be applied to arbitrarily curved shell structures. © 1997 by John Wiley & Sons, Ltd.     

Direct estimation of generalized stress intensity factors using a three‐scale concurrent multigrid X‐FEM

A concurrent multigrid method is devised for the direct estimation of stress intensity factors and higher‐order coefficients of the elastic crack tip asymptotic field. The proposed method bridges three characteristic length scales that can be present in fracture mechanics: the structure, the crack and the singularity at the crack tip. For each of them, a relevant model is proposed. First, a truncated analytical reduced‐order model based on Williams' expansion is used to describe the singularity at the tip. Then, it is coupled with a standard extended finite element (FE) method model which is known to be suitable for the scale of the crack. A multigrid solver finally bridges the scale of the crack to that of the structure for which a standard FE model is often accurate enough. Dedicated coupling algorithms are presented and the effects of their parameters are discussed. The efficiency and accuracy of this new approach are exemplified using three benchmarks. Copyright © 2010 John Wiley & Sons, Ltd.     

Three‐dimensional steady thermal stress analysis by triple‐reciprocity boundary element method

The steady thermal stress problems without heat generation can be solved easily by the boundary element method. However, for the case with arbitrary heat generation, the domain integral is necessary. In this paper, it is shown that the problems of three‐dimensional steady thermal stress with heat generation can be approximately solved without the domain integral by the triple‐reciprocity boundary element method. In this method, an arbitrary distribution of heat generation is interpolated by boundary integral equations. In order to solve the problem, the values of heat generation at internal points and on the boundary are used. Copyright © 2005 John Wiley & Sons, Ltd.     

Transient response of FG higher-order nanobeams integrated with magnetostrictive layers using modified couple stress theory

The present paper studies the transient response of a functionally graded nanobeam integrated with magnetostrictive layers. The material properties of sandwich nanobeam are temperature dependent and assumed to vary in the thickness direction. In order to consider small-scale effects, the modified couple stress theory is also taken into consideration. Using a unified beam theory that contains various beam models and energy method as well as Hamilton's principle, the governing motion equations and related boundary conditions are obtained. The obtained results in this paper can be used as sensors and actuators in sensitive applications.     

T‐stress evaluations for nonhomogeneous materials using an interaction integral method

A new equivalent domain integral of the interaction integral is derived for the computation of the T‐stress in nonhomogeneous materials with continuous or discontinuous properties. It can be found that the derived expression does not involve any derivatives of material properties. Moreover, the formulation can be proved valid even when the integral domain contains material interfaces. Therefore, the present method can be used to extract the T‐stress of nonhomogeneous materials with complex interfaces effectively. The interaction integral method in conjunction with the extended FEM is used to solve several representative examples to show its validity. Finally, using this method, the influences of material properties on the T‐stress are investigated. Numerical results show that the mechanical properties and their first‐order derivatives affect the T‐stress greatly, while the higher‐order derivatives affect the T‐stress slightly. Copyright © 2012 John Wiley & Sons, Ltd.     

Analysis of singular stress fields at multi‐material corners under thermal loading

The scaled boundary finite‐element method is extended to the modelling of thermal stresses. The particular solution for the non‐homogeneous term caused by thermal loading is expressed as integrals in the radial direction, which are evaluated analytically for temperature changes varying as power functions of the radial coordinate. When applied to model a multi‐material corner, only the boundary of the problem domain is discretized. The boundary conditions on the straight material interfaces and the side‐faces forming the corner are satisfied analytically without discretization. The stress field is expressed semi‐analytically as a series solution. The stress distribution along the radial direction, including both the real and complex power singularity and the power‐logarithmic singularity, is represented analytically. The stress intensity factors are determined directly from their definitions in stresses. No knowledge on asymptotic expansions is required. Numerical examples are calculated to evaluate the accuracy of the scaled boundary finite‐element method. Copyright © 2005 John Wiley & Sons, Ltd.     

Wide‐range and accurate mixed‐mode weight functions and stress intensity factors for cracked discs

The Wu‐Carlsson displacement‐based weight function method is extended to obtain the mode I and mode II weight functions for the edge‐ and centre‐cracked discs. Compared with Fett's direct adjustment weight functions for the edge‐cracked discs, the present weight functions are more accurate and are applicable for a wider range of crack lengths. Using the present weight functions, extensive and highly accurate mixed‐mode stress intensity factors are obtained for the cracked discs subjected to diametrically compressive forces. Assuming perfect contact and using Coulomb friction law and the present weight functions, the mode II stress intensity factors for the cracked discs with consideration of friction are obtained and widely compared with the corresponding results from finite element analyses.     

Superconvergent Patch Recovery for plate problems using statically admissible stress resultant fields

In this paper, we study an approach for recovery of an improved stress resultant field for plate bending problems, which then is used for a posteriori error estimation of the finite element solution. The new recovery procedure can be classified as Superconvergent Patch Recovery (SPR) enhanced with approximate satisfaction of interior equilibrium and natural boundary conditions. The interior equilibrium is satisfied a priori over each nodal patch by selecting polynomial basis functions that fulfil the point‐wise equilibrium equations. The natural boundary conditions are accounted for in a discrete least‐squares manner. The performance of the developed recovery procedure is illustrated by analysing two plate bending problems with known analytical solutions. Compared to the original SPR‐method, which usually underestimates the true error, the present approach gives a more conservative error estimate. Copyright © 1999 John Wiley & Sons, Ltd.     

A mixed solid‐shell element for the analysis of laminated composites

This paper presents a versatile low order locking‐free mixed solid‐shell element that can be readily employed for a wide range of linear elastic structural analyses, that is, from thick isotropic structures to multilayer anisotropic composites. This solid‐shell element has eight nodes with only displacement degrees of freedom and few assumed stress parameters that provide very accurate interlaminar stress calculations through the element thickness. These elements can be stacked on top of each other to model multilayer structures, fulfilling the interlaminar stress continuity at the interlayer surfaces and zero traction conditions on the top and bottom surfaces of the laminate. The element formulation is based on the well‐known Fraeijs de Veubeke–Hu–Washizu mixed variational principle with enhanced assumed strains formulation and assumed natural strains formulation to alleviate the different types of locking phenomena in solid‐shell elements. The distinct feature of the present formulation is its ability to accurately calculate the interlaminar stress field in multilayer structures, which is achieved by the introduction of a constraint equation on the interlaminar stresses in the Fraeijs de Veubeke–Hu–Washizu principle‐based enhanced assumed strains formulation. The intelligent computer coding of the present formulation makes the present element appropriate for a wide range of structural analyses. To assess the present formulation's accuracy, a variety of popular numerical benchmark examples related to element convergence, mesh distortion, and shell and laminated composite analyses are investigated and the results are compared with those available in the literature. These benchmark examples reveal that the proposed formulation provides very good results for the structural analysis of shells and multilayer composites. Copyright © 2011 John Wiley & Sons, Ltd.     

Near‐crack contour behaviour and extraction of log‐singular stress terms of the self‐regular traction boundary integral equation

This work contains an analytical study of the asymptotic near‐crack contour behaviour of stresses obtained from the self‐regular traction‐boundary integral equation (BIE), both in two and in three dimensions, and for various crack displacement modes. The flat crack case is chosen for detailed analysis of the singular stress for points approaching the crack contour. By imposing a condition of bounded stresses on the crack surface, the work shows that the boundary stresses on the crack are in fact zero for an unloaded crack, and the interior stresses reproduce the known inverse square root behaviour when the distance from the interior point to the crack contour approaches zero. The correct order of the stress singularity is obtained after the integrals for the self‐regular traction‐BIE formulation are evaluated analytically for the assumed displacement discontinuity model. Based on the analytic results, a new near‐crack contour self‐regular traction‐BIE is proposed for collocation points near the crack contour. In this new formulation, the asymptotic log‐singular stresses are identified and extracted from the BIE. Log‐singular stress terms are revealed for the free integrals written as contour integrals and for the self‐regularized integral with the integration region divided into sub‐regions. These terms are shown to cancel each other exactly when combined and can therefore be eliminated from the final BIE formulation. This work separates mathematical and physical singularities in a unique manner. Mathematical singularities are identified, and the singular information is all contained in the region near the crack contour. Copyright © 2003 John Wiley & Sons, Ltd.     

Analyse the role of the non‐singular stress in brittle fracture by BEM coupled with eigen‐analysis

A coupled model resulting from the boundary element method and eigen‐analysis is proposed in this paper to analyse the stress field at crack tip. This new combine method can yield several terms of the non‐singular stress in the Williams asymptotic expansion. Then the maximum circumferential stress (MCS) criterion taken the non‐singular stress into account is introduced to predict the brittle fracture of cracked structures. Two earlier experiments are re‐examined by the present numerical method and the role of the non‐singular stress in the brittle fracture is investigated. Results show that if more terms of non‐singular stress are taken into account, the predicted crack propagation direction and the critical loading by MCS criterion are much closer to the existing experimental results, especially for dominating mode II loading conditions. Moreover, numerical results manifest that Williams series expansion can describe the stress field further from the crack tip if more non‐singular stress terms are adopted.     

Determination of stress intensity factors of 2D fracture mechanics problems through a new semi‐analytical method

This study presents a novel development of a new semi‐analytical method with diagonal coefficient matrices to model crack issues. Accurate stress intensity factors based on linear elastic fracture mechanics are extracted directly from the semi‐analytical method. In this method, only the boundaries of problems are discretized using specific subparametric elements and higher‐order Chebyshev mapping functions. Implementing the weighted residual method and using Clenshaw–Curtis numerical integration result in diagonal Euler's differential equations. Consequently, when the local coordinates origin is located at the crack tip, the stress intensity factors can be determined directly without further processing. In order to present infinite stress at the crack tip, a new form of nodal force function is proposed. Validity and accuracy of the proposed method is fully demonstrated through four benchmark problems, which are successfully modeled using a few numbers of degrees of freedom. The numerical results agree very well with the analytical solution, experimental outcomes and the results from existing numerical methods available in the literature.     

The method of fundamental solutions for inverse source problems associated with the steady‐state heat conduction

This paper presents the use of the method of fundamental solutions (MFS) for recovering the heat source in steady‐state heat conduction problems from boundary temperature and heat flux measurements. It is well known that boundary data alone do not determine uniquely a general heat source and hence some a priori knowledge is assumed in order to guarantee the uniqueness of the solution. In the present study, the heat source is assumed to satisfy a second‐order partial differential equation on a physical basis, thereby transforming the problem into a fourth‐order partial differential equation, which can be conveniently solved using the MFS. Since the matrix arising from the MFS discretization is severely ill‐conditioned, a regularized solution is obtained by employing the truncated singular value decomposition, whilst the optimal regularization parameter is determined by the L‐curve criterion. Numerical results are presented for several two‐dimensional problems with both exact and noisy data. The sensitivity analysis with respect to two solution parameters, i.e. the number of source points and the distance between the fictitious and physical boundaries, and one problem parameter, i.e. the measure of the accessible part of the boundary, is also performed. The stability of the scheme with respect to the amount of noise added into the data is analysed. The numerical results obtained show that the proposed numerical algorithm is accurate, convergent, stable and computationally efficient for solving inverse source problems in steady‐state heat conduction. Copyright © 2006 John Wiley & Sons, Ltd.     

Trefftz boundary elements—multi‐region formulations

This paper is concerned with an effective numerical implementation of the Trefftz boundary element method, for the analysis of two‐dimensional potential problems, defined in arbitrarily shaped domains. The domain is first discretized into multiple subdomains or regions. Each region is treated as a single domain, either finite or infinite, for which a complete set of solutions of the problem is known in the form of an expansion with unknown coefficients. Through the use of weighted residuals, this solution expansion is then forced to satisfy the boundary conditions of the actual domain of the problem, leading thus to a system of equations, from which the unknowns can be readily determined. When this basic procedure is adopted, in the analysis of multiple‐region problems, proper boundary integral equations must be used, along common region interfaces, in order to couple to each other the unknowns of the solution expansions relative to the neighbouring regions. These boundary integrals are obtained from weighted residuals of the coupling conditions which allow the implementation of any order of continuity of the potential field, across the interface boundary, between neighbouring regions. The technique used in the formulation of the region‐coupling conditions drives the performance of the Trefftz boundary element method. While both of the collocation and Galerkin techniques do not generate new unknowns in the problem, the technique of Galerkin presents an additional and unique feature: the size of the matrix of the final algebraic system of equations which is always square and symmetric, does not depend on the number of boundary elements used in the discretization of both the actual and region‐interface boundaries. This feature which is not shared by other numerical methods, allows the Galerkin technique of the Trefftz boundary element method to be effectively applied to problems with multiple regions, as a simple, economic and accurate solution technique. A very difficult example is analysed with this procedure. The accuracy and efficiency of the implementations described herein make the Trefftz boundary element method ideal for the study of potential problems in general arbitrarily‐shaped two‐dimensional domains. Copyright © 1999 John Wiley & Sons, Ltd.