Boundary element method for shear deformable plates with combined geometric and material nonlinearities

In this paper a new boundary element formulation for shear deformable plate theory with combined geometric and material nonlinearities is presented. The material is assumed to undergo large deflection with small strains. The von Mises criteria is used to evaluate the plastic zone and an elastic perfectly plastic material behaviour is assumed. An initial stress formulation is used to formulate the boundary integral equations. The domain integrals involving geometrical and material nonlinear terms are evaluated using a cell discretization technique. A total incremental method is applied to solve the nonlinear boundary integral equations. Numerical examples are presented to demonstrate the validity and the accuracy of the proposed formulation.     

BEM formulation for von Kármán plates

This work deals with nonlinear geometric plates in the context of von Kármán's theory. The formulation is written such that only the boundary in-plane displacement and deflection integral equations for boundary collocations are required. At internal points, only out-of-plane rotation, curvature and in-plane internal force representations are used. Thus, only integral representations of these values are derived. The nonlinear system of equations is derived by approximating all densities in the domain integrals as single values, which therefore reduces the computational effort needed to evaluate the domain value influences. Hyper-singular equations are avoided by approximating the domain values using only internal nodes. The solution is obtained using a Newton scheme for which a consistent tangent operator was derived.     

Nonlinear Free Vibration of In-Plane Functionally Graded Rectangular Plates

Nonlinear free vibration of functionally graded (FG) plates with in-plane material inhomogeneity subjected to different boundary conditions is presented. The nonlinear equations of motion and the related boundary conditions are extracted based on the classical plate theory. Green's strain tensor together with von Kármán assumptions is employed to model the geometrical nonlinearity. The differential quadrature method as an efficient and accurate numerical tool is employed to discretize the governing equations in spatial domain. After validating the presented approach, parametric studies are performed to clarify the effects of different parameters on the nonlinear frequency parameters of the in-plane FG plates.     

Boundary element analysis of two-dimensional elastoplastic contact problems

A general boundary element formulation for contact problems, capable of dealing with local elastoplastic effects and friction, is presented. Both conforming and non-conforming problems may be analysed. The contact problem is solved by means of a direct constraint technique, in which compatibility and equilibrium conditions are directly enforced in the general system of equations. The contact areas are modelled with linear interpolation functions, and quadratic interpolation functions are used everywhere else. Elastoplasticity is solved by a BEM initial strain approach The Von Mises yield criterion with its associated flow rule is adopted. Both perfectly plastic and work hardening materials are studied in the proposed formulation.

An incremental loading technique is proposed, which allows accurate development of the loading history of the problem. The non-linear nature of these problems demands the use of an iterative procedure, to determine the correct frictional conditions at every node of the contact area and the value of the plastic strains at selected points where local yielding may have occurred. Several numerical examples are presented to demonstrate the efficiency of the proposed formulation.     

Least squares finite element analysis of laminar boundary layer flows

Based on the least squares error criterion, a class of finite element is formulated for the numerical analysis of steady state viscous boundary layer flow problems. The method is essentially a discrete element-wise minimization of square and weighted residuals which arise from the attempts in approximately satisfying boundary layer equations. An iterative linearization scheme is developed to circumvent the mathematical difficulties posed by the non-linear boundary layer equations. It results in a process of successive least squares minimizations of residual errors arising from satisfying a set of linear differential equations. A mathematical justification for the method is presented. A major feature of the method lies in the linearization approach which renders non-linear differential equations amenable to linear least squares finite element analysis. Another important feature rests on the proposed finite element formulation which preserves the symmetric nature of finite element matrix equations through the use of the least squares error criterion. Numerical examples of viscous flow along a flat plate are presented to demonstrate the applicability of the method as well as to illuminate discussions on the theoretical aspects of the method.     

Multiple random crack propagation using a boundary element formulation

This paper proposes a boundary element method (BEM) model that is used for the analysis of multiple random crack growth by considering linear elastic fracture mechanics problems and structures subjected to fatigue. The formulation presented in this paper is based on the dual boundary element method, in which singular and hyper-singular integral equations are used. This technique avoids singularities of the resulting algebraic system of equations, despite the fact that the collocation points coincide for the two opposite crack faces. In fracture mechanics analyses, the displacement correlation technique is applied to evaluate stress intensity factors. The maximum circumferential stress theory is used to evaluate the propagation angle and the effective stress intensity factor. The fatigue model uses Paris’ law to predict structural life. Examples of simple and multi-fractured structures loaded until rupture are considered. These analyses demonstrate the robustness of the proposed model. In addition, the results indicate that this formulation is accurate and can model localisation and coalescence phenomena.     

Nonlinear analysis of forced vibration of nonlocal third-order shear deformable beam model of magneto–electro–thermo elastic nanobeams

This paper deals with the forced vibration behavior of nonlocal third-order shear deformable beam model of magneto–electro–thermo elastic (METE) nanobeams based on the nonlocal elasticity theory in conjunction with the von Kármán geometric nonlinearity. The METE nanobeam is assumed to be subjected to the external electric potential, magnetic potential and constant temperature rise. Based on the Hamilton principle, the nonlinear governing equations and corresponding boundary conditions are established and discretized using the generalized differential quadrature (GDQ) method. Thereafter, using a Galerkin-based numerical technique, the set of nonlinear governing equations is reduced into a time-varying set of ordinary differential equations of Duffing type. The pseudo-arc length continuum scheme is then adopted to solve the vectorized form of nonlinear parameterized equations. Finally, a comprehensive study is conducted to get an insight into the effects of different parameters such as nonlocal parameter, slenderness ratio, initial electric potential, initial external magnetic potential, temperature rise and type of boundary conditions on the natural frequency and forced vibration characteristics of METE nanobeams.     

Non-linear analysis of wave propagation using transform methods

Non-linear wave propagation/transient dynamics in lattice structures is modeled using a technique which combines the Laplace transform and the Finite element method. The first step in the technique is to apply the Laplace transform to the governing differential equations and boundary conditions of the structural model. The non-linear terms present in these equations are represented in the transform domain by making use of the complex convolution theorem. Then, a weak formulation of the transformed equations yields a set of element level matrix equations. The trial and test functions used in the weak formulation correspond to the exact solutions of the linear parts of the transformed governing differential equations. Numerical results are presented for a viscoelastic rod and von Karman type beam.     

A dual reciprocity boundary element approach for the problems of large deflection of thin elastic plates

 A new numerical method, which is based on the dual reciprocity boundary element method, is developed for the large deflection of thin elastic plates whose behaviour is governed by von Kármán equations. In the proposed method, the nonlinear and coupled parts of von Kármán equations are transformed to a set of boundary integrals, and only are the boundary discretized into elements. Therefore, a `pure' boundary element approach for the problems of large deflection of thin elastic plates can be achieved. On the other hand, benefiting from the present method, the plate stresses can be calculated directly without integral and singularity. Several examples are given to demonstrate the efficiency and accuracy of the present method. Received 11 October 1999     

Post-buckling analysis of skew plates subjected to combined in-plane loadings

Post-buckling behavior of laminated composite, sandwich and functionally graded skew plates is analyzed in the present work. The problem formulation is based on higher-order shear deformation theory and von Kármán’s nonlinear kinematics. Linear mapping is used to transform the physical domain into the computational domain. Chebyshev polynomials are used for spatial discretization of governing differential equations and boundary conditions. The nonlinear terms are linearized using quadratic extrapolation technique. The effect of the skew angle on the buckling and post-buckling response of the composite, sandwich and FGM-clamped skew plates is investigated for different combinations of in-plane compressive loadings.     

Thermal buckling and postbuckling analysis of functionally graded annular plates with temperature-dependent material properties

As a first endeavor, the thermal buckling and postbuckling analysis of functionally graded (FG) annular plates with material properties graded in the radial direction is presented. The formulation is derived based on the first-order shear deformation theory (FSDT) and the geometrical nonlinearity is modeled using Green’s strain in conjunction with von Karman’s assumptions. The material properties are temperature-dependent and graded according to the power law distribution. It is assumed that the temperature varies along the radial direction. Using the virtual work principle, the pre-buckling and postbuckling equilibrium equations and the related boundary conditions are derived. Differential quadrature method (DQM) as an efficient numerical technique is adopted to solve the governing equations. The presented formulation and the method of solution are validated by performing convergence and comparison studies with available results in the literature. Finally, the effects of volume fraction index, geometrical parameters, mechanical/thermal properties of the constituent materials and boundary conditions on the thermal buckling and postbuckling behavior of the radially graded annular plate are evaluated and discussed.     

A fast boundary cloud method for 3D exterior electrostatic analysis

An accelerated boundary cloud method (BCM) for boundary‐only analysis of 3D electrostatic problems is presented here. BCM uses scattered points unlike the classical boundary element method (BEM) which uses boundary elements to discretize the surface of the conductors. BCM combines the weighted least‐squares approach for the construction of approximation functions with a boundary integral formulation for the governing equations. A linear base interpolating polynomial that can vary from cloud to cloud is employed. The boundary integrals are computed by using a cell structure and different schemes have been used to evaluate the weakly singular and non‐singular integrals. A singular value decomposition (SVD) based acceleration technique is employed to solve the dense linear system of equations arising in BCM. The performance of BCM is compared with BEM for several 3D examples. Copyright © 2004 John Wiley & Sons, Ltd.     

A stable Galerkin reduced order model for coupled fluid–structure interaction problems

A stable reduced order model (ROM) of a linear fluid–structure interaction (FSI) problem involving linearized compressible inviscid flow over a flat linear von Kármán plate is developed. Separate stable ROMs for each of the fluid and the structure equations are derived. Both ROMs are built using the ‘continuous’ Galerkin projection approach, in which the continuous governing equations are projected onto the reduced basis modes in a continuous inner product. The mode shapes for the structure ROM are the eigenmodes of the governing (linear) plate equation. The fluid ROM basis is constructed via the proper orthogonal decomposition. For the linearized compressible Euler fluid equations, a symmetry transformation is required to obtain a stable formulation of the Galerkin projection step in the model reduction procedure. Stability of the Galerkin projection of the structure model in the standard L² inner product is shown. The fluid and structure ROMs are coupled through solid wall boundary conditions at the interface (plate) boundary. An a priori energy linear stability analysis of the coupled fluid/structure system is performed. It is shown that, under some physical assumptions about the flow field, the FSI ROM is linearly stable a priori if a stabilization term is added to the fluid pressure loading on the plate. The stability of the coupled ROM is studied in the context of a test problem of inviscid, supersonic flow past a thin, square, elastic rectangular panel that will undergo flutter once the non‐dimensional pressure parameter exceeds a certain threshold. This a posteriori stability analysis reveals that the FSI ROM can be numerically stable even without the addition of the aforementioned stabilization term. Moreover, the ROM constructed for this problem properly predicts the maintenance of stability below the flutter boundary and gives a reasonable prediction for the instability growth rate above the flutter boundary. Copyright © 2013 John Wiley & Sons, Ltd.     

The boundary element‐free method for elastoplastic implicit analysis

A meshless procedure, based on boundary integral equations, is proposed to analyze elastoplastic problems. To cope with non‐linear problems, the usual boundary element method introduces domain discretization cells, often considered a ‘drawback’ of the method. Here, to get rid of the standard element and cell, i.e. boundary and domain discretization, the orthogonal moving least squares (also known as improved moving least squares) method is used. The algorithm adopted to solve these particular inelastic non‐linear problems is a well‐established, criterion‐independent implicit procedure, previously developed by the authors. Comparative results are presented at the end to illustrate the effectiveness of the proposed techniques. Copyright © 2008 John Wiley & Sons, Ltd.     

Regularization of hypersingular boundary integral equations: a new approach for axisymmetric elasticity

A hypersingular boundary integral equation (HBIE) formulation, for axisymmetric linear elasticity, has been recently presented by de Lacerda and Wrobel [Int. J. Numer. Meth. Engng 52 (2001) 1337]. The strongly singular and hypersingular equations in this formulation are regularized by de Lacerda and Wrobel by employing the singularity subtraction technique. The present paper revisits the same problem. The axisymmetric HBIE formulation for linear elasticity is interpreted here in a ‘finite part’ sense and is then regularized by employing a ‘complete exclusion zone’. The resulting regularized equations are shown to be simpler than those by de Lacerda and Wrobel.     

Boundary value techniques for finite element solutions of the diffusion-convection equation

In this paper, the formulation of six-point and nine-point finite element equations for the solution of the diffusion-convection equation is presented. The six-point equation requires the solution of a tridiagonal system of equations and the nine-point centred equation is treated as a solution of a boundary value problem which leads to a large linear system of equations. Some numerical experiments are presented and the comparison with existing methods is included.     

Design and analysis of mechanically fastened composite joints and repairs

In this paper, a boundary element formulation is developed to analyze mechanically bolted composite joints and repairs. Boundary equations are formulated for all the member panels of the composite joints/repairs. These equations are solved together with the fastener equations to get the resultant contact forces for all the fasteners involved. The fasteners are modeled as 1-D springs that are governed by linear relationships between the fastener forces and the displacements of member panels at the respective fastener centers. After obtaining the fastener forces from the so-called global analysis, detailed stress analyses are conducted for regions around individual fasteners. The stress distributions around fastener holes are then used to evaluate the margin of safety of the composite panels. The numerical predictions on the fastener forces, failure modes and failure loads of two composite joints using this method agree very well with experimental results. The boundary element formulation presented here is especially suitable for design and analysis of aircraft battle damage repairs where time and facility availability is the prime concern.     

Failure analysis of composite laminates under large deflection

This paper presents the results of the application of a procedure, developed by Reddy & Reddy [ Composites Science and Technology 1992, 44, 227–255], for the evaluation of the first ply-failure load in multilayered composite plates. The procedure, which is based on the use of the finite element method (FEM) and which is suitable for the analysis of generally loaded plates, uses the non-linear von Karman formulation and, therefore, allows comparison of the failure loads in both the linear and the geometrically non-linear behaviour. Nevertheless, the use of the Newton-Raphson technique in searching the non-linear equilibrium points restricts its application to the case of plates without limit-point behaviour. The displacement model adopted in the FEM formulation is the traditional firstorder Reissner-Mindlin plate model that takes the shear deformation effect into consideration. Concerning the failure criteria, the analysis is based on a tensor polynomial criterion to which all other polynomial and independent criteria are brought back as particular cases. The study refers to the failure analysis of thin and thick plates under a uniformly distributed transverse load. Furthermore, a comparison of the failure criteria when the shear stresses are evaluated by means of the constitutive equations and by means of the local equilibrium equations is carried out. Finally, adopting a very simple degradation model of the mechanical properties to account for the stiffness decrease consequent to the failure, the qualitative behaviour of plates after the first non-catastrophic failure is also presented.     

Nonlinear oscillation of unsymmetric angle-ply plate on elastic foundation having nonuniform edge supports

This paper is analytically concerned with nonlinear flexural oscillation of an unsymmetrically laminated angle-ply rectangular plate resting on a Pasternak-type elastic foundation. The plate edges are subjected to the varying rotational constraints. Based on dynamic von Kármán-type nonlinear plate theory a single-mode analysis is carried out. In the formulation of a solution the force function and bending moments along the four edges are expanded into generalized Fourier series. These moments are also replaced by an equivalent lateral pressure near these edges. Galerkin's procedure and the perturbation technique are applied to the equation of motion and the time equation respectively. Numerical results for the amplitude-frequency response of the plate are presented graphically for various high-modulus composite materials, geometries of lamination, aspect ratios, moduli of elastic foundation and boundary conditions. Present results are compared with the existing values.     

Stress and velocity in 2D transient elastodynamic analysis by the boundary element method

This paper is mainly concerned with the development of integral equations to compute stress and velocity components in transient elastodynamic analysis by the boundary element method. All expressions required are presented explicitly. The boundary is discretized by linear isoparametric elements whereas linear and constant time interpolation are assumed, respectively, for the displacement and traction components. Time integration is carried out analytically and the resulting expressions are presented. An assessment of the accuracy of the results provided by the present formulation can be seen at the end of the article, where two examples are presented.