Internal stress analysis for single and multilayered coating systems using the boundary element method

Various thin-coating films are designed and utilized for industrial applications to improve machining performance due to better temperature and wear resistant properties than their substrate counterparts. However, the widespread experimental research on thin coatings underlies a general lack of modeling efforts, which can accurately and efficiently predict the coating and thin film performance. In this paper, the boundary element method (BEM) for 2D elastostatic problems is studied for the analysis of single and multilayered coating systems. The nearly singular integrals, which is the primary obstacle associated with the BEM formulations, are dealt efficiently by using a general nonlinear transformation technique. For the test problems studied, very promising results are obtained when the thickness of coated layers is in the orders of 1.0E?6 to 1.0E?9, which is sufficient for modeling most coated systems in the micro- or nano-scales.     

On the analysis of composite structures with material and geometric non-linearities

If major weight saving is to be realised it is essential that composites be used in “primary” structural components, i.e., wing and fuselage skins. To this end it is essential that analytical tools be developed to ensure that composite structures meet the FAA damage tolerance certification requirements. For stiffened composite panels one potential failure mechanism is the separation of the skin from the stiffeners; resulting from excessive “through the thickness” stresses. This failure mechanism is also present in bonded composite joints and composite repairs. Currently failure prediction due to in-plane loading appears to be relatively well handled. Unfortunately, this is not yet true for matrix-dominated failures. Consequently, it is essential that a valid analysis methodology capable of addressing all of the possible failure mechanisms, including failure due to interlaminar failure, be developed. To aid in achieving this objective the present paper outlines the results of a series of experimental, analytical and numerical studies into the matrix-dominated failures of rib stiffened structures.     

Boundary element model of electrochemical dissolution with geometric non-linearities

Large scale changes in geometry due to corrosion of polycrystalline pure copper are modeled using the Boundary Element Method. Axisymmetric geometries are considered. A quasi-static analysis is performed and a nonlinear polarization curve is determined from experiments to impose third kind boundary conditions. Super-elements, based on a cubic spline fit interpolation, are introduced to model the motion of the boundaries. Faraday's law is used to relate boundary motion to the surface flux. An set of controlled experiments using nearly pure (99.99% and 99.9%) copper with aerated NH₄OH electrolyte was used to test the experimental methods developed for this study and to verify the functionality of the numerical code in predicting large changes in geometry due to long duration dissolution. Polarization curves were measured and input into the BEM code and recession profiles were predicted. Comparison between experiment and predictions reveal that, given the polarization curves measured in the lab, the BEM code predicts accurate recession profiles.     

Stress concentration analysis of fibre-reinforced multilayered composites with pin-loaded holes

The problem of stress concentrations in the area of pin-loaded holes is of particular significance in the design of multilayered fibre-reinforced composite structures. For the purpose of simulating such problem zones in anisotropic multilayered composites, analytical methods offer decisive advantages since they, in comparison to numerical methods, allow weighting of influencing parameters and in this way permit a physical interpretation of complicated notch phenomena. At the Institut für Leichtbau und Kunststofftechnik (ILK) sophisticated analytical solution methods for the stress concentration problem in multilayered composites with pin-loaded holes were developed on the basis of layer-related solutions and have been confirmed in numerical calculations.     

Stress analysis of multilayered plates around circular holes

In this paper, a technique to study the 3-dimensional stress state around a circular hole in laminated plates is developed. First, the 3-dimensional elasticity problem for a thick plate with a circular hole is formulated in a systematic fashion by using the z-component of the Galerkin vector and that of Muki's harmonic vector function. This problem was originally solved by Alblas[1]. The reasons for reconsidering it are to introduce a technique which may be used in solving the elasticity problem for a multilayered plate and to verify and extend the results given by Alblas. Among the additional results of particular interest, one may mention the significant effect of the Poisson's ratio on the behavior and the magnitude of the stresses. Secondly, the elasticity problem for a laminated thick plate, which consists of two bonded dissimilar layers and which contains a circular hole, is considered. The problem is formulated for arbitrary axisymmetric tractions on the hole surface. Through the expansion of the boundary conditions into Fourier series, the problem is reduced to an infinite system of algebraic equations which is solved by the method of reduction. Of particular interest in the problem are the stresses along the interface as they relate to the question of delamination failure of the composite plate. These stresses are calculated and are observed to become unbounded at the hole boundary. An approximate treatment of the singular behavior of the stress state is presented, and the stress intensity factors are calculated. It is also observed that, the results compare rather well with those obtained from the finite element method.     

Stress analysis of spherical reservoirs with thin coating

The article explains an engineering method of stress analysis of spherical reservoirs with thin coating subjected to the weight of the liquid, internal pressure, and changes of temperature. The method is based on dividing the state of stress and strain into momentless and edge effects. For finding the edge effects, the method of asymptotic integration of the respective equations is used. Particular attention is given to the determination of stresses. It is shown that the thermal strain of the bearing reinforcing rib has to be taken into account. A numerical example is discussed.     

Prediction of freezing and thawing times for foods of two-dimensional irregular shape by using a semi-analytical geometric factor

Semi-analytical solutions for conduction and convection heat transfer problems with phase change were used to derive formulae for a geometric factor that describes the effect of product shape on freezing and thawing times for two-dimensional irregular products. These formulae depend only on the Biot number and three geometric parameters: the characteristic dimension, the volume and the surface area of the product. The accuracy of these formulae is demonstrated by comparisons with numerically calculated data and experimental freezing and thawing data for irregular objects. The new formulae out-perform all geometric factors previously proposed.     

2D analysis for geometrically non-linear elastic problems using the BEM

A boundary element method (BEM) approach for the solution of the elastic problem with geometrical non-linearities is proposed. The geometrical non-linearities that are considered are both finite strains and large displacements. Material non-linearities are not considered in this paper, so the constitutive law employed is Hooke's elastic one and the fundamental solution introduced in the integral equations is the usual one for isotropic linear elasticity. In order to deal with the intricate non-linear equations that govern the problem, an incremental–iterative method is proposed. The equations are linearized and a Total Lagrangian Formulation is adopted. The integral equations of the BEM are developed in an incremental form. The iterative process is necessary in order to achieve a good approximation to the governing equations. The problem of a slab under homogeneous deformation is solved and the results obtained agree with the analytical solution. The problem of a hollow cylinder under internal pressure is also solved and its solution compared with that obtained by a standardized finite element method code.     

Three-dimensional BEM for piezoelectric fracture analysis

A boundary element approach with quadratic isoparametric elements, quarter-point elements and singular quarter-point elements for three-dimensional crack problems in piezoelectric solids under mechanical and electrical loading conditions, is presented in this paper for the first time. The procedure is based on Deeg's fundamental solution for anisotropic piezoelectric materials, and the classical extended displacement boundary integral equation. Stress and electric displacement intensity factors are directly evaluated as system unknowns, and also as functions of the computed nodal displacements and electric potentials at crack faces. Special attention is paid to the fundamental solution evaluation. Several three-dimensional crack problems in transversely isotropic bodies under mechanical and electrical loading conditions are analysed. Numerical solutions computed for prismatic cracked 3D plate problems with a plane strain behaviour are in very good agreement with their corresponding 2D BE solutions. Results for a penny shape crack in a piezoelectric cylinder are presented for the first time. The proposed approach is shown to be a simple, robust and useful tool for stress and electric displacement intensity factors evaluation in piezoelectric media.     

A semi-analytical simulation method for reliability assessments of structural systems

A semi-analytical simulation method is proposed in this paper to assess system reliability of structures. Monte Carlo simulation with variance-reduction techniques, systematic and antithetic sampling, is employed to obtain the samples of the structural resistance in this method. Variance-reduction techniques make it possible to sufficiently simulate the structural resistance with less runs of structural analysis. When resistance samples and its moments determined, exponential polynomial method (EPM) is used to fit the probability density function of the structural resistance. EPM can provide the approximate distribution and statistical characteristic of the structural resistance and then the first-order second-moment method can be carried out to calculate the structural failure probability. Numerical examples are provided for a structural component and two ductile frames, which illustrate the method proposed facilitates the evaluation of system reliability in assessments of structural safety.     

An instrument for measuring coating thickness and geometric dimensions

Translated from Izmeritel'naya Tekhnika, No. 11, pp. 28–30, November, 1991.     

Frequency domain analysis of elastic bounded domains using a new semi-analytical method

In this paper, a new semi-analytical method is developed for solving two-dimensional elastodynamic problems in the frequency domain, employing Fast Fourier Transform. Using specific non-isoparametric elements, the boundary of the problem’s domain is discretized. By employing higher-order Chebyshev mapping functions, special shape functions, Clenshaw–Curtis quadrature, and implementing a weak form of weighted residual method, coefficient matrices of equation system become diagonal. This fact results in a set of decoupled Bessel differential equations to be used for solving the whole system. This means that the governing Bessel differential equation for each degree of freedom (DOF) becomes independent from other DOFs of the domain. For each DOF, the Bessel differential equation is solved for a specific frequency. Finally, the time history of responses may be obtained by using Inverse Fast Fourier Transform. Three numerical examples are presented to demonstrate the accuracy of the present new method.     

Gateable nanofluidic interconnects for multilayered microfluidic separation systems

The extension of microfluidic devices to include three-dimensional fluidic networks allows complex fluidic and chemical manipulations but requires innovative methods to interface fluidic layers. Externally controllable interconnects, employing nuclear track-etched polycarbonate membranes containing nanometer-diameter capillaries, are described that produce hybrid three-dimensional fluidic architectures. Controllable nanofluidic transfer is achieved by controlling applied bias, polarity, and density of the immobile nanopore surface charge and the impedance of the nanocapillary array relative to the microfluidic channels. Analyte transport between vertically separated microchannels has three stable transfer levels, corresponding to zero, reverse, and forward bias. The transfer can even depend on the properties of the analyte being transferred such as the molecular size, illustrating the flexible character of the analyte transfer. In a specific analysis implementation, nanochannel array gating is applied to capillary electrophoresis separations, allowing selected separated components to be isolated for further manipulation, thereby opening the way for preparative separations at attomole analyte mass levels.     

A new inverse analysis approach for multi-region heat conduction BEM using complex-variable-differentiation method

This paper presents a new inverse analysis approach for identifying material properties and unknown geometries for multi-region problems using the Boundary Element Method (BEM). In this approach, the material properties and coordinates of an unknown region boundary are taken as the optimization variables, and the sensitivity coefficients are computed by the Complex-Variable-Differentiation Method (CVDM). Due to the use of CVDM, the sensitivity coefficients can be accurately determined in a way that is as simple to use as the Finite Difference Method (FDM) and an inverse analysis for a complex composite structure can be easily performed through a similar procedure to the direct computation. Although basic integral equations are presented for heat conduction problems, the application of the proposed algorithm to other problems, such as elastic problems, is straightforward. Two numerical examples are given to demonstrate the potential of the proposed approach.     

Quasi-static analysis of multilayered domains with viscoelastic layer using incremental-layerwise finite element method

This paper presents a layerwise finite element formulation for quasi-static analysis of laminated structures with embedded viscoelastic material whose constitutive behavior is represented by the Prony series. To account the time dependence of the constitutive relations of linear viscoelastic materials, the incremental formulation in the temporal domain is used. This approach avoids the use of relaxation functions and mathematical transformations. A computer code based on the presented formulation has been developed to provide the numerical results. The high accuracy of the method is exhibited by comparing the results with existing solutions in the literature and also with those obtained using the ABAQUS software. Finally, and as an application of the presented formulation, the effects of time and load rate on the quasi-static structural response of asphalt concrete (AC) pavements are studied.     

Conjugate gradient methods for three-dimensional BEM systems of equations

A fundamental advantage of the boundary element method (BEM) is that the dimensionality of the problems is reduced by one. However, this advantage has to be weighted against the difficulty in solving the resulting systems of algebraic linear equations whose matrices are dense, non-symmetric and sometimes ill conditioned. For large three-dimensional problems the application of the classical direct methods becomes too expensive.This paper studies the comparative performance of iterative techniques based on conjugate gradient solvers as bi-conjugate gradient (Bi-CG), generalized minimal residual (GMRES), conjugate gradient squared (CGS), quasi-minimal residuals (QMR) and bi-conjugate gradient stabilized (Bi-CGStab) for potential and exterior problems. Preconditioning is also considered and assessed.Two examples, one from electrostatics and other from fluid mechanics, were employed to test these methods, which proved to be effective and competitive as solvers for BEM linear algebraic systems of equations.     

Constructing efficient substructure-based preconditioners for BEM systems of equations

In this work, a generic substructuring algorithm is employed to construct global block-diagonal preconditioners for BEM systems of equations. In this strategy, the allowable fill-in positions are those on-diagonal block matrices corresponding to each BE subregion. As these subsystems are independently assembled, the preconditioner for a particular BE model, after the LU decomposition of all subsystem matrices, is easily formed. So as to highlight the efficiency of the preconditioning proposed, the Bi-CG solver, which presents a quite erratic convergence behavior, is considered. In the particular applications of this paper, 3D representative volume elements (RVEs) of carbon-nanotube (CNT) composites are analyzed. The models contain up to several tens of thousands of degrees of freedom. The efficiency and relevance of the preconditioning technique is also discussed in the context of developing general (parallel) BE codes.     

A BEM formulation for inelastic transient dynamic analysis using domain decomposition and particular integrals

This study deals with the domain decomposition method and particular integrals for multi-region inelastic transient dynamic analysis. The particular integral formulation for single-region inelastic transient dynamic analysis is obtained by eliminating the acceleration volume integral and treating the initial stress term by volume cell. The Houbolt time integration scheme is used for the time- marching process. The Newton-Raphson algorithm for plastic multiplier is used to solve the system equation. In order to extend to multi-region problems, the domain decomposition method is examined. The domain of the original problem is subdivided into subregions. The interface boundary conditions are updated by using the iterative coupling employing Schwarz algorithm. Numerical results of two example problems are given to demonstrate the validity and accuracy of the present formulation.     

A time domain BEM for 3-D elastodynamic analysis using the B-spline fundamental solutions

A Boundary Element for 3-D elastodynamic analysis is introduced in detail. The method uses a new generation of the Stokes fundamental solutions that utilize the B-spline family of polynomials. The integration techniques of the boundary element kernels are also discussed for both the singular and non-singular cases. A number of numerical examples are presented for the validation of the proposed methodology.