A boundary element approach to buckling of laminated plates subjected to arbitrary in-plane loading

Laminated plate buckling is analyzed by the boundary element method (BEM). Ignoring bending stretching coupling, a solution is first sought for the membrane stresses due to arbitrary in plane loading. Using the stress function concept, it is shown that this problem is mathematically equivalent to the plate-bending problem. Based on this similarity, a new boundary element formulation is developed for the prediction of the pre-buckling membrane state of stress in an anisotropic plate. The integral equations for the buckling mode are then derived from a variational principle using the fundamental solution of the plate-bending problem. An irreducible domain integral depending on plate deflection rather than curvatures is numerically accounted for by adopting deflection modeling over the plate in addition to boundarymodeling. Linear discontinuous boundary elements as well as domain cells are used along with special schemes for the approximation of jump term at corners.Analytical integration of singular integrals is performed over elements containing the source point. Thus a set of integral equations is transformed into an eigenvalue problem from which the critical load is evaluated. The reliability of the proposed analysis is established by comparing BEM predictions with solutions available from the literature or obtainable through a general purpose finite element program

Evaluation of polymer fracture parameters by the boundary element method

The boundary element method (BEM) for two-dimensional linear viscoelasticity is applied to polymer fracture. The time-dependence of stress intensity factors is assessed for various viscoelastic models as well as loading and support conditions. Various representations of the energy release rate under isothermal conditions are adopted. Additional boundary integral equations for the displacement gradient in the domain are linked to algorithms for the evaluation of path-independent J-integrals. The consistency of BEM predictions and their good agreement with other analytical results confirms BEM as a valid modelling tool for viscoelastic fracture characterisation and failure assessment under complex geometric and loading conditions

Evaluation of various schemes for quasi-static boundary element analysis of polymers

The behaviour of polymers under quasi-static load is analysed by various boundary element schemes. Linear viscoelasticity, for which the correspondence principle applies, is assumed. The problem is first solved in the Laplace transform domain with the time-dependent response determined by numerical inversion. A solution is also obtained directly in the time domain using fundamental solutions for unit step load excitation. Two alternative time-domain schemes, applied until recently only to dynamic problems, are adapted to quasi-static conditions. Both are based on a reciprocity relation involving Riemann convolutions and use fundamental solutions for a Dirac impulse excitation. The second of those schemes, however, uses only the Laplace transforms of these fundamental solutions, which are directly formed from the corresponding elasticity solutions and thus not specific to the viscoelastic model used. Rapid derivation of time-dependent fundamental solutions for general standard linear solids enhances the applicability of time domain methods. Computer codes based on the different algorithms are developed and applied to benchmark problems in order to assess their relative accuracy, versatility and efficiency. The various BEM predictions are generally consistent and reliable. The numerical instability of the last, so called, mixed method is minimised through appropriate choice of modelling parameters

Comparison of current methods for polymer analysis by boundary element

In this paper, quasi-static analyses of polymers, based on the boundary element method, are reviewed and implemented. Linear viscoelasticity, for which the correspondence principle applies, is assumed. Thus, one of the adopted BEM approaches solves the problem in the Laplace transform domain and relies on numerical inversion for the determination of the time-dependent response. The second solves directly in the time domain using fundamental solutions specific to the solid geometry and the viscoelastic model used. A third, recently proposed method also produces directly the time-dependent response but relies on the different algorithms are developed and applied to benchmark problems in order to assess their relative accuracy and efficiency. Particular attention is given to the effectiveness of the methods to predict fracture parameters in cracked plate problems. The versatility, computational efficiency and accuracy of the different schemes are compared. In general, good agreement is achieved between various BEM predictions and other published numerical results. Schemes for possible extension of the method to account for more complex viscoelastic models are briefly discussed

Stress amplification in three-dimensional narrow zones created by cavities

The paper is concerned with a particular case of stress amplification arising from the proximity of a spherical cavity to the boundary of a loaded elastic solid. The performed approximate analysis yields distributions of stresses and displacements in the narrow region formed between a spherical cavity and the faces of a thin flat layer subjected to a far field uniform radial tension. The narrow region is modelled as a circular plate of non-uniform thickness undergoing coupled membrane and flexural deformation. Series solutions are obtained for both membrane forces and bending moments leading to estimates for the stress concentration factor at minimum thickness. These predictions are found consistent with those obtained from both the exact analytical solution and finite element modelling of the problem. Cross-validated results from the two latter methods also provide trends for the stress amplification due to the narrowness of the region

Boundary element applications to polymer fracture

This paper reports on applications of the boundary element method (BEM) to polymer fracture. Both Laplace transformed and time domain two-dimensional analyses, based on linear viscoelasticity, are developed and applied to centre-cracked plates under tension in order to assessed for various viscoelastic models as well as loading conditions. Various approaches for direct assessment of the energy release rate are proposed; its representation through path-independent J-type integrals is also explored. Systematic procedures for J-integral evaluation are developed requiring the derivation of the additional boundary integral equations for the displacement gradient distributions in the solid domain. The BEM formulation is extended to the determination of the energy dissipation rate for the incremental crack growth. Then, the application of conservation of energy, combined with the knowledge of the critical and current values of strain energy release rate, leads to the assessment of the crack growth rate. Numerical results are compared with other analytical solutions and some experimental measurements. The consistency between BEM predictions and other published results confirms the method as a valid modelling tool for polymer fracture characterisation and investigation under complex conditions

Numerical conformal mapping based on the generalised conjugation operation

An iterative procedure for numerical conformal mapping is presented which imposes no restriction on the boundary complexity. The formulation involves two analytically equivalent boundary integral equations established by applying the conjugation operator to the real and the imaginary parts of an analytical function. The conventional approach is to use only one and ignore the other equation. However, the discrete version of the operator using the boundary element method (BEM) leads to two non-equivalent sets of linear equations forming an over-determined system. The generalised conjugation operator is introduced so that both sets of equations can be utilised and their least-square solution determined without any additional computational cost, a strategy largely responsible for the stability and efficiency of the proposed method. Numerical tests on various samples including problems with cracked domains suggest global convergence, although this cannot be proved theoretically. The computational efficiency appears significantly higher than that reported earlier by other investigators