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• Numerical investigations on 3D planar cracks in 1D hexagonal piezoelectric QCs with thermal effect are carried out. • A direct and effective method is proposed to derive Green's functions for point and element EDDs. • A numerical approach, EDD boundary element method, is developed and employed for the analysis. • Validation of the study is made by comparing results from theoretical and numerical analyses. Theoretical and numerical investigations on three-dimensional (3D) planar crack problems in one-dimensional (1D) hexagonal piezoelectric quasicrystals (QCs) with thermal effect are carried out systematically. Part II of the work aims to develop a general numerical approach to study 3D planar crack problems in 1D hexagonal piezoelectric QC media. Based on the theoretical formulations presented in Part I, a direct and effective method is proposed to derive the Green's functions for point extended displacement discontinuities (EDDs). These Green's functions are presented explicitly by a series of potential functions in a compact form. Using the superposition principle, the Green's functions for uniformly distributed EDDs over the crack elements are obtained. Related element solutions are used to construct the numerical approach, known as EDD boundary element method, for 1D hexagonal piezoelectric QCs. The proposed numerical method can be applicative to many complicated planar crack problems, such as multiple cracks, and cracks with non-uniform loadings, for 3D media composed of 1D hexagonal piezoelectric QCs with thermal effect. A comparison of the results obtained from the theoretical solutions given in Part I of the work with those obtained from the numerical method proposed here validates of the present investigation. [ABSTRACT FROM AUTHOR]

An analytical solution is developed for a simply supported multilayer magneto-electro-elastic plate, which is adhesively bonded by viscoelastic interlayer and subjected to transverse loading. The three-dimensional equations of magneto-electro-elasticity are used to describe the mechanical behavior in each layer of the plate, while the mechanical property of the viscoelastic interlayer is simulated by the standard linear solid model with strain memory effect. The imperfect electric conditions between adjacent layers are considered. Making use of the Fourier series expansion as well as the state-space method, a linear equation system to the solution of the problem considered is derived. By means of the Laplace transformation, the undetermined coefficients contained in the equation system are obtained analytically. The present solution can be used as the benchmark to assess the other numerical solutions. The comparison study shows a trend that the finite element solution converges to the proposed theoretical results as the mesh density increases; in contrast to the theoretical solution, the finite element solution is, however, time-consuming, in terms of mesh division and calculation. In this study, the effects of the time; interlayer thickness; and interlayer electric coefficient on the mechanical, electric, and magnetic fields are also investigated. [ABSTRACT FROM AUTHOR]

This paper presents a new hybrid finite element approach with fundamental solutions (HFS-FEM) as a trial function for modeling thermal behavior in perforated or cellular solids containing multiple randomly distributed circular holes with arbitrary sizes and locations, using special elements to reduce mesh effort. Based on the independent intra-element field in the element consisting of fundamental solutions and the frame field defined on the element boundary, the approach has characteristic features of elementary boundary integrals and versatile element construction by virtue of combining the new hybrid functional. Special purpose hole elements and regular elements are constructed using the special fundamental solution satisfying the specified hole boundary conditions and the conventional fundamental solution, respectively, such that the circular hole region can be modeled with a much smaller number of elements. Numerical examples including problems with single, double and randomly distributed multiple holes are considered and the results demonstrate the versatility, accuracy, and efficiency of the approach presented. [ABSTRACT FROM AUTHOR]

In this paper, a new type of hybrid fundamental solution-based finite element method (HFS-FEM) is developed for analyzing plane piezoelectric problems with defects by employing fundamental solutions (or Green's functions) as internal interpolation functions. The hybrid method is formulated based on two independent assumptions: an intra-element field covering the element domain and an inter-element frame field along the element boundary. Both general elements and a special element with a central elliptical hole or crack are developed in this work. The fundamental solutions of piezoelectricity derived from the elegant Stroh formalism are employed to approximate the intra-element displacement field of the elements, while the polynomial shape functions used in traditional FEM are utilized to interpolate the frame field. By using Stroh formalism, the computation and implementation of the method are considerably simplified in comparison with methods using Lekhnitskii's formalism. The special-purpose hole element developed in this work can be used efficiently to model defects such as voids or cracks embedded in piezoelectric materials. Numerical examples are presented to assess the performance of the new method by comparing it with analytical or numerical results from the literature. [ABSTRACT FROM AUTHOR]

In this article a new hybrid boundary integral-based (HBI) finite element method (FEM) is presented for analyzing two-dimensional (2D) and three-dimensional (3D) thermoelastic problems with arbitrary distribution of body force and temperature changes. The method of particular solution is used to decompose the displacement field into homogeneous part and particular part. The homogeneous solution is obtained by using the HBI-FEM with fundamental solutions, yet the particular solution related to the body force and temperature change is approximated by radial basis function (RBF). The detailed formulation for both 2D and 3D HBI-FEM for thermoelastic problems are given, and two different approaches for treating the inhomogenous terms are presented and compared. Five numerical examples are presented to demonstrate the accuracy and performance of the proposed method. When compared with the existing analytical solutions or ABAQUS results, it is found that the proposed method works well for thermoelastic problems and also when using a very coarse mesh, results with satisfactory accuracy can be obtained. [ABSTRACT FROM AUTHOR]

Based on the concept of eigenstrain, a straightforward computational model of the inverse approach is proposed for determining the residual stress field induced by welding using the eigenstrain formulations of boundary integral equations (BIEs). The eigenstrains are approximately expressed in terms of low-order polynomials in the local area around welded zones. The domain integrals with polynomial eigenstrains are transformed into the boundary integrals to preserve the favourable features of the boundary-only discretization. The sensitivity matrices in the inverse approach for evaluating the eigenstrain fields are constructed by either the measured deformations on the boundary or the measured stresses in the domain after welding over selected measuring points, or by both the measured information. It shows from the numerical examples that the results of residual stresses from measured deformations are always better than those from measured stresses but they are sensitive to experimental noises. The results from stress measurements can be improved by introducing a few deformation measuring points while reducing the number of points for stress measuring to reduce the cost in practice. [ABSTRACT FROM AUTHOR]

The present paper develops a new type of hybrid finite element model with regular and special fundamental solutions (also known as Green's functions) as internal interpolation functions for analyzing plane elastic problems in structures weakened by circular holes. A variational functional used in the proposed model is first constructed, and then, the assumed intra-element displacement fields satisfying a priori the governing partial differential equations of the problem under consideration is constructed using a linear combination of fundamental solutions at a number of source points outside the element domain, as was done in the method of fundamental solutions. To ensure continuity of fields over inter-element boundaries, conventional shape functions are employed to construct the independent element frame displacement fields defined over the element boundary. The linkage of these two independent fields and the element stiffness equations in terms of nodal displacements are enforced by the minimization of the proposed variational functional. Special-purpose Green's functions associated with circular holes are used to construct a special circular hole element to effectively handle stress concentration problems without complicated local mesh refinement or mesh regeneration around the hole. The practical efficiency of the proposed element model is assessed via several numerical examples. [ABSTRACT FROM AUTHOR]

Abstract: The objective of this work is to investigate the dynamic large deflection response of fully clamped metal foam core sandwich beam struck by a low-velocity heavy mass. Analytical solution and ‘bounds’ of dynamic solutions are derived, respectively. Also, finite element analysis is carried out to obtain the numerical solution of the problem. Comparisons of the dynamic, the quasi-static and numerical solutions for the non-dimensional maximum deflection of the sandwich beam with non-dimensional initial kinetic energy of the striker are presented for different cases of mass ratio, impact velocity and location. It is seen that the dynamic solution approaches the quasi-static one as the mass ratio of the striker to the beam is large enough, the quasi-static solution is in good agreement with the numerical results and both solutions lie in the ‘bounds’ of dynamic solutions. The quasi-static and numerical results for the impact force against the maximum deflection of the sandwich beam are obtained. It shows that the quasi-static solution can offer adequate accuracy to predict the low-velocity heavy-mass impact response of fully clamped sandwich beam. [Copyright &y& Elsevier]

The low-order polynomial-distributed eigenstrain formulation of the boundary integral equation (BIE) and the corresponding definition of the Eshelby tensors are proposed for the elliptical inhomogeneities in two-dimensional elastic media. Taking the results of the traditional subdomain boundary element method (BEM) as the control, the effectiveness of the present algorithm is verified for the elastic media with a single elliptical inhomogeneity. With the present computational model and algorithm, significant improvements are achieved in terms of the efficiency as compared with the traditional BEM and the accuracy as compared with the constant eigenstrain formulation of the BIE. [ABSTRACT FROM AUTHOR]

Abstract: A new hybrid finite element formulation is presented for solving two-dimensional orthotropic elasticity problems. A linear combination of fundamental solutions is used to approximate the intra-element displacement fields and conventional shape functions are employed to construct elementary boundary fields, which are independent of the intra-element fields. To establish a linkage between the two independent fields and produce the final displacement-force equations, a hybrid variational functional containing integrals along the elemental boundary only is developed. Results are presented for four numerical examples including a cantilever plate, a square plate under uniform tension, a plate with a circular hole, and a plate with a central crack, respectively, and are assessed by comparing them with solutions from ABAQUS and other available results. [Copyright &y& Elsevier]

Abstract: Many synthetic and natural media which are often described as multifunctional smart materials demonstrate thermo-electro-chemo-mechanical coupling behavior and are sensitive to external environmental stimuli. This paper presents a set of basic equations, a variational principle and a finite element procedure for investigating the coupled behavior of thermo-electro-chemo-elastic media. Emphasis here is placed on introducing chemical effects into the coupled equation system. Using the governing equations of thermal conduction, electric flow, ionic diffusion and momentum balance, a variational principle is deduced for a linearly coupled system by means of the extended Gibb’s free energy function. The variational principle is then used to derive a fully coupled multi-field finite element formulation for simulating the coupled thermo-electro-chemo-elastic behavior of biological tissues. Numerical examples are considered to illustrate the coupled phenomena of the materials and to verify the proposed variational theory and numerical procedure. [Copyright &y& Elsevier]

Abstract: The present investigation provides a hybrid Trefftz finite element approach for analysing elastoplastic problems. A dual variational functional is constructed and used to derive hybrid Trefftz finite element formulation for elastoplasticity of bulky solids. The formulation is applicable to either strain hardening or elastic-perfectly plastic materials. A solution algorithm based on initial stress formulation is introduced into the new element model. The performance of the proposed element model is assessed by three examples and comparison is made with results obtained by other approaches. The hybrid Trefftz finite element approach is demonstrated to be particularly suited for nonlinear analysis of two-dimensional elastoplastic problems. [Copyright &y& Elsevier]

Applications of boundary element method (BEM) to piezoelectric composites in conjunction with homogenization approach for determining their effective material properties are discussed in this paper. The composites considered here consist of inclusion and matrix phases. The homogenization model for composites with inhomogeneities is developed and introduced into a BE formulation to provide an effective means for estimating overall material constants of two-phase composites. In this model, a representative volume element (RVE) is used whose volume average stress and strain are calculated by the boundary tractions and displacements of the RVE. Thus BEM is suitable for performing calculations on average stress and strain fields of the composites. Numerical results for a piezoelectric plate with circular inclusions are presented to illustrate the application of the proposed micromechanics––BE formulation. [Copyright &y& Elsevier]

This work presents a reliable and efficient technique for displaying color mapping of data for post-processing of the Hybrid-Trefftz (HT) finite element method (FEM). The isoparametric mapping fill method developed in conventional FE models is generalized to HT FE models. Several steps of the procedure for HT FEM are demonstrated, as well as the aspects to be modified. For illustration purposes, a computer program has been written in VC++ and two 2D examples discretized by HT FEM are provided. The results are found to agree well with the analytical solutions although there are some discrepancies. Finally, conclusions are inferred and extension of this work to the 3D case is discussed. [Copyright &y& Elsevier]

The paper presents a family of variational formulations of Trefftz finite elements wherein the assumed internal displacement and electric potential fields a priori fulfil the governing differential equations of the problem over the element sub-domain, while the inter-element continuity and the boundary conditions are enforced using a modified variational principle together with an independent frame field defined on each element boundary. It is based on four free energy densities, each with two kinds of independent variables as basic independent variables, i.e. (σ,D), (ϵ,E), (ϵ,D), and (σ,E). Based on the assumed intra-element and frame fields, an element stiffness matrix equation is obtained which is implemented into computer programs for numerical analysis. Some numerical examples are considered to show the application of the proposed formulation. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the effects of the hierarchical wavy morphology of an interface on the apparent interfacial fracture toughness. First, the influence of two-level hierarchical sinusoidal geometry under mode-I and mode-II far-field loadings on crack propagation behavior is numerically studied by using a cohesive zone model. Second, the effects of interfacial friction on the mode-II effective fracture toughness are examined. The results show that an interface with a hierarchical sinusoidal structure has relatively higher fracture resistance ability, especially for mode-II cracks, than an interface with a flat or pure sinusoidal interface. Moreover, it is found that interfacial friction notably enhances the mode-II fracture toughness of a hierarchical sinusoidal interfacial crack. This study is helpful not only for understanding the superior mechanical properties of some biological materials, but also for optimal design of advanced composites with enhanced fracture toughness. [Copyright &y& Elsevier]

Abstract: Based on the concepts of eigenstrain and equivalent inclusion of Eshelby for inhomogeneity problems, a computational model and its solution procedure are presented using the proposed three-dimensional (3D) eigenstrain formulation of boundary integral equations (BIE) for simulating ellipsoidal particle-reinforced (and/or void-weakened) inhomogeneous materials. In the model, the eigenstrains characterizing deformation behaviors of each particle embedded in the matrix are determined using an iterative scheme with the aid of the corresponding Eshelby tensors, which can be obtained beforehand either analytically or numerically. With the proposed numerical model, the unknowns of the problem appear only on the boundary of the solution domain, since the interface condition between particles/voids and the matrix is satisfied naturally. The solution scale of the inhomogeneity problem can thus be significantly reduced. Using the algorithm, the stress distribution and the overall elastic properties are identified for ellipsoidal particle-reinforced/void-weakened inhomogeneous materials over a representative volume element (RVE). The effects of a variety of factors on the overall properties of the materials as well as the convergence behavior of the algorithm are studied numerically, showing the validity and efficiency of the proposed algorithm. [Copyright &y& Elsevier]

Abstract: Based on the boundary integral equations and stimulated by the work of Young et al. [J Comput Phys 2005;209:290–321], the boundary point method (BPM) is a newly developed boundary-type meshless method enjoying the favorable features of both the method of fundamental solution (MFS) and the boundary element method (BEM). The present paper extends the BPM to the numerical analysis of linear elasticity. In addition to the constant moving elements, the quadratic moving elements are introduced to improve the accuracy of the stresses near the boundaries in the post processing and to enhance the analysis for thin-wall structures. Numerical tests of the BPM are carried out by benchmark examples in the two- and three-dimensional elasticity. Good agreement is observed between the numerical and the exact solutions. [Copyright &y& Elsevier]

Abstract Efficient evaluation of near-boundary and boundary solutions for the Helmholtz equation with wideband wavenumbers by the boundary collocation method has been a difficult task for a long time. This study provides a regularized approach to bypass this limitation. The singular boundary method avoids the near singularity by using the nearly singular factors to replace the corresponding nearly singular terms. The core idea of the regularized approach is to substitute an artificially constructed general solution of the Helmholtz equation into the boundary integral equation or hyper boundary integral equation to determine the nearly singular factors. The core difficulty is the construction of the appropriate general solutions. The proposed regularized approach is free of integrations, easy-to-use and independent with particular wavenumbers. Numerical experiments show that accuracy of the near-boundary and boundary solutions of the singular boundary method improved by several orders of magnitude through application of the proposed regularized approach. [ABSTRACT FROM AUTHOR]