Finite element analysis of piezoelectric corner configurations and cracks accounting for different electrical permeabilities

Based on eigenfunctions of asymptotic singular electro-elastic fields obtained from a kind of ad hoc finite element method [Chen MC, Zhu JJ, Sze KY. Finite element analysis of piezoelectric elasticity with singular inplane electroelastic fields. Engng Fract Mech 2006;73(7):855-68], a super corner-tip element model is established from the generalized Hellinger-Reissner variational functional and then incorporated into the regular hybrid-stress finite element to determine the coefficients of asymptotic singular electro-elastic fields near a corner-tip. The focus of this paper is not to discuss the well-known behavior of electrically impermeable and permeable (usually it means fully permeable, hereinafter the same) cracks but analyze the limited permeable crack-like corner configurations embedded in the piezoelectric materials, i.e., study the influence of a dielectric medium inside the corner on the singular electro-elastic fields near the corner-tip. The boundary conditions of the impermeable or permeable corner can be considered as simple approximations representing upper and lower bounds for the electrical energy penetrating the corner. Benchmark examples on the piezoelectric crack problems show that present method yields satisfactory results with fewer elements than existing finite element methods do. As application, a piezoelectric corner configuration accounting for the limited permeable boundary condition is investigated, and it is found that the limited permeable assumption is necessary for corners with very small notch angles.     

Electroelastic singularities in piezoelectric-elastic wedges and junctions

This paper concerns the determination of the order and angular variation of inplane singular electroelastic states due to material and geometric discontinuities in piezoelectric-elastic wedges and junctions. The mathematical complexity required for deriving the order and angular variation of singular electroelastic fields is avoided by an ad hoc developed one-dimensional finite element formulation. The polarization orientation of the piezoelectric material may be arbitrary. To illustrate the simplicity, accuracy and efficiency of the suggested procedure, the order and angular variation of singular electroelastic fields for practically useful piezoelectric-elastic wedges and junctions are computed and compared with the existing analytical solutions.     

Modelling and analysis of the dynamic behaviour of piezoelectric materials containing interacting cracks

This study is concerned with the treatment of the dynamic behaviour of piezoelectric materials containing interacting cracks under antiplane mechanical and inplane electric loading. A general electrical boundary condition is used to enable the treatment of both permeable and impermeable conditions along the crack surfaces. The theoretical solution of the problem is formulated using integral transform techniques and an appropriate pseudo-incident wave method. The resulting singular integral equations are solved using Chebyshev polynomials to provide the dynamic stress and electric fields. Numerical examples are provided to show the effect of the geometry of the cracks, the piezoelectric constant of the material and the frequency of the incident wave upon the dynamic stress intensity factors. The results show the significant effect of electromechanical coupling upon local stress distribution.     

用切口尖端单元分析各向异性材料中多边形孔奇异性应力场

该文提出了一种基于全数值方法的新型杂交元方法, 用于研究各向异性复合材料中多边形孔奇异性应力场干涉问题。该方法的建立分3 个步骤:首先, 用一维有限元方法求解各向异性材料切口尖端奇异性应力场数值特征解;然后, 采用杂交有限元列式构造一种超级切口尖端单元, 其中, 假设应力场和位移场是利用上述奇异性场数值特征解推导出来的;最后, 将上述超级切口尖端单元与传统4 结点杂交应力元组装, 得到新型杂交元方法。算例中, 将裂纹问题作为考核例, 并进一步考察双菱形孔和双矩形孔的奇异性应力干涉问题。算例表明:当前模型能降低单元数, 且精度好;与传统有限元法和积分方程方法相比, 该模型更具有通用性和高效性, 为各向异性材料的细观力学分析打下了基础。     

Boundary element analysis of 2D thin walled structures with high-order geometry elements using transformation

Thin structures have been widely designed and utilized in many industries. However, the analysis of the mechanical behavior of such structures represents a very challenging and attractive task to scientists and engineers because of their special geometrical shapes. The major difficulty in applying the boundary element method (BEM) to thin structures is the coinstantaneous existence of the singular and nearly singular integrals in conventional boundary integral equation (BIE). In this paper, a non-linear transformation over curved surface elements is introduced and applied to the indirect regularized boundary element method for 2-D thin structural problems. The developed transformation can remove or damp out the nearly singular properties of the integral kernels, based on the idea of diminishing the difference of the orders of magnitude or the scale of change of operational factors. For the test problems studied, very promising results are obtained when the thickness to length ratio is in the orders of 1E?01 to 1E?06, which is sufficient for modeling most thin structures in industrial applications.     

Singular stress analysis of an anisotropic elastic medium containing polygonal holes using a novel hybrid finite element method

A novel hybrid finite element method based on a numerical procedure is proposed to compute singular field near V-shaped notch corners in an anisotropic material containing polygonal holes. The finite element method is established by the following three steps: (1) an ad hoc one-dimensional finite element formulation is employed to determined numerical eigensolutions of the singular field near an V-shaped notch corner; (2) a super corner tip element is constructed to determine the strength of the singular field, in which the independent assumed stress fields are extracted from the eigensolutions; (3) a novel hybrid finite element equation is obtained by coupling the super corner tip element with the conventional hybrid stress elements. In numerical examples, generalized stress intensity factors for interactions between two polygonal holes with various geometry, space position and material property are mainly discussed. All the numerical results show that present method yields satisfactory singular stress field solutions with fewer elements. Compared with the conventional finite element methods and integral equation methods, the present method is more suitable for dealing with micromechanics of anisotropic materials.     

Stress analysis of ply drop-off in composite structures

A combination of analytical and numerical method is employed to solve the ply drop-off problem. The singular stress fields in the drop-off region are characterized by using the eigenfunction expansion method. A global element capable of capturing the singular behavior is developed and incorporated into a general finite element analysis to provide an overall accurate solution of the stress fields in the ply drop-off region. This facilitates the use of a coarse mesh at the ply drop-off location even there are singular fields. The validation of the global–local finite element method has been carried out with the help of numerical examples.     

Effect of finite cracking on the electromechanical coupling properties of piezoelectric materials

Cracks and porosities inside the piezoelectric materials can weaken the electromechanical coupling effect, and hence influence the electromechanical coupling behavior of piezoelectric materials considerably. This paper studies the effect of internal cracking on the effective properties of piezoelectric media. It focuses on the piezoelectric medium of finite size with finite crack. The mechanical and electric fields in the piezoelectric material and the crack are formulated by singular integral method. Effects of crack size, medium border, and electric permeability of the crack on the overall electromechanical properties of the piezoelectric material are obtained and displayed graphically. In addition, the crack tip coupling electromechanical field intensity factors are also presented as they are not available in open literature for a finite crack in a finite piezoelectric media.     

Analysis of damaged material containing periodically distributed elliptical microcracks by using the homogenization method based on the superposition method

The presence of multiple microcracks in a structural component causes material degradation such as reduction in the stiffness or reduction in the fracture toughness of the component. In this paper, the homogenization method is used to evaluate mechanical properties of the damaged material. The adaptation of the superposition method to the homogenization method is also presented. The proposed method makes use of the finite element solution of uncracked solid and the analytical solution. The effective elastic moduli of damaged materials containing lattice-distribution microcracks are estimated by the proposed method. Furthermore, the stress fields and the stress intensity factors of the elliptical microcracks in the damaged material at a micro-mechanics scale are evaluated to illustrate microscopic behavior such as crack interaction.     

Hybrid stress finite element analysis of bending of a plate with a through flaw

A hybrid stress finite element procedure for the solution of bending stress intensity factors of a plate with a through-the-thickness crack is presented. Reissner's sixth-order plate theory including the effects of transverse shear deformation is used. The dominant singular crack tip stress field is embedded in the crack tip singular elements and only regular polynomial functions are assumed in the far field elements. The stress intensity factors can be calculated directly from the crack tip singular stress solution functions. The effects of the plate thickness, the ratio between the crack size and the inplane dimension of the plate, and the singular element size on the stress intensity factor solution are investigated. The effects of the explicit enforcement of traction-free conditions along crack surfaces, which are the natural boundary conditions in the present hybrid stress finite element model, are also investigated. The numerical results of bending of a plate with a straight central crack compare favourably with analytical solutions. It is also found that the explicit enforcement of traction-free conditions along crack surfaces is mandatory to obtain meaningful results for the Mode I type of bending stress intensity factor.     

Energy flux associated with a plane crack along an (hyper)elastic bimaterial interface

A numerical procedure, incorporated with the finite element solutions, is developed to evaluate the energy flux vector for a crack located along the interface of 2-D hyperelastic bimaterial solids. The formulation is considered with finite strains for use with both linear and nonlinear material behavior. The formulation is verified to be path-independent in a modified sense and so the near-tip region, where singular mechanical behavior dominates, is always included. Special attention is hence addressed on appropriate modeling of the singular behavior. The numerical results show good accuracy without using any particular singular finite elements. This revised version was published online in July 2006 with corrections to the Cover Date.     

A physically and geometrically nonlinear scaled‐boundary‐based finite element formulation for fracture in elastomers

This paper is devoted to the formulation of a plane scaled boundary finite element with initially constant thickness for physically and geometrically nonlinear material behavior. Special two‐dimensional element shape functions are derived by using the analytical displacement solution of the standard scaled boundary finite element method, which is originally based on linear material behavior and small strains. These 2D shape functions can be constructed for an arbitrary number of element nodes and allow to capture singularities (e.g., at a plane crack tip) analytically, without extensive mesh refinement. Mapping these proposed 2D shape functions to the 3D case, a formulation that is compatible with standard finite elements is obtained. The resulting physically and geometrically nonlinear scaled boundary finite element formulation is implemented into the framework of the finite element method for bounded plane domains with and without geometrical singularities. The numerical realization is shown in detail. To represent the physically and geometrically nonlinear material and structural behavior of elastomer specimens, the extended tube model and the Yeoh model are used. Numerical studies on the convergence behavior and comparisons with standard Q1P0 finite elements demonstrate the correct implementation and the advantages of the developed scaled boundary finite element. Copyright © 2014 John Wiley & Sons, Ltd.     

C/C-SiC缎纹编织复合材料孔隙缺陷的建模及其拉伸性能仿真

主要研究了随机孔隙缺陷在C/C-SiC缎纹编织复合材料中的有限元建模方法及其对拉伸性能的影响。基于C/C-SiC缎纹编织复合材料的细观结构和实验观察所得的微观形貌,得出孔隙缺陷具有随机分布特征,提出了一种三维随机碰撞算法模拟孔隙在复合材料中的分布,建立了含随机孔隙缺陷的C/C-SiC缎纹编织复合材料的有限元模型。采用有限元软件ABAQUS模拟了其在拉伸载荷下的力学行为,讨论了孔隙缺陷的尺寸和分布形式对材料拉伸性能的影响,并对试样进行了单轴拉伸实验测试,验证了数值模拟的有效性。结果表明,用本文方法建立的有限元模型符合含孔隙缺陷C/C-SiC缎纹编织复合材料的真实细观结构,相应的数值模拟结果也与试验数据吻合较好。本文的研究结果为含孔隙缺陷的缎纹编织复合材料及具有相似结构特征的复合材料的力学分析与优化设计提供了一种有效的方法。      

Modelling and analysis of piezoelectric actuators in anisotropic structures

Summary In this paper, we examine the coupled electromechanical behaviour of a piezoceramic actuator bonded to a finite elastic medium under inplane mechanical and electric loading. The purpose of the current work is to study the suitability of using a simple actuator model to simulate the load transfer between actuators and the host medium. The actuator is characterized by an electroelastic line model with the poling direction being perpendicular to its length. The solution of this electromechanically couple problem is provided by solving singular integral equations in terms of an interfacial shear stress and conducting finite element analysis. The results show that the transfer of the actuation energy between the actuator and the host structure can be effectively simulated using the developed theoretical model. Typical examples are provided to show the effects of the geometry, the material combination and anisotropy upon the load transfer. The study is further extended to treat the interfacial debonding between the actuator and the host material.     

Performance of variable order singular elements in analyzing interfacial fractures

This work concerns the development of singular boundary elements and the investigation of their numerical performance in analyzing interfacial cracks. In the vicinity of such cracks arise singular stress fields with variable order of singularity depending on the material characterizing parameters. The development of these elements which approximate displacement and traction functions is accomplished through controlled relocation of the mid-side node determined by compatibility and continuity requirements which must obey shape functions. These elements were applied to simulate the elastic behavior of cracks which are perpendicular and terminate on the interface of a bimaterial structure. Their efficiency in conjunction to the boundary only element method, are demonstrated in crack opening displacement diagrams and crack tip stress tabulated results.     

Buckling behavior of a central cracked thin plate under tension

The buckling characteristics of cracked plates subject to uniaxial tensile loads are analysed by the aid of the finite element method. Owing to the fact that crack buckling behavior is affected by the in-plane stress distribution around a crack, to get more accurate results, pre-buckling in-plane stress fields are analysed by the finite element method. For the critical loads calculation, the finite element approach adopted is based on Von Karman's linearize theory for buckling of plates subjected to pre-buckling state of plane stress. Several singular elements based on the Willian series are used in this plate bending approach. In this study, the effect of crack length, the effect of boundary condition, the effect of Poison's ratio and the effect of biaxial force on critical loads are analysed and discussed. Furthermore, the effect of initial imperfection is also discussed. There is a good agreement between other researcher's work and present results.     

Determination of high‐order fields for multianisotropic material antiplane V‐notches and inclusions by the asymptotic expansion technique and an overdeterministic method

In this paper, the singular behavior for anisotropic multimaterial V‐notched plates is investigated under antiplane shear loading condition. Firstly, the elastic governing equations are transformed into eigen ordinary differential equations through introducing the asymptotic expansions of displacements near the notch tip. The stress singularity exponents, including the higher‐order terms, and the corresponding eigen angular functions are then obtained by solving the established equations by using the interpolating matrix method. Thus, using the combination of the results from finite element analyses and the derived asymptotic expansion, an overdeterministic method is employed to calculate the amplitudes of the coefficients in the asymptotic expansions. Finally, the stress and displacement fields in the vicinity of the notch tip, consisting of both singular terms and higher‐order terms, are determined. The effects of material properties and geometry characteristic on the singular behaviour of the notch tip are discussed in detail.     

二维机织复合材料弹性常数的有限元法预测

为了预测二维机织复合材料的弹性性能,建立了有限元力学分析模型。基于二维机织复合材料的几何特征,建立了参数化的单胞模型;考虑了织物纤维束呈现出的各向异性材料特征,将有限元中材料主方向转化到纤维屈曲方向,建立其力学分析有限元模型;分析了单胞边界面保持平面假设的不足,提出了对于二维机织复合材料通用的周期边界条件,获得了更为准确的二维机织复合材料的工程弹性常数。结果表明:织物衬垫单胞边界面,在单向拉伸载荷和纯剪切载荷下,呈凹凸翘曲变形,即为周期边界;应用给出的织物参数化几何建模方法与有限元求解方法,可以精确地获得工程弹性常数,数值计算结果与实验值吻合较好。      

Compressive response of circular cell polycarbonate honeycombs under inplane biaxial static and dynamic loading—Part II: simulations

The static and dynamic experimental results of a polycarbonate circular cell honeycomb subjected to inplane biaxial loading are simulated through numerical analysis using the finite element method. The experimental results were presented in Part I of this two part paper (Chung and Waas, Int. J. Impact Eng. (2001), in preparation). Through several comparisons between the experimental and numerical results, the biaxial inplane crushing mechanisms of the circular cell honeycomb material are studied. The finite element simulations are able to accurately capture the essential features observed and measured in the experiments.     

Investigation of anisotropic reinforcement patch corners

Corners at symmetrically attached reinforcement patches of in-plane loaded laminate plates constitute a source for stress localizations. The mechanical in-plane fields of such stress localizations are obtained by means of the boundary finite element method (BFEM). Corresponding in-plane investigations provide the singular in-plane force exponents and their displacement modes characterizing the corner situation. Due to the availability of the in-plane fields in a semi-analytical way their gradients are readily evaluated then. The interlaminar stresses inside the plate are retrieved by equilibrium considerations on a ply-by-ply basis.