Characteristic Angular Functions Research Articles

Determination of singular and higher order non-singular stress for angularly heterogeneous material notch

The complete stress distribution around the tip of angularly heterogeneous material V-notch includes singular stress term and higher order non-singular stress term. The latter has important role on the prediction of the fracture, while it is usually neglected due to the evaluation difficulty. The idea of combing the singularity analysis near the notch vertex with the finite element analysis is established to deal with this difficulty. Firstly, a set of variable coefficient differential equations for the singular analysis of angularly heterogeneous material notch are built. The interpolating matrix method is introduced to solve the singular characteristic equations to prepare the singularity order and characteristic angular function. The V-notched structure is then analyzed by the coarse finite element meshes. The displacements from the finite element analysis are substituted into the singularity asymptotic expression to form a set of over-determined equation. The least square method is applied to yield the amplitude coefficients in the series asymptotic expansion. The complete stress field near the angularly heterogeneous material V-notch tip is then constructed according to the obtained amplitude coefficients, singularity orders, characteristic angular functions, together with their first order derivatives. The singular and non-singular stress term can be respectively calculated according to the truncated series terms. The reconstructed stresses have the same precision of the displacement from the view point of the finite element analysis. The role of the non-singular stress on the stress intensity factors of angularly heterogeneous material notch is then investigated.

Singularity of Electromechanical Coupling Field in Piezoelectric Composite Junctions

The double singularities including singular stress field and singular electric displacement field, in the tips of piezoelectric composite junctions, are analyzed by the interpolation matrix method (IMM). The double singularity analysis problem of piezoelectric composite junctions is converted into eigenvalue solution problem of ordinary differential equations with variable coefficients under corresponding boundary conditions. In numerical examples, the first couple of singularity orders and the corresponding characteristic angular functions of displacement and electric potential for the electromechanical coupling field are obtained and comparisons are presented to validate the accuracy of the proposed method. The singularity of the electromechanical coupling field at the tip of piezoelectric composite material junctions is closely related to the bonding angle and fiber direction. According to the numerical results, the best scheme can be configured for the combination of dissimilar materials.

Determination of stress intensity factors of V-notch structures by characteristic analysis coupled with isogeometric boundary element method

To describe the physical fields of the V-notch structure accurately, the structure is first divided into two parts, the notch tip sector containing singular stress field and the non-singular remained part. A twofold procedure is then carried out to obtain the stress intensity factors: (I) the asymptotic expansions are introduced to describe physical fields of the singular region and transform the elastic governing equation to the characteristic ordinary differential equations. The interpolating matrix method is then implemented to provide stress singular orders and related characteristic angular functions of this singular area. (II) The boundary integral equation for the left part is then discretized by the non-uniform rational B-splines (NURBS) elements and is solved under the isogeometric framework. The amplitudes of the asymptotic expansions in process (I) are calculated by coupling procedures (I) and (II) through utilizing the interfacial continuity conditions. The stress intensity factors are finally obtained and compared with the referenced results for both symmetric and inclined V-notch structures. Benefiting from the isogeometric analysis, satisfied results can be provided by discretizing the artificial arc boundary between the singular and non-singular parts with fewer NURBS elements. This method can also be integrated with the preprocessing of computer aided design (CAD) conveniently.

Evaluation of notch stress intensity factors by the asymptotic expansion technique coupled with the finite element method

• The asymptotic expansion technique coupled with the FEM is proposed to evaluate notch stress intensity factors . • The singular order, stress angular function and amplitude coefficient in stress asymptotic field of a V-notch can be yielded. • There no special element is needed while the high computational accuracy is obtained. • The proposed method can be implanted into the commercial FE software to analyze V-notches in engineering structures. A characteristic analysis method coupled with the finite element method is proposed to calculate stress intensity factors for V-notches, in which the notched structure is divided into a singular stress sector and the remained part. The asymptotic expansions are introduced into the singular stress region to transform elastic governing equations into characteristic ordinary differential equations. The established equations are solved by the interpolating matrix method to provide stress singular orders and characteristic angular functions. The asymptotic solution in the singular stress region is then coupled with finite element equations built on the remained structure by interfacial continuous conditions to establish the system equations. Thus, the amplitude coefficients in the stress asymptotic expansions can be determined. The notch stress intensity factors can be evaluated after the stress angular functions and amplitude coefficients being obtained. A symmetric V-notch and an inclined V-notch are respectively investigated to verify the accuracy of calculated notch stress intensity factors by the proposed method through comparing with the reference ones. Due to only conventional finite element method is needed, the proposed method can be freely implanted into commercial finite element software to analyze the singular stress for the V-notched engineering structures. This method can be also easily generalized to analyze the singular physical field for V-notches in composite materials.

Singularity analysis for 2D and 3D thermoelastic media containing V-notches

ABSTRACTThe stress field and heat flux singularities at the vertex of two-dimensional (2D) and three-dimensional (3D) V-notches are, respectively, investigated. By introducing typical terms in the series asymptotic expansions of displacement and temperature fields near the notch tip, the thermoelastic governing equations and radial boundary conditions of a V-notched structure are transformed into characteristic ordinary differential equations with respect to the singular order. These equations can be solved by the interpolating matrix method developed earlier by part of the authors. The singular orders related to the stress field and heat flux can be derived simultaneously and can be specified by the corresponding characteristic angular functions in the present method. The singularities are compared between 2D and 3D thermoelastic specimen containing V-notches. It is found that one singular order will be discarded if a 3D medium is simplified into a 2D one.

Singularity characteristic analyses for magneto-electro-elastic V-notches

Abstract Due to the material or geometrical discontinuities, the stress, electric displacement and magnetic induction may become theoretically infinite and singular at the vertexes of magneto-electro-elastic (MEE) V-notches, where the mechanical failure or dielectric breakdown may initiate from. Based on the assumption of the asymptotic expansions of the physical fields near the vertex, the characteristic differential equations with respect to the singularity order are derived from the equilibrium equations and Maxwell equations. After a set of variable replacement, these non-linear differential equations are transformed into the linear ones. The traditional iterative method for solving the transcendental equation is avoided. The mechanical, electric and magnetic boundary conditions together with interfacial continuity conditions are also expressed by the combination of the singularity order and characteristic angular functions. The singularity characteristic analyses for MEE V-notches are transformed into a problem of solving characteristic ordinary differential equations with variable coefficients. The singularity orders and characteristic angular functions can be derived by introducing the interpolating matrix method to solve the established characteristic equations. Herein, the singularities for the in-plane and anti-plane MEE V-notches are respectively investigated. The influence of the poling direction on the singularities of MEE V-notches is discussed. The role of the volume fraction of the BaTiO 3 inclusions on the singularities of MEE V-notches is studied. The obtained results can be used to design MEE products for reducing the singularity induced by the V-notch.

Singularity analysis for a V-notch with angularly inhomogeneous elastic properties

Stress singularity of a V-notch with angularly inhomogeneous elasticity is studied in this work. The characteristic differential equations with respect to the orders of the stress singularity for the V-notch problem are derived by using of asymptotic expansion technique. The boundary conditions on the two radial edges of the V-notch are then expressed by the combination of the orders of the stress singularity and the characteristic angular functions. With the manipulation, the problem is transformed into solving the characteristic ordinary differential equations with variable coefficients under the corresponding boundary conditions, which are solved by the interpolating matrix method developed by us before. The method and treatments presented are suitable for the singularity analyses of the V-notches with any angularly inhomogeneous material properties under the plane and anti-plane loading. As examples, the singularities of the V-notches with three differential types of angularly variable elasticity moduli are evaluated and compared for their orders of the stress singularity.

Failure initiation criterion in bonded joints with composites based on singularity parameters and practical procedure for design purposes

Composite to metal adhesively bonded joints generates critical points where geometry and material properties change abruptly. These points (multimaterial corners) are potential locations for failure initiation. There exist proposals predicting the initiation of failure at these multimaterial corners using a Fracture Mechanics approach, in which the Generalized Stress Intensity Factors (GSIFs) at the multimaterial corner control the failure initiation. The calculation of these GSIFs at anisotropic multimaterial corners involves not-straightforward calculations of the stress singularities and characteristic angular functions, numerical modeling of the joint and a careful postprocessing of the results. In this study a parametric Finite Element Analysis has been carried out allowing the generation of plots to calculate the GSIFs for unit values of the axial force, shear force and bending moment at one end of the overlap length. These results allow calculating the GSIFs at the multimaterial corners by a simple beam analysis of the joint, the use of these plots and the application of the superposition principle, for their use in the prediction of failure initiation by means of the singular parameters of the joint. Additionally, experiments have been carried out to propose an explicit failure criterion based on GSIF and Generalized Fracture Toughness values which fit very well with double-lap joint test results.

Stress singularity analysis for orthotropic V-notches in the generalised plane strain state

The singularity for the V-notch under the generalised plane deformation is investigated by the combination of the asymptotic analysis with the interpolating matrix method developed by part of the authors before. The displacement asymptotic expansions at the vicinity of the V-notch vertex are introduced into the equilibrium equations, which are transformed into a set of characteristic ordinary differential equations with respect to the notch singularity orders. The boundary conditions and interfacial compatibility conditions are also represented by the combination of the singularity orders and characteristic angular functions. The determination of the singularity orders and characteristic angular functions are transformed into solving the ordinary differential equations with variable coefficients, which are solved by the interpolating matrix method. The present method is suitable for the singularity analysis for isotropic and orthotropic V-notches. It is versatile for analysing the stress singularity of single material V-notches, bi-material V-notches, interface edges and cracks. The correctness of the results by the proposed method is ensured by the comparison with the published ones.

Evaluation of the singularity exponents and characteristic angular functions for piezoelectric V-notches under in plane and out of plane conditions

Based on the assumption that the physical fields close to the V-notch vertex are expressed by the series asymptotic expansions, the evaluation of singularity exponents for piezoelectric V-notches is turned into solving the characteristic values of ordinary differential equations under given boundary conditions, which are a set of equations with variable coefficients and solved by the interpolating matrix method developed by part of the authors before. The singularity analysis for V-notches under in plane and out of plane conditions is taken into account. Numerical results show that all the singularity exponents and the characteristic angular functions can be evaluated synchronously without the need of solving the transcendental equations iteratively. The present method is not only suitable for the singularity analysis of V-notches, but also for cracks and interface ends, without encountering the problem of ill-posed equations when the complex potential function method for the singularity analysis of V-notches is degenerated to analyze the singularity of cracks.

Singularity analysis near the vertex of the V‐notch in Reissner's plate by coupling the asymptotic expansion technique with the interpolating matrix method

ABSTRACTA new numerical method for calculating the singularity orders of V‐notches in Reissner's plate is proposed in this paper. By introducing the asymptotic expansion of the generalised displacement field at the notch tip into the equilibrium equations of a plate, a set of characteristic ordinary differential equations with respect to the singularity order are established. In addition, by adopting the variable substitution technique, the obtained non‐linear characteristic equations are transformed into linear ones, which are solved by the interpolating matrix method. The singularity orders of moments and shear forces can be obtained simultaneously and can be distinguished from the corresponding characteristic angular functions conveniently. Four types of boundary conditions are proposed to investigate the influence of boundary conditions on the singularity order values. The effect of the Poisson's ratio on the singularity orders of the V‐notch in Reissner's plate is discussed. The present method is versatile for the singularity analysis of single material V‐notches and bi‐material V‐notches, and can be easily extended to multi‐material V‐notches.

Analysis of heat flux singularity at 2D notch tip by singularity analysis method combined with boundary element technique

It is known that the heat flux becomes infinite near the notch tip due to material or geometric discontinuities. Numerical methods such as boundary element method and finite element method cannot adequately analyze the accurate singular heat flux field since they traditionally employ piecewise polynomials. In this paper, the singularity analysis method coupled with the boundary element technique is proposed for the accurate analysis of two-dimensional singular heat flux field near the notch tip. The V-notched structure is departed into two parts, which are the near tip singular sector and far tip non-singular section. The singularity analysis is executed on the near notch tip sector for searching singularity orders and corresponding characteristic angular functions by introducing the heat flux asymptotic expansions into heat conduction governing equations. The conventional boundary element method is applied to modeling the far notch tip region because there is no heat flux singularity. The asymptotic expansions of the near tip physical field and the boundary integral equations established on the far notch tip region are combined together for solving the expansion coefficients in heat flux asymptotic expansions. Thus, the complete heat flux field near the notch vertex can be accurately determined.

A least squares procedure for the evaluation of multiple generalized stress intensity factors at 2D multimaterial corners by BEM

Detailed characterization of linear elastic stress states at corners and crack tips requires knowledge of the stress singularity orders, the characteristic angular functions and the generalized stress intensity factors (GSIF). Typically a high accuracy is found in the literature for the evaluation of the stress singularity orders and characteristic angular functions (numerically computed from analytical expressions in most cases). Nevertheless, GSIF values, evaluated by means of a numerical model using FEM or BEM and usually by postprocessing the results, are often reported with a lower level of confidence. A robust procedure is presented in this work for the evaluation of the GSIF at multimaterial corners. The procedure is based on a simple least squares technique involving stresses and/or displacements, computed by BEM, at the neighborhood of the corner tip. A careful verification of the robustness and accuracy of the procedure using a few benchmark problems in the literature has been carried out. Applications of the procedure developed to the evaluation of GSIFs appearing at corners in metal-composite adhesive lap joints are presented.