Interpolating Matrix Method 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.

Application of Interpolating Matrix Method to Study Dynamics of Axially Moving Beams Made of Functionally Graded Materials

In this paper, the divergent instability and coupled flutter characteristics of axially moving beams made of functionally graded materials (FGM) are studied using the interpolation matrix method. The material property of the beam is designed to change smoothly and continuously along the thickness direction. In considering the Euler-Bernoulli beam theory, Hamilton’s principle is used to derive the differential equation of the transverse vibration kinematics of axially moving FGM beams. In addition, the calculation model for solving the complex frequency of the beam based on the interpolation matrix method has been established. The presented solutions are compared with those in the literature to illustrate the effectiveness of the interpolation matrix method. The results show that the divergence and flutter velocities of axially moving FGM beams tend to decrease with the increase of the material gradient index, and there is a very narrow stability region between the first static instability region (divergence) and the first dynamic instability region (first- and second-order coupled flutter).

Analysis of multiple plastic stress singularities for antiplane V-notches in hardening materials by using the interpolating matrix method

Analysis of multiple plastic stress singularities for antiplane V-notches in hardening materials by using the interpolating matrix method

Singularity analysis of linear hardening V-notch by asymptotic expansion coupled BEM

In this paper, a general numerical method by the asymptotic expansion coupled the boundary element method (CBEM), considering the singular term and the higher order terms, is developed to analyze the complete singular stress field of linear hardening elastic–plastic material with a V-notch. The method firstly transforms the singularity analysis for the governing equation of the V-notch tip into an ordinary differential eigenvalue equation, which is then solved by the interpolating matrix method. Then, the complete singular stress field and the notch stress intensity factors (SIFs) can be obtained by coupling the asymptotic expansion of the elastic–plastic singular stresses at the V-notch tip with the conventional boundary elements method for the remaining region of the structure. Two types of examples are analyzed for the V-notch problem of linear hardening material. It is shown that the elastic–plastic singular stress field can be evaluated by the equivalent Young’s modulus and Poisson’s ratio. The proposed new approach is found to be accurate and effective in evaluating singular stress fields in a V-notch structure with linear hardening material.

Calculation of Critical Load of Axially Functionally Graded and Variable Cross-Section Timoshenko Beams by Using Interpolating Matrix Method

In this paper, the interpolation matrix method (IMM) is proposed to solve the buckling critical load of axially functionally graded (FG) Timoshenko beams. Based on Timoshenko beam theory, a set of governing equations coupled by the deflection function and rotation function of the beam are obtained. Then, the deflection function and rotation function are decoupled and transformed into an eigenvalue problem of a variable coefficient fourth-order ordinary differential equation with unknown deflection function. According to the theory of interpolation matrix method, the eigenvalue problem of the variable coefficient fourth-order ordinary differential equation is transformed into an eigenvalue problem of a set of linear algebraic equations, and the critical buckling load and the corresponding deflection function of the axially functionally graded Timoshenko beam can be calculated by the orthogonal triangular (QR) decomposition method, which is the most effective and widely used method for finding all eigenvalues of a matrix. The numerical results are in good agreement with the existing results, which shows the effectiveness and accuracy of the method.

High-order Asymptotic Analysis for the V-notch in a Power-law Hardening Material

The aim of this study is to determine the complete elastic–plastic stress asymptotic solutions at the plane V-notch tip with the two edges being clamped–clamped, free-clamped and free-friction. Firstly, the displacement and stress fields around the notch tip are expressed as asymptotic expansions, and then, these asymptotic expansions are substituted into the displacement–strain relation and the equilibrium equations to establish the ordinary differential equations (ODEs). Finally, the interpolating matrix method is employed to solve the eigenvalue problem of the ODEs, and consequently, the leading-order and higher-order stress and displacement eigen-solutions at the notch tip are obtained. Numerical examples demonstrate that the presented method has the advantages of great versatility and high accuracy.

Asymptotic Solutions of Plastic Stress and Displacement at V-Notch Tips Under Anti-Plane Shear

An efficient method was developed to determine the first- and high-order terms of asymptotic solutions of plastic stress and displacement near V-notch tips under anti-plane shear in power-law hardening materials. Through introduction of the asymptotic series expansions of stress and displacement fields around the V-notch tip into the fundamental equations of the elastoplastic theory, the governing ordinary differential equations (ODEs) with the stress and displacement eigen-functions were established. Then the interpolating matrix method was employed to solve the resulting nonlinear and linear ODEs. Consequently, the high-order stress exponents and the associated eigen-solutions were obtained. The presented method, being capable of dealing with the V-notches with arbitrary opening angles and strain hardening indexes under anti-plane shear, has the advantages of great versatility and high accuracy. Typical examples were given to demonstrate the accuracy and effectiveness of this method.

Analyze the singularity of the heat flux with a singular boundary element

In the steady state heat conduction problem, the heat flux could be infinite at the re-entrant corner, at the point where the boundary condition is discontinuous, or at the place where the material properties are changing abruptly. The conventional numerical methods, such as the finite element method (FEM) and the boundary element method (BEM), have difficulties in analyzing the singular heat flux field. Herein, a new singular element employed in the boundary integral equation is developed to interpolate the heat flux field near the singular point. The shape function of the singular element is set as an asymptotic expansion model with respect to the distance from the singular point. The singular order in the asymptotic expansion can be determined by solving the singularity eigen equation on the basis of the interpolating matrix method. The self-adaptive co-ordinate transformation method is introduced to deal with the weakly singular integral in the proposed method. Benefited from the singular element, more accurate temperature and heat flux results near the singular point are obtained with fewer elements. In addition, the proposed method is easy to be coupled with the conventional BEM without too much modifications.

The calculation of singular orders for composite material anti-plane propagating V-notches

Because of geometric or material discontinuities, stress singularities can occur around the vertex of a V-notch. The singular order is an important parameter for characterizing the degree of the stress singularity. The present paper focuses on the calculation of the singular order for the anti-plane propagating V-notch in a composite material structure. Starting from the governing equation of elastodynamics and the displacement asymptotic expansion, an ordinary differential eigen equation with respect to the singular order is proposed. The interpolating matrix method is then employed to solve the established eigen equation to conduct the singular orders. The effects of the material principal axis direction, shear modulus, and propagation velocity and acceleration on the singular orders of the V-notch are respectively investigated, and some conclusions are drawn. The singular orders of the V-notches decrease with an increase in the material principal axis direction angle, except for the crack, whose singular orders do not change with the principal axis direction. The singular orders increase with an increase in the shear modulus $$G_{13} $$ , while they decrease with an increase in the shear modulus $$G_{23} $$ . The singular orders increase as the magnitude of the propagation velocity increases, while they decrease as the direction of the propagation velocity increases. The singular orders increase with an increase in the value of the propagation acceleration, while they decrease with an increase in the direction of the propagation acceleration. The singular orders become larger when the ratio of $$G_{13}^{(2)} /G_{13}^{(1)} $$ increases, while they become smaller when the ratio of $$G_{23}^{(2)} /G_{23}^{(1)} $$ increases, and they increase with an increase in the mass density ratio for the bi-material V-notch.

Complete elastic–plastic stress asymptotic solutions near general plane V-notch tips

An efficient method is developed to determine the multiple term eigen-solutions of the elastic–plastic stress fields at the plane V-notch tip in power-law hardening materials. By introducing the asymptotic expansions of stress and displacement fields around the V-notch tip into the fundamental equations of elastic–plastic theory, the governing ordinary differential equations (ODEs) with the stress and displacement eigen-functions are established. Then the interpolating matrix method is employed to solve the resulting nonlinear and linear ODEs. Consequently, the first four and even more terms of the stress exponents and the associated eigen-solutions are obtained. The present method has the advantages of greater versatility and high accuracy, which is capable of dealing with the V-notches with arbitrary opening angle under plane strain and plane stress. In the present analysis, both the elastic and the plastic deformations are considered, thus the complete elastic and plastic stress asymptotic solutions are evaluated. Numerical examples are shown to demonstrate the accuracy and effectiveness of the present method.

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

AbstractIn 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.

Two dimensional displacement and stress fields for tri-material V-notches and sharp inclusions in anisotropic plates

The singular behavior for multi-material V-notch notched plates is studied in this paper, considering both isotropic and anisotropic materials. Firstly, by introducing the asymptotic expansions of displacements near the apex of the sharp notch, the elastic governing equations are transformed into eigen ordinary differential equations. The stress singularity orders and eigen angular functions are then obtained solving the established equations by using the interpolating matrix method. Secondly, using in combination the results from proper finite element analyses and the asymptotic analytical solution derived, an over-deterministic method is employed to calculate the amplitudes of the coefficients in the asymptotic expansions of displacements and stresses near the V-notch tip. The efficiency of the proposed method is assessed analyzing several notched specimens made of different materials. The accuracy of the notch stress intensity factors and the displacement and stress fields are evaluated by comparison with results available in the literature, as well as the results obtained from a number of finite element analyses.

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.

Singularity characteristic analysis for anti-plane accelerated propagating V-notches

By introducing the displacement asymptotic expansion around the vertex of an anti-plane V-notch into the elasto-dynamics equation, the singularity characteristic analysis for an accelerated propagating V-notch is transformed into a problem of solving a characteristic ordinary differential equation. The singularity can be generated once the interpolating matrix method is applied to solve the established characteristic equation. The singularities for anti-plane single material and bi-material propagating V-notches are investigated under different initial propagating velocities and accelerations respectively. For the single material anti-plane V-notch, the singularity becomes stronger with the decrease of notch opening angle, and it becomes stronger with the increase of the direction angle of initial propagating velocity and acceleration. It turns stronger with the increase of propagating speed when the propagating direction of initial velocity is greater than 45°, otherwise, the opposite conclusion may be drawn on the other hand. For the bi-material anti-plane V-notch, the singularity becomes weaker with the increase of propagating acceleration and it becomes stronger with the increase of the ratio of transverse wave speed.

Transverse free vibration analysis of a tapered Timoshenko beam on visco-Pasternak foundations using the interpolating matrix method

The characteristics of transverse free vibration of a tapered Timoshenko beam under an axially conservative compression resting on visco-Pasternak foundations are investigated by the interpolating matrix method. The research is executed in view of a three-parameter foundation which includes the effects of the Winkler coefficient, Pasternak coefficient and damping coefficient of the elastic medium. The governing equations of free vibration of a non-prismatic Timoshenko beam under an axially conservative force resting on visco-Pasternak foundations are transformed into ordinary differential equations with variable coefficients in light of the bending rotation angle and transverse displacement. All the natural frequencies orders together with the corresponding mode shapes of the beam are calculated at the same time, and a good convergence and accuracy of the proposed method is verified through two numerical examples. The influences of foundation mechanical characteristics together with rotary inertia and shear deformation on natural frequencies of the beam with different taper ratios are analyzed. A comprehensive parametric numerical study is carried out emphasizing the primary parameters that describe the dynamic property of the beam.

Singularity evaluation for single material and bi-material propagating V-notches

By introducing displacement asymptotic expansion around the vertex of a propagating V-notch into elasticity governing equation, radial boundary conditions and interfacial continuity conditions, the singularity eigen analysis for a propagating V-notch is transformed into solving a set of eigen ordinary differential equations. The singularity orders and corresponding eigen angular functions for a propagating V-notch can be obtained after applying interpolating matrix method in order to solve the established eigen equation. The singularities for single material and bi-material propagating V-notch are respectively analyzed under different material properties, propagating directions and propagating velocities. It is found that the relationship between stress singularity and propagating velocity depends on the propagating direction as well as the singularity variation with respect to propagating velocity is not monotonous. For a single material V-notch, the singularity will be the weakest when the propagating direction is along the bisector of V-notch, and it will be the strongest when the propagating direction is perpendicular to the notch bisector. For a bi-material V-notch, the strongest singularity for a bi-material V-notch occurs when the propagating direction is along the harder material.

Electric and elastic singularity eigenanalysis of the V-notch in a piezoelectric Reissner plate

The singular eigenstate with respect to the stress and electric displacement at the vertex of a V-notch in a piezoelectric plate is investigated. The displacement and electric potential near the vertex of a V-notch are expressed as series asymptotic expansions, and their typical terms are substituted into the governing equations to establish the singularity eigenequations, from which the singularity orders and angular eigenfunctions can be yielded by the interpolating matrix method. The coupling effect of elastic and electric singularities can be found by distinguishing the singularity orders from their angular eigenfunctions. There are most four singularity orders for the V-notch in a piezoelectric Reissner plate, which is one more than the one in a homogeneous plate. Two orders are corresponding to the singularity of bending moment and electric displacement in the thickness direction, and another two orders are with respect to the singularity of shear force and electric displacement in the mid-plane. The free and electrically opened, and clamped and electrically closed boundary conditions are respectively introduced to investigate the influence of boundary conditions on the singularity. It is found that the singularity orders with respect to bending moment and electric displacement in the thickness direction change with boundary conditions, while the singularity orders with respect to shear force and electric displacement in the mid-plane are identical under these two kinds of boundary conditions.

Analysis of plastic stress singularities of cracks and wedges under the plane stress conditions

The aim of this paper is to analyze the plastic stress singularities of the V-shaped notches and cracks under the plane stress conditions and mixed mode. Based on the formulation of assuming an asymptotic displacement field with respect to the radial coordinate centered at the notch tip, the stress components near the notch tip are expressed as the asymptotic series expansions in term of the radial coordinate by satisfying the plastic yield criterion and the nonlinear stress-strain relation. Then, the series expansions of the stress and displacement components are substituted into the governing differential equations of the plastic theory. Hence, the nonlinear second-order system of ordinary differential equations (ODEs) with the plastic stress singularity orders and the associated eigenfunctions are deduced. The interpolating matrix method is applied to solving the nonlinear eigenvalue problems of the ODEs by an iteration procedure. Thus, the leading plastic stress singularity orders and the associated eigenvectors of the displacement and stress fields in the notch tip region are achieved simultaneously. In addition, the numerical examples are given to demonstrate the accuracy and the effectiveness of the present approach to solve the plastic stress singularities of the plane V-shaped notches and cracks. Furthermore, some mechanical behaviors related to the plastic stress singularities in the notch tip region are investigated.

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.

Investigation of singularity orders and eigen-angular functions for V-notches in radially inhomogeneous materials

ABSTRACTThe singularities of V-notches in a material whose modulus of elasticity varies as a power function of the r-coordinate are investigated. The eigen-equations are derived by substituting the asymptotic expansions of displacement fields around a notch vertex into the elasticity equilibrium equations. The radial boundary conditions are also presented by a combination of stress singularity orders and eigen-angular functions. The singularity orders and eigen-angular functions can be calculated simultaneously by solving a set of eigen ordinary differential equations with variable coefficients, which are solved by the interpolating matrix method developed by some of the authors before without the iterative process. The accuracy of the proposed method is verified by comparing the present results with the reference ones when the inhomogeneous material is degraded into a homogeneous one. Then, the stress singularities of V-notches in the radially inhomogeneous material under the plane and anti-plane loadings are investigated, respectively. The results show that the stress singularity of a V-notch in the radially inhomogeneous material under the plane loading is more serious than the one under the anti-plane loading. The plane V-notch under the clamped–clamped boundary condition presents the stress singularity at a smaller notch angle α than the one under the free–free boundary condition. The radially inhomogeneous bulgy V-notch even presents singularity. In addition, the stress singularity becomes stronger with the increase of the exponent c in the variation function of the elasticity modulus.