Generalized Stress Intensity Factors Research Articles

Dynamic fracture analysis of the soil-structure interaction system using the scaled boundary finite element method

Dynamic fracture analysis of the soil-structure interaction system by using the scaled boundary finite element method is presented in this paper. The polygon scaled boundary finite elements, which have some salient features to model any star convex polygons, are employed for modelling the near-field bounded domains. A procedure for coupling the bounded domains with an improved continued-fraction-based high-order transmitting boundary is established. The formulations of the soil-structure interaction system are coupled via the interaction force vector at the interface. The dynamic stress intensity factors and T-stress are extracted according to the definition of the generalized stress intensity factors. The dynamic stress intensity factors of the coupled system are evaluated accurately and efficiently. Two numerical examples are demonstrated to validate the developed method.

An efficient enrichment strategy for modeling stress singularities in isotropic composite materials with X-FEM technique

An efficient enrichment strategy is presented for modeling stress singularities in multi-material problems and crack-tips terminating at a bi-material interface within the X-FEM framework. The eigen-function expansion method is employed to evaluate eigen-values and asymptotic fields around singular points. The generalized stress intensity factors are defined on the basis of direct and interaction integral techniques. Numerical examples are solved for different types of eigen-values, and the convergence study is performed to investigate the accuracy of the proposed enrichment strategy. The results indicate that the proposed method has superior accuracy, higher convergence rate and lower condition numbers over the other methods.

Incomplete contacts in partial slip subject to varying normal and shear loading, and their representation by asymptotes

We develop a method for the solution of partial slip contact problems suffering complex loading cycles where, generally, the normal load, shear force and, potentially, differential bulk tensions are all functions of time, using an edge-asymptote approach. The size of the slip zone and local shear traction distribution are revealed as functions of time. The results are then re-worked in asymptotic form, so that they do not hinge on inherent symmetry and anti-symmetry conditions for the contact overall, and are of general applicability. The multipliers on the local solutions (generalised stress intensity factors) are also appropriate as a means of taking laboratory tests quantifying fretting fatigue and employing them to wholly different prototypical problems.

Investigation on Filippi’s stress equation for V-shaped notch with rounded vertex Part I: Mode I stresses

In the present paper, in order to investigate systematically the effectiveness of Filippi’s stress equation applied at evaluating stresses generated in a rounded V-shaped notch with a finite depth, the stresses generated in a semi-infinite plate with rounded V-shaped notch subjected to a uniform tension load at remote are used as a base data to examine the applicability of Filippi’s stress equation. It is found that Filippi’s equation gives an excellent approximation to the stresses along the notch bisector for rounded V-shaped notches. The generalized stress intensity factor Kρ,IV(r) is almost constant along the notch bisector in the region where the stresses are governed by the singular stress field in a sharp notch; the peak discrepancy between Filippi’s equation and the numerical analysis is less than 7% when the notch opening angle γ=0°,2.5% when γ=30° and 5% when γ=90° and 120°. Also it is found that Filippi’s equation is effective in evaluating the stresses along direction different from the notch bisector, however, as the calculation angle θ with respect to the notch bisector increases, the discrepancy between Filippi’s equation and the numerical analysis becomes larger. Therefore, it is advisable to limit calculation points of stress within a range of θ⩽30°. Moreover, values of the maximum stress σmax predicted by various techniques are also examined using the numerical-analysis results of the semi-infinite plate with a V-shaped notch, and an analytical approach for predicting the constant a1 in Filippi’s equation is proposed.

Analytical-Numerical Determination of Stress Distribution around a Tip of Polygon-Like Inclusion

A stress distribution in vicinity of a tip of polygon-like inclusion exhibits a singular stress behaviour. Singular stresses at the tip can be a reason for a crack initiation in composite materials. Knowledge of stress field is necessary condition for reliable assessment of such composites. A stress field near the general singular stress concentrator can be analytically described by means of Muskhelishvili plane elasticity based on complex variable functions. Parameters necessary for the description are the exponents of singularity and Generalized Stress Intensity Factors (GSIFs). The stress field in the closest vicinity of the SMI tip is thus characterized by 1 or 2 singular exponents (1 - λ) where, 0<Re (λ)<1, and corresponding GSIFs that follow from numerical solution. In order to describe stress filed further away from the SMI tip, the non-singular exponents for which 1<Re (λ), and factors corresponding to these non-singular exponents have to be taken into account. Analytical-numerical procedure of determination of stress distribution around a tip of sharp material inclusion is presented. Parameters entering to the procedure are varied and tuned. Thus recommendations are stated in order to gain reliable values of stresses and displacements.

Sharp notch roots subject to in-plane and anti-plane loading: An alternative normalisation of the eigenfields & their application to example problems

The state of stress at a sharp notch is characterised by the classical Williams eigenfunction expansion for in-plane loading, and a corresponding anti-plane solution. Excluding the case when the internal notch angle is 2π (a crack), the solutions have differing orders of singularity, and hence, between any pair of modes of loading, there is an internal implied length scale. Here we bring out those length scales, and show how the three generalised stress intensity factors may be replaced by equivalent parameters having a much more direct physical interpretation. The revised form of the semi-infinite notch solutions are then applied to example geometries, showing how the characteristics of the notch-root process zone may be inferred without further calculation.

Exact solutions for anti-plane deformation of a cylindrically monoclinic wedge under concentrated loads

This article studies anti-plane deformation of a cylindrically monoclinic wedge under concentrated loads. To the best of the author's knowledge, exact solutions to this type of wedge problem under concentrated loads are not available in the literature. By applying a newly defined argument to the displacements in terms of a holomorphic function in cooperation with a complex analogous Mellin transform, the exact solutions for the considered problems are obtained. The novel arrangements greatly simplify the formulation and result in concise complex shear stress equations that can be solved. With prescribed boundary settings, the closed-form solutions for Green's functions can be derived conveniently. Exact solutions are obtained for two kinds of boundary conditions. The stress fields obtained from the two cases are presented and discussed for certain combinations of anisotropic parameters. Contours of the generalized stress intensity factor, which is related to the direction of approach to the wedge apex and the material properties, are also shown. In addition, the results of a problem with distributed loads agree well with numerical solutions. The proposed method clarifies and simplifies the analysis and solution of related wedge problems. In determining the reduced orthotropic and isotropic cases to solutions under anti-plane deformation, the results are generated naturally.

On the stress state in rectilinear anisotropic thick plates with blunt cracks

AbstractIn this paper the analytical solution for the stress fields in the close neighbourhoods of blunt cracks in thick anisotropic plates is provided. As a first step the equations of the three‐dimensional theory of elasticity are successfully reduced to two un‐coupled equations in the two‐dimensional space. Later on, the 3D stress field solution for anisotropic plates with blunt cracks is presented and its degree of accuracy discussed comparing theoretical results and numerical data from 3D FE analyses. Eventually, relevant expressions for the Generalised Stress Intensity Factors for blunt cracks in orthotropic plates under mode I, II and III are provided.

Modeling crack propagation with the extended scaled boundary finite element method based on the level set method

The extended scaled boundary finite element method (X-SBFEM) based on the level set method (LSM) is proposed in this paper to combine the advantages of the scaled boundary finite element method (SBFEM) and the extended finite element method (XFEM). The level set method (LSM) algorithm is applied to further develop the X-SBFEM, especially for the crack propagation problem. The Heaviside enrichment function is used to represent a jump across a discontinuity surface in a split element, and the non-smooth behavior around the crack tip is described using the semi-analytical SBFEM. The stiffness of the region containing the crack tip is computed directly, and the generalized stress intensity factors of many types of singularities are obtained directly from their definitions using consistent formulas. In the numerical simulations, a square plate with an edge crack under tension, a three-point bending beam, a four-point shear beam and a dam (the Koyna dam) with a single propagating crack are modeled. The results show that the proposed X-SBFEM is capable of calculating the stress intensity factors of cracks and predicting crack trajectories and load–displacement relations accurately. An analysis of the sensitivity of the parameters is employed to demonstrate that various mesh densities and crack propagation step lengths led to consistent results.

Reconstruction of a 2D stress field around the tip of a sharp material inclusion

The stress distribution in the vicinity of a sharp material inclusion (SMI) tip exhibits a singular stress behavior. The strength of the stress singularity depends on material properties and geometry. The SMI is a special case of a general singular stress concentrator (GSSC). The stress field near a GSSC can be analytically described by means of Muskhelishvili plane elasticity based on complex variable function methods. Parameters necessary for the description are the exponents of singularity and generalized stress intensity factors (GSIFs). The stress field in the closest vicinity of an SMI tip is thus characterized by 1 or 2 singular exponents (λ-1), for which 0<Re(λ)<1, and corresponding GSIFs. In order to describe a stress field further away from an SMI tip, the non-singular exponents, 1<Re(λ), and factors corresponding to these non-singular exponents have to be taken into account. For given boundary conditions of the SMI, the exponents are calculated as an eigenvalue problem. Then, by formation of corresponding eigenvectors, the stress or displacement angular functions for each stress or displacement series term are constructed. The contribution of each stress or displacement series term function to the total stress and displacement field is given by the corresponding GSIF. The GSIFs are calculated by the over deterministic method (ODM), which finds a solution of an over-determined system of linear equations by the least squares method. On the left-hand side of the system are the displacement series term functions multiplied by unknown GSIFs, while the right-hand is formed by results of finite element analysis (FEA). Thus the results of FEA, namely nodal displacements in the radial and tangential direction, are employed in order to obtain the GSIFs. In the numerical example, the stress field for particular bi-material configurations and geometries is reconstructed using i) singular terms only ii) singular and non-singular terms. The reconstructed stress field polar plots are compared with FEA results.

Numerical and experimental investigation of a complete contact problem by comparing with an asymptotic analysis

The stress and displacement fields of a complete contact problem are investigated using a finite element analysis (FEA) and a Moiré experiment, respectively. Asymptotic analysis of an adhered contact condition is preceded for comparison. Aluminum and copper alloys are used as the contact materials, from which specimens of different contact angles are fabricated. The FEA scheme is validated through comparison with the displacement distributions that are measured in the Moiré experiment. In turn, the generalized stress intensity factors are calculated using the validated FE models. These factors are used to evaluate the dimensional stress distributions in the asymptotic analysis, which are then compared with the FEA results. Excellent agreement in stress components is found between the asymptotic and FE methodologies. The relative error of the stress components σ rr and σ θθ between the methodologies was less than 3.8% within the range from the contact edge to 1 mm, and it increased gradually as the distance from the contact edge increased. In addition, a slight discrepancy is found in the contact of an elastically dissimilar case.

Elastic Equilibrium of a Space Containing a Thin Curved Elastic Inclusion in Longitudinal Shear

We study the problem of antiplane deformation of an isotropic medium containing a thin elastic curved inhomogeneity. The methods used for the solution of this problem are based on the application of the method of jump functions and the conditions of interaction of the matrix containing a thin curvilinear inclusion and the solution of the resultant system of singular integral equations with Cauchy-type kernels by the collocation method. Numerous examples are considered. The results of evaluation of the stress intensity factors for a crack and an absolutely rigid inclusion along a circular arc are compared with the corresponding analytic results. For a crack along a symmetric parabolic arc, the stressed state is thoroughly investigated. We also study the influence of the modulus of elasticity and the shape of the curvature of inhomogeneity (circular arc, parabola, or a half of a cosine curve) on the generalized stress intensity factors. It is shown that, for the stress intensity factors near the tip of the inhomogeneity, the inclination of the tangent at the tip to the plane of application of the shear forces is of determining importance.

Critical value of the generalized stress intensity factor for a crack perpendicular to an interface

When a crack tip impinges upon a bi-material interface, the order of the stress singularity will be equal to, less than or greater than one-half. The generalized stress intensity factors have already been determined for some such configurations, including when a finite-length crack is perpendicular to the interface. However, for these non-square-root singular stresses, the determination of the conditions for crack growth are not well established. In this investigation, the critical value of the generalized stress intensity factor for tensile loading is related to the work of adhesion by using a cohesive zone model in an asymptotic analysis of the separation near the crack tip. It is found that the critical value of the generalized stress intensity factor depends upon the maximum stress of the cohesive zone model, as well as on the Dundurs parameters (αandβ). As expected this dependence on the cohesive stress vanishes as the material contrast is reduced, in which case the order of the singularity approaches one-half.

Closed-form field in an infinite space of transversely isotropic multiferroic composite medium with an elliptical or penny-shaped crack: 3D exact analysis

This paper studies the mode I problem of an elliptical or penny-shaped crack contained in an infinite space of multiferroic composite material with transverse isotropy. Closed-form analytical solutions and numerical results are presented in this work in a systematic way. The crack is postulated to be either electrically/magnetically permeable or impermeable. For the first time, boundary integro-differential equations, corresponding to the crack with different magneto-electric properties, are established using the generalized method of potential theory in conjunction with the static general solutions. For four physical cases, closed-form coupled magneto-electro-elastic field variables in the space are explicitly obtained. Crack surface displacement, generalized stress intensity factor and crack compliance matrix generalized from the rock mechanics all turn out to be an intrinsic combination of a material factor and a geometrical factor. The weak and strong interactions in multiferroic composite media are exhaustively accounted for. The obtained analytical solutions may serve as benchmarks for numerous forthcoming simplified and numerical studies.

Screw dislocation near a piezoelectric oblique edge crack

The interaction between a screw dislocation and an oblique edge crack in a semi-infinite piezoelectric medium under remote anti-plane mechanical and in-plane electrical loadings is studied in this article. The screw dislocation suffers a finite discontinuity in the displacement and electric potential across the slip plane. The dislocation also has a line force and a line charge along its core. In the framework of linear piezoelectricity theory, the analytical solutions are derived from the complex variable and Schwarz–Christoffel transformation. Furthermore, the electric field, stress field, generalized stress intensity factor, image force and crack extension force are formulated exactly. Numerical calculations are then given graphically for studying the effects of crack length, crack angle and load position on these electric mechanical parameters. Besides, the dislocation solutions proposed here can be served as Green’s functions for the analyses of corresponding piezoelectric cracking or distributed loading problems.

2D fracture analysis of magnetoelectroelastic composites by the SBFEM

This paper extends the scaled boundary finite element method (SBFEM) to 2D fracture analysis of magnetoelectroelastic composites. The generalized extended stress intensity factors, i.e., the stress intensity factors, the electric displacement intensity factor and the magnetic induction intensity factor, for different crack configurations and material combinations are evaluated. The governing equations of the SBFEM for magnetoelectroelastic materials are derived based on the virtual work principle. The generalized extended stress intensity factors are determined from the solution of the singular stress, electric displacement and magnetic induction fields by extending the definition of the generalized stress intensity factors. The electrically and magnetically impermeable and permeable crack face boundary conditions are implemented and discussed. Numerical examples including the cracks in magnetoelectroelastic materials, the interface cracks between magnetoelectroelastic bimaterials and magnetoelectroelastic bimaterial wedge are presented to validate the proposed technique. The accuracy of the proposed technique is verified by comparison with the reference results.

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.

Mutual Influence of an Interface Tunnel Crack and An Interface Tunnel Inclusion in a Piecewise Homogeneous Anisotropic Space

The problem of an interface tunnel crack and an interface tunnel rigid inclusion perfectly coupled with a piecewise homogeneous anisotropic space is reduced to a system of six singular integral equations. To solve this system, we proposed an efficient numerical-analytic method. The mutual influence of the crack and the inclusion is investigated. In particular, we establish the dependences of the generalized stress intensity factors at the tips of the crack and the inclusion on the relative distance between them.

응착조건의 완전접촉문제 해석: 실험 및 수치해석과 이론해의 비교

Asymptotic method has been often used to theoretically analyze the complete contact problem. The error of the asymptotic results increases as the distance from the contact edge increases. The singularity cannot be properly obtained from a finite element (FE) analysis owing to the finiteness of the element size. In the present work, the complete contact problem in bonded condition is analyzed using a combined experimental-numerical approach to assist and/or compare with the asymptotic results. Al and Cu alloys are used for the material combination of the punch and substrate. 120 and 135 degrees are used for the punch angle. The FE models are validated by comparison of displacement distributions obtained by the FE analysis and experiment. Generalized stress intensity factors are evaluated using the validated FE models. Stress field in the vicinity of the sharp contact edges obtained from the FE and asymptotic analyses are compared. The discrepancies are also discussed.

Analytical solutions of stress singularities in a sectorial plate based on Reissner–Mindlin plate theory

This paper aims at the stress singularities in a sectorial plate based on Reissner–Mindlin plate theory. The Mellin transform is applied to solve the governing equations of the plate. Both the bending stress singularities due to concentrated moments at boundaries and transverse shear stress singularities due to concentrated transverse shear forces at boundaries are investigated. The application of inverse Mellin transform gives the exact forms of singular stress fields near the corner, including the stress singularity orders and the generalized stress intensity factors. For various plate angles and loading conditions, the singular stress parameters are computed by finite element models and the results are compared to the analytical solutions. The comparisons reveal that the stresses obtained by the analytical procedure proposed in this paper are correct.