On the calculation of derivatives of stress intensity factors for multiple cracks

In this paper, the work of Lin and Abel [Lin SC, Abel JF. Variational approach for a new direct-integration form of the virtual crack extension method. Int J Fract 1988;38:217-35] is further extended to the general case of multiple crack systems under mixed-mode loading. Analytical expressions are presented for stress intensity factors and their derivatives for a multiply cracked body using the mode decomposition technique. The salient feature of this method is that the stress intensity factors and their derivatives for the multiple crack system are computed in a single analysis. It is shown through two-dimensional numerical examples that the proposed method gives very accurate results for the stress intensity factors and their derivatives. It is also shown that the variation of mode I and II displacements at one crack-tip influence the mode I and II stress intensity factors at any other crack. The computed errors were about 0.4-3% for stress intensity factors, and 2-4% for their first order derivatives for the mesh density used in the examples.     

Reduction in energy release rate for mode I fracture of a fibre with a cracked coating layer due to small-scale interfacial debonding

In order to predict the effect of small-scale interfacial debonding on the energy release rate at a crack tip for mode I fracture of a fibre with a cracked coating layer, an approximate calculation method has been presented. The relation of debonding length, thickness of the coating layer and ratio of elastic modulus of the coating layer to that of the fibre, to the energy release rate of the fibre was calculated for some examples. It was demonstrated that small-scale debonding reduces the energy release rate and, therefore, effectively prevents reduction in fibre strength.     

Energy release rate computation by an inverse flexibility method

This paper presents the “Inverse Flexibility Method” which allows substantial gain in time when computing energy release rates (or stress intensity factors) for various lengths of a growing crack. Reduction ratio is of order of ten compared with classical methods. Despite the relative coarseness of meshes used, when practicing finite element analysis, accuracy of results remains good as shown by examples presented (metallic cracked plate and composite specimens).     

On the calculation of energy release rates for cracked laminates with residual stresses

Prior methods for calculating energy release rate in cracked laminates were extended to account for heterogeneous laminates and residual stresses. The method is to partition the crack tip stresses into local bending moments and normal forces. A general equation is then given for the total energy release rate in terms of the crack-tip moments and forces and the temperature difference experienced by the laminate. The analysis method is illustrated by several example test geometries. The examples were verified by comparison to numerical calculations. The residual stress term in the total energy release rate equation was found to be essentially exact in all example calculations.     

Eigenfunction expansion variational method for stress intensity factor and T-stress evaluation of a circular cracked plate

Summary In this paper, the eigenfunction expansion variational method (Abbreviated as EEVM) is developed to solve the T-stress problem of the circular cracked plate. In the traction boundary value problem, EEVM is equivalent to the theorem of least potential energy in elasticity. Therefore, EEVM possesses a clear physical meaning. EEVM does not need any boundary collocation scheme. For the circular cracked plate, the following boundary value problems are solved: (a) with a uniform normal loading on the boundary, (b) with a partial loading on the boundary, (c) under mixed boundary condition. For the circular cracked plate with applied concentrated forces, after using the superposition principle and EEVM, the boundary value problem is solved. In the numerical examples, many computed results for stress intensity factor (SIF) and T-stress are presented. Some of computed results for T-stress are first presented in this paper.     

Stress intensity factors for cracked triangular cross‐section thin‐walled tubes

For one kind of finite‐boundary crack problems, the cracked equilateral triangular cross‐section tube, an analytical and very simple method to determine the stress intensity factors has been proposed based on a new concept of crack surface widening energy release rate and the principle of virtual work. Different from the classical crack extension energy release rate, the crack surface widening energy release rate can be defined by the G*‐integral theory and expressed by stress intensity factors. This energy release rate can also be defined easily by the elementary strength theory for slender structures and expressed by axial strains and loads. These two forms of crack surface widening energy release rate constitute the basis of a new analysis method for cracked tubes. From present discussions, a series of stress intensity factors are derived for cracked equilateral triangular cross‐section tubes. Actually, the present method can also be applied to cracked polygonal tubes.     

A modelling technique for calculating stress intensity factors for structures reinforced by bonded straps. Part II: Validation

In this second part of the two-part paper validation of the 2D FE modelling technique described in the first part is presented for a range of test configurations. Each mechanism that influences crack growth behaviour of strap reinforced structures is modelled for different substrate geometries, strap materials and dimensions in order to test the accuracy and robustness of the methodology. First, calculated through-thickness strain energy release rate distribution is compared with the result of a 3D FE model to validate this 2D model. Second, calculated disbond areas, thermal residual stresses and their redistribution with crack propagation are validated against experimental measurements. Third, influence of geometric nonlinearity and the need to use the alternate analysis method described in part I are demonstrated by examples, and errors generated by not following this analysis rule are given. Finally, using the 2D model calculated stress intensity factors, fatigue crack growth rates and lives are predicted for different specimens, strap materials and applied stress levels and are compared with the experimental tests. Good or acceptable agreement has been achieved for each case.     

Calculation of eigenpair derivatives for asymmetric damped systems with distinct and repeated eigenvalues

An algorithm is derived for the computation of eigenpair derivatives of asymmetric quadratic eigenvalue problem with distinct and repeated eigenvalues. In the proposed method, the eigenvector derivatives of the damped systems are divided into a particular solution and a homogeneous solution. By introducing an additional normalization condition, we construct two extended systems of linear equations with nonsingular coefficient matrices to calculate the particular solution. The method is numerically stable, and the homogeneous solutions are computed by the second‐order derivatives of the eigenequations. Two numerical examples are used to illustrate the validity of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.     

Fractal finite element method based shape sensitivity analysis of multiple crack system

This paper presents fractal finite element based continuum shape sensitivity analysis for a multiple crack system in a homogeneous, isotropic, and two dimensional linear-elastic body subjected to mixed-mode (modes I and II) loading conditions. The salient feature of this method is that the stress intensity factors and their derivatives for the multiple crack system can be obtained efficiently since it only requires an evaluation of the same set of fractal finite element matrix equations with a different fictitious load. Three numerical examples are presented to calculate the first-order derivative of the stress intensity factors or energy release rates.     

Stress intensity factors for cracked rectangular cross-section thin-walled tubes

For cracked structural rectangular thin-walled tubes, an exact and very simple method to determine the stress intensity factors has been proposed based on a new concept of crack surface widening energy release rate. Unlike the classical crack extension energy release rate, the crack surface widening energy release rate can be expressed by the G^*-integral and elementary strength theory of materials for slender cracked structures. From present discussions, a series of new and exact solutions of stress intensity factors are derived for cracked rectangular and square tubes. The present method can also be applied to cracked polygon thin-walled tubes.     

Equivalent homogeneous finite element for composite materials via reissner principle. Part I: Finite element for plates

The stiffness matrix in the finite element method for multi-layered materials is generally computed by expressing the strain energy in each layer and adding them together. In order to lower the computing time, which may be prohibitive if the number of layers is high, and to get accurate information on the stresses, especially on transverse shear stresses, we present a new finite element using the Reissner principle. In the first part the case of plates will be detailed: extensions to shell problems will be presented in the second part. The efficiency of the method is tested on a special analytic solution, and some examples are given.     

Computation of stress intensity factors of interface cracks based on interaction energy release rates and BEM sensitivity analysis

Stress intensity factors of bimaterial interface cracks are evaluated based on the interaction energy release rates. The interaction energy release rate is defined based on the energy release rates of a cracked body, corresponding to two independent loading conditions, actual field and an auxiliary field, and is related to the sensitivities of the potential energies for crack extensions. The potential energy of a cracked body is expressed with a domain integral, which is converted to a boundary integral expression by applying the divergence theorem. By differentiating this expression with the crack length, a boundary integral expression for the interaction energy release rate is obtained. The boundary integral representation for the interaction energy release rate involves the displacement, the traction, and their sensitivity coefficients with respect to the crack length. The boundary element sensitivity analyses are used to calculate these quantities accurately. A regularized boundary integral equation relating the boundary displacement and traction is differentiated with respect to an arbitrary shape parameter to derive the regularized boundary integral equation for the sensitivity coefficients of the boundary displacement and traction. The proposed approach is applied to several cracks in dissimilar media and the results are compared with those obtained by the conventional approach based on the extrapolation method. The analytical displacement and stress solutions for an interface crack between two infinite dissimilar media subjected to uniform stresses at infinity are used to give the auxiliary field, in which the values of the stress intensity factors are known. It is demonstrated that the present method can give accurate results for the stress intensity factors of various bimaterial interface cracks under coarse mesh discretizations.     

Stress intensity factors for cracked T-sections and dynamic behaviour of T-beams

This paper presents an extension of a simple and convenient method proposed by Kienzler and Herrmann [An elementary theory of defective beams. Acta Mech 1986;62:37-46] to estimate the stress intensity factors of cracked beams and bars. This method is based on an elementary beam theory estimation of the strain energy release as the crack is widened into a fracture band. As an extension, the power of the simple beam theory analysis is demonstrated by application to cracked T-beams subjected to a bending moment, shear forces and a torsion. Moreover, the present work addresses the coupled bending-torsional vibration of cracked T-beams within the context of the dynamic stiffness matrix method of analysing structures.     

Crack-tip force method for computing energy release rate in delaminated plates

A new method called the crack-tip force method (CTFM) is derived for computing the energy release rate in delaminated beams and plates. In this method the delaminated plate is divided into two laminates on either side of the plane of delamination. The interaction forces, called crack-tip forces, between the sub-laminates at the crack-tip are computed. The energy release rate is expressed as a quadratic function of the crack-tip forces and the plate compliance coefficients. The CTFM is compared to the virtual crack closure technique (VCCT) as well as to a previously derived method called the strain energy density method using double cantilevered beam specimens as examples. The CTFM is found to be very efficient as the crack-tip forces are part of the solution of finite element analysis of delaminated plates, and they can be readily used to compute the point-wise energy release rate along the delamination front.     

Moving least squares differential quadrature method and its application to the analysis of shear deformable plates

A moving least squares differential quadrature (MLSDQ) method is developed and employed for the analysis of moderately thick plates based on the first‐order shear deformation theory (FSDT). To carry out the analysis, the governing equations in terms of the generalized displacements (transverse deflection and two rotations) of the plate are formulated by employing the moving least squares approximation. The weighting coefficients used in the MLSDQ approximation are computed through a fast computation of shape functions and their derivatives. Numerical examples illustrating the accuracy, stability and convergence of the MLSDQ method are presented. Effects of support size, order of completeness and node irregularity on the numerical accuracy are investigated. Copyright © 2003 John Wiley & Sons, Ltd.     

Free vibration analysis of uniform and stepped cracked beams with circular cross sections

This paper presents a novel numerical technique applicable to analyse the free vibration analysis of uniform and stepped cracked beams with circular cross section. In this approach in which the finite element and component mode synthesis methods are used together, the beam is detached into parts from the crack section. These substructures are joined by using the flexibility matrices taking into account the interaction forces derived by virtue of fracture mechanics theory as the inverse of the compliance matrix found with the appropriate stress intensity factors and strain energy release rate expressions. To reveal the accuracy and effectiveness of the offered method, a number of numerical examples are given for free vibration analysis of beams with transverse non-propagating open cracks. Numerical results showing good agreement with the results of other available studies, address the effects of the location and depth of the cracks on the natural frequencies and mode shapes of the cracked beams. Modal characteristics of a cracked beam can be employed in the crack recognition process. The outcomes of the study verified that presented method is appropriate for the vibration analysis of uniform and stepped cracked beams with circular cross section.     

Time‐domain vector potential technique for the meshless radial point interpolation method

A time‐domain meshless algorithm based on vector potentials is introduced for the analysis of transient electromagnetic fields. The proposed numerical algorithm is a modification of the radial point interpolation method, where radial basis functions are used for local interpolation of the vector potentials and their derivatives. In the proposed implementation, solving the second‐order vector potential wave equation intrinsically enforces the divergence‐free property of the electric and magnetic fields. Furthermore, the computational effort associated with the generation of a dual node distribution (as required for solving the first‐order Maxwell's equations) is avoided. The proposed method is validated with several examples of 2D waveguides and filters, and the convergence is empirically demonstrated in terms of node density or size of local support domains. It is further shown that inhomogeneous node distributions can provide increased convergence rates, that is, the same accuracy with smaller number of nodes compared with a solution for homogeneous node distribution. A comparison of the magnetic vector potential technique with conventional radial point interpolation method is performed, highlighting the superiority of the divergence‐free formulation. Copyright © 2015 John Wiley & Sons, Ltd.     

Probabilistic fracture mechanics by Galerkin meshless methods – part I: rates of stress intensity factors

 This is the first in a series of two papers generated from a study on probabilistic meshless analysis of cracks. In this paper (Part I), a Galerkin-based meshless method is presented for predicting first-order derivatives of stress-intensity factors with respect to the crack size in a linear-elastic structure containing a single crack. The method involves meshless discretization of cracked structure, domain integral representation of the fracture integral parameter, and sensitivity analysis in conjunction with a virtual crack extension technique. Unlike existing finite-element methods, the proposed method does not require any second-order variation of the stiffness matrix to predict first-order sensitivities, and is, consequently, simpler than existing methods. The method developed herein can also be extended to obtain higher-order derivatives if desired. Several numerical examples related to mode-I and mixed-mode problems are presented to illustrate the proposed method. The results show that first-order derivatives of stress-intensity factors using the proposed method agree very well with reference solutions obtained from either analytical (mode I) or finite-difference (mixed mode) methods for the structural and crack geometries considered in this study. For mixed-mode problems, the maximum difference between the results of proposed method and finite-difference method is less than 7. Since the rates of stress-intensity factors are calculated analytically, the subsequent fracture reliability analysis can be performed efficiently and accurately. Received 20 February 2001 / Accepted 19 December 2001     

Free out-of-plane vibration of cracked curved beams on elastic foundation by estimating the stress intensity factor

Abstract

This paper is concerned with evaluating stress intensity factors (SIFs), for a cracked curved beam of rectangular cross section, applying an approach which allows us to estimate the strain energy release rate. The beam is located on an elastic foundation. The out-of-plane vibration of the beam is investigated. This approach requires an additional factor namely correction factor, on the basis of the energy release zone slope to approximate the SIFs. The initial curvature of the beam, however, adds some complication in using this factor. The second part of this study is investigating a numerical approach, namely differential quadrature element method (DQEM), to gain the natural frequencies of the cracked beam. This method is applied to show the application of the SIFs to calculate the compliance of the cracked section for modeling the crack. The other method which is used to obtain the natural frequencies is the finite element method (FEM). The results of these two methods are found to be in good agreement, which shows the precision of the stress intensity factors of the cracked beam.     

Strain energy release rates for a straight-fronted edge crack in a circular bar subject to bending

The strain energy release rate for a straight-fronted edge crack in a bar of circular cross section subjected to pure bending is determined. The cracked bar is modelled with two-dimensional plane-stress finite elements and strain energy release rates, determined from this model, are shown to be in close agreement with existing results for a bar subjected to three-point bending in which strain energy release rates were determined by measuring the compliance of the bar experimentally. The strain energy release rates for a crack in the circular cross section bar are found to be lower than those in a rectangular cross section bar having the same relative crack length and subjected to the same bending moment. Previously determined results for uniform tension are superimposed to obtain strain energy release rates for a circular cross section bar which is subjected simultaneously to a tensile load and a bending moment.