The mode I crack problem for layered piezoelectric plates

The plane strain singular stress problem for piezoelectric composite plates having a central crack is considered. For the case of the crack which is normal to and ends at the interface between the piezoelectric plate and the elastic layer, the order of stress singularity around the tip of the crack is obtained. The Fourier transform technique is used to formulate the problem in terms of a singular integral equation. The singular integral equation is solved by using the Gaus–Jacobi integration formula. Numerical calculations are carried out, and the main results presented are the variation of the stress intensity factor as functions of the geometric parameters, the piezoelectric material properties and the electrical boundary conditions of the layered composites.     

Fatigue strength of steel and aluminium welded joints based on generalised stress intensity factors and local strain energy values

Weld bead geometry cannot, by its nature, be precisely defined. Parameters such as bead shape and toe radius vary from joint to joint even in well-controlled manufacturing operations. In the present paper the weld toe region is modelled as a sharp, zero radius, V-shaped notch and the intensity of asymptotic stress distributions obeying Williams’ solution are quantified by means of the Notch Stress Intensity Factors (NSIFs). When the constancy of the angle included between weld flanks and main plates is assured and the angle is large enough to make mode II contribution non-singular, mode I NSIF can be directly used to summarise the fatigue strength of welded joints having very different geometry. By using a large amount of experimental data taken from the literature and related to a V-notch angle of 135°, two NSIF-based bands are reported for steel and aluminium welded joints under a nominal load ratio about equal to zero. A third band is reported for steel welded joints with failures originated from the weld roots, where the lack of penetration zone is treated as a crack-like notch and units for NSIFs are the same as conventional SIF used in LEFM. Afterwards, in order to overcome the problem related to the variability of the V-notch opening angle, the synthesis is made by simply using a scalar quantity, i.e. the mean value of the strain energy averaged in the structural volume surrounding the notch tips. This energy is given in closed form on the basis of the relevant NSIFs for modes I and II and the radius R_C of the averaging zone is carefully identified with reference to conventional arc welding processes. R_C for welded joints made of steel and aluminium considered here is 0.28 mm and 0.12 mm, respectively. Different values of R_C might characterise welded joints obtained from high-power processes, in particular from automated laser beam welding. The local-energy based criterion is applied to steel welded joints under prevailing mode I (with failures both at the weld root and toe) and to aluminium welded joints under mode I and mixed load modes (with mode II contribution prevailing on that ascribable to mode I). Surprising, the mean value of ΔW related to the two groups of welded materials was found practically coincident at 2 million cycles. More than 750 fatigue data have been considered in the analyses reported herein.     

Dynamic stress intensity factors due to scattering of harmonic waves by a crack in an infinite medium

An analytical method for calculating dynamic stress intensity factors in the mixed mode (combination of opening and sliding modes) using complex functions theory is presented. The crack is in infinite medium and subjected to the plane harmonic waves. The basis of the method is grounded on solving the two‐dimensional wave equations in the frequency domain and complex plane using mapping technique. In this domain, solution of the resulting partial differential equations is found in the series of the Hankel functions with unknown coefficients. Applying the boundary conditions of the crack, these coefficients are calculated. After solving the wave equations, the stress and displacement fields, also the J‐integrals are obtained. Finally using the J‐integrals, dynamic stress intensity factors are calculated. Numerical results including the values of dynamic stress intensity factors for a crack in an infinite medium subjected to the dilatation and shear harmonic waves are presented.     

A direct method for calculating the stress intensity factor in BEM

The proposed algorithm employs singular crack tip elements in which the stress intensity factor appears as a degree of freedom. The additional degrees of freedom are compensated by constraint conditions which originate from imposing continuity across elements and a contour integration formula. The two benchmark problems indicate the proposed algorithm can accurately predict the stress intensity factor and the distribution of the primary and secondary variables in fracture problems.     

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.     

Stress intensity factor analysis for an interface crack between dissimilar isotropic materials under thermal stress

Thermal stresses, one of the main causes of interfacial failure between dissimilar materials, arise from different coefficients of linear thermal expansion. Two efficient numerical procedures in conjunction with the finite element method (FEM) for the stress intensity factor (SIF) analysis of interface cracks under thermal stresses are presented. The virtual crack extension method and the crack closure integral method are modified using the superposition method. The SIF analyses of some interface crack problems under mechanical and thermal loads are demonstrated. Very accurate mode separated SIFs are obtained using these methods.     

Piezoelectric sensor for stress intensity factor measurement of two dimensional cracks

A piezoelectric sensor for the measurement of stress intensity factors (SIFs) of two dimensional cracks induced in a structure is developed. Two small pieces of piezoelectric elements are adhered near the crack tip so that the piezoelectric elements are placed close to each other and the crack tip’s position is between them. The electric currents from the piezoelectric elements are integrated by integration circuits and the output voltages which are proportional to the electric charge induced in the piezoelectric elements are measured. The SIFs of Mode I (K_I) as well as of Mode II (K_II) based on the piezoelectric constitutive law and fracture mechanics are calculated.     

Generalized stress intensity factors of V-shaped notch in a round bar under torsion, tension, and bending

In this study, generalized stress intensity factors K_I,λ1, K_II,λ2, and K_III,λ4 are calculated for a V-shaped notched round bar under tension, bending, and torsion using the singular integral equation of the body force method. The body force method is used to formulate the problem as a system of singular integral equations, where the unknown functions are the densities of body forces distributed in an infinite body. In order to analyze the problem accurately, the unknown functions are expressed as piecewise smooth functions using three types of fundamental densities and power series, where the fundamental densities are chosen to represent the symmetric stress singularity and the skew-symmetric stress singularity. Generalized stress intensity factors at the notch tip are systematically calculated for various shapes of V-shaped notches. Normalized stress intensity factors are given by using limiting solutions; they are almost determined by notch depth alone, and almost independent of other geometrical parameters. The accuracy of Benthem-Koiter’s formula proposed for a circumferential crack is also examined through the comparison with the present analysis.     

Evaluation of stress intensity factor under pure shear stress in orthotropic material

Various types of composite materials are currently being developed and used for automobiles, airplanes, ships and other structures in response to required service conditions which are getting increasingly more severe. Of growing importance under such circumstances is the study of stress analysis and fracture mechanics for these composite material structures. Particularly, the primary concern in design of structures and machines should be the initiation of cracks due to excessive deformation, delamination in material or other material defects. In evaluating safety, it is indispensable from the structural design point of view that K value should be known by an analysis conducted in advance. In this study, stress intensity factor (mode II) under a pure shear stress was obtained using the photoelastic method and caustic method and applying an isotropic material and orthotropic material (copper fibre epoxy composite (CFEC) developed by the authors), each containing the crack. Results were compared with theoretical values. As a result, this method was found useful and the effect of the direction of the primary axis of this material on the stress intensity factor was clarified.     

An interaction integral method for computing mixed mode stress intensity factors for curved bimaterial interface cracks in non-uniform temperature fields

In finite element analysis the interaction integral has been a useful tool for computing the stress intensity factors for fracture analysis. This work extends the interaction integral to account for non-uniform temperatures in the calculation of stress intensity factors for three dimensional curvilinear cracks either in a homogeneous body or on a bimaterial interface. First, the derivation of the computational algorithm, which includes the additional terms developed by the non-zero gradient of the temperature field, is presented in detail. The algorithm is then implemented in conjunction with commercial finite element software to calculate the stress intensity factors of a crack undergoing non-uniform temperatures on both a homogeneous and a bimaterial interface. The numerical results displayed path independence and showed excellent agreement with available analytical solutions.     

Estimation of mixed-mode stress intensity factors using digital image correlation and an interaction integral

This paper presents a technique for the experimental measurement of stress intensity factors in cracked specimens under mixed-mode loading. This technique is based on full-field measurement using digital image correlation and an interaction integral. Such domain-independent integrals are often used in the finite element method to calculate stress intensity factors. The main advantage of this technique is that the errors made in the estimation of the measured displacement field near the cracks tip do not affect the measurement of the stress intensity factors. The capabilities of the method are illustrated through fracture measurements on compact tension specimens made of maraging steel. Another test under mixed-mode loading is presented.     

Theoretical and experimental determination of magnetic stress intensity factors of a crack in a double cantilever beam specimen

This paper presents the results of an experimental and theoretical investigation of the magnetic fracture behaviour of double cantilever beam (DCB) specimens. DCB tests were conducted on ferritic stainless steel SUS430 in the bore of a superconducting magnet at room temperature. A simple experimental technique using strain gauges was used to determine the stress intensity factor. The experiments show the predicted increase in the stress intensity factor with increasing magnetic field. The theoretical analysis is based on a beam‐plate theory for magnetoelastic interactions in a soft ferromagnetic material. Numerical calculations are carried out, and the stress intensity factor is obtained for several values of magnetic field. A comparison of the stress intensity factor is made between theory and experiment, and the agreement is good for the magnetic field considered.     

Effects of CFRP bond locations on the Mode I stress intensity factor of centre‐cracked tensile steel plates

The fatigue life of cracked steel members can be greatly extended by externally attached carbon fibre reinforced plastics (CFRP), which reduces the stress intensity factors (SIFs) at the crack tip. Access to cracks is sometimes limited and the CFRP has to be attached away from the cracks. There is a lack of knowledge on SIFs for such strengthening scheme. This paper presents the effects of CFRP bond locations on the Mode I SIF of centre‐cracked tensile (CCT) steel plate. The Mode I SIF at the crack tip is calculated using the finite element (FE) models. A correction factor is introduced as a function of CFRP bond location and crack length. The FE results are compared and agree well with experimental tests conducted by the authors. By combining with another two factors (one considering CFRP mechanical properties and the other considering CFRP bond width) derived previously by the authors, SIF formulae are proposed for CFRP reinforced CCT steel plates.     

Computing a-posteriori bounds for stress intensity factors in elastic fracture mechanics

This paper mainly focuses on computing the lower and upper bounds on stress intensity factors in elastic fracture mechanics with an efficient finite element output bound procedure on quantities of interest in engineering. The bounds procedure is obtained by minimizing the quadratic energy functional of output with constraints of equilibrium conditions of mechanics and continuity conditions of finite element space. The computation is based on solving the elemental Neumann residual problems for the bounds on energy norm of error in finite element solutions. The lower and upper bounds on the intensity factors of an open mode and a shear mode elastic fracture problems are computed in this paper.     

Direct estimation of generalized stress intensity factors using a three‐scale concurrent multigrid X‐FEM

A concurrent multigrid method is devised for the direct estimation of stress intensity factors and higher‐order coefficients of the elastic crack tip asymptotic field. The proposed method bridges three characteristic length scales that can be present in fracture mechanics: the structure, the crack and the singularity at the crack tip. For each of them, a relevant model is proposed. First, a truncated analytical reduced‐order model based on Williams' expansion is used to describe the singularity at the tip. Then, it is coupled with a standard extended finite element (FE) method model which is known to be suitable for the scale of the crack. A multigrid solver finally bridges the scale of the crack to that of the structure for which a standard FE model is often accurate enough. Dedicated coupling algorithms are presented and the effects of their parameters are discussed. The efficiency and accuracy of this new approach are exemplified using three benchmarks. Copyright © 2010 John Wiley & Sons, Ltd.     

XFEM analysis of the effects of voids,inclusions and other cracks on the dynamic stress intensity factor of a major crack

This paper is devoted to the extraction of the dynamic stress intensity factor (DSIF) for structures containing multiple discontinuities (cracks, voids and inclusions) by developing the extended finite element method (XFEM). In this method, four types of enrichment functions are used in the framework of the partition of unity to model interface discontinuity within the classical finite element method. In this procedure, elements that include a crack segment, the boundary of a void or the boundary of an inclusion are not required to conform to discontinuous edges. The DSIF is evaluated by the interaction integral. After the effectiveness of the implemented XFEM program is verified, the effects of voids, inclusions and other cracks on the DSIF of a stationary major crack are investigated by using XFEM. The results show that the dynamic effects have an influence on the path independence of the interaction integral, and these voids, inclusions and other cracks have a significant effect on the DSIF of the major crack.     

Generalized stress intensity factors due to steady and transient thermal loads with applications to welded joints

This paper deals with the elastic and plastic stress fields induced by thermal loads in the vicinity of sharp V‐notch tips in plates. Under the hypothesis of steady‐state heat transfer and plane‐strain conditions, the thermal and mechanical problem requires the numerical solution of an ordinary differential equation (ODE) system, obtained by extending the ‘stress function approach’. The intensity of the stress distributions ahead of V‐notch tips can be expressed in terms of thermal notch stress intensity factors (thermal NSIFs), as for external loads. The problem becomes much more demanding in the presence of transient thermal loads. The residual asymptotic stress distribution arising from the solidification of a fusion zone during an arc welding process is obtained by considering different boundary conditions. An aluminium butt‐welded joint is analysed after having modelled the weld toe region as a sharp V‐notch. A finite element (FE) simulation of the welding process is carried out by means of SYSWELD code (version 2004.1) modelling the arc welding torch by means of Goldak's source. Near the weld toe, the intensity of the residual stress field is given in terms of elastic or elastic—plastic generalized NSIFs.     

Multiple penny-shaped cracks interaction in a finite body and their effect on stress intensity factor

In this paper we investigate the stress intensity factors (SIFs) of multiple penny-shaped cracks in an elastic solid cylinder under mode I (axial tension) loading. The cracks are located symmetrically and in parallel to one another in the isotropic cylinder. The fractal-like finite element method (FFEM) is employed to study the interaction of multiple cracks and to demonstrate the efficiency of the FFEM for multiple crack problems. The results show that the SIF values of the inner cracks, which are denoted as crack number 1,2,3,…,(n+1)/2 of a stack of n parallel cracks, are lower than the SIF values of a single crack by between 16% and 48%. Also, the outermost crack, that is the crack closest to the boundaries of a multiple cracked body, has the highest SIF values and is, therefore, likely to fail first.     

Stress intensity factor computation using the method of fundamental solutions: mixed‐mode problems

The method of fundamental solutions is applied to the computation of stress intensity factors in linear elastic fracture mechanics. The displacements are approximated by linear combinations of the fundamental solutions of the Cauchy–Navier equations of elasticity and the leading terms for the displacement near the crack tip. Two algorithms are developed, one using a single domain and one using domain decomposition. Numerical results are given. Copyright © 2006 John Wiley & Sons, Ltd.     

Dynamic stress intensity factors of two 3D rectangular permeable cracks in a transversely isotropic piezoelectric material under a time‐harmonic elastic P‐wave

The dynamic behavior of two 3D rectangular permeable cracks in a transversely isotropic piezoelectric material is investigated under an incident harmonic stress wave by using the generalized Almansi's theorem and the Schmidt method. The problem is formulated through double Fourier transform into three pairs of dual integral equations with the displacement jumps across the crack surfaces as the unknown variables. To solve the dual integral equations, the displacement jumps across the crack surfaces are directly expanded as a series of Jacobi polynomials. Finally, the relations among the dynamic stress field and the dynamic electric displacement filed near the crack edges are obtained, and the effects of the shape of the rectangular crack, the characteristics of the harmonic wave, and the distance between two rectangular cracks on the stress and the electric intensity factors in a piezoelectric composite material are analyzed. Copyright © 2015 John Wiley & Sons, Ltd.