The stress intensity factor for grooved DCB specimens loaded by splitting forces

An approximate analysis is used to obtain the stress intensity factors K for DCB specimens with a groove, loaded by splitting forces P. A review is given for the various approaches that have been taken to obtain the stress intensity factor for an ungrooved DCB specimen. The accuracy of the results is evaluated using the exact solution of K, which was finally obtained by Foote and Buchwald [1]. A so-called improved beam analysis is used to obtain K for a rectangular grooved DCB specimen. When the specimen is loaded by bending moments M, the solution of the stress intensity factor is given by Wu et al. [4].     

Analytical modelling of dynamic fracture and crack arrest in DCB specimens under fixed displacement conditions

An analytical solution via the beam theory considering shear deformation effects is developed to solve the static and dynamic fracture problem in a bounded double cantilever beam (DCB) specimen. Fixed displacement condition is prescribed at the pin location under which crack arrest occurs. In the static case, at first, the compliance function of a DCB specimen is obtained and shows good agreement with the experimental results cited in the literature. Afterward, the stress intensity factor is determined at the crack tip via the energy release rate formula. In the dynamic case, the obtained governing equations for the model are solved supposing quasi‐static treatment for unstable crack propagation. Finally, a closed form expression for the crack propagation velocity versus beam parameters and crack growth resistance of the material is found. It is shown that the reacceleration of crack growth appears as the crack tip approaches the finite boundary. Also, the predicted maximum crack propagation velocity is significantly lower than that obtained via the Euler–Bernoulli theory found in the literature.     

Mode I weight functions for an orthotropic double cantilever beam

Approximate weight functions are proposed and validated numerically for an orthotropic double cantilever beam (DCB) loaded in mode I. They define the stress intensity factor at the crack tip due to a pair of point forces acting on the crack surfaces and have been deduced from the corresponding isotropic result using an orthotropy rescaling technique. The weight functions allow mode I large scale bridging problems in beams and plates to be formulated as integral equations, in terms of stress intensity factors at the crack tip, without the limitations imposed on accuracy by beam theory approximations. The proposed functions are applied to investigate the influence of the orthotropy of the material on the fracture behavior of DCBs in the presence of large scale bridging.     

COHESIVE ZONE MODELLING OF DAMAGE AT THE TIP OF CRACKS IN SLENDER BEAM STRUCTURES

The use of simple beam theory for cohesive zone modelling of the damage response at the crack tip in linear elastic isotropic double cantilever beam (DCB) specimens has been investigated. Damage resistance curves (DR-curves) relating the applied stress intensity factor to the growth of the cohesive zones for beam theory modelling has been compared with two-dimensional elasticity calculations for different material parameters and specimen dimensions. A substantial difference is observed between DR-curves for the two types of models. As expected this difference vanishes for decreasing beam heights. For large beam heights the DR-curves calculated by two-dimensional elasticity are approaching small-scale yielding DR-curves, i.e. DR-curves for an edge crack in an infinite plate. The beam height for which beam theory is applicable could be up to 10^-3 times the height for which small scale bridging DR-curves are applicable.     

An analytical analysis of energy release rate in bonded composite joints in a mode I condition

A complete analytical solution of mode I strain energy release rate, G_I, was derived for bonded composite joints based on an augmented double cantilever beam (DCB) model. Good agreement was obtained between current and existing comparable theoretical solutions for this joint with a long adhesive bond. For a short bond length joint, the current solution can greatly reduce the degree of the mathematical singularity encountered in analyses of thick, short beams and avoid it entirely for thin, long beams. A correlation between the current theoretical and associated ASTM solutions was established. A bonded DCB laminate test case was conducted, and good agreement was obtained between the experimental and current theoretical results. Commentary was included regarding the tested critical strain energy release rate and the deduced critical adhesive peel stress.     

Stress intensity factor solutions for estimation of fatigue lives of laser welds in lap-shear specimens

Fatigue behavior of laser welds in lap-shear specimens of high strength low alloy (HSLA) steel is investigated based on experimental observations and two fatigue life estimation models. Fatigue experiments of laser welded lap-shear specimens are first reviewed. Analytical stress intensity factor solutions for laser welded lap-shear specimens based on the beam bending theory are derived and compared with the analytical solutions for two semi-infinite solids with connection. Finite element analyses of laser welded lap-shear specimens with different weld widths were also conducted to obtain the stress intensity factor solutions. Approximate closed-form stress intensity factor solutions based on the results of the finite element analyses in combination with the analytical solutions based on the beam bending theory and Westergaard stress function for a full range of the normalized weld widths are developed for future engineering applications. Next, finite element analyses for laser welded lap-shear specimens with three weld widths were conducted to obtain the local stress intensity factor solutions for kinked cracks as functions of the kink length. The computational results indicate that the kinked cracks are under dominant mode I loading conditions and the normalized local stress intensity factor solutions can be used in combination with the global stress intensity factor solutions to estimate fatigue lives of laser welds with the weld width as small as the sheet thickness. The global stress intensity factor solutions and the local stress intensity factor solutions for vanishing and finite kinked cracks are then adopted in a fatigue crack growth model to estimate the fatigue lives of the laser welds. Also, a structural stress model based on the beam bending theory is adopted to estimate the fatigue lives of the welds. The fatigue life estimations based on the kinked fatigue crack growth model agree well with the experimental results whereas the fatigue life estimations based on the structural stress model agree with the experimental results under larger load ranges but are higher than the experimental results under smaller load ranges.     

Analysis of unidirectional (0°) fiber-reinforced laminated composite double cantilever beam specimen using higher order beam theories

Mathematical models, for the stress analysis of unidirectional (0°) fiber-reinforced laminated composite double cantilever beam (DCB) specimen using classical beam theory, first and higher order shear deformation beam theories, have been developed to determine the mode I strain energy release rate (SERR) for unidirectional composites. In the present study, appropriate matching conditions at the crack tip of the DCB specimen have been derived by using variational principles. SERR has been calculated using compliance method. In general, the performance of shear deformation beam models of DCB specimen with variationally derived matching conditions at the crack tip is good in determining the SERR for medium to long crack lengths. Performance of higher order shear deformation beam model (having quadratically varying transverse displacement over the thickness) of DCB specimen, with non-variationally derived matching conditions at the crack tip, is good in determining the SERR for all the crack lengths in comparison with the available theoretical and finite element solutions in the literature. Higher order shear deformation beam theories having varying transverse displacement over the thickness are more appropriate to analyze DCB specimen as they predict the appropriate nature of the interlaminar normal stress at the crack tip and its distribution ahead of the crack tip.     

Fracture and deformation properties of Ni-Fe superalloy in cryogenic high magnetic field environments

The present paper includes experimental and analytical data on the fracture properties of a nickel-iron superalloy, a ferromagnetic austenite, at 4 K in magnetic fields of 0 and 6 T. The tensile, notch tensile and small punch tests are employed. A finite element analysis is also performed to convert the experimentally measured load-displacement data into useful engineering information. To interpret the results we review the available theory of the influence of magnetic field on the stress intensity factor for a crack in ferromagnetic materials.     

Magneto stress analysis of a strip with a semi elliptical notch under uniform magnetic field

Two dimensional solutions of the magnetic field and magneto elastic stress are presented for a magnetic material of a thin strip with a semi-elliptical notch subjected to uniform magnetic field. The strip is a finite plate of a simply connected region. A linear constitutive equation is used for the stress analysis. According to the electro-magneto theory, only Maxwell stress is caused as a body force in a plate. Therefore, the magneto elastic stress is analyzed using Maxwell stress. In the present problem, as a result, the plane stress state does not arise, and the σ_z in the direction of the plate thickness and the shear deflection (anti-plane shear stress) arise for soft ferromagnetic material. The stress σ_z in the plate is strong compressive stress for a soft ferromagnetic material. A rational mapping function is used for the stress analysis, and the each solution is obtained as a closed form. No further assumption of the plane stress state that the plate is thin is made for the stress analysis, though Maxwell stress components are expressed by nonlinear terms. The rigorous boundary condition is completely satisfied without any linear assumptions on the boundary. The anti-plane shear stress causes Mode III stress intensity factor when the notch is a crack. Stress concentration values are investigated for a notch problem, of which expression is given. Figures of the anti-plane shear stress distribution, Mode III stress intensity factor, and stress concentration values are shown.     

Effect of magnetostriction on fracture of a soft ferromagnetic medium with a crack-like flaw

The magnetic‐induction field in the vicinity of an elliptical inclusion embedded in an infinite soft ferromagnetic medium is determined based on complex potential theory. By using a constitutive relation of magnetostriction for isotropic materials, the stress field in the vicinity of an elliptical flaw is obtained. Furthermore, the stress field at the tip of a slender elliptical crack is determined for the case in which only an external magnetic field perpendicular to the major axis of the ellipse is applied at infinity. The results indicate that the stress field in the neighbourhood of the tip is governed by the magnetostriction and permeability of the soft ferromagnetic material. The induction magnetostrictive modulus is a key parameter in determining which of the two mechanisms, i.e., magnetostriction and magnetic‐force‐induced deformation, is dominant in determining the stress field in the neighbourhood of the tip of a crack‐like flaw. With regard to the influence of the magnetic field on the apparent toughness of a soft ferromagnetic body with a crack‐like flaw, soft ferromagnetic materials can be roughly divided into two categories: one possesses a large induction magnetostrictive modulus and the other has a small modulus. An approximate criterion for categorizing the materials is presented. For the benefit of engineering design, the expressions of the stress‐intensity factor for these two categories of soft ferromagnetic materials are presented. The results show that the stress‐intensity factor is affected not only by the flaw geometry, but also by the permeability of the medium inside the flaw.     

压电-压磁板条中反平面裂纹的电-磁-弹性分析

基于线性电磁弹性理论,获得了压电-压磁板条中反平面裂纹尖端附近的奇异应力、电场和磁场。假设裂纹位于和板条边界平行的中心位置,并且裂纹是电磁渗透型的。利用Fourier变换,将裂纹面的混合边值问题化为对偶积分方程,即而归结为第二类Fredholm积分方程。通过渐近分析,得到了裂纹尖端附近应力、应变、电位移、电场、磁场和磁感的封闭表达式。结果表明,对于电磁渗透裂纹,电场强度因子和磁场强度因子总为0;板条的宽度对应力强度因子有显著的影响;能量释放率总为正值。     

Study on crack propagation behavior in a quenched glass plate

Crack pattern transition and crack propagation behavior in a quenched glass plate are investigated. Theoretical analysis indicates that the distance between the crack tip and the cold front is closely related to the crack pattern transition. This theoretical result is examined experimentally using instantaneous phase-stepping photoelasticity. As expected theoretically, when the crack tip remains close enough to the cold front, crack propagation remains straight. When this distance reaches a given value, the crack oscillates. These experimental results are in good agreement with the theory of crack pattern transition. Therefore, present theoretical analysis is valid in predicting the instability of crack propagation. The crack tip stress field is also examined by the present experimental method. In particular, in the oscillating regime, the mode-I stress intensity factor frequently becomes larger than the fracture toughness, and the mode-II stress intensity factor has a nonzero value during propagation. For the former result, some reasons are discussed, but the cause of this problem is still unknown. However, the latter result can be explained by the theoretical analysis of an infinitesimal kinked edge crack just after crack initiation.     

An exact solution for the stress intensity factor for a double cantilever beam

An exact solution for the stress intensity factor for a double cantilever beam (DCB) specimen loaded by opposing point loads has been obtained by the Wiener-Hopf technique. A simple approximate formula is given for the stress intensity factor that is accurate to within 1.1 percent for all crack lengths with a large uncracked ligament.

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Résumé En utilisant la technique de Wiener-Hopf, on a obtenu une solution exacte pour le facteur d'intensité d'entaille correspondant à une éprouvette en forme de poutre double Cantilever sollicitée par des charges ponctuelles opposées. On a déterminé le facteur d'intensité de contraintes sous forme d'une formulation approchée simple dont la précision est inférieure à 1,1%, pour toute longueur de fissures compatible avec une zone ligament non fissurée d'assez grande dimension.

     

Debonding and crack kinking in foam core sandwich beams—I. Analysis of fracture specimens

Sandwich beam specimens, recently developed for the study of facing/core debond fracture, were analyzed using the finite element method. Peel fracture was approached using a modified double cantilever beam (DCB) sandwich specimen with a precrack between the facing and core, while shear fracture employed a modification of the ASTM block shear test to include a facing/core precrack. Complex and conventional stress intensity factors were calculated for bimaterial cracks located between facing and bondlayer and bondlayer and core over a large range of core moduli. Overall, much larger stress intensity factors were observed for an interfacial crack between the facing and bondlayer than for a crack between the bondlayer and core for both types of specimens. Crack kinking analysis of the DCB specimen revealed that the debond tends to remain interfacial for stiff core materials, but may deflect into the core for compliant core materials. In shear loading of a debonded sandwich beam it was demonstrated that crack kinking is possible for any core material.     

A test specimen with constant stress intensity factor for prescribed displacement

A test specimen has been developed that gives a constant value of the stress intensity factor in mode I under fixed displacement loading conditions. The test specimen is a double contoured cantilever beam specimen whose shape is derived from engineering beam theory. A final shape for a range of crack extensions where stress intensity is constant was established using a finite element analysis. It is believed that this specimen could be useful for many applications where a constant stress intensity factor is needed over a range of crack extension for constant displacement loading e.g. viscoelastic crack growth in an adhesive layer or stress corrosion cracking.     

Impact response of a cracked soft ferromagnetic medium

A solution is given for the problem of an infinite soft ferromagnetic solid containing a central crack subjected to normal impact load. The solid is permeated by a uniform magnetostatic field normal to the crack surface. Laplace and Fourier transforms are employed to reduce the transient problem to the solution of integral equations in the Laplace transformed plane. A numerical Laplace inversion technique is used to compute the values of the dynamic stress-intensity factor, and the results are compared with the corresponding elastodynamic values to reveal the influence of magnetic field on the dynamic stress-intensity factor. The dynamic stress intensity factor is found to increase with increasing values of the magnetic field.With 4 Figures     

Applicability of classical isotropic fracture mechanics specimens to wood crack propagation studies

Different specimen types (double cantilever beam: DCB, compact tension: CT, single edge notched in tension: SENT) have been numerically studied (with special finite elements) for ten wood species and for cracks situated in a material plane of symmetry (the crack is denoted xy with x the direction of the normal to the crack plane and y the direction of propagation). Single-edge notched speciments used in RL, TL directions appear to be insensitive to the orthotropic properties of the material. So, calibration known for isotropic materials can be used in stress intensity factor calculations. In transverse directions (LT, LR), such specimens could be used with a double height. Small differences, but depending on the species tested, have also been obtained in TR and RT directions. Calibration of CT specimens of an isotropic material cannot be used even in TL and RL directions. A calibration acceptable for all the studied species is proposed. A different calibration is also given in the TR direction. It is suggested, however, not to use specimens in the RT direction without a calibration specific for the wood species tested. DCB specimens could be used in RL, TL and TR directions with a calibration which is a function of elastic moduli. This calibration is obtained from analytical calculations. In other directions, these specimens do not offer any experimental interest.     

Effect of magnetic properties of a composite superconductor on the losses in a variable magnetic field: Part 3 collective interaction of turns

The results obtained in Parts 1 and 2 of this paper were for a solitary conductor in an external magnetic field. However, the magnetization of the other winding turns has a considerable effect on losses in a variable magnetic field. A theoretical treatment of the phenomena associated with the collective interaction of turns is presented. The experimental results obtained on samples of different geometry are in a good agreement with theory.