Delamination dynamics in through-thickness reinforced laminates with application to DCB specimen

Bridged crack models using beam theory formulations have proved to be effective in modeling quasi-static delamination crack growth under large scale bridging conditions in through-thickness reinforced structures. In this paper, beam theory is used to study dynamic mode I crack propagation in through-thickness reinforced laminar structures. In particular, steady state dynamic crack growth for a Double Cantilever Beam (DCB) loaded with a flying wedge is examined. The steady state crack propagation characteristics are mapped out in terms of controllable loading and material parameters including the crack velocity and the properties of the through-thickness reinforcement. For small crack velocities, the through-thickness reinforcement considerably enhances the delamination resistance of the structure. At higher velocities, the kinetic energy term dominates the overall energetics and the relative effect of the reinforcement on the delamination resistance is insignificant. The model suggests a simple fracture test for estimating the properties of the through-thickness reinforcement under dynamic loading conditions.     

A simplified improved beam analysis of the DCB specimen

Large deviations from the simple cantilever beam model have been observed when analysing double cantilever beam tests in composite materials. In the present paper a simplified improved beam model is proposed. The cracked part of the specimen is analysed by means of shear-corrected classical beam theory. The uncracked part is analysed by considering Saint Venant effects and deformation of a beam on an elastic foundation. Superposition of compliances results in a simple closed form expression which can be used to isolate the effects of different material parameters and analytical simplifications. Crack length corrections are discussed as well as the corrections that have to be made when calculating the strain energy release rate. An extensive numerical comparison is made with the more elaborate solutions of Whitney and Williams. Finally, a discussion is made on the relevance of one-dimensional analysis as compared with two- and three-dimensional analyses. The discussion is supported by numerical comparisons.     

Analysis of the sandwich DCB specimen for debond characterization

Analysis of the compliance and energy release rate of the sandwich double cantilever beam (DCB) specimen is presented. It is assumed that there is a starter crack at the upper face/core interface and that the crack remains at or near this interface during crack propagation. Beam, elastic foundation, and finite element analyses are presented and compared to experimentally measured compliance data, and compliance calibrated energy release rate over a range of crack lengths for foam cored sandwich DCB specimens. It is found that the beam analysis provides a conservative estimate on the compliance and energy release rate. The elastic foundation model is in agreement with finite element analysis and experimental compliance data. Recommendations for specimen design and an expression for an upper limiting crack length are provided.     

Crack growth resistance evaluation of ductile material using DCB specimen

An analytical method is derived to determine the tearing resistance of a material using the double-cantilever-beam specimen. The analysis makes use of a beam-on-elastic foundation model to determine the elastic deflection of the specimen and the Dugdale model to simulate the stress field at the crack tip.J resistance curve of 2024-T3 aluminium material is obtained based on the experimentally observed load and crack extension and is compared with those deduced using compliance method and finite element method.

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Résumé On déduit une méthode analytique permettant de déterminer les résistances d'un matériau à l'arrachement, en utilisant une éprouvette en double poutre Cantilever.L'analyse fait appel à un modèle de poutre sur appuis élastiques en vue de déterminer la flèche élastique de l'éprouvette, et au modèle de Dugdale en vue de simuler le champ de contraintes à l'extrémité de la fissure.Une courbe de résistance, exprimée parJ, est obtenue dans le cas de l'alliage d'aluminium 2024-T.3, en se basant sur les observations expérimentales de la charge appliquée et de l'extension de la fissure; cette courbe est comparée avec celles que l'on obtient en utilisant une méthode de compliance et une méthode par éléments finis.

     

Simulations of fast fracture in the DCB specimen using Kanninen's model

Kanninen's beam model for the DCB specimen is used to analyze several fast fracture problems. First, the model is studied under two loading conditions to which it has not been applied previously: constant-force (dead) loading and rapid-wedge (constant velocity) loading. The predictions of crack propagation under both of these loading conditions are similar to those obtained by Bilek and Burns. Second, the implications of using different methods of simulating the bluntness of a starter-notch are investigated for conditions typical of Kanninen's analyses and experiments conducted at the Battelle Laboratories. The predictions of crack behavior are in general agreement with Kanninen's results; however, it appears that quantitative predictions are sensitive to the specific manner in which the bluntness is simulated in the analytical model.

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Résumé Le modèle en poutre de Kanninen pour les éprouvettes double Cantilever est utilisé en vue d'analyser plusieurs problèmes de rupture rapide. En premier lieu, le modèle est étudié sous 2 conditions de charge pour lesquelles il n'avait pas été appliqué précédemment: la charge constante (point mort) et le chargement rapide de côté (à vitesse constante). Les prédictions de propagation de fissure sous ces 2 conditions de charge sont similaires à celles obtenues par Bilek et Burns. En deuxième lieu, les implications de l'utilisation de différentes méthodes pour simuler l'arrondissement d'une entaille initiale sont étudiées pour des conditions typiques des analyses de Kanninen et des expériences conduites aux Laboratoires du Battelle. Les prédictions du comportement de la fissure sont en général en accord avec les résultats de Kanninen. Toutefois, il apparait que les prédictions quantitatives sont sensibles à la manière spécifique suivant laquelle l'arrondissement est simulé dans le modèle analytique.

     

Finite element simulation of fast fracture in steel DCB specimen

Results of a numerical simulation, based on an energy consistent moving singularity dynamic finite element procedure, of fast crack propagation and arrest in a high strength steel DCB specimen are presented. The influence of material properties of high strength steel on dynamic crack propagation and arrest is investigated. The influence of the loss of constact of specimen with the loading wedge is also critically examined. The present numerical results are compared with available experimental data. It is found that the present results agree well with available experimental data, and the crack arrest toughness values obtained in the present analysis correlate well with the ratio of the maximum kinetic energy of the specimen to the input energy.     

Modal analysis of the dynamic crack growth and arrest in a DCB specimen

This paper discusses dynamic crack growth and arrest in an elastic double cantilever beam (DCB) specimen, simulated using the Bernoulli–Euler beam theory. The specimen is made from two different materials. The section of interest, where the dynamic crack growth takes place, is made from a material, the fracture energy of which will be denoted \(2\Gamma _1 \) . The initial crack grows slowly in a starter material with a fracture energy \(2\Gamma _{0} \;(\Gamma _{0} >\Gamma _1 )\) , while opposed displacements on both arms of the specimen are continuously increased. As the crack reaches the material interface at \(\ell =\ell _c \) , the loading displacement is instantly suspended, and the crack suddenly propagates through the test zone, until it stops at \(\ell =\ell _A \) . During this process, the energy \(2\Gamma _1 (\ell _A -\ell _c )\) is dissipated. The beam motion and the fracture process during the fast crack growth stage are investigated, based on the balance energy associated to the Griffith criterion. The motion equations are approximated using a modal decomposition up to order \(N\) of the beam deflection (the analysis has been performed up to \(N=10\) but in most cases \(N=5\) is sufficient to obtain an accurate solution). This process leads to a set of N second order differential equations whose unknowns are the mode amplitudes and their derivatives, and another equation the unknowns of which are the current crack length \(\ell (t)\) , velocity \(\dot{\ell }(t)\) and acceleration \(\ddot{\ell }(t)\) . To demonstrate the accuracy of this method, it is first tested on a one dimensional peeling stretched film problem, with an insignificant bending energy. An exact solution exists, accurately approximated by the modal solution. The method is then applied to the DCB specimen described above. Despite the rather crude nature of the Bernoulli–Euler model, the results crack kinematics, and specially the arrest length, correspond well to those obtained by the combined use of finite elements and cohesive zone models, even for a few modes. Moreover, for the basic mode \(N=0\) (also referred to as Mott solution), even if the crack kinematics is not accurately reproduced, the prediction of the crack arrest length remains correct for moderate ratios. Some parametric studies about the beam geometry and the initial crack velocity are performed. The relative crack arrest \(\ell _A /\ell _c \) appears to be almost insensitive to these parameters, and is mainly governed by the ratio \(R=\Gamma _{0} /\Gamma _1 \) which is the key parameter to predict the crack arrest.     

Compliances of a tapered DCB specimen configurations by a finite element method

A series of finite element calculations is carried out to investigate the influence of the crack configuration on the compliance of a tapered double cantilever beam specimen. The results of the calculations are compared with experimental data.     

A dynamic analysis of unstable crack propagation and arrest in the DCB test specimen

A simple analytical model is developed to accompany experimental work on rapid crack propagation and arrest in the DCB test specimen. The present work extends the beam-on-elastic foundation model used previously by taking account of shear deformation and of both translational and rotary inertia. Crack speeds predicted with the model are found to be in good agreement with the constant-speed behavior observed experimentally. It is demonstrated that kinetic energy makes an important contribution to maintaining unstable crack propagation and to crack arrest.     

Delamination buckling of FRP layer in laminated wood beams

An analytical solution for predicting delamination buckling and growth of a thin fiber reinforced-plastic (FRP) layer in laminated wood beams under bending is presented. Based on a strength-of-materials approach, displacement functions for a delaminated beam under four-point bending are derived. Using force and displacement compatibility conditions, an explicit form relating the applied transverse load with the delamination buckling load is established. An explicit form of the strain-energy release rate is presented to study the delamination growth in beams under bending. The analytical solution is evaluated using experimental data for glued-laminated timber (glulam) beams reinforced with a thin fiber-reinforced plastic composite on the compression face. The delamination growth in bending is shown to behave differently to that of the in-plane loading case.     

One-dimensional description of cylindrically symmetric laser beams: application to bessel-type nondiffracting beams

We introduce a new representation of coherent laser beams that are usually described in circular cylindrical coordinates. This representation is based on the decomposition of a laser beam of a given azimuthal order into beams exhibiting Cartesian symmetry. These beams, which we call constituent waves, diffract along only one of their transverse dimensions and propagate noncollinearly with the propagation axis. A cylindrically symmetric laser beam is then considered a coherent superposition of constituent waves and is represented by an integral over an angular variable. Such a representation allows for the introduction of the propagation factor M2, defined in terms of one-dimensional root-mean-square (rms) quantities, in the treatment of two-dimensional beams. The representation naturally leads to the definition of a new rms parameter that we call the quality factor Q. It is shown that the quality factor defines in quantitative terms the nondiffracting character of a laser beam. The representation is first applied to characterize Laguerre-Gauss beams in terms of these one-dimensional rms parameters. This analysis reveals an asymptotic link between Laguerre-Gauss beams and one-dimensional Hermite-Gauss beams in the limit of high azimuthal orders. The representation is also applied to Bessel-Gauss beams and demonstrates the geometrical and one-dimensional characters of the Bessel-Gauss beams that propagate in a nondiffracting regime. By using two separate rms parameters, Q and M2, our approach gives an alternative way to describe laser beam propagation that is especially well suited to characterize Bessel-type nondiffracting beams.     

Dynamic fracture detection using the force-displacement reciprocity: application to the compact compression specimen

Dynamic fracture toughness is determined by impacting fracture specimens and determining the onset of crack propagation. In such experiments characteristic impedance (actuator-specimen) matching problems can affect the accuracy of the measured forces. In addition, fracture time is mostly determined by means of surface rather than bulk devices (e.g. gages). In this paper we address these issues in linear-elastic materials. Based on the H-integral we show that either the measured forces or the displacements (whichever is more accurate) can be used to calculate the evolutions of the stress intensity factors. These evolutions must be identical by virtue of the reciprocity between forces and displacements, at least until some bulk fracture process develops. Examples are presented to illustrate how these observations can be applied to fine-tune dynamic fracture experiments and complement fracture gage readings.     

Delamination detection in laminated composite beams using the empirical mode decomposition energy damage index

Fiber reinforced plastic composite (FRPC) beams have been widely used as effective stand-alone structural elements and/or reinforcing agents in various engineering applications. Such lightweight, stiff and strong structural elements may suffer, however, from inter-ply defects (referred to as delaminations), arising from various causes and factors. A delamination in such structural components could propagate into severely large damage zone, thus compromising the structural integrity. Therefore, detection of delamination at an early stage is of paramount importance.     

Finite element modelling of delamination growth in the DCB and edge delaminated DCB specimens

This paper presents the results of a finite element investigation of delamination growth in a conventional Mode I double cantilever beam (DCB) specimen and in an edge-delaminated version of this specimen. The investigation was performed using a recently developed FE model for delamination growth prediction to which an approximate contact area detection method has been added. The results of the FE analyses are used to evaluate the existing data reduction techniques for calculation of interlaminar fracture toughness. It is shown that the conventional DCB data reduction schemes can be applied to the edge delaminated specimen but the results are dependent upon the difference between the measured apparent delamination length and the actual length being constant. An alternative data reduction method is presented in which the critical energy release rate is determined by comparison of the experimental data with finite element results and does not require measurement of the delamination length.     

An application of the single specimen technique to a double cantilever beam specimen to determine GIC from a single test record

The single specimen technique is effectively used in classical fracture mechanics for evaluating the J-integral from a single test record. It was previously shown using a cracked lap shear specimen that the technique can be applied to composite structures and the energy release rate of composite laminates can be determined from a single test specimen. The objective of this investigation is to determine the η_el form of a double cantilever beam (DCB) specimen, and to compare it with that of a compact tension (CT) specimen. The result showed that while η_el is linearly related to crack length for the CT specimen, it is proportional to the reciprocal of the crack length for the DCB composite specimen. Also, G_Ic determined from the compliance method was compared with that determined from the η_el approach. The result showed that the η_el form determined from a single beam analysis can be used to determine G_Ic from a single test record.     

Self-imaging properties of polychromatic Gaussian beams: application to wavelength demultiplexing and space routing

We propose an analytical model of the self-imaging of Gaussian beams and discuss the impact of the finite aperture and the beam truncation on the quality of the self-image reconstruction. Extension to polychromatic operation is then presented in the context of wavelength division multiplexing systems. From the first point we derive conditions for good self-imaging, compatible with the requirements of the telecommunication environment. A basic optical setup is then proposed to implement both wavelength demultiplexing and routing in a lensless configuration.     

A novel application of the FPM to the buckling differential equation of non-uniform beams

In this study the buckling load (P) of non-uniform, deterministic and stochastically heterogeneous beams, is found by applying the Functional Perturbation Method (FPM) directly to the Buckling (eigenvalue) Differential Equation (BDE). The FPM is based on considering the unknown P and the transverse deflection (W) as functionals of heterogeneity, i.e., the elastic bending stiffness “K” (or the compliance S=1/K). The BDE is expanded functionally, yielding a set of successive differential equations for each order of the (Frèchet) functional derivatives of P and W. The obtained differential equations differ only in their RHS, and therefore a single modified Green function is needed for solving all orders. Consequently, an approximated value for the buckling load is obtained for any given morphology. Four examples of simply supported columns are solved and discussed. In the first three, deterministic realizations of K are considered, whereas in the fourth, K is assumed to be the stochastic field. The results are compared with solutions found in the literature for validation.     

Plastic bending of an edge-cracked specimen

The plastic zones and the rotation of the cracked planes of an edge-cracked bend specimen are investigated by a numerical three-dimensional analysis. For an alloy-steel, for bending moments up to the first yielding moment of the uncracked specimen, it is shown that the crack tip is in small scale yielding.