Notch Flexure Research Articles

On the failure analysis of bondlines: Stress or energy based fracture criteria?

Bonded structure, can be either considered a multilayer material system, where its sublayers are held together through media that provide adhesion, or an adhesive joint designated for the assembly of different parts involved in a structure. For the assessment of the adhesion strength that characterizes a given bondline, both its cohesive strength and fracture toughness material parameters must be experimentally defined. Based on these properties, failure analysis of the bondline can be done either through stress- or energy-based criteria. The aim of this work is to investigate the effectiveness of each criteria type to effectively predict debonding initiation/propagation of different bondlines (brittle, quasi-brittle, ductile), which are eventually utilized for the evaluation of the joint’s failure load. By representing a bondline according to its cohesive length scale, an effort is made to classify the wide range of bondlines with respect to the failure theory that best describes the debonding process. Cohesive length scale effects are first demonstrated by modeling end notch flexure geometries and later by modeling double strap joint geometries within the framework of a wide numerical experimentation programme. Fracture initiation and propagation of the bondlines was numerically simulated by cohesive zone models.

Generalized Equations for Estimating Stress Concentration Factors of Various Notch Flexure Hinges

The flexure hinges are the most vulnerable parts in a flexure-based mechanism due to their smaller dimensions and stress concentration characteristics, therefore evaluating the maximum stresses generated in them is crucial for assessing the workspace and the fatigue life of the mechanism. Stress concentration factors characterize the stress concentrations in flexure hinges, providing an analytical and efficient way to evaluate the maximum stress. In this work, by using the ratio of the radius of curvature of the stress-concentrating feature to the minimum thickness as the only fitting variable, generalized equations for both the bending and tension stress concentration factors were obtained for two generalized models, the conic model and the elliptic-arc-fillet model, through fitting the finite element results. The equations are applicable to commonly used flexure hinges including circular, elliptic, parabolic, hyperbolic, and various corner-fillet flexure hinges, with acceptable errors. The empirical equations are tractable and easy to be employed in the design and optimization of flexure-based mechanisms. The case studies of the bridge-type displacement amplifiers demonstrated the effectiveness of the generalized equations for predicting the maximum stresses in flexure-based mechanisms.

Effect of Starter Defect to G<sub>IIC</sub> of Unidirectional CFRP Composite

Critical strain energy release rate in CFRP composites characterizes the delamination resistance. More study is still needed to measure the critical strain energy release rate in sliding shear mode (GIIC) considering various factors that influence its measurement. This study evaluates one of the influencing factors, the starter defect. Two types of on thin, unidirectional CFRP composites with one having thin film insert as starter defect and another one with pre-crack under Mode II loading were prepared and tested in three point bending end notch flexure (3ENF) test. It was found that the (GIIC) of the former was more than twice higher than that of the latter, supposedly due to the presence of resin rich region in the former.

Planar Compliances of Symmetric Notch Flexure Hinges: The Right Circularly Corner-Filleted Parabolic Design

This study proposes a general analytical compliance model for symmetric notch flexure hinges composed of segments with constant width and analytically defined variable thicknesses. Applying serial combination and longitudinal/transverse mirroring of base segments, the in-plane compliances of the full flexure are obtained as functions of one quarter-flexure compliances. The new right circularly corner-filleted parabolic flexure hinge is introduced as an illustration of the general analytical modeling algorithm. Experimental testing and finite element simulation confirm the analytical model predictions for an aluminum flexure prototype. The planar compliances sensitivity to the relevant geometric parameters is also studied.

Influence of the Matrix Toughness in Carbon-Epoxy Composites Subjected to Delamination under Modes I, II, and Mixed I/II

In this work, the fracture behavior under modes I, II, and mixed mode I/II has been studied for two different AS4 carbon fiber epoxy laminates. One of the laminates was produced with a Hexcel 3501-6 epoxy resin while the other was laminated with a tougher modified Hexcel 8552 epoxy resin. Both laminates were experimentally tested in modes I, II, and mixed I/II with different mixity ratios by means of DCB (double cantilever beam), ENF (end notch flexure), and MMB (mixed mode bending) specimens, respectively. Finite element modeling (FEM) was used in order to analyze modes I, II, and mixed I/II and to compare experimental and numerical results. The modified 8552 resin matrix presented the best behavior in mode I and mixed mode I/II as the critical energy release rate was higher than that for the 3501-6 matrix composites. In mode II, the best performance was reached for the 3501-6 matrix laminates. It was also found that the critical energy Gc and the scatter increased as the mode ratio GII/Gc increased. Finally, experimental and numerical results showed a good agreement as the differences obtained from both procedures were generally lower than 10%.

Hybrid flexure hinges

This paper designs and analyzes the hybrid flexure hinge composed of half a hyperbolic flexure hinge and half a corner-filleted flexure hinge. As it is transversely asymmetric, it has different performance when the fixed and free ends switch. Considering the diversion of rotation center from midpoint, closed-form equations are formulated to characterize both the active rotation and all other in-plane parasitic motion by the Castigliano's second theorem. The maximum stress is evaluated as well. These equations are verified by the finite element analysis and experimentation. The compliance precision ratios are proposed to indicate flexure hinges' ability of preserving the rotation center when they have the same displacement at the free end. The hybrid flexure hinges are compared with five kinds of common notch flexure hinges (circular, corner-filleted, elliptical, hyperbolic, and parabolic flexure hinges) quantitatively based on compliance, precision, compliance precision ratios, and the maximum stress. Conclusions are drawn regarding the performance of these six kinds of flexure hinges.

Crack propagation analysis in composite bonded joints under mixed-mode (I+II) static and fatigue loading: a damage-based model

In this paper, a first step towards the definition of a unifying model based on the process zone concept has been attempted. The aim of the proposed model is to summarize the fatigue behaviour of composite bonded joints under mixed-mode (I + II) fatigue loading conditions focusing as close as possible on the actual damage mechanisms observed during the fatigue tests, which were found strongly dependent on the mode mixity. The proposed model has been verified on the experimental results obtained from double cantilever beam (DCB), end notch flexure (ENF) and mixed-mode bending (MMB) tests as well as from single-lap bonded joints. Finally, the model is successfully applied to the description of joint behaviour under static mixed-mode loading.

A nonlinear cohesive model for mixed-mode delamination of composite laminates

An anisotropic nonlinear cohesive model based on continuum damage mechanics is proposed to predict the mixed-mode delamination initiation and growth of composite laminates numerically. A damaged exponential traction–displacement jump relationship is proposed for the cohesive model and an exponential damage evolution law on the evolvement of displacement jump is proposed to describe the irreversible delamination crack propagation. The implementation of cohesive model uses zero-thickness cohesive element and middle-plane isoparametric interpolation technique. The mesh size effect is solved by regulating the delamination fracture toughness by the cohesive length, and the numerical convergence problem for the snap instability is addressed by viscous regularization. Four representative delamination cases of composite laminates are used to validate the proposed cohesive model: (a) double cantilever beam (mode-I), (b) end notch flexure (mode-II), (c) mixed-mode bending fracture, and (d) fixed ratio mixed mode. The effects of the cohesive strengths and mesh sizes on the global load–displacement response for four cases are studied. In addition, the numerical results using proposed cohesive model are also compared with the results obtained by the virtual crack closure technique and other existing cohesive models. The proposed cohesive model is demonstrated to predict the mixed-mode delamination response of composite laminates accurately and effectively.

Crack propagation analysis in composite bonded joints under mixed-mode (I+II) static and fatigue loading: experimental investigation and phenomenological modelling

This paper illustrates the results of an extensive experimental investigation on composite bonded joints under mixed-mode (I+II) static and cyclic loading conditions oriented to understand the influence of the mode mixity condition on the crack propagation resistance at the bondline. The double cantilever beam (DCB), end notch flexure (ENF) and mixed-mode bending (MMB) tests were conducted on pre-cracked samples and both fracture toughness and crack propagation resistance were seen to increase, both for static and fatigue loading, respectively, as the mode II contribution increases. The crack propagation and damage evolution were carefully investigated and documented, and a strong dependence of the propagation mode on the mode mixity was found. Fatigue data under the different loading conditions are then described by a phenomenological model based on the strain energy release rate contributions, which represents a useful engineering tool for preliminary design. After that a damage-based model, developed on the basis of the actual damage mechanisms, is presented in a companion paper.

Interlaminar toughness characterisation of 3D woven carbon fibre composites

The addition of reinforcement in the through thickness direction using three-dimensional (3D) weaving techniques has been shown to improve the delamination toughness of composite materials, mitigating the reduced out of plane performance of traditional composite materials. At present, the optimum architecture for improving delamination resistance is uncertain. To address this, three geometries of 3D woven carbon fibre reinforced epoxy composites were evaluated in mode I using the double cantilever beam test method. Mode II testing was also carried out using the end loaded split and four-point end notch flexure test methods. For large delaminations (i.e. when the R curve reaches its plateau value), an orthogonal weave is found to be most effective in resisting delamination propagation in mode I and is comparable to the layer to layer architecture in mode II. In all cases, an angle interlock weave appears to be less effective than either the orthogonal or layer to layer weaves.

High-order B-splines based finite elements for delamination analysis of laminated composites

Finite element analysis of the delamination process in delaminated composites is addressed using high order Bézier elements relied on the Bernstein polynomials which are commonly used in Computer Aided Design. Laminae of a composite laminate are modeled with two dimensional Bézier elements. Zero-thickness cohesive elements, which are inserted along the interface in between the laminae to model the interface cracking, employ the high order univariate Bernstein basis functions. Quadratic, cubic and quartic Bernstein basis functions are examined. An open source pre-processing code is presented to generate Bézier meshes and interface elements. Implementation of a simple yet highly efficient arc-length solver which is able to trace equilibrium paths having multiple snap-backs is given. Numerical examples are presented for the analyses of benchmark composite delamination problems which include the double cantilever beam, the end notch flexure and the mixed mode bending tests. Additionally multiple delamination analysis of a composite specimen is also presented. Additionally multiple delamination analysis of a composite specimen is also investigated. For all given examples, the highly efficient performance of the proposed formulation is found.

Planar Compliances of Thin Circular-Axis Notch Flexure Hinges with Midpoint Radial Symmetry

The new class of flexure hinges with circular longitudinal axis and midpoint radial symmetry is introduced. Using rotation and mirroring, the symmetric flexure hinge is obtained from one half flexure. The six planar-bending compliances of the full hinge are determined analytically for small deformations by combining only three compliances of the half flexure. To illustrate the general flexure hinge category, the novel circular-axis, right circularly corner-filleted design is introduced. Experimental and finite element results correlate well with the analytical model predictions. The new flexure hinge design is compared to the circular-axis, constant-thickness flexure and the straight-axis, right circularly corner-filleted hinge.

In-plane elastic response of two-segment circular-axis symmetric notch flexure hinges: The right circular design

Abstract This work presents a general analytical model of the planar compliances for two-segment circular-axis symmetric notch flexure hinges. Compared to a similar straight-axis flexure design, the circular-axis configuration enhances the design parameter domain by adding the median circle radius. Six compliances are formulated with respect to a central reference frame and to an end reference frame in terms of a smaller number of half-flexure compliances. The model covers both relatively large radius-to-thickness ratio designs (thin flexures) and small radius-to-thickness ratio configurations (thick flexures). The new circular-axis right circular flexure is introduced to illustrate the general model. Predictions of the analytical compliance model for this flexure were confirmed by finite element analysis and experimental testing. The analytical model is further utilized to investigate the influence of normal and shear forces in the thick-member model and to assess the accuracy of the simpler thin-beam model, which considers only bending. A comparison is also performed between the corresponding compliances of the new circular-axis flexure design and the existing straight-axis right circular flexure hinge in terms of defining geometric parameters.

Bone fracture characterization using the end notched flexure test

A miniaturized version of the end notch flexure test was used in the context of pure mode II fracture characterization of bovine cortical bone. To overcome the difficulties intrinsic to crack length monitoring during its propagation an equivalent crack method was employed as data reduction scheme. The proposed method was validated numerically by means of a finite element analysis including a cohesive zone modeling and subsequently applied to experimental results to determine the fracture energy of bone under pure mode II loading. Finally, a cohesive law representative of fracture behavior of each specimen was determined employing an inverse method, considering a trapezoidal shape for the softening law. The consistency of the obtained results leads to the conclusion that the trapezoidal law is adequate to simulate fracture behavior of bone under mode II loading. The proposed testing setup and the employed data reduction scheme constitute powerful tools in which concerns fracture characterization of bone under pure mode II loading and can be viewed as the main outcomes of this work.

Cohesive zone modeling of coupled buckling – Debond growth in metallic honeycomb sandwich structure

Buckling-induced skin–core debond growth in honeycomb sandwich cantilever beam is demonstrated using a cohesive zone model. The input parameters for the analysis are interfacial bond strength, mode I and mode II interfacial fracture toughness values, obtained from flatwise tension tests, drum-peel tests and three-point end notch flexure tests, respectively. Debonded honeycomb specimens are tested and the acoustic emission technique was used to observe the initiation of the debond growth. The load-displacement response from the cohesive zone model model shows a good agreement with the experimental results. The conventional analysis without cohesive zone model overestimates failure load by 56%. Cohesive zone model is able to predict the coupled debond growth and buckling failure in honeycomb sandwich structures.

Fractographic analysis of delamination in glass/fibre epoxy composites

Unidirectional laminates of reinforced glass/epoxy composites UE400/ET443 were tested by the use of fracture mechanics tests. Mode I, Mode II, Mode III and Mixed-Mode I-II were, respectively, obtained from specimens submitted to double cantilever beam, end notch flexure, Edge Crack Torsion and mixed-mode bending tests. Two different loadings were applied to the edge crack torsion specimens: the original and the modified edge crack torsion test. Scanning electron microscopy, analytical and numerical analysis were used to identify and separate fractographic features and to establish the differences between the various modes of fracture. Scanning electron micrographs of fractured surfaces showed that the most predominant fractographic features in Mode I and Mode II are the large amount of fibre pull-out and the cusps markings, respectively. Mode III characterisation shows that some limited Mixed-Mode II-III appear in both ends of the edge crack torsion specimen on delamination initiation and growth, but a predominant Mode III delamination exists in the middle of the specimens.

Symmetry-Based Compliance Model of Multisegment Notch Flexure Hinges

An analytical procedure is proposed to calculate the compliances of planar-motion notch flexure hinges with transverse or/and axial symmetry by employing base segments of known compliances. Mirroring, translation, and serial combination of individual segments are used to obtain the flexure hinge compliances that are necessary to evaluate its capacity of deformation and precision of rotation. Two case studies are illustrated: one derives the compliances of the known right circular flexure hinge while the other introduces a new flexure hinge with both axial and transverse symmetry. For the latter flexure design, the numeric values of compliances are confirmed by finite element simulation.

Mode II Fracture Toughness of Adhesively Bonded Joints as a Function of Temperature: Experimental and Numerical Study

In this work, an experimental and numerical study is performed to evaluate the effect of temperature on the Mode II fracture toughness of a high temperature epoxy adhesive. Mode II fracture testing on end notch flexure (ENF) test specimens was performed at room temperature (RT) and high temperatures (100, 150, and 200°C) and the Mode II fracture toughness, G IIc, as a function of temperature was obtained. It is shown that at temperatures well below the glass transition temperature, the fracture toughness, G IIc, increases with temperature and decreases as the temperature approaches T g, while above T g a drastic decrease in fracture toughness was observed. Furthermore, cohesive zone models, in which the failure behaviour is expressed by a bilinear traction–separation law, were used to define the adhesive behaviour in Mode II and to predict the adhesive load-displacement (P–δ) curves as a function of temperature. The simulation response matched the experimental results very well at temperatures below T g . The sensitivity of the various cohesive zone parameters in predicting the overall mechanical response of the joint was examined as well for a deeper understanding of this predictive method.

Note: Bending compliances of generalized symmetric notch flexure hinges

The bending compliances of generalized notch flexure hinges with transverse or transverse-and-axial symmetry are studied in two particular reference frames. For an end-point reference frame, the cross compliance and the rotary compliance are proportional. When the reference frame is placed at the flexure's midpoint, the cross compliance is zero. The translatory and rotary compliances of only half the flexure hinge are sufficient to calculate the overall compliances of a transverse-symmetry flexure configuration. Similarly, the overall bending compliances of a flexure hinge with transverse-and-axial symmetry require prior calculation of the translatory and rotary compliances of a quarter flexure solely.

Interlaminar Fracture Toughness and Low-Velocity Impact Resistance of Woven Glass Epoxy Composite Laminates of EP3 Grade

This work presents the delamination resistance of woven glass fibre reinforced polymers (GFRP) and its influence on GFRP’s resistance to point impact. Two different types of laminates were fabricated by hand lay-up technique; (i) woven glass fibre epoxy matrix laminates and (ii) woven glass fibre epoxy matrix laminates with 3% graphite particulate fillers. End Notch Flexure (ENF) test was adopted for the measurement of delamination resistance. The two GFRPs laminates show similar mode II delamination resistance. At impact velocities between 2.215 and 4.429 m/sec, the GFRP developed a smaller damage size than the graphite-based GFRP laminates, indicating higher impact toughness. Difference of the impact resistance between the two GFRPs is mainly on the impact damage size developed. The total energy absorbed during the impact remains the same, which is independent of mode II delamination resistance of the GFRP. The history of relevant dynamic and energetic quantities, both to synthesize the dependency of the energy parameters and force threshold values on the impact velocity are discussed.