Determination of mixed mode cohesive laws

A novel approach is proposed for the determination of mixed mode cohesive laws for large scale crack bridging problems. The approach is based on a plane, two-dimensional analysis utilizing the J integral applied a double cantilever beam specimens loaded with uneven bending moments. The normal and shear stresses of the cohesive laws are obtained from consecutive values of the fracture resistance, the normal and tangential displacements of the end of the cohesive zone. The data analysis involves fitting and determination of partial differentials. This is done by a numerical method using Chebyshev polynomials. The accuracy of the numerical procedure is investigated by the use of synthetic data. It is found that both the shape and peak stress of the cohesive law can be determined with high accuracy, providing that the data possess low noise and a sufficiently high number of datasets are used. The investigation leads to some practical guidelines for experimental use of the proposed approach.     

Crack – fiber sensor interaction and characterization of the bridging tractions in mode I delamination

Fiber bridging is regularly encountered in mode I delamination tests of unidirectional fiber reinforced composites. However, characterization of the bridging tractions is rather difficult. One way to indirectly evaluate the bridging traction distribution is to embed a fiber Bragg grating (FBG) sensor close to the crack tip and to measure the distributed strain along this FBG. The strain measurements from the FBG sensor are used to characterize the fiber bridging tractions by an identification method. In this work, the sensor is embedded in a unidirectional carbon/epoxy composite. Firstly, it is treated as an inclusion near the crack plane and a numerical analysis is performed to study its effect on the measured strain field and energy release rate. The results demonstrate that the sensor, located at about two fiber diameters from the crack plane, has a negligible effect on the fracture process. Secondly, among the identified linear, bilinear, and exponential bridging traction distributions, the exponential one is found to be a suitable model. Characterization of the bridging tractions allows to calculate the energy release due to the bridging fibers which is similar to the difference between the initiation energy release and the propagation value . The results also agree with the bridging tractions evaluated from the conventional energy release rate – crack opening displacement method.     

Temperature effects on mixed mode I/II delamination under quasi-static and fatigue loading of a carbon/epoxy composite

This paper addresses the effect of temperature on the mixed-mode interlaminar fracture toughness and fatigue delamination growth rate of a carbon-fibre/epoxy material, namely IM7/8552. Quasi-static and fatigue characterisation tests were carried out at −50 °C, 20 °C, 50 °C and 80 °C, using asymmetric cut-ply coupons. The experimental results show that temperature may have an accelerating or delaying effect on delamination growth, depending on the loading regime, i.e. either quasi-static or fatigue. Fractographic examinations were also carried out in order to assist the interpretation of the experimental data. A semi-empirical equation is introduced to describe the experimentally observed fatigue delamination growth rates at elevated temperatures.     

Failure criteria for mixed mode delamination in glass fibre epoxy composites

Critical strain energy release rate of glass/epoxy laminates using the virtual crack closure technique for mode I, mode II, mixed-mode I + II and mode III were determined. Mode I, mode II, mode III and mixed-mode I + II fracture toughness were obtained using the double cantilever beam test, the end notch flexure test, the edge crack torsion test and the mixed-mode bending test respectively. Results were analysed through the most widely used criteria to predict delamination propagation under mixed-mode loading: the Power Law and the Benzeggagh and Kenane criteria. Mixed-mode fracture toughness results seem to represent the data with reasonable accuracy.     

Fracture simulation of CFRP laminates in mixed mode bending

This paper analyses the progressive mixed mode delamination failure in unidirectional and multidirectional composite laminates using fracture experiments, finite element (FE) simulations and an analytical solution. The numerical model of the laminate is described as an assembly of damageable layers and bilinear interface elements subjected to mixed mode bending. The analytical approach is used to estimate the total mixed mode and decomposed fracture energies for laminates with different stacking sequences, which is also validated through experiments. It is concluded that the interlaminar fracture toughness of multidirectional laminates is considerably higher than that of the unidirectional ones. The effect of initial interfacial stiffness and element size is studied and it is also shown that their value must not exceed a definite limit for the numerical simulations to converge. The model can also be further extended to simulate the mixed mode fracture in hybrid fiber metal laminates.     

Bridging effect on mode I fatigue delamination behavior in composite laminates

This paper discusses the bridging effect of fibres on mode I fatigue delamination growth in unidirectional and multidirectional polymer composite laminates based on a series of double cantilever beam (DCB) tests. From the results, there is sufficient evidence that fibre bridging can decrease the crack growth rate da/dN significantly, and using only one fatigue resistance curve to determine the delamination behavior in composite materials with large-scale fibre bridging may be inadequate. The bridging created in fatigue delamination is different from that of quasi-static delamination at the same crack length. So it is incorrect to use the resistance curve (R-curve) from quasi-static delamination tests to normalize fatigue delamination results.     

Specimen proposals for mixed mode testing of adhesive layer

The experimental methods to determine the fracture properties for adhesives under mixed mode loading is not as well established as compared to such methods for adhesives under pure mode loading. Some controversies exist regarding the decomposition of the mode mixity. For a flexible linear elastic adhesive, the mode mixity of a single-layer adhesive joint is directly related to the deformation of the adhesive layer at the crack tip. The governing equations for linear elastic single-layer adhesive joints show that the mode mixity depends on the external loads, the properties of the adherends and often on the flexibility of the adhesive layer. This implies some fundamental problems that have to be addressed before an experimental method can be established. The purpose of this paper is to investigate different specimen configurations for mixed mode loading. Requirements for the design of a specimen configuration are given. A new specimen configuration is proposed based on the geometry of a semi-infinite symmetric DCB-specimen. According to this study, the proposed test specimen offers exceptional flexibility, variety and stability.     

An engineering solution for mesh size effects in the simulation of delamination using cohesive zone models

A methodology to determine the constitutive parameters for the simulation of progressive delamination is proposed. The procedure accounts for the size of a cohesive finite element and the length of the cohesive zone to ensure the correct dissipation of energy. In addition, a closed-form expression for estimating the minimum penalty stiffness necessary for the constitutive equation of a cohesive finite element is presented. It is shown that the resulting constitutive law allows the use of coarser finite element meshes than is usually admissible, which renders the analysis of large-scale progressive delamination problems computationally tractable.     

Fracture of U-notched specimens under mixed mode: Experimental results and numerical predictions

The first part of the paper gives an account of 153 fracture tests on blunted notched specimens (with notches of root radius ranging from 0.3 to 4.0 mm), loaded under mixed mode (ranging from almost pure mode I to mode II, and beyond). Maximum loads and initial crack angles were measured as a function of notch root radius and loading mixity. Such results can help in evaluating numerical models of the fracture of notched components. The second part of the paper deals with the suitability of the cohesive crack concept for predicting fracture loads under mixed mode. Use of local mode I was considered for numerical computations. Comparison of experimental results with numerical predictions was significantly accurate. Diagrams of fracture loci for notched components loaded under mixed mode are discussed.     

A new cohesive zone model for mixed mode interface fracture in bimaterials

A new two-dimensional cohesive zone model which is suitable for the prediction of mixed mode interface fracture in bimaterials is presented. The model accounts for the well known fact that the interfacial fracture toughness is not a constant, but a function of the mode mixity. Within the framework of this model, the cohesive energy and the cohesive strength are not chosen to be constant, but rather functions of the mode mixity. A polynomial cohesive zone model is derived in light of analytical and experimental observations of interface cracks. The validity of the new cohesive law is examined by analyzing double cantilever beam and Brazilian disk specimens. The methodology to determine the parameters of the model is outlined and a failure criterion for a pair of ceramic clays is suggested.     

Accurate simulation of delamination growth under mixed-mode loading using cohesive elements: Definition of interlaminar strengths and elastic stiffness

A methodology for predicting accurately the propagation of delamination under mixed-mode fracture with cohesive elements is proposed. It is shown that changes in the local mode ratio during the evolution of damage under mixed-mode loading can cause errors in the determination of the energy dissipation and result in inaccurate predictions of the global load–displacement response – even under conditions where, according to Linear Elastic Fracture Mechanics, the global mode ratio is constant. To address this difficulty, relations between the interlaminar strengths and the penalty stiffness are proposed which ensure a correct energy dissipation when delamination propagates. The validity of the proposed methodology is demonstrated for different mode ratios by comparison with the corresponding analytical solutions.     

In-situ characterization of interface delamination by a new miniature mixed mode bending setup

A new miniature mixed mode bending (MMMB) setup for in-situ characterization of interface delamination in miniature multi-layer structures was designed and realized. This setup consists of a novel test configuration to accomplish the full range of mode mixities and was specially designed with sufficiently small dimensions to fit in the chamber of a scanning electron microscope (SEM) or under an optical microscope for detailed real-time fracture analysis during delamination. Special care was taken to minimize the effects of friction, the influence of gravity, and non-linearities due to the geometry of the setup. The performance of the setup was assessed using specially-designed test samples supported by finite element analyses. Delamination experiments conducted on homogeneous bilayer samples in mode I and mixed mode loading were visualized with a scanning electron microscope and showed the formation of small micro cracks ahead of the crack tip followed by crack bridging and a full crack, thereby demonstrating the advantages of in-situ testing to reveal the microscopic delamination mechanism.     

A phantom node formulation with mixed mode cohesive law for splitting in laminates

A phantom node method with mixed mode cohesive law is proposed for the simulation of splitting in laminates. With this method, a discontinuity in the displacement field can be modeled at arbitrary locations. The micromechanical phenomenon that splitting cracks grow parallel to the fiber, is incorporated on the mesolevel, i.e., in the homogenized ply, by setting the direction of the crack propagation equal to the fiber direction. A new mixed mode cohesive law is introduced for increased robustness of the incremental-iterative solution procedure. The model is validated with mixed mode bending tests, and its utility is illustrated with examples for a single ply and for a laminate.     

An interface element for the simulation of delamination in unidirectional fiber-reinforced composite laminates

Unidirectional fiber-reinforced composite laminates are widely used in aerospace industry for a great variety of structural parts. In order to enhance the exploitation of material reserves, there is a need for the integration of progressive damage scenarios in the design phase. Due to their hazardous effects on the load-carrying capacity of composite structures, this work focusses on the simulation of delaminations. A finite element based on a cohesive zone approach is developed. Two constitutive laws are proposed. One is characterized by linear degradation after delamination onset, the other is governed by exponential softening response. The damage process is history-dependent leading to an irreversible stiffness degradation in damaged zones. The practicability of the proposed model and the assets and drawbacks of the two material laws are shown by some numerical examples.     

Fracture assessment of U-notches under mixed mode loading: two procedures based on the ‘equivalent local mode I’ concept

Two fracture criteria are proposed and applied to blunt-notched components made of brittle materials loaded under mixed mode; the former is based on the averaged strain energy density over a given control volume, the latter on the cohesive crack zone model. In both instances use of the equivalent local mode I hypothesis is made. Only two material properties are needed: the ultimate tensile strength and the fracture toughness. Numerical predictions of rupture loads from the two criteria are compared with experimental measurements from more than 160 static tests with notched beams. The samples are made of PMMA and tested at − 60°C to assure a bulk behaviour almost linear elastic up to rupture. Notch root radii range from 0.2 to 4.0 mm and load mixicity varies from pure mode I to a prevailing mode II. The good agreement between theory and experimental results adds further confidence to the proposed fracture criteria.     

Mixed mode cohesive law

A traction-separation relation to model the fracture process is presented. The cohesive law captures the linear elastic and softening behaviour prior to fracture. It also allows for different fracture parameters, such as fracture energy, strength and critical separation in different mode mixities. Thus, the fracture process in mode I (peel), in mode II (shear) or in mixed mode (a combination of peel and shear) can be modelled without the limitation of a common fracture energy in peel and shear. Examples are given in form of FE- implementations of the normalised cohesive law, namely for the Unsymmetrical Double Cantilever Beam (UDCB) specimen and the Mixed-mode double Cantilever Beam (MCB) specimen. Both specimens are adhesively bonded and loaded in mixed-mode.     

Cohesive zone length in numerical simulations of composite delamination

Accurate analysis of composite delamination using interface elements relies on having sufficient elements within a softening region known as the cohesive zone ahead of a crack tip. The present study highlights the limitations of existing formulae used to predict numerical cohesive zone length and demonstrates modifications necessary for improved accuracy. Clarification is also provided regarding the minimum number of interface elements within the cohesive zone. Finally, appropriate values of numerical interfacial strength are examined. The results presented will aid the application of mesh design techniques that both preserve numerical accuracy, whilst minimising computational expense.     

Cohesive zone with continuum damage properties for simulation of delamination development in fibre composites and failure of adhesive joints

A new approach is developed to implement the cohesive zone concept for the simulation of delamination in fibre composites or crack growth in adhesive joints in tension or shear mode of fracture. The model adopts a bilinear damage evolution law, and uses critical energy release rate as the energy required for generating fully damaged unit area. Multi-axial-stress criterion is used to govern the damage initiation so that the model is able to show the hydrostatic stress effect on the damage development. The damage material model is implemented in a finite element model consisting of continuum solid elements to mimic the damage development. The validity of the model was firstly examined by simulating delamination growth in pre-cracked coupon specimens of fibre composites: the double-cantilever beam test, the end-notched flexure test and the end-loaded split test, with either stable or unstable crack growth. The model was then used to simulate damage initiation in a composite specimen for delamination without a starting defect (or a pre-crack). The results were compared with those from the same finite element model (FEM) but based on a traditional damage initiation criterion and those from the experimental studies, for the physical locations of the delamination initiation and the final crack size developed. The paper also presents a parametric study that investigates the influence of material strength on the damage initiation for delamination.     

Determination of fracture energy and tensile cohesive strength in Mode I delamination of angle-ply laminated composites

The energy release rate in delamination of angle-ply laminated double cantilever composite beam specimens was calculated using the compliance equation, and interlaminar cohesive strengths were obtained. Instead of the traditional approach of a beam on an elastic foundation, a second-order shear-thickness deformation beam theory (SSTDBT) was considered. The equilibrium equations were obtained using the principle of minimum total potential energy and the system of ordinary differential equations were solved analytically. The problem was solved for [0°]₆ , [±30°]₅, and [±45°]₅ laminates with mid-plane delaminations and the results were verified using experimental evidence available in the literature.