Bilinear approximations to the mode II delamination cohesive law using an inverse method

Cohesive zone modelling has proved to be a powerful tool to model delamination problems in high performance composites. Application to mode II delamination is of particular interest due to the large fracture process zone. In this paper, bilinear approximations to the mode II delamination cohesive law were determined for two carbon/epoxy composites. This was achieved by an inverse method that consisted of making Finite Element Analyses fit experimental load–displacement curves measured in well-known End-Notched Flexure tests. The optimal parameters of the bilinear cohesive law were determined by a genetic algorithm. The results showed the adequacy of the bilinear cohesive law and of the methodology employed.     

On the notch sensitivity of flax fibre metal laminates under static and fatigue loading

Damage progression and failure characteristics of open‐hole flax fibre aluminium laminate (flax‐FML) specimens subjected to quasi‐static tensile or tension‐tension fatigue loading were experimentally investigated. Notched and unnotched flax‐FML composites exhibited brittle fracture with little or no fibre pull‐out and minimal delamination at the aluminium/adhesive interface. The flax‐FMLs were tested to failure under tension‐tension fatigue loading conditions (R ratio of 0.1; frequency of 10 Hz; applied fatigue stresses ranging between 30% and 80% of the respective ultimate tensile strength values). The fatigue cycles to failure decreased with the increase in the applied fatigue stress and hole diameter. A phenomenological modelling technique was developed to evaluate the fatigue life of an open‐hole flax‐FML composite. Fatigue tests on specimens subjected to a maximum load equivalent to 35% of the respective tensile failure strength were interrupted at around 85% of the corresponding fatigue life. The accumulated fatigue damage in these specimens was characterised using X‐ray computed tomography. For benchmarking purposes, the fatigue performance and related damage progression in the flax‐FML composite were compared with those of the glass‐FMLs.     

Simulation of delamination in composites under high-cycle fatigue

A damage model for the simulation of delamination propagation under high-cycle fatigue loading is proposed. The basis for the formulation is a cohesive law that links fracture and damage mechanics to establish the evolution of the damage variable in terms of the crack growth rate dA/dN. The damage state is obtained as a function of the loading conditions as well as the experimentally-determined coefficients of the Paris law crack propagation rates for the material. It is shown that by using the constitutive fatigue damage model in a structural analysis, experimental results can be reproduced without the need of additional model-specific curve-fitting parameters.     

A cohesive zone model for fatigue and creep–fatigue crack growth in single crystal superalloys

A numerical analysis using cohesive zone model under cyclic loading is proposed to develop a coupled predictive approach of crack growth in single crystal. The process of material damage during fatigue crack growth is described using an irreversible cohesive zone model, which governs the separation of the crack flanks and eventually leads to the formation of free surfaces. The cohesive zone element is modeled to accumulate fatigue damage during loadings and no damage during unloadings. This paper presents the damage model and its application in the study of the crack growth for precracked specimens. The use of cohesive zone approach is validated through a convergence study. Then, a general procedure of parameters calibration is presented in pure fatigue crack growth. In the last section, an extension of the cohesive zone model is presented in the case of creep–fatigue regime at high temperature. The model showed its capability to predict with a good agreement the crack growth in the case of complex loading and complex specimen geometries.     

A level set model for simulating fatigue-driven delamination in composites

This paper proposes a level set model for simulating delamination propagation in composites under high-cycle fatigue loading. For quasi-static loading conditions, interface elements with a cohesive law are widely used for the simulation of delamination. However, basic concepts from fatigue analysis such as the notion that the crack growth rate is a function of energy release rate cannot be embedded in existing cohesive laws. Therefore, we propose a model in which the cohesive zone is eliminated from the computation while maintaining the flexibility that the crack shape is not bound to element edges. The model is able to predict the delamination growth rate and its front shape accurately. To demonstrate the validity of the model, several tests under different fracture modes are conducted and the results are compared with experimental data, analytical solutions and results from cohesive zone analysis.     

Cohesive zone beam modelling of mixed-mode I–II delamination

Two cohesive zone beam models were developed for mixed-mode I–II delamination. The models were assessed by application to the moment-loaded double cantilever beam specimen, for which exact solutions are available. The main difficulties were caused by the very different distributions of the normal and shear tractions in the vicinity of the crack tip. This generated considerable local mode-mix variations along the cohesive zone. The separation controlled damage formulation was more sensitive to those difficulties, which could even compromise the accuracy of results. Instead, the energy controlled damage formulation proved to be robust and thus best suited for cohesive zone beam modelling of delamination.     

Interaction of Matrix Cracking and Delamination in Cross-ply Laminates: Simulations with Stochastic Cohesive Zone Elements

Two main damage mechanisms of laminates—matrix cracking and inter-ply delaminationare closely linked together (Joshi and Sun 1). This paper is focussed on interaction between matrix cracking and delamination failure mechanisms in CFRP cross-ply laminates under quasi-static tensile loading. In the first part of the work, a transverse crack is introduced in 90o layers of the cross-ply laminate [01/904/01], and the stresses and strains that arise due to tensile loading are analyzed. In the second part, the cohesive zone modelling approach where the constitutive behaviour of the cohesive elements is governed by traction-displacement relationship is employed to deal with the problem of delamination initiation from the matrix crack introduced in the 90o layers of the laminate specimen. Additionally, the effect of microstructural randomness, exhibited by CFRP laminates on the damage behaviour of these laminates is also accounted for in simulations. This effect is studied in numerical finite-element simulations by introducing stochastic cohesive zone elements. The proposed damage modelling effectively simulated the interaction between the matrix crack and delamination and the variations in the stresses, damage and crack lengths of the laminate specimen due to the microstructural randomness.     

Fatigue crack growth and delamination behaviours of advanced Al‐Li alloy laminate under single tensile overload

In this paper, fatigue crack growth and delamination behaviours of a new fibre metal laminate (FML) named as Al‐Li alloy laminate were tested under different single tensile overloads and compared with those of glass laminate aluminium reinforced epoxy. The results indicate that the crack growth rate of Al‐Li alloy laminate after overload applied can quickly get back to its original level when the crack grows outside of the overload plastic zone. The overload has no influence on the delamination shape and size of Al‐Li alloy laminate. These results are obviously different from those found in the present study for GLARE, in which the crack growth rate cannot recover after overload, even though the crack is far beyond the overload plastic zone. A kink nearby the location of overload applied was found in the obtained delamination shape. This study provides some new results for better understanding the damage tolerance mechanism of FMLs.     

Fatigue simulation for titanium/CFRP hybrid laminates using cohesive elements

This paper presents a new numerical approach for predicting fatigue crack growth in fiber-metal laminate (FML). Cohesive elements are used to express the complicated damage consisting of transverse cracking, splitting, and interlaminar delamination. The damage growth in the cohesive elements due to cyclic loading is represented by the conventional damage-mechanics model. The simulation was applied to notched Ti/CFRP hybrid laminates of two stacking configurations. In both cases, the crack growth rate in the titanium layer and the delamination shape agreed well with experiments reported in the literature. Complementary analysis for crack extension in the metal sheet is performed out of consideration of the damage in internal FRP layers. The numerical results demonstrated that the underlying damage modes in the FRP layer must be taken into account to predict the fatigue crack growth at the metal layer in FMLs.     

Modeling delamination growth in composites under fatigue loadings of varying amplitudes

A numerical model was developed to simulate the progressive delamination of a composite subjected to mode I fatigue loading regimes of varying amplitude. The model employs a cohesive zone approach, which combines damage mechanics and fracture mechanics, and requires only standard material data as input, namely the delamination toughness and the fatigue delamination growth curve. The proposed model was validated against delamination growth data obtained from a fatigue test conducted on a DCB specimen. The model predictions agree very well with the experimental results. This model is an initial step toward life prediction of composite structures subjected to complex fatigue regimes.     

A generic approach to constitutive modelling of composite delamination under mixed-mode loading conditions

A generic approach to constitutive modelling of composite delamination under mixed mode loading conditions is developed. The proposed approach is thermodynamically consistent and takes into account two major dissipative mechanisms in composite delamination: debonding (creation of new surfaces) and plastic/frictional deformation (plastic deformation of resin and/or friction between crack surfaces). The coupling between these two mechanisms, experimentally observed at the macro scales through the stiffness reduction and permanent crack openings, is usually not considered in depth in many cohesive models in the literature. All model parameters are shown to be identifiable and measurable from experiments. The model prediction of mixed-mode delamination is in good agreement with benchmarked mixed-mode bending experiments. It is further shown that accounting for all major dissipative mechanisms in the modelling of delamination is the key to the accurate prediction of both resistance and damage of the interface.     

Analysis on the fatigue damage evolution of notched specimens with consideration of cyclic plasticity

This paper presents a damage mechanics method applied successfully to assess fatigue life of notched specimens with plastic deformation at the notch tip. A damage‐coupled elasto‐plastic constitutive model is employed in which nonlinear kinematic hardening is considered. The accumulated damage is described by a stress‐based damage model and a plastic strain‐based damage model, which depend on the cyclic stress and accumulated plastic strain, respectively. A three‐dimensional finite element implementation of these models is developed to predict the crack initiation life of notched specimens. Two cases, a notched plate under tension‐compression loadings and an SAE notched shaft under bending‐torsion loadings including non‐proportional loadings, are studied and the predicted results are compared with experimental data.     

Numerical simulation of fatigue‐driven delamination using interface elements

This paper presents a computational technique for the prediction of fatigue‐driven delamination growth in composite materials. The interface element, which has been extensively applied to predict delamination growth due to static loading, has been modified to incorporate the effects of cyclic loading. Using a damage mechanics formulation, the constitutive law for the interface element has been extended by incorporating a modified version of a continuum fatigue damage model. The paper presents details of the fatigue degradation strategy and examples of the predicted fatigue delamination growth in mode I, mode II and mixed mode I/II are presented to demonstrate that the numerical model mimics the Paris law behaviour usually observed in experimental testing. Copyright © 2005 John Wiley & Sons, Ltd.     

Plastic zone size estimation under cyclic loadings using in situ optical microscopy fatigue testing

An in situ optical microscopy fatigue testing is proposed in this paper to investigate the forward and reversed plastic zone size under cyclic loadings for Al‐7075‐T6. This experimental study is used to verify the hypotheses in a recently developed small time scale formulation of fatigue crack growth. During the testing, the entire cyclic loading cycle is divided into a certain number of steps. Images of the crack tip are taken at each step. The full strain field around the crack tip is determined using the digital image correlation (DIC) technique. The plastic zone size is obtained by combining the DIC results and the material constitutive relationship. Experimental measurements from the proposed study are compared with theoretical predictions. It is observed that the crack closure has a large effect on the reversed plastic zone size. The plastic zone size remains almost constant when the unloading path is below a certain stress level, which is one of the hypotheses used in a previous crack growth model. Discussions are given for the modelling of plastic zone size variation under cyclic loadings and several conclusions are drawn based on the current investigation.     

A cohesive crack model coupled with damage for interface fatigue problems

An semi-analytical formulation based on the cohesive crack model is proposed to describe the phenomenon of fatigue crack growth along an interface. Since the process of material separation under cyclic loading is physically governed by cumulative damage, the material deterioration due to fatigue is taken into account in terms of interfacial cohesive properties degradation. More specifically, the damage increment is determined by the current separation and a history variable. The damage variable is introduced into the constitutive cohesive crack law in order to capture the history-dependent property of fatigue. Parametric studies are presented to understand the influences of the two parameters entering the damage evolution law. An application to a pre-cracked double-cantilever beam is discussed. The model is validated by experimental data. Finally, the effect of using different shapes of the cohesive crack law is illustrated     

A computationally efficient cohesive zone model for fatigue

A cohesive zone model has been developed for the simulation of both high and low cycle fatigue crack growth. The developed model provides an alternative approach that reflects the computational efficiency of the well‐established envelop‐load damage model yet can deliver the accuracy of the equally well‐established loading‐unloading hysteresis damage model. A feature included in the new cohesive zone model is a damage mechanism that accumulates as a result of cyclic plastic separation and material deterioration to capture a finite fatigue life. The accumulation of damage is reflected in the loading‐unloading hysteresis curve, but additionally, the model incorporates a fast‐track feature. This is achieved by “freezing in” a particular damage state for one loading cycle over a predefined number of cycles. The new model is used to simulate mode I fatigue crack growth in austenitic stainless steel 304 at significant reduction in the computational cost.     

Multiaxial creep–fatigue under anisothermal conditions

Because creep–fatigue is mainly studied in uniaxial tension, it is shown here how to proceed to perform both experiments and calculations under multiaxial loading and when the temperature varies both in time and space. The constitutive equations used are those of elasto‐visco‐plasticity coupled or not, to damage, with isotropic and kinematic hardening. It is shown that the unified damage law first proposed for ductile failure and then for fatigue may also be applied to multiaxial creep–fatigue interactions with a new expression for the damage threshold. The procedure for the identification of material parameters is described in detail. Finally, it is shown that the uncoupled calculation procedure, where damage is calculated as a post‐processing of an elasto‐visco‐plastic computation, gives satisfactory results in comparison to the fully coupled analysis; the latter being more accurate but very expensive in computer time.     

Mixed‐mode delamination in 2D layered beam finite elements

In this work, a 2D finite element (FE) formulation for a multi‐layer beam with arbitrary number of layers with interconnection that allows for mixed‐mode delamination is presented. The layers are modelled as linear beams, while interface elements with embedded cohesive‐zone model are used for the interconnection. Because the interface elements are sandwiched between beam FEs and attached to their nodes, the only basic unknown functions of the system are two components of the displacement vector and a cross‐sectional rotation per layer. Damage in the interface is modelled via a bi‐linear constitutive law for a single delamination mode and a mixed‐mode damage evolution law. Because in a numerical integration procedure, the damage occurs only in discrete integration points (i.e. not continuously), the solution procedure experiences sharp snap backs in the force‐displacements diagram. A modified arc‐length method is used to solve this problem. The present model is verified against commonly used models, which use 2D plane‐strain FEs for the bulk material. Various numerical examples show that the multi‐layer beam model presented gives accurate results using significantly less degrees of freedom in comparison with standard models from the literature. Copyright © 2015 John Wiley & Sons, Ltd.     

Evaluating a strain energy fatigue method using cyclic plasticity models

For the development of constitutive equations that describe the behaviour of materials under cyclic plastic strains, different kinds of formulations can be adopted. Recently, an energy‐based fatigue damage parameter has been developed to present energy‐fatigue life curves using a calculation of the total strain energy. In this study, the damage criterion is examined by calculation of the plastic strain energy from stress–strain hysteresis loops in the cyclic plasticity models under condition of multi‐axial fatigue. These cyclic plasticity models are the Garud multi‐surface model and the Chaboche nonlinear kinematic hardening model. The models are briefly explained and the general features of their computational procedure are presented. Then, the hysteresis loops of these models will be obtained and the fatigue lives are predicted and compared to experimental data by the ratio of predicted life to experimental life. Consequently, a weighting factor on shear plastic work is presented to decrease the life factors.