Trapezoidal Traction-separation Law Research Articles

A new cohesive-based modelling of moisture-dependent mode II delamination of carbon/epoxy composites

The cohesive zone model (CZM) is widely employed for simulating delamination and debonding behaviour in engineering structures. However, the choice of CZM shape impacts the accuracy of simulation results. Therefore, the selection of a suitable Traction-Separation Law (TSL) with appropriate cohesive parameters is crucial. This study focuses on simulating the mode II delamination of unidirectional carbon/epoxy composites under varying moisture content levels (0%, 2.2%, 3.8%, and 5.3%) using the End Notched Flexure (ENF) test. Trapezoidal TSL (TTSL) with varying pseudo-plasticity parameter (Γ) was implemented and compared. A guideline was proposed in this study to aid in the selection of cohesive parameters, and a relationship between these parameters and moisture content was established. The results indicate that Γ = 0.99 yields a more accurate force-displacement response compared to Γ = 0. The estimated cohesive zone length falls within the range of 2.0–2.4 mm. During crack growth, the analysis reveals that, at the peak force, a region approximately 0.5–0.6 mm ahead of the crack tip undergoes total failure. Simultaneously, damage initiation occurs in the region 2.3–2.4 mm beyond the complete failure zone.

An equivalent cohesive zone model for a wide range of ductile and brittle adhesives

AbstractThis paper aims to propose an equivalent trapezoidal traction–separation law (ET‐TSL) of the cohesive zone model applicable to a wide range of ductile and brittle adhesives using equivalent and fictitious material concepts. The ET‐TSL can reduce the analysis time and cost of complex problems without degrading the prediction accuracy. To experimentally validate the proposed ET‐TSL, double‐cantilever beam (DCB) samples were fabricated with glass fiber‐reinforced polymer substrates and a highly ductile adhesive and then tested. The fracture behavior of the specimens was modeled using the proposed ET‐TSL. Moreover, the accuracy and comprehensiveness of the proposed model were verified by employing the proposed ET‐TSL for different adhesive joints made of different adhesives ranging from brittle to ductile adhesives and different substrates found in the literature. It was found that the proposed ET‐TSL can provide reasonable accuracy in simulating the fracture response of different types of adhesive joints with brittle and ductile adhesives.

Characterization and prediction of hygrothermally aged CFRP adhesive joint subjected to mode II load

Carbon fiber composite adhesives joints provide higher specific strength and fatigue life than traditional joints. However, carbon fiber composites and adhesives are polymeric materials that may lose their integrity in hygrothermal environment. Predicting the environmental degradation behavior of composite joints is challenging because of limited understanding of the complex failure behavior. Moreover, the experiments are time-consuming and therefore it is essential to use accelerated aging methods such as the open faced method. However, the challenge is how to incorporate the accelerated fracture test data into the fracture prediction of real joints. To address this, a methodology is presented for predicting mode II cohesive law parameters of degraded CFRP epoxy adhesive joints using a direct method, and an accelerated aging test. End notched flexure (ENF) specimens made with CFRP laminate and epoxy adhesive were used to study the mode II fracture. Accelerated and uniform aging was achieved using open-faced specimens aged at 40 °C and 82% relative humidity (RH). Fracture testing of aged open-faced specimens along with crack-tip images were used to determine the degradation in trapezoidal traction-separation law (TSL) with aging. These variations in trapezoidal TSL with aging were used to develop a 3D finite element (FE) model of a closed ENF specimen that captures the non-uniform degradation across the specimen width. The TSL parameters degraded significantly at the specimen edges compared to width center for the closed ENF specimens. Validation of the FE model with aged closed ENF joint showed good agreement.

Hydrogen-Assisted Brittle Fracture Behavior of Low Alloy 30CrMo Steel Based on the Combination of Experimental and Numerical Analyses.

Compact-tension (CT) specimens made of low alloy 30CrMo steels were hydrogen-charged, and then subjected to the fracture toughness test. The experimental results revealed that the higher crack propagation and the lower crack growth resistance (CTOD-R curve) are significantly noticeable with increasing hydrogen embrittlement (HE) indexes. Moreover, the transition in the microstructural fracture mechanism from ductile (microvoid coalescence (MVC)) without hydrogen to a mixed quasi-cleavage (QC) fracture and QC + intergranular (IG) fracture with hydrogen was observed. The hydrogen-enhanced decohesion (HEDE) mechanism was characterized as the dominant HE mechanism. According to the experimental testing, the coupled problem of stress field and hydrogen diffusion field with cohesive zone stress analysis was employed to simulate hydrogen-assisted brittle fracture behavior by using ABAQUS software. The trapezoidal traction-separation law (TSL) was adopted, and the initial TSL parameters from the best fit to the load-displacement and J-integral experimental curves without hydrogen were calibrated for the critical separation of 0.0393 mm and the cohesive strength of 2100 MPa. The HEDE was implemented through hydrogen influence in the TSL, and to estimate the initial hydrogen concentration based on matching numerical and experimental load-line displacement curves with hydrogen. The simulation results show that the general trend of the computational CTOD-R curves corresponding to initial hydrogen concentration is almost the same as that obtained from the experimental data but in full agreement, the computational CTOD values being slightly higher. Comparative analysis of numerical and experimental results shows that the coupled model can provide design and prediction to calculate hydrogen-assisted fracture behavior prior to extensive laboratory testing, provided that the material properties and properly calibrated TSL parameters are known.

Primal interface debonding formulation for finite strain isotropic plasticity

A framework is developed for modeling ductile damage of nonlinear materials whose plastic deformation is characterized using rate independent classical plasticity. This method relies on the assumption that the free energy can be decomposed into elastic, plastic and damage parts. A thermodynamically consistent method is derived which satisfies the second law of thermodynamics in the Clausius–Duhem inequality form. The dissipation associated with plasticity takes place in the domain only, while damage dissipation is localized to the interface. The method is developed using Variational Multiscale ideas to obtain definitions of the interface fluxes within a primal formulation analogous to the Discontinuous Galerkin method, which ensures weakly vanishing interface gap prior to reaching a damage initiation criterion. The local nonlinear problem to calculate both plastic deformation gradient and damage variable follows an incremental approach similar to classical plasticity return mapping algorithm. This elastoplastic damage formulation is developed for material undergoing finite strain, and it naturally accommodates a trapezoidal traction separation law (TSL) whose shape can be varied to model either ductile interface behavior or brittle interface behavior. The formulation's performance is assessed through modeling a patch test and a compact tension specimen.

Influence of geometrical parameters on the strength of Hybrid CFRP-aluminium tubular adhesive joints

Tubular adhesive joints, used in truss structures to join pultruded carbon fibre-reinforced polymer members to aluminium nodes, are modelled with varying dimensions. The numerical model uses a Cohesive Zone Modelling formulation with a trapezoidal traction-separation law for the adhesive layer, and experimental tests are carried to validate it. The results showed that the joint strength increases significantly with the bonding area, with a limit on the overlap length above which it stops increasing. This upper limit is affected by the thickness and tapering angle of the adherends, due to their influence on the shear stress distribution along the overlap. On the other hand, the adhesive thickness has only a marginal influence on the joint strength.

A modified cohesive zone model for fatigue delamination in adhesive joints: Numerical and experimental investigations

A modified cohesive zone model (CZM) has been developed to simulate damage initiation and evolution in Fibre-Metal Laminates (FMLs) manufactured in-house but based on the Glare® material specifications. Specimens containing both splice and doubler features were analysed under high cycle fatigue loading. The model uses a novel trapezoidal traction-separation law to describe the elastic-plastic behaviour of this material under monotonic and high-cycle fatigue loading. The model is implemented in the software Abaqus/Explicit via an user-defined cohesive material subroutine. Several models of increasing complexity were investigated to validate the proposed approach. A two-stage experimental testing programme was then conducted to validate the numerical analyses. Firstly, quasi-static tests were used to determine the ultimate tensile strength (UTS) of a series of specimens with and without internal features. Secondly, high-cycle fatigue tests were conducted on both laminate types with variable load amplitude so that S-N curves could be built. Tests were monitored using digital image correlation (DIC) for full-field strain mapping and acoustic emission (AE) sensing to detect the initiation and propagation of damage during quasi-static and fatigue tests. Good correlation was observed between predicted onset and growth of delaminations and the history of cumulative AE energy during the tests, which supports the validity of the cohesive modelling approach for FMLs.

Simulations of fracture tests of uncharged and hydrogen-charged additively manufactured 304 stainless steel specimens using cohesive zone modeling

Fracture tests of uncharged and hydrogen-charged single edge bend specimens of additively manufactured 304 stainless steels are simulated using the cohesive zone modeling (CZM) approach. Two-dimensional plane strain finite element analyses without cohesive elements are conducted to identify the values of cohesive energy. Similar analyses using CZM with the trapezoidal traction-separation laws are then conducted. The best-fit cohesive parameters show the values of cohesive strength for the uncharged specimens are higher than those for the hydrogen-charged ones whereas the value of cohesive energy for the uncharged specimens can be either slightly lower or higher than that for the hydrogen-charged ones.

Trapezoidal traction–separation laws in mode II fracture in nano-composite and nano-adhesive joints

The main focus of this paper is on the experimental investigation and comparison between different bridging laws. For mode II fracture in the presence of nano-particles, these laws are calculated from three data reduction schemes for describing the bridging zone and trapezoidal traction–separation law parameters. For the calculation of the energy release rate in mode II fracture, three corresponding data reduction schemes, compliance calibration method, corrected beam theory and compliance-based beam method, have been utilized for different percentages of nano-particles in the adhesives and the adherents.

Analysis of the fracture process zone and effective crack length in the adhesively bonded end-notched flexure specimen

ABSTRACTThe mode II fracture of adhesive joints is well-known to involve large fracture process zones. Their effect in fracture energy measurements can be taken into account by the effective crack length approach. Moreover, fracture process zones can be simulated by cohesive zone models, which are increasingly used for structural analysis of adhesive joints. This paper aimed at evaluating the influence of the traction-separation law on the fracture process zone and on the effective crack length in end-notched flexure tests. Novel analytical cohesive zone models were developed for the bilinear and trapezoidal traction-separation laws. The latter were shown to affect significantly the energy dissipation rate versus effective crack length curve prior to crack initiation. Therefore, this effect seems to provide a simple approach for evaluating approximate traction-separation laws. The models here developed are easy to apply and provide simple approximate expressions useful for specimen selection.

Simulation of ductile fracture in pipeline steels under varying constraint conditions using cohesive zone modeling

In this study, the effect of non-singular T-stress (a measure of crack tip constraint) on crack growth resistance curves (R-curves) and crack tip opening angle (CTOA) was investigated. The investigation proceeded in two parts; first using a modified boundary layer (MBL) model to systematically assess the effect of T-stress on the R-curves and CTOA, and second comparing the CTOA under different loading modes, namely, bending and tension. Three steels were examined in this work; the first was a typical high strength steel referred to as TH, the second and third were pipeline steels referred to as X70 and X100. Surface-based cohesive zone models with four sets of bilinear traction-separation (TS) laws were used for TH steel. The models of X70 and X100 were computed using element-based cohesive zone modeling with a trapezoidal TS law. All simulations were conducted using the finite element (FE) program ABAQUS. It was assumed in these simulations that there was no effect of T-stress on the TS laws per se. With this assumption, it was found that the T-stress and loading mode have only a small effect (1–2°) on the CTOA for the materials studied.

A cohesive zone element for mode I modelling of adhesives degraded by humidity and fatigue

A finite element is proposed, based on the cohesive zone model approach and implemented as a user element, for the modelling of adhesively bonded joints subjected to degradation by humidity and fatigue using software ABAQUS®. Functionality included in this element that is not available using standard cohesive zone elements includes: (a) various types of traction-separation laws, such as triangular with exponential softening, trapezoidal and exponential, (b) an intuitive and easy to use graphical interface built in MATLAB that helps visualize all traction-separation laws, create the mesh, run the simulation and visualize the results, (c) custom degradation laws for both humidity and fatigue which allow the user to easily model the effects of said degrading parameters. It is shown that the trapezoidal traction-separation law is the most appropriate to model the experimental data in both unaged and aged specimens. The proposed fatigue degradation approach correctly predicts the number of cycles until failure of all unaged and aged conditions, thus proving itself as a very useful tool capable of modelling a vast array of experimental conditions and details that adhesive joints are subjected to in real world applications.

Numerical evaluation of buckling behaviour induced by compression on patch-repaired composites

A progressive damage model is proposed to predict buckling strengths and failure mechanisms for both symmetric and asymmetric patch repaired carbon-fibre reinforced laminates subjected to compression without lateral restrains. Solid and cohesive elements are employed to discretize composite and adhesive layers, respectively. Coupling with three dimensional strain failure criteria, an energy-based crack band model is applied to address the softening behaviour in composites with mesh dependency elimination. Both laminar and laminate scaled failure are addressed. Patch debonding is simulated by the cohesive zone model with a trapezoidal traction–separation law applied for the ductile adhesive. Geometric imperfection is introduced into the nonlinear analysis by the first-order linear buckling configuration. Regarding strengths and failure patterns, the simulation demonstrates an accurate and consistent prediction compared with experimental observations. Though shearing is the main contributor to damage initiation in adhesive, stress analysis shows that lateral deformation subsequently reverses the distribution of normal stresses which stimulates patch debonding at one of the repair sides. The influence of patch dimensions on strengths and failure mechanisms can be explained by stress distributions in adhesive and lateral deformation of repairs. Comparison between symmetric and asymmetric regarding strength and failure modes shows that structural asymmetry can intensify lateral flexibility. This resulted in earlier patch debonding and negative effects on strengths.

Dislocation shielding of a cohesive crack

Dislocation interaction with a cohesive crack is of increasing importance to computational modelling of crack nucleation/growth and related toughening mechanisms in confined structures and under cyclic fatigue conditions. Here, dislocation shielding of a Dugdale cohesive crack described by a rectangular traction-separation law is studied. The shielding is completely characterized by three non-dimensional parameters representing the effective fracture toughness, the cohesive strength, and the distance between the dislocations and the crack tip. A closed form analytical solution shows that, while the classical singular crack model predicts that a dislocation can shield or anti-shield a crack depending on the sign of its Burgers vector, at low cohesive strengths a dislocation always shields the cohesive crack irrespective of the Burgers vector. A numerical study shows the transition in shielding from the classical solution of Lin and Thomson (1986) in the high strength limit to the solution in the low strength limit. An asymptotic analysis yields an approximate analytical model for the shielding over the full range of cohesive strengths. A discrete dislocation (DD) simulation of a large (>10 3) number of edge dislocations interacting with a cohesive crack described by a trapezoidal traction-separation law confirms the transition in shielding, showing that the cohesive crack does behave like a singular crack at very high cohesive strengths (∼7 GPa), but that significant deviations in shielding between singular and cohesive crack predictions arise at cohesive strengths around 1GPa, consistent with the analytic models. Both analytical and numerical studies indicate that an appropriate crack tip model is essential for accurately quantifying dislocation shielding for cohesive strengths in the GPa range.

Tensile behaviour of three-dimensional carbon-epoxy adhesively bonded single- and double-strap repairs

An experimental and numerical study of the tensile behaviour of three-dimensional carbon-epoxy adhesively bonded strap repairs is presented. Experimentally, the failure mode, elastic stiffness and strength were evaluated for different overlap lengths and patch thicknesses. The numerical simulations, performed in ABAQUS ®, allowed obtaining the elastic stiffness and the patch debonding load, used to understand the repairs behaviour. The adhesive layer was simulated with cohesive elements including a mixed-mode cohesive damage model with trapezoidal traction-separation laws in pure modes I and II, to account for the ductile behaviour of the adhesive used. These laws were determined by an inverse method, which consists on the estimation of the cohesive parameters with a fitting procedure of the experimental and numerical load–displacement curves of the respective fracture characterization test. The pure mode III cohesive law was equalled to the pure mode II one. This numerical methodology was found adequate to reproduce the experimentally observed behaviour of these repairs.

Buckling Behaviour of Carbon–Epoxy Adhesively-Bonded Scarf Repairs

The present work is dedicated to the experimental and numerical study of the buckling behaviour under pure compression of carbon–epoxy adhesively-bonded scarf repairs, with scarf angles varying from 2 to 45°. The experimental results were used to validate a numerical methodology using the Finite Element Method and a mixed-mode cohesive damage model implemented in the ABAQUS® software. The adhesive layer was simulated using cohesive elements with trapezoidal traction–separation laws in pure modes I and II to account for the ductility of the adhesive used. The cohesive laws in pure modes I and II were determined with Double Cantilever Beam and End-Notched Flexure tests, respectively, using an inverse method. Since in the experiments interlaminar and transverse intralaminar failures also occurred, cohesive laws to simulate these failure modes were also obtained experimentally following a similar procedure. Good correlations were found between the numerical predictions and experimental results for the elastic stiffness, maximum load and the corresponding displacement, plateau displacement and failure mode of the repairs.