Residual strength prediction of a complex structure using crack extension analyses

The residual strength of a flat panel (thickness 7.6 mm) with five stringers, machined from a monolithic block of Al2024-T351 material, which contained a crack that divided the central stringer, was to be predicted during a Round Robin organised by ASTM. The initial crack tips were right ahead of the stringers #2 and #4, respectively, so that crack branching along the skin and into the stringers occurred after initiation. The prediction has been achieved using finite element simulations including crack extension, for which a cohesive model was utilised. Conventional material properties, yield and ultimate strength as well as experimental results from M(T) specimens in terms of force, COD and Δa, were given. The residual strength prediction was performed in two-steps: First the crack extension parameters for the cohesive model, the cohesive strength, T₀, and the cohesive energy, Γ₀, were determined by numerical reproduction of the results of the M(T) specimen. With the optimised parameters, the five-stringer panel was modelled. These steps were conducted by two different finite element models: by a shell and a 3D finite element mesh. It turned out that it is possible to analyse the structure with both models. In the 3D case, the residual strength prediction was conservative and the deviation of the predicted from the experimental value was below 9%. The results of the shell simulation were even closer to the experiment (deviation approximately 3%), but the simulation was non-conservative.     

Simulation of hydrogen assisted stress corrosion cracking using the cohesive model

This paper investigates the effect of hydrogen diffusion on stable crack propagation by using numerical finite element simulations based on the cohesive model. The model with its two common parameters, cohesive strength, T₀, and critical separation, δ₀, and its two additional parameters for stress corrosion cracking, the effective diffusivity, D_eff, and a material parameter, α, which represents the reduction of the cohesive strength, is described. This model is then employed to predict the stable crack propagation in C(T) specimens made from a high strength structural steel which were tested under hydrogen charging conditions in rising displacement tests using various deformation rates. It is shown that, in general, the prediction of stable crack propagation is promising, but may be further improved by the use of a more sophisticated diffusion equation. Finally, the influence of variations of the effective diffusivity and the cohesive strength reduction on the thus simulated crack growth resistance curves is studied.     

Application of the cohesive model for predicting the residual strength of a large scale fuselage structure with a two-bay crack

The residual strength of a curved and stiffened panel containing a two-bay crack was assessed using the cohesive model. This panel represents a section of a wide-body aeroplane fuselage. The tests were conducted at IMA GmbH Dresden in cooperation with Airbus Industries Germany. The structural panel was modelled using 3D finite elements and a layer of cohesive elements ahead of each crack tip allowing for 70 mm crack extension. Identification of the cohesive parameters was done on small laboratory test pieces. Special effort was made for the transfer of these parameters to the structure. Reasonably conservative predictions of the residual strength of the panel were achieved. The boundary conditions of the loading devices of the test rig are shown to have substantial influence on the predictions.     

Fracture and damage mechanics modelling of thin-walled structures - An overview

This paper reviews the most important current approaches for residual strength prediction of thin-walled structures. Crack driving force parameters such the linear elastic stress intensity factor and its plastic zone corrected extension for contained yielding conditions, the crack tip opening displacement δ₅, the crack tip opening angle CTOA, the cohesive zone model parameters, separation energy, critical tensile stress and critical separation and the parameters of the damage models of Gurson-Tvergaard-Needleman type are introduced and discussed with respect to their benefits and limitations for the simulation of plane and stiffened panels. In addition, specific aspects of modern non-integral and integral structures which pose a challenge are addressed. These comprise multi-site damage, crack deviation and branching, welding residual stresses, strength mismatch in material compounds and problems in bonded structures, such as delamination. A number of examples are provided to illustrate the potential of the various approaches.     

Residual strength analysis using CTOA criteria for fuselage structures containing multiple site damage

An extensive experimental program was conducted by the Boeing Company under the funding of the Federal Aviation Administration (FAA), National Aeronautics and Space Administration (NASA), and the United States Air Force Research Laboratory (USAF/RL) to investigate the effects of multiple-site damage (MSD) on the residual strength of typical fuselage splice joints. The experimental results were used to validate the analytical prediction using various methodologies, including STAGS (a generalized shell finite element code) with the crack-tip-opening angle and T^* fracture criteria.The test specimens consisted of large flat panels, curved panels, and an aft pressure bulkhead. The flat panel specimens included three types of longitudinal splice joints and one type of circumferential splice joint. For each type, one panel contained only a lead crack and the other two panels contained MSD 1.3 and 2.5 mm in size, respectively, at the fastener holes ahead of the lead crack. The curved panels were tested under simulated loads of combined cabin pressure and fuselage down bending. Two skin splice types were tested. For each splice type, one panel contained a lead crack only and the other had a lead crack with various sizes of MSD. A section of an aft fuselage containing a large lead crack and MSD in the pressure dome was also tested to demonstrate the capabilities of the methodologies in analyzing actual aircraft structures. This paper presents the analytical approaches and the comparison of predictions with the experimental results in terms of crack linkup stress and residual strength.     

Fully-automatic modelling of cohesive crack growth using a finite element-scaled boundary finite element coupled method

This study develops a method coupling the finite element method (FEM) and the scaled boundary finite element method (SBFEM) for fully-automatic modelling of cohesive crack growth in quasi-brittle materials. The simple linear elastic fracture mechanics (LEFM)-based remeshing procedure developed previously is augmented by inserting nonlinear interface finite elements automatically. The constitutive law of these elements is modelled by the cohesive/fictitious crack model to simulate the fracture process zone, while the elastic bulk material is modelled by the SBFEM. The resultant nonlinear equation system is solved by a local arc-length controlled solver. The crack is assumed to grow when the mode-I stress intensity factor K_I vanishes in the direction determined by LEFM criteria. Other salient algorithms associated with the SBFEM, such as mapping state variables after remeshing and calculating K_I using a “shadow subdomain”, are also described. Two concrete beams subjected to mode-I and mixed-mode fracture respectively are modelled to validate the new method. The results show that this SBFEM-FEM coupled method is capable of fully-automatically predicting both satisfactory crack trajectories and accurate load-displacement relations with a small number of degrees of freedom, even for problems with strong snap-back. Parametric studies were carried out on the crack incremental length, the concrete tensile strength, and the mode-I and mode-II fracture energies. It is found that the K_I ? 0 criterion is objective with respect to the crack incremental length.     

Residual stresses and fracture mechanics analysis of a crack in welds of high strength steels

This study presented the characteristics of residual stresses in welds of high strength steels (POSTEN60, POSTEN80) whose tensile strengths were 600 MPa and 800 MPa, respectively. Three-dimensional thermal elastic-plastic analyses were conducted to investigate the characteristics of welding residual stresses in welds of high strength steels through the thermal and mechanical properties at high temperatures obtained from the elevated temperature tensile tests. A finite element analysis method which can calculate the J-integral for a crack in a residual stress field was developed to evaluate the J-integral for a centre crack when mechanical stresses were applied in conjunction with residual stresses.The results show that the volumetric changes associated with the austenite to martensite phase transformation during rapid cooling after welding of high strength steels significantly influence on the development of residual stresses in the weld fusion zone and heat-affected zone. For a centre crack in welds of high strength steels where only residual stresses are present, increased tensile strength of the steel, increased the J-integral values. The values of the J-integral for the case when mechanical stresses are applied in conjunction with residual stresses are larger than those for the case when only residual stresses are present.     

Fracture characterization of sandwich structures interfaces under mode I loading

The accurate prediction of failure of sandwich structures using cohesive mixed-mode damage models depends on the accurate characterization of the cohesive laws under pure mode loading. In this work, a numerical and experimental study on the asymmetric double cantilever beam (DCB) sandwich specimen is presented with the objective to characterize the debonding fracture between the face sheet and the core under pure mode I. A data reduction method based on beam theory was formulated in such a way to incorporate the complex damaging phenomena of the debonding due to the material and geometric asymmetry of the specimen, via the consideration of an equivalent crack length (a_e). Experimental DCB tests were performed and the proposed methodology was followed to obtain the debonding fracture energy (G_Ic). The experimental tests were numerically simulated and a cohesive damage model was employed to reproduce crack propagation. An inverse method was followed to obtain the local cohesive strength (σ_u,I) based on the fitting of the numerical and experimental load–displacement curves. With the value of fracture energy and cohesive strength defined, the cohesive law for interface mode I fracture is characterized. Good agreement between the numerical and the experimental R-curves validates the accuracy of the proposed data reduction procedure.     

On the determination of the cohesive zone parameters for the modeling of micro-ductile crack growth in thick specimens

The paper deals with the determination of the cohesive zone parameters (separation energy, , and cohesive strength, T _max) for the 3D finite element modeling of the micro-ductile crack growth in thick, smooth-sided compact tension specimens made of a low-strength steel. Since the cohesive zone parameters depend, in general, on the local constraint conditions around the crack tip, their values will vary along the crack front and with crack extension. The experimental determination of the separation energy via automated fracture surface analysis is not accurate enough. The basic idea is, therefore, to estimate the cohesive zone parameters, and T _max, by fitting the simulated distribution of the local crack extension values along the crack front to the experimental data of a multi-specimen J _IC-test. Furthermore, the influence of the cohesive zone parameters on the crack growth behavior is investigated. The point of crack growth initiation is determined only by the magnitude of . Both and T _max affect the crack growth rate (or the crack growth resistance), but the influence of the cohesive strength is much stronger than that of the separation energy. It turns out that T _max as well as vary along the crack front. In the center of the specimen, where plane strain conditions prevail, the separation energy is lower and the cohesive strength is higher than at the side-surface.     

Delamination buckling and propagation analysis of honeycomb panels using a cohesive element approach

The cohesive element approach is proposed as a tool for simulating delamination propagation between a facesheet and a core in a honeycomb core composite panel. To determine the critical energy release rate (G _c) of the cohesive model, Double Cantilever Beam (DCB) fracture tests were performed. The peak strength (_c) of the cohesive model was determined from Flatwise Tension (FWT) tests. The DCB coupon test was simulated using the measured fracture parameters, and sensitivity studies on the parameters for the cohesive model of the interface element were performed. The cohesive model determined from DCB tests was then applied to a full-scale, 914×914 mm (36×36 in.) debond panel under edge compression loading, and results were compared with an experiment. It is concluded that the cohesive element approach can predict delamination propagation of a honeycomb panel with reasonable accuracy.     

Numerical analysis of the Edge Crack Torsion test for mode III interlaminar fracture of composite laminates

A detailed numerical analysis of the Edge Crack Torsion test was performed to verify its adequacy for the mode III interlaminar fracture characterization of composite laminates. A new data reduction scheme based on specimen compliance was proposed. Three-dimensional finite element analyses including a cohesive damage model were performed in order to evaluate the accuracy of perceived G_IIIc values obtained with the proposed method. Despite some degree of crack length dependency of perceived G_IIIc, acceptable errors could be achieved within a certain crack length range, which is thus recommended for experimental tests.     

Numerical Modeling of Combined Matrix Cracking and Delamination in Composite Laminates Using Cohesive Elements

Sub-laminate damage in the form of matrix cracking and delamination was simulated by using interface cohesive elements in the finite element (FE) software ABAQUS. Interface cohesive elements were inserted parallel to the fiber orientation in the transverse ply with equal spacing (matrix cracking) and between the interfaces (delamination). Matrix cracking initiation in the cohesive elements was based on stress traction separation laws and propagated under mixed-mode loading. We expanded the work of Shi et al. (Appl. Compos. Mater. 21, 57–70 2014) to include delamination and simulated additional [45/?45/0/90]_s and [0₂/90_n]_s {n?=?1,2,3} CFRP laminates and a [0/90₃]_s GFRP laminate. Delamination damage was quantified numerically in terms of damage dissipative energy. We observed that transverse matrix cracks can propagate to the ply interface and initiate delamination. We also observed for [0/90_n/0] laminates that as the number of 90° ply increases past n?=?2, the crack density decreases. The predicted crack density evolution compared well with experimental results and the equivalent constraint model (ECM) theory. Empirical relationships were established between crack density and applied stress by linear curve fitting. The reduction of laminate elastic modulus due to cracking was also computed numerically and it is in accordance with reported experimental measurements.     

Residual strength analyses of stiffened and un-stiffened panels––Part I: laboratory specimens

This paper presents the results of residual strength analyses on stiffened and un-stiffened panels using the STructural Analysis of General Shells (STAGS) finite-element shell code and the critical crack-tip-opening angle (CTOA) fracture criterion. Previous analyses of wide, flat panels have shown that high-constraint conditions around a crack front must be modeled in order for the critical CTOA fracture criterion to predict wide panel failures from small laboratory tests. Thus, the STAGS code with the “plane-strain” core option was used in all analyses. In the present study, the critical CTOA (Ψ_c) value and the plane-strain core height were determined from a fit to the experimental load-against-crack-extension results from a series of middle-crack tension specimens (76–1016 mm wide) tested with anti-buckling guides. In the residual strength analyses of the 305-mm wide stiffened panels with a single crack, modeling of the sheet, stiffeners, rivet flexibility and buckling were based on methods and criteria, like that currently used in industry. STAGS and the CTOA criterion were used to predict load-against-crack extension for the single stiffened panels for both intact and cut stiffeners. Analyses were able to predict stable crack growth and residual strength of the single stiffened panels within about ±5% of the test failure loads.     

Determination of cohesive laws of composite bonded joints under mode II loading

In this work, mode II cohesive laws of carbon–epoxy composite bonded joints were obtained using the direct method applied to the end notched flexure (ENF) test. The direct method is based on the differentiation of the relation between the evolution of the fracture energy (J_II) and the crack tip opening displacement in mode II (CTOD_II) during the test. A data reduction scheme based on equivalent crack length concept was used to obtain the evolution of the fracture energy during the test. The method allows overcoming problems related to identification of crack tip in mode II tests and the presence of a non-negligible fracture process zone (FPZ), which both difficult the right estimate of J_II. The digital image correlation technique (DIC) was used to monitor the CTOD_II, which was synchronized with the load–displacement data. A trapezoidal cohesive law was fitted to the experimental one in order to perform numerical simulations using finite element analysis. The main goal was to validate all the procedure used to get the cohesive laws. The good agreement obtained between the numerical and experimental load-CTOD_II curves and between the cohesive laws demonstrates the adequacy of the proposed procedure concerning the evaluation of the composite bonded joints cohesive laws under mode II loading.     

Effect of residual stresses on ductile crack growth resistance

It is well known that residual stresses influence the ductile fracture behaviour. In this paper, a numerical study was performed to assess the effect of residual stresses on ductile crack growth resistance of a typical pipeline steel. A modified boundary layer model was employed for the analysis under plane strain, Mode I loading condition. The residual stress fields were introduced into the finite element model by the eigenstrain method. A sharp crack was embedded in the center of the weld region. The complete Gurson model has been applied to simulate the ductile fracture by microvoid nucleation, growth and coalescence. Results show that tensile residual stresses can significantly reduce the crack growth resistance when the crack growth is small compared with the length scale of the tensile residual stress field. With the crack growth, the effect of residual stresses on the crack growth resistance tends to diminish. The effect of residual stress on ductile crack growth resistance seems independent of the size of geometrically similar welds. When normalized by the weld zone size, the ductile crack growth resistance collapses into one curve, which can be used to assess the structural integrity and evaluate the effect of residual stresses. It has also been found that the effect of residual stresses on crack growth resistance depends on the initial void volume fraction f₀, hardening exponent n and T-stress.     

Post-buckling and debond propagation in sandwich panels subject to in-plane compression

Nonlinear finite element analysis is conducted to predict initiation of debond propagation in compression loaded foam cored sandwich panels containing a circular face/core debond embedded at the panel center. A three-dimensional geometrically nonlinear finite element model of the debonded sandwich panel combined with linear elastic fracture mechanics is used to determine the stress intensity factors K_I and K_II and energy release rate at the debond (crack) front parallel and perpendicular to the applied load. A range of core densities and debond sizes are analyzed. The opening mode (mode I) was found to dominate the fracture process. The critical load for crack propagation predicted using fracture mechanics concepts was found to agree with measured collapse loads for smaller debonds, but fell below measured debond propagation loads for larger debonds. In all cases the predicted direction of crack propagation was perpendicular to the loading direction, in agreement with experimental observations.     

Application of the substructured finite element/extended finite element method (S-FE/XFE) to the analysis of cracks in aircraft thin walled structures

The substructured finite element/extended finite element (S-FE/XFE) approach is used to compute stress intensity factors in large aircraft thin walled structures containing cracks. The structure is decomposed into a ‘safe’ domain modeled with classical shell elements and a ‘cracked’ domain modeled using three-dimensional extended finite elements. Two applications are presented and discussed, supported by validation test cases. First a section of stiffened panel containing a through-thickness crack is investigated. Second, small surface cracks are simulated in the case of a generic ‘pressure membrane’ with realistic crack configurations. These two semi-industrial benchmarks demonstrate the accuracy, robustness and computational efficiency of the substructured finite element/extended finite element approach to address complex three-dimensional crack problems within thin walled structures.     

Crack growth resistance behavior of a functionally graded material: computational studies

This paper describes crack growth resistance simulation in a ceramic/metal functionally graded material (FGM) using a cohesive zone ahead of the crack front. The plasticity in the background (bulk) material follows J₂ flow theory with the flow properties determined by a volume fraction based, elastic-plastic model (extension of the original Tamura-Tomota-Ozawa model). A phenomenological, cohesive zone model with six material-dependent parameters (the cohesive energy densities and the peak cohesive tractions of the ceramic and metal phases, respectively, and two cohesive gradation parameters) describes the constitutive response of the cohesive zone. Crack growth occurs when the complete separation of the cohesive surfaces takes place. The crack growth resistance of the FGM is characterized by a rising J-integral with crack extension (averaged over the specimen thickness) computed using a domain integral (DI) formulation. The 3-D analyses are performed using WARP3D, a fracture mechanics research finite element code, which incorporates solid elements with graded elastic and plastic properties and interface-cohesive elements coupled with the functionally graded cohesive zone model. The paper describes applications of the cohesive zone model and the DI method to compute the J resistance curves for both single-edge notch bend, SE(B), and single-edge notch tension, SE(T), specimens having properties of a TiB/Ti FGM. The numerical results show that the TiB/Ti FGM exhibits significant crack growth resistance behavior when the crack grows from the ceramic-rich region into the metal-rich region. Under these conditions, the J-integral is generally higher than the cohesive energy density at the crack tip even when the background material response remains linearly elastic, which contrasts with the case for homogeneous materials wherein the J-integral equals the cohesive energy density for a quasi-statically growing crack.     

Improvement of interlaminar mechanical properties of CFRP laminates using VGCF

In order to improve the interlaminar mechanical properties of CFRP laminates, hybrid CFRP/VGCF laminates have been fabricated by using a newly-developed method, i.e., powder method, where the powder of vapor grown carbon fiber (VGCF) is added at the mid-plane of [0°/0°]₁₄ CFRP laminates. Experimental results of double cantilever beam (DCB) tests indicate the improvement on the interlaminar mechanical properties of Mode-I fracture behavior with much higher critical load P_C and fracture toughness G_IC with VGCF interlayer. Crack propagation and fracture surface have also been observed to interpret this improvement mechanism. Moreover, based on experimental G_IC, numerical simulations using finite element method (FEM) with cohesive elements have been carried out to analyze the delamination propagation. The interlaminar tensile strength of hybrid CFRP/VGCF laminates, which is obtained by matching the numerical load–COD (crack opening displacement) curves to experimental ones, is higher than that of base CFRP laminates.     

Effect of residual stress on cleavage fracture toughness by using cohesive zone model

This study presents the effect of residual stresses on cleavage fracture toughness by using the cohesive zone model under mode I, plane stain conditions. Modified boundary layer simulations were performed with the remote boundary conditions governed by the elastic K‐field and T‐stress. The eigenstrain method was used to introduce residual stresses into the finite element model. A layer of cohesive elements was deployed ahead of the crack tip to simulate the fracture process zone. A bilinear traction–separation‐law was used to characterize the behaviour of the cohesive elements. It was assumed that the initiation of the crack occurs when the opening stress drops to zero at the first integration point of the first cohesive element ahead of the crack tip. Results show that tensile residual stresses can decrease the cleavage fracture toughness significantly. The effect of the weld zone size on cleavage fracture toughness was also investigated, and it has been found that the initiation toughness is the linear function of the size of the geometrically similar weld. Results also show that the effect of the residual stress is stronger for negative T‐stress while its effect is relatively smaller for positive T‐stress. The influence of damage parameters and material hardening was also studied.