J-integral Method Research Articles

Comparison of J-integral methods to experimentally determine cohesive laws in shear for adhesives

High-quality simulation methods demand accurate material models. In simulations an adhesive can be represented by a cohesive layer. A cohesive layer model utilizes a cohesive law to represent the homogenized mechanical behaviour of a layer with a thickness. In the current paper we use three experimental methods to measure the cohesive law in shear using the ENF-specimen; one of the methods is novel and is also useful for evaluation of experiments with the ELS-specimen. Two sets of experiments are performed, one with elastic substrates and one with plastically deforming substrates. Each experiment is evaluated using all three methods. The evaluation shows that all methods provide reasonable data; the results are similar if the substrates are elastic. With smaller specimens, the substrates deform plastically and one of the methods is identified as the most accurate.

Effect of notch length and pre-crack size on mode II fracture energy of brittle adhesives

In this study, the effects of notch length and pre-crack size on mode II fracture energy of brittle adhesives are investigated. These effects are not considered by the direct beam theory (DBT) and the compliance-based beam method (CBBM) for the end-notched flexure (ENF) specimens. The sum of the lengths of the notch and the pre-crack are known as the pre-crack size in common data reduction schemes. In the current work, experiments were performed on ENF adhesive joints using digital image correlation (DIC) technique. Then, the fracture energy was obtained experimentally based on the J-integral method, the classic beam theory, and the compliance-based beam method. Numerical analyses were carried out to measure the stress distribution ahead of the pre-crack tip. Based on the obtained results, it is shown that there is a specific pre-crack length and notch size in which the pure mode II condition in an ENF test is obtained. Other combinations of the notch to pre-crake size ratio result in mixed mode conditions where the obtained fracture energy significantly changes.

Experimental study of the dynamic fracture toughness of anisotropic black shale using notched semi-circular bend specimens

This paper investigates the dynamic fracture toughness anisotropy of black shale (BS). Notched semi-circular bend (NSCB) BS specimens are prepared with different loading angles (the angle between the stratification plane and the loading direction). NSCB specimens are dynamically loaded using a modified split Hopkinson pressure bar (SHPB) system. Formulae for calculating mode-I fracture toughness of transversely isotropic material are obtained by finite element method (FEM) in combination with the J-integral method. Dynamic fracture toughness of the NSCB sample is calculated using stress waves recorded in SHPB tests and the obtained formulae. The results show that the fracture toughness of specimens for all loading angles exhibits obvious loading rate dependence, with the fracture toughness increasing with the loading rate. Similar to the static case, the specimens with higher loading angle have larger fracture toughness values as compared with ones with lower loading angles under similar loading rates. We proposed two envelope lines to enclose the fracture toughness of specimens under different loading rate. The difference between two envelope lines decreases with the increase of the loading rate. The anisotropy of black shale diminishes under extreme high loading rate. Anisotropy index, defined as the ratio of the maximum to the minimum of fracture toughness, is equal to 1.75 under static loading and 1.52 under smaller dynamic loading. These conclusions may have significant implications to the design of hydraulic fracturing in shales, e.g., fracture fluids pumping could be considered as static loading and perforation can be considered as dynamic loading. The anisotropy of shales should be considered in both these cases.

Influence of loading direction on the static and fatigue fracture properties of the long term operated metallic materials

Abstract In this paper, the problem of fatigue crack growth resistance and fracture toughness in steel structural components have been discussed. For investigation were extracted parts of bridges made of 19th century puddled steel and early 20th century mild steel. The object of interest of the authors were fragments of steel structures from ancient bridges located in Warsaw (Poland): Most pod Cytadelą (marked as “C” material); Most Kierbedzia (marked as “W”); and, a parts from the Main Railway Station hall in Wroclaw, Poland (marked as “B”). In all cases, the metallographic analyses were performed. It has been proofed the presence of degradation processes in such old steel. The mentioned processes consisted mainly in precipitations of the brittle phases inside the ferrite grains, as well on the grain boundaries. The mechanical properties and crack growth resistance, as well as fracture toughness expressed by the J-integral method have been determined for the specimens collected from the Main Railway Station hall in Wroclaw (Poland). It has been shown that crack resistance – expressed with the critical value of J-integral – in the rolling direction is significantly higher than in the direction perpendicular to rolling. For fatigue analysis, fatigue crack growth experiments were performed for all materials in rolling direction as well as in transversal direction. Based on the experimental results is worth to notice that, the obtained values are far lower than the critical values of the J-integrals characterizing the modern low-carbon constructional steels (S235JR or S355JR steels).

Effect of high strain rate and low temperature on mode II fracture toughness of ductile adhesive

Adhesive joints have a great potential use in different industrial applications which may encounter low temperature and high strain rate conditions, the fracture behavior of adhesive needs to be reliably predicted under such conditions. This paper experimentally studies the mode II fracture toughness of a polyurethane (PU) adhesive at room temperature (RT) under quasi-static condition with J-integral method and at low temperatures under high strain rate conditions with compliance-based beam method (CBBM), which does not need the observation of crack propagation. Typical R-curves of the PU ductile adhesive under the loading rate of 2 m/s, 3 m/s and at RT, −20°C, −40°C respectively are obtained. From the experimental results, low temperature and high strain rate has great effect on mode II fracture toughness of adhesive. The mode II fracture toughness of this PU adhesive decreases by at least one order of magnitude when conditions vary from quasi-static to high rate. Under the high strain rate conditions, the mode II fracture toughness of adhesive increases with the decrease of the temperature. At −40°C under the loading rate of 3 m/s, mode II fracture toughness of adhesive increases by 130% compared with the data at RT under the loading speed of 2 m/s. Through the experiment, the relationship between the mode II fracture toughness of adhesive, the nominal shear strain rate and the temperature is obtained. It is concluded that the mode II fracture toughness of adhesive at RT under quasi-static condition cannot be used to design and analyze adhesive joints at low temperatures under high strain rate conditions.

Micromechanical evaluation of DP1000-GI dual-phase high-strength steel resistance spot weld

In situ micro-cantilever bending tests were carried out on resistance spot welded DP1000-GI dual-phase high-strength steel in order to derive the mechanical response of the welds. Notched micro-cantilevers were milled using focused ion beam milling at the base metal, inter-critical, fine-grained and coarse-grained heat affected zones, and fusion zone. It was shown that due to large plastic yielding, linear-elastic fracture mechanics are inapplicable. To evaluate the fracture toughness of different weld zones, cyclic loading was applied to track the crack size and the conditional fracture toughness of weld zones was measured using crack tip opening displacement and J-integral methods. It was found that micro-cantilever bending method provides insight to the fracture toughness and local mechanical response of different weld zones. The results obtained can be used to make an accurate correlation between resistance spot welding process, microstructure and mechanical response of DP1000-GI dual-phase high-strength steel welds.

A microstructure-sensitive driving force for crack growth

A stored energy based methodology for calculating the driving force for crack growth is introduced which can capture the highly local microstructural sensitivity. This has been implemented in the context of crystal plasticity finite element simulations with explicit representation of the crack with the eXtended Finite Element Method (XFEM), with non-local approaches for both stored energy and J-integral calculation. The model is shown to have good agreement with discrete dislocation plasticity (DDP) models in terms of the crack-tip dislocation configurational energy, and with experimental observations of long and very short (microstructurally-sensitive) cracks for both fracture toughness and crack growth rate data. The method is shown to capture the microstructural sensitivity, in contrast with the widely used J-Integral method. By modelling different crack lengths, the diminution of the microstructural sensitivity with increasing crack length is quantified and a critical length defined above which the microstructural sensitivity is insignificant.

Influence of sustained loading on fracture properties of concrete

To investigate the effects of sustained loading on the fracture properties of concrete, basic creep and three-point bending (TPB) tests were conducted on the pre-notched beams. The specimens were first subjected to two sustained loading levels, i.e. 30% peak load and the initial cracking load over 115 days. Then, they were moved out from the loading frames and tested under TPB loading until failure. The critical crack propagation length (Δac), the peak load (Pmax) and the fracture energy (Gf) were measured in the tests, and the unstable fracture toughness (KICun) was calculated accordingly. Furthermore, based on the load-displacement curves obtained in the TPB tests, the energy dissipation was derived using the modified J-integral method. By enforcing balance between the energy dissipated and the energy caused by the fictitious cohesive force acting on the fracture process zone, the tension-softening constitutive laws under the two sustained loading levels were established and also simplified as bilinear forms for practical applications. Finally, the effects of sustained loading on the fracture properties were examined by comparing with the tested results from the aging specimens in the static TPB tests. The test results indicate that low sustained loading had no effects on all fracture properties of concrete investigated in this study, while under high sustained loading, Δac and KICun increased and Gf and Pmax almost remained unchanged. Meanwhile, a smaller free-stress crack opening displacement was obtained under the high sustained loading level, which indicates a shorter FPZ length formed, resulting in the increase in brittleness of concrete.

Effects of strain rate on the cohesive properties and fracture process of a pressure sensitive adhesive

Pressure sensitive adhesives provide beneficial properties for bi-material bonding and we here focus on influences of the loading rate. The cohesive law for a tape is measured using the J-integral method. In-situ studies of the fracture process are made. The fracture energy and cohesive law levels off below an engineering strain rate of about 2 s−1. All the cohesive laws include two peak stresses, the first is associated with nucleation of cavities and occurs at a stress level comparable to the critical stress for cavitation in rubber. The second peak stress occurs just before final fracture when walls formed between the cavities break down. Slower rates give more time for cavities to nucleate. A similarity of the effects of loading rate and ageing is also identified.

Comparison of J-integral methods for the characterization of tough polypropylene grades close to the glass transition temperature

The fracture mechanical characterization of very tough Polypropylene at room temperature is of great interest for many technical applications. Precision and validity of obtained values can significantly be influenced by notching and crack length determination, which has thoroughly been shown in literature. Additionally, the proximity to the glass transition temperature can induce further uncertainties.The aim of this study is to compare different J-integral methods of elastic plastic fracture mechanics to determine J-R curves for tested materials. Methods consist of the classical multi-specimen, a single specimen method (load separation) and a statistical approach (Weibull). Specific data reduction procedures based on physical principles were applied to reduce scattering and achieve reliable values. In this way, a reliable fracture mechanics characterization of Polypropylene at room temperature was achieved with a critical verification of the used methods. Finally, the determined fracture toughness value showed a good agreement with the surface texture of inspected fracture surfaces.

Steady-state crack growth in polymer gels: A linear poroelastic analysis

Based on a linear poroelastic formulation, we present an asymptotic analysis of the crack tip fields for steady-state crack growth in polymer gels. A finite element method is then developed for numerical analysis. A semi-infinite crack in a long strip specimen subject to plane-strain loading is studied in detail. The crack-tip fields from the numerical analysis agree with the asymptotic analysis with a poroelastic stress intensity factor, which is generally smaller than the stress intensity factor predicted by linear elasticity due to poroelastic shielding. Similarly, the crack-tip energy release rate calculated by a modified J-integral method is smaller than the applied energy release rate. The size of the poroelastic crack-tip field (or K-field) is characterized by a diffusion length scale that is inversely proportional to the crack speed. For relatively fast crack growth, the diffusion length is small compared to the strip thickness (small-scale diffusion), and the poroelastic K-field transitions to an elastic K-field at a distance proportional to the diffusion length. In this case, the crack-tip energy release rate decreases with increasing crack speed under the same loading condition. For relatively slow crack growth, the diffusion length is greater than the strip thickness (large-scale diffusion), and the poroelastic crack-tip field is confined and transitions to a one-dimensional diffusion zone ahead of the crack tip. In this case, the energy release rate increases with increasing crack speed. Both immersed and not-immersed crack face conditions are considered. Under the same loading conditions, the poroelastic stress intensity factor and the modified J-integral are higher for the immersed case than not-immersed, but they approach the same values at the fast and slow crack limits. In general, if the crack-tip energy release rate is taken to be the intrinsic fracture toughness of the gel, the applied energy release rate as the apparent fracture toughness is greater due to energy dissipation associated with solvent diffusion, which is referred to as poroelastic toughening. It is proposed that the modified J-integral can be used to determine the intrinsic fracture toughness of the gel in experiments, which may or may not depend on crack speed. Moreover, the present results are found to be qualitatively consistent with previous experiments on the effects of solvent viscosity and crack-tip soaking.

An experimental-finite element method based on beach marks to determine fatigue crack growth rate in thick plates with varying stress states

The relation between the fatigue crack growth rate and stress state is significant for fatigue life estimation, especially for thick-welded structures with complex stress states. This paper proposes a hybrid Experimental-Finite Element Method aimed to obtain the fatigue crack growth rate corresponding to plane stress and plane strain states. A static load marking method is proposed to mark a series of crack fronts in a single edge notched bending specimen at known numbers of cycles, which is analysed by nonlinear finite element method to evaluate the stress state by a local constraint factor. Stress intensity factor at the center of the thickness and the surface position where plane strain and plane stress states approximately exist and remain unchanged with crack propagation, is calculated by the 1/4-node displacement method with consideration of non-orthogonal mesh influence. The J-integral method and the virtual crack closure technique are also applied to check the performance of the methods in dealing with the actual cracked body problem. Crack length increments are determined based on the crack front profiles. The da/dN-ΔK curves of plane strain and plane stress states are determined and compared with the results from normal fatigue crack growth rate tests according to Chinese standard: GB/T 6398-2000. The fatigue crack growth behavior of a semi-elliptical surface crack located at the weld toe of a T-plate welded joint subjected to cyclic tension loading is assessed based on the curves and the analytical results are compared with the experimental results. The asymptotic crack shape observed in experiments is successfully predicted by using the curves determined by the hybrid method.

Finite Element Modelling of Bi-Material Interface for Crack Growth Evaluation: Technical Note

Finite element (FE) method is commonly used to study cracks in structures. In this paper, J-integral method is applied over FE model of a cracked body to determine stress intensity factor (SIF) in the domain of linear elastic fracture mechanics (LEFM). This paper formulates the J-integral methodology for 2D FE model using a coarse mesh with less degrees of freedom. Two cases , a finite plate with edge cracks and a normal crack growth in fiber metal laminated plate, are demonstrated. Numerical implementation and mesh refinement issues to maintain path independent J-integral values are explored.

On the validity of the J-integral as a measure of the transverse intralaminar fracture energy of glass fiber-reinforced polyurethanes with nonlinear material behavior

The determination of the fracture toughness of fiber-reinforced polymers (FRPs) exhibiting significant nonlinear material behavior requires the use of methods that take into account the contribution of nonlinear phenomena on the computation of the crack driving force. While the J-integral method has been standardized (ASTM E1820) and extensively used to quantify the fracture toughness of homogeneous nonlinear materials, its validity to unidirectional FRPs remains uncertain. This work addresses this issue by comparing different methods to quantify the energy dissipation during fracture of glass fiber-reinforced polymers with a nonlinear polyurethane matrix. To this end, the J-integral under mode I transverse intralaminar crack growth is evaluated for the first time by directly computing J versus load-line displacement curves using strain maps obtained around the crack tip with digital image correlation. This direct approach is then compared with 2 established data reduction methods and data obtained from virtual tests to determine the suitability of standardized procedures to quantify the fracture energy of such heterogeneous materials. The results show that the best agreement among the values obtained with the different experimental methods is reached when the initial crack length accounts for 40 to 50% of the specimen's width. Accounting for the experimental uncertainties, the procedure described in the standard ASTM E1820 is suitable to quantify the fracture energy of unidirectional laminates displaying nonlinear mechanical response.

Fatigue crack growth in a compressor stage of a turbofan engine by FEM-DBEM approach

In this work, the fatigue crack-growth process in a rotating disk of an aircraft gas turbine engine has been simulated. The considered crack nucleated in the attachment between a blade and the disk of a compressor stage, both made of a two-phase titanium alloy.The fatigue crack-growth process of such crack has been simulated by means of two codes, ABAQUS and BEASY, based on Finite Element Method (FEM) and Dual Boundary Element Method (DBEM) respectively. In particular, a variant of the submodelling technique, based on the superposition principle, has been used for coupling the two codes in order to exploit simultaneously their peculiar strength points. The FEM code has been used to compute the global stress field whereas the DBEM code has been used to calculate the fracture parameters, useful to predict the crack-growth evolution. The J-integral method and the Minimum Strain Energy Density Criterion (MSED) have been used for calculating K values and predicting crack kinking respectively.In this work, the FEM-DBEM crack path is compared with both the path obtained by a full-scale experimental test and the path predicted via a full FEM approach: having in an initial stage considered the only centrifugal load, with no allowance e.g. for the fluid pressure on the blades and for the blade dynamic behaviour, some discrepancies are found between numerical and experimental results.The computational advantages of the proposed submodelling approach are highlighted, in addition to a preliminary fatigue assessment provided for the considered compressor disk (further analyses are under development).

Strain rate effects on the intralaminar fracture toughness of composite laminates subjected to compressive load

This paper presents an experimental and numerical study focused on the mode-I intralaminar toughness characterization of a woven carbon/epoxy composite loaded in compression and subjected to high strain rates. Simulations for non-standardized Single Edge Notch Bending (SENB) and Double Edge Notch (DEN) specimens were carried out using a continuum damage mechanics based failure model implemented as an user defined material model within ABAQUS software. A Finite Element Model was used in order to produce an optimal specimen for intralaminar fracture toughness tests. A new data reduction scheme based on the numerical evaluation of the strain energy release rate using the J-integral method is proposed to determine the stress intensity factor for composites. The proposed methodology accounts for finite geometry and material anisotropy effects. The dynamic tests were carried out at strain rates of 560s-1,690s-1,770s-1 using an adapted version of the Split Hopkinson Pressure Bar. A high-speed camera was used for monitoring the crack propagation. A Scanning Electron Microscope (SEM) was used to aid the fractographic analyses on the damaged surface of the tested samples searching for the possible failures mechanisms within the material. The experimental results indicated that the composite laminates studied herein are very sensitive to the strain rate effects.

Numerical and Experimental Investigation on I-type Damage Evolution in Glass Fiber Reinforced Laminates Subjected to Quasi Static Loading

Glass fiber reinforced polymer composites has become more popular material used in aerospace and marine structural applications due to their specific strength and anti-corrosion properties. The present work discusses the numerical and experimental investigation on I-type damage evaluation in glass fiber reinforced laminates under quasi-static loading is presented. Three volume fraction of glass and resin (40/60, 50/50 and 60/40) plies were selected for both numerical and experimental damage analysis. The damage analysis was conducted for double cantilever beam as per ASTM standard using a universal testing machine. The J-integral method of numerical investigation carried out by virtual crack closure technique and cohesive zone modelling. The fracture toughness has increased with the resin content and dependent on the ductility caused at the crack tip due to the plastic zone around the crack tip. The numerical results were found to possess good agreement with the experimental values. Cite this Article Nikhil Rangaswamy, Shivakumar S, Kallol Anupama N. Numerical and Experimental Investigation on I-Type Damage Evolution in Glass Fiber Reinforced Laminates Subjected to Quasi Static Loading. Emerging Trends in Chemical Engineering. 2017; 4(3): 27–33p.

Evaluation of stress intensity factors for bi-material interface cracks using displacement jump methods

The aim of the present work is to investigate the numerical modeling of interfacial cracks that may appear at the interface between two isotropic elastic materials. The extended finite element method is employed to analyze brittle and bi-material interfacial fatigue crack growth by computing the mixed mode stress intensity factors (SIF). Three different approaches are introduced to compute the SIFs. In the first one, mixed mode SIF is deduced from the computation of the contour integral as per the classical J-integral method, whereas a displacement method is used to evaluate the SIF by using either one or two displacement jumps located along the crack path in the second and third approaches. The displacement jump method is rather classical for mono-materials, but has to our knowledge not been used up to now for a bi-material. Hence, use of displacement jump for characterizing bi-material cracks constitutes the main contribution of the present study. Several benchmark tests including parametric studies are performed to show the effectiveness of these computational methodologies for SIF considering static and fatigue problems of bi-material structures. It is found that results based on the displacement jump methods are in a very good agreement with those of exact solutions, such as for the J-integral method, but with a larger domain of applicability and a better numerical efficiency (less time consuming and less spurious boundary effect).

Role of crystallographic texture, delamination and constraint on anisotropy in fracture toughness of API X70 line pipe steels

Fracture toughness studies of line-pipe steels are extremely crucial as line-pipes are exposed to extreme conditions in application. The effects of crystallographic texture and constraint on fracture toughness of API X70 line pipe steel were investigated in this study. The J-integral method was considered to quantify the fracture toughness across orientations and thicknesses of the investigated steel. The difference in the crack initiation toughness clearly demonstrated the presence of strong anisotropy. Further the counteracting effects of thickness and delamination on constraint were found to govern fracture mechanism as well as the fracture toughness of the specimens. The anisotropy in toughness was investigated with macro-texture measurements of the specimens. The major crystallographic textures and their influences on fracture toughness were analyzed. Amongst the observed texture components, strong intensities of {113}<110>component of α-fiber was identified as the major reason for anisotropy in fracture toughness. The distribution of volume fractions of {112}<110> component also showed a strong correlation with the fracture toughness across orientations.

A review of fatigue and fracture mechanics with a focus on rubber-based materials

Prediction of how cracks nucleate and develop is a major concern in fracture mechanics. The purpose of this study is to provide an overview of the state of the art on fracture mechanics with primary focus on different methodologies available for crack initiation and growth prediction in rubber-based materials under the static and fatigue loading conditions. The concept of fracture mechanics applied to rubber-based materials and concern of finite element analysis for J-integral estimation in elastomers are discussed in this paper. The strain energy release rate is commonly used to describe the energy dissipated during fracture per unit of fracture surface area and can be calculated by J-integral method, which represents a path-independent integral around the crack tip. As fatigue crack growth most commonly occurs in structures, the high-cycle fatigue life of components needs to be predicted by using extended finite element, strain energy density, finite fracture mechanics, and other techniques which will be covered in this review paper. In addition, some recent testing and numerical results reported in the literature and their applications will be discussed.