J-integral Approach Research Articles

Experimental and numerical investigation of the influence of temperature on the fracture behavior of high impact polystyrene evaluated by the J-integral approach using multiple specimen method

The aim of the present work is to study the mechanical behavior in fracture of high impact polystyrene (HIPS) subjected to tensile tests under the effect of temperature to determine the R-curves. The theory of the J-integral contour has been used for the development of a characterization method of the fracture strength appropriate to the case of this non-linear elastoplastic polymer material. To this end, we used the method of multiple specimens (Single edge notch tension SENT) of thin thickness; we used several identical test pieces containing cracks of the same lengths. The tensile tests were conducted under different temperatures, ranging from 25°C to 100°C. The results suggested a very temperature-dependent behavior due to significant increase in crack resistance as demonstrated by a comparison of JIC values related to initiation of crack propagation. The fracture energy absorbed as a function of the temperature suggested that the increase of the temperature is marked by the brittle-ductile transition in the material, which we confirmed numerically via a CASTEM software simulation.

Determination of the bridging law for mixed-mode I/II delamination without measuring the crack length and crack relative displacements

To accurately predict the delamination behavior, bridging effect should be introduced into the numerical model so that an effective method to determine the bridging law is required. The main effort of the current study is to propose an efficient procedure for determining the mixed-mode I/II bridging law of composite laminates. The crack opening and shear displacements at the pre-crack tip are derived via a numerical method based on equivalent crack length approach. In addition, the formula for energy release rate can also be obtained based on equivalent crack length approach. Then, the bridging law can be derived according to the J-integral approach. The derived bridging laws have been implemented into the cohesive zone model (CZM) for numerical simulation, good agreement with experimental results validate the feasibility of the proposed procedure for determining mixed-mode I/II bridging law. Furthermore, this procedure does not require the measurement of the crack length and relative displacements at the pre-crack tip during the test.

Elastic plastic fracture mechanics investigation of toughness of wet colloidal particulate materials: Influence of saturation

HypothesisPrevious use of linear elastic fracture mechanics to estimate toughness of wet particulate materials underestimates the toughness because it does not account for plastic deformation as a dissipation mechanism. Plastic deformation is responsible for the majority of energy dissipated during the fracture of wet colloidal particulate materials. Plastic deformation around the crack tip increases with saturation of the particulate body. The toughness of the body increases with increasing saturation. ExperimentsElastic plastic fracture mechanics using the J-integral approach was used for the first time to measure the fracture toughness (JIC) of wet micron sized alumina powder bodies as a function of saturation. The samples were prepared by slip casting. The saturation was controlled by treatment in a humidity chamber. The elastic modulus (E) and the energy dissipated by plastic flow (Apl) were measured in uniaxial compression. The critical stress intensity factor (KIC) was measured using a diametral compression sample with a flaw of known size. The fracture toughness (JIC) was calculated from these measured quantities and the geometry of the specimen. FindingsElastic plastic fracture mechanics was used for the first time to quantitively account for plastic deformation of wet particulate materials. The linear elastic fracture mechanics approach previously used accounted for less than 1% of the total energy dissipated in fracture. Toughness (JIC) was found to increase with increasing saturation due to plastic deformation that increased with saturation level. The improved understanding of toughness as a function of saturation will aid in providing quantitative analysis of cracking in drying colloidal films and bodies.

Closed form solution for the energy release rate and mode partitioning of the single cantilever beam sandwich debond from an elastic foundation analysis

The goal of this paper is to derive closed form expressions for the energy release rate and mode partitioning offace/core debonds for the Single Cantilever Beam Sandwich Composite testing configuration, which is loaded with an applied shear force and/or bending moment. This is achieved by an elastic foundation approach, in which a finite length sandwich beam is treated as having a free “debonded” section and a “joined” section where a series of springs exists between the face and the substrate (core and bottom face). The elastic foundation analysis is done for a general asymmetric sandwich construction. A J-integral approach is subsequently used to derive a closed form expression for the energy release rate. It is also shown that the energy release rate is very close to the differential energy stored in the springs at the beginning of the elastic foundation, i.e. the energy released by the “broken” differential spring element as the debond propagates. In the context of this elastic foundation model, a mode partitioning measure is defined based on the transverse and axial displacements at the beginning of the elastic foundation. Since the normal springs account for the transverse compressibility of the core but not for the shear, a correction for the shear of the substrate is included by deriving the expression for the corresponding shear angle and accounting for the additional horizontal “tip” displacement. The results are compared with finite element results for a range of core materials and show very good agreement.

Applicability of the mixed-mode controlled double cantilever beam test and related evaluation methods

Recently, a novel methodology has been proposed to study non-linear fracture in mixed mode I + III, aimed at directly identifying constitutive laws regarding cohesive zone modeling from experimental data. In the case of adhesive joints, cohesive laws in the mixed mode have been derived by applying this methodology.This study discusses theoretical aspects of the novel methodology and investigates the proposed test setup by numerical simulations. This provides information on the sensitivity and limitations of the methodology, which is based on the J-integral approach.

Estimation of the fracture toughness of tungsten fibre-reinforced tungsten composites

Tungsten fibre-reinforced tungsten composites (Wf/W) have been developed to overcome the inherent brittleness of tungsten, which is a promising candidate for the plasma-facing material in a future fusion power plant. As the development of Wf/W evolves, the fracture toughness of the composite is in the focus of interest for further component design. In this contribution fracture mechanical tests on two different types of chemical vapour deposited (CVD) Wf/W are presented. Three-point bending tests according to ASTM E399 as a standard method for brittle materials were used to get a first estimation of the toughness. A provisional fracture toughness value of up to 241 MPa m1/2 was calculated for the as-fabricated and of up to 20.5 MPa m1/2 for a heat-treated and thus embrittled state. As the material does not show a brittle fracture in the as-fabricated state, the J-Integral approach based on the ASTM E1820 was additionally applied for this state. A maximum value of the J-integral of 7.5 kJ/m2 (57.6 MPa m1/2) was determined. A detailed post mortem investigations was used to obtain the active mechanisms.

Failure behavior of sandwich honeycomb composite beam containing crack at the skin.

Skin crack defects can develop in sandwich honeycomb composite structures during service life due to static and impact loads. In this study, the fracture behavior of sandwich honeycomb composite (SHC) beams containing crack at the skin was investigated experimentally and numerically under four-point loading. Three different arrangements of unidirectional (UD) carbon fiber composite and the triaxially woven (TW) fabric were considered for the skins. The presence of a 10 mm crack at mid-span of the top skin, mid-span of the bottom skin, and mid-way between load and support of the top skin, respectively, were considered. Failure load equations of the load initiating the skin crack extension were analytically derived and then numerically developed using the J-integral approach. The crack extension failure mode dominated all cracked specimens except those with low-stiffness skin which were controlled by the compressive skin debonding and core shear failures.

Determination of the bridging law for mode I delamination via elastic restraint beam model and equivalent crack method

This article presented a simple procedure to determine the Mode I bridging law for both unidirectional and multidirectional composite laminates. It was based on an ‘Elastic Restrained Beam Model’ which assumed that the boundary conditions at the end of the cracked arms of the double cantilever beam (DCB) are elastically restrained by torsion springs. In this way, the closed-form solution of the DCB has been derived, without the need for a crack-length correction used in the corrected beam theory (CBT). We also derived very accurate data reduction formulas for the energy release rate (GI) and the opening displacement at the pre-crack tip (δ*). Then, the bridging law was obtained by numerical differentiation of GI with respect to δ* using the J-integral approach. Two major advantages of the proposed procedure are that it does not require the measurement of the crack length and separation δ* during the test. The derived bridging laws were implemented into the cohesive zone model (CZM) for numerical simulations and the results were proved to have a great agreement with experimental results, which validated that the proposed procedure is suitable for the determination of Mode I bridging law.

Fracture analysis of inhomogeneous beam with internal longitudinal crack

The present paper is focused on fracture analysis of an inhomogeneous beam with an internal longitudinal crack. The beam exhibits continuous material inhomogeneity along the height of the cross-section. The material of the beam has non-linear elastic behavior. The beam is clamped in its right-hand end. One axial force and one bending moment are applied at the left-hand end of the beam. The longitudinal crack is located arbitrary along the height of the beam. Since the crack is internal, the bending moments and the axial forces in the two crack arms can not be determined directly. The beam is treated as a static undetermined structure with two internal redundants in order to derive the bending moments and axial forces in the crack arms. The longitudinal fracture behavior is studied in terms of the strain energy release rate. The J-integral approach is used for verification of the solution to the strain energy release rate. The fracture is analyzed also for the case when the crack reaches the left-hand end of the beam.

Lengthwise fracture analysis of an inhomogeneous beam in aggressive environment

Lengthwise fracture analysis of a continuously inhomogeneous beam is performed by using linear-elastic fracture mechanics. The beam under consideration is subjected to four-point bending. The material is continuously inhomogeneous in both height and length directions of the beam. The upper surface of the beam is in contact with aggressive environment. As a result of this, a damage zone appears in the beam. The depth of the damage zone increases with the time. Fracture is analyzed in terms of the strain energy release rate. The compliance method is applied to derive a solution to the strain energy release rate with taking into account the effect of the damage zone. The solution derived is time-dependent since the damage zone depth is a function of the time. The J-integral approach is applied to verify the solution to the strain energy release rate. The analytical solution obtained is used to evaluate the influence of the damage zone on the strain energy release rate in the inhomogeneous beam configuration.

Poroelastic Effects on the Time- and Rate-Dependent Fracture of Polymer Gels

AbstractFracture of polymer gels is often time- and rate-dependent. Subject to a constant load, a gel specimen may fracture immediately or after a delay (time-dependent, delayed fracture). When a crack grows in a gel, the fracture energy may depend on the crack speed (rate-dependent). The underlying mechanisms for the time- and rate-dependent fracture of gels could include local molecular processes, polymer viscoelasticity, and solvent diffusion coupled with deformation (poroelasticity). This paper focuses on the effects of poroelasticity. A path-independent, modified J-integral approach is adopted to define the crack-tip energy release rate as the energetic driving force for crack growth in gels, taking into account the energy dissipation by solvent diffusion. For a stationary crack, the energy release rate is time-dependent, with which delayed fracture can be predicted based on a Griffith-like fracture criterion. For steady-state crack growth in a long-strip specimen, the energy release rate is a function of the crack speed, with rate-dependent poroelastic toughening. With a poroelastic cohesive zone model, solvent diffusion within the cohesive zone leads to significantly enhanced poroelastic toughening as the crack speed increases, rendering a rate-dependent traction-separation relation. While most of the results are based on a linear poroelastic formulation, future studies may extend to nonlinear theories with large deformation. In addition to the poroelastic effects, other mechanisms such as viscoelasticity and local fracture processes should be studied to further understand the time and rate-dependent fracture of polymer gels.

Elastic Foundation Solution for the Energy Release Rate and Mode Partitioning of Face/Core Debonds in Sandwich Composites

AbstractThe goal of this paper is to derive closed form expressions for the energy release rate and mode partitioning of face/core debonds in sandwich composites, which include loading in shear. This is achieved by treating a finite length sandwich beam as having a “debonded” section where the debonded top face and the substrate (core and bottom face) are free and a “joined” section where a series of springs (elastic foundation) exists between the face and the substrate. The elastic foundation analysis is comprehensive and includes the deformation of the substrate part (unlike other elastic foundation studies in the literature) and is done for a general asymmetric sandwich construction. A J-integral approach is subsequently used to derive a closed form expression for the energy release rate. In the context of this elastic foundation model, a mode partitioning approach based on the transverse and axial displacements at the beginning of the elastic foundation (“debond tip”) is proposed. The results are compared with finite element results and show very good agreement.

A simple procedure for the determination of the cohesive law in 4-ENF test with consideration of the friction and R-curve effect

The measurement of initiation and propagation fracture toughness is important for characterizing the delamination behavior of composite laminates, especially in the case with distinct R-curve behavior which can be caused by fiber bridging, shear deformation of matrix, etc. Four-Point End Notched Flexure (4-ENF) is one of the widely adopted tests to measure the mode II toughness values, while the friction between the crack surfaces may have a significant impact on the results. In this paper, a simple procedure to determine the cohesive law in 4-ENF test with consideration of the friction and R-curve effect is proposed. Firstly, determining the equivalent crack advance according to the compliance variation of the specimen is performed, without the need for monitoring the crack length during the test. Next, the Energy Release Rate in mode II (GII) and crack sliding displacement (δt) are derived based on the point-friction assumption and beam theory. Finally, the cohesive law is obtained via the J-integral approach. To obtain suitable values of the penalty stiffness and interface strength, the influence of these parameters on the prediction of 4-ENF test is studied. Comparisons of the numerical results and the experimental data show a good correlation. The effect of variation of friction coefficients on derived cohesive stress and initiate energy release rate is discussed as well.

On the bondline thickness effect in mixed-mode fracture characterisation of an epoxy adhesive

ABSTRACT In-situ testing as a method for fracture characterisation of adhesives has received increased attention in recent years. Previous studies show that the fracture properties of the adhesive determined by in-situ testings are influenced by the bond-line thickness. In this work, the effect of the bond-line thickness on the fracture properties of a two-component epoxy ductile adhesive under mixed-mode loading is investigated. Double Cantilever Beam (DCB) and Mixed-Mode Bending (MMB) tests with different mixed-mode ratios were conducted. A recently proposed data reduction technique based on the J-integral approach was employed for the determination of strain energy release rate (SERR) of the epoxy adhesive under mixed-mode loading. Results showed that the fracture properties – particularly critical SERR GC- in in-situ testing is not only a function of bond-line thickness but also the ratio of mode mixity. Increase in the bond-line thickness led to an increase in GIC under pure mode I tests up to 40%. However, this trend was not observed under mixed-mode loadings due to the different type of constraints existing in the MMB tests.

Recycled Stone/ABS particulate composite: Micromechanical finite element fracture analysis

New stone composite samples, fabricated using nano- to micro-sized recycled granite particles with irregular shapes and random distribution within a ABS matrix, demonstrate a highly nonlinear and complex fracture behavior. To model this behavior, a mixed mode cohesive zone finite element model is identified using the single-leg bending (SLB), in order to represent the granite particles-ABS interfacial debonding. An inverse methodology is then proposed to determine the parameters of the cohesive zone (CZM). A direct method based on J-integral approach is employed to determine the effective parameters of the traction-separation law. A two dimensional micromechanical extended finite element model (XFEM) of the composite is generated using X-ray micro-computed tomography (XMT) to mimic the actual shape of the particles. Finally, the identified cohesive model parameters have been employed to simulate the crack growth within the granite particulates/ABS composite. The comparison of the numerical and experimental results of the SLB test demonstrated the effectiveness of the proposed simulation framework.

Mode I interlaminar fracture toughness of woven glass/epoxy composites with mat layers at delamination interface

Abstract In this study, double cantilever beam (DCB) specimen was employed to take into account the effects of mat layer at delamination interface on mode I fracture toughness of woven glass/epoxy laminated composites. DCB specimens were made using hand lay-up method assisted by the vacuum bagging technique. The experiments revealed that the mat-interlayer changed crack propagation mechanism, reducing the initiation and propagation fracture toughness during mode I interlaminar tests, by 80 and 69%, respectively. Also, the results show that fiber bridging is not the main mechanism for increasing fracture toughness in DCB samples without mat interlayer. Moreover, for numerical simulation of delamination propagation in DCB samples, the cohesive traction was measured in tests using J-integral approach. Comparing load-displacement response curves of numerical and experimental results, proved that the proposed cohesive zone model capable of precisely predicting the experimental load-displacement curve.

Isothermal and thermomechanical fatigue interaction in fatigue crack propagation behavior of a low-carbon nitrogen-controlled 316 stainless steel

In this work, the effect of superimposing of isothermal low cycle fatigue loading to the thermomechanical fatigue loading on the crack propagation behavior of the naturally initiated short crack in Low-carbon nitrogen-controlled 316 stainless steel was investigated. The experimental results indicated that the crack propagation path depends on the loading condition; the cracks appear to be initiated and propagated at grain boundary perpendicular to the loading axis which might be a relatively week region at elevated temperature under the in-phase type TMF loading and the LCF loading at high temperature, on the other hand, the cracks initiated and propagated by the transgranular mode under the out-of-phase type TMF loading and the LCF loading at middle temperature. The short crack growth rate has also affected the microstructure, i.e., the intergranular crack exhibits higher growth rate compared with the transgranular crack. In addition, the crack growth rate was accelerated by superimposing of the isothermal low cycle fatigue loading to the main thermo-mechanical fatigue loading. The short crack growth rate could be predicted according to summation law of crack growth behavior based on the J-integral approach considering with crack propagation path.

Experimental and numerical characterization of Mode I fracture in unidirectional CFRP laminated composite using XIGA-CZM approach

This article presents an effective study for characterization of interlaminar Mode I fracture process zone (FPZ) in unidirectional CFRP laminated composite structure. Mode I fracture toughness testing has been carried out on double cantilever beam (DCB) specimens. Experimental studies have been carried out on non-precraked DCB specimens in order to obtain the amount of energy released during crack initiation from the natural crack front. The energy release rate (ERR) is obtained from experimental data as a function of crack growth (Δa) and pre-crack tip opening displacement (δ∗) by utilizing five data reduction schemes. The analytical function between ERR and δ∗ is used to construct fibre bridging law using J-integral approach. The obtained traction separation laws from each data reduction scheme is implemented into cohesive zone modelling (CZM) for numerical simulation. Fractured surfaces have been examined by means of Scanning electron microscope (SEM). Numerical studies have been carried out for both non-precracked and precracked specimens using combined framework of fracture mechanics using extended isogeometric analysis (XIGA) and damage mechanics using CZM approach. It has been observed that crack travels catastrophically in non-precracked (NPC) specimens due to fast dissipation of excess energy. Introduction of fibre bridging law, obtained from experimental investigation, into CZM in combined framework of XIGA methodology shows a good agreement with experimental results.

3D SSY Estimate of EPFM Constraint Parameter under Biaxial Loading for Sensor Structure Design.

Conventional sensor structure design and related fracture mechanics analysis are based on the single J-integral parameter approach of elastic-plastic fracture mechanics (EPFM). Under low crack constraint cases, the EPFM one-parameter approach generally gives a stress overestimate, which results in a great cost waste of labor and sensor components. The J-A two-parameter approach overcomes this limitation. To enable the extensive application of the J-A approach on theoretical research and sensor engineering problem, under small scale yielding (SSY) conditions, the authors developed an estimate method to conveniently and quickly obtain the constraint (second) parameter A values directly from T-stress. Practical engineering application of sensor structure analysis and design focuses on three-dimensional (3D) structures with biaxial external loading, while the estimate method was developed based on two-dimensional (2D) plain strain condition with uniaxial loading. In the current work, the estimate method was successfully extended to a 3D structure with biaxial loading cases, which is appropriate for practical sensor design. The estimate method extension and validation process was implemented through a thin 3D single edge cracked plate (SECP) specimen. The process implementation was completed in two specified planes of 3D SECP along model thickness. A wide range of material and geometrical properties were applied for the extension and validation process, with material hardening exponent value 3, 5 and 10, and crack length ratio 0.1, 0.3 and 0.7.

The elastic-plastic delamination analysis of layered beam configurations

The elastic-plastic delamination fracture in layered beams was studied theoretically. Two Four Point Bend (FPB) beam configurations (the Double Leg Four Point Bend (DLFPB) and the Single Leg Four Point Bend (SLFPB)) were analyzed. An elastic-plastic constitutive model with power law hardening was used in the analysis. Fracture behavior was studied by applying the J-integral approach. The analytical solutions of the J-integral were obtained at characteristic levels of the external load. The solutions obtained were verified by analyzing the strain energy release rate with taking into account the material non-linearity. The variation of J-integral value in a function of crack location along the beam dept was investigated. The effect of material non-linearity on the fracture was evaluated. The analysis revealed that the J-integral value decreased with increasing the lower crack arm thickness. It was also found that the material non-linearity has to be taken into account in fracture mechanics based safety design of structural members and components made of layered materials. The analytical solutions obtained are very useful for non-linear investigations, since the simple formulae derived capture the essentials of non-linear fracture in the layered beams under consideration.