J resistance behavior in functionally graded materials using cohesive zone and modified boundary layer models

This paper describes elastic–plastic crack growth resistance simulation in a ceramic/metal functionally graded material (FGM) under mode I loading conditions using cohesive zone and modified boundary layer (MBL) models. For this purpose, we first explore the applicability of two existing, phenomenological cohesive zone models for FGMs. Based on these investigations, we propose a new cohesive zone model. Then, we perform crack growth simulations for TiB/Ti FGM SE(B) and SE(T) specimens using the three cohesive zone models mentioned above. The crack growth resistance of the FGM is characterized by the J-integral. These results show that the two existing cohesive zone models overestimate the actual J value, whereas the model proposed in the present study closely captures the actual fracture and crack growth behaviors of the FGM. Finally, the cohesive zone models are employed in conjunction with the MBL model. The two existing cohesive zone models fail to produce the desired K–T stress field for the MBL model. On the other hand, the proposed cohesive zone model yields the desired K–T stress field for the MBL model, and thus yields J _R curves that match the ones obtained from the SE(B) and SE(T) specimens. These results verify the application of the MBL model to simulate crack growth resistance in FGMs.     

Cohesive fracture modeling of elastic-plastic crack growth in functionally graded materials

This work investigates elastic-plastic crack growth in ceramic/metal functionally graded materials (FGMs). The study employs a phenomenological, cohesive zone model proposed by the authors and simulates crack growth by the gradual degradation of cohesive surfaces ahead of the crack front. The cohesive zone model uses 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) to describe the constitutive response of the material in the cohesive zone. A volume fraction based, elastic-plastic model (extension of the original Tamura-Tomota-Ozawa model) describes the elastic-plastic response of the bulk background material. The numerical 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. Numerical values of volume fractions for the constituents specified at nodes of the finite element model set the spatial gradation of material properties with isoparametric interpolations inside interface elements and background solid elements to define pointwise material property values. The paper describes applications of the cohesive zone model and the computational scheme to analyze crack growth in a single-edge notch bend, SE(B), specimen made of a TiB/Ti FGM. Cohesive parameters are calibrated using the experimentally measured load versus average crack extension (across the thickness) responses of both Ti metal and TiB/Ti FGM SE(B) specimens. The numerical results show that with the calibrated cohesive gradation parameters for the TiB/Ti system, the load to cause crack extension in the FGM is much smaller than that for the metal. However, the crack initiation load for the TiB/Ti FGM with reduced cohesive gradation parameters (which may be achieved under different manufacturing conditions) could compare to that for the metal. Crack growth responses vary strongly with values of the exponent describing the volume fraction profile for the metal. The investigation also shows significant crack tunneling in the Ti metal SE(B) specimen. For the TiB/Ti FGM system, however, crack tunneling is pronounced only for a metal-rich specimen with relatively smaller cohesive gradation parameter for the metal.     

The effect of notch depth on J-integral and critical fracture load in plates made of functionally graded aluminum–silicone carbide composite with U-notches under bending

The present paper deals with the effect of notch depth on J-integral and critical fracture load in a plate made of functionally graded aluminum–silicone carbide composite (Al–SiC) with U-notch under bending. The weight fraction of SiC particles varies from 0% to 20% through the specimen width. Using three criteria namely mean stress (MS), point stress (PS), and averaged strain-energy density (ASED), the critical fracture load has been predicted and its variation with respect to the notch depth has been studied. A comparison of the J-integral between functionally graded and homogeneous Al–SiC composite were made, where the notch tip in the functionally graded material is situated in a layer with same mechanical properties as the homogeneous composite. The results indicated that in the case where the notch scene is toward brittleness increment the critical J-integral in functionally graded material (FGM) is larger than that of in homogeneous material with the same mechanical properties at the notch tip. Therefore, FGM is more convenient than homogeneous material against fracture.     

Analytical and numerical study of cohesive zones for a crack in an adhesive layer between identical isotropic materials

A crack in a thin adhesive elastic-perfectly plastic layer between two identical isotropic elastic half-spaces is considered. Uniformly distributed normal stress is applied to the substrates at infinity. First, stress distribution in the cohesive zones and the J-integral values are defined numerically by the finite element method (FEM). Further, a mathematical formulation of the problem is given and its analytical solution is proposed. It is assumed that, at the crack continuations, there exist cohesive zones. The interlayer thickness is neglected since it is much smaller than the crack length. The distribution of the normal stress, which was obtained by means of the FEM, is now approximated by a piecewise-constant function and assumed to be applied at the faces of the cohesive zones. The formulated problem is solved analytically and an equation for determination of the cohesive zone lengths is derived. Also, closed expressions for the crack tip opening displacement and for the J-integral are obtained in an analytical form. These parameters are found with respect to the values of the normal stress applied at infinity. Finally, a universal approximating function, which describes the stress distribution in the cohesive zones, is constructed. This function depends on the ratio between the interlayer thickness and the crack length and on the ratio between the normal stress applied at infinity and the yield limit of the interlayer’s material. Once again, the problem is solved analytically, but this time for the stress distribution prescribed by the universal approximating function. The cohesive zone lengths, the values of the crack tip opening displacement and of the J-integral are calculated. A comparative analysis of the obtained results is carried out. A good agreement of the J-integral values calculated by means of the developed analytical models and by the associated finite element analysis is demonstrated.     

J-integral evaluation of austenitic-martensitic functionally graded steel in plates weakened by U-notches

In this paper, a numerical method for J-integral evaluation of plates weakened by U-notches for brittle or quasi-brittle functionally graded steel (FGS) has been proposed. The material contains austenite phase in addition to martensite layer produced by electroslag remelting (ESR). The Young’s modulus and the Poisson’s ratio have been assumed to be constant, while other mechanical properties vary exponentially along the specimen width. The effect of notch depth on the J-integral and the critical fracture load has been studied. A comparison of the J-integral between functionally graded and homogeneous steels was made, where the notch tip in the functionally graded steel is situated in a layer with same mechanical properties as the homogeneous steel.     

A boundary layer framework considering material gradation effects

This paper describes the development and application of a novel modified boundary layer (MBL) model for graded nonhomogeneous materials, e.g. functionally graded materials (FGMs). The proposed model is based on a middle-crack tension, M(T), specimen with traction boundary conditions applied to the top and lateral edges of the model. Finite element analyses are performed using WARP3D, a fracture mechanics research finite element code, which incorporates elements with graded elastic and plastic properties. Elastic crack-tip fields obtained from the proposed MBL model show excellent agreement with those obtained from full models of the cracked component for homogeneous and graded nonhomogeneous materials. The K-T dominance of FGMs is investigated by comparing the actual stress fields with the asymptotic stress fields (the Williams’ solution). The examples investigated in the present study consider a crack parallel to the material gradient. Results of the present study provide insight into the K-T dominance of FGMs and also show the range of applicability of the proposed MBL model. The MBL model is applied to analyze the elastic-plastic crack-tip response of a Ti/TiB FGM SE(T) specimen. The numerical results demonstrate that the proposed MBL model captures the effect of T-stress on plastic zone size and shape, constraint effects, etc. for such configurations.     

Prediction of mode I crack growth resistance based on a comparative investigation of J-integral and energy dissipation rate concept

An energy dissipation rate concept is employed in conjunction with the J-integral to calculate crack growth resistance of elastic-plastic fracture. Different from Rice’s J-integral, the free energy density is employed in place of the stress working density to define an energy-momentum tensor, which yields that the slightly changed J-integral is path dependent regardless of incremental plasticity and deformational plasticity. The J-integral over the remote contour is split into the plastic influence term and the J _FPZ-integral over the fracture process zone which is an appropriate estimate of the separation work of fracture. Finite element simulations are carried out to predict the plane strain mode I crack growth behavior by an embedded fracture process zone. It can be concluded that J-integral characterization is in essence a stress intensity-based fracture resistance similar to the K criterion of linear elastic fracture, and energy dissipation rate fracture resistance can be taken as an extension of the Griffith criterion to the elastic-plastic fracture.     

Cohesive modeling of dynamic fracture in functionally graded materials

The dynamic fracture of functionally graded materials (FGMs) is modeled using an explicit cohesive volumetric finite element scheme that incorporates spatially varying constitutive and failure properties. The cohesive element response is described by a rate-independent bilinear cohesive failure model between the cohesive traction acting along the cohesive zone and the associated crack opening displacement. A detailed convergence analysis is conducted to quantify the effect of the material gradient on the ability of the numerical scheme to capture elastodynamic wave propagation. To validate the numerical scheme, we simulate dynamic fracture experiments performed on model FGM compact tension specimens made of a polyester resin with varying amounts of plasticizer. The cohesive finite element scheme is then used in a parametric study of mode I dynamic failure of a Ti/TiB FGM, with special emphasis on the effect of the material gradient on the initiation, propagation and arrest of the crack.     

Fatigue growth of short cracks in Ti-17: Experiments and simulations

The fatigue behaviour of through thickness short cracks was investigated in Ti-17. Experiments were performed on a symmetric four-point bend set-up. An initial through thickness crack was produced by cyclic compressive load on a sharp notch. The notch and part of the crack were removed leaving an approximately 50 μm short crack. The short crack was subjected to fatigue loading in tension. The experiments were conducted in load control with constant force amplitude and mean values. Fatigue growth of the short cracks was monitored with direct current potential drop measurements. Fatigue growth continued at constant R-ratio into the long crack regime. It was found that linear elastic fracture mechanics (LEFM) was applicable if closure-free long crack growth data from constant K_Imax test were used. Then, the standard Paris’ relation provided an upper bound for the growth rates of both short and long crack.The short crack experiments were numerically reproduced in two ways by finite element computations. The first analysis type comprised all three phases of the experimental procedure: precracking, notch removal and fatigue growth. The second analysis type only reproduced the growth of short cracks during fatigue loading in tension. In both cases the material model was elastic-plastic with combined isotropic and kinematic hardening. The agreement between crack tip opening displacement range, cyclic J-integral and cyclic plastic zone at the crack tip with ΔK_I verified that LEFM could be extended to the present short cracks in Ti-17. Also, the crack size limits described in the literature for LEFM with regards to plastic zone size hold for the present short cracks and cyclic softening material.     

Crack growth characteristics of carbon nanotube-based polymer composites subjected to cyclic loading

This paper presents the characterization of crack growth in carbon nanotube (CNT)-based polymer composites under fatigue loading. Fatigue crack growth tests were performed on single-edge cracked plate specimens of CNT/polycarbonate composites at room temperature and liquid nitrogen temperature (77 K). An elastic–plastic finite element analysis was also conducted to determine the J-integral range. The crack growth rate data were expressed in terms of the J-integral range, and the effect of nanotube addition on the fatigue crack growth behavior was examined. In addition, possible mechanisms of the crack growth in the nanocomposites are discussed based on microscopic observations of the specimen fracture surfaces.     

Crack resistance curves of GLARE laminates by elastic compliance

The objective of this work was to study the applicability of the elastic compliance technique for crack resistance curves evaluation of commercial GLARE laminates using small SE(B) and C(T) specimens. The experimental evaluation of R-curves of 25.0 mm wide SE(B) specimens of unidirectional GLARE 1 3/2 0.3 and 50.0 mm wide C(T) specimens of bidirectional GLARE 3 5/4 0.3 was performed. Fracture toughness was measured through a recently proposed experimental methodology based on standardized specimens and elastic-plastic methodologies (J-integral and CTOD δ₅), whereas crack growth was measured optically and estimated by elastic compliance. According to the results the elastic compliance technique seemed to be applicable to GLARE laminates, accurately predicting stable crack growth during the tests.     

Calculation of the energy J-integral for bodies with notches and cracks

The approximate solutions for calculation of the energy J-integral of a body both with a notch and with a crack under elastic-plastic loading have been obtained. The crack is considered as the limit case of a sharp notch. The method is based on stress concentration analysis near a notch/crack tip and the modified Neuber's approach. The HRR-model and the method based on an equation of equilibrium were also employed to calculate the J-integral. The influence of the strain hardening exponent on the J-integral is discussed. New aspects of the two-parameter J ^* _c-fracture criterion for a body with a short crack are studied. A theoretical investigation of the effect of the applied critical stress (or the crack length) on the strain fields ahead of the crack tip has been carried out.     

Use of J-integral to predict static failures in sharp V-notches and rounded U-notches

In the present paper, the physical meaning of J_V (namely, the classic J-integral applied to either sharp V-notch) is discussed. Consider a Cartesian reference frame having the x-axis parallel to the notch bisector, each mode of J_V, for a given circular path, is proportional to the correspondent mode of the classic J-integral of a virtual crack having length equal to the path radius and emanating from the tip of the V-notch. Analytical and numerical results have been performed for linear elastic materials. Additionally, in order to verify the formulations of J_V, experimental result of embedded cracks of sharp V-notch was considered.Then, by introducing a characteristic path radius ρ^∗, assumed to be dependent only on the material properties, the J_V parameter was used for the estimation of the static failure load of sharp V-notches specimens under mode I loading.Furthermore, the J_Vρ parameter (namely, the classic J-integral applied to U-rounded notches) was used to analyze the static failure of two new series of specimens with double U-notches made of brittle material (PMMA and PVC glass) subjected to tensile loading. This method allowed us to prove that when the ratio between the notch tip radius and ρ^∗ is small the approach agrees with the classic J-integral, whereas when ρ^∗ becomes small with respect to the notch tip radius, the J_Vρ method agrees with the classic peak stress approach.     

Fatigue life prediction of notched members based on local strain and elastic-plastic fracture mechanics concepts

Fatigue life predictions for notched members are made using local strain and elastic-plastic fracture mechanics concepts. Crack growth from notches is characterized by J-integral estimates made for short and long cracks. The local notch strain field is determined by notch geometry, applied stress level and material properties. Crack initiation is defined as a crack of the same size as the local notch strain field. Crack initiation life is obtained from smooth specimens as the life to initiate a crack equal to the size of cracks in the notched member. Notch plasticity effects are included in analyzing the crack propagation phase. Crack propagation life is determined by integrating the equation that relates crack growth rate to ΔJ from the initiated to final crack size. Total fatigue life estimates are made by combining crack initiation and crack propagation phases. These agree within a factor of 1.5 with measured lives for the two notch geometries.     

Estimation procedure of J-resistance curves for SE(T) fracture specimens using unloading compliance

This work provides an estimation procedure to determine J-resistance curves for pin-loaded and clamped SE(T) fracture specimens using the unloading compliance technique and the η-method. A summary of the methodology upon which J and crack extension are derived sets the necessary framework to determine crack resistance data from the measured load vs. displacement curves. The extensive plane-strain analyses enable numerical estimates of the nondimensional compliance, μ, and parameters η and γ for a wide range of specimen geometries and material properties characteristic of structural and pipeline steels. Laboratory testing of an API 5L X60 steel at room temperature using pin-loaded SE(T) specimens with side-grooves provide the load-displacement data needed to validate the estimation procedure for measuring the crack growth resistance curve for the material. The results presented here produce a representative set of solutions which lend further support to develop standard test procedures for constraint-designed SE(T) specimens applicable in measurements of crack growth resistance for pipelines.     

J-Resistance curve testing of HY80 steel using SE(B) specimens and normalization method

The normalization method is adopted for standard and nonstandard specimens in this paper to develop J-R curves for HY80 steel directly from load versus load-line displacement records without use of automatic crack length measurement. The standard specimens usually contain high crack-tip constraints, while the nonstandard specimens involve low crack-tip constraints. To obtain J-R curves with different constraints, a series of single edge notched bend (SE(B)) specimens with different crack lengths for an HY80 steel are tested in accordance with ASTM standard E1820. The normalization method is then used for determining crack extension and J-R curves for these SE(B) specimens.To validate the normalization method, the J-R curves determined using the normalization method are compared with those obtained by the elastic unloading compliance method for the SE(B) specimens. The comparison shows that good agreements exist between the two methods, and the normalization method is a viable tool to be used to determine J-R curves of the HY80 steel for the standard as well as nonstandard SE(B) specimens. In the J-integral calculations, the resistance curve test method, the basic test method and the modified basic test method specified in ASTM E1820 are evaluated. The results indicate that the modified basic method can be equivalent to the resistance curve method.     

Evaluation of R-curve behavior of ceramic-metal functionally graded materials by stable crack growth

This paper deals with the fracture toughness and R-curve behavior of ceramic-metal functionally graded materials (FGMs). A possibility of stable crack growth in a three-point-bending specimen is examined based on the driving force and resistance for crack growth in FGMs, and the distribution of fracture toughness or R-curve behavior is evaluated on FGMs fabricated by powder metallurgy using partially stabilized zirconia (PSZ) and stainless steel (SUS 304). The materials have a functionally graded surface layer (FGM layer) with a thickness of 1 mm or 2 mm on a SUS 304 substrate. Three-point-bending tests are carried out on a rectangular specimen with a very short crack in the ceramics surface. On the three-point-bending test, a crack is initiated from a short pre-crack in unstable manner, and then it propagates in stable manner through the FGM layer with an increase in the applied load. From the relationship between applied load and crack length during the stable crack growth in the FGM layer, the fracture toughness is evaluated. The fracture toughness increases with an increase in a volume fraction of SUS 304 phase.     

Influence of notch root radius on ductile rupture fracture toughness evaluation with Charpy-V type specimens

The blunt notch fracture toughness of four types of carbon-manganese steel (ASTM A516 grade 70) has been determined by J-integral tests on Charpy-V type samples with different values of notch root radius, ρ. J-ρ plots, determined using specimens with a notch depth to width ratio, a/w, equal to 0.5, have shown the existence of a limiting ρ value (ρ_eff) below which applied J-intergral values at fracture initiation are constant. These ρ_eff values have been seen to depend only on second-phase particle distribution and not on their volume fraction or on the steel ferritic grain size. The procedure for deriving J-integral values at the onset of stable crack growth from J resistance curves in the case of notches has also been discussed. Experiments with Charpy specimens with a/w = 0.2 do not allow the derivation of meaningful J-ρ plots. In all cases, a ductile fracture criterion based on the constancy of the notch tip strain at rupture initiation has been proved when ρ >ρ_eff.     

Advantages of the J-integral approach for calculating stress intensity factors when using the commercial finite element software ABAQUS

A predictive method for remaining component lifetime evaluation consists in integrating the crack growth law of the material considered in a finite element step-by-step process. So, as part of a linear elastic fracture mechanics analysis, the determination of the stress intensity factor distribution is a crucial point. The aim of the present work is to test several existing numerical techniques reported in the literature. Both the crack opening displacement extrapolation method and the J-integral approach are applied in 2D and 3D ABAQUS finite element models. The results obtained by these various means on CT specimens and cracked round bars are in good agreement with those found in the literature. Nevertheless, since the knowledge of the field near the crack tip is not required in the energetic method, the J-integral calculations seem to be a good technique to deal with the fatigue growth of general cracks.     

Quasi-static and dynamic fracture behavior of composite sandwich beams with a viscoelastic interface crack

Fracture analysis of sandwich beams with a viscoelastic interface crack under quasi-static and dynamic loading has been studied. Firstly, a three-parameter standard solid material model was employed to describe the viscoelasticity of the adhesive layer. And a novel interfacial fracture analysis model called three material media model was established, in which an interface crack was inserted in the viscoelastic layer. Secondly, a finite element procedure based on Rice J-integral and Kishimoto J-integral theories was used to analyze quasi-static and dynamic interface fracture behavior of the sandwich beam, respectively. Finally, the influence of viscoelastic adhesive layer on the quasi-static J-integral was discussed. In addition, comparison of quasi-static Rice J-integral with Kishimoto J-integral under various loading rates was carried out. The numerical results show that the oscillating characteristic of dynamic J-integral is more evident with shorter loading rise time.