Subcritical crack growth,surface energy,fracture toughness,stick-slip and embrittlement

It is proposed that the difficulties encountered with the meaning of subcritical crack growth arose from a misunderstanding of the Griffith equation. This equation is G=2γ for an equilibrium crack (stable or unstable) where γ is the intrinsic surface energy. When G>2γ the crack has a velocity v depending on the crack extension force G−2γ, even in a vacuum, and the following equation, well verified for adherence of elastomers, G−2γ=2γφ _T(v) where φ _T(v) is related to viscoelastic losses or internal friction at the crack tip, is generalized to other materials. At a critical speed v _c, dφ/dv becomes negative; as a negative branch cannot be observed the velocity jumps to high values on a second positive branch, so that G=G _c is a criterion for crack speed discontinuity, not the Griffith criterion. The multiplicative factor 2γ on the right-hand side accounts for the shift of the v-K curves with environment. No stress corrosion is needed to explain subcritical crack growth. Subcritical crack growth in glasses and ceramics and velocity jump in brittle polymers are shown to agree with this proposal. This model can also explain stick-slip motion when a mean velocity is imposed in the negative branch. Occurrence of velocity jump or stick-slip depends on the geometry tested and the stiffness of the apparatus. A second kind of stick-slip associated with cavitation in liquid-filled cracks is discussed. When the surrounding medium can reach the crack tip and reduce the surface energy, even at the critical speed v _c, the critical strain energy release rate G _c is reduced in the same proportion as γ, and a loading which would have given subcritical growth will give a catastrophic failure. Reduction of surface energy in the Rehbinder effect and in embrittlement by segregation is discussed. Finally, the evolution of ideas concerning the Irwin-Orowan formula and fracture toughness is examined.     

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.     

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.     

Quasi-static extension of cohesive crack described by the energy partition technique

Nonlinear differential equation governing mode I fracture under small scale yielding condition has been derived on the basis of the energy partition concept. This technique is associated with a cohesive crack model. The nonlinear zone which precedes a propagating crack has been assumed to have a structured nature. In addition to this microstructural assumption, it has been postulated that the energy dissipated within the process zone (Δ), embedded in a larger nonlinear zone (R), remains invariant to the extent of crack growth. Upper and lower bounds of the tearing modulus have been related to the material ductility via closed form expressions. It has been demonstrated that the energy screening, measured by the ratio of the true fracture energy (Ŵ) to the total work expended in the cohesive zone during the process of irreversible deformation, is a monotonic function of the crack growth increment, resembling a reciprocal of the apparent material resistance to cracking described by anR-curve.     

On the failure of cracks under mixed-mode loads

Fracture of plates containing a crack under mixed-mode, I and II, loading conditions is investigated. Fracture mechanisms are first examined from fracture surface morphology to correlate with the macroscopic fracture behavior. Two distinct features are observed and they are typical of shear and tensile types of failure. From this correlation, a fracture criterion based on the competition of the attainment of a tensile fracturing stress σ_{_C} and a shear fracturing stress τ_{_C} at a fixed distance around the crack tip is proposed. Material ductility is incorporated using τ_{_C}/σ_{_C} determined from classical material failure theories. The type of fracture is predicted by comparing τ_{_max}/σ_{_max} at r=r_{_C} for a given mixed mode loading to the material ductility_{τ_C/σ_C} , i.e. τ_{_max}/σ_{_max})<(τ_{_C} /σ_{_C}) for tensile type of fracture and (τ_{_max}/σ_{_max}) r (τ_{_C}/ σ_{_C}) for shear type of fracture. It is found that, for typical engineering structural metals with certain ductility, (1) crack propagation initiates according to the maximum hoop stress criterion when the the mode mixity is near mode I and according to the maximum shear stress criterion when the mode mixity is near mode II, and (2) the transition of the failure from tensile to shear type can be predicted by the proposed criterion. For brittle materials the maximum hoop (opening) stress always reaches the tensile fracturing stress before the maximum shear stress reaches the shear fracturing stress of the material at a crack tip. Therefore, specimens made of brittle materials tend to fail under the maximum hoop stress criterion, as demonstrated by Erdogan and Sih (1963) and others. This revised version was published online in August 2006 with corrections to the Cover Date.     

Three-dimensional modeling of ductile crack growth: Cohesive zone parameters and crack tip triaxiality

For 10 mm thick smooth-sided compact tension specimens made of a pressure vessel steel 20MnMoNi55, the interrelations between the cohesive zone parameters (the cohesive strength, T_max, and the separation energy, Γ) and the crack tip triaxiality are investigated. The slant shear-lip fracture near the side-surfaces is modeled as a normal fracture along the symmetry plane of the specimen. The cohesive zone parameters are determined by fitting the simulated crack extensions to the experimental data of a multi-specimen test. It is found that for constant cohesive zone parameters, the simulated crack extension curves show a strong tunneling effect. For a good fit between simulated and experimental crack growth, both the cohesive strength and the separation energy near the side-surface should be considerably lower than near the midsection. When the same cohesive zone parameters are applied to the 3D model and a plane strain model, the stress triaxiality in the midsection of the 3D model is much lower, the von-Mises equivalent stress is distinctly higher, and the crack growth rate is significantly lower than in the plane strain model. Therefore, the specimen must be considered as a thin specimen. The stress triaxiality varies dramatically during the initial stages of crack growth, but varies only smoothly during the subsequent stable crack growth. In the midsection region, the decrease of the cohesive strength results in a decrease of the stress triaxiality, while the decrease of the separation energy results in an increase of the triaxiality.     

Determination of the effective mode-I toughness of a sinusoidal interface between two elastic solids

A finite element model of crack propagation along a sinusoidal interface with amplitude A and wavelength λ between identical elastic materials is presented. Interface decohesion is modeled with the Xu and Needleman (J Mech Phys Solid 42(9):1397, 1994) cohesive traction–separation law. Ancillary calculations using linear elastic fracture mechanics theory were used to explain some aspects of stable and unstable crack growth that could not be directly attained from the cohesive model. For small aspect ratios of the sinusoidal interface (A/λ ≤ 0.25), we have used the analytical Cotterell–Rice (Intl J Fract 16:155–169, 1980) approximation leading to a closed-form expression of the effective toughness, K _Ic , given by where is the work of separation, E is Young’s modulus, and ν is Poisson’s ratio. For A/λ > 0.25, both the cohesive zone model and numerical J-integral estimates of crack tip stress intensity factors suggest the following linear relationship: Parametric studies show that the length of the cohesive zone does not significantly influence K _Ic , although it strongly influences the behavior of the crack between the initiation of stable crack growth and the onset of unstable fracture. An erratum to this article can be found at     

Cohesive Fracture Model Based on Necking

A linear hardening model together with a linear elastic background material is first used to discuss some aspects of the mathematical and physical limitations and constraints on cohesive laws. Using an integral equation approach together with the cohesive crack assumption, it is found that in order to remove the stress singularity at the tip of the cohesive zone, the cohesive law must have a nonzero traction at the initial zero opening displacement. A cohesive zone model for ductile metals is then derived based on necking in thin cracked sheets. With this model, the cohesive behavior including peak cohesive traction, cohesive energy density and shape of the cohesive traction–separation curve is discussed. The peak cohesive traction is found to vary from 1.15 times the yield stress for perfectly plastic materials to about 2.5 times the yield stress for modest hardening materials (power hardening exponent of 0.2). The cohesive energy density depends on the critical relative plate thickness reduction at the root of the neck at crack initiation, which needs to be determined by experiments. Finally, an elastic background medium with a center crack is employed to re-examine the shape effect of cohesive traction–separation curve, and the relation between the linear elastic fracture mechanics (LEFM) and cohesive zone models by considering the cohesive zone development and crack growth in the infinite elastic medium. It is shown that the shape of the cohesive curve does affect the cohesive zone size and the apparent energy release rate of LEFM for the crack growth in the elastic background material. The apparent energy release rate of LEFM approaches the cohesive energy density when the crack extends significantly longer than the characteristic length of the cohesive zone.     

A nonlinear viscoelastic fracture analysis of concrete/FRP delamination in aggressive environments

In this paper, the durability of the bondline between concrete and FRP reinforcement was characterized at various temperature and humidity levels. The linear and nonlinear viscoelastic constitutive behavior of the epoxy bondline was characterized and used for a nonlinear viscoelastic fracture analysis of delamination. A hygrothermal nonlinear viscoelastic pseudo-stress model was developed and calibrated in order to compute a generalized J integral. Driven wedge tests were conducted for examining the fracture behavior of the interface. A finite element analysis was developed for determining the cohesive zone size and the generalized J integral at various temperature and humidity levels. The fracture energy obtained from these parameters greatly depended upon crack growth rate, temperature and humidity.     

Damage Dependent Constitutive Behavior and Energy Release Rate for a Cohesive Zone in a Thermoviscoelastic Solid

In this paper a cohesive zone is introduced ahead of a crack tip in order to avoid the singularity at the crack tip. By applying thermodynamics to the cohesive zone and the surrounding body, a fracture criterion will be established so that the inelastic energy dissipation both in the cohesive zone and the surrounding bulk material can be distinguished from the energy released by fracture, and the propagation of crack can be predicted. In addition, the cohesive zone constitutive equation is constructed utilizing the Helmholtz free energy in the form of a single hereditary integral for a nonlinear viscoelastic material. The resulting constitutive model for the cohesive zone contains an internal state variable which represents the damage state within the cohesive zone. When the cohesive zone opening displacement is known, the energy release rate is thus history dependent, which is expressed in terms of the damage state, the length of separation in the cohesive zone and the geometric configuration of the cohesive zone opening displacement. Example results contained herein demonstrate this effect. This revised version was published online in July 2006 with corrections to the Cover Date.     

Determination of double-K criterion for crack propagation in quasi-brittle fracture, Part II: Analytical evaluating and practical measuring methods for three-point bending notched beams

In this paper, an attempt is made to determine the double-K fracture parameters K _Ic ⁱⁿⁱ and K _Ic ^un using three-point bending notched beams. First, based on the knowledge from extensive investigations which showed that the nonlinearity of P-CMOD curve is mainly associated with crack propagation, a linear asymptotic superposition assumption is proposed. Then, the critical effective crack length a is analytically evaluated by inserting the secant compliance c into the formula of LEFM. Furthermore, an analytical result of a fictitious crack with cohesive force in an infinite strip model was obtained. The double-K fracture parameters K _Ic ⁱⁿⁱ and K _Ic ^un as well the critical crack tip opening displacement CTOD_c were analytically determined. The experimental evidence showed that the double-K fracture parameters K _Ic ⁱⁿⁱ and K _Ic ^un are size-independent and can be considered as the fracture parameters to describe cracking initiation and unstable fracture in concrete structures. The testing method required to determine K _Ic ⁱⁿⁱ and K _Ic ^un is quite simple, without unloading and reloading procedures. So, for performing this test, a closed-loop testing system is not necessary.     

The cohesive zone model in a problem of delayed hydride cracking of zirconium alloys

The crack tip model with the cohesive zone ahead of a finite crack tip has been presented. The estimation of the length of the cohesive zone and the crack tip opening displacement is based on the comparison of the local stress concentration, according to Westergaard's theory, with the cohesive stress. To calculate the cohesive stress, von Mises yield condition at the boundary of the cohesive zone is employed for plane strain and plane stress. The model of the stress distribution with the maximum stress within the cohesive zone is discussed. Local criterion of brittle fracture and modelling of the fracture process zone by cohesive zone were used to describe fracture initiation at the hydride platelet in the process zone ahead of the crack tip. It was shown that the theoretical K _IH-estimation applied to the case of mixed plane condition within the process zone is qualitatively consistent with experimental data for unirradiated Zr-2.5Nb alloy. In the framework of the proposed model, the theoretical value of K ^H _IC for a single hydride platelet at the crack tip has been also estimated.     

Modeling of crack tip high inertia zone in dynamic brittle fracture

A phenomenological modification is proposed to the existing cohesive constitutive law of Roy and Dodds to model the crack tip high inertia region proposed by Gao. The modification involves addition of a term which is attributed to fracture mechanisms that result in high energy dissipation around the crack tip. This term is assumed to be a function of external energy per unit volume input into the system. Finite element analysis is performed on PMMA with constant velocity boundary conditions and mesh discretizations based on the work of Xu and Needleman. The cohesive model with the proposed dissipative term is only applied in the high inertia zone and the traditional Roy and Dodds model is applied on cohesive elements in the rest of the domain. The results show that crack propagates in three phases with a speed of 0.35c_R before branching, confirming experimental observations. The modeling of high inertia zone is one of the key aspects to understanding brittle fracture.     

A multiscale parametric study of mode I fracture in metal-to-metal low-toughness adhesive joints

The failure of adhesively-bonded joints, consisting of metallic adherends and epoxy-based structural adhesive with a relatively low toughness ~200 J/m², has been studied. The failure was via quasi-static mode I, steady-state crack propagation and has been modelled numerically. The model implements a ‘top-down approach’ to fracture using a dedicated steady-state, finite-element formulation. The damage mechanisms responsible for fracture are condensed onto a row of cohesive zone elements with zero thickness, and the responses of the bulk adhesive and of the adherends are represented by continuum elements spanning the full geometry of the joint. The material parameters employed in the model are first quantitatively identified for the particular epoxy adhesive of interest, and their validity is verified by comparison with experimental results. The model is then used to conduct a detailed study on the effects of (a) large variations in the geometrical configuration of the different types of specimens and (b) the adherend stiffness on the predicted value of the adhesive fracture energy, G _a . These numerical modelling results reveal that the adhesive fracture energy is a strong nonlinear function of the thickness of the adhesive layer, the other variables being of secondary importance in influencing the value of G _a providing the adhesive does not contribute significantly to the bending stiffness of the joint. These results which fully agree with experimental observations are explained in detail by identifying, and quantifying, the different sources of energy dissipation in the bulk adhesive contributing to the value of G _a . These sources are the locked-in elastic energy, crack tip plasticity, reverse plastic loading and plastic shear deformation at the adhesive/adherend interface. Further, the magnitudes of these sources of energy dissipation are correlated to the degree of constraint at the crack tip, which is quantified by considering the opening angle of the cohesive zone at the crack tip.     

Proposal of a new concept on creep fracture remnant life for a precracked specimen

Abstract

The creep life time of a smooth specimen can be predicted using existing laws for creep deformation and steady state creep rate. When crack growth behaviour is involved, it is necessary to construct a law of creep crack growth rate to predict creep fracture life. Creep fracture life can be measured by integrating the law of creep crack growth rate. One example is the creep crack growth rate, represented by the parameter Q*. In this study, we investigated the applicability of this prediction method to creep fracture remnant life for a cracked specimen. The Ω criterion is proposed to predict creep fracture remnant life for a smooth specimen for creep ductile materials. In this study, the correlation between Q*_L derived from the paremeters Q* and Ω is investigated. The correlation between Q_L* and Ω provided a unified theoretical prediction law of creep fracture remnant life for high-temperature creep-ductile materials in the range from smooth to precracked specimens.     

Viscoelastic Creep Crack Growth: A Review of Fracture Mechanical Analyses

The study of time dependent crack growth in polymers using a fracture mechanics approach has been reviewed. The time dependence of crack growth in polymers is seen to be the result of the viscoelastic deformation in the process zone, which causes the supply of energy to drive the crack to occur over time rather than instantaneously, as it does in metals. Additional time dependence in the crack growth process can be due to process zone behavior, where both the flow stress and the critical crack tip opening displacement may be dependent on the crack growth rate. Instability leading to slip-stick crack growth has been seen to be the consequence of a decrease in the critical energy release rate with increasing crack growth rate due to adiabatic heating causing are duction in the process zone flow stress, a decrease in the crack tip opening displacement due to a ductile to brittle transition at higher crack growth rates, or an increase in the rate of fracture work due to more rapid viscoelastic deformation. Finally, various techniques to experimentally characterize the crack growth rate as a function of stress intensity have been critiqued. This revised version was published online in July 2006 with corrections to the Cover Date.     

Stacking sequence effects on fracture of notched carbon fibre/epoxy composites

The purpose of this study was to investigate the ability of the so-called damage zone model (DZM) to predict the influence of stacking sequence on the strength of notched carbon fibre/epoxy composites. The DZM is in essence based on the unnotched tensile strength, σ₀, and the apparent fracture energy, G_c^*, and the damage zone is modelled as a crack with cohesive forces acting on the crack surfaces. The DZM predicts fracture loads for three-point bend (TPB) specimens and specimens with circular holes quite accurately. As an attempt to explain the difference in strengths, the damage zone extension in the TPB specimens with different stacking sequence was examined.     

Generalization of <Emphasis Type="Italic">T</Emphasis> and <Emphasis Type="Italic">A</Emphasis> integrals to time-dependent materials: analytical formulations

This paper deals with the generalization of T-integral to crack growth process in viscoelastic materials. In order to implement this expression in a finite element software, a modelling form of this integral, called Aθ, is developed. The analytical formulation is based on conservative law, independent path integral, and a combination of real, virtual displacement fields, and real, virtual thermal fields introducing, in the same time, a bilinear form of free energy density F. According to the generalization of Noether’s method, the application of Gauss Ostrogradski’s theorem combined with curvilinear cracked contour, T _v is obtained. By introducing a volume domain around crack tip, the modelling expression Aθ is also defined.. Finally, the viscoelastic generalization through a thermodynamic approach, called A _v , is introduced by using a discretisation of the creep tensor according to a generalized Kelvin Voigt representation.     

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.     

Vacancy concentration in strain ageing of substitutional f c c alloys

The relation between vacancy concentration, C _v and tensile plastic strain, ɛ, has been constantly expressed as C _v ∞ ɛ ^m . To take into account the grain-size effect, we have recently proposed that C∞ ɛβ_v ∞ϱ_m, where ϱ_m is the mobile dislocation density. With the conventional expression that ϱ_m d ⁻ⁿ , where d is the average grain size, the strain and grain-size dependence of vacancy concentration appears to be C _v ∞ ɛ^βγ d ^-nγ. This equation has proved effective in rationalizing the onset strain, ɛ_c, and stress amplitude, Δσ, of flow instability associated with the Portevin-LeChatelier effect of substitutional f c c alloys, where ɛ_c and Δσ are described by