Temperature dependence of fracture toughness J IC and ductility for BCC materials in the transition region

The temperature dependence of ductility, strength and fracture toughness for a BCC material undergoing predominantly linear elastic behavior at low temperatures and elastic-plastic behavior at higher temperatures is examined. A model, based on ductile fracture mechanisms involving void nucleation followed by cavity growth and void coalescence, is developed to relate the fracture toughness parameter J _IC with temperature. Two general equations for linear elastic and elastic plastic regimes of J _IC versus temperature T, are obtained. Applications of this model to experimental data obtained on a carbon steel show that J IC varies with T ² at low temperatures and with T at higher temperatures, thus defining a transition temperature.     

Limitations of elastic definitions in Al-Si-Mg cast alloys with enhanced plasticity: linear elastic fracture mechanics versus elastic-plastic fracture mechanics

Linear elastic fracture mechanics describes the fracture behavior of materials and components that respond elastically under loading. This approach is valuable and accurate for the continuum analysis of crack growth in brittle and high strength materials; however it introduces increasing inaccuracies for low-strength/high-ductility alloys (particularly low-carbon steels and light metal alloys). In the case of ductile alloys, different degrees of plastic deformation precede and accompany crack initiation and propagation, and a non-linear ductile fracture mechanics approach better characterizes the fatigue and fracture behavior under elastic-plastic conditions.To delineate plasticity effects in upper Region II and Region III of crack growth an analysis comparing linear elastic stress intensity factor ranges (ΔK_el) with crack tip plasticity adjusted linear elastic stress intensity factor ranges (ΔK_pl) is presented. To compute plasticity corrected stress intensity factor ranges (ΔK_pl), a new relationship for plastic zone size determination was developed taking into account effects of plane-strain and plane-stress conditions (“combo plastic zone”). In addition, for the upper part of the fatigue crack growth curve, elastic-plastic (cyclic J based) stress intensity factor ranges (ΔK_J) were computed from load-displacement records and compared to plasticity corrected stress intensity factor ranges (ΔK_pl). A new cyclic J analysis was designed to compute elastic-plastic stress intensity factor ranges (ΔK_J) by determining cumulative plastic damage from load-displacement records captured in load-control (K-control) fatigue crack growth tests. The cyclic J analysis provides the true fatigue crack growth behavior of the material. A methodology to evaluate the lower and upper bound fracture toughness of the material (J_IC and J_max) directly from fatigue crack growth test data (ΔK_FT(JIC) and ΔK_FT(Jmax)) was developed and validated using static fracture toughness test results. The value of ΔK_FT(JIC) (and implicitly J_IC) is determined by comparing the plasticity corrected elastic fatigue crack growth curve with the elastic-plastic fatigue crack growth curve. A most relevant finding is that plasticity adjusted linear elastic stress intensity factor ranges (ΔK_pl) are in remarkably good agreement with cyclic J analysis results (ΔK_J), and provide accurate plasticity corrections up to a ΔK corresponding to J_IC (i.e. ΔK_FT(JIC)). Towards the end of the fatigue crack growth test (above ΔK_FT(JIC)) when plasticity is accompanied by significant tearing, the cyclic J analysis provides a more accurate way to capture the true behavior of the material and determine ΔK_FT(Jmax). A procedure to decouple and partition plasticity and tearing effects on crack growth rates is given.Three cast Al-Si-Mg alloys with different levels of ductility, provided by different Si contents and heat treatments (T61 and T4) are evaluated, and the effects of crack tip plasticity on fatigue crack growth are assessed. Fatigue crack growth tests were conducted at a constant stress ratio, R = 0.1, using compact tension specimens.     

Experimental correlation between ductility and J-integral in the transition region of 1045 steel

A study of the variation of fracture toughness j_IC and ductility, measured under both tensile loading and biaxial plane strain (bulge) loading, of AISI 1045 steel in the annealed condition, in the transition temperature range of ?60 to 25°C was carried out. This temperature range delineates the changes in behavior from linear elastic to elastic-plastic behavior for this steel. It was found that the variation of j_ic with temperature shows a transition at about ?20°C while the bulge ductility only has a marked transition below ?40°C. These trends are explained in terms of the effect of material properties, namely the flow stress, on crack blunting, while the bulge ductility is correlated with the total strain to cause significant crack growth to take place. In the elastic-plastic region, a linear relationship between j_IC and bulge ductility was found to occur.     

Crack growth on elastic-plastic bimaterial interfaces

Finite element calculation based on finite strain theory is carried out to simulate the crack growth on bimaterial interfaces under the assumption of small scale yielding and plane strain condition. The modified Gurson's constitutive equation and the element vanish technique introduced by Tvergaard et al. are used to model the final formation of an open crack. The crack growths in homogeneous material and in bimaterials are compared. It is found from the calculation that the critical macroscopic fracture toughness for crack growth J _IC is much lower in bimaterials than in homogeneous material. For bimaterial cases, the J _IC of a crack between two elastic-plastic materials which have identical elastic properties with different yield strength is lower than that of a crack between an elastic-plastic material and a rigid substrate. It seems that the difference in yield strength between the dissimilar materials has more significant influence on the void nucleation and crack growth than the difference in hardening exponent.     

Effect of microstructure on the relationship between fracture toughness and ductility

The relationship between fracture toughness, expressed as J_IC, and ductility, given by measurements of the bulge ductility, was studied experimentally for two microstructures of 1045 steel. These microstructures consist of annealed pearlitic structure with hardness R_c = 15 and a tempered matensitic structure with R_c = 23. J_IC and bulge ductility measurements were performed in the temperature range of −100-25°C. The tensile properties at these temperatures were also obtained. The results show that the variation in flow stress with temperature is similar for both microstructures. However, the flow stress is higher, and the bulge ductility is lower, at a given temperature, for the martensitic structure. Also shifts in the transition temperature from linear elastic to elastic-plastic behavior are observed. Previously developed models by the authors describing the variation of J^IC with temperature and ductility are used to account for the behavior of the different microstructures examined.     

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.     

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.     

An alternative formulation of the Ritchie-Knott-Rice local fracture criterion

In the paper an alternative formulation of the RKR local fracture criterion is proposed. It is based on the features of the stress distribution in front of a blunted crack in an elastic-plastic material. The stress distribution is computed using the finite strain option in the finite element method. It is postulated that the opening stress in front of the crack should be greater than the critical one, σ_c, over the distance l ? l_c, where l_c is considered as a material parameter. The hypothesis is applied to estimate the influence of the in-plane constraint on fracture toughness. New formulas to compute the critical value of the J-integral are derived both for the small scale yielding and large plastic deformations in front of the crack. The results obtained are compared with the Sumpter and Forbes experimental results and with the O’Dowd analytical formula concerning the J_c = J_c(J_IC,Q) relation.     

Fracture toughness and fracture mechanisms of PBT/PC/IM blend

The static and impact fracture toughnesses of a polybutylene terephthalate/polycarbonate/impact modifier (PBT/PC/IM) blend were studied at different temperatures. The static fracture toughness of the blend was evaluated via the specific fracture work concept and the J-integral analysis. A comparison of these two analytical methods showed that the specific essential fracture work, W _e, was equivalent to the obtained by the ASTM E813-81 procedure, representing the crack initiation resistance of the material. The discrepancy between W _e and of ASTM E813-89 was caused by the extra energy component in consumed by a 0.2 mm crack growth. Impact fracture toughness was also analysed using the specific essential fracture work approach. When the fracture was elastic, W _e was equivalent to the critical potential energy release rate, G _IC, obtained via LEFM analysis. Temperature and strain-rate effects on the fracture toughness were also studied. The increase in impact toughness with temperature was attributed to two different toughening mechanisms, namely, the relaxation processes of the rubbery particles and the parent polymers in a relatively low-temperature range and thermal blunting of the crack tip at higher temperatures. The enhancement in static fracture toughness at temperatures below — 60 °C was thought to be caused by plastic crack-tip blunting, but the monotonic reduction in yield stress was largely responsible for the toughness decreasing with higher temperatures. The temperature-dependent fracture toughness data obtained in static tests could be horizontally shifted to match roughly the data for the impact tests, indicating the existence of a time-temperature equivalence relationship.     

The jump-like crack growth model, the estimation of fracture energy and JR curve

In this paper the jump-like crack growth model for monotonic loading is applied to re-examine both the onset of crack growth and process of stable crack growth. In the former case the fracture energy associated with a new surface creation is estimated and the in-plane constraint influence on this quantity is examined using the J-A₂ approach. In the later case the formula to compute the J-resistance curve is re-examined and compared with the one known from the standards. In the analysis the plane strain model of a structural element made of elastic-plastic material is assumed.     

A mechanical model to predict elastic-plastic fracture toughness in high strength materials

A mechanical model of crack initiation and propagation, which is based on the actual mechanism of ductile fracture in high strength materials, is proposed. Assuming that a crack initiates when the equivalent stress at a distance ρ from the crack tip reaches a critical value $\text{}\text{?}$gsf, an equation for predicting fracture toughness J_IC is obtained. From comparison between the predicted values and the experimental results, it is found that the distance ρ corresponds to the spacing of micro-inclusions. The temperature dependence of fracture toughness J_IC estimated according to the derived equation is given in an Arrhenius form of equation and is nearly consistent with the experimental results.     

Brittle and ductile crack propagation using automatic finite element crack box technique

In this paper, a numerical automatic crack box technique (CBT) is developed to perform fine fracture mechanics calculations in various structures without complete re-meshing. This technique aims to simulate the fatigue crack growth under mixed mode loading in 2D medium and shell structures calculated with the ABAQUS code, for elastic and for elastic-plastic materials. Using this method, series of numerical calculations by FEM of the mixed mode crack growth are carried out and compared with experimental tests such as a special cracked specimen subjected to different mixed mode loads. The crack growth paths are determined by using different elastic and elastic-plastic crack extension criteria. It is shown that the proposed technique is an efficient tool to simulate the crack extension angle in elastic and elastic-plastic materials. Nevertheless further experiments are needed to confirm conclusions deduced from elastic-plastic calculations.Using this technique, several phenomena influencing the crack extension are analyzed: the overload during fatigue, the fracture toughness of the material in relation with its critical J integral and its behaviour law.     

DAMAGE MECHANICS APPROACH TO REMOVE THE CONSTRAINT DEPENDENCE OF ELASTIC–PLASTIC FRACTURE TOUGHNESS

It is now generally agreed that the applicability of a one-parameter J-based ductile fracture approach is limited to so-called high constraint crack geometries, and that the elastic-plastic fracture toughness J_1c, is not a material constant but strongly specimen geometry constraint-dependent. In this paper, the constraint effect on elastic-plastic fracture toughness is investigated by use of a continuum damage mechanics approach. Based on a new local damage theory for ductile fracture(proposed by the author) which has a clear physical meaning and can describe both deformation and constraint effects on ductile fracture, a relationship is described between the conventional elastic-plastic fracture toughness, J_1c, and crack tip constraint, characterized by crack tip stress triaxiality T. Then, a new parameter J_dc (and associated criterion, J_d=J_dc) for ductile fracture is proposed. Experiments show that toughness variation with specimen geometry constraint changes can effectively be removed by use of the constraint correction procedure proposed in this paper, and that the new parameter J_dc is a material constant independent of specimen geometry (constraint). This parameter can serve as a new parameter to differentiate the elastic-plastic fracture toughness of engineering materials, which provides a new approach for fracture assessments of structures. It is not necessary to determine which laboratory specimen matches the structural constraint; rather, any specimen geometry can be tested to measure the size-independent fracture toughness J_dc. The potential advantage is clear and the results are very encouraging.     

Fracture toughness and crack morphology in indentation fracture of brittle materials

The relationship between the indentation fracture toughness, K _c, and the fractal dimension of the crack, D, has been examined on the indentation-fractured specimens of SiC and AIN ceramics, a soda-lime glass and a WC-8%Co hard metal. A theoretical analysis of the crack morphology based on a fractal geometry model was then made to correlate the fractal dimension of the crack, D, with the fracture toughness, K _IC, in brittle materials. The fractal dimension of the indentation crack, D, was found to be in the range 1.024–1.145 in brittle materials in this study. The indentation fracture toughness, K _c, increased with increasing fractal dimension, D, of the crack in these materials. According to the present analysis, the fracture toughness, K _IC, can be expressed as the following function of the fractal dimension of the crack, D, such that $$In K_{IC} = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}\{ In[2\Gamma E/(1 - \nu ^2 )] - (D - 1)In r_L \}$$ Where Γ is the work done in creating a unit crack surface, E is Young's modulus, v is Poisson's ratio, and r _L is r _min/r _max, the ratio of the lower limit, r _min, to the upper limit, r _max, of the scale length, r, between which the crack exhibits a fractal nature (r _min ?r?r _max). The experimental data (except for WC-8%Co hard metal) obtained in this study and by other investigators have been fitted to the above equation. The factors which affect the prediction of the value of K _IC from the above equation have been discussed.     

On the fundamental basis of fracture mechanics

Based on the crack tip stress and strain fields, the linear and the non-linear fracture mechanics have been developed. Their applications to the studies of fracture initiation and stable crack growth may differ because of the difference in the basic postulates of various fracture theories. The correct postulates will help to develop non-linear fracture mechanics for valid fracture toughness measurements and to extend fracture mechanics beyond the realms of K and J.The basic postulates of the linear elastic fracture mechanics are examined. The theory of global energy balance, the theory of sharp notch, and the theory of the characterization of crack tip stress and strain fields by K are analyzed. Fracture initiation and stable crack growth are local fracture phenomena. Therefore the global energy balance theory for crack initiation and stable crack growth without the study of the detailed fracture processes is fortuitous. The capability of the stress intensity factor to characterize the crack tip stress and strain fields for the localized fracture process is the basis for the validity of the linear elastic fracture mechanics.The concept of the characteristic crack tip field can be directly extended to the non-linear fracture mechanics. The fracture toughness and the tearing modulus of a tough material are measures of the fracture ductility of the material. The possibility to extend fracture mechanics beyond the realms of K and J are discussed.     

Size effects and ductile-brittle transition of polypropylene

The influence of specimen width on fracture parameters has been investigated. The range examined was sufficiently large to obtain ductile and brittle fractures. With reference to previously published work, the phenomenology has been analysed by combining BCS model and Carpinteri's brittleness number approach.Nomenclature a crack length - f(a/W) shape function according to ASTM specification [16] - F(a/W) shape function according to Tada Paris notation [21] - E elastic modulus - K _IC plane strain fracture toughness - K _IC ^f fictitious plane strain fracture toughness - K _IC2 plane stress fracture toughness - J _IC ^f J-integral at maximum load - L span - weight average molecular weight - number average molecular weight - polydispersity - P _M maximum load - P _F load of brittle fracture - p _P load of plastic collapse - s brittleness number - V machine cross speed - W specimen width - _y yield stress - strain rate     

Study on the behavior of elastic-plastic fracture under mixed I+II mode loading in aluminum alloy Ly12-J-integral analysis

The elastic-plastic fracture behavior of aluminum alloy Ly12 under mixed I+II mode loading was studied by finite element method and fracture test. A mixed mode elastic-plastic fracture criterion of J-integral was proposed by using the J-resistance curve, and the maximum fracture effective plastic strain _p max of different mixed ratios at crack tip were also calculated. The results show that(1) the initiation J-integral values of different mixed ratios have the equation

Use of COD in KIC determination

It has been theoretically postulated that the crack opening displacement (COD) technique is applicable in characterising fracture of material in the generally yielded as well as linear elastic regimes. Hence, it is possible to determine the plane strain fracture toughness, K_IC, of a material using the COD method if the onset of crack propagation and the location of the centre of rotation can be evaluated. In the present investigation, COD values at crack initiation, δ_i, for Rochling Moulrex A steel were obtained with 3-point bend specimens at six different tempering temperatures using the multiple specimens load-unload extrapolation technique. It is found that by using the BS 5762:1979 analysis, K_IC values were grossly over-estimated. The use of a constant rotational factor is observed to be inadequate. Under small scale yielding conditions, the centre of rotation is postulated from the present results to be located at approximately the end of the stretched zone. This supposition was applied to estimate the K_IC values of standard compact tension specimens made from Rochling Moulrex A, Assab 25X and Comsteel En25 steels. Where the material was linear elastic, the estimation was good with discrepancy of less than 7%. Overestimation was seen at high tempering temperatures due to the increasing amount of plasticity in the material that shifted the location of the centre of rotation from the end of the stretched zone to a position closer to that suggested in the British Standard.     

Model analysis of stable crack growth

The results of a numerical model study of quasi-static crack growth are reported. This study made use of an elastic-plastic stringer sheet in which the stringers were broken in an appropriate manner to simulate crack growth. The stringer-sheet model was used to demonstrate the qualitative effect of material properties, specimen geometry, initial crack length, and type of applied load on crack growth. In addition, stringer sheet analogs were constructed for both the R-curve concept of linear elastic fracture mechanics and for a modification of this concept in which K is replaced by the J-integral. The modified R curve was called the R^p curve. Calculated stringer sheet R and R^p curves were not material properties, but were influenced by the extent of plastic yielding during stable crack growth. In general, however, the R^p curve provided a somewhat better correlation of crack growth than the R curve.     

J-integral evaluation for cases involving non-proportional stressing

Calculation of J for cases where the proportional stressing condition cannot be satisfied is investigated. A modified J definition is derived and implemented into an ABAQUS post-processing program for both 2-D and axisymmetric problems. The modified J-integral is path independent for cases of proportional and non-proportional stressing. For cases with proportional stressing, the modified integral gives the same value as does the standard ABAQUS J function. It is also found that the modified J is equivalent to the stress intensity factor for a linear elastic material and provides a measure of the intensity of the crack-tip fields for non-linear elastic and elastic-plastic materials. The modified J formulation is applied to the case of a cylinder with an external circumferential crack under various load conditions.