Specimen Size Requirements for Two-parameter Fracture Toughness Testing

Using a single parameter fracture mechanics theory, a minimum specimen size requirement of min(a, b, B) >200J/σ₀ in tension and min(a, b, B) >25J/σ₀ in bending, where B is the thickness, b the remaining ligament and a is the crack length of the specimen, were derived [Shih and German (1981), International Journal of fracture 17, 27–43] which have provided the basis for modern fracture toughness testing procedures. Two-parameter fracture toughness testing including the constraint, on the other hand, is desirable since it offers a solution to the transferability issue. A size requirement for a valid two-parameter fracture toughness testing based on the J-A₂ three-term solution was determined as min(a, b, B) > 11J/σ₀ [Chao and Zhu (1998), International Journal of fracture 89, 285–307] in which the limiting case is bend specimens under large scale yielding (LSY). Recent work by Chao et al. (2004, International Journal of fracture, 27, 283–302) has shown that the J-A₂ dominance at a crack tip can be significantly enhanced for bending specimens under LSY if a modified J-A₂ solution is adopted. This current paper further studies the size of the J-A₂ dominant zone using the modified J-A₂ solution for deep bend specimens with hardening from low to high and loading from SSY to LSY using finite element analysis. Based on the results, a rather relaxed specimen size requirement min(a, b, B) >6J/σ₀ is developed and recommended for a valid two-parameter fracture toughness testing using the J-A₂ fracture criterion. Validity of the size requirement is demonstrated by using the experimental J-R curves from non-standard bending specimens for A285 steel.     

Analytic and numerical studies on mode I and mode II fracture in elastic-plastic materials with strain gradient effects

This paper presents a study on fracture of materials at microscale (∼1 μm) by the strain gradient theory (Fleck and Hutchinson, 1993; Fleck et al., 1994). For remotely imposed classical K fields, the full-field solutions are obtained analytically or numerically for elastic and elastic-plastic materials with strain gradient effects. The analytical elastic full-field solution shows that stresses ahead of a crack tip are significantly higher than their counterparts in the classical K fields. The sizes of dominance zones for mode I and mode II near-tip asymptotic fields are 0.3l and 0.5l,while strain gradient effects are observed within land 2l to the crack tip, respectively, where l is the intrinsic material length in strain gradient theory and is on the order of microns in strain gradient plasticity (Fleck et al., 1994; Nix and Gao, 1998; Stolken and Evans, 1997). The Dugdale–Barenblatt type plasticity model is obtained to provide an estimation of plastic zone size for mode II fracture in materials with strain grain effects. The finite element method is used to investigate the small-scale-yielding solution for an elastic-power law hardening solid. It is found that the size of the dominance zone for the near-tip asymptotic field is the intrinsic material lengthl. For mode II fracture under the small-scale-yielding condition, transition from the remote classical K _IIfield to the near-tip asymptotic field in strain gradient plasticity goes through the HRR field only when K _IIis relatively large such that the plastic zone size is much larger than the intrinsic material length l. For mode I fracture under small-scale-yielding condition, however, transition from the remote classical K _I field to the near-tip asymptotic field in strain gradient plasticity does not go through the HRR field, but via a plastic zone. This revised version was published online in July 2006 with corrections to the Cover Date.     

Applications of the element-free Galerkin method for singular stress analysis under strain gradient plasticity theories

It is known that the plasticity models affect characterization of the crack tip fields. To predict failure one has to understand the crack tip stress field and control the crack. In the present work the element-free Galerkin methods for gradient plasticity theories have been developed and implemented into the commercial finite element code ABAQUS and used to analyze crack tip fields. Based on the modified boundary layer formulation it is confirmed that the stress singularity in the gradient plasticity theories is significantly higher than the known HRR solution and seems numerically to equal to 0.78, independently of the strain-hardening exponent. The strain singularity is much lower than the known HRR one. The crack field in gradient plasticity under small-scale yielding condition consists of three zones: The elastic K-field, the plastic HRR-field dominated by the J-integral and the hyper-singular stress field. Even under gradient plasticity there exists an HRR-zone described by the known J-integral, whereas the hyper-singular zone cannot be characterized by J. The hyper-singular zone is very small (r ? J/σ₀) and contained by the HRR zone in the infinitesimal deformation framework. The finite strains under the gradient plasticity will not eliminate the stress singularity as r → 0, in contrast to the known finite strain results under the Mises plasticity. Numerically no significant changes in characterization of the stress field were found in comparison with the infinitesimal deformation theory. Since the hyper-singular stress field is much smaller than the HRR zone and in the same size as the fracture process zone, one may still use the known J concept to control the crack in the gradient plasticities. In this sense the gradient plasticity will not change characterization of the crack.     

Analysis of high temperature fatigue crack growth behavior

High temperature fatigue crack growth has been examined in the light of the new concepts developed by the authors. We observe that the high temperature crack growth behavior can be explained using the two intrinsic parameters ΔK and K_max, without invoking crack closure concepts. The two-parameter requirement implies that two driving forces are required simultaneously to cause fatigue cracks to grow. This results in two thresholds that must be exceeded to initiate the growth. Of the two, the cyclic threshold part is related to the cyclic plasticity, while the static threshold is related to the breaking of the crack tip bonds. It is experimentally observed that the latter is relatively more sensitive to temperature, crack tip environment and slip mode. With increasing test temperature, the cycle-dependent damage process becomes more time-dependent, with the effect that crack growth is dominated by K_max. Thus, in all such fracture processes, whether it is an overload fracture or subcritical crack growth involving stress corrosion, sustained load, creep, fatigue or combinations thereof, K_max (or an equivalent non-linear parameter such as J_max) remains as one essential driving force contributing to the final material separation. Under fatigue conditions, cyclic amplitude ΔK (or an equivalent non-linear parameter like ΔJ) becomes the second necessary driving force needed to induce the characteristic cyclic damage for crack growth. Cyclic damage then reduces the role of K_max required for crack growth at the expense of ΔK.     

Determination of double-Determination of double-K criterion for crack propagation in quasi-brittle fracture Part I: experimental investigation of crack propagation

The results of experimental investigations using laser speckle interferometry on small size three-point bending notched beams and using photoelastic coating and the strain gauges on very large size compact tension specimens of concrete are presented in detail. The investigations showed that there exists a stage of stable crack propagation before unstable fracture occurs. The results are in agreement with other researchers' investigations using moire interferometry, holographic interferometry, dye-impregnation method and microscope. Further detailed study shows that the three different states, i.e., crack initiation, stable crack propagation and unstable fracture can be distinguished in the fracture process in concrete structures. In order to predict the crack propagation during the fracture process in quasi-brittle materials a double-K criterion is proposed. The double-K criterion consists of two size-independent parameters. Both of them are expressed in terms of the stress intensity factors. One of them reflects the initial cracking toughness, denoted with Kⁱⁿⁱ, which can be directly evaluated by the initial cracking load, P_ini, and the precast crack length, a₀, using a formula of LEFM. The other one refers to the unstable fracture toughness, denoted with K^un, which can be obtained inserting the maximum load, P_max, and the effective crack length, a, into the same formula of LEFM. The values of the two parameters, K _Ic ⁱⁿⁱ and K _Ic ^un , obtained from the small size three-point bending notched beams and the large size compact tension specimens show that K _Ic ⁱⁿⁱ and K _Ic ^un are size-independent. Evaluating with the K-resistance curves obtained from the same test data, it is found that the proposed double-K criterion is equivalent to it in basic principle, but, the double-K criterion can be applied more easily than the K-resistance curve. Finally, as a practical example, the application of the double-K criterion to the prediction of the crack propagation in a concrete dam is discussed.     

Crack-tip constraint in mode II deformation

Studies of cracked specimens loaded in mode I have shown that the stresses near the crack tip depend significantly on the level of constraint. The stresses can be determined near the crack tip using the HRR solution, but only for high constraint specimens. For other levels of constraint, O'Dowd and Shih's Q parameter may be used to adjust the stresses derived from the HRR solution. Only limited research has been carried out to study the effect of constraint in mode II. In this paper a mode II boundary layer formulation is used to study the effect of far field elastic stresses on the size and shape of the plastic zone around the crack tip and on the stresses inside the plastic zone. It is shown that in mode II, both positive and negative values of remote T-stress influence the tangential stress along the direction of maximum tangential stress. In the spirit of O'Dowd and Shih, a dimensionless parameter Q _II is introduced to quantify the constraint for mode II specimens failing by brittle fracture. The relation between Q _II and T/₀ is determined for different values of the strain hardening coefficient n. To investigate the range of validity of the Q–T diagram for real specimens, the constraint parameter Q _II is calculated directly from finite element analysis for three mode II specimens and compared with the evaluation using the Q–T diagram.     

On the evaluation of the J-integral by a plastic crack tip element

Two crack tip elements are formulated for a stationary, mode I plastic crack in planar structures using hybrid assumed stress approach, based on the secant modulus and the Newton-Raphson schemes, respectively. The stress distribution in the crack tip element is assumed to be the HRR field superimposed by the regular polynomial terms. The formulated (hybrid) crack tip elements are compatible with the isoparametric element so that they can be used conveniently along with the conventional displacement-based finite elements. The intensity of the HRR stress field, the J-integral, is determined directly from the finite element equations together with the nodal displacements. The dominance of the HRR stress field at the crack tip is pertinent to the present approach, which depends on geometry and loading conditions. Since the J-integral is globally path-independent for nonlinear elastic materials (deformation plasticity model), in order to assess the accuracy and efficiency of the methodology as compared to the contour integration approach, numerical studies of common plane-stress cracked configurations are performed for these materials. The results indicate that for a sufficiently small crack tip element size, J from the present approach correlates well, within 6 percent difference, with that from the contour integration for a wide range of material hardening coefficients if the HRR zone exists at the crack tip. These highly accurate results for J from the crack tip stresses could not be achieved without using (newly) modified variational principles and a refined numerical technique. It should be emphasized that the present methodology also can be applied to cracks in J ₂ flow materials under HRR dominance. In such case, the J integral may not be globally path independent, and hence it now must be determined from the stress and strain fields near the crack tip.     

J-Dominance in Mixed Mode Ductile Fracture Specimens

The asymptotic mixed mode crack tip fields in elastic-plastic solids are scaled by the J-integral and parameterized by a near-tip mixity parameter, M _{_p} . In this paper, the validity and range of dominance of these fields are investigated. To this end, small strain elastic-plastic finite element analyses of mixed mode fracture are first performed using a modified boundary layer formulation. Here, a two term expansion of the elastic crack tip field involving the stress intensity factor |K| the elastic mixity parameter M _{_e} as well as the T-stress is prescribed as remote boundary conditions. The analyses are conducted for different values of M _{_e} and the T-stress. Next, several commonly used mixed mode fracture specimens such as Compact Tension Shear (CTS), Four Point Bend (4PB), and modified Compact Tension specimen are considered. Here, the complete range of loading from contained yielding to large scale yielding is analyzed. Further, different crack to width ratios and strain hardening exponents are considered. The results obtained establish that the mixed mode asymptotic fields dominate over physically relevant length scales in the above geometries, except for predominantly mode I loading and under large scale yielding conditions. This revised version was published online in August 2006 with corrections to the Cover Date.     

The boundary-layer effect on the crack tip field in mechanism-based strain gradient plasticity

The theory of mechanism-based strain gradient (MSG) plasticity involves two material length parameters, namely the intrinsic material length land the mesoscale cell size l , which are on the order of a few microns and 0.1 m, respectively. Prior studies suggest that l has essentially no effect on the macroscopic quantities, but it may affect the local stress distribution. We demonstrate in this paper that there is a boundary layer effect associated with l in MSG plasticity, and the thickness of the boundary layer is on the order of   l ² big/l. By neglecting this boundary layer effect, a stress-dominated asymptotic field around a crack tip in MSG plasticity is obtained. This asymptotic field is valid at a distance to the crack tip between l and l(i.e., from 0.1 m to a few microns). The stress in this asymptotic field has an approximate singularity of r ^–2/3, which is more singular than not only the HRR field in classical plasticity but also the classical elastic Kfield (r ^–1/2). The stress level in this asymptotic field is two to three times higher than the HRR field, which provides an alternative mechanism for cleavage fracture in ductile materials observed in experiments.     

Residual strength of unidirectional fibre-metal laminates based on JC toughness of C(T) and SE(B) specimens: comparison with M(T) test results

Fibre‐metal laminates (FMLs) are structural composites designed with the aim of producing very low fatigue crack‐propagation rate, damage‐tolerant and high‐strength materials, if compared to aeronautical Al alloys. Their application in aeronautical structures demands a deep knowledge of a wide set of mechanical properties and technological values, including both fracture toughness and residual strength. The residual strength of FMLs have been traditionally determined by using wide centre‐cracked tension panels M(T). The use of this geometry requires large quantities of material and heavy laboratory facilities. In this work, fracture toughness ( J_C) of some unidirectional FMLs laminates was measured using a recently proposed methodology for critical fracture toughness evaluation on compact tension C(T) and single‐edge bend SE(B) specimens. Additionally, residual strength values of wider M(T) specimens with different widths (W from 150 to 200 mm) and several crack to width ratios (2a/W) were experimentally obtained. Some experimental residual strength values of M(T) specimens (W from 150 to 400 mm and different 2a/W ratios) of Arall were also obtained from the bibliography. Based on J_C results from C(T) and SE(B) specimens, and either using or not using crack‐tip plasticity corrections, the residual strengths of the M(T) specimens were predicted and compared to the experimental ones. The results showed good agreement, especially when crack‐tip plasticity corrections were applied.     

<Emphasis Type="Italic">T</Emphasis><Subscript><Emphasis Type="Italic">z</Emphasis></Subscript> constraints of semi-elliptical surface cracks in elastic plates subjected to uniform tension loading

The out-of-plane constraints T_z around the semi-elliptical surface cracks in an elastic plate subjected to uniform tension loading have been investigated through detailed three-dimensional (3D) finite element (FE) analyses. The distributions of T_z are obtained in the vicinity of the crack border with aspect ratios of 0.2, 0.4, 0.5, 0.6, 0.8 and 1.0. T_z drops from Poissons ratio at the crack tip to approximate zero beyond certain radial distance in the normal plane of the crack front line, and increases gradually from the free surface to the mid-plane at the same radial distance. By fitting the numerical results, empirical formulae are obtained to describe the 3D distributions of T_z for semi-elliptical surface cracks with a sufficient accuracy in the wide aspect ratio range of 0.2a/c 1.0 except very near the free surface, where T_z is extremely low. T_z, combining with the corresponding K and T or J and Q, can be applied to establish the three-parameter dominated stress field, which can characterize the 3D crack front field completely as an attempt.     

A simplified method for determining double-K fracture parameters for three-point bending tests

A simplified method for determining the double-K fracture parameters K _Ic ⁱⁿⁱ and K _Ic ^un for three-point bending tests is proposed. Two empirical formulae are used to describe the crack mouth opening displacement CMOD and the stress intensity factor K _I ^c caused by the cohesive force (x) on the fictitious crack zone for three-point bending beams. It has been found that the two empirical formulae are accurate for a large practical region of a/D. Experiments carried out by many researchers showed that the new formula of CMOD for three-point bending beams can be directly used to predict the initial crack length for precracked beams, the notch depth and the critical effective crack length, as well as the crack length in the post-critical situation with a satisfactory accuracy. Further verification is demonstrated to determine the double-K parameters K _Ic ⁱⁿⁱ and K _Ic ^un. They are very close to those determined by the method proposed in our previous work. Using the simplified procedure, the experiments can be performed even without a closed-loop testing facility and the calculation can be carried out on a pocket calculator.     

Mechanical heterogeneity and validity of J-dominance in welded joints

Fracture criterion of the J-integral finds wide application in the integrity evaluation of welded components, but there exist some confused problems such as the dependence of the fracture toughness on the strength mis-matching and specimen geometry which need to be clarified. It is rough and unsuitable to attribute the variation of J-integral fracture parameter simply to the effect of mechanical heterogeneity. In the present paper, a two-dimensional finite element method is employed to analyze the distribution and variation of crack tip field of welded joints with different strength mis-matching in four kinds of specimen geometry, and then the validity of J-dominance in welded joints is investigated. It is found that the crack tip field of mis-matched joint is different from that of either the weld metal or base metal of which the joint is composed, but it is situated between those of weld metal and base metal. Under the plane strain, there is obvious difference in stress triaxiality for different strength mis-matched joints. The validity of J-dominance in welded joint can not be obtained by comparing whether the stress triaxiality meets that required by the HRR solution because of the existence of mechanical inhomogeneity. By ascertaining if the stress triaxiality of welded joint near the crack tip is dependent of specimen geometry, the conclusion can be arrived at: for plane stress the validity of J-dominance is valid, whilst for plane strain the validity of J-dominance is lost. Based on the above, attempt has been made to point out that the influence of mechanical heterogeneity on the fracture toughness of weldment arises from the variation of constraint intensity-crack tip stress triaxiality. Compared with the effect of mechanical heterogeneity on the stress triaxiality, the losing of validity of J-dominance in mis-matched joint under plane strain may play a more critical role in the variation of J-integral fracture parameter of weldment.     

Measurement of Specific Fracture Energy and Surface Tension of Brittle Materials in Powder Form

Studies of the influence of specimen geometry and size–effect on the K _R –curves and the related fracture parameters were carried out by the authors (Kumar and Barai 2008b). The present paper is a supplementary contribution and reports interesting results related to the effect of the loading condition and size–effect studies on the K _R –curves associated with the cohesive stress distribution for complete fracture process, the double–K fracture parameters, the CTOD–curves and the process zone length using two different loading conditions (i.e., three–point bending test and four–point bending test). The laboratory size specimen with initial–notch length/depth ratios 0.3 and 0.5 are considered in the work. The load–crack opening displacement curves for these loading conditions are obtained using well known version of fictitious crack model.     

Influence of specimen geometry and size-effect on the <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">R</Emphasis></Subscript>-curve based on the cohesive stress in concrete

Comparative study for determining the K _R -curves associated with the cohesive stress distribution for complete fracture process for two standard specimen geometries i.e., three-point bending test and compact tension test specimen geometries of concrete using analytical method and weight function approach is reported in the paper. The laboratory size specimen (100 ≤  D  ≤  400 mm) with initial-notch length/depth ratios 0.3 and 0.5 are considered in the investigation. The load-crack opening displacement curves for these specimens are obtained using well known version of Fictitious Crack Model (FCM). It is found from the numerical results that the weight function method improves computational efficiency without any appreciable error. The stability analysis on the K _R -curves and the influence of specimen geometry and the size-effect on the K _R -curves, the CTOD-curves and the process zone length during crack propagation of complete fracture process are also described.     

Study on cleavage fracture criteria of the quasi-brittle and micro-inhomogeneous materials

On the bases of recent achievements on the micro-mechanism of cleavage this paper analyses the inherent deficiencies of the stress intensity factor K _I which is used to evaluate the fracture toughness of quasi-brittle and micro-inhomogeneous materials. The K _I parameter can uniquely determine the field intensity ahead of a crack tip in the condition of elastic and small scale yielding (SSY). However, the K _I cannot uniquely determine the critical condition triggering the cleavage fracture in a quasi-brittle and inhomogeneous steel where the cleavage fracture process is not a direct extension of the precrack but is initiated at a variable distance from the precrack tip. The variable distances of cleavage initiations invoke varied critical values of K _I. On the bases of authors' experiments, this paper analyses the physical meaning of the local fracture stress _f, its stability and the feasibility to be used as an engineering parameter for assessing the fracture toughness.     

Mixed mode fracture in power law hardening materials near Mode I

Mixed mode fracture in power law hardening materials near mode I loading conditions is investigated for the case of plane strain. It is demonstrated by application of full-field finite element analysis and two-parameter asymptotic analysis that the asymptotic mixed mode solution of Shih (1974), where the effects of arbitrary loading combine to enter the leading term of the asymptotic series, does not apply close to mode I. Instead, a `mode I dominant' higher order asymptotic form applies, where the leading term is the pure mode I symmetric HRR solution and the second term is antisymmetric. The significant difference between the two asymptotic solutions is the singular nature of the antisymmetric part of the stresses and strains. Therefore, near mode I the antisymmetric part of the local fields does not contribute to the J-integral. The transition from the mixed mode asymptotic solution to this two-term solution occurs when the ratio of the radial stress on the upper crack surface to that on the lower crack surface, i.e., σ<sub>rr</sub>(r,+π)/σ<sub>rr</sub>(r,?π), for the limit as r approaches zero, switches from ?1 to +1. Budiansky and Rice (1973) predicted this change in sign somewhere between mode I and mode II. This sign change corresponds to the singular nature of the antisymmetric fields switching from the leading HRR eigenvalue to a weaker second term eigenvalue. Based on full-field finite element results, the far-field loading for which this `jump' occurs is very difficult to identify, as it originates deep within the plastic zone at unrealistically small distances from the crack tip. At the physical length scale on the order of the crack tip opening displacement, the transition from the mode II dominant asymptotic solution to the mode I dominant solution appears to be continuous, not abrupt. Identification of this transition is estimated from the full-field results by making use of the two asymptotic solutions. These results identify a breakdown in the HRR theory for the representation of the antisymmetric part of the stresses for mixed mode fracture near mode I. This breakdown is explained by the switching from a mixed mode solution to a higher order solution. Unfortunately, the practical application of this higher-order solution is limited to hardening powers of approximately 2<n<3.     

A non-singular boundary integral formula for determining the T-stress for cracks of arbitrary geometry

A simple, yet accurate 2-D boundary integral equation (BIE) for determining the T-stress for cracks of arbitrarily geometry is introduced in this paper. The formulation is based upon the asymptotic expansion for the stress field in the vicinity of a crack tip. It can be conveniently implemented in the post-processing stage of a boundary element fracture analysis. As demonstrated in this work, the proposed BIE is non-singular, and thus it can directly be collocated at the crack tip under consideration. The technique requires a similar computational effort as that used in calculating the stress components at an interior point of a domain. Consequently, this new approach is very computationally effective and accurate for evaluating the elastic T-stress. Five test examples, involving straight, kink and curved cracks, are studied to validate the proposed technique and to assess its accuracy.     

Constraint-modified J−R curves and its application to ductile crack growth

The concept of J-controlled crack growth is extended to J–A ₂ controlled crack growth using J as the loading level and A ₂ as the constraint parameter. It is shown that during crack extension, the parameter A ₂ is an appropriate constraint parameter due to its independence of applied loads under fully plastic conditions or large-scale yielding. A wide range of constraint level is considered using five different types of specimen geometry and loading configuration; namely, compact tension (CT), three-point bend (TPB), single edge-notched tension (SENT), double edge-notched tension (DENT) and centre-cracked panel (CCP). The upper shelf initiation toughness J _IC, tearing resistance T _R and J–R curves tested by Joyce and Link (1995) for A533B steels using the first four specimens are analysed. Through finite element analysis at the applied load of J _IC, the values of A ₂ for all specimens are determined. The framework and construction of constraint-modified J–R curves using A ₂ as the constraint parameter are developed and demonstrated. A procedure of transferring the J–R curves determined from standard ASTM procedure to non-standard specimens or practical cracked structures is outlined. Based on the test data, the constraint-modified J–R curves are presented for the test material of A533B steel. Comparison shows the experimental J–R curves can be reproduced or predicted accurately by the constraint-modified J–R curves for all specimens tested. Finally, the variation of J–R curves with the size of test specimens is produced. The results show that larger specimens tend to have lower crack growth resistance curves.