Fracture mechanics model for subthreshold indentation flaws

In Part II of this two-part study we extend the shear-fault-microcrack model to non-equilibrium fracture, to allow for rate effects in the critical instability configurations in chemically interactive environments. The calibratedK-fields of Part I are combined with independently evaluated crack velocity functions to determine kinetic conditions for microcrack extension. The analysis enables evaluation of (i) a time delay in radial crack pop-in from a subthreshold flaw; (ii) a time dependence in the strength characteristics, in both the subthreshold and postthreshold domains. Comparisons with delayed pop-in and strength-stressing-rate literature data for silicate glasses in moist environments indicate that the analysis is capable of quantitative predictions of kinetic characteristics. In the strength data, the model accounts for the relatively high magnitudes, scatter and fatigue susceptibilities in the subthreshold region. Guest Scientists: On leave from the Department of Materials Science and Engineering, Lehigh University, Bethlehem, PA 18015, USA.     

Fracture mechanics transfer function for two-dimensional noncircular flaws

A transfer function that converts an irregularly shaped two-dimensional crack into an equivalent circular crack of equal propagation lifetime is proposed. The methodology extends the exact elliptical to circular crack transfer function to other noncircular cracks with geometries that slightly deviate from the elliptical. Results for rectangular, symmetric dumbbell, and asymmetric dumbbell cracks are presented.     

Fracture of ceramics with surface flaws introduced by Knoop indentation

The crack opening behaviour of surface flaws in ceramic materials, produced by Knoop indentation tests, was investigated. The damaged region below the indenter tip was replaced by a wedge. The influence of this wedge is reduced, if tensile stresses are acting at the damaged surface. For alumina the wedging force in moment of fracture was found approximately 50% of its value without additional bending.     

Fracture mechanics transfer function for irregularly shaped two dimensional flaws

In an elastic medium, transfer functions converting a two-dimensional nondrcular flaw to a circular flaw with equivalent fatigue properties are developed for the simple case of uniaxial stress applied normal to the flaw plane. Elliptical and rectangular flaws are specifically discussed. The method outlined may have important applications in predicting the kinetics of growth of irregularly shaped flaws in conjunction with nondestructive measurements.     

Fracture mechanics analysis of thin coatings under spherical indentation

Spherical indentation of a thin, hard coating bonded to a thick substrate is investigated. The bending of the coating over the softer substrate induces concentrated tensile stresses on the lower and upper coating surfaces, from which transverse cracks may ensue. This work is primarily concerned with ring cracks originating from the top surface of the coating. In-situ indentation tests are carried out on a model glass/polycarbonate bi-layer, with the coating thickness and the indenter radius being the main test variables. As the coating thickness is decreased, the critical load to initiate ring cracks progressively departs from that associated with a critical surface stress, the effect that increases with increasing the indenter radius. A fracture mechanics approach in conjunction with the FEM technique is used to elucidate the onset of cylindrical ring cracks in thin-film bi-layer structures due to spherical indentation. The analysis, conducted as a function of the coating thickness and the indenter radius, reveals the existence of bending-induced compression stress regions ahead of the crack tip, which tend to shield the crack or increase the fracture resistance. The specific behavior is dictated by a complex interplay between the contact radius, a, the coating thickness, d, and the crack length, c. An interesting manifestation of this shielding mechanism is that when the coating surface contains flaws of various sizes, small flaws in this population may be more detrimental than large ones. Incorporation of this aspect into the analysis led to a good correlation with the experimental results. In the limit case of point-load, a closed-form, approximate solution for the stress intensity factors and the critical loads is obtained. This solution constitutes a lower bound for the critical loads, and is furthermore directly applicable to finite size indenters provided da. In the limit c/d/to0, a failure stress criterion may be used irrespective of the ball radius, r. The analysis in this case reveals that decreasing either d/r or the coating/substrate modulus ratio tend to favor ring cracking over radial type cracking. The transition between these two failure modes is identified explicitly as a function of the system parameters.     

Fracture mechanics of two dimensional noncircular flaws—the equivalent area approximation

The equivalent area method of converting a two dimensional irregularly shaped flaw into a circular one of same fracture mechanics propagation lifetime appears to be promising. It is shown to be a good approximation for the simpler elliptical and square flaw geometries. Whether the equivalent area method also holds for the more complex flaw geometries remains to be verified. The importance of this study to nondestructive lifetime evaluation technology of structural components is discussed.     

Fatigue analysis of brittle materials using indentation flaws

A two-part study has been made of the fatigue characteristics of brittle solids using controlled indentation flaws. In this part a general theory is developed, with explicit consideration being given to the role played by residual contact stresses in the fracture mechanics to failure. The distinctive feature of the formulation is a stress intensity factor for well-defined indentation cracks, suitably modified to incorporate the residual component. Taken in conjunction with a standard power-law crack velocity function, this leads to a differential equation for the dynamic fatigue response of a given material/ environment system. Reduced variables are then introduced to facilitate generation of universal fatigue curves, determined uniquely by the crack velocity exponent,n. A scheme for using these curves to evaluate basic fracture parameters from strength data is outlined. In this way the foundation is laid for lifetime predictions of prospective brittle components, as well as for reconstruction of the crack velocity function. One of the major advantages of the analysis is the manner in which the residual stress parameters are accommodated in the normalized fracture mechanics equations: whereas it is understood thatall strength data are to be taken from test pieces in their as-indented state, so making it unnecessary to have to resort to inconvenient stress-removal procedures between the contact and failure stages of testing,a priori knowledge of the residual stress level is not required. The method is proposed as an economical route to materials evaluation and offers physical insight into the behaviour of natural flaws.     

Fatigue analysis of brittle materials using indentation flaws

The results of an experimental dynamic fatigue study on glass-ceramic specimens containing indentation flaws are analysed in terms of the theory developed in Part 1. A Vickers indenter is used to introduce the flaws, and a conventional four-point bend apparatus to break the specimens. Base-line data for testing the essential theoretical predictions and for evaluating key material/environment parameters are obtained from polished surfaces, i.e. surfaces prepared to a sufficient finish to ensure removal of any pre-existing spurious stresses. The fatigue tests are carried out in water. Inert strength tests in dry nitrogen are used to calibrate appropriate equilibrium fracture parameters, with dummy indentations on selected control specimens providing a convenient measure of the critical crack dimensions at failure. Regression analysis of the dynamic fatigue data yields values for apparent kinetic parameters, which are converted to true kinetic parameters via the transformation equations of Part I. Regeneration of the fatigue function from the theory using the parameters thus determined gives a curve which passes closely through the experimental data points, thereby providing a self-consistent check of the formalism. The implications of the results in relation to the use of macroscopic fracture parameters in the prediction of strength properties for materials with small-scale flaws is an important adjunct to this work. Finally, a recommended procedure for the general testing of dynamic fatigue properties of ceramics using indentation flaws is described.     

Dynamic fatigue of brittle materials containing indentation line flaws

A study is made of the dynamic fatigue response of brittle materials containing indentation-induced line flaws. The theoretical fracture mechanics of median crack evolution to failure under applied tension are first developed, with special emphasis on the role of residual contact stresses. In particular, it is shown that use of fatigue curves to evaluate the exponent in an assumed power-law crack velocity function may result in systematic error, by as much as a factor of two, if proper account is not taken of this residual contact contribution. Data from strength tests on soda-lime glass bars in water, using a tungsten carbide cutting wheel to introduce the median pre-cracks, confirm the basic predictions. The results suggest that extreme care needs to be exercised when using surfaces with a contact history, e.g. as with machining damage, in fatigue test programmes for materials analysis.     

Fracture from inherent flaws in polymers

Fracture data on polymethyl methacrylate, polyvinyl chloride, polyacetal and polypropylene in both tension and bending are given for a range of temperatures using unnotched specimens. It is concluded that all the failures occur at the gross crazing or yield stress and are thus stress-controlled, and the flaw size has little effect. The propagation behaviour of the flaws under these conditions is discussed in terms of stability criteria.     

Strength of poly(methyl methacrylate) with indentation flaws

Controlled flaws were introduced into poly(methyl methacrylate) samples in the presence of liquid acetone using a Vickers indenter over a range of indentation loads from 100 to 1400 N. Due to the large plastic zone underneath the indenter, the radial crack formed by indentation consisted of two halves, known as Palmqvist cracks, instead of a single semicircular crack. The strengths of the samples were measured in air either immediately following indentation or after a stress-relief anneal. The strength of the as-indented samples was about 6% less than that of the annealed samples; however, the dependence of strength on indentation load was similar for both sets of samples. These results were interpreted in terms of an indentation fracture mechanics model. The analysis is consistent with poly(methyl methacrylate) having a rising fracture toughness with increasing crack size.     

Fracture mechanics

Book Review

Fracture mechanicsT.L. Anderson, CRC press, inc., Boca Raton, FL     

Fracture indentation beneath flat and spherical punches

The mechanics of crack initiation and propagation beneath an axisymmetric flat punch are investigated. The stress tensor given by Sneddon in 1946 is described. Numerical integration along stress trajectories gives the strain energy release rate as a function of both the crack length and its position relative to the indenter. Comparison with Hertzian fracture is made. The initiation of crack outside the circle of contact is shown to be due to the steepest gradient of stresses along the flaws near the circle of contact. The meaning of Auerbach's law is discussed. The Auerbach range is shown to correspond to the relatively flat maximum of the envelope of theG againstc/a curves for various starting radii. The influence of subcritical crack growth is also discussed. The model proposed in 1978 by Maugis and Barquins for kinetics of crack propagation between punches and viscoelastic solids is used. It is assumed that the static fatigue limit corresponds to the true Griffith criterion with intrinsic surface energy , and that the critical strain energy release rateG _c corresponds to a criterion for crack speed instability and velocity jump, so that no stress corrosion is needed to explain subcritical crack growth for 2<G<G _c. The 1971 experimental results of Mikosza and Lawn are easily interpreted by this model. Finally, experiments performed on a borosilicate glass give results that agree satisfactorily with the theory. Due to kinetic effects, an apparent surface energy of about 4.5 J m^–2 is obtained, larger than the intrinsic surface energy and slightly lower than the fracture energy derived from high-speed experiments.     

On the mechanics of edge chipping from spherical indentation

Edge chipping is a basic failure mode in brittle materials which is dictated by a wealth of material and geometric variables. Here we examine the effect of indenter bluntness on chipping load and chip dimensions. Soda-lime glass and YTZP plates are subject to surface-normal loading near an edge by a W/C ball or a Vickers tool. The ball radius r is varied from 0.2 to 8.7 mm while the indent distance h is varied from several millimeters down to a few microns. Although cone cracks are a common feature under spherical indentation, the chipping event is dominated by median-radial cracks formed under the contact. The fracture behavior is characterized by a “large” indent distance regime where the median cracks progress stably up to chipping and a “small” one where they grow unstably to form a chip once initiated. Closed-form relations for chipping load P _F under spherical indentation is developed with the aid of the test data and non-dimensional arguments. While in the “large” distance regime P _F is proportional to h ^3/2 irrespective of tool bluntness, in the “small” regime P _F is proportional to r ^1/2 h ^3/4. Interestingly, the chip dimensions are virtually independent of ball radius, varying linearly with h. Beyond relevance to structural integrity, the chipping test facilitates a simple means for determining fracture toughness K _C as well as the load needed to initiate median cracks in opaque brittle materials. An attempt is made to extend the static analysis to low-velocity impact. The results show that the damage formed during the fracture process has a major influence on dynamic chipping.     

Fracture mechanics of ion-nitrided steel

Materials Science - The results show that ion nitriding has a positive effect on the cracking resistance of the examined tool steel. The critical thickness of the layer, above which nitriding...     

Fracture mechanics of piezoelectric materials

This paper presents an analysis of crack problems in homogeneous piezoelectrics or on the interfaces between two dissimilar piezoelectric materials based on the continuity of normal electric displacement and electric potential across the crack faces. The explicit analytic solutions are obtained for a single crack in an infinite piezoelectric or on the interface of piezoelectric bimaterials. For homogeneous materials it is found that the normal electric displacement D₂, induced by the crack, is constant along the crack faces which depends only on the remote applied stress fields. Within the crack slit, the perturbed electric fields induced by the crack are also constant and not affected by the applied electric displacement fields. For bimaterials, generally speaking, an interface crack exhibits oscillatory behavior and the normal electric displacement D₂ is a complex function along the crack faces. However, for bimaterials, having certain symmetry, in which an interface crack displays no oscillatory behavior, it is observed that the normal electric displacement D₂ is also constant along the crack faces and the electric field E₂ has the singularity ahead of the crack tip and has a jump across the interface. Energy release rates are established for homogeneous materials and bimaterials having certain symmetry. Both the crack front parallel to the poling axis and perpendicular to the poling axis are discussed. It is revealed that the energy release rates are always positive for stable materials and the applied electric displacements have no contribution to the energy release rates. This revised version was published online in July 2006 with corrections to the Cover Date.