Process Zone Size Research Articles

The prediction of crack growth rates from total endurances in high strain fatigue—thirty years on

McClintock1, 2 was one of the first to model continuous-cycling fatigue crack growth by assuming a succession of miniature low-cycle-fatigue (LCF) specimens at the crack tip. Elements ahead of the crack amass damage until the arrival of the tip itself. Such models had been summarized by Majumdar and Morrow,3 but the author was unaware of these papers at the time. The ideas were pursued further by Chakrabortty4 whose paper, again unknown to the author, was published in the same issue (1979, Vol. 2) of FEMS. In the original paper5 crack propagation is represented by successive regeneration at the crack tip, the process becoming progressively easier as the crack grows owing to an increase in strain concentration. These ‘initiation’ cycles were related to the ‘Coffin’ expression for crack initiation, thereby introducing two empirical constants k and α. The paper is of the class ‘ρ/N’ where ρ is the assumed size of a crack-tip process zone and N is the cycles required to traverse that zone. Expressions had been previously derived linking LCF with linear-elastic fracture-mechanics (LEFM) crack growth, using the parameter ΔJ.6 (It was later shown that this was identical to using an equivalent stress intensity parameter.7) It has been shown that the approach does not apply for crack depths < 180 μm.6 At elevated temperature the ΔJ approach was successful for describing crack growth in the range 0.2–1.2 mm for the cobalt-based alloy MAR-M509 at 600 °C.8 Other studies have in fact shown9 that crack growth rates are approximately constant below a depth of 200 μm. Even to this day short crack growth relations in LCF and their practical use are not as familiar as those in LEFM. It may justifiably be argued that for every paper published on LCF short crack growth, several hundred have appeared dealing with LEFM crack growth in terms of the ‘Paris’ law. One of the original referees' comments was that the paper5 should have been presented in terms of LEFM. The author nevertheless defended his approach because for small cracks it was thought useful to retain the crack depth explicitly. It will be shown that LCF crack growth data have a definite part to play in the assessment of components and structures. The paper assumes that the elastic stress concentration at the tip of a crack of depth a is of the form 2(a/r)1/2 where r is an effective crack tip radius. Because it was known that the crack opening displacement increases with increasing crack depth, so too does the effective radius. This was achieved via a power law, which introduced two empirical constants G and n. Exponential crack growth in 316 steel at 600 °C, see Eq. (7). Adapted from tabulated data in Ref. 13. Values of parameter B (circles) adapted from Fig. 1 compared with upper bound relation given by Eq. (6). It is appropriate to record that the original paper5 and this last reviewed paper13 were both published by permission of the former Central Electricity Generating Board.

Toughening mechanism and frontal process zone size of ceramics

In order to improve the fracture toughness of intrinsically brittle ceramics, it is essential to expand the frontal process zone (FPZ) ahead of a crack tip. Nanocomposites proposed by Niihara utilize dislocations generated around the dispersed nano-sized particles in matrix and expand the FPZ. In this paper, we discussed the toughening mechanism of ceramics using our experimental data on alumina-based nanocomposites, focusing on fundamental theories of fracture mechanics, such as Griffith-Irwin energy equilibrium and the local fracture criterion. We estimated the FPZ size using an indirect technique proposed recently, and clarified that the FPZ expansion toughening mechanism is achieved in nanocomposites by means of dislocation activities. The results revealed that ceramics with a larger FPZ size had higher fracture toughness and properly annealed nanocomposites had the largest FPZ size. The FPZ expansion mechanism in nanocomposites was considered that sessile dislocations generated around a crack tip in matrix, served as nano-crack nuclei, and hence expanded the FPZ size. It is clarified that the critical local stress is the strength of an infinite plate with no crack, and must be the true strength with no size effect. The fracture toughness, the critical local stress, and the FPZ size are the mutually dependent material properties, and the relation among them is clarified.

Limiting state of an orthotropic plate weakened by a periodic system of collinear cracks

On the basis of a modified δc-model of cracks, we study the limiting state of an orthotropic plate made of a material satisfying the general strength condition and weakened by a system of collinear cracks. The relations for the determination of major parameters of the model of cracks (the size of process zones, stresses in these zones, and the crack-tip opening displacements) are deduced. The mechanism of fracture of the plate containing a periodic system of collinear cracks is investigated. The influence of the degree of anisotropy and geometric parameters of the problem on the formation of the process zones and limiting state of the plate is revealed. The region of safe loading of an orthotropic viscoelastic plate with cracks is determined. The influence of the rheological parameters of the material on the region of safe loading is analyzed. Numerous modern works in the field of fracture mechanics are devoted to the development of new approaches to the investigation of fracture processes in various materials (metals, polymers, composites, rocks, etc.) under the action of external loads of various types [1–5]. The extensive application of elements made of anisotropic materials explains the necessity of investigation of their behavior under loads close to critical [6 – 8]. In [9], on the basis of a modified Leonov – Panasyuk – Dugdale model, the solutions of the problems of fracture were constructed for orthotropic plates of elastic, elastoplastic, and viscoelastic materials weakened by rectilinear cracks. In a similar statement, the solution of the problem of fracture of an orthotropic plate weakened by a periodic system of collinear cracks was obtained in [10]. In what follows, we investigate the limiting state of an orthotropic plate of viscoelastic material weakened by a periodic system of collinear cracks.

Transient damage spreading and anomalous scaling in mortar crack surfaces

The scaling properties of a post-mortem mortar crack surface are investigated. The root mean square of the height fluctuations is found to obey anomalous scaling properties, but with three exponents, two of them characterizing the local roughness ( zeta approximately 0.79 and zetae approximately 0.41 ) and the third one driving the global roughness (zetag approximately 1.60) . The critical exponent zeta approximately 0.79 is conjectured to reflect damage screening occurring for length scales smaller than the process zone size, while the exponent zetae approximately 0.41 characterizes roughness at larger length scales, i.e., at length scales where the material can be considered as linear elastic. Finally, we argue that the global roughness exponent could be material dependent contrary to both local roughness exponents ( zeta approximately 0.8 and zetae approximately 0.4 ) which can be considered as universal.

Evaluation of Nano-Sized Nickel Dispersed Alumina Materials

In order to develop new ceramics for a power electric device, alumina/nickel (Ni) nanocomposites were fabricated using the soaking method and the pulse electric current sintering technique. The volume fractions of Ni were 0.5, 1, and 5% in γ–alumina powder. 10mass% α–alumina powder was added as a sintering seed. SEM and TEM observations revealed that the nanocomposites exhibited an intra-granular fracture mode, and agglomerated Ni particles existed mostly along alumina grain boundaries especially for 5vol% Ni. The experimental results showed that the strength and fracture toughness values of the nanocomposites were higher than those of monolithic alumina, and the thermal conductivity decreased slightly with increase in Ni content. Moderate improvements in the strength and fracture toughness seemed to result from the fact that the nano-sized Ni particles dispersed within alumina grains creating dislocations around the Ni particles, which reduced sintering residual stresses in alumina grains and expanded the frontal process zone size. It was concluded that nanocomposites with a lower Ni content had better mechanical, electrical, and thermo-mechanical properties.

Study on Interfacial Crack Propagation and Fracture between Embedded Steel and HSHPC

Steel reinforced high strength and high performance concrete (SRHSHPC) specimens were experimented to study the mechanical behaviors between steel and concrete interface. In experiment, interfacial bond softening process was observed, which can be explained in terms of damage along the interface, leading to progressive reduction of shear transfer capability between steel and high strength and high performance concrete (HSHPC). In this paper, bond softening process along the interface is considered in the analysis of crack-induced debonding. Interfacial bond-slip mechanism between steel and HSHPC is studied in detail based on fracture mechanics. With the help of acoustic emissions technology, the crack propagation in the interlayer was observed, thus the interfacial crack propagation and fracture model is set up. Under the assumption that the interlayer is weak concrete compared with concrete matrix, the stress field as well as displacement field around the crack tip is deduced. The characteristics of interfacial fracture process are discussed and a model for interfacial fracture process zone is built up. With this model, the size of fracture process zone can be derived. At last, the influence of the fracture process zone on interfacial fracture toughness is determined using critical fracture toughness. All these may contribute to improvement of theory for SRHSHPC composite structure.

Fabrication and Characterization of Alumina/Silver Nanocomposites

Alumina/silver nanocomposites were fabricated using a soaking method through a sol-gel route to construct an intra-type nanostructure. The pulse electric-current sintering (PECS) technique was used to sinter the nanocomposites. Several specimens were annealed after sintering. The microstructure, mechanical properties, critical frontal process zone (FPZ) size, and thermo-mechanical properties of the nanocomposites were estimated. The relative densities of the specimens sintered at 1350 and 1450℃ were 95% and 99%, respectively. The maximum value of the three-point bending strength was found to be 780 ㎫ for the 2×2×10 ㎜ specimen sintered at 1350℃. The fracture toughness of the specimen sintered at 1350℃ was measured to be 3.60 ㎫·m 1/2 using the single-edge V-notched beam (SEVNB) technique. The fracture mode of the nanocomposites was transgranular, in contrast to the intergranular mode of monolithic alumina. The fracture morphology suggested that dislocations were generated around the silver nanoparticles dispersed within the alumina matrix. The specimens sintered at 1350℃ were annealed at 800℃ for 5 min, following which the maximum fracture strength became 810 ㎫ and the fracture toughness improved to 4. 21 ㎫m 1/2 . The critical FPZ size was the largest for the specimen annealed at 800℃ for 5 min. Thermal conductivity of the alumina/silver nanocomposites sintered at 1350℃ was 38 W/mK at room temperature, which was higher than the value obtained with the law of mixture.

Crack tip process zone domain switching in a soft lead zirconate titanate ceramic

Non-180° domain switching leads to fracture toughness enhancement in ferroelastic materials. Using a high-energy synchrotron X-ray source and a two-dimensional detector in transmission geometry, non-180° domain switching and crystallographic lattice strains were measured in situ around a crack tip in a soft tetragonal lead zirconate titanate ceramic. At K I = 0.71 MPa m 1/2 and below the initiation toughness, the process zone size, spatial distribution of preferred domain orientations, and lattice strains near the crack tip are a strong function of direction within the plane of the compact tension specimen. Deviatoric stresses and strains calculated using a finite element model and projected to the same directions measured in diffraction correlate with the measured spatial distributions and directional dependencies. Some preferred orientations remain in the crack wake after the crack has propagated; within the crack wake, the tetragonal 0 0 1 axis has a preferred orientation both perpendicular to the crack face and toward the crack front.

Simulations of dynamic crack propagation in brittle materials using nodal cohesive forces and continuum damage mechanics in the distinct element code LDEC

Experimental data indicates that the limiting crack speed in brittle materials is less than the Rayleigh wave speed. One reason for this is that dynamic instabilities produce surface roughness and microcracks that branch from the main crack. These processes increase dissipation near the crack tip over a range of crack speeds. When the scale of observation (or mesh resolution) becomes much larger than the typical sizes of these features, effective-medium theories are required to predict the coarse-grained fracture dynamics. Two approaches to modeling these phenomena are described and used in numerical simulations. The first approach is based on cohesive elements that utilize a rate-dependent weakening law for the nodal cohesive forces. The second approach uses a continuum damage model which has a weakening effect that lowers the effective Rayleigh wave speed in the material surrounding the crack tip. Simulations in this paper show that while both models are capable of increasing the energy dissipated during fracture when the mesh size is larger than the process zone size, only the continuum damage model is able to limit the crack speed over a range of applied loads. Numerical simulations of straight-running cracks demonstrate good agreement between the theoretical predictions of the combined models and experimental data on dynamic crack propagation in brittle materials. Simulations that model crack branching are also presented.

Influence of paste volume on shrinkage cracking and fracture properties of self-compacting concrete

Self-compacting concrete (SCC) mixtures are usually designed with higher volumes of paste than vibrated concrete mixtures. The results reported in this paper come from a study of nine SCC concrete mixtures. Volume of paste was varied between 291 and 457 l/m3. One of the mixtures had already been used in a large scale test, and the others were designed by varying several parameters of the reference concrete mixture. Mechanical properties, shrinkage, fracture parameters and fracture process zone (FPZ) size were measured. Fracture behavior was characterized by means of three-point bending tests and acoustic emission analysis. From the experimental results, increasing the volume of paste has a restricted effect on strength, unless water content varies. Strength, elastic modulus and fracture resistance slightly decrease with an increase in paste content. Volume of paste causes an increase in shrinkage and cracking due to shrinkage. Fracture and acoustic emission analysis show that increasing the volume of paste tends to make SCC more brittle.

Nanocomposites – Toughened Ceramics

In order to improve fracture toughness of ceramics, an intrinsically small frontal process zone (FPZ) size must be expanded. An intra-type nano-structure, where nano-particles are embedded within matrix grains, yields dislocations around the dispersed particles due to residual stresses. These dislocations become sessile dislocations at room temperature, operate as origins of small stress concentration in the matrix, and create nano-cracks in the FPZ. To produce the intra-type nano-structure, we developed a soaking method. This method makes it possible to produce nano-sized metallic particles dispersed within ceramic powders. In this research, alumina-nickel nanocomposite powder was obtained using the soaking method. The powder mixed with α-alumina as a seed was sintered using a pulse electric current sintering technique. The sintered nanocomposites are then annealed to disperse dislocations around the nanoparticles into alumina grains. Results showed that the maximum fracture toughness was 7.6 MPam1/2, which was two times higher than that of alumina.

On a ferroelastic mechanism governing the toughness of metastable tetragonal-prime ( t ′) yttria-stabilized zirconia

This article investigates the toughness of yttria-stabilized zirconia (YSZ) with the tetragonal-prime ( t ′) structure. Such materials are used as durable thermal barriers in gas turbines. Their durability has been attributed to high toughness, relative to materials in the cubic phase field. Based on prior literature, a ferroelastic toughening mechanism is hypothesized and this assertion is examined by characterizing the material in the wake of an indentation-induced crack. Assessment by transmission electron microscopy, Raman spectroscopy and optical interferometry has affirmed the existence of a process zone, approximately 3 μm in width, containing a high density of nano-scale domains, with equal proportions of all three crystallographic variants. Outside the zone, individual grains contain a single variant (no domains) implying that the toughening mechanism is controlled by domain nucleation (rather than the motion of pre-existing twin boundaries). The viability of the ferroelastic toughening mechanism is assessed using a process zone model that relates the observed toughening to the stress/strain hysteresis accompanying domain formation. Based on the measured process zone size, the known tetragonality of t ′-7 wt% YSZ and the enhancement in toughness relative to cubic YSZ, consistency between the model and the observed toughening is demonstrated for a reasonable choice of the coercive stress.

Novel Estimation of Critical Frontal Process Zone of Ceramics by a Single-Edge V-Notched-Beam Technique

A novel estimation for the critical size of the frontal process zone of ceramics is proposed using a single-edge V-notched beam (SEVNB) technique. A three-point flexure test is carried out on aluminum titanate ceramics containing a sharp V-shaped notch with different depth. An exact solution of the critical local stress is analyzed at a critical distance from the notch tip. The critical frontal process zone size is determined as the distance between the notch tip and the point where the critical local stress equals the flexural strength of specimens without notches, based on the local fracture criterion and the Griffith-Irwin criterion. The critical size of the frontal process zone, the fracture toughness and the flexural strength were also estimated for several materials, such as, alumina, porous alumina, and alumina-based nanocomposites. The relationship between these mechanical properties indicated that there was an almost linear relationship between the fracture toughness and the resultant of strength and square root of the critical frontal process zone size, and that both of them must be increased to improve the fracture toughness of ceramics.

The effect of flaw shape on fracture initiation at a blunt flaw

The author is involved in a wide-ranging research programme, the objective being to extend the fracture mechanics methodology for sharp cracks to blunt flaws, so as to take credit for the blunt flaw geometry. The approach is based on the cohesive process zone representation of the micro-mechanistic processes that are associated with fracture. An earlier paper has derived a blunt flaw fracture initiation relation which gives the critical elastic flaw-tip peak stress σpcr (a “signifier” of a critical condition in the process zone) in terms of the process zone material parameters, subject to the proviso that the process zone size s is small compared with the flaw depth (length) and any characteristic dimension other than the flaw root radius ρ. The relation has been derived using a “two-extremes” procedure, whereby the separate σpcr solutions for small and large s/ρ are blended together to give an all-embracing relation that is valid for all s/ρ. A key feature of the relation is that σpcr essentially depends on only one geometrical parameter: the flaw root radius ρ. Though the relation has evolved from a consideration of the characteristics of one model, i.e. that of an elliptical flaw in an infinite solid that is subjected to an applied tensile stress, it is anticipated that the relation can be applied equally well for a wide range of geometrical configurations involving different flaw shapes. It is against this background that the present paper demonstrates that the relation also applies to the behaviour of an intrusion type flaw in the surface of a semi-infinite solid subjected to an applied tensile stress.

Interface crack propagation in porous and time-dependent materials analyzed with discrete models

A model describing the crack propagation at the interface between a rigid substratum and a beam is considered. The interface is modeled using a fiber bundle model (i.e. using a discrete set of elements having a random strength). The distribution of avalanches, defined as the distance over which the crack is propagated under a fixed force, is studied in order to capture the effect of ageing and time-dependent response of the interface. The avalanches depend not only on the statistical distribution of strength but more importantly on time (or displacement) correlations. Namely, local fiber breakage kinetics is related to a correlation length, which sets the size of the fracture process zone which occurs ahead of the crack due to progressive failure. First, a variation of porosity of the interface is considered. It corresponds for instance to diffusion controlled dissolution processes. Interpreting the results in Delaplace et al. [Delaplace A, Roux S, Pijaudier-Cabot G (2001) J Eng Mech 127:646–652], it is shown that the size of the fracture process zone increases with increasing porosity in accordance with experimental observations [Haidar K, Pijaudier-Cabot G, Dubé J-F, Loukili A (2005) Mater Struct 38:201–210]. The creep–fracture interaction is analyzed in the second part of the paper. It is found based on a Maxwell model that the size of the process zone depends on the fracture propagating velocity and on the distribution of forces in the interface due to the interaction between the interface and the rest of the specimen. The observed decrease of the size of the process zone, in creep experiments, compared to the size of the process zone in a time-independent process, is justified by the proposed model for an interface that is less viscous than the rest of the material.

Scaling Exponents for Fracture Surfaces in Homogeneous Glass and Glassy Ceramics

We investigate the scaling properties of postmortem fracture surfaces in silica glass and glassy ceramics. In both cases, the 2D height-height correlation function is found to obey Family-Viseck scaling properties, but with two sets of critical exponents, in particular, a roughness exponent zeta approximately 0.75 in homogeneous glass and zeta approximately 0.4 in glassy ceramics. The ranges of length scales over which these two scalings are observed are shown to be below and above the size of the process zone, respectively. A model derived from linear elastic fracture mechanics in the quasistatic approximation succeeds to reproduce the scaling exponents observed in glassy ceramics. The critical exponents observed in homogeneous glass are conjectured to reflect the damage screening occurring for length scales below the size of the process zone.

A theory of rupture pulse on softening interface with application to the Chi‐Chi earthquake

The rupture process at the near‐front zone of a propagating crack, being strongly nonlinear and irreversible, a detailed Fourier analysis becomes complicated, often impractical. Meanwhile, a simplified approach, employing formally derived dispersion equation, may serve for obtaining meaningful information. Data on the rupture velocity, the size of the fracture process zone (FPZ), the possibility of exponential growth in time, and also on decaying with the distance from the rupture surface become available. The approach actually presumes that the size of the FPZ defines a dominant wavelength in the Fourier spectrum of a propagating rupture pulse. We verify this suggestion by revisiting the classical problem of a propagating shear crack having a cohesive FPZ with linear softening. It appears that in this particular problem, the simplified approach provides results which remarkably agree with the exact solution. This suggests using the approach for the analysis of the unique data on a rupture pulse obtained in the Chi‐Chi (Taiwan, 20 September 1999) earthquake. We conclude that the approach allows easy interpretation of the observed phenomena. The importance of the softening modulus as an independent characteristic of initiation and propagation of rupture is emphasized.

Critical Frontal Process Zone Size and R-Curve Behavior of Porous Ceramics

Although porous ceramics are materials with high potential for helping conserve the environment, the characteristics of pore-related mechanical properties have not yet been examined sufficiently. The R-curve behavior of porous ceramics was estimated using the crack stabilizer technique developed by Nojima et al. Also, the critical frontal process zone (CFPZ) size for porous ceramics was estimated from the strength and fracture toughness of the materials used. The results revealed that the R-curve behavior was almost flat in porous ceramics, in contrast with a steeply rising R-curve behavior for porous silicon carbide observed previously, and that the CFPZ size of porous ceramics was larger than that of dense ceramics. A schematic explanation for the crack extension in porous materials was presented to discuss the R-curve behavior of porous ceramics.

On the fracture resistance of adhesively jointing structures

The interface toughness of adhesively bonded structural members is one of the critical parameters for adhesive joint design. It is often assumed that the joint toughness is a material constant so that its value can be obtained from fracture tests of simple geometries such as DCB for Mode-I, ENF for Mode-II, using linear elastic fracture mechanics (LEFM). However, the LEFM assumption of point-wise crack-tip fracture process is overly simplistic and may cause significant error in interpreting fracture test data. In this paper, the accuracy and applicability of various traditional beam-bending-theory based methods for fracture toughness evaluation, such as simple beam theory (SBT), corrected beam theory (CBT) and experimental compliance method (ECM), were assessed using the cohesive zone modelling (CZM) approach. It was demonstrated that the fracture process zone (FPZ) size has profound influence on toughness calculation and unfortunately, all the classic beam-bending theories based methods fail to include this important element and are erroneous especially when the ratio of crack length to FPZ size is relatively small (<5.0). It has also been demonstrated that after the FPZ size is incorporated into simple beam formulations, they provide much improved evaluation for fracture toughness. Formulation of first order estimate of FPZ size is also given in this paper.

Temperature dependence of mechanical properties of aluminum titanate ceramics

Aluminum titanate (Al 2TiO 5: AT) is a synthetic ceramic material of potential interest for many structural applications. A critical feature, which greatly limits the mechanical properties of polycrystalline AT, is the considerable intergranular microcracking, which occurs due to the large thermal anisotropy of individual grains during cooling after sintering. This study discusses the temperature dependence of mechanical properties, and presents observations on the microstructure morphology. Both the fracture strength and fracture toughness increased considerably with increasing temperature. The critical frontal process zone (FPZ) size was estimated from the mechanical properties. A decrease in FPZ size was observed with increasing temperature in the first thermal treatment. These phenomena were explained on the basis of the stress redistribution and unique microscopic feature on the fracture surface of AT ceramics. The experimental results revealed that further thermal treatment increased the fracture strength and reduced the fracture toughness, while it had almost no effect on the FPZ size.