Wide range data for crack tip parameters in two disc-type specimens under mixed mode loading

Disc-type specimens are among favorite test samples for determining mode I and mixed mode fracture toughness in brittle materials like rocks, brittle polymers, ceramics, etc. In this research, the finite element method is used to analyze two disc-type specimens: a semi-circular disc specimen containing an edge crack and subjected to three-point-bend loading (SCB specimen), and a centrally cracked circular disc subjected to diametral compressive loading, often called the Brazilian disc specimen. The crack parameters K_I, K_II and T are calculated for different mode mixities from pure mode I to pure mode II. Although the stress intensity factors K_I and K_II are presented mainly for validation of the analyses, they are also used for determining the crack angle corresponding to pure mode II for each specimen. It is shown that in general the T-stress increases for larger crack angles. While the T-stress in the Brazilian disc specimen is always negative for any combinations of mode I and mode II, the sign of T-stress in the SCB specimen depends on the mode mixity. A very good agreement is shown to exist between the calculated results for T and those very limited data presented in other papers.     

Transformation zones, crack shielding, and crack-growth resistance of Ce-TZP/alumina composite in mode II and combined mode II and mode I loading

Crack-tip transformation zones, crack shielding and crack-growth-resistance (R-curve) behaviors of a transformation-toughened ceria-partially stabilized zirconia–alumina (Ce-TZP/alumina) composite were studied in mode II and combined mode I and mode II loading using compact-tension-shear (CTS) specimens. The mode II and mode I stress intensities for both the initial straight cracks and the subsequent kinked cracks were assessed by the method of caustics using geometrically equivalent specimens of polymethyl methacrylate (PMMA). The angle of formation of the transformation zones as well as of extension of the cracks increased systematically with increasing ratio of the mode II and the mode I stress intensities and approached a value of θ^*=−72° in pure mode II loading. This angle was close to the angle for maximum hoop tension in the stress field of a mode II crack (θ^*=−70.5°). A crack-initiation toughness envelope was constructed on a K_I–K_II diagram using the critical loads for incremental crack extension. The crack-initiation toughness in pure mode II loading was less than the corresponding toughness in mode I loading. This result was consistent with calculations that indicated no shielding from the asymmetric and elongated zones developed in mode II loading. The fracture toughness measured for the kinked cracks at long kink lengths approached the maximum fracture toughness measured for a mode I crack.     

Mixed-mode two-dimensional crack problem by fractal two level finite element method

Fractal two level finite element method (F2LFEM) has been extended to calculate the mixed mode stress intensity factor in a two-displacement cracked body. The complete eigenfunction expansion of displacement by Williams is employed for the global interpolation function, the factors K_I and K_II can be easily computed for any arbitrary loading on any boundary. Results are obtained for some slant crack problems in finite sheets and are compared with known results where available.     

Investigation of near-tip displacement fields of a crack normal to and terminating at a bimaterial interface under mixed-mode loading

The displacement fields near the tip of a crack in a bimaterial joint under mixed-mode loading have been investigated by using a highly sensitive moiré interferometry technique. With a scheme adopted for data reduction, the study find that the near-tip displacements due respectively to opening and slipping loads are non-coupled and separable. The study on the parameters characteristic of the crack-tip deformation include: (1) the strength of stress singularity; (2) the angular distribution; and (3) the stress intensity factors K_I and K_II. The characteristic parameters determined experimentally are compared with the theoretical ones for the problem given by Zak and Williams [J. Appl. Mech. 30 (1963) 142], and Chen [Eng. Fract. Mech. 49 (4) (1994) 517].     

Crack surface relative displacement analysis of mixed-mode fracture mechanics problems

In this paper a unique criteria, crack surface relative displacement, is used to evaluate mixed-mode (mode I and mode II) fracture mechanics problems. Using a conic-section simulation of a crack surface, relationships among the energy release rate G, the stress intensity factors (K₁ and K₂), and crack surface relative displacement are developed. Because the crack surface relative displacement criterion makes direct use of the displacements on the crack surface, instead of the stress field in the region of the crack tip, it simplifies numerical analysis of crack problems. A finite element model of a slant-center-cracked plate is employed to demonstrate the applicability of crack surface relative displacement to mixed-mode problems. The numerical results obtained agree well with analytical solutions. In addition, it is illustrated that similar to K₁, K₂, and G (J in LEFM), crack surface, relative displacement can serve as a fracture criterion for general mixed-mode I and II fracture mechanics problems.     

Probabilistic description of fatigue crack growth in polycrystalline solids

A stochastic model describing the crack evolution and scatter associated with the crack propagation process has been built on the basis of the discontinuous Markovian process. The evolution and scatter are identified in terms of constant probability curves whose equation is derived as In P_r(i) = B(e^KI₀ − e^Ki), i ≥ I₀, where i is the number of cycles, B and K are crack-length-dependent variables, P_r(i) is the probabiliity of the crack being at position r along the fracture surface after i cycles elapse and I₀ is the minimum number of cycles required for the crack to advance from one position on the fracture surface to the next. The validity of the model is established by comparing the crack growth curves generated for Al 2024-T3 at a specific loading condition with those experimentally obtained.     

Near-tip dynamic fields for a crack advancing in a power-law elastic-plastic material: Modes I, II and III

Near-tip dynamic asymptotic stress fields of a crack advancing in an incompressible power-law elastic-plastic material are presented. It is shown that the stress- and strain-singularity are, respectively, of the order (In(R₀/r))^1/(n−1) and (In(R₀/r))^n/(n−1), where R₀ is a length parameter, r measures distance from the crack tip, and n is the power-law exponent. The angular variations of these fields are identical with those corresponding to dynamic crack growth in an elastic-perfectly-plastic material (Gao and Nemat-Nasser, 1983a,b).     

Growth mechanism of stress corrosion cracking in high strength steel

Stress corrosion crack growth rates are measured at sveral stress intensity levels for low-tempered 4340 steel in 0.1N H₂SO₄ solution. The characteristics of the growth rates are divided into three regions of stress intensity factors: Region I near K_1SCC; Region III near unstable fracture toughness, K_1SC; and Region II, which lies between the two. K_1SCC is the value of K at which no crack growth can be detected after 240 hr.

In order to explain these experimental results, the crack initiation analysis reported in a previous paper is extended to the growth rates. A detached crack initiates and grows at the tip of an already existing crack. When the detached crack reaches the tip of the main crack, the process repeats as a new existing crack.

A relationship between crack growth rate, v, and stress intensity factor, K, is obtained as a function of b/a and a = b + d, where b is the distance from the tip of the main crack to the detached crack, and d is the ydrogen atom saturated domain.

The experimental data are in good agreement with the theoretical values in Region II when a = 0.02 mm, b/a = 0.8, c₁/c₀ = 2.8 for 200°C tempered specimens and a = 0.015 mm, b/a = 0.7, c₁/c₀ = 3.0, ρ_b = 0.055 mm for 400°C tempered specimens, where ρ_b is a fictitious notch radius. The plateau part in Region II for 400°C tempered specimens is also successfully explained by the present theory. For Region III, the value of b/a will be almost equal to 1 because v → ∞ for b/a → 1. On the other hand, for Region I, b/a will be zero, since the value of v becomes negligibly small and no crack growth is observable.     

Stress intensity factors for an inclined edge crack in a semiplane

Mode I and II Stress Intensity Factors under uniform general biaxial loadings were derived for an inclined edge crack in a semiplane. By interpolating Finite Element results in the angular range [0°÷80°], analytical expressions were obtained for both K_I and K_II with an accuracy better than 1%. Influence coefficients were defined in the crack reference frame thus highlighting the coupling effects between Modes I and II due to the loss of symmetry when the crack is not normal to the surface.     

The method of caustics for mixed-mode crack loading with higher order term effects

The method of caustics or shadow-spot method for mixed-mode crack loading is investigated within the framework of Westergaard stress function analysis. The effect of higher order terms in the stress functions onto the shape and size of the caustic is studied for optically isotropic materials. The classical evaluation procedure is generalized and it is shown that the relationship between stress intensity factor and caustic diameter is of the general form K∞ D^(5/2)+n.     

A theory of caustics for mixed-mode fast running cracks

The method of caustics (shadow spot method) has proven to be a powerful optical method to measure stress intensity factors in static and dynamic fracture mechanics problems. In this paper, a theory of caustics was developed for elastodynamically propagating cracks under inplane mixed-mode conditions. Complex potentials for the general solutions of a near-tip field which have been previously derived by the authors were used in this theoretical development. Completely analytical expressions were derived for the caustic curves as well as for the initial curves for fast running cracks under inplane mixed-mode conditions. The effects of crack velocity and mixed-mode condition on the caustic pattern and the initial curve were investigated. New procedures were also proposed for the evaluation of the dynamic stress intensity factors K_I and K_II using the overall dimensions of the caustic pattern. The method of caustics developed here enables one to study quantitatively various mixed-mode dynamic fracture phenomena such as crack branching, crack curving, and crack kinking.     

Combination of finite element analysis and analytical crack tip solution for mixed mode fracture

A finite element program was developed which combines the analytical crack tip solution with a conventional finite element analysis and evaluates various crack tip parameters as part of the solution. This program was used to analyze cracked specimens subjected to mixed mode loading. The importance of retaining the second term of the series expansion for local stress, a contribution which is independent of the distance from the crack tip, was demonstrated. It was first shown analytically that the presence of a load applied parallel to the crack reveals itself only through this constant second term, which vanishes only for specific loading conditions. The results of the numerical analysis demonstrate that the stress intensity factor K_I is independent of the load applied parallel to the crack only when this term is included in the analytical crack tip solutions. Failure to include the constant term has the effect that K_I varies with the horizontal load. The parameter K₁₁ is independent of this load in both cases. This indicatesonce again that it is this constant term which accounts solely and entirely for the presence of a load applied parallel to the crack.     

A general method for determining mixed-mode stress intensity factors from isochromatic fringe patterns

A general method is presented for determining mixed-mode stress intensity factors K_I and K_II from isochromatic fringes near the crack tip. The method accounts for the effects of the far-field, non-singular stress, σ_ox. A non-linear equation is developed which relates the stress field in terms of K_I, K_II, and σ_ox to the co-ordinates, r and θ, defining the location of a point on an isochromatic fringe of order N.

Four different approaches for the solution of the non-linear equation are given. These include: a selected line approach in which data analysis is limited to the line θ = π and the K---N relation can be linearized and simplified, the classical approach in which two data points at (r_m, θ_m) are selected where r_m/θ = 0; a deterministic method where three arbitrarily located data points are used; and an over-deterministic approach where m (>3) arbitrarily located points are selected from the fringe field.

Except for the selected line approach, the method of solution involves an iteractive numerical procedure based on the Newton-Raphson technique. For the over-deterministic approach, the method of least squares was employed to fit the K-N relation to the field data.

All four methods provide solutions to 0.1% providing that the input parameters r, θ, and N describing the isochromatic field are exact. Convergence of the iterative methods is rapid (3–5 iterations) and computer costs are nominal. When experimental errors in the measurements of r and θ are taken into consideration, the over-deterministic approach which utilizes the method of least squares has a significant advantage. The method is global in nature and the use of multiple-point data available from the full-field fringe patterns permits a significant improvement in accuracy of K_I, K_II, and σ_ox determinations.     

Characterization of delamination behavior of unidirectional graphite/PEEK laminates using cracked lap shear (CLS) specimens

Delamination failure criterion is an important tool for characterizing the fracture behavior of laminated composites under mixed loading. In this paper, a fracture envelope was built based on the energy release rate as a fracture criterion of graphite/PEEK laminates. Unidirectional cracked lap shear (CLS) specimens were employed to calculate mode I and mode II energy release rates (G_I, G_II. Static fracture tests were conducted using the specimens with two different lap to strap thickness ratios in order to obtain a wide range of G_I/G_II values. The G_I/G_II values for each thickness ratio were calculated numerically using finite element analysis. The results showed that as a delamination length changes, the G_I/G_II varies from 0.13 to 0.48 depending on the lap to strap thickness ratio. It was also found that a linear fracture envelope may be appropriate for a CLS composite specimen.     

Fracture mechanical characterisation of mixed-mode toughness of thermoplast/glass interfaces

According to studies conducted by, e.g. Liechti and Chai [J. Appl. Mech. 58 (1991) 680], Yuuki et al. [Eng. Fract. Mech. 47 (3) (1994) 367] and Ikeda and Miyazaki [Eng. Fract. Mech. 59 (6) (1998) 725], a significant increase of interfacial toughness is observed, whenever the magnitude of the bond tangential shear load of the asymptotic elastic mixed-mode state is increased in either direction. Between these extremes the interfacial toughness curve exhibits a pronounced minimum, which, according to Hutchinson and Suo [Mater. Sci. Eng. A 107 (1989) 135] is believed to represent the so-called intrinsic adhesion, i.e. the failure toughness under pure local mode I loading. Within linear elasticity, the biaxial, singular near-tip solution for an open interface crack may be employed for characterising the local stress state as long as non-linearities such as, e.g. crack-wall contact and plastic flow are contained within a zone small enough compared to the extension of the singular opening-dominated fields. Then, the critical stress state is given in terms of bimaterial stress intensity factors K_1,c, K_2,c and the fracture toughness under mixed-mode loading may be expressed in terms of the critical energy release rate as a function of the mode-mixity ψ=tan⁻¹K_2,c/K_1,c. The stress intensities have to be extracted from a stress analysis of the specimen under the critical load, which in the present work is performed by means of an FE-model of the loaded sample.     

Determination of stress intensity factors and J-integrals using the method of caustics

Applications of the optical shadow method of reflective caustics to the measurement of the stress intensity factor and J-integral in various specimens are investigated. The necessary experimental requirements to help in determining an accurate stress intensity factor and J-integral are described. The ratios of r₀ (radius of initial curve)/r_p, (plastic zone size) and r₀/t (thickness of specimen) are found to be very important experimental parameters with which to obtain meaningful stress and/or strain intensities surrounding crack tips. The appropriate ranges to determine accurate values of stress intensity factor and J-integral for polycarbonate (compact tension) and aluminum (c-shaped tension) specimens are presented.     

Fracture toughness of multilayer silicon nitride with crack deflection

The fracture resistance of a multilayer silicon nitride consisting of alternate dense and porous layers was investigated by a single-edge-V-notched beam (SEVNB) technique. Since silicon nitride whiskers were aligned parallel to the laminar direction in the porous layer, the crack deflected macroscopically along the whiskers, resulting in high apparent K_I values, 15–25 MPa m^1/2. The crack then propagated in mode I, and was arrested when K_I was reduced to the fracture resistance without the crack deflection effects. These fracture resistance behaviors were well-explained in terms of the notch-insensitivity and the shielding effects of pull-out of the aligned whiskers.     

A fracture criterion for three-dimensional crack problems

A criterion for predicting the growth of three-dimensional cracks is developed on the basis of the strain energy density concept which has been used successfully for treating two-dimensional crack problems. Fracture is assumed to initiate from the nearest neighbor element located by a set of spherical coordinates (r, θ, φ) attached to the crack border. The new fracture surface is described by a locus of these elements whose locations correspond to the strain energy function, dW/dV, being a minimum. The function dW/dV is found to be singular of the type 1/r and is of quadratic form in the three stress intensity factors k₁, k₂ and k₃ expressed through the strain energy density factor S. It is postulated that unstable crack propagation initiates from a region where S reaches a critical value S_cr = r₀(dW/dV)_cr. The locations of failure lying on the fracture surface is determined by holding (dW/dV)_cr = S_min/r₀ constant. The quantity S_min stands for the value of S minimized with respect to θ and φ and r₀ is a radial distance measured from the crack border.

An example of failure prediction for an embedded elliptical crack subjected to both normal and shear loads is presented. According to the S-criterion, fracture initiation takes place at the ends of the minor axis. An unexpected result is that for a narrow elliptical crack and Poisson's ratio of 1/3 the lowest failure load occurs when the uniaxial tensile load makes an angle of approximately 60° with the crack surface and is in the plane of the major axis. This is in contrast to the expectation that the lowest critical load occurs when the uniaxial tension is perpendicular to the crack surface. In the limit as the elliptical crack becomes increasingly narrower, the result reduces to the two dimensional line crack case of Mode I and III loading. The S-criterion is also applied to the failure prediction of three dimensional cracks under compressive loads.     

Growth and coalescence of two collinear indentation cracks in glass

Subcritical growth and coalescence of two collinear cracks of different lengths were investigated using small Knoop indentation cracks in glass. Indentation cracks subjected to bending in water showed anomalous crack growth in terms of the stress intensity factor, K_I. The crack growth velocity, dc/dt, was initially high, decreased and thereafter increased with increasing K_I. The effective stress intensity factor, K_I,eff, was calculated by adding a term describing the state of residual stress to explain this anomalous growth. Before crack coalescence, a large crack showed a crack velocity higher than expected from the coalescent crack. The coalescent crack velocity increased with K_I,eff and the slope of dc/dt−K_I,eff curves differed from that for a single crack, depending on the crack length.     

Fatigue crack growth characteristics of rotor steels

Room temperature fatigue crack growth rate data were generated for Ni-Mo-V (ASTM A469, Cl-4), Cr-Mo-V (ASTM A470, Cl-8) and Ni-Cr-Mo-V (ASTM A471, Cl-4 and a 156,000 psi yield strength grade) rotor forging steels. Testing was conducted with WOL type compact toughness specimens and the results presented in terms of fracture mechanics parameters. Data show that the Ni-Cr-Mo-V steels exhibit slower fatigue crack growth rates at a given stress intensity range (ΔK) than do the Ni-Mo-V steels. In addition, the Cr-Mo-V steel was found to exhibit slower growth rates than the other alloys at ΔK levels below 40 ksi √in but somewhat foster rates at ΔK levels in excess of 45 ksi √in. The fatigue crack growth rate properties of the alloys studied conform to the generalized fracture mechanics crack growth rate law where da/dN = C₀ΔK^R. It was noted that the fatigue crack growth rate parameters n and C₀ tend to decrease and increase, respectively, with increasing material toughness, K_ic.