Fracture angle and strain-energy-density-factor of a crack at hole at an arbitrary angle

For both the maximum stress criterion and strain-energy-density-factor (S) theory, fracture angle (the initial angle of crack growth) − θ_o is predicted by using opening and sliding mate stress intensity factors, k₁ and k₂. These theoretical predictions are consistent with experimental fracture angles.

For the S theory, the crack spreads in the negative θ_o-direction in a plane for which S is a minimum, S_min. This quantity was obtained analytically. The experimental data of the critical S (S_cr) on plexiglass fracture specimens remains essentially constant.     

Crack growth resistance characterized by the strain energy density function

Stable ductile fracture of a typical metal alloy is found to be governed by the condition dS/da = const., i.e. the rate change of the strain energy density S with crack length 2a (or a) remained constant. Since fracture and/or yielding are load rate dependent, the incremental theory of plasticity is employed for analyzing crack growth where unloading in the material near the crack can take place. Attention is focused on the energy per unit volume, dW/dV, stored along the prospective path of crack growth. The nearest neighbor continuum element must necessarily be at a finite distance r from the crack front. This leads to the general relation dW/dV = S/R. The critical value (dW/dV)_c representing the area under the uniaxial true stress and strain curve is assumed to correspond with failure of material elements. If yielding and unloading occurred locally, a certain amount of irrecoverable energy will not be available for dissipation during macrocracking. Hence, the threshold energy density must be modified to read as (dW/dV)_c^* < (dW/dV)_c. The quantity (dW/dV)_c may be regarded as the crack growth resistance whose magnitude decreases with increasing distance from the crack tip at which point yielding is most intensified.

The results are displayed graphically and shown that the condition dS/da = const. provides a rational means of collating and interpreting ductile fracture data.     

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.     

Strain energy density fracture criterion in elastodynamic mixed mode crack propagation

Sih's fracture criterion based on strain energy density, S, for mixed mode crack extension under static loading is extended to dynamic mixed mode, K_I and K_II, crack propagation. Influence of the second order term, σ_ox, which represents the non-singular constant stress acting parallel to the direction of crack propagation, on the S distribution surrounding the crack tip, is demonstrated. Numerical studies show that positive σ_ox enhances the fracture angle and negative σ_oxreduces the fracture angle irrespective of the sign of K_II/K_I, when S is measured at a critical distance r_c from the crack tip. This fracture criterion is verified by the crack curving results of dynamic photoelastic fracture specimens. Omission of σ_ox term leads to predicted fracture angles which are at variance with experimental data.     

Energy-density concept in fracture mechanics

A theory of fracture mechanics is proposed in which attention is focused on the intensity of the energy field in the crack tip region. This energy field possesses a 1/r-type of singularity for both elastic and plastic materials. The strength or amplitude of this field will be referred to as the “energy-density factor”, S. Unlike the stress-intensity factor k in classical fracture mechanics which is only a measure of the local stress amplitude, the energy-density factor is also direction sensitive. The difference between k and S is analogous to the difference between a scalar and vector quantity. In this sense, the critical value S_cr specifies the direction of crack initiation as well as the fracture toughness of the material.     

A higher-order approximation for the T-criterion of fracture in biaxial fields

The T-criterion of fracture is based on the principle that crack propagates when the maximum value of the distribution of the dilatational component of strain energy density T_v, evaluated along contour lines of constant distortional energy density T_D around the crack tip, attains a limiting value T_vo The angle of this maximum defines also the direction of initiation of crack propagation. Then, the study of the distribution of T_v around the crack tip presents a special interest for understanding mechanisms of fracture.

In this investigation an exhaustive theoretical analysis of the distribution of t_v-component around the tip of crack under in-plane modes of loading was undertaken. The T_v-distribution was evaluated along the elastic-plastic boundary, developed around the crack tip for impending plasticity, according to the Mises yield condition (T_D = T_D0 = const.). The mode of loading of the cracked plate was assumed biaxial with different biaxiality ratios k and a two-term approximation for the respective complex stress function was considered, according to the studies of Liebowitz et al.[1], instead of only the singular term considered up-to-now.

It was found that the T_v-distribution along the Mises initial elastic-plastic boundary presents always a maximum in front of the crack tip, whose position and magnitude depend on the biaxiality factor k and the angle of loading β. The position and the magnitude of this maximum for the two-term approximation of φ(z) showed differences in some regions with the respective values for the singular solution.     

Computer simulation of martensitic textures

We consider a Ginzburg-Landau model free energy F(ε, e₁, e₂) for a (2D) martensitic transition, that provides a unified understanding of varied twin/tweed textures. Here F is a triple well potential in the rectangular strain (ε) order parameter and quadratic e₁², e₂² in the compressional and shear strains, respectively. Random compositional fluctuations η(r) (e.g. in an alloy) are gradient-coupled to ε, ˜ − ∑_rε(r)[(Δ_x² − Δ_y²)η(r)] in a “local-stress” model. We find that the compatibility condition (linking tensor components ε(r) and e₁(r), e₂(r)), together with local variations such as interfaces or η(r) fluctuations, can drive the formation of global elastic textures, through long-range and anisotropic effective ε-ε interactions. We have carried out extensive relaxational computer simulations using the time-dependent Ginzburg-Landau (TDGL) equation that supports our analytic work and shows the spontaneous formation of parallel twins, and chequer-board tweed. The observed microstructure in NiAl and Fe_xPd_{1 − x} alloys can be explained on the basis of our analysis and simulations.     

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.     

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.     

An analysis of crack propagation velocity in delayed failure under repeating load using an internal friction model

Frequency and temperature dependency of crack propagation velocity in delayed failure under superposed repeating load was analyzed using an internal friction model which assumes the interaction between hydrogen atoms and the cyclic moving of the position with tri-axial tensile stress at crack tip.

The decrease of crack propagation velocity (da/dt)_R by the superposition of repeating load, the appearance of minimum value in (da/dt)_R at a certain frequency ƒ₀, and the shift of ƒ₀ by the change of temperature, are well explained by the internal friction model. Another reason for the decrease of (da/dt)_R by the superposition of repeating load appears to be the decrease of effective stress intensity at crack tip, though this cannot explain the appearance of minimum value of (da/dt)_R.     

A model for high cycle fatigue

Failure due to fatigue consists of such macroscopic events as crack initiation and propagation. Microscopic events including microcrack nucleation, microcrack growth and coalescence of some of the microcracks are also important in that such crack interactions can be considered to contribute to the development of a critical defect, i.e. a defect which can self propagate and lead to failure. Since crack initiation is important at high cycles, this paper considers a microcrack and computes its growth to advance a high cycle stress-life (S-N_f) formulation for metals and alloys based on crack initiation. In addition to a material's Burgers' vector and grain size, an indirect effort is also made to include the role of its propensity to cracking through a ratio S(σ_I)/S(σ_f), where S(σ_I) and S(σ_f) simply represent steady state crack spacings at stress amplitudes σ_I and σ_f (endurance limit), respectively. Conceptually, a decrease of this ratio suggests an increased tendency for cracking and vice versa. As shown in the text, the model predicts that the crack initiation period varies increasingly with this ratio, and the value of the steady state crack spacing ratio is in and of itself quite sufficient to model the experimental stress-life data in instances where life is controlled by crack initiation period and not the stage II crack propagation. Because of this limitation, extreme care must thus be taken with regard to its application.     

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.     

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.     

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).     

Prediction of constant-amplitude fatigue life to failure under pulsating tension by use of the local-strain approach

The aim of this investigation was to compare local-strain approximation (LSA) life predictions with pulsating-tension tests to failure. It was shown that, in the case of pulsating tension, the ratio N_i/N_f does not change significantly and therefore an LSA calculation can be of use for predicting A-M-N lines for R0. The accuracy of the prediction is associated with the choice of the k-value in Neuber's relation K_σK_ε=k². The -values that give the best life predictions were obtained close to those of the corresponding notch factors for pulsating tension (R=S_min/Smax=0). Because of the size and surface effects (which are not taken into account in LSA), and in view of the difficulty in knowing whether the specimen with a given K_F-value has exactly the same cyclic parameters as those used in the LSA calculations, it is then preferable to use a k-value proved to give prediction results in agreement with some pulsating-tension tests. This k-value may be used for all combinations of S_a and S_m (S_aS_m and S_max<S_y). The results are valid for a specific specimen material, geometry, size and surface finish.     

Griffith's theory of brittle fracture in three dimensions

The Griffith theory of brittle fracture is extended to the three-dimensional problem of a flat elliptical crack in an otherwise uniform field of tensile and shear stresses. A method for finding the correct expressions of the change in strain energy due to the elliptical crack is developed. This is done by expressing the stresses and displacements in terms of the radius R₀ of a large sphere around the crack and by imposing the condition of equilibrium that the stresses or displacements across the spherical surface should agree with the prescribed boundary conditions as R₀ → ∞. The strain energy due to the presence of the elliptical crack is found to be independent of the tension applied parallel to the crack plane at infinity. On the basis of the thermodynamic argument of Griffith, it is also observed that the critical tensile and shear stresses increase rapidly as the ratio of major to minor semi-axes of the ellipse approaches unity.     

Impact fatigue properties of epoxy resin filled with SiO2 particles

Impact fatigue tests were carried out on epoxy resin filled with SiO₂ particles. The effects of the percentage of SiO₂ particles and the impact cyclic loading frequency on the impact fatigue strength was investigated. The micromechanism of impact fatigue failure was examined and correlated with the morphology of the fracture surface. The impact stress amplitude, σ_t, can be estimated by the formula, σ₂(N_f · Te)^mt = D_t where (N_f· Te) is the cumulative duration time, and mt and D_t, are parameters describing impact fatigue characteristics. The impact fatigue strength and the static strength are governed by the percent of SiO₂ particles. Crack initiation under monotonie cyclic impact loading was attributed to decision of the epoxy-SiO₂ interface. Unstable crack propagation occurs when the crack passes through the SiO₂ particles.     

Exploring the fracture toughness KIE and KIC of a medium-Strength steel plate using the surface-flaw test

The fracture toughness of a 30 CrMnSiA steel plate of three thicknesses (10,8 and 5 mm) and three widths (110,80 and 56 mm) has been investigated by using surface-flaw method under room temperature. It is not easy to compute the value of K_IE by the maximum applied load. But the values of K_IE and K_IC could be obtained easily, if the computation of the conditional applied load P₁₀ and P₅ based on the relative effective extension Δa/a₀ = 10% and 5% were adopted, together with the conditions of P_max/P₁₀ 1.2 and P_max/P₅ 1.3. The K_R — Δ_a curve, i.e. the resistance-curve described by the parameter K, has been plotted. The values of K_IC and K_IE are then the resistances corresponding to the real extensions of flaws of Δ/a₀ = 2 and 7%, respectively. These values so obtained are in good agreement with the computed values of K_IC and K_IE by using the conditional applied loads. The values of K_IC and K_IE so obtained are also in agreement with the value of K_IC converted from the J-integral and the effective value of K_IE computed by the maximum applied load, respectively.

An approximate relation between K_IC and K_IE has been found to be: K_IC = (0.85˜0.95)K_IE.

The requirements for the dimensions of specimens are: Thickness of plate: B 1.0(K_IC/σ_0.2)² or 1.25(K_ICσ_0.2)²]; Width of plate: 8 W/B 10, 4 W/2c 5; Effective length: l 2W.     

Weight function calculations for mixedmode fracture problems with the virtual crack extension technique

An efficient finite element method is presented for calculating the stress intensity factors (K_I and K_II) and the weight functions for mixed-mode cracks with one virtual crack extension. The computational efficiency is enhanced through the use of singular elements and the application of colinear virtual crack extension (VCE) technique to symmetric mesh in cracktip neighborhood. This symmetric mesh in crack-tip vicinity permits the analytical separation of strain energy release rate into G_I for Mode I and G_II for Mode II for the mixed fracture problems with the colinear virtual crack extension.

Rice's displacement derivative representation of weight function vector for symmetric crack has been extended to the mixed fracture mode at nodal location (x_i,y_i) with crack length (a) and inclination angle (β) as h_I(II)(x_i, y_i, a, β) = (H/2K_I(II)(∂U_I(II)(x_i, y_i, a, β/∂a).

This equation permits explicit determination of weight functions for the entire structure of a given asymmetric crack geometry with colinear VCE technique. The explicit weight functions for mixed fracture mode depend strongly on the constraint conditions. The method of obtaining the required stress intensity factors of a given asymmetric crack geometry, from the weight function concept under the selected constraint conditions, which are different from constraint conditions used in the available weight functions for the same crack geometry, is also presented in this paper. This is accomplished by combining the predetermined explicit weight functions with the self-equilibrium forces at their application locations. These self-equilibrium forces include both the applied surface tractions and the reaction forces induced from the constraint conditions.     

Influence of stress ratio at baseline loading on delayed retardation phenomena of fatigue crack growth in A553 steel and A5083 aluminium alloy

The delayed retardation phenomena of fatigue crack growth following a single application of tensile overload were investigated under the baseline loading with the stress ratio, R = σ_min/σ_max, ranging from −1 to 0.5 for A553 steel and A5083 aluminium alloy. Two different overload cycles were applied; the one is the case that the ratio of peak stress range to baseline stress range, r = Δσ₂/Δσ₁, is equal to two and the other is the case that the ratio of maximum peak stress to maximum baseline stress, σ_2max/σ_1max, is equal to two. The retardation took place stronger in aluminium than in steel. Under the condition of r = 2 the normalized number of cycles, N_D/N_C, (N_D: the number of cycles during retardation, N_C: the number of cycles required for propagation through the overload-affected-zone size) decreased slightly as the R ratio increased from −1 to 0.5, while under the condition of σ_2max/σ_1max = 2 the N_D/N_C-values increased drastically as the R ratio increased from −1 to 0 (or the overload ratio, r, increased from 1.5 to 2) in both the materials. These retardation behaviors were expressed theoretically according to the model proposed by Matsuoka and Tanaka [1, 3] by using four parameters: the overload ratio, r, the exponent in Paris equation, m, the overload-affected-zone size, ω_D, and the distance at the inflection point, ω_B.