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.     

Use of double clip gauge method to measure the plastic rotational factor rp

The value of the plastic rotational factor r_p at initiation in 3-point bend specimens with deep and shallow notches has been studied by using the double clip gauge method and it is shown that r_p is decreasing with decreasing of a/W, the experimental results agreeing well with the experimental results by using a clip gauge and catalytically hardening silicone rubber Unitex Xantropren Blue dental impression material.     

Experimental evaluation of hinge phenomenon in notched three point bend bars using laser speckle metrology

In elastic-plastic fracture, material behavior is often characterized by the J-integral or crack tip opening displacement (CTOD) parameters. In order to evaluate these parameters accurately, the location of the plastic hinge point, and subsequently the value of the plastic rotational factor, r_p, must be determined. Traditionally, hinge point location and r_p have been inferred through crack opening displacement (COD) measurements. However, this work indicates that laser speckle metrology can be used to analyse directly the hinging phenomenon. Using notched three point bend bars fabricated from a high stretch low alloy steel, a full-field map of in-plane displacements was generated over the course of each test. The results indicate the existance of a hinge region, rather than an explicit hinge point. The hinge region appears to contain the computed hinge point location using the range of cited r_p values. This indicates that it may be appropriate to use the centroid of the hinge region in subsequent CTOD and J-integral calculations.     

A comparison on the effect of the ratio a/w on the fracture behaviour between ductile and brittle materials

The effect of the ratio a/W on the fracture behaviour of ductile and brittle materials has been studied by measuring the crack-opening displacement and J-integral for ductile material and the stress intensity factor for brittle material in three-point bend specimens with shallow and deep notches. It is shown that, for ductile material, the values of δ_i and J_i, for specimens with shallow notches are larger than those of deep notches. On the contrary, for brittle material, the values of K_IC for specimens with shallow notches are smaller than those of deep notches. The reason for this is explained.     

The calculation of thermal stresses in cylinders

The thermal stresses in a long cylinder are calculated on the assumption that the metal behaves below a critical temperature T_c as a perfectly elastic isotropic medium, while above T_c it presents a negligible resistance to shear. Two cases are considered. In the first, that of isolated billets, the cold ends prevent longitudinal flow of the plastic core, which remains under hydrostatic tension. In the second, that of continuous casting, the hot top allows more metal to flow into the core and relieve the hydrostatic tension.     

A model for predicting fatigue crack growth behaviour of a low alloy steel at low temperatures

In the present study, a model to predict the fatigue crack growth (FCG) behaviour at low temperatures is proposed for a low alloy steel (16 Mn). The experimental results indicate that fatigue ductile-brittle transition (FDBT) occurs in 16 Mn steel and the FDBT temperature (T_FDBT) is about 130 K. When T > T_FDBT, the FCG mechanism in the intermediate region is the formation of ductile striation and the FCG rates decrease with decreasing temperature. When T ≈ T_FDBT, the FCG mechanism changes into microcleavage and the fatigue fracture toughness K_fc of the steel decreases sharply. The FCG rates tend to increase as the temperature is further reduced. The test data of the FCG rates are well fitted by the formula developed by Zheng and Hirt. An approximate method to predict ΔK_th of the steel at low temperatures is proposed and then a general expression of the FCG rates is given at temperatures ranging from room temperature to T_FDBT. By means of the expressions proposed in this paper, the FCG rates at low temperatures can be predicted from the tensile properties if the endurance limit σ₋₁ and δk_th, at room temperature are known. Finally, a model for FDBT is tentatively proposed. Using this model, one can predict T_FDBT from the ductile-brittle transition curve determined from impact or slow bending tests of cracked Charpy specimens.     

Algebraic Relationship Between Symmetrical and Asymmetrical Capability Indices

There have been many investigations on the capability indices C_p, C_pk, C_pm, and C_pmk, for the common situation in which the target is the midpoint of the tolerance interval. However, only a few investigations deal with the specific case of asymmetrical tolerances. In that particular case, a number of symmetrical and asymmetrical indices are put forward, but there is no full literature treatment or synthesis showing the similarity between those indices and the common ones. We intend here, to demonstrate that the algebraic links between the indices C_p, C_pk, C_pm, and C_pmk, are similar to the ones which relate the symmetrical indices proposed in the case of asymmetrical tolerances. In that case, the algebraic structure allows us to propose asymmetrical indices families. An example based on a pharmaceutical filling operation is used to illustrate the application.     

Inference of critical cod from Charpy V test result

Correlation between the Charpy absorbed energy and critical COD is investigated to obtain a useful method for estimating critical COD from Charpy V data. The round bar tension test, Charpy V-notch test and static 3-point bend test with fatigue notched specimen are carried out using mild steel, 785 MPa grade high strength steel and A5083 aluminum alloy. Correlation is found between W'_c/σ_Y and δ_c as well as between EW'_c/σ_Y² and Eδ_c/σ_Y, where W'_c is the Charpy absorbed energy obtained by considering temperature difference between the Charpy transition temperature and COD transition temperature. The symbols σ_Y, δ_c and E are yield strength, critical COD and Young's modulus, respectively. The correlations are established for various kinds of metals and over a wide temperature range including not only upper shelf range but also the transition range.     

A path-independent integral for 2-dimensional cracks in homogeneous isotropic conductive plate

A path-independent integral which is denoted by j_e, is introduced for the 2-dimensional crack problems in the homogeneous isotropic conductor in which the steady current flows. By utilizing the j_e-integral the distributions of the electric potential, current density and the Joule heating rate near the crack tip are derived. It is shown that the j_e-integral provides a parameter which dominates the distributions of the electric potential and current density near the crack tip as the square of the amplitude.     

Similarity solutions of a strong shock wave propagation in a mixture of a gas and dusty particles

The similarity solutions of a strong shock wave propagation in a mixture of a gas and small solid particles have been investigated. Similarity solution exists only when the shock is very strong and the surrounding medium is of a constant density and at rest and with negligible counterpressure. The non-dimensional fundamental equations are derived and studied. The results depend on three non-dimensional parameters; i.e. (i) the ratio of the specific heats of the gas γ, (ii) the mass concentration of the solid particles k_p in the mixture and (iii) the ratio of the density of the solid to that of initial density of the gas G. Numerical solutions for various values of γ k_p and G are presented and discussed. The speeds of the shock wave front and its location with various energy releases are given.     

An analysis of crack extension in creeping solids

A theoretical analysis for the crack growth under creep conditions in ductile materials has been carried out based on the stress distribution in the ligament section and the plastic deformation around the crack zone. The two extreme cases namely: (a) large creep deformation over the entire ligament and; (b) negligible creep deformation even near the crack tip, are considered based on the model proposed. The resulting expressions are compared with the experimental data carried out on a 6061 Al alloy in the temperature range of 0.55–0.65 T_m.     

Calculation of the ferrite volume in some dual phase steels

The relation between the γ/γ + boundary temperature, T, and the equivalent values of [Cr] and [Ni], as well as the variation of the ferrite volume, V_f, with the temperature in + γ dual-phase steels have been studied. With the aid of a computer, the regressive expressions derived from the experimental results are: T (°C) = T₃ + 21.2 [Cr] − 15.8 [Ni] + 223; V_f (%) = 0.715 exp [0.015(T − T_δ)] − exp[0.015(T_c − T_δ)] + 1.85 exp [0.0083(T − T_c]).     

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.     

Effect of Testing Normality on Estimating Process Capability Indices

Process capability indices are used to measure whether a manufacturing process meets the specifications. The studies of these indices are usually based on the assumption that the process follows a normal distribution. These include the indices C_p, C_pk, C_pm, and C_pmk. When the investigator is uncertain whether the process follows a normal distribution, a test of normality may be used to resolve the uncertainty. If the test accepts the null hypothesis that the process follows a normal distribution, the investigator uses C_p, C_pk, C_pm, or C_pmk. If the test rejects the null hypothesis, the investigator uses indices under non-normal distributions. Therefore the test of normality is a preliminary test that determines the form of the distribution and the index to use. In this paper we study the effect of the preliminary test of normality on the estimation of the four process capability indices mentioned above.     

Magneto-mechanical properties evolution of Fe–C alloy during precipitation process

Evolution of magneto-mechanical properties of 160 ppm Fe–C alloy due to carbides precipitation during isothermal annealing at 473 K (up to t=50×10³ s) was studied by means of classical Barkhausen noise (HBN) and mechanical Barkhausen noise (MBN) effects. The MBN was measured for the torsion mode of load with a torque motor. Also the B(H) hysteresis loop and the coercive field H_c were evaluated using a low-frequency magnetisation set. Magnetic hysteresis losses ΔW₁ were compared with the integral ΔW₂ of HBN intensity over one period of magnetisation and the integral ΔW₃ of MBN intensity over one period of the mechanical load. The internal stress distribution function and the resulting mean level of internal stress parameter σ_i were evaluated from the MBN ‘first load’ data. It was revealed that a correlated increase of ΔW₁ and ΔW₂ parameters exists. However, the relative increase of ΔW₃ is much lower than the relative increase of ΔW₁. The relationship between H_c and σ_i was found to be parabolic. This dependence explained by Néel’s model of the impact of the residual stress level on H_c. The presence of precipitates of type was confirmed by scanning electron microscopy.     

The effect of pipe bends on the stability of a circumferential through-wall crack in a type 304 stainless steel piping system

The paper analyses simple models that simulate the effect of pipe bends on the stability of a circumferential through-wall crack in a Type 304 stainless steel piping system that is subject to fixed displacement loadings, which are appropriate for an accident condition. The instability criterion is expressed in terms of the applied and material tearing moduli, i.e. T_APP and T_MAT or equivalently in terms of an effective pipe length L_EFF. By expressing T_APP (or L_EFF) as a function of the position of the cracked section in the piping system, it is possible to specify the positions where T_APP is a maximum, and identify them in relation to the pipe bends.     

Relaxation of residual stress with fatigue loading

Investigations have been carried out to study the relaxation of the surface residual stress in 0.23% C steel due to the application of fatigue loading. The residual stress was induced in the specimen by pre-straining and was measured by X-ray back reflection method using Cr-K radiation. The surface residual stress induced, depends on the plastic strain and appears to bear a relation of the type σ_R = σ_o(e_p)^0.78. The decay of the residual stress appears to depend on log N, given by the relation σ_R1 = σ_RO−K log N, where N is the number of fatigue cycles. The constant K depends on the initial value of the residual stress.     

Comparison of fracture toughness test procedures for aluminum alloys

The Dynamic Tear (DT) test permits the measurement of fracture propagation energy across the toughness spectrum for metals which are definable by linear elastic analyses to those requiring gross plastic strains for fracture. The linear elastic fracture mechanics parameter K_ic provides a relationship between critical flaw size and stress level at which crack instability will occur. Unlike the DT test, the K_ic toughness test cannot be utilized for fracture under conditions of elastic-plastic or gross plastic strain.

A correlation has been developed between the DT test and the K_IC parameter for ahuminum alloys. The relationship may also be expressed in terms of β_ic-DT and _ic-DT. The K_ic values were determined with several specimen types and a comparison of the values for different specimens is provided.

The correspondence between K_ic and DT serves several purposes. It provides a frame of reference for DT values obtained from frangible metals that fracture under linear elastic conditions. Accordingly, it permits utilization of the inexpensive DT test to approximate the flaw size-stress instability conditions which otherwise must be determined by the more expensive K_ic test. Furthermore, through extrapolation, it is possible to utilize the DT test to estimate the critical flaw size under an elastic-plastic strain field.     

Statistical investigation of surface fatigue cracks in large-sized turbine rotor shaft steel

Distribution properties of an initiation life N_i and a propagation life N_p of surface cracks, statistical characteristics of a crack growth rate dl/dN, and a relationship between a scatter of the distributions and a gradient a of S-log N curves in rotating bending fatigue tests were investigated for Ni-Cr-Mo-V steel, using for a large-sized turbine rotor shaft. The distributions of N_i and N_p were expressed as Weibull distributions, and the scatter of them for smooth specimens and for lower stress amplitude σ_a tests were larger than those for notched ones and for higher σ_a tests, respectively. The statistical properties of crack propagation rate were almost similar in both smooth and notched specimens. The relationship between the a and a coefficient of variation η for the distributions of N_i, N_p and a final fracture life N_f was expressed as η = c(a)^−b, where c and b are constants.     

Evaluating Process Capability for Geometrically Toleranced Parts: A Practical Approach

The practice of geometric tolerancing has gained in industry popularity since the 1990s. This approach has advantages over conventional tolerancing in defining both geometry and associated tolerance and, thus, generating a more realistic “acceptable design space.” There have been a number of highly mathematical treatments of the subject over the years which have not found their way into popular usage. We look at a specific example of geometric tolerancing and derive a simple approach based on the geometry of the situation and standard C_p and C_pk calculations. The study describes an approach based on understanding the limiting conditions of acceptable operation. In the example of a pin and clearance hole, we derive the limiting condition as a zero radial gap for a hole and a perfectly centered pin at maximum metal condition. We then performed a standard C_pk calculation with the limiting condition acting as the effective tolerance limit. Because we are dealing with radial gaps, we have an effective one-sided tolerance with a minimum acceptable value of 0 for the radial gaps. We tested the approach using Normally distributed simulated data and found that it provides an accurate evaluation of process capability and projected scrap levels. With minor cautions, we conclude that this methodological approach could be extrapolated to other geometrically toleranced situations.