Stress concentration formulae useful for any shape of notch in a round test specimen under tension and under bending

In this work stress concentration factors, K_t , for a round bar with a circular-arc or V-shaped notch are considered on the basis of exact solutions for special cases and accurate numerical results. Then, a set of K_t formulae useful for any shape of notch is proposed. The conclusions can be summarized as follows. (i) For the limiting cases of deep (d) and shallow (s) notches, the body force method is used to calculate the K_t values. Then, the formulae are obtained as K_td and K_ts . (ii) On the one hand, upon comparison of K_t and K_td it is found that K_t is nearly equal to K_td if the notch is deep or blunt. (iii) On the other hand, if the notch is sharp or shallow, K_t is mainly controlled by K_ts and the notch depth. (iv) The notch shape is classified into several groups according to the notch radius and notch depth. Then, the least-squares method is applied for the calculation of K_t /K_td and K_t /K_ts . (v) Finally, a set of convenient formulae is proposed that are useful for any shape of notch in a round test specimen. The formulae give SCFs with <1% error for any shape of notch.     

Energy density zone model and fatigue life prediction considering microscopic effects

An energy density zone (EDZ) model is developed for the prediction of fatigue life. The microscopic effects can be involved in the EDZ model. Three scale transitional functions in the model are utilized to describe the trans‐scale behaviours of fatigue failure from micro‐scale to macro‐scale. Fatigue failure behaviours of a low‐alloy and ultra‐high‐strength steel material (i.e. 40CrNi2Si2MoVA steel) is investigated. Two fatigue parameters in the model are determined from the experimental S–N curves for the smooth cylinder specimens (the stress concentration factor, SCF, K_t = 1). Then, fatigue lives of notched specimens with SCFs K_t = 2 and K_t = 3 are predicted respectively by the proposed model. The predicted S–N curves are satisfactory in comparison with the experimental results. Scatter of the fatigue test data can be depicted when the microscopic effects are considered. Influences of microscopic effects on the fatigue behaviours are explored by means of numerical simulations.     

Solution of Certain Problems in the Thermal Diffusion of Binary Gases

Two variants of the experimental method of unique determination of the thermal-diffusion ratio (K _t) of binary gases are described. A formula expressing the temperature and concentration dependence of K _t and obtained by a thermodynamic method is proposed.     

Measurement of mode I and mode II rock dynamic fracture toughness with cracked straight through flattened Brazilian disc impacted by split Hopkinson pressure bar

In order to find an effective and convincing method to measure rock dynamic fracture toughness for mode I and mode II, cracked straight through flattened Brazilian disc specimens of marble, which were geometrically similar for three size, were diametrically impacted by split Hopkinson pressure bar on the flat end of the specimen with three load angle respectively. History of stress intensity factors (K_I(t) for opening mode I, and K_II(t) for sliding mode II), mode mixture ratio (K_I(t)/K_II(t)), as well as mode I and mode II dynamic fracture toughness at crack initiation (K_Id and K_IId) were determined with the experimental–numerical method. It is found that there is a unique size effect for dynamic fracture test with the specimens, the mode mixture ratio is not solely determined by load angle (the angle between load direction and crack line) as in the static loading; the pure mode II load angle is 19° for the ?50 mm specimen, however it is 10° for the ?130 mm and ?200 mm specimens; the mode II load angle decreases with increment of specimen size. Realization of pure mode II is justified by the mode mixture ratio approaching zero, it can be realized under certain load angle and loading rate for the specimen of specified size. K_IId is generally greater than K_Id. Both K_Id and K_IId increase with increment of specimen size, and this trend for K_IId is more remarkable than that for K_Id.     

Notched fatigue testing of Inconel 718 prepared by selective laser melting

Selective laser melting (SLM) was used to prepare notched high‐cycle fatigue test specimens made from nickel‐based superalloy Inconel 718. Samples were designed to have 1 of 3 different notch geometries, including V notches with K_t of 2.2 or 3.1, a U notch with K_t of 2.0, and were printed in either vertical or horizontal orientations. Samples were tested with as‐printed dimensions and surfaces after heat treatment, but a separate set of SLM samples were printed as plates and machined to final dimensions comporting to the V‐notch specimen with K_t = 3.1. High‐cycle fatigue testing showed that machined SLM specimens behaved similar to wrought Inconel 718 plate specimens, but testing with as‐produced surfaces led to a decrease in fatigue life. The explanation for this difference is based on approximations of linear elastic fracture mechanics solutions for short cracks emanating from notch roots, with intrinsic surface features of SLM materials serving as the cracks. Analysis of the actual notch geometries after SLM fabrication indicates that stress intensity in the presence of these features plays a prominent role in determining number of cycles before fatigue crack initiation and propagation occurs.     

Effect of small scale notches on the very high cycle fatigue of AISI 310 stainless steel

Fatigue behaviour of AISI 310 stainless steel has been investigated up to very high cycles. The fatigue crack initiation sites were found at the surface of the material. Persistent slip bands developed at the surface of the specimens led to the crack initiation. At lower stress levels, shallow persistent slip bands were found at the surface of the specimens, and the fatigue limit was obtained. Notched specimens showed lower fatigue lives. Notched specimens with higher stress concentration factor (K_t) showed higher fatigue strength reduction factor (K_f). It was found that shallow notches of depth ~100 µm may reduce the fatigue life substantially.     

Three‐dimensional stress concentrations at circular pin holes in clearance‐fit lugs

The influences of thickness and bonding clearance on stress concentration factors (SCFs) at circular holes in pin‐loaded straight lugs are systematically investigated using the finite element method. The three‐dimensional effect on stress concentration at pin hole is strong when the thickness B of lug is higher than the radius R of pin hole. The maximum tensional SCF K_max normalized by its corresponding plane stress solution K_p–σ increases with increasing B/R when B/R is higher than 2 for all of r/R (the radius of pin to that of lug), and also increases with decreasing r/R for a given B/R. It is also found that the plane stress SCF K_p–σ nearly remains a constant when r/R < 0.98, but is strongly sensitive to r/R and increases by 30% with r/R changing from 0.98 to 1. On the other hand, the friction coefficient, Young's modulus and the load level have also influences on stress concentrations, which should not be neglected in design of structures. An empirical formula of the maximum SCF is obtained for convenience of engineering applications.     

A simple method to analyse the notch sensitivity of specimens in fatigue tests

A simple method to analyse the notch sensitivity of specimens in fatigue tests is presented. The parameter m, which can be used to measure the notch sensitivity, the nominal stress and the stress concentration factor (K_t) are used to establish the method. In order to verify the feasibility of the method, notch fatigue test results from our group and literatures were collected. The results reveal that an optimal value of parameter m does exist for each material. Life predictions indicated that the model is able to describe the life evolution for notched specimens under high cycle fatigue and low cycle fatigue tests. Because the geometry effect is accounted for K_t, the method is suitable for the conditions when the notch geometries and the absolute dimensions are similar to the tested specimens.     

Effects of test procedures and lithology on estimating the mode I fracture toughness of rocks using empirical relations

Empirical estimation is a common method for getting mode I fracture toughness K_IC of rock. By collecting data from tests in this study and literature, 204 sets of K_IC and tensile strength σ_t test data are obtained for new empirical K_IC‐σ_t relations regression. The empirical relations make the estimation of K_IC values from σ_t conveniently, but test procedures and lithology will influence its reasonability and reliability. Results indicate that the empirical K_IC‐σ_t relations obtained from the four different suggested K_IC test methods are all in good but obviously different linear relationship. The analyses show that cracked chevron notch Brazilian disc specimen (CCNBD) test‐based empirical relation is more accurate for estimating K_IC than the other three test‐based empirical relations. As to different lithology, isotropic rocks such as sandstone and carbonatite may be more appropriate for the application of empirical estimation method. However, for coarse grained or anisotropic rocks such as granite and marble, estimation method should be applied carefully because of possibly weak K_IC‐σ_t relations.     

Evaluation of dynamic fracture toughness KId by Hopkinson pressure bar loaded instrumented Charpy impact test

A novel method for measuring the dynamic fracture toughness, K_Id, using a Hopkinson pressure bar loaded instrumented Charpy impact test is presented in this paper. The stress intensity factor dynamic response curve (K_I(t)−t) for a fatigue-precracked Charpy specimen is evaluated by means of an approximate formula. The onset time of crack initiation is experimentally detected using the strain gauge method. The value of K_Id is determined from the critical dynamic stress intensity factor at crack initiation. A K_Id value for a high-strength steel is obtained using this method at a stress-intensity-factor rate () greater than 10⁶ MPa .     

Poling process and piezoelectric properties of lead zirconate titanate/sulphoaluminate cement composites

The sulphoaluminate cement was, for the first time, used to fabricate 0-3 piezoelectric composites by compressing technique. The dependences of piezoelectric properties of composites on poling conditions, particle size and contents of Lead Zirconate Titanate (PZT) were discussed especially. The results show that the piezoelectric properties of composites are improved by increasing poling field E, poling time t and poling temperature t. The optimum E, t and T are 4.0 kV/mm, 45 min and 120°C respectively in this work. The piezoelectric strain factor d₃₃ is found to increase with increasing particle size of PZT. When the particle size of PZT is larger than about 130 m, the d₃₃ is nearly independent of particle size. The d₃₃, g₃₃ and the electromechanical coupling coefficient K _p, K _t of composites increase rapidly with increasing content of PZT. When the content of PZT reaches 85%, K _p and K _t are 28.54 and 28.19%, respectively.     

High cycle fatigue behaviour of microsphere Al2O3–Al particulate metal matrix composites

The high-cycle stress-life (S–N) curve and fatigue crack growth threshold (ΔK_th) behaviour of COMRAL-85^TM, a 6061 aluminium–magnesium–silicon alloy reinforced with 20 vol.% Al₂O₃-based polycrystalline ceramic microspheres, and manufactured by a liquid metallurgy route, have been investigated for a stress ratio of R = −1 (fully reversed loading). Fatigue testing was conducted on both smooth round bar (S–N) specimens and notched round bar (fatigue threshold) specimens. Unreinforced Al 6061-T6 also processed by a liquid metallurgy route and six powder metallurgy processed composites with particle volume fractions ranging between 5% and 30% were also studied. S–N data revealed that the powder metallurgy processed composites generally gave longer fatigue lives than the matrix alloy, whereas COMRAL-85^TM exhibited a reduced fatigue life. The fatigue threshold results were very similar for all the composites, being lower than for Al 6061-T6. Fatigue failure mechanisms were determined from examination of the fracture surfaces and the crack profiles.     

SHEAR LIPS ON FATIGUE FRACTURES OF ALUMINIUM ALLOY SHEETS SUBJECTED TO BIAXIAL CYCLIC LOADS AT VARIOUS R-RATIOS

An analysis is made of shear lip measurements as observed on fatigue crack surfaces of an aluminium alloy sheet material. It is shown that, for biaxial cyclic loads at stress ratios 1 and various R-ratios, shear lips were controlled by an equivalent intensity factor K_e=K₁F(λ,R). For crack growth in air the shear lip width t_s was approximately proportional to K²_e at t_s < t*_s= 2.5 × 10^?4 m and K⁴_E, at t_s > t*_s where t*_s is the critical value of the shear lip dimension. The initiation of shear lips and the orientation of crack growth for different parameters of the loading cycle are briefly considered in the discussion.     

An integrated local energy density approach to notch low cycle fatigue life prediction

An approach to notch low cycle fatigue (LCF) life prediction based on hysteresis loop energy is described. The method employs an elastic-plastic finite element model of a notched component and a supplemental post-processor to calculate, order and summate elemental energies. These energies, combined with smooth bar LCF data, are used to predict notch LCF life. Predictions using the method are compared to test results on circumferentially notched René 80 specimens tested at 760 and 982°C with stress concentration factors (K_t) of 3.49 and 2.22. Excellent correlation was obtained for K_t = 3.49 at 982°C when a critical notch root volume equal to the notch root plastic zone was assumed. Correlation at 760°C, K_t = 3.49 and 982°C, K_t = 2.22 was not as good. Limitations due to analytic assumptions and micro-mechanistic features as determined by optical, scanning and transmission electron microscopy are discussed and recommendations for further research indicated.     

破损铝电解槽阴极钢棒的冲击回波法检测

电解槽的寿命及修复关系着整个电解铝行业的经济效益,是不容忽视的技术问题。为能对破损电解槽进行及时修复从而延长电解槽寿命,提出应用冲击回波法对电解槽破损状况进行评估。首先利用数值模拟方法探讨了冲击回波激振源特性对应力波脉冲宽度及应力峰值的影响,得出激振源直径D与脉冲宽度tc呈线性相关：tc=0.004D,且与应力峰值(P)呈指数相关P=2e~8D^{1.928 8},而冲击速度v不改变脉冲宽度tc,但与应力峰值P呈线性相关：P=0.0751v-0.068 6。进而基于数值分析结果,运用冲击回波法检测破损电解槽中的阴极钢棒,通过对检测结果的时频域分析,得出了阴极钢棒的破损状况及位置,并与刨槽后真实的阴极钢棒对比。结果表明：冲击回波法能很好地实现电解槽破损部位的定位,可为电解槽寿命评估与修复提供参考。     

Fatigue strength of carbon steel specimens having an extremely shallow notch

In order to elucidate how the principal dimensions of a notch as small as grains affect the fatigue strength of carbon steel, rotating bending fatigue tests were carried out on annealed 0.45% C steel specimens having an extremely small artificial notch whose depth is either 5 or 10μm. The fatigue processes at notched parts were observed successively.In these extremely shallow notches, the parameter controlling fatigue limits for fracture σ_w was the notch depth, independently of notch sharpness. On the other hand, the crack initiation limits in the notches σ_W1 were determined by a single K_tσ_W1 — χ relation obtained from the tests on ordinary notches, where K_t is the stress concentration factor and χ is the stress gradient at the notch root.These results were studied on the basis of the characteristics of crack initiation process, the stress intensity variations at the crack tip and the stress distribution near the notch root.     

Standard bases,critical tropisms and flatness

Using the standard basis notion, we adapt to the affine algebraic geometry the notion of the critical tropisms of Lejeune-Teissier and we give some effective criteria for testing geometrical properties for the generic projectionX=SpecK[t₁,...,t_m][x₁,...,x_n]/I y=SpecK[t₁,...,t_m].     

Evaluation by indentation of fracture toughness of ceramic materials

A transition fracture mode from Palmqvist to median has been observed in a number of ceramic materials. A new expression to determine the fracture toughness (K _IC) by indentation is presented. The K _IC values calculated by this formula are independent of the crack profile (median or Palmqvist) and of the applied load. This formula has been obtained by modifying the universal curve of Evans and Charles to incorporate Palmqvist and median cracks over a wide range of loads in the case of brittle materials with different mechanical properties (elastic properties: E, v, K _IC).     

State‐of‐the‐art review on extended stress intensity factor concepts

The stress intensity factor concept for describing the stress field at pointed crack or slit tips is well known from fracture mechanics. It has been substantially extended since Williams' basic contribution (1952) on stress fields at angular corners. One extension refers to pointed V‐notches with stress intensities depending on the notch opening angle. The loading‐mode‐related simple notch stress intensity factors K₁, K₂ and K₃ are introduced. Another extension refers to rounded notches with crack shape or V‐notch shape in two variants: parabolic, elliptic or hyperbolic notches (‘blunt notches’) on the one hand and root hole notches (‘keyholes’ when considering crack shapes) on the other hand. Here, the loading‐mode‐related generalised notch stress intensity factors K_1ρ, K_2ρ and K_3ρ are defined. The concepts of elastic stress intensity factor, notch stress intensity factor and generalised notch stress intensity factor are extended into the range of elastic–plastic (work‐hardening) or perfectly plastic notch tip or notch root behaviour. Here, the plastic notch stress intensity factors K_1p, K_2p and K_3p are of relevance. The elastic notch stress intensity factors are used to describe the fatigue strength of fillet‐welded attachment joints. The fracture toughness of brittle materials may also be evaluated on this basis. The plastic notch stress intensity factors characterise the stress and strain field at pointed V‐notch tips. A new version of the Neuber rule accounting for the influence of the notch opening angle is presented.     

Berechnung der Dauerschwingfestigkeit unter Berücksichtigung der spannungsmechanischen und statistischen Stützziffer

Fatigue Limit and Geometrical and Statistical Size Effect The ratio of the fatigue limit of an unnotched specimen to that of a notched one, the fatigue notch factor K_f, is usually smaller than the theoretical stress concentration factor K_t. With the assumption of a plastic cyclic deformation ?_apW at the level of the fatigue limit the fatigue limit for a notched specimen can be calculated. According to formula (4) this fatigue limit σ_naD is a function of K_t, ?_apW, the notch stress amplitude σ_a, the cyclic strengthening exponent n′ and the fatigue limit of a smooth specimen σ_W. Moreover, taking into account the statistical size effect with the known “weakest link concept”, see flow chart Fig. 9, the calculation is in a good agreement with 77 test results for steel.