A finite fracture mechanics approach to structures with sharp V-notches

Criteria assuming that failure of quasi-brittle materials is affected by the stresses acting over a finite distance from the crack tip are widely used inside the scientific community. For instance, they have been applied to predict the failure load of specimens containing sharp V-notches, assuming as a critical parameter the average stress ahead the notch tip. However, this kind of approaches disregards energy balance considerations, which, as well known, are the basis of linear elastic fracture mechanics (LEFM). In order to overcome these drawbacks, the present paper uses a recently introduced finite fracture mechanics (FFM) criterion, i.e. a fracture criterion assuming that crack grows by finite steps. The length of this finite extension is determined by a condition of consistency of both energy and stress requirements; as a consequence, the crack advancement is not a material constant but a structural parameter. The criterion is applied to structures with sharp V-notches. The expression of the generalized fracture toughness, which is a function of material tensile strength, fracture toughness and notch opening angle, is given analytically. Finally, we provide comparisons with: (i) the experimental data we obtained from testing Polystyrene specimens under three point bending; (ii) some experimental data available in the literature. The agreement between theoretical predictions and experimental results is generally satisfactory and, for most of the cases analyzed, the FFM predictions are better than the ones provided by the simple average stress approach.     

Data-driven identification for approximate analytical solution of first-passage problem

The first-passage problem plays a significant role in engineering performance evaluation and design optimization. To address general stochastic dynamical systems, a data-driven method is proposed to identify approximate analytical solutions for the first-passage problem which explicitly includes parameters of the system, excitation, and those related to the initial and boundary conditions. The method consists of two successive processes. First, the probability density of the first-passage time is assumed to satisfy the modified Weibull distribution and its expansion expression is constructed by using the rule of dimensional consistency. Second, by comparing the expansion with the probability density of the first-passage time estimated from random state data, the coefficients are determined by solving a set of overdetermined linear algebraic equations. Two representative examples, including the Duffing oscillator and a 2-DOF nonlinear dynamical system, are discussed in detail to illustrate the application and efficiency of the data-driven method. The efficacies of the approximate analytical solutions for the external parameters are also verified.     

Fictitious notch rounding concept applied to sharp V-notches: Evaluation of the microstructural support factor for different failure hypotheses. Part I: Basic stress equations

With the aim to perform a comprehensive and accurate evaluation of the microstructural support factor of sharp V-notches (Neuber’s notch rounding concept), in Part I of this contribution, the indispensable theoretical tools, especially the basic stress equations, are reconsidered and amended in respect of accuracy of results. First, the analytical solution derived by Neuber [Neuber H. Kerbspannungslehre. 2nd ed. Berlin: Springer-Verlag; 1958] for sharp rounded V-notches with an arbitrary flank angle under tension loading is considered. The equation of the normal stress has been obtained with the restriction to the notch bisector. Using the Airy stress function suggested by Neuber, this solution is extended to the region outside the notch bisector, and the complete stress field is derived in this manner. A comparison between Neuber’s solution, a more recent solution due to Filippi et al. [Filippi S, Lazzarin P, Tovo R. Developments of some explicit formulas useful to describe elastic stress fields ahead of notches in plates. Int J Solids Struct 2002;39:4543-65] and highly accurate FE results is performed. Filippi’s equations which include Williams’ solution [Williams ML. Stress singularities resulting from various boundary conditions in angular corners on plates in tension. J Appl Mech 1952;19:526-8] for pointed V-notches, are shown to be superior.     

A numerical study of the elastic stress field arising from sharp and blunt V-notches in a SENT-specimen

The elastic stress field arising from a V-notch in a Single Edge Notched Tension (SENT) specimen is studied using the Sherman–Lauricella integral equation. Accurate values of the generalized stress intensity factor, here denoted Q _I, are determined and compared with values from the literature. A Q _I-dominated distance ahead of the notch tip is defined and determined for several notch angles and different notch depth to specimen width ratios. It is found that for depth to width ratios over 0.45 the Q _I-dominated distance is approximately constant while below this value the distance can show a great variation with the notch angle. Additionally, a general formula for the maximum stress in a V-notch with a finite root radius is derived. Its applicability on the SENT-specimen is studied and presented as a family of curves. If used within these curves, the formula can accurately predict stress concentrations, even for very smooth notches.     

Use of J-integral to predict static failures in sharp V-notches and rounded U-notches

In the present paper, the physical meaning of J_V (namely, the classic J-integral applied to either sharp V-notch) is discussed. Consider a Cartesian reference frame having the x-axis parallel to the notch bisector, each mode of J_V, for a given circular path, is proportional to the correspondent mode of the classic J-integral of a virtual crack having length equal to the path radius and emanating from the tip of the V-notch. Analytical and numerical results have been performed for linear elastic materials. Additionally, in order to verify the formulations of J_V, experimental result of embedded cracks of sharp V-notch was considered.Then, by introducing a characteristic path radius ρ^∗, assumed to be dependent only on the material properties, the J_V parameter was used for the estimation of the static failure load of sharp V-notches specimens under mode I loading.Furthermore, the J_Vρ parameter (namely, the classic J-integral applied to U-rounded notches) was used to analyze the static failure of two new series of specimens with double U-notches made of brittle material (PMMA and PVC glass) subjected to tensile loading. This method allowed us to prove that when the ratio between the notch tip radius and ρ^∗ is small the approach agrees with the classic J-integral, whereas when ρ^∗ becomes small with respect to the notch tip radius, the J_Vρ method agrees with the classic peak stress approach.     

An approximate analytical 2D-solution for the stresses and strains in eigenstrained cubic materials

Summary Continuous and discrete Fourier transforms (CFT and DFT, respectively) are used to derive a formal solution for the Fourier transforms of stresses and strains that develop in elastically homogeneous but arbitrarily eigenstrained linear-elastic bodies. The solution is then specialized to the case of a dilatorically eigenstrained cylindrical region in an infinite matrix, both of which are made of the same cubic material with the same orientation of principal axes. In the continuous case all integrations necessary for the inverse Fourier transformation can be carried out explicitly provided the material is slightly cubic. This results in an approximate but analytical expression for the stresses and strains in physical space. Moreover, the stress-strain fields inside of the inclusion prove to be of the Eshelby type, i.e., they are homogeneous and isotropic. The range of validity of the analytical solution is assessed numerically by means of discrete Fourier transforms (DFT). It is demonstrated that even for strongly cubic materials the stresses and strains are quite well represented by the aforementioned approximate solution. Moreover, the total elastic energy of two eigenstrained cylindrical inclusions in slightly cubic material with the same orientation of their principal axes is calculated analytically by means of CFT. The minimum of the energy is determined as a function of the relative position of the two inclusions with respect to the crystal axes and it is used to explain the formation of textures in cubic materials.     

An approximate approach for the calculation ofM s in iron-base alloys

Having estimated the critical driving force associated with martensitic transformation,ΔG ^α→M, as $$\Delta G^{\alpha \to M} = 2.1 \sigma + 900$$ whereσ is the yield strength of austenite atM _s, in MN m^?2, we can directly deduce theM _s by the following equation: $$\Delta G^{\gamma \to {\rm M}} |_{M_S } = \Delta G^{\gamma \to \alpha } + \Delta G^{\alpha \to M} = 0.$$ The calculatedM _s are in good agreement with the experimental results in Fe-C, Fe-Ni-C and Fe-Cr-C, and are consistent with part of the data in Fe-Ni, Fe-Cr and Fe-Mn alloys. Some higher “M _s” determined in previous works may be identified asM _a,M _s of surface martensite or bainitic temperature. TheM _s of pure iron is about 800 K. TheM _s in Fe-C can be approximately expressed as $$M_S (^\circ {\text{C}}) = 520 {\text{--- }}\left[ {{\text{\% C}}} \right]{\text{ }}x 320.$$ In Fe-X, the effect of the alloying element onM _s depends on its effect onT ₀ and on the strengthening of austenite. An approach for calculation of ΔG ^γ→α in Fe-X-C is suggested. Thus dM _s/dx _c in Fe-X-C is found to increase with the decrease of the activity coefficient of carbon in austenite.     

An analytical approach for free vibration and transient response of functionally graded piezoelectric cylindrical panels subjected to impulsive loads

In this article, the free vibration and dynamic response of simply supported functionally graded piezoelectric cylindrical panel impacted by time-dependent blast pulses are analytically investigated. Using Hamilton’s principle, the equations of motion based on the first-order shear deformation theory are derived. Also, Maxwell’s electricity equation is taken as one of the governing equations. Three sets of electric surface conditions including closed circuit and two mixtures of closed and open circuit surface conditions are considered. By introducing an analytical approach and using the Fourier series expansions, the Laplace transform and Laplace inverse method, the solution of unknown variables are obtained in the real time domain based on a combination of system frequencies. Finally, the effects of various electric surface conditions, geometric parameters and the material power law index on the free vibration and transient response of functionally graded piezoelectric cylindrical panels subjected to various impulsive loads are examined in detail.     

An approximate dynamic programming approach for a product distribution problem

This paper proposes an approximate dynamic programming-based method for optimizing the distribution operations of a company manufacturing a certain product in numerous plants and distributing it to different regional markets for sale. The production at each plant and in each time period follows a nonstationary stochastic process. The product can be stored at the plants or be shipped to a regional market to serve the random demand. The proposed solution method formulates the problem as a dynamic program and uses approximations of the value function. We develop a tractable procedure to compute the marginal worth of a unit of inventory at a production plant. This quantity is used to update the value function approximations in an iterative improvement scheme. We numerically show that our method yields high quality solutions.     

Fictitious notch rounding concept applied to sharp V-notches: Evaluation of the microstructural support factor for different failure hypotheses: Part II: Microstructural support analysis

On the basis of the comprehensive and accurate stress field equations for sharp rounded V-notches derived in Part I of this contribution, the microstructural support factor of these notches is determined which quantifies the fictitious notch radius in Neuber’s elastic microstructural support concept. By means of Filippi’s equations and considering different failure criteria (Rankine, von Mises and Beltrami) the fictitious notch radius is evaluated for different notch opening angles as a function of the actual notch radius and the microstructural support length. Plane stress and, alternatively, plane strain conditions are introduced. Once the fictitious radius has been found, the support factor s is derived from the expression: fictitious notch radius minus actual notch radius divided by microstructural support length. The support factor s is found to be very sensitive to the notch opening angle, but constant ‘plateau values’ are determined for an actual radius greater than the microstructural support length. The dependence of s on the failure criterion and the multiaxiality conditions (plane stress or plane strain) is also investigated. Various numerical analyses using the FE method have been carried out to compare the theoretical stress concentration factor to the effective stress concentration factor, the former obtained by considering fictitiously rounded notches under tension loading using the plateau values of s, the latter obtained by integrating the relevant stress over the microstructural support length along the bisector of the pointed V-notch. Finally, dealing with out-of-plane shear loading, Neuber’s corresponding solution valid for sharp rounded notches is re-evaluated and the numerical analysis described above is extended to this loading case. All the comparisons above are preceded by elementary solutions for pointed notches in general. It is shown that the plateau values of s are well suited for engineering usage in structural strength assessments.     

一种高速列车引起的地基振动解析解

基于波函数展开法表示柱面波源自由场,首次提出了一种求解柱面波源下管桩屏障对弹性波隔振效应的解析方法。方法首先考虑柱面波源与管桩屏障的位置关系,采用波函数展开法对入射柱面波进行了0阶Hankel展开表示,并采用任意坐标系间变换的Graf加法定理在任意桩体坐标系中表示,将入射波场和散射波场叠加后通过满足所有桩体的边界条件以求解所有桩体的散射波场。该解析方法分析了柱面波源入射下群桩屏障的隔振效果,首次在排桩隔振问题中考虑了入射波曲率的影响,为柱面波的散射问题提供了理论解答。该文重点讨论了入射波曲率、管桩个数和桩排数等因素对管桩隔振效果的影响,结果表明：整体上管桩屏障对柱面P波的隔振效果优于柱面SV波;柱面波源距离管桩屏障更近时排桩后场地的位移反应显著增大;相比平面波,柱面波源作用下排桩数量的提升对隔振效果的影响较小;三排管桩屏障比两排管桩屏障隔振效果更强,更宜采用三排管桩屏障进行柱面波源的隔振。     

An implicit gradient application to fatigue of sharp notches and weldments

This paper addresses the problem of stress singularities at the tip of sharp V-notches by means of a non-local implicit gradient approach. A non-local equivalent stress is defined as a weighted average of a local stress scalar quantity computed on the assumption of linear elastic material behaviour. In the case of a crack, we propose an analytical solution for the non-local equivalent stress at the crack tip when the local equivalent stress assumes the analytical form proposed by Irwin. For open notches, several numerical procedures are possible.For welded joints, we assume that the material obeys a linear elastic constitutive law. In this case, the non-local equivalent stress obtained from the implicit gradient approach is assumed as the effective stress for assessments of joint fatigue. Using the principal stress as local equivalent stress and a notch tip or weld toe radius equal to zero, we analyse many series of arc welded joints made of steel and subjected to either tensile or bending loading, and we propose a unifying fatigue scatter band. If the welded joints are subjected only to mode I loading, an analytical relationship between the relevant Notch Stress Intensity Factors (NSIF) of mode I and the effective stress is established; otherwise, the effective stress is evaluated by means of a simplified numerical analysis. For complex welded structures, however, a completely numerical solution is proposed; when different crack initiation sites are present (i.e. either weld toes or roots), the proposed approach correctly estimates the actual critical point.     

An approximate analytical solution for a three-dimensional heat-conduction problem in an air-radiation heating system

We offer a method for the calculation of the heat transferred from a system, this method being based on replacement of the three-dimensional process by a combination of a two-dimensional and a one-dimensional process in various cross-sectional planes of the heater channels.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 57, No. 6, pp. 1010–1015, December, 1989.     

An engineering analytical approach to the design of cold wire drawing processes for strain-hardening materials

Improved analytical solutions developed to design and optimize cold wire drawing processes are proposed. They account for strain-hardening, which could be accounted for, in the equations usually adopted in the engineering practice, only in a semi-empirical way. All the new proposed solutions can be expressed in terms of the coefficient of friction between the wire and the die, thus avoiding the need to calibrate physically meaningless coefficients as was necessary for the previous literature solutions. The effectiveness of the proposed results is demonstrated by comparison with existing experimental results.     

An approximate analytical model of flow in the first barrel of an overexpanded jet

A simple analytical model of flow in the first barrel of a plane overexpanded jet is proposed. The adequacy of the model is confirmed by comparison with the results of numerical calculations. The proposed model is used to solve the problem of determining the Mach disk height in a jet.     

A finite-volume-energy based approach to predict the static and fatigue behavior of components with sharp V-shaped notches

The paper presents an energetic approach useful to predict of the static and fatigue behavior of components weakened by sharp re-entrant corners. Despite the fact that stresses and strain energy density tend toward infinity at the point of singularity, the energy in a small volume of material surrounding the notch tip has obviously a finite value and such a value is thought of as the entity that controls the failure. The energy, averaged in a volume of radius R (which depends on the material properties), is a precise function of the Notch Stress Intensity Factors and is given in closed form for plane stress and plane strain conditions, the material being thought of as isotropic and linear elastic. The method is validated taking into account experimental data already reported in the literature, concerning both static tests carried out on polymethyl metacrylate (PMMA)and Duraluminium specimens and fatigue tests on welded joints and notched components in structural steels. As a matter of fact, the method proposed here is the re-formulation, on one hand, of some recent area/volume criteria (in which averaged values of the maximum principal stress are used to predict component fatigue limits) and, on the other, of N-SIF-based criteria, where the Notch Stress Intensity Factors are thought of as the parameters that control static and fatigue failures.     

An approximate technique for evaluating the closures due to mining

Summary A modified form of the Kerr foundation model is used to obtain closed form analytical expressions for the closure between opposing points in the roof and the floor of a mining excavation. The results are found to closely agree with both a finite element analysis, and also with exprimentally determined data.

---

Eine Näherungsmethode für die Abschätzung des Verschließens im Bergbau

Zusammenfassung Eine modifizierte Form des Kerrschen Gründungsmodells wird verwendet, um einen geschlossenen analytischen Ausdruck für das Verschließen zwischen entgegengesetzten Punkten der Decke und des Bodens eines Bergwerkschachtes zu erhalten. Die Ergebnisse stimmen gut mit einer Finite-Element-Methode und ebenso mit experimentell ermittelten Daten überein.

---

---

With 5 Figures     

New analytical approximate solution of the generalised temperature integral for kinetic reactions

Simple closed-form expression of the generalised temperature integral in the basic equation to describe kinetic reactions for solid materials in linear heating process is always suitable for use in determining parameters. Many developed solutions only can give high accuracies on the general conditions. A new analytical approximate solution was deduced in this work. The deviations of this solution from the true value are fully analysed. This solution takes advantage in broader application conditions than other known solutions. The application of the new analytical approximate solution in austenite kinetic reaction in this work reinforces that austenite reaction rate is feasible to be given a priori.     

An analytical approach for fracture and fatigue in functionally graded materials

The problem of brittle crack propagation and fatigue crack growth in functionally graded materials (FGMs) is addressed. The proposed analytical approach can be used to estimate the variation of the stress-intensity factor as a function of the crack length in FGMs. Furthermore, according to the Paris’ law, the fatigue life and the crack-tip velocity of crack propagation can be predicted in the case of fatigue crack growth. A comparison with numerical results obtained according to the Finite Element method will show the effectiveness of the proposed approach. Detailed examples are provided in the case of three-point bending beam problems with either a FGM interlayer, or a FGM external coating. A comparison is presented between two types of grading in the elastic modulus: a continuous linear variation in the FGM layer and a discrete approximation with a multi-layered beam and a constant Young’s modulus in each layer.