Nao-Aki Noda

Stress concentration factors for round and flat test specimens with notches

Abstract The stress concentration problem of round and flat bars with V-shaped notches under various loadings is especially important for test specimens used to investigate the fatigue strength of materials. Accurate stress concentration factors have been given in a recent analysis of the body force method. However, the results of the solutions have been presented in tabular form, which is not suitable for engineering applications. In this paper convenient formulae, which give the stress concentration factors with better than 1% accuracy, are proposed using the Neuber formula and the solution of a V-shaped notch in a semi-infinite plate. The stress concentration factors are also provided in a graphical way on the basis of the formulae.

Stress concentration of a cylindrical bar with a V-shaped circumferential groove under torsion, tension or bending

Abstract The stress concentration of a cylindrical bar with a V-shaped circumferential groove is analyzed by the body force method. The stress field due to a ring force in an infinite body is used to solve this problem. The solution is obtained by superposing the stress fields of ring forces in order to satisfy the given boundary conditions. The present results for semi-circular notches are in close agreement with Hasegawas results. As a result of the systematic calculation of a 60° V-shaped notch, it is found that the stress concentration factors obtained by Neubers trigonometric rule used currently have non-conservative errors of about 10% for a wide range of notch depths. The stress concentration factors are illustrated in charts so they can be used easily in design or research.

Generalized stress intensity factors of V-shaped notch in a round bar under torsion, tension, and bending

Abstract In this study, generalized stress intensity factors K I, λ 1 , K II, λ 2 , and K III, λ 4 are calculated for a V-shaped notched round bar under tension, bending, and torsion using the singular integral equation of the body force method. The body force method is used to formulate the problem as a system of singular integral equations, where the unknown functions are the densities of body forces distributed in an infinite body. In order to analyze the problem accurately, the unknown functions are expressed as piecewise smooth functions using three types of fundamental densities and power series, where the fundamental densities are chosen to represent the symmetric stress singularity and the skew-symmetric stress singularity. Generalized stress intensity factors at the notch tip are systematically calculated for various shapes of V-shaped notches. Normalized stress intensity factors are given by using limiting solutions; they are almost determined by notch depth alone, and almost independent of other geometrical parameters. The accuracy of Benthem–Koiter’s formula proposed for a circumferential crack is also examined through the comparison with the present analysis.

Numerical solutions of the singular integral equations in the crack analysis using the body force method

In this paper, numerical solutions of the singular integral equations of the body force method in the crack problems are discussed. The stress fields induced by ‘two kinds of displacement discontinuity’ are used as fundamental solutions. Then, the problem is formulated as a hypersingular integral equation with the singularity of the form r2. In the numerical calculation, two kinds of unknown functions are approximated by the products of the fundamental density function and the Chebyshev polynomials. As examples, the stress intensity factors of the oblique edge crack, kinked crack, branched crack and zig-zag crack are analyzed. The calculation shows that the present method gives accurate results even for the extremely oblique edge crack and kinked crack with extremely short bend which has been difficult to analyze by the previous method using the approximation by the products of the fundamental density function and the stepped functions etc.

Analysis of newly-defined stress intensity factors for angular corners using singular integral equations of the body force method

In this study, numerical solutions of singular integral equations are discussed in the analysis of angular corners. The problems are formulated as a system of singular integral equations on the basis of the body force method. In the numerical solutions, two types of fundamental density functions are chosen to express the symmetric type stress singularity of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca% aIXaaabaGaamOCamaaCaaaleqajqwaacqaaiaaigdacqGHsislcqaH% 7oaBdaWgaaqcKjaGaeaacaaIXaaabeaaaaaaaaaa!3CE1!\[{1 \mathord{\left/ {\vphantom {1 {r^{1 - \lambda _1 } }}} \right. \kern-\nulldelimiterspace} {r^{1 - \lambda _1 } }}\] and the skew-symmetric type stress singularity of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca% aIXaaabaGaamOCamaaCaaaleqajqwaacqaaiaaigdacqGHsislcqaH% 7oaBdaWgaaqcKjaGaeaacaaIYaaabeaaaaaaaaaa!3CE2!\[{1 \mathord{\left/ {\vphantom {1 {r^{1 - \lambda _2 } }}} \right. \kern-\nulldelimiterspace} {r^{1 - \lambda _2 } }}\] then the unknown functions are expressed as a linear combination of the fundamental density functions and power series. The calculation shows that the present method gives rapidly converging numerical results for angular corners as well as ordinary cracks in homogeneous materials. The stress intensity factors of angular corners are calculated for various geometrical and loading conditions.

Variation of stress intensity factor along the front of a 3D rectangular crack by using a singular integral equation method

In this paper a singular integral equation method is applied to calculate the distribution of stress intensity factor along the crack front of a 3D rectangular crack. The stress field induced by a body force doublet in an infinite body is used as the fundamental solution. Then, the problem is formulated as an integral equation with a singularity of the form of r−3. In solving the integral equation, the unknown functions of body force densities are approximated by the product of a polynomial and a fundamental density function, which expresses stress singularity along the crack front in an infinite body. The calculation shows that the present method gives smooth variations of stress intensity factors along the crack front for various aspect ratios. The present method gives rapidly converging numerical results and highly satisfied boundary conditions throughout the crack boundary.

Stress intensity factors of a rectangular crack meeting a bimaterial interface

Using the hypersingular integral equation method based on body force method, a planar crack meeting the interface in a three-dimensional dissimilar materials is analyzed. The singularity of the singular stress field around the crack front terminating at the interface is analyzed by the main-part analytical method of hypersingular integral equations. Then, the numerical method of the hypersingular integral equation for a rectangular crack subjected to normal load is proposed by the body force method, which the crack opening dislocation is approximated by the product of basic density functions and polynomials. Numerical solutions of the stress intensity factors of some examples are given.

Stress concentration factors for shoulder fillets in round and flat bars under various loads

Abstract The stress concentration problem of shoulder fillets in round and flat bars under various loads is often encountered in machine design of shafts. Also it is important for test specimens used to investigate the mechanical properties of materials. Accurate stress concentration factors (SCFs) have been given in a recent analysis of the body force method. However, the results of the solutions have been presented in tabular form which is not suitable for engineering applications. For notched bars, Neuber proposed the simple approximate formula K tN which is useful for a wide range of notch shape: 1/( K tN − 1) m = 1/( K ts − 1) m + 1/( K td − 1) m and m = 2. Here, K ts and K td are exact solutions for shallow and deep notches, respectively. Neubers simple formula has been used for >40 years in the design of notched bars because of its convenience. In this study, similar convenient equations K tN are initially proposed as an extension of Neubers formula to the problem of shoulder fillet. In this formula new definitions of K ts and K id are used corresponding to two extreme cases of shoulder fillet in round and flat bars. Next, the most suitable exponent m is determined so as to minimize the difference between K tN and accurate K t , that is, the results of the body force method. Next, by applying the least squares method to the ratio K t / K tN more accurate formulas are proposed. The formulas proposed in this paper are found to give the stress concentration factors with better than 1 % accuracy. In addition, the stress concentration factors are also provided in a graphical way on the basis of the formula so they can be used easily in design or research.

Variation of stress intensity factor and crack opening displacement of semi-elliptical surface crack

In this paper a singular integral equation method is applied to calculate the stress intensity factor along crack front of a 3D surface crack. Stress field induced by body force doublet in a semi infinite body is used as a fundamental solution. Then the problem is formulated as an integral equation with a singularity of the form of r-3. In solving the integral equations, the unknown functions of body force densities are approximated by the product of a polynomial and a fundamental density function; that is, the exact density distribution to make an elliptical crack in an infinite body. The calculation shows that the present method gives the smooth variation of stress intensity factors along the crack front and crack opening displacement along the crack surface for various aspect ratios and Poissons ratio. The present method gives rapidly converging numerical results and highly satisfactory boundary conditions throughout the crack boundary.

Singular Integral Equation Method for Interaction Between Elliptical Inclusions

This paper deals with numerical solutions of singular integral equations in interaction problems of elliptical inclusions under general loading conditions. The stress and displacement fields due to a point force in infinite plates are used as fundamental solutions. Then, the problems are formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where the unknowns are the body force densities distributed in infinite plates having the same elastic constants as those of the matrix and inclusions. To determine the unknown body force densities to satisfy the boundary conditions, four auxiliary unknown functions are derived from each body force density. It is found that determining these four auxiliary functions in the range 0 ≤ Φ k ≤ π/2 is equivalent to determining an original unknown density in the range 0 ≤ Φ k ≤ 2π. Then, these auxiliary unknowns are approximated by using fundamental densities and polynomials. Initially, the convergence of the results such as unknown densities and interface stresses are confirmed with increasing collocation points. Also, the accuracy is verified by examining the boundary conditions and relations between interface stresses and displacements. Randomly or regularly distributed elliptical inclusions can be treated by combining both solutions for remote tension and shear shown in this study.