Fracture dynamics problem on mode I semi-infinite crack

By means of the theory of complex functions, fracture dynamics problems concerning mode I semi-infinite crack were studied. Analytical solutions of stress, displacement and dynamic stress intensity factor under the action of moving increasing loads Pt ³/x ³, Px ³/t ², respectively, are very easily obtained using the ways of self-similar functions. The correlative closed solutions are attained based on the Riemann–Hilbert problems.     

Fracture dynamics problems of orthotropic solids under anti-plane shear loading

By application of the approaches of the theory of complex functions, fracture dynamics problems of orthotropic solids under anti-plane shear loading were researched. Universal representation of analytical solutions was obtained by means of self-similar functions. The problems dealt with can be facilely transformed into Riemann–Hilbert problems by this technique, and analytical solutions of the stress, the displacement and dynamic stress intensity factor under the actions of moving increasing loads Px ²/t ² and Pt ³/x ² for the edges of asymmetrical mode III crack, respectively, were acquired. In the light of corresponding material properties, the variable rule of dynamic stress intensity factor was illustrated very well.     

Analytical solutions of dynamic mode I crack under the conditions of displacement boundary

By the approaches of the theory of complex variable functions, the problems of dynamic mode I crack under the condition of displacement boundary are investigated. For this kind of dynamic crack extension problems with arbitrary index of self-similarity, the universal representations of analytical solutions are facilely deduced by the methods of self-similar functions. Analytical solutions of the stresses, displacements and stress intensity factors are readily acquired using the methods of self-similar functions. The problems studied can be very easily translated into Riemann–Hilbert problems and their closed solutions are gained rather straightforward in terms of this technique. According to corresponding material properties, the mutative rule of stress intensity factor was illustrated very well. Using those solutions and superposition theorem, the solutions of arbitrarily complex problems can be attained.     

正交异性复合材料界面上反平面动态自相似扩展裂纹问题的解

采用复变函数论的方法，对复合材料界面上的裂纹扩展问题进行研究。并根据任意的自相似指数的断裂动力学问题，进行自相似求解，导出解析解的一般表示。应用该法可以迅速地将所论问题转化为Riemann-Hil-bert问题，并可以相当简单地得到问题的闭合解。文中分别对裂纹中心受阶跃载荷，裂纹面受到瞬时脉冲载荷作用下的界面裂纹扩展问题进行求解。得到了裂纹的位移。尖端的应力和动态应力强度因子的解析解。应用该解并通过叠加原理。就可以很容易的求得任意复杂问题的解。     

动态扩展裂纹的若干反平面问题的研究

采用复变函数论,对反平面条件下的动态裂纹扩展问题进行研究。通过自相似函数的方法可以获得解析解的一般表达式。应用该法可以很容易地将所讨论的问题转化为Riemann—Hilbert问题,并可以相当简单地得到问题的闭合解。文中分别对裂纹面受均布载荷、坐标原点受集中增加载荷、坐标原点受瞬时冲击载荷以及裂纹面受运动集中载荷Px／t作用下的动态裂纹扩展问题进行求解,得到了裂纹扩展位移、裂纹尖端的应力和动态应力强度因子的解析解。应用该解并通过叠加原理,就可以求得任意复杂问题的解。     

Problem on circular-arc cracks between bonded dissimilar materials under concentrated force and moment

The method is very efficient by applying extended Schwarz principle integrated with the analysis of the singularity of complex stress functions to solve some plane-elastic problems under concentrated loads, in Ref.[1], this method is used to deal with the elastic problems of homogeneous plane. In this paper, it is extended to the case of dissimilar materials with co-circular cracks under concentrated force and moment. For several typical cases the solutions of complex stress function in closed form are built up and the stress intensity factors are given. From these solutions, we provide a series of particular results, in which two of them coincide with those in Refs. [1] and [6].     

Dynamic propagation problems concerning surfaces of asymmetrical mode Ⅲ crack subjected to moving loads

With the theory of complex functions, dynamic propagation problems concerning surfaces of asymmetrical mode Ⅲ crack subjected to moving loads are investigated. General representations of analytical solutions are obtained with self-similar functions. The problems can be easily converted into Riemann-Hilbert problems using this technique. Analytical solutions to stress, displacement and dynamic stress intensity factor under constant and unit-step moving loads on the surfaces of asymmetrical extension crack, respectively, are obtained. By applying these solutions, together with the superposition principle, solutions of discretionarily intricate problems can be found.     

Problems on collinear cracks between bonded dissimilar materials under concentrated loads

Following Ref. [6], this paper deals with the problem on collinear cracks between bonded dissimilar materials under a concentrated force and moment at an arbitrary point. Several typical solutions of complex stress functions in closed form are formulated and the stress intensity factors are given. These solutions include a series of results of previous researchers, and redress some errors in the researches of problems containing semi-infinite cracks^[3],[4].     

Analytical Solution for Radial Deformations of Functionally Graded Isotropic and Incompressible Second-Order Elastic Hollow Spheres

We analytically analyze radial expansion/contraction of a hollow sphere composed of a second-order elastic, isotropic, incompressible and inhomogeneous material to delineate differences and similarities between solutions of the first- and the second-order problems. The two elastic moduli are assumed to be either affine or power-law functions of the radial coordinate R in the undeformed reference configuration. For the affine variation of the shear modulus μ, the hoop stress for the linear elastic (or the first-order) problem at the point R=(R _ou R _in (R _ou +R _in )/2)^1/3 is independent of the slope of the μ vs. R line. Here R _in and R _ou equal, respectively, the inner and the outer radius of the sphere in the reference configuration. For μ(R)∝R ⁿ , for the linear problem, the hoop stress is constant in the sphere for n=1. However, no such results are found for the second-order (i.e., materially nonlinear) problem. Whereas for the first-order problem the shear modulus influences only the radial displacement and not the stresses, for the second-order problem the two elastic constants affect both the radial displacement and the stresses. In a very thick homogeneous hollow sphere subjected only to pressure on the outer surface, the hoop stress at a point on the inner surface depends upon values of the two elastic moduli. Thus conclusions drawn from the analysis of the first-order problem do not hold for the second-order problem. Closed form solutions for the displacement and stresses for the first-order and the second-order problems provided herein can be used to verify solutions of the problem obtained by using numerical methods.     

Dynamic fracture of a kane-mindlin plate

We study dynamic crack problems for an elastic plate by using Kane-Mindlin's kinematic assumptions. The general solutions of the Laplace transformed displacements and stresses are first derived. Path independent integrals for stationary cracks subjected to transient loads and steadily growing cracks are deduced. For a stationary crack in a very thin plate subjected to impact loads, the crack tip dynamic stress intensity factor (DSIF), K₁(t), is related to the far field plane stress one, K₁⁰(t), by where ν is Poisson's ratio. For a crack steadily growing with speed V, the crack tip DSIF, K₁(V), is given by where K₁⁰(V) is the plane stress DSIF and A(V) and B(V) are known functions of V. These results are applied to compute the DSIF for a semi-infinite stationary crack in an unbounded plate subjected to impact pressure on the crack faces. The results of DSIF for a finite crack in an infinite plate under uniform impact pressure on the crack surfaces show that for each plate thickness, the maximum DSIF is higher than that for the plane stress case.     

Some particular solutions for penny-shaped crack problem by using hypersingular integral equation or differential-integral equation

Summary A hypersingular integral equation or a differential-integral equation is used to solve the penny-shaped crack problem. It is found that, if a displacement jump (crack opening displacement COD) takes the form of (a ²−x ²−y ²)^1/2 x ^m y ⁿ , where a denotes the radius of the circular region, the relevant traction applied on the crack face can be evaluated in a closed form, and the stress intensity factor can be derived immediately. Finally, some particular solutions of the penny-shaped crack problem are presented in this paper. Received 1 July 1997; accepted for publication 13 October 1997     

Variation of the stress intensity factor along the front of a 3-D rectangular crack subjected to mixed-mode load

Summary  The singular integral equation method is applied to the calculation of the stress intensity factor at the front of a rectangular crack subjected to mixed-mode load. The stress field induced by a body force doublet is used as a fundamental solution. The problem is formulated as a system of integral equations with r ⁻³-singularities. In solving the integral equations, unknown functions of body-force densities are approximated by the product of polynomial and fundamental densities. The fundamental densities are chosen to express two-dimensional cracks in an infinite body for the limiting cases of the aspect ratio of the rectangle. The present method yields rapidly converging numerical results and satisfies boundary conditions all over the crack boundary. A smooth distribution of the stress intensity factor along the crack front is presented for various crack shapes and different Poisson's ratio. Received 5 March 2002; accepted for publication 2 July 2002     

Dynamic propagation problems concerning surfaces of asymmetrical mode III crack subjected to moving loads

With the theory of complex functions, dynamic propagation problems concerning surfaces of asymmetrical mode III crack subjected to moving loads are investigated. General representations of analytical solutions are obtained with self-similar functions. The problems can be easily converted into Riemann-Hilbert problems using this technique. Analytical solutions to stress, displacement and dynamic stress intensity factor under constant and unit-step moving loads on the surfaces of asymmetrical extension crack, respectively, are obtained. By applying these solutions, together with the superposition principle, solutions of discretionarily intricate problems can be found. Project supported by the Post-Doctoral Science Foundation of China (No. 2005038199) and the Natural Science Foundation of Heilongjiang Province of China (No. ZJG04-08)     

Variation of the stress intensity factor along the crack front of interacting semi-elliptical surface cracks

Summary  In this study, the interaction between two semi-elliptical co-planar surface cracks is considered when Poisson's ratio ν = 0.3. The problem is formulated as a system of singular integral equations, based on the idea of the body force method. In the numerical calculation, the unknown density of body force density is approximated by the product of a fundamental density function and a polynomial. The results show that the present method yields smooth variations of stress intensity factors along the crack front very accurately, for various geometrical conditions. When the size of crack 1 is larger than the size of crack 2, the maximum stress intensity factor appears at a certain point, β₁=177^∘, of crack 1. Along the outside of crack 1, that is at β₁=0∼90^∘, the interaction can be negligible even if the two cracks are very close. The interaction can be negligible when the two cracks are spaced in such a manner that their two closest points are separated by a distance exceeding the small crack's major diameter. The variations of stress intensity factor of a semi-elliptical crack are tabulated and charted. Received 30 August 1999; accepted for publication 22 February 2000     

Interaction between elliptical and ellipsoidal inclusions under bending stress fields

Summary  This paper deals with interaction problems of elliptical and ellipsoidal inclusions under bending, using singular integral equations of the body force method. The problems are formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where unknown functions are densities of body forces distributed in the x,y and r,θ,z directions in infinite bodies having the same elastic constants as those of the matrix and inclusions. In order to satisfy the boundary conditions along the elliptical and the ellipsoidal boundaries, the unknown functions are approximated by a linear combination of fundamental density functions and polynomials. The present method is found to yield the exact solutions for a single elliptical or spherical inclusion under a bending stress field. It yields rapidly converging numerical results for interface stresses in the interaction of inclusions. Received 9 September 1999; accepted for publication 15 January 2000     

On solutions of the problem in stresses with the use of Maxwell stress functions

The spatial problems of elasticity are mainly solved in displacements [1, 2], i.e., the Lamé equations are taken as the initial equations. This is related to the lack of general solutions for the system of basic equations of elasticity expressed in stresses. In this connection, a new variational statement of the problem in stresses was developed in [3, 4]; this statement consists in solving six generalized equations of compatibility for six independent components of the stress tensor, while the three equilibrium equations are transferred to the set of boundary conditions. This method is more convenient for the numerical solution of problems in stresses and has been tested when solving various boundary value problems. In the present paper, analyzing the completeness of the Saint-Venant identities and using the Maxwell stress functions, we obtain a new resolving system of three differential equations of strain compatibility for the three desired stress functions φ, ξ, and ψ. This system is an alternative to the three Lamé equilibrium equations for three desired displacement components u, v, w and is simpler in structure. Moreover, both of these systems of resolving equations can be solved by the new recursive-operator method [5, 6]. In contrast to well-known methods for constructing general solutions of linear differential equations and their systems, the solutions obtained by the recursive-operator method are constructed as operator-power series acting on arbitrary analytic functions of real variables (not necessarily harmonic), and the series coefficients are determined from recursive relations (matrix in the case of systems of equations). The arbitrary functions contained in the general solution can be determined directly either from the boundary conditions (the obtained system of inhomogeneous equations with a right-hand side can also be solved by the recursive-operator method [6]) or by choosing them from various classes of analytic functions (elementary, special); a complete set of particular solutions can be obtained in the same function classes, and the coefficients of linear combinations of particular solutions can be determined by the Trefftz method, the least-squares method, and the collocation method.     

Analytical and experimental determination of dynamic impact stress intensity factor for 40 Cr steel

A linear elastic three-dimensional finite element analysis is made to analyze the near field stress behavior of an edge cracked rectangular bar simply supported and subjected to central impact at the back side of the crack. The material is made of 40 Cr steel. Determined numerically are the local time histories of the stress wave, displacement near load point, crack tip strain, and dynamic stress intensity factor K_(d)¹. The above quantities were also measured experimentally by performing impact tests; they agreed well with the analytical results and determine the load at fracture initiation and hence the critical dynamic stress intensity factor K_(d)^1c. The interaction effect between the loading bar and specimen appears to be negligible.     

Gearhart–Prüss Theorem and Linear Stability for Riemann Solutions of Conservation Laws

We study the spectral and linear stability of Riemann solutions with multiple Lax shocks for systems of conservation laws. Using a self-similar change of variables, Riemann solutions become stationary solutions for the system u _t + (Df(u) − x I)u _x = 0. In the space of O((1 + |x|)^−η) functions, we show that if , then λ is either an eigenvalue or a resolvent point. Eigenvalues of the linearized system are zeros of the determinant of a transcendental matrix. On some vertical lines in the complex plane, called resonance lines, the determinant can be arbitrarily small but nonzero. A C ⁰ semigroup is constructed. Using the Gearhart–Prüss Theorem, we show that the solutions are O(e ^{γ t} ) if γ is greater than the real parts of the eigenvalues and the coordinates of resonance lines. We study examples where Riemann solutions have two or three Lax-shocks. Dedicated to Professor Pavol Brunovsky on his 70th birthday.     

An appraisal of rheological models as applied to polymer melt flow

Summary An attempt is made at giving an appraisal of some representative rheological models of both differential and integral type, using the standard rheological measurements of six polymer melts. Experimental data obtained were the steady shear viscosity and the first normal stress difference by means of aWeissenberg rheogoniometer over the range of shear rates: 10^–2 ~ 10 sec^–1, and by means of aHan slit/capillary rheometer over the range of shear rates: 10 ~ 10³ sec^–1. Also measured by means of theWeissenberg rheogoniometer were the dynamic viscosity and dynamic elastic modulus over the range of frequencies: 0.3 × 10^–2 ~ 3 × 10² sec^–1. Rheological models chosen for an appraisal are theSpriggs 4-constant model, theMeister model, and theBogue model.It is found that the capability of the three models considered is about the same in their prediction of the rheological behavior of polymer melts in simple shearing flow. It is pointed out however that, due to the ensuing mathematical complexities, the usefulness of these models is limited to the study of flow problems associated with simple flow situations. Therefore, in analysing the complex flow situations often encountered with various polymer processings, the authors suggest use of the empirical models of the power-law type for both the viscosity and normal stress functions.With 11 figures, 4 schemas and 1 table     

The analysis of thermal residual stresses near the apex in bonded dissimilar materials

By employing the complex variable method and constructing the particular solution sequences in the form of complex functions, all the cases of the thermal residual stress field near the apex in dissimilar materials bonded with two arbitrary angles are researched theoretically, and the corresponding classical solutions are obtained. Moreover, the primary paradox, the secondary paradox and even the triple paradox are discovered in the classical solutions and also resolved here, thereby it is confirmed that thermal residual stresses near the apex in bonded dissimilar materials probably possess the singularities of lnr (when the primary paradox occurs) , ln²r (when the secondary paradox occurs) and even ln³r (when the triple paradox occurs) . In addition, the systematic method to solve multiple paradox problems is put forward. © 1999 Elsevier Science Ltd. All rights reserved.