analysis for t stress of cracks in 3d anisotropic elastic media by weakly singular integral equation method abstract this paper proposes an efficient numerical procedure for computing t stresses of cracks in three dimensional linearly elastic infinite media the technique is established in a broad framework allowing a medium made of generally anisotropic materials and cracks of arbitrary shape and under general loading conditions to be treated a pair of weakly singular weak form displacement and traction boundary integral equations is utilized to formulate the key equations governing the unknown crack face fields besides the basic benefits such as the reduction of spatial dimensions of the solution space and the efficient treatment of unbounded domains and remote boundary data use of such integral equations in the formulation offers additional positive features including the computational simplicity resulting from the weakly singular nature of all involved integrals and the involvement of a complete set of crack face displacement fields a weakly singular symmetric galerkin boundary element method together with the special near front approximation is utilized to solve for the unknown relative crack face displacement whereas the sum of the crack face displacement is obtained from the displacement boundary integral equation for cracks via the galerkin technique the latter step is one of the essential aspects of the present study that provides the direct means for determining the t stress data in terms of the sum of the crack face displacement in the neighborhood of the crack front an extensive numerical study is conducted for various scenarios and a selected set of results is reported to demonstrate both the accuracy and capability of the proposed technique

an extended boundary element method formulation for the direct calculation of the stress intensity factors in fully anisotropic materials copyright 2016 john wiley sons ltd we propose a formulation for linear elastic fracture mechanics in which the stress intensity factors are found directly from the solution vector of an extended boundary element method formulation the enrichment is embedded in the boundary element method formulation rather than adding new degrees of freedom for each enriched node therefore a very limited number of new degrees of freedom is added to the problem which contributes to preserving the conditioning of the linear system of equations the stroh formalism is used to provide boundary element method fundamental solutions for any degree of anisotropy and these are used for both conventional and enriched degrees of freedom several numerical examples are shown with benchmark solutions to validate the proposed method copyright 2016 john wiley sons ltd

on the elastic fields of an elliptical inhomogeneity under plane deformation a generalized and unified approach is presented for the two dimensional plane problem of an elliptical inhomogeneity in an isotropic elastic medium the analysis is based upon the use of conformal mapping and laurent series expansion of muskhelishvili s complex potentials the resulting elastic fields are derived explicitly in both transformed and physical planes for the inhomogeneity and the surrounding matrix the associated expressions are universal in the sense of being applicable to generalized geometries and applied loads the application of the general solution is illustrated by several examples and the resulting solutions are verified with those existing in the literature the results advanced here can be used as building blocks for dealing with more complex problems involving a wide variety of geometry and loading conditions

kinked crack analysis by a hybridized boundary element boundary collocation method in this research a two dimensional displacement discontinuity method which is a kind of indirect boundary element method using higher order elements i e a source element with a cubic variation of displacement discontinuities having four sub elements is used to obtain the displacement discontinuities along each boundary element in this paper three kinds of the higher order boundary elements are used the ordinary elements the kink elements and the special crack tip elements the boundary collocation technique is used for the calculation of the displacement discontinuities at the center of each sub elements again a special boundary collocation technique is used to treat the kinked source elements occur in the crack analysis considering the two source elements each having four sub elements joined at a corner kink point the collocation points in the cubic element model which are outside of the kink point are moved to the crack kink then the displacement discontinuities on the left and right sides of the kink are calculated the displacement discontinuities of the kink point are obtained by averaging the corresponding values of its left and right sides the special crack tip elements are also treated by the boundary displacement collocation technique considering the singularity variation of the displacements and stresses near the crack tip some simple example problems are solved numerically by the proposed method the numerical results are compared with the corresponding results obtained by the previous methods cited in the literature this comparison shows a very good agreement between the results and verify the accuracy and validity of the proposed method

on interface crack in laminated anisotropic medium in this paper the interface crack problems of a multilayered anisotropic medium under a state of generalized plane deformation are considered within the framework of anisotropic theory a general solution procedure is introduced such that it can be uniformly applied to media with transversely isotropic orthotropic monoclinic etc layers the problem is reduced to the solution of a system of singular integral equations by means of fourier transform method and the stiffness matrix formulation a jacobi polynomial technique is then used to solve the integral equations numerically the stress intensity factors are provided the stress intensity factors have been calculated numerically and displayed graphically

wave propagation in a split timoshenko beam abstract wave propagation in a split beam is analyzed by treating each section separately as a waveguide and imposing appropriate connectivities at their joints the dynamics are treated by way of spectral analysis while the justification of the connectivities is sought through the exact solution of an analogous two dimensional static problem the wave analysis is verified experimentally by use of strain gauges finally a novel application of the dynamic j integral is used to calculate the stress intensity factor history

contour regression a general approach to dimension reduction we propose a novel approach to sufficient dimension reduction in regression based on estimating contour directions of small variation in the response these directions span the orthogonal complement of the minimal space relevant for the regression and can be extracted according to two measures of variation in the response leading to simple and general contour regression scr and gcr methodology in comparison with existing sufficient dimension reduction techniques this contour based methodology guarantees exhaustive estimation of the central subspace under ellipticity of the predictor distribution and mild additional assumptions while maintaining sqrt n consistency and computational ease moreover it proves robust to departures from ellipticity we establish population properties for both scr and gcr and asymptotic properties for scr simulations to compare performance with that of standard techniques such as ordinary least squares sliced inverse regression principal hessian directions and sliced average variance estimation confirm the advantages anticipated by the theoretical analyses we demonstrate the use of contour based methods on a data set concerning soil evaporation

numerical solution of singular integral equations via initial value methods quadrature methods for solving singular integral equations especially with logarithmic kernels have been known to break down completely even if they succeed the results are often inaccurate and the accuracy difficult to estimate product integration methods to eliminate the singularity in the kernel function can tackle the problem effectively if the required solution is a sufficiently smooth function but at a prohibitive cost of computational time the author proposes a different type of method for solving singular integral equations by reducing them into equivalent fredholm integral equations whenever possible and then imbedding the latter in a class of equivalent initial value problems which tend to be stable and can be solved in a routine manner the validity of the method is also discussed

rational function approximations in the numerical solution of cauchy type singular integral equations cauchy type singular integral equations of the second kind with constant coefficients are solved via rational and polynomial approximations rational functions similar to that of polynomials have the property that for r t rational and for many of the weight functions w t encountered in practice r x 11w t r t t x dt is also rational hence approximations by rational functions is feasible rational function approximations in the solution of the dominant equation results in a linear algebraic system which possesses block diagonal structure it is further shown that the determinant of the coefficient matrix is bounded below away from zero and stability is ensured under fairly non restrictive conditions for the complete cauchy type singular integral equation i e the equation with both the principal and regular parts gaussian quadrature in conjunction with the rational function method is synthesized in the construction of a hybrid scheme error estimates and convergence are established a variety of problems from aerodynamics and fracture mechanics are solved and presented as a basis of comparison to polynomial based schemes

alternative solution methods for crack problems in plane anisotropic elasticity with examples two dimensional crack problems of homogeneous anisotropic linear elasticity are solved using the riemann hilbert method to this end the riemann hilbert problem of line discontinuity is formulated for anisotropic plane problems and the necessary parameters and functions are identified for illustration the method is applied to obtain the complete stress field and the stress intensity factors for a crack in an infinite anisotropic plate which is loaded on a part of one of its faces then the well established method of continuously distributed edge dislocations is considered and illustrated via some example problems e g an infinite anisotropic plate under uniform farfield loads containing 1 a closed frictional crack and a pair of arbitrarily located single edge dislocations and 2 an infinite row of equally spaced parallel open cracks the illustrative examples reveal that the first method oaers an eaective solution technique for problems where unbalanced tractions are applied on crack surfaces whereas for problems with self equilibrating loads applied on the crack faces the second method is generally well suited in addition the method of resultant forces along the crack is discussed and its formulation in terms of the dislocation density functions and also the crack opening displacements which is new is presented the solutions to some of the example problems are provided in some detail and for others just the key formulae e g stress functions and stress intensity factors are calculated and analyzed in brief this paper presents the generalization of the riemann hilbert method from isotropic to anisotropic in plane elasticity problems and also provides a collection of certain basic two dimensional anisotropic crack problems some of the results here are also new 7 2000 elsevier science ltd all rights reserved

on the unified approach to anisotropic and isotropic elasticity for singularity interface and crack in dissimilar media abstract proposed in this paper is the equivalence between anisotropic and isotropic elasticity for two dimensional deformation under certain conditions that is the isotropic elasticity can be reconstructed in the same framework of the anisotropic elasticity when the interface between dissimilar media lies along a straight line therefore many known solutions for an anisotropic bimaterial are valid for a bimaterial of which one or both of the constituent materials are isotropic the usefulness of the equivalence is that the solutions for singularities and cracks in an anisotropic isotropic bimaterial can easily be obtained without solving the boundary value problems directly the interaction solutions of singularities interfaces and cracks in infinite anisotropic bimaterial are summarized to be used for the cases of isotropic isotropic and anisotropic isotropic bimaterials conservation integrals also have the similar analogy between anisotropic and isotropic elasticity so that j integral and j based mutual integral m are expressed in the same complex forms for anisotropic and isotropic materials when both end points of the integration paths are on the straight interface the use of j and m integrals together with the present equivalence are exemplified to obtain energy release rate stress intensity factors and t stresses of interfacial cracks lying in the interface of anisotropic anisotropic isotropic isotropic or anisotropic isotropic solids

stress analysis of quasi orthotropic elastic plane abstract the quasi orthotropic elastic plane in which the characteristic roots of the fundamental differential equation for the orthotropic elastic plane are doubled is investigated for the associated stress analysis a rational mapping function is used and a closed form solution is obtained therefore the stress analysis is rigorous for the rational mapping function the stress functions can be obtained without any integration as a demonstration of the stress analysis a half plane with an oblique edge crack is analyzed stress distributions and stress intensity factors are investigated the relationships between the quasi orthotropic elastic plane and isotropic plane with respect to the stress intensity factor are investigated it is also determined that the stress intensity factors for the quasi orthotropic elastic plane can be calculated from those of the isotropic elastic plane using a similar configuration and loading condition

common canonical variates canonical correlation analysis measures the linear relationship between two random vectors x 1 and x 2 as the maximum correlation between linear combinations of x 1 and linear combinations of x 2 several generalisations of canonical correlation analysis to k 2 random vectors x 1 x k have been proposed in the literature kettenring 1971 1985 based on the principle of maximising some generalised measure of correlation in this paper we propose an alternative generalisation called common canonical variates based on the assumption that the canonical variates have the same coefficients in all k sets of variables this generalisation is applicable in situations where all x i have the same dimension we present normal theory maximum likelihood estimation of common canonical variates and illustrate their use on a morphometric data set

stress distribution at sharp and rounded v notches in quasi orthotropic plane abstract the eigensolution of the quasi orthotropic wedge problem is presented the degenerate orthotropic material of the wedge is characterized by doubled roots of the characteristic equation using the method of singular integral equations the distribution of the stress field in the quasi orthotropic plane weakened by semi infinite rounded v notch was obtained the edge of the notch was free of load and the asymptotic stress field defined by a stress intensity factor was set at the infinity on this basis the relationship between stress intensity factor at the sharp notch vertex and the normal stress at the vertex of corresponding rounded v notch with the same opening angle was established as it was shown in the example this relationship is asymptotic for finite notched bodies and it can be used to calculate a notch stress intensity factor based on a stress concentration results obtained for a notch rounded with small radius of curvature

on the solution of the minimal rational interpolation problem it is shown that the reachability and the observability indices of appropriately defined pairs of matrices provide the key for solving the minimal rational bitangen tial interpolation problem

a new numerical approximation for volterra integral equations combining two quadrature rules abstract this paper describes a numerical approximation to the solution of volterra integral equations of the second kind this algorithm combines trapezoidal and simpson rules we prove the convergence of the method numerical examples are provided to illustrate the accuracy of the method

energy release rate and crack kinking abstract a closed form solution is presented for the energy release rate at the onset of kinking of a straight crack in an infinite elastic medium subjected to a predominantly mode i loading the solution is accurate to the second order of kink angle and is carried out by the method which models the kink as a continuous distribution of infinitesimal edge dislocations on the basis of the maximum energy release rate criterion simple expressions are obtained for the critical kink angle and the critical applied stress the examination of the results shows that the second order solutions are in very good agreement up to fairly large kink angles with numerical results reported by others it is also analytically established that the irwin formula for the energy release rate remains valid under the predominantly mode i loading provided that the stress intensity factors in the formula are appropriately interpreted

effects of pre stress on crack tip fields in elastic incompressible solids abstract a closed form asymptotic solution is provided for velocity fields and the nominal stress rates near the tip of a stationary crack in a homogeneously pre stressed configuration of a nonlinear elastic incompressible material in particular a biaxial pre stress is assumed with stress axes parallel and orthogonal to the crack faces two boundary conditions are considered on the crack faces namely a constant pressure or a constant dead loading both preserving an homogeneous ground state starting from this configuration small superimposed mode i or mode ii deformations are solved in the framework of biot s incremental theory of elasticity in this way a definition of an incremental stress intensity factor is introduced slightly different for pressure or dead loading conditions on crack faces specific examples are finally developed for various hyperelastic materials including the j2 deformation theory of plasticity the presence of pre stress is shown to strongly influence the angular variation of the asymptotic crack tip fields even if the nominal stress rate displays a square root singularity as in the infinitesimal theory relationships between the solution with shear band formation at the crack tip and instability of the crack surfaces are given in evidence

infinite sequence of parallel cracks in an anisotropic piezoelectric solid in this paper the problem of an infinite sequence of parallel cracks in an infinite extended piezoelectric solid is analyzed a system of singular integral equations is formulated for general anisotropic piezoelectric materials of which the kernel functions developed are in complex form for commonly used transversely isotropic piezoelectric materials the kernel functions are given in real forms furthermore the obtained real kernel functions may be reduced respectively to those kernel functions for purely elastic and purely electric problems when the coupled mechanical and electric effects disappear the system of singular integral equations is solved numerically and the coupling effects of the mechanical and electric phenomena are presented by the generalized stress intensity factors

spline approximation for cauchy principal value integrals abstract in this paper we prove the uniform convergence of some quadrature formulas based on spline approximation for cauchy principal value integrals of the type f b a w x f k x x y d x k 0 1 and we present some numerical applications in particular we apply our rules to the well known prandtl s integral equation

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

a riemann hilbert boundary value problem in a triangle abstract in this article a riemann hilbert boundary value problem on an isosceles orthogonal triangle is considered using explicit schwarz poisson type formulae for the triangle schwarz type and pompeiu type operators are obtained boundary behaviors of these operators are discussed in detail finally we investigate the riemann hilbert boundary value problem for both homogeneous and inhomogeneous cauchy riemann equations an explicit solvability of the problem is obtained

boundary layers in wedges of laminated composite strips under generalized plane deformation part i asymptotic solutions abstract based upon lekhnitskii s formulation and the stroh formalism the structure of the asymptotic solutions has been examined for the boundary layer on the wedge type cross section of a laminated composite strip the composite strip is assumed under the so called generalized plane deformation which includes tension bending and or torsion by the terminal tractions as well as the generalized plane strain problem the solution structures are obtained with the aid of numerical calculation for various kinds of wedge geometry including the free edge and the delamination cracks with the crack faces opened or closed finally the nature of the asymptotic solutions is discussed including the mode mixity of singular stress field ahead of the wedge tip it is found that for a free edge problem the mode mixity of the singular asymptotic traction vector on the interfacial plane near the free edge remains invariant under varying types of remote loadings once a pair of adjacent materials or ply orientations is given and that accordingly one single scaling parameter governs the near field response

model reduction by moment matching for linear and nonlinear systems the model reduction problem for single input single output linear and nonlinear systems is addressed using the notion of moment a re visitation of the linear theory allows to obtain novel results for linear systems and to develop a nonlinear enhancement of the notion of moment this in turn is used to pose and solve the model reduction problem by moment matching for nonlinear systems to develop a notion of frequency response for nonlinear systems and to solve model reduction problems in the presence of constraints on the reduced model connections between the proposed results projection methods the covariance extension problem and interpolation theory are presented finally the theory is illustrated by means of simple worked out examples and case studies

galerkin methods with splines for singular integral equations over 0 1 in this paper a convergence analysis of galerkin methods with splines for strongly elliptic singular integral equations over the interval 0 1 is given as trial functions we utilize smoothest polynomial splines on arbitrary meshes and continuous splines on special nonuniform partitions multiplied by a weight function using inequalities of garding type for singular integral operators in weightedl 2 spaces and the complete asymptotics of solutions at the endpoints we provide error estimates in certain sobolev norms

an internal crack in a half plane solid with clamped boundary abstract an internal crack in a half plane solid with clamped boundary is analyzed a system of singular integral equations is derived for the material whose response is general anisotropic numerical method is then used to investigate the solutions to the system of equations results are presented for the stress intensity factors for orthotropic material the effects of the orientation of the crack the material angle and the degree of anisotropy are also discussed in some detail in particular when the crack is perpendicular to the clamped boundary and when one of the material principal axes is along the crack faces our numerical analysis shows that the result for the stress intensity factor k i under uniform pressure loading is exactly the same as that for k ii under uniform shear loading furthermore these stress intensity factors are invariant under the 90 degrees of rotation of the material principal axes

piecewise cubic interpolatory polynomials and approximate solution of singular integral equations abstract a method based on piecewise cubic interpolatory polynomials for the numerical solution of singular integral equations of cauchy type is considered the class of interpolatory functions under consideration is the class of x splines of particular interest is the size of the matrix of linear algebraic equations it is shown that one can get matrices of order n n where n is the number of node points on the interval of integration the analysis becomes considerably simpler for the case when the node points are equally spaced numerical results from the x spline method are compared with those obtained using other methods

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

optimal hankel model reduction for nonminimal systems a basis free descriptor system representation is shown to facilitate the computation of all minimum degree and optimal kth order all pass extensions and hankel norm approximants the descriptor representation has the same simple form for both the optimal and suboptimal cases the method makes hankel model reduction practical for nonminimal and nearly nonminimal systems by eliminating the ill conditioned calculation of a minimal balanced realization a simple numerically sound method based on singular value decomposition enables the results to be expressed in state space form

singularities interacting with interfaces and cracks abstract solutions to singularities such as point force point moment edge dislocation and transformation strain spot embedded in bonded elastic blocks of dissimilar materials are found to relate to the solutions to the same singularities in an infinite homogeneous plane by a formula independent of the nature of the singularities this universal result is then used to analyze the interactions between singularities and interface cracks the complete solutions and stress intensity factors are presented for two important interface crack configurations

spline collocation for singular integro differential equations over 0 1 this paper analyses the convergence of spline collocation methods for singular integro differential equations over the interval 0 1 as trial functions we utilize smooth polynomial splines the degree of which coincides with the order of the equation depending on the choice of collocation points we obtain sufficient and even necessary conditions for the convergence in sobolev norms we give asymptotic error estimates and some numerical results

statistical shape analysis clustering learning and testing using a differential geometric treatment of planar shapes we present tools for 1 hierarchical clustering of imaged objects according to the shapes of their boundaries 2 learning of probability models for clusters of shapes and 3 testing of newly observed shapes under competing probability models clustering at any level of hierarchy is performed using a minimum variance type criterion and a markov process statistical means of clusters provide shapes to be clustered at the next higher level thus building a hierarchy of shapes using finite dimensional approximations of spaces tangent to the shape space at sample means we implicitly impose probability models on the shape space and results are illustrated via random sampling and classification hypothesis testing together hierarchical clustering and hypothesis testing provide an efficient framework for shape retrieval examples are presented using shapes and images from eth surrey and amcom databases

explicit solutions for collinear interface crack problems abstract by combining stroh s formalism the method of analytical continuation and the translating technique introduced in this paper the explicit full domain solutions for the problems of collinear interface cracks between dissimilar anisotropic media have been presented to define a proper bimaterial stress intensity factor the full domain solution is applied to the near tip of an interface crack the results show that the effect of material properties is totally reflected through the oscillation index e and the bimaterial constants d 11 d 22 and d 33 which have inverse relations with young s moduli e 1 e 2 and shear modulus g 23 g 31 respectively for the purpose of illustration three typical examples are solved explicitly such as a semi infinite interface crack a finite interface crack and two collinear interface cracks

invariance identifiability and morphometrics the form of an object is that characteristic that remains invariant under a group of transformations comprising translation rotation and possibly reflection group invariance thus naturally plays an important role in the statistical analysis of forms we examine the existing methods for the statistical analysis of form from the invariance perspective we begin with a review of the important basic ideas behind invariance and derive a maximal invariant under the group of transformations consisting of rotation reflection and translation and its distribution under the gaussian and elliptically symmetric perturbation models we first consider the single sample case and discuss the issue of identifiability of the parameters we show that method of moments estimators based on the distances between landmarks and maximum likelihood estimators mles based on the size and shape coordinates are invariant and estimate identifiable parameters however a number of commonly used methods do not we compare the stati

the use of spline on spline for the approximation of cauchy principal value integrals in this paper the spline on spline interpolatory polynomials are used to approximate cauchy principal value integrals and their derivatives this technique yields better results than using the derivatives of the traditional single spline functions which has been considered by orsi improved order of convergence for these approximations are discussed

consistency of procrustes estimators lele has shown that the procrustes estimator of form is inconsistent and raised the question about the consistency of the procrustes estimator of shape in this paper the consistency of estimators of form and shape is studied under various assumptions in particular it is shown that the procrustes estimator of shape is consistent under the assumption of an isotropic error distribution and that consistency breaks down if the assumption of isotropy is relaxed the relevance of these results for practical shape analysis is discussed as a by product some new results are derived for the offset uniform distribution from directional data

model reduction of neutral linear and nonlinear time invariant time delay systems with discrete and distributed delays the problem of model reduction by moment matching for linear and nonlinear differential time delay systems is studied the class of models considered includes neutral differential time delay systems with discrete delays and distributed delays the description of moment is revisited by means of a sylvester like equation for linear time delay systems and by means of the center manifold theory for nonlinear time delay systems in addition the moments at infinity are characterized for both linear and nonlinear time delay systems parameterized families of models achieving moment matching are given the parameters can be exploited to derive delay free reduced order models or time delay reduced order models with additional properties e g interpolation at an arbitrary large number of points finally the problem of obtaining a reduced order model of an unstable system is discussed and solved

slightly curved or kinked cracks in anisotropic elastic solids abstract slightly curved or kinked cracks in anisotropic elastic solids are studied via a perturbation analysis valid for the second order accuracy in the deviation of the crack surfaces from a straight line the analysis is based on complex variable representations in the stroh formalism and known solutions for a perfectly straight reference crack first and second order perturbation solutions are given for the stress intensity factors at the tip of a finite curved crack under remote stresses in a solid with arbitrary anisotropy and are used to establish an approximate relationship between the apparent and local stress intensities at a crack tip in view of the possible shielding effects of the crack surface morphology near the tip the role of anisotropy is examined for a number of problems including circular arc cracks slightly jogged or kinked cracks cosine wavy cracks etc in several cases further insight into the anisotropic effect on crack curving or kinking is provided by specializing the general perturbation formulae to cracks in materials with orthotropic symmetry in the limiting case of a crack with an infinitesimal branch length comparison with numerical results reported in the literature indicates that our perturbation solutions are accurate over the full range of practically important branch angles up to nearly 150 for anisotropic elastic solids various fracture criteria such as those based on energy release rate g and stress intensity factors k 1 and k 11 cease to give consistent predictions on crack behaviour by investigating the effects of mode mixity material anisotropy and non singular t stress on the behaviour of a nearly symmetric crack path in an orthotropic solid we show that the k based fracture criteria lead to very peculiar perhaps even physically unreasonable predictions such as i branch angle at a mixed mode crack tip becomes infinite as the compliances perpendicular and parallel to the crack s vv s vv reach a four fold difference ii the sign of the branch angle is reversed when s vv s vv 4 with the consequence of reversing the role of the non singular t stress on the stability of a symmetric fracture path namely the path becomes stable for t 0 and unstable for t k t becomes a local minimum with respect to crack angle once s vv s vv 4 in contrast the g based criterion gives reasonable predictions such as i cracks under mixed mode loading always tend to branch toward symmetric orientations ii material and loading asymmetries play an equivalent role in affecting crack branching near a symmetric orientation iii a compressive t stress always tends to stabilize a symmetric fracture path while a tensile t stress destabilizes such a path for general crack path stability the role of anisotropy can be manifested through i the value of t stress at the crack tip and ii the variation of fracture resistance with respect to crack orientation

fractional balanced reduction model reduction via fractional representation a method for the model reduction of finite dimensional linear time invariant fdlti plants is proposed which uses normalized fractional representations is proposed the method dubbed fractional balanced reduction applies balance and truncate to a special representation of the graph operator of the plant this operation yields the graph operator of a reduced order plant the method has such properties as existence of an a priori error bound in the graph metric and preservation of sector containment coupling fractional representations with principal component analysis gives a model reduction method that is capable of producing in a numerically stable way a good reduced order model using the whole full order model sector properties are also preserved these are useful for deciding stability when nonlinearities are in the loop

spline product quadrature rules for cauchy singular integrals abstract in this paper product quadrature rules based on cubic spline interpolation are obtained for the numerical evaluation of cauchy singular integrals and an error bound is proposed some numerical examples that test the performance of the rules are given

moments of random variables a systems theoretic interpretation moments of continuous random variables admitting a probability density function are studied we show that under certain assumptions the moments of a random variable can be characterized in terms of a sylvester equation and of the steady state output response of a specific interconnected system this allows to interpret well known notions and results of probability theory and statistics in the language of systems theory including the sum of independent random variables the notion of mixture distribution and results from renewal theory the theory developed is based on tools from center manifold theory the theory of the steady state response of nonlinear systems and the theory of output regulation our formalism is illustrated by means of several examples and can be easily adapted to the case of discrete and multivariate random variables

3 d boundary element method for cracks in a plate new fundamental solutions which automatically satisfy boundary conditions at the interfaces of an elastic plate perfectly bonded to two elastic halfspaces are implemented in a 3 d boundary element method bem for crack problems the bem features a new integration scheme for highly singular kernels the capability is achieved through a part analytic and part numerical integration procedure such that the analytic part of the integration is similar for all slip opening variations part through elliptic cracks in an elastic plate with traction free surfaces are analysed and the stress intensity factor sif values along the crack front are found to compare favourably with widely accepted numerically obtained sif results by raju and newman 1

dimension reduction for multivariate response data this article concerns the analysis of multivariate response data with multivariate regressors methods for reducing the dimensionality of response variables are developed with the goal of preserving as much regression information as possible we note parallels between this goal and the goal of sliced inverse regression which intends to reduce the regressor dimension in a univariate regression while preserving as much regression information as possible a detailed discussion is given for the case where the response is a curve measured at fixed points the problem in this setting is to select basis functions for fitting an aggregate of curves we propose that instead of focusing on goodness of fit attention should be shifted to the problem of explaining the variation of the curves in terms of the regressor variables a data adaptive basis searching method based on dimension reduction theory is proposed simulation results and an application to a climatology problem are given

on the numerical treatment of the singular integral equation of the second kind here a numerical treatment for solving the integral equation of the second kind with cauchy kernel is presented the singular term has been removed and the solution in the legendre polynomial form has been used to obtain a system of linear algebraic equation this system is solved numerically

effects of change of reference coordinates on the stress analyses of anisotropic elastic materials abstract when the displacements are independent of the x 3 coordinate the eigenvalues and eigenvectors of the anisotropic elasticity constants depend on the orientation of the x 1 x 2 axes it is shown that the components of the eigenvectors which are complex values transform according to the law of transformation for tensors of order one the transformation of the eigenvalues is more complicated the effects of change of the reference coordinates on the form of general solution are discussed also discussed is the form of general solution when the eigenvalue p is a multiple root finally we show that as the angle of rotation of the coordinate axes varies from 0 to 2 each p traverses a circle in the complex plane which is orthogonal to the unit circle with center at the origin a graphical solution of the eigenvalue p for a given is presented some functions of p which are invariant to the rotation of the coordinate axes are obtained

an exact correspondence between plane piezoelectricity and generalized plane strain in elasticity we consider an anisotropic body bounded by a cylindrical surface which is infinitely long in the axial direction suppose the body is loaded in such a way that the field variables do not vary along the generators an exact correspondence is established between the plane piezoelectric equations and those of generalized plane strain in elasticity in particular we show that by setting a linkage between the two sets of material constants any problem of a plane deformation in piezoelectricity may be solved as a generalized plane strain in elasticity and vice versa the assertation is true for rectilinearly as well as for cylindrically anisotropic solids the equivalence is found for the most general anisotropic case which links the fields between a monoclinic piezoelectric body of class m and a fully anisotropic triclinic elastic solid a few degenerate systems are also identified in addition these correspondences can be extended to inhomogeneous media applied to composite materials or polycrystalline aggregates they imply that results for effective elastic tensors immediately give the formulae for the effective electroelastic tensors and vice versa we also demonstrate that the connection can be used to extend the scope of the invariant stress theorem proved by cherkaev lurie and milton in illustration we present solutions for the plane problem of an elliptical inhomogeneity in an unbounded piezoelectric medium subjected to a uniform loading at infinity lekhnitskii9s complex potential approach together with the conformal mapping technique are employed general solutions of the fields inside the inhomogeneity and the matrix are obtained the results are analytically proven to be identical with the existing solutions of the corresponding purely elastic boundary value problem

a finite element approach to three dimensional singular stress states in anisotropic multi material wedges and junctions abstract a finite element formulation is developed to determine the order and angular variation of singular stress states due to material and geometric discontinuities in anisotropic materials the formulation applies to any two dimensional geometry that is prismatic in the third direction and has three dimensional displacement fields in some special cases the three dimensional fields become uncoupled antiplane and inplane fields and this formulation yields the uncoupled results the formulation provides for the determination of the asymptotic stress and displacement fields present at interior singular points of three dimensional structures the displacement field of the sectorial finite element is quadratic in the angular coordinate direction and asymptotic in the radial direction measured from the singular point the formulation of yamada and okumura 1983 hybrid and mixed finite element methods pp 325 343 wiley chichester for inplane problems is adapted for this purpose the simplicity and accuracy of the formulation are demonstrated by comparison with several analytical solutions for both isotropic and anisotropic multi material wedges and junctions the nature and speed of convergence associated with the element suggests that it could be employed in developing two dimensional and three dimensional enriched elements for use along with standard elements to yield accurate and computationally efficient solutions to problems having complex global geometries leading to singular stress states

a model free test for reduced rank in multivariate regression we propose a test of dimension in multivariate regression this test is in the spirit of tests on the rank of the coefficient matrix in a multivariate linear model but it does not require a prespecified model the test may be particularly useful at the outset of an analysis before a multivariate model is posited because it can lead to low dimensional summary plots that are inferred to contain all of the sample information on the multivariate mean function

an exact and explicit treatment of an elliptic hole problem in thermopiezoelectric media abstract this paper presents an exact solution for the problem of an elliptic hole or a crack in a thermopiezoelectric solid first based on the extended version of eshelby stroh s formulation the generalized 2d problems of an elliptical hole in a thermopiezoelectric medium subject to uniform heat flow and mechanical electrical loads at infinity are studied according to exact boundary conditions at the rim of the hole the complex potentials in the medium and the electric field inside the hole are obtained in closed form respectively then when the hole degenerates into a crack the explicit solutions for the field intensity factors near the crack tip and the electric field inside the crack are presented it is shown that the singularities of all the field are dependent on the material constants the applied heat load and mechanical loads at infinity but not on the applied electric loads it is also found that the electric field inside the crack is linearly variable which is different from the result based on the impermeable crack model

partition of unity enrichment for bimaterial interface cracks partition of unity enrichment techniques are developed for bimaterial interface cracks a discontinuous function and the two dimensional near tip asymptotic displacement functions are added to the finite element approximation using the framework of partition of unity this enables the domain to be modelled by finite elements without explicitly meshing the crack surfaces the crack tip enrichment functions are chosen as those that span the asymptotic displacement fields for an interfacial crack the concept of partition of unity facilitates the incorporation of the oscillatory nature of the singularity within a conforming finite element approximation the mixed mode complex stress intensity factors for bimaterial interfacial cracks are numerically evaluated using the domain form of the interaction integral good agreement between the numerical results and the reference solutions for benchmark interfacial crack problems is realized copyright 2004 john wiley sons ltd

a generalized entropy criterion for nevanlinna pick interpolation with degree constraint we present a generalized entropy criterion for solving the rational nevanlinna pick problem for n 1 interpolating conditions and the degree of interpolants bounded by n the primal problem of maximizing this entropy gain has a very well behaved dual problem this dual is a convex optimization problem in a finite dimensional space and gives rise to an algorithm for finding all interpolants which are positive real and rational of degree at most n the criterion requires a selection of a monic schur polynomial of degree n it follows that this class of monic polynomials completely parameterizes all such rational interpolants and it therefore provides a set of design parameters for specifying such interpolants the algorithm is implemented in a state space form and applied to several illustrative problems in systems and control namely sensitivity minimization maximal power transfer and spectral estimation

a unified definition for stress intensity factors of interface corners and cracks abstract based upon linear fracture mechanics it is well known that the singular order of stresses near the crack tip in homogeneous materials is a constant value 1 2 which is nothing to do with the material properties for the interface cracks between two dissimilar materials the near tip stresses are oscillatory due to the order of singularity being 1 2 i e and 1 2 the oscillation index e is a constant related to the elastic properties of both materials while for the general interface corners their singular orders depend on the corner angle as well as the elastic properties of the materials owing to the difference of the singular orders of homogeneous cracks interface cracks and interface corners their associated stress intensity factors are usually defined separately and even not compatibly since homogenous cracks and interface cracks are just special cases of interface corners in order to build a direct connection among them a unified definition for their stress intensity factors is proposed in this paper based upon the analytical solutions obtained previously for the multibonded anisotropic wedges the near tip solutions for the general interface corners have been divided into five different categories depending on whether the singular order is distinct or repeated real or complex to provide a stable and efficient computing approach for the general mixed mode stress intensity factors the path independent h integral based on reciprocal theorem of betti and rayleigh is established in this paper the complementary solutions needed for calculation of h integral are also provided in this paper to illustrate our results several different kinds of examples are shown such as cracks in homogenous isotropic or anisotropic materials central or edge notches in isotropic materials interface cracks and interface corners between two dissimilar materials

model reduction by matching the steady state response of explicit signal generators model reduction by moment matching for interpolation signals which do not have an implicit model i e they do not satisfy a differential equation is considered particular attention is devoted to discontinuous possibly periodic signals the notion of moment is reformulated using an integral matrix equation it is shown that under specific conditions the new definition and the one based on the sylvester equation are equivalent new parameterized families of models achieving moment matching are given the results are illustrated by means of a numerical example

singularities interfaces and cracks in dissimilar anisotropic media for a non pathological bimaterial in which an interface crack displays no oscillatory behaviour it is observed that apart possibly from the stress intensity factors the structure of the near tip field in each of the two blocks is independent of the elastic moduli of the other block collinear interface cracks are analysed under this non oscillatory condition and a simple rule is formulated that allows one to construct the complete solutions from mode iii solutions in an isotropic homogeneous medium the general interfacial crack tip field is found to consist of a two dimensional oscillatory singularity and a one dimensional square root singularity a complex and a real stress intensity factors are proposed to scale the two singularities respectively owing to anisotropy a peculiar fact is that the complex stress intensity factor scaling the oscillatory fields however defined does not recover the classical stress intensity factors as the bimaterial degenerates to be non pathological collinear crack problems are also formulated in this context and a strikingly simple mathematical structure is identified interactive solutions for singularity interface and singularity interface crack are obtained the general results are specialized to decoupled antiplane and in plane deformations for this important case it is found that if a material pair is non pathological for one set of relative orientations of the interface and the two solids it is non pathological for any set of orientations for bonded orthotropic materials an intuitive choice of the principal measures of elastic anisotropy and dissimilarity is rationalized a complex variable representation is presented for a class of degenerate orthotropic materials throughout the paper the equivalence of the lekhnitskii and stroh formalisms is emphasized the article concludes with a formal statement of interfacial fracture mechanics for anisotropic solids

homogeneity analysis with k sets of variables an alternating least squares method with optimal scaling features homogeneity analysis or multiple correspondence analysis is usually applied tok separate variables in this paper we apply it to sets of variables by using sums within sets the resulting technique is called overals it uses the notion of optimal scaling with transformations that can be multiple or single the single transformations consist of three types nominal ordinal and numerical the corresponding overals computer program minimizes a least squares loss function by using an alternating least squares algorithm many existing linear and nonlinear multivariate analysis techniques are shown to be special cases of overals an application to data from an epidemiological survey is presented

the degeneration of image singularities from anisotropic materials to isotropic materials for an elastic half plane abstract the degeneration of image singularities from an anisotropic material to an isotropic material for a half plane is discussed in this study the green s functions for anisotropic and isotropic half planes with traction free boundary subjected to concentrated forces and dislocations have been obtained by many authors it was commonly accepted that the solution of isotropic problem cannot be derived from anisotropic solutions however we believe that this possibility exists as we will demonstrate in this paper anisotropic materials include only image singularities of order o 1 r i e forces and dislocations existing on image points there are many image points for anisotropic materials and the locations of these image points depend on the material constants however isotropic materials have only one image point with higher order image singularities o 1 r2 o 1 r3 from the analysis provided in this study it is found that the higher order image singularities for an isotropic half plane are generated by combining the concentrated forces and dislocations when an anisotropic material degenerates to an isotropic material the solutions of higher order image singularities for isotropic material are dependent therefore these image singularities can be combined to form only three or four simpler image singularities acting on an image point of the isotropic material

sylvester equations and projection based model reduction in this paper we establish a connection between krylov subspace techniques for multipoint pade interpolation and the use of sylvester equations for constructing reduced order models we also briefly point out that this connection partly extends to adi type techniques and to the smith iteration for computing approximate solutions of lyapunov equations

multirate modeling and control design for switched mode power converters a class of pulse width modulated pwm systems that includes many switched mode power converters is considered a new multirate low frequency sampling pwm scheme is introduced that not only provides the possibility for new types of operation but also unifies the switching logic corresponding to established pwm schemes an exact nonlinear discrete time model is used to develop multirate nonlinear controllers and observers based on map inversion concepts this nonlinear design methodology is widely applicable requiring only that the continuous time small signal average model be controllable and observable

common canonical variates in k independent groups abstract suppose that data are collected from k independent but closely related populations and that a canonical correlation analysis is to be performed within each group in such a situation it may turn out that the transformation to canonical variates is similar across all k groups the common canonical variates model studied in this article assumes that the transformation to canonical variates is actually identical across groups while the canonical correlations may vary we develop some algebra of the common canonical variates model discuss normal theory maximum likelihood estimation and likelihood ratio testing give illustrative examples

numerical solution of fredholm volterra integral equation in one dimension with time dependent in this paper we consider fredholm volterra integral equation in one dimension with time dependent the kernel of the fredholm term is assumed to be singular while the kernel of volterra term is continuous the existence of the solution is proved and a numerical method is presented to get an approximate to such solution

numerical treatment of solving singular integral equations by using sinc approximations abstract in this paper we propose numerical treatment for singular integral equations the methods are developed by means of the sinc approximation with smoothing transformations such approximation is an effective technique against the singularities of the equations and achieves exponential convergence therefore the methods improve conventional results where only polynomial convergence have been reported the resulting algebraic system is solved by least squares approximation and leap frog algorithm estimation of errors of the approximate solution is presented some experimental tests are presented to show the efficient of the proposed methods

boundary integral equations for 2d elasticity and its application in discrete dislocation dynamics simulation in finite body 1 general theory abstract a unified approach originating from cauchy integral theorem is presented to derive boundary integral equations for two dimensional elasticity problems several sets of boundary integral equations are derived and their relations are revealed explicit expressions for materials with different symmetry planes are listed special attention is given to the formulation that is based on the tractions and the tangential derivatives of displacements along solid boundary since its integral kernels have the weakest singularities the formulation is further extended to include singular points such as dislocations and line forces in a finite body so that the singular stress field can be directly obtained from solving the integral equations on the external boundary without involving the linear superposition technique that was often used in the literature its application in simulating discrete dislocation motion in a finite solid body is discussed

the complex bingham distribution and shape analysis summary a complex version of the bingham distribution is defined on the unit complex sphere in ck various statistical properties including asymptotic normality under high concentration are derived symmetries in the distribution make it a natural tool for the analysis of the shape of landmark data in two dimensions strengths and weaknesses of this approach are investigated links and contrasts with bookstein co ordinates are discussed a hybrid approach to principal component analysis based on complex and real co ordinates is suggested

an informational measure of association and dimension reduction for multiple sets and groups with applications in morphometric analysis in this article we propose a new general index that measures relationships between multiple sets of random vectors this index is based on kullback leibler kl information which measures linear or nonlinear dependence between multiple sets using joint and marginal densities of affine transformations of the random vectors estimates of the matrixes are obtained by maximizing the kl information and are shown to be consistent the motivation for introducing such an index comes from morphological integration studies a topic in biological science as a special case of this index we define an overall measure of association and two other measures for dimension reduction the use of these measures is illustrated through real data analysis in morphometric studies and extensive simulations and their performance is compared with that of approaches based on canonical correlation analysis extensions of the aforementioned measures to multiple groups are also discussed in contrast to canonical correlation analysi

a piecewise linear quadrature of cauchy singular integrals the aim of this paper is to develop an algorithm for the quadrature weights induced by the piecewise polynomial quadrature of cauchy singular integrals we present a simple calculation scheme for the quadrature weights some numerical test problems are given

the use of piecewise quadratic polynomials for the solution of singular integral equations of cauchy type abstract a method for the numerical solution of singular integral equations of cauchy type is developed the unknown function is expressed as a product of a weight function and a continuous function t the continuous function t is approximated by piecewise quadratic polynomials and the singular integral equation is reduced to a linear algebraic system numerical examples are given and comparisons are made with the widely used gauss type methods

relative entropy measures of multivariate dependence abstract there has been a lot of work on measures of dependence or association for bivariate probability distributions or bivariate data these measures usually assume that the variables are both continuous or both categorical in comparison there is very little work on multivariate or conditional measures of dependence the purpose of this article is to discuss measures of multivariate dependence and measures of conditional dependence based on relative entropies these measures are conceptually very general as they can be used for a set of variables that can be a mixture of continuous ordinal categorical and nominal categorical variables for continuous or ordinal categorical variables a certain transformation of relative entropy to the interval 0 1 leads to generalizations of the correlation multiple correlation and partial correlation coefficients if all variables are nominal categorical the relative entropies are standardized to take a maximum of 1 and then transformed so that in the bivar

multivariate association and dimension reduction a generalization of canonical correlation analysis summary in this article we propose a new generalized index to recover relationships between two sets of random vectors by finding the vector projections that minimize an l 2 distance between each projected vector and an unknown function of the other the unknown functions are estimated using the nadaraya watson smoother extensions to multiple sets and groups of multiple sets are also discussed and a bootstrap procedure is developed to detect the number of significant relationships all the proposed methods are assessed through extensive simulations and real data analyses in particular for environmental data from los angeles county we apply our multiple set methodology to study relationships between mortality weather and pollutants vectors here we detect existence of both linear and nonlinear relationships between the dimension reduced vectors which are then used to build nonlinear time series regression models for the dimension reduced mortality vector these findings also illustrate potential use of our method in many other applications a comprehensive assessment of our methodologies along with their theoretical properties are given in a web appendix