Unbounded Matrix Research Articles

Nonsymmetric cavity formation at a circular inclusion under remote equibiaxial load

This paper examines the phenomenon of cavity formation through an analysis of the problem of a plane circular elastic inclusion embedded in an unbounded elastic matrix subject to a remote equibiaxial load. Consistent with infinitesimal strain kinematics, nonlinear behavior is confined to a cohesive zone so that inclusion-matrix interaction is characterized by a nonlinear interface force-interface separation law requiring a characteristic length for its prescription. Equilibrium solutions for both rotationally symmetric and nonsymmetric cavity shapes are sought based on an integral equation formulation together with known elasticity solutions for circular domains. For values of remote load, interface strength and elastic moduli within certain bounds only rotationally symmetric cavities occur under decreasing characteristic length-inclusion radius ratio. At other parameter values the existence of nonsymmetric cavities is studied by performing a linearized bifurcation analysis about the rotationally symmetric equilibrium state. A post bifurcation analysis is carried out by reducing the governing integral equations to a truncated set of nonlinear algebraic equations and analysing those. Stability of equilibrium states is assessed with the Hadamard stability definition. Calculations for the interfacial tractions are carried out as well. The study reveals that rotationally symmetric cavities must give way to the abrupt formation of stable nonsymmetric cavities when the interface force attains its maximum value. Thus, the phenomenon of ductile decohesion, or the gradual opening of a cavity coincident with an unloading of the interface, cannot occur (for the system being studied) without artificially constraining the inclusion against rigid displacement.

Dislocation Inside, Outside, or on the Interface of an Anisotropic Elliptical Inclusion

The plane elasticity problem of dislocation inside, outside, or on the interface of an anisotropic elliptical inclusion in an unbounded anisotropic matrix is considered. A general solution for the stresses and deformations in the entire domain is obtained by applying the Stroh’s formalism and the method of analytical continuation. Since the general solutions have not been found in the literature, in order to verify the present solution, comparison is made with some special cases of which the analytical solutions exist, which shows that our results are exact and universal.

Separation at a circular interface under biaxial load

T his p aper is concerned with the plane problem of the decohesion of a smooth circular elastic inclusion from an unbounded elastic matrix subject to a remote biaxial load. The phenomenon of brittle decohesion is treated as a problem of bifurcation of equilibrium separation at the inclusion-matrix interface. In accordance with other cohesive zone type models the interface is characterized by a constitutive equation relating tractions across the interface to the displacement discontinuity that develops there. The specific form considered in this work is the physically based nonlinear force law of Ferrante, Smith and Rose that couples the normal component of displacement jump to the normal component of interface traction and which requires a characteristic length for its prescription. Under decreasing values of characteristic length to inclusion radius ratio, three types of separation behavior (ductile decohesion, brittle decohesion, closure) can occur provided the remote load, interface strength and elastic moduli of inclusion and matrix are within the required bounds. Within this context of interfacial separation, the phenomenon of void nucleation is briefly examined and a connection is made between the classical nucleation criteria of critical interfacial stress and critical energy release and the criteria for ductile and brittle decohesion. Interaction effects are analysed in an approximate way by considering the effects of decohesion on a hypothetical, rigidly bonded inclusion with the same elastic moduli as the matrix.

Spectral Measures, Orthogonal Polynomials, and Absolute Continuity

This paper studies the spectral measure of an unbounded tridiagonal matrix operator for which the matrix entries satisfy a certain growth condition, and presents a sufficient condition for the existence of an absolutely continuous part. The results are related to a class of orthogonal polynomials with exponential weights.

The edge dislocation inside an elliptical inclusion

The plane elasticity problem of an edge dislocation located within an elliptical inclusion in an unbounded matrix is considered. A general solution to this problem is obtained in terms of complex potential functions and all coefficients in the series representation of these potential functions are explicitly obtained. Convergence of these series is thus assured. Two specific cases of this general solution are considered in detail. The first case considers a very long, thin inclusion corresponding physically to the geometry of typical crazes in glassy polymers. the second case considers the almost circular inclusion representative of a number of crystal defects and imperfections. For the second case, expressions for the resultant force on the dislocation agree with results previously obtained for the limiting circular inclusions.

Singular and unbounded matrix solutions for both time-dependent matrix and vector differential systems

Singular and unbounded matrix solutions for both time-dependent matrix and vector differential systems

Multiple Circular Inclusion Problems in Plane Elastostatics

The problem of an unbounded elastic matrix containing any number of elastic inclusions is considered. The inclusions can have any radii and elastic moduli. Furthermore, the spacing of the inclusions can be arbitrary. The solution for the cases of uniaxial tension and in-plane bending is found by the Schwarz alternating method. Graphical results are presented for a number of examples.

On the application of Newton's method in a singular case

ONE of the most frequently used methods for solving sets of non-linear equat ions g(x) = d, where g(x) is a fairly smooth vector function of a vector argument x with N components g i(x), is Newton's method (see [1, 2]). The computational scheme for this method can be written as: Γ( y ( k) ) z ( k) = g( y ( k) ) − d, y ( k + 1) = y ( k) − z ( k) ( k = 0,1,2,…). Here Γ( y ( k) ) is the Jacobian matrix Γ(y) = ( ∂g ∂y 1 , ∂g ∂y 1 ,2., ∂g ∂y N ) , taken with y = y ( k) . In many cases in practice, in particular in thesolution of non-linear boundary value problems, some other computational scheme is used for the method which replaces the matrix Γ(y (k) in (2) by the difference matrix (R k,y (k=(g(y (k)+h ke 1)−g(y (k)/h k;…;(g(y (k)+h ke N)−g(y (k)/h k where e i is the i-th coordinate unit vector. This scheme takes the form Rh k , y ( k)) z( suk) = g( y ( k) )− d, y ( k+1) = y ( k) − z ( k) (3) ( k = 0,1,2,…), h k are real numbers from some given convergent sequence. In future when we refer to Newton's method we shall mean its modification (3). The convergence of the iterative processes (2) and (3) is generally proved on the assumption that the matrices Γ( y) and R( h, y) are nonsingular in the neighbourhood of the solution x ∗ of (1) (see for instance [1]). In [3] the proof of the convergence of process (2) is given also for the case where the matrix Γ( y) can be singular in the neighbourhood of the point x ∗, but has a constant rank, and all systems in (2) are solvable. In the present paper we shall consider the question of the realization of the iterative process (3) and its convergence in the case where the matrix R( h, y) has a variable rank within the limits 1 to n in the neighbourhood of the solution x ∗, i.e. we shall assume only that the matrix R( h, y) is non-zero in this neighbourhood. In a less general statement this question has been considered for process (2) in [4]. We shall explain what computational difficulties arise in using Newton's method with an unbounded matrix RR −1( h, y) in the neighbourhood of the solution x ∗. Let x ∗ be an isolated singular point of the matrix Γ( y) (i.e. the matrixΓF( x ∗) is singular), and let limR(h k,y (k)) = Γ(x∗) . k→∞ Two cases are then possible. In the first the matrices R( h k , y ( k) ) tend to the singular matrix, remaining non-singular matrices on each iterative process (3). Here we encounter the need to solve with large k systems, with ill-conditioned matrices. The second case consists of the possibility of the appearance of singular matrices R( h k , y ( k) ) on some iterations in (3) (or on all with large k). Consequently in the realization of process (3) in this case we must be able to solve sets of linear equations with a singular matrix, if these systems are solvable, or to reduce the system to solvable form if it has no solution.

On Boundaries and Lateral Conditions for the Kolmogorov Differential Equations

Kolmogorov [13] has shown that under appropriate regularity conditions such a P(t) will satisfy a pair of matrix differential equations (adjoint to each other). Since then a great many investigations have been concerned with problems of non-uniqueness, with solutions which fail to satisfy the adjoint equation, and with other pathological situations. It is the purpose of the present paper to show that the unbounded matrix operator Q on which the Kolmogorov equations depend shares the essential properties of second order elliptic differential operators, such as the SturmLiouville operator in one dimension or the Laplacian A in two dimensions. The Kolmogorov equations are then the exact counterpart of the heat equation ut = Au, except that due to the essential lack of symmetry of Q the adjoint equation bears little resemblance to the original. Our solutions will depend on boundary conditions in which one easily recognizes the classical boundary conditions of, say, the theory of harmonic functions. Of course, the latter depend on normal derivatives and no analogue to partial derivatives exists in our case. However, our boundary conditions are expressed in terms of functionals which remain meaningful under the most general circumstances, and can be applied to harmonic functions. There they reduce to the normal derivatives whenever the boundary is sufficiently smooth, but they give the proper expression for an arbitrary boundary. The perfect analogy of the Kolmogorov equations with diffusion equations explains the many phenomena and problems connected with the non-uniqueness of the solutions, and leads to new insights. Consider, for simplicity, an isolated point X of the boundary. It is possible to extend Q (that is, the Kolmogorov equations) and the matrix P(t) to the countable set E + a. This extension is not uniquely determined, but it involves very little arbitrariness. It adds a new row and a new column to Q, and these bear no similarity to the remaining rows and columns of Q. This leads in the most natural manner to generalizations of the