Unbounded Matrix Research Articles

The prime spectrum of algebras of quadratic growth

We study prime algebras of quadratic growth. Our first result is that if A is a prime monomial algebra of quadratic growth then A has finitely many prime ideals P such that A / P has GK dimension one. This shows that prime monomial algebras of quadratic growth have bounded matrix images. We next show that a prime graded algebra of quadratic growth has the property that the intersection of the non-zero prime ideals P such that A / P has GK dimension 2 is non-zero, provided there is at least one such ideal. From this we conclude that a prime monomial algebra of quadratic growth is either primitive or has non-zero locally nilpotent Jacobson radical. Finally, we show that there exists a prime monomial algebra A of GK dimension two with unbounded matrix images and thus the quadratic growth hypothesis is necessary to conclude that there are only finitely many prime ideals such that A / P has GK dimension 1.

Electrostatic interactions of arbitrarily dispersed spheres with spherically transversely isotropic constituents

We propose a theoretical framework for evaluation of electrostatic potentials in an unbounded isotropic matrix containing many arbitrarily dispersed spherical inclusions subjected to a remotely prescribed potential field. The inclusions could be isotropic or spherically transversely isotropic, and may have different sizes with different conductivities. The approach is based on a multipole expansion formalism, together with a construction of consistency conditions and translation operators. This procedure generalizes the approach of the classic work of Lord Rayleigh [L. Rayleigh, On the influence of obstacles arranged in rectangular order upon the properties of a medium, Philos. Mag. 34 (1892) 481–502] for a periodic array to an arbitrary dispersion of spheres. In the formulation, we expand the potential field versus various local coordinates with origins positioned at each inclusion’s center. We show that the coefficients of field expansions can be written in the form of an infinite set of linear algebraic equations. Numerical results are presented for a few different configurations. For the case of an infinite space containing two spheres subjected to a uniform intensity, we have verified our solutions with those obtained from the bispherical coordinate transformation. The derived field solutions can be used to assess the effective conductivity of a random heterogeneous medium.

Defect propagation at a circular interface

In this paper a nonlinear, nonuniform cohesive zone is employed to study the detailed features of quasi-static defect evolution in a simple, planar elastic system consisting of a circular inclusion embedded in an unbounded matrix subject to different remote loading configurations. The inclusion–matrix interface is assumed to be described by Needleman-type force-separation relations characterized by an interface strength, a characteristic force length and a shear stiffness parameter. Interface defects are modeled by an interface strength which varies with interface coordinate. Infinitesimal strain equilibrium solutions, which allow for rigid body inclusion displacement, are sought by eigenfunction approximation of the solution of the governing interfacial integral equations. For equibiaxial tension, quasi-static defect initiation and propagation occur under increasing remote load. For decreasing characteristic force length, a transition occurs from more or less uniform decohesion along the bond line to propagation of a crack-like defect. In the later case a critical failure load is well defined and interface failure is shown to be defect dominated (brittle decohesion). For interfaces with large characteristic force length, the matrix “lifts off” the inclusion accompanied by a delay in defect propagation (ductile decohesion). The decohesion modes ultimately give rise to a cavity with the inclusion situated within it on the side opposite to the original defect. Results for small characteristic force length show consistency with England’s results for the sharp arc crack on a circular interface (England AH (1966) ASME J Appl Mech 33:637–640) Stress oscillation and contact at the tip of the defect are observed primarily for small characteristic force lengths under extremely small loading. Results for remote tension, compression and pure shear loading are discussed as well. In the final section of the paper the results obtained in the first part are utilized to estimate the plane effective bulk response of a composite containing a dilute distribution of inclusions with randomly oriented interface defects.

New phenomena concerning a screw dislocation interacting with two imperfect interfaces

Dislocation mobility and stability in nanocrystals and electronic materials are influenced by the material composition and interface conditions. Its mobility and stability then affect the mechanical behaviors of the composites. In this paper, we first address, in detail, the problem of a screw dislocation located in an annular coating layer which is imperfectly bonded to the inner circular inhomogeneity and to the outer unbounded matrix. Both the inhomogeneity–coating interface and coating–matrix interface are modeled by a linear spring with vanishing thickness to account for the possible damage occurring on the interface. An analytic solution in series form is derived by means of complex variable method, with all the unknown constants being determined explicitly. The solution is then applied to the study of the dislocation mobility and stability due to its interaction with the two imperfect interfaces. The most interesting finding is that when the middle coating layer is more compliant than both the inner inhomogeneity and the outer unbounded matrix and when the interface rigidity parameters for the two imperfect interfaces are greater than certain values, one stable and two unstable equilibrium positions can exist for the dislocation. Furthermore, under certain conditions an equilibrium position, which can be either stable or unstable (i.e., a saddle point), can exist, which has never been observed in previous studies. Results for a screw dislocation interacting with two parallel straight imperfect interfaces are also presented as the limiting case where the radius of the inner inhomogeneity approaches infinity while the thickness of the coating layer is fixed.

Antiplane time-harmonic green’s functions for a circular inhomogeneity with an imperfect interface

Two-dimensional antiplane time-harmonic Green’s functions for a circular inhomogeneity with an imperfect interface are derived. Here the linear spring model with vanishing thickness is employed to characterize the imperfect interface. Explicit expressions for the displacement and the stress fields induced by time-harmonic antiplane line forces located both in the unbounded matrix and in the circular inhomogeneity are presented. When the circular frequency approaches zero, our results reduce to those for the static case. Numerical results are presented to show the influence of the frequency and the imperfection of the interface on the stress and displacement fields.

Effect of Inter-Fibre Distance on Energy Transfer in Unidirectional Composites Containing Ultrasonic Waves

Interaction of time harmonic monochromatic anti-plane (SH) waves with a pair of parallel unidirected elastic cylindrical fibres embedded in an unbounded elastic matrix is considered. The classical method of separation of variables along with the appropriate wave field expansions, the pertinent boundary conditions, and the translational addition theorems for cylindrical wave functions are employed to obtain a series solution in terms of Fourier coefficients. The presented development is illustrated with numerical examples in which the fibres are insonified at end-on/broadside incidence for a relatively wide range of nondimensional frequencies. Particular attention has been focused on the effects of lateral fibre spacing on energy transfer between the fibres and matrix. The numerical results reveal that, while inter-fibre scattering (interference) has a small influence on the normalized fibre energies for broadside incidence at all frequencies, it has a considerable impact on the energy distribution for end-on incidence (fibres positioned in the eclipse configuration) at intermediate and high frequencies. The proposed model can complement acousto-ultrasonic (AU) techniques in non-destructive evaluation of unidirectional fibre-reinforced composite materials which is generally hampered by the highly dense concentration of the fibres. Limiting case is considered and fair agreement with a well-known solution is established.

The scattering of electro-elastic waves by a spherical piezoelectric particle in a polymer matrix

This paper is devoted to the study of scattering of plane harmonic waves by a piezoelectric sphere with spherical isotropy embedded in an unbounded isotropic polymer matrix. The scattered displacement field and the electric potential in the matrix are expressed in terms of spherical vector wave functions and spherical harmonic functions, respectively. For the field points inside the inhomogeneity, new displacement functions are introduced. Expansion of the new displacement functions and the electric potential in terms of spherical harmonic functions, the equations of motion and electrostatic lead to four second order ordinary differential equations (odes), where three of them are coupled. The coupled system of odes is solved by the generalized Frobenius series. This approach is readily used to handle low and high frequencies. Three different types of piezoelectric inhomogeneities, PZT-4, PZT-5H, and BaTiO 3 are considered and the associated piezoelectric effects on the electro-mechanical fields, differential and total scattering cross-sections are addressed.

A piezoelectric screw dislocation interacting with a nonuniformly coated circular inclusion

This paper investigated the interaction of a piezoelectric screw dislocation with a nonuniformly coated circular inclusion in an unbounded piezoelectric matrix subjected to remote antiplane shear and electric fields. In addition to having a discontinuous displacement and a discontinuous electric potential across the slip plane, the dislocation is subjected to a line force and a line charge at the core. The alternating technique in conjunction with the method of analytical continuation is applied to derive the general solutions in an explicit series form. This approach has a clear advantage in deriving the solution to the heterogeneous problem in terms of the solution for the corresponding homogeneous problem. The presented series solutions have rapid convergence which is guaranteed numerically. The image force acting on the piezoelectric screw dislocation is calculated by using the generalized Peach–Koehler formula. The numerical results show that the varying thickness of the interphase layer will exert a significant influence on the shear stress and electric field within the circular inclusion, and on the direction and magnitude of the image force. This solution can be used as Green’s function for the analysis of the corresponding piezoelectric matrix cracking problem.

Antiplane study on confocally elliptical inhomogeneity problem using an alternating technique

This paper presents a novel efficient procedure to analyze the two-phase confocally elliptical inclusion embedded in an unbounded matrix under antiplane loadings. The antiplane loadings considered in this paper include a point force and a screw dislocation or a far-field antiplane shear. The analytical continuation method together with an alternating technique is used to derive the general forms of the elastic fields in terms of the corresponding problem subjected to the same loadings in a homogeneous body. This approach could lead to some interesting simplifications in solution procedures, and the derived analytical solution for singularity problems could be employed as a Green's function to investigate matrix cracking in the corresponding crack problems. Several specific solutions are provided in closed form, which are verified by comparison with existing ones. Numerical results are provided to show the effect of the material mismatch, the aspect ratio, and the loading condition on the elastic field due to the presence of inhomogeneities.

Piezoelectric study on confocally multicoated elliptical inclusion

Within the framework of linear piezoelectricity, an effective method is developed and used to derive an analytical solution for the problem of a confocally multicoated elliptical inclusion embedded in an unbounded matrix which is subjected to arbitrary electromechanical loadings. Each layer of the composite is assumed to have different material constants, but has the same material orientation with x 3 being the poling direction. The alternating technique in conjunction with the method of analytical continuation is applied to derive the general series solution of electric field and displacement field for each layer of the composite. This approach has a clear advantage in deriving the solution to the heterogeneous problem in terms of the solution to the corresponding homogeneous problem. Several specific solutions are provided in closed form, which are verified by comparison with the existing ones. Some numerical results are provided to investigate the influence of material combinations and different electromechanical loadings on the shear stress and electric field. Besides, they also show that the derived solutions are well satisfied with the boundary conditions.

Analysis of a piezoelectric screw dislocation in the interphase layer between a circular inclusion and an unbounded matrix

The electroelastic interaction of a piezoelectric screw dislocation in the interphase layer with a circular inclusion and an unbounded matrix is dealt with. An efficient and concise method for complex multiply connected region is developed by combining the sectional holomorphic function, Schwartz symmetric principle, Cauchy-type integral and Laurent series expansion techniques, in terms of which explicit series form solutions of the complex potentials in the matrix, the interphase layer and the inclusion regions are derived. The image force acting on the piezoelectric screw dislocation is also calculated by using the generalized Peach–Koehler formula. The influence of the elctroelastic properties of the materials and the relative thickness of the interphase layer on the image force is discussed and shown in graphs. The results show that the above factors can drastically affect the motion of the dislocation in the interphase layer. In addition, a fundamental solution is provided in this paper to further investigate the problem involving interfacial defects and the dislocation mechanism. The present solutions contain a number of novel and previously known results which can be shown to be special cases.

이방성 섬유의 배열이 복합재료의 응력에 미치는 영향

In order to investigate effects of anisotropic fiber packing on stresses in composites, a Volume Integral Equation Method is applied to calculate the elastostatic field in an unbounded isotropic elastic medium containing multiple orthotropic inclusions subject to remote loading, and a Mixed Volume and Boundary Integral Equation Method is introduced for the solution of elastostatic problems in unbounded isotropic materials containing multiple anisotropic inclusions as well as one void under uniform remote loading. A detailed analysis of stress fields at the interface between the isotropic matrix and the central orthotropic inclusion is carried out for square, hexagonal and random packing of orthotropic cylindrical inclusions, respectively. Also, an analysis of stress fields at the interface between the isotropic matrix and the central orthotropic inclusion is carried out, when it is assumed that a void is replaced with one inclusion adjacent to the central inclusion of square, hexagonal and random packing of orthotropic cylindrical inclusions, respectively, due to manufacturing and/or service induced defects. The effects of random orthotropic fiber packing on stresses at the interface between the isotropic matrix and the central orthotropic inclusion are compared with the influences of square and hexagonal orthotropic fiber packing on stresses. Through the analysis of plane elastostatic problems in unbounded isotropic matrix with multiple orthotropic inclusions and one void, it will be established that these new methods are very accurate and effective for investigating effects of general anisotropic fiber packing on stresses in composites.

Scattering of anti-plane shear waves by a partially debonded magneto-electro-elastic circular cylindrical inhomogeneity

This paper presents an analysis of the scattering of anti-plane shear waves by a single piezoelectric cylindrical inhomogeneity partially bonded to an unbounded piezomagnetic matrix. The anti-plane governing equations for linearly magneto-electro-elastic medium are reduced to one Helmholtz and two Laplace's equations. The fields of scattered waves are obtained by means of the wave function expansion method on condition that the bonded interface is perfect. The region of the debonding is modeled as an interface crack with noncontacting faces. The magneto-electric permeable boundary conditions are adopted, i.e. the normal electric displacement, electric potential, normal magnetic induction and magnetic potential are continuous across the crack faces. The crack opening displacement is represented by Chebyshev polynomials and a system of equations is derived and solved for the unknown coefficients. Some examples are calculated and the results are figured. The results show that the COD increases as k 1 a becomes larger initially, then the COD decreases as k 1 a becomes larger still, where k 1 is the incident wave number and a the inhomogeneity radius. The COD gets smaller as the piezoelectric coefficient of the inhomogeneity increases.

A Confocally Multicoated Elliptical Inclusion under Antiplane Shear: Some New Results

The problem of a confocally multicoated elliptical inclusion in an unbounded matrix subjected to an antiplane shear is studied. Making use of the complex potentials and conformal mapping techniques, we show that the multiple coatings can be analyzed through a recurrence procedure in the transformed domain, while remaining explicit in detail and transparent overall. Particularly, the effect of the multiple confocal coatings is mathematically represented by a (2×2) array alone, resulting from a serial multiplication of matrices of the same order. Further we prove the following proposition. If the displacement prescribed at the remote boundary of the matrix is a polynomial of degree j in the position coordinates x i , the stresses at the innermost core are polynomials of degree j−1,j−3,..., in x i . This result is universally true provided that all elliptical interfaces are confocal, while no regard is paid to the number of coatings, their constituent properties and area fractions. Explicit expressions for the stresses at the innermost core are obtained in simple, closed forms.

혼합 체적-경계 적분방정식법을 이용한 응력확대계수 계산

A recently developed numerical method based on a mixed volume and boudary integral equation method is applied to calculate the accurate stress intensity factors at the crack tips in ungounded isotropic solids in the presence of multiple anisotropic inclusions and cracks subject to external loads.firstly,it should be noted that this newly developed numerical method does not require the Green's function for anisotropic inclusions to solve this class of problems since only Green's function for the unbounded isotropic mareix is involved intheir formulation for the analysis. Secondly, this method takes full advantage of the capabilities developed in FEM and BIEM.In this paper, a detailed analysis of the stress intensity factors are carried out for an unbounded isotropic matrix containing an orthotropic cylindrical inclusion and a crack. The accuracy and effectiveness of the new method are examined through comparison with results obtained from analytical method and valume interal equation method. It is demonstrated that this new method is very accurate and effective for solving plane elastostatic problems in unbounded solids containing anisotropic inclusions and cracks.

Symmetric Problem of Stable Interaction of Cracks with Thin Rigid Inclusions in a Three-Dimensional Matrix

By the method of boundary integral equations, we solve the problem of interaction of coplanar circular cracks with thin rigid inclusions in an unbounded elastic matrix under the action of normal loads applied to the crack surfaces and harmonic as functions of time. The inertial effects of the distribution of stresses along the crack contour are analyzed with regard for the presence of neighboring macroinclusions of fixed mass. The corresponding static solutions are obtained in the limiting case where the wave numbers are equal to zero.

A circular inclusion with a nonuniform interphase layer in anti-plane shear

An analytical solution is derived for the problem of a nonuniformly coated circular inclusion in an unbounded matrix under anti-plane deformations. The inclusion/interphase/matrix system is subject to (1) remote uniform shear and uniform eigenstrain imposed on the circular inclusion, and (2) a screw dislocation or a point force in the matrix. It is found that the varying interphase thickness will exert a significant influence on the nonuniform stress field within the circular inclusion, and on the direction and magnitude of the image force acting on a screw dislocation. In the course of development, it is found that the presence of certain coated inclusions, which are termed stealth, will not cause change of elastic energy in the body. The derived analytical solution for a screw dislocation is then employed as Green’s function to investigate a radial matrix crack interacting with the nonuniformly coated inclusion. The numerical results show that the varying interphase thickness will also affect the stress intensity factors.

Effective Conductivity of an Isotropic Heterogeneous Medium of Lognormal Conductivity Distribution

The study aims at deriving the effective conductivity Kef of a three-dimensional heterogeneous medium whose local conductivity K(x) is a stationary and isotropic random space function of lognormal distribution and finite integral scale IY. We adopt a model of spherical inclusions of different K, of lognormal pdf, that we coin as a multi-indicator structure. The inclusions are inserted at random in an unbounded matrix of conductivity K 0 within a sphere $\Omega $, of radius R 0, and they occupy a volume fraction n. Uniform flow of flux $% U_{\infty }$ prevails at infinity. The effective conductivity is defined as the equivalent one of the sphere $\Omega ,$ under the limits $n\rightarrow 1$ and $R_{0}/I_{Y}\rightarrow \infty .$ Following a qualitative argument, we derive an exact expression of Kef by computing it at the dilute limit $% n\rightarrow 0.$ It turns out that Kef is given by the well-known self-consistent or effective medium argument. The above result is validated by accurate numerical simulations for $\sigma _{Y}^{2}$ $\leq $ 10 and for spheres of uniform radii. By using a faced-centered cubic lattice arrangement, the values of the volume fraction are in the interval 0<n < 0.7. The simulations are carried out by the means of an analytic element procedure. To exchange space and ensemble averages, a large number N=10000 of inclusions is used for most simulations. We surmise that the self-consistent model is an exact one for this type of medium that is different from the multi-Gaussian one.

A conducting arc crack between a circular piezoelectric inclusion and an unbounded matrix

The present paper investigates the problem of a conducting arc crack between a circular piezoelectric inclusion and an unbounded piezoelectric matrix. The original boundary value problem is reduced to a standard Riemann–Hilbert problem of vector form by means of analytical continuation. Explicit solutions for the stress singularities δ=−(1/2)±i ε are obtained, closed form solutions for the field potentials are then derived through adopting a decoupling procedure. In addition, explicit expressions for the field component distributions in the whole field and along the circular interface are also obtained. Different from the interface insulating crack, stresses, strains, electric displacements and electric fields at the crack tips all exhibit oscillatory singularities. We also define a complex electro-elastic field concentration vector to characterize the singular fields near the crack tips and derive a simple expression for the energy release rate, which is always positive, in terms of the field concentration vector. The condition for the disappearance of the index ε is also discussed. When the index ε is zero, we obtain conventionally defined electro-elastic intensity factors. The examples demonstrate the physical behavior and the correctness of the obtained solution.

Scattering of anti-plane shear waves by a partially debonded piezoelectric circular cylindrical inclusion

This paper presents an analysis of the scattering of anti-plane shear waves by a single piezo-electric cylindrical inclusion partially bonded to an unbounded matrix. The anti-plane governing equations for piezoelectric materials are reduced to Helmholtz and Laplacian equations. The fields of scattered waves are obtained by means of the wave function expansion method when the bonded interface is perfect. When the interface is partially debonded, the region of the debonding is modeled as an interface crack with non-contacting faces. The electric permeable boundary conditions are adopted, i.e. the normal electric displacement and electric potential are continuous across the crack faces. The crack opening displacement is represented by Chebyshev polynomials and a system of equations is derived and solved for the unknown coefficients.