Unbounded Matrix Research Articles

Many-body interactions between contracting living cells.

The organization of live cells into tissues and their subsequent biological function involves inter-cell mechanical interactions, which are mediated by their elastic environment. To model this interaction, we consider cells as spherical active force dipoles surrounded by an unbounded elastic matrix. Even though we assume that this elastic medium responds linearly, each cell's regulation of its mechanical activity leads to nonlinearities in the emergent interactions between cells. We study the many-body nature of these interactions by considering several geometries that include three or more cells. We show that for different regulatory behaviors of the cells' activity, the total elastic energy stored in the medium differs from the superposition of all two-body interactions between pairs of cells within the system. Specifically, we find that the many-body interaction energy between cells that regulate their position is smaller than the sum of interactions between all pairs of cells in the system, while for cells that do not regulate their position, the many-body interaction is larger than the superposition prediction. Thus, such higher-order interactions should be considered when studying the mechanics of multiple cells in proximity.

Browder spectra of closed upper triangular operator matrices

<abstract><p>Let $ T_{B} = \left[\begin{array}{ll} A & B \\ 0 & D \end{array}\right] $ be an unbounded upper operator matrix with diagonal domain, acting in $ \mathcal H \oplus\mathcal K $, where $ \mathcal H $ and $ \mathcal K $ are Hilbert spaces. In this paper, some sufficient and necessary conditions are characterized under which $ T_{B} $ is Browder (resp., invertible) for some closable operator $ B $ with $ \mathcal D(B)\supset \mathcal D(D) $. Further, a sufficient and necessary condition is given under which the Browder spectrum (resp., spectrum) of $ T_{B} $ coincides with the union of the Browder spectrum (resp., spectrum) of its diagonal entries.</p></abstract>

Weakened interfaces in Cosserat bi-materials with constrained rotation

A continuum mechanics–based model is proposed to describe the mechanical behavior of a weakened interface lying between materials with microstructure. The weakened interface is simulated by a surface elastic medium adhering on either side with bulk elastic continua exhibiting behavior of Cosserat materials with constrained rotation. Under general frame invariance considerations, it is shown that the deformation state of the surface continuum is completely specified by an extension and a rotation vector with the latter being tangential to the surface continuum. Conjugate to these deformation quantities are a surface force and a surface couple being in local equilibrium with the reduced force traction and the reduced couple traction of the bulk continua acting on their connected surfaces. The constitutive equations describing the mechanical behavior of the surface continuum within the context of linear elasticity of small deformation are five relationships that linearly relate each component of the surface force and couple to the corresponding component of the extension and rotation vector. It can be said that the surface continuum represents a spring-type interface since its behavior can been seen as that of linear and torsional springs of vanishing thickness distributed continuously over its surface. Then, the proposed model is used to investigate the effect of a weakened interface on the stress concentration around a spherical inhomogeneity embedded in an unbounded matrix both of which consist of Cosserat materials with constrained rotation.

Harmonic And Neutral Spherical Elastic Inhomogeneities with A Functionally Graded Interphase Layer

Summary We study the elastic field in a three-phase composite composed of an internal spherical homogeneous elastic inhomogeneity, an intermediate functionally graded interphase layer and an outer unbounded homogeneous elastic matrix subjected to an arbitrary uniform remote loading. The shear modulus of the interphase layer obeys a power law distribution along the radial direction. We accomplish the design of harmonic and neutral spherical elastic inhomogeneities. Specifically, the shear modulus of the matrix can be judiciously chosen in such a way that the insertion of the harmonic spherical inhomogeneity does not disturb the original constant mean stress in the surrounding matrix. The shear modulus of the matrix and relative thickness of the interphase can also be suitably chosen such that the insertion of the neutral spherical inhomogeneity does not disturb the original uniform deviatoric stresses in the surrounding matrix.

B-essential spectra of 2 × 2 block operator matrix pencils

Abstract The paper is devoted to the study of the relative B-essential spectra of 2 × 2 {2\times 2} unbounded operator matrix pencils defined with nondiagonal domain. The main tool used to prove the results is the Frobenuis–Schur factorization associated with this kind of operator matrix which has proven very useful in dealing with stability analysis in the theory of B-Fredholm operators pencils. Therefore, the theoretical results are tested in pencil transport integro-differential equation with specific boundary conditions.

On bounded solutions of linear systems of differential equations with unbounded coefficients

This paper deals with a problem of finding a bounded solution of a system of nonhomogeneous linear differential equations with an unbounded matrix of coefficients on a finite interval. The right-hand side of the equation belongs to a space of continuous functions bounded with some weight; the weight function is chosen taking into account the behavior of the coefficient matrix. The problem is studied using a modified version of the parameterization method with non-uniform partitioning. Necessary and sufficient conditions of well-posedness of the problem are obtained in terms of a bilaterally infinite matrix of special structure.

Acousto-elastic radiation force on a fluid cylindrical inclusion embedded in a linear elastic medium

In the framework of linear elasticity, the longitudinal acousto-elastic time-averaged radiation force exerted on a cylindrical fluid-filled inclusion of arbitrary size embedded in an unbounded elastic matrix is derived based upon the integration of the time-averaged power flow density over a surface enclosing the particle. The cases of compressional and shear plane progressive elastic waves are considered. Mathematical expressions for the longitudinal radiation force efficiencies are derived and cast in exact series expansions in cylindrical coordinates. The elastic scattering solution, which is utilized to derive the acousto-elastic radiation force expressions, is determined. The analytical expressions for the acousto-elastic radiation force clearly show the contributions related to mode conversion P → S for the incident compressional waves, and S → P for shear wave incidence (where P and S denote, respectively, the compressional and shear waves). Another method, based on the extinction energy efficiency and the scattering asymmetry parameter, is shown to provide the same exact expression of the force obtained based upon the power flow density formalism. Computations for the dimensionless radiation force efficiencies and their cross-polarized (P → S and S → P) components demonstrate that mode conversion contributes negatively, positively or neutrally depending on the mass density of the particle, its size, and the properties of the surrounding elastic medium. In addition, elastic resonances are manifested in the acousto-elastic radiation force plots. This analysis provides a rigorous exact formalism using exact partial-waves series to compute the acousto-elastic radiation forces of compressional or shear waves on an elongated cylindrical particle encased in a linear elastic medium. The analytical series expressions and numerical predictions can be used as a priori information in optimizing and designing acousto-elastic radiation force experiments in materials science, elastography, vibro-acoustography, nondestructive evaluation, and the activation/excitation of a fluid cylinder embedded in an elastic medium, to name a few examples. The electromagnetic/optical analogue for materials exhibiting circular dichroism is also noted.

Stress fields for a three-phase spherical inclusion problem

In this paper, a systematic approach is presented to obtain the stress fields in a linear elastic body where a spherical inclusion surrounded by a spherical annulus is embedded in an unbounded matrix phase. The entire body is subject to far-field uniform strain. A formulation is carried out for the hydrostatic part and the deviatoric part of the far field strain separately. The obtained formula for the stress field is in closed form, which is new.

Boundary Charge Layer and Current for Spherical Inclusion in Conducting System

The steady-state current in heterogeneous conducting system is considered. One of the goals is to generate a battery of establish exact solutions for the analysis of steady-state currents in the current-conducting heterogeneous system. These solutions are to be used a) for Validation and Verification purposes the electromagnetic software and b) for deeper understanding of some features that were not explored sufficiently so far. The system under study consists of an isotropic unbounded isotropic matrix containing spherical inclusion (the inclusion can be either isotropic or anisotropic, and even nonlinear). This was analyzed in the classical textbooks and monographs. However, the solutions of previous studies miss analysis of the boundary charge layer and the current in this layer. In this paper, we establish these elements of the exact solution.

Instabilities of embedded cylindrical inclusions undergoing isotropic swelling or growth

Infinite circular cylindrical elastic inclusions, or rods, embedded in an unbounded elastic matrix display various modes of instability when they undergo sufficiently large expansion due to either swelling or volumetric growth. In this letter two modes of instability are examined: sinusoidal axisymmetric modes and sinusoidal bending modes. The rod and the matrix are neo-Hookean materials, and the full range of the modulus ratio of rod to matrix is considered. In the primary case examined, deformation is driven by an isotropic volumetric expansion, or transformation, of the rod. A three-dimensional bifurcation analysis of the rod constrained by the matrix reveals the onset of the critical instability mode as dependent on the modulus ratio. Comparisons with related results are discussed, including the compressive buckling of a stiff rod in a compliant matrix and the other limit when the modulus of the rod is very small compared to that of the matrix and behaves effectively as a fluid exerting pressure on the wall of the matrix cavity.11Contributed to celebrate the awarding of the Prager medal to Prof. Horacio Espinosa.

The Combined Effects of Gravitational and Thermocapillary Driving Forces on the Interactions of Slightly Deformable, Surfactant - Free Drops

By an asymptotic approach previously employed for gravitational or thermocapillary motion alone, collision efficiencies are calculated for slightly deformable drops in combined gravitational and thermocapillary motion with negligible inertia and thermal convection. The constant imposed temperature gradient may be aligned with gravity in either the same or opposite direction. In the dimensionless parameter space, deformation becomes important at a smaller drop size ratio when the temperature gradient and gravity are aligned in the same direction, because the driving force is larger and induces dimple formation earlier. For the same reason, in a physical system of ethyl salicylate (ES) drops in an unbounded matrix of diethylene glycol (DEG), deformation becomes important for smaller drops when the driving forces have a parallel, rather than anti-parallel, arrangement. In developing the population dynamics for slightly deformable drops, a new, simplified expression for the collision efficiency for spherical drops in the absence of van der Waals forces is presented, which successfully separates the contributions of the two driving forces. Two collision-forbidden regions can occur for opposed driving forces leading to a shark-fin shaped collision efficiency curve for two slightly deformable drops. As shown in population dynamics, if the drop distribution is broad enough, it is possible for drops to jump the first collision-forbidden region.

Ratio asymptotics for biorthogonal matrix polynomials with unbounded recurrence coefficients

In this paper we study matrix biorthogonal polynomials sequences that satisfy a nonsymmetric three term recurrence relation with unbounded matrix coefficients. The outer ratio asymptotics for this family of matrix biorthogonal polynomials is derived under quite general assumptions. Some illustrative examples are considered.

Parallel Volume Integral Equation Method for Two-Dimensional Elastodynamic Analysis

The parallel volume integral equation method (PVIEM) is applied for the analysis of two-dimensional elastic wave scattering problems in an unbounded isotropic solid containing various types of multiple multilayered anisotropic inclusions. It should be noted that the volume integral equation method (VIEM) does not require the use of the Green’s function for the anisotropic inclusion — only the Green’s function for the unbounded isotropic matrix is needed. A detailed analysis of the SH wave scattering problem is presented for various types of multiple multilayered orthotropic inclusions. Numerical results are presented for the elastic fields at the interfaces for square and hexagonal packing arrays of various types of multilayered orthotropic inclusions in a broad frequency range of practical interest. Standard parallel programming was used to speed up computation in the VIEM. The PVIEM enables us to investigate the effects of single/multiple scattering, fiber packing type, fiber volume fraction, single/multiple layer(s), multilayer’s shapes and geometry, isotropy/anisotropy, and softness/hardness of various types of multiple multilayered anisotropic inclusions on displacements and stresses at the interfaces of the inclusions and far-field scattering patterns. Also, powerful capabilities of the PVIEM for the analysis of general two-dimensional multiple scattering problems are investigated.

On the Optimal Control Problem for a Linear Stochastic System with an Unstable State Matrix Unbounded at Infinity

We consider a control problem over an infinite time horizon with a linear stochastic system with an unstable asymptotically unbounded state matrix. We extend the notion of anti-stability of a matrix to the case of non-exponential anti-stability, and introduce an antistability rate function as a characteristic of the rate of growth for the norm of the corresponding fundamental matrix. We show that the linear stable feedback control law is optimal with respect to the criterion of the adjusted extended long-run average. The designed criterion explicitly includes information about the rate of anti-stability and the parameters of the disturbances. We also analyze optimality conditions.

Stability of essential B-spectra of unbounded linear operators and applications

In this paper, we study the stability of some essential B-spectra of closed linear operators on a Banach space X, under polynomially finite rank operators and we give the characterization of some essential B-spectra of a $$2\times 2$$ of unbounded matrix operator acting in the product of Banach spaces $$X\times Y$$ . Then, using the functional calculus, we prove that a spectral mapping type theorem holds for these essential B-spectra. As an application, we study the effect of the functional calculus on the class of meromorphic operators, and on the class of isoloid operators with sable sign index, satisfying generalized Weyl theorem.

Inclusion problem for a generalized strain gradient elastic continuum

This paper treats an inclusion problem in the framework of a generalized strain gradient theory (GSGT) of elasticity. The GSGT used here consists of three material characteristic length scales for the evaluation of the elastic response. The Green’s function for the governing differential equations, involving stresses and higher-order stresses, is obtained considering the strain gradient model mentioned above. Then, Eshelby’s tensor is derived in the current framework for an inclusion of any arbitrary shape, under uniform eigenfields. This general form of the Eshelby’s tensor is used to derive closed-form expressions for a spherical inclusion and an infinitely long cylindrical inclusion, each embedded in an unbounded matrix to ignore the boundary effects. To determine the homogenized properties of the composites considering size-effects, the volume averages of the position-dependent Eshelby’s tensor are evaluated for both the inclusions. All the results from the current study match the corresponding results from classical elasticity, when the strain gradient effects are ignored. The homogenized properties of composites with either of the inclusions are strongly influenced by the gradient effects when the dimensions of the inclusions are comparable to the characteristic length scales. They approach the volume averages evaluated using classical elasticity, when the dimensions of the inclusion are very high in comparison with the characteristic lengths. The current study presents the importance of considering the size-effects using GSGT, without any assumption, for the evaluation of the effective properties of the micro- and nano-composites.

Near and far field scattering of SH waves by multiple multilayered anisotropic circular inclusions using parallel volume integral equation method

PurposeThe purpose of this paper is to calculate near field and far field scattering of SH waves by multiple multilayered anisotropic circular inclusions using parallel volume integral equation method (PVIEM) quantitatively.Design/methodology/approachThe PVIEM is applied for the analysis of elastic wave scattering problems in an unbounded solid containing multiple multilayered anisotropic circular inclusions. It should be noted that this numerical method does not require the use of the Green’s function for the inclusion – only the Green’s function for the unbounded isotropic matrix is needed. This method can also be applied to solve general elastodynamic problems involving inhomogeneous and/or anisotropic inclusions whose shape and number are arbitrary.FindingsA detailed analysis of the SH wave scattering problem is presented for multiple multilayered orthotropic circular inclusions. Numerical results are presented for the displacement fields at the interfaces and the far field scattering patterns for square and hexagonal packing arrays of multilayered circular inclusions in a broad frequency range of practical interest.Originality/valueTo the best of the authors’ knowledge, the solution for scattering of SH waves by multiple multilayered anisotropic circular inclusions in an unbounded isotropic matrix is not currently available in the literature. However, in this paper, calculation of displacements on interfaces and far field scattering patterns of multiple multilayered anisotropic circular inclusions using PVIEM as a pioneer of numerical modeling enables us to investigate the effects of single/multiple scattering, fiber packing type, fiber volume fraction, single/multiple layer(s), the multilayer’s geometry, isotropy/anisotropy and softness/hardness.

On optimal controls in coefficients for ill-posed non-Linear elliptic Dirichlet boundary value problems

We consider an optimal control problem associated to Dirichlet boundary value problem for non-linear elliptic equation on a bounded domain $Ω$ . We take the coefficient $u(x)∈ L^∞(Ω)\cap BV(Ω)$ in the main part of the non-linear differential operator as a control and in the linear part of differential operator we consider coefficients to be unbounded skew-symmetric matrix $A_{skew}∈ L^q(Ω;\mathbb{S}^N_{skew})$ . We show that, in spite of unboundedness of the non-linear differential operator, the considered Dirichlet problem admits at least one weak solution and the corresponding OCP is well-possed and solvable. At the same time, optimal solutions to such problem can inherit a singular character of the matrices $A^{skew}$ . We indicate two types of optimal solutions to the above problem and show that one of them can be attained by optimal solutions of regularized problems for coercive elliptic equations with bounded coefficients, using the two-parametric regularization of the initial OCP.

Debonded arc-shaped interface conducting rigid line inclusions in piezoelectric composites

We consider an arc-shaped conducting rigid line inclusion located at the interface between a circular piezoelectric inhomogeneity and an unbounded piezoelectric matrix subjected to remote uniform anti-plane shear stresses and in-plane electric fields. Moreover, one side of the rigid line inclusion has become fully debonded from the matrix or the inhomogeneity leading to the formation of an insulating crack. After the introduction of two sectionally holomorphic vector functions, the problem is reduced to a vector Riemann–Hilbert problem, which can be decoupled sequentially by repeated application of the orthogonality relations between the eigenvectors for two corresponding generalized eigenvalue problems.

The Newtonian potential inhomogeneity problem: non-uniform eigenstrains in cylinders of non-elliptical cross section

Understanding the fields that are set up in and around inhomogeneities is of great importance in order to predict the manner in which heterogeneous media behave when subjected to applied loads or other fields, e.g., magnetic, electric, thermal, etc. The classical inhomogeneity problem of an ellipsoid embedded in an unbounded host or matrix medium has long been studied but is perhaps most associated with the name of Eshelby due to his seminal work in 1957, where in the context of the linear elasticity problem, he showed that for imposed far fields that correspond to uniform strains, the strain field induced inside the ellipsoid is also uniform. In Eshelby’s language, this corresponds to requiring a uniform eigenstrain in order to account for the presence of the ellipsoidal inhomogeneity, and the so-called Eshelby tensor arises, which is also uniform for ellipsoids. Since then, the Eshelby tensor has been determined by many authors for inhomogeneities of various shapes, but almost always for the case of uniform eigenstrains. In many application areas in fact, the case of non-uniform eigenstrains is of more physical significance, particularly when the inhomogeneity is non-ellipsoidal. In this article, a method is introduced, which approximates the Eshelby tensor for a variety of shaped inhomogeneities in the case of more complex eigenstrains by employing local polynomial expansions of both the eigenstrain and the resulting Eshelby tensor, in the case of the potential problem in two dimensions.