Numerical determination of a class of generalized stress intensity factors

A modification of the collocation method for the numerical solution of Cauchy-type singular integral equations appearing in plane elasticity and, especially, crack problems is proposed. This modification, based on a variable transformation, applies to the case when the unknown function of the singular integral equation behaves like A(x ? c)^α + B(x ? c)^β, where α < 0, 0 < β ? α < 1, near an endpoint c of the integration interval. In plane elasticity such a point is either a crack tip or a corner point of the boundary of the elastic medium. Thus the method seems to be quite efficient for the numerical evaluation of generalized stress intensity factors near such points. A successful application of the method to the classical plane elasticity problem of an antiplane shear crack terminating at a bimaterial interface was also made.     

Global existence and blow-up analysis for a nonlinear diffusion equation with inner absorption and boundary flux

This article deals with a nonlinear diffusion equation with inner absorption u _t ?=?(log^σ(1?+?u)u _x ) _x ???λ(1?+?u)log ^p (1?+?u), in ?₊?×?(0,?+∞), subject to a logarithmic boundary flux ? log^σ(1?+?u)u _x (0,?t)?=?(1?+?u)log ^q (1?+?u)(0,?t), t?∈?(0,?+∞). We establish the critical global existence curve and give the asymptotic behaviour close to the blow-up time.     

Transformation of continuous‐time state equations to discrete‐time state equations

Abstract

A method is proposed for transforming a continuous‐time state equation, x(t)=Ax(t)+Bu(t), to a discrete‐time state equation, x[(k+1)τ]=ø(τ)x(kτ)+B(τ)u(kτ). It is based on expanding the matrix exponential exp (Aτ) into a shifted matrix Chebyshev series. An example is given to demonstrate the superiority of the method over other methods.     

Semiconductor quantum dots: Theory and phenomenology

Research in semiconductor quantum dots (q-dots) has burgeoned in the past decade. The size (R) of these q-dots ranges from 1 to 100 nm. Based on the theoretical calculations, we propose energy and length scales which help in clarifying the physics of this mesoscopic system. Some of these length scales are: the Bohr exciton radius (α_B*), the carrier de Broglie and diffusion length (λ_D andl _D), the polaron radius (α_p), and the reduction factor modulating the optical matrix element (M _x).R<α_B is an individual particle confinement regime, whereas the larger ones are exciton confinement regime wherein Coulomb interaction play an important role. Similarly a size-dependent dielectric constantε(R) should be used forR<α_p<α_B. An examination ofM _x reveals that an indirect gap material q-dot behaves as a direct gap material in the limit of very small dot size. We have carried out effective mass theory (EMT) calculations to estimate the charge density on the surface of the quantum dot. We present tight binding (TB) calculation to show that the energy upshift scales as 1/R ^x, wherex is less than 2 and the exponent depends on the orientation of the crystallite.     

Linear differential-algebraic equations with properly stated leading term: B -critical points

We examine in this article so-called B-critical points of linear, time-varying differential-algebraic equations (DAEs) of the form A(t)(D(t)x(t))′ + B(t)x(t) = q(t). These critical or singular points, which cannot be handled by classical projector methods, require adapting a recently introduced framework based on Π-projectors. Via a continuation of certain invariant spaces through the singularity, we arrive at a scenario which accommodates both A- and B-critical DAEs. The working hypotheses apply in particular to standard-form analytic systems although, in contrast to other approaches to critical problems, the scope of our approach extends beyond the analytic setting. Some examples illustrate the results.     

Solvable Brioschi Resolvents

 The Brioschi resolvent y ⁵ −10Zy ³ +45Z ² y−Z ² is a key component of an algorithm for calculating the roots of a general quintic polynomial. It is obtained from the general quintic polynomial by applying two Tschirnhausen transformations. In this paper it is shown that if the quintic polynomial is a solvable polynomial, then its associated parameter Z in the Brioschi resolvent satisfies Z=g(t) where g(t) is a rational function in ℚ(t) and t is chosen from an appropriate field. Received: July 23, 2001; revised version: October 14, 2002 Keywords: Solvable quintic, Brioschi quintic, resolvent sextic, parametrization. The author was supported by a research grant from the Natural Sciences and Engineering Research Council of Canada     

Asymptotic analytical solutions of the two-degree-of-freedom strongly nonlinear van der Pol oscillators with cubic couple terms using extended homotopy analysis method

This paper adopts the extended homotopy analysis method (EHAM) to obtain the asymptotic analytical series solutions of the strongly nonlinear two-degree-of-freedom (2DOF) van der Pol oscillators with cubic couple terms. It turns out that the oscillators occur essentially in only two variations: If the system has periodic solutions, then it either has only one solution which is out-of-phase (i.e., x ₁(t) = −x ₂(t)) or has two solutions that are not only in-phase (i.e., x ₁(t) = x ₂(t)) but also out-of-phase. Two examples, as two types of the problem have been raised, correspondingly. Employing the EHAM for those two problems, the explicit analytical solutions of frequency ω and displacements x ₁(t) and x ₂(t) are well formulated, but the conventional homotopy analysis method (HAM) can hardly do it if the cubic couple terms are complex. Thus, the EHAM is rather general. Moreover, the fifth-order analytical solutions are then compared with those derived from the established Runge–Kutta method in order to verify the accuracy and validity of this approach. It is shown that there is excellent agreement between the two sets of results, even if the time variable t progresses to a comparatively large domain in the time-history responses. Finally, the convergence theorem for the present method is also presented and discussed. All these results confirm that the EHAM can solve the presented problem successfully and completely, and that the EHAM will be a powerful and efficient tool for solving other multi-degree-of-freedom (MDOF) dynamical systems in engineering and physical sciences.     

Fast Parallel Algorithms for Matrix Reduction to Normal Forms

 We investigate fast parallel algorithms to compute normal forms of matrices and the corresponding transformations. Given a matrix B in ℳ _n,n (K), where K is an arbitrary commutative field, we establish that computing a similarity transformation P such that F=P ^-1 BP is in Frobenius normal form can be done in ?C ² _K . Using a reduction to this first problem, a similar fact is then proved for the Smith normal form S(x) of a polynomial matrix A(x) in ℳ _n,m (K[x]); to compute unimodular matrices U(x) and V(x) such that S(x)=U(x)A(x)V(x) can be done in ?C ² _K . We get that over concrete fields such as the rationals, these problems are in ?C ². Using our previous results we have thus established that the problems of computing transformations over a field extension for the Jordan normal form, and transformations over the input field for the Frobenius and the Smith normal form are all in ?C ² _K . As a corollary we establish a polynomial-time sequential algorithm to compute transformations for the Smith form over K[x]. Received: February 29, 1996; revised version: August 29, 1997     

A variant of the heat balance integral method and a new proof of the exponentially fast asymptotic behavior of the solutions in heat conduction problems with absorption

We give a new and explicit estimate for the asymptotic behavior of the solutions of the problem u_t − u_xx + u₊^P = 0, x > 0, t> 0, with conditions u(0, t) = 1, t > 0 and u(x, 0) = U₀(x) ≥ 0, x > 0, for a class of functions U₀ and parameter 0 < p < 1. We use an approximate solution given by the heat balance integral method with the innovation property which fixes appropriately the asymptotic limit of the corresponding approximate free boundary.     

g-Jitter mixed convection adjacent to a vertical stretching sheet

This paper studies the effect of periodical gravity modulation, or g-jitter induced mixed convection, on the flow and heat transfer characteristics associated with a stretching vertical surface in a viscous and incompressible fluid. The velocity and temperature of the sheet are assumed to vary linearly withx, wherex is the distance along the sheet. It is assumed that the gravity vector modulation is given byg*(t)=g_o [1+ɛ cos(πωt)]k, and the resulting non-similar boundary layer equations are solved numerically using an implicit finite-difference scheme. The effects of the amplitude of modulation, frequency of the single-harmonic component of oscillation, mixed convection parameter and Prandtl number on the skin friction and Nusselt number are discussed in detail.     

Viscosity and supercooling of Fe-Cr (≤40 at % Cr) melts

We have studied the composition dependences of the kinematic viscosity (ν) and supercooling (Δt) of Fe-Cr melts at chromium contents of up to 40 at %. The results indicate that there is a minimum in viscosity at 5 at % Cr, a maximum at 12 at % Cr, and a sharp rise in the crystallization tendency of the solid solution in this composition range. The Δt(x) and ν(x) data obtained can be understood in terms of the composition dependence of the bonding energy, which is governed by the geometric and chemical short-range order in the atomic arrangement. Using a probabilistic approach to evaluation of various short-range order configurations and the experimental viscosity and supercooling data at chromium contents of 2.5 and 30 at %, we have calculated Δt(x) and ν(x) up to 40 at % Cr.     

Effect of Impurities on Supersolid Condensate: A Ginzburg–Landau Approach

Basing our arguments on a wavefunction that contains both positional and superfluid order, we propose a Ginzburg–Landau functional for a supersolid with the two order parameters necessary to describe such a phase: density n _B (r) and supersolid order parameter ψ _V . We argue that adding lighter. ³He atoms to a ⁴He supersolid produces attractive regions for vacancies, leading to patches of higher T _c. On the other hand, the supersolid stiffness decreases in this granular state with increased ³He disorder. Both effects are linear in ³He concentration.     

Study of lithium–zinc borophosphate glasses

Mixed lithium–zinc borophosphate glasses were prepared and studied in three compositional series xLi₂O–(50−x)ZnO–50P₂O₅, xLi₂O–(50−x)ZnO–10B₂O₃–40P₂O₅ and xLi₂O–(50−x)ZnO–20B₂O₃–30P₂O₅ with x = 0, 10, 20, 30, 40 and 50 mol% Li₂O. The obtained glasses were characterized by the measurements of the density (ρ), molar volume (V _M), glass transition temperature (T _g) and thermal expansion coefficient (α). For the investigation of structural changes ¹¹B and ³¹P MAS NMR and Raman spectroscopy were applied. The replacement of zinc by lithium in borophosphate glasses slightly decreases V _M and T _g, while α increases. In Li–Zn metaphosphate glasses the compositional dependence of T _g reveals a minimum, while at the borophosphate series T _g decreases monotonously with increasing Li₂O content. Chemical stability of Li–Zn borophosphate glasses is very good for glasses with x = 0–30 mol% Li₂O. Spectral studies showed in the glass series with 10 mol% B₂O₃ only the presence of BO₄ sites. In the glasses with 20 mol% B₂O₃ the presence of BO₃ and two BO₄ sites was revealed in ZnO-rich glasses and only one BO₄ site in Li₂O-rich glasses; the number of BO₃ groups decreases with increasing Li₂O content which is ascribed to the formation of P–O–Zn covalent bonds in ZnO-rich glasses.     

A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method

The Element free Galerkin method, which is based on the Moving Least Squares approximation, requires only nodal data and no element connectivity, and therefore is more flexible than the conventional finite element method. Direct imposition of essential boundary conditions for the element free Galerkin (EFG) method is always difficult because the shape functions from the Moving Least Squares approximation do not have the delta function property. In the prior literature, a direct collocation of the fictitious nodal values & u circ; used as undetermined coefficients in the MLS approximation, u ^h (x) [u ^h (x)=Φ·& u circ;], was used to enforce the essential boundary conditions. A modified collocation method using the actual nodal values of the trial function u ^h (x) is presented here, to enforce the essential boundary conditions. This modified collocation method is more consistent with the variational basis of the EFG method. Alternatively, a penalty formulation for easily imposing the essential boundary conditions in the EFG method with the MLS approximation is also presented. The present penalty formulation yields a symmetric positive definite system stiffness matrix. Numerical examples show that the present penalty method does not exhibit any volumetric locking and retains high rates of convergence for both displacements and strain energy. The penalty method is easy to implement as compared to the Lagrange multiplier method, which increases the number of degrees of freedom and yields a non-positive definite system matrix.     

Elastodynamic Response of a Periodic Layer of Spherical Particles Containing Vacancies

This paper is concerned with the influence of vacancies on the elastodynamic response of a periodic (square) array of identical spherical elastic inclusions embedded in an unbounded elastic matrix. A response function of the array is defined as H ^* (ω)\equiv R ^* (ω)/T ^* (ω) , where R ^* (ω) and T ^* (ω) are, respectively, the reflection and transmission spectra of the lattice. H ^* (ω) was measured for a ``perfect lattice,' i.e., one without any vacancies, and was found to be characterized by lattice resonances. H ^* (ω) was also measured for lattices containing one and three vacancies within the 72 lattice site area insonified by the ultrasonic beam. A counter-intuitive observation is that the presence of even one vacancy significantly reduces the amplitude of the fundamental lattice resonance. Furthermore, in the case of the specimen containing three closely spaced vacancies, the reduction is not three times the reduction due to one vacancy; it is significantly less than that.     

The influence of surface kinetics in modelling chemical vapour deposition processes in porous preforms

The isothermal chemical vapour infiltration (ICVI) process is a well known technique for the production of composites and the surface modification of porous preforms. Mathematical modelling of the process can provide a better understanding of the influence of individual process parameters on the deposition characteristics such as final porosity or deposition profiles in the pore network. The influence of different rate expressions for several binary compounds on the ICVI process is discussed. Experimental work is used to validate the importance of correct kinetic expressions in a continuous ICVI model for cylindrical pores. The predicted infiltration characteristics are compared with experimental results. The final densification and Thiele modulus, i.e. a number which is a measure for the diffusion limitations in a pore, are used for the evaluation of the presented model, and conditions are given for an optimal densification of a porous preform by the ICVI process for several binary compounds. The deposition profiles as predicted by the model calculations are in agreement with the experimentally determined deposition profiles of TiN and TiC in small tubes. Moreover, it can be concluded that the shape of the deposition profiles is determined by the heterogeneous reaction kinetics. There is only a qualitative agreement between the predicted densification and measured densification for the synthesis of TiN and TiB₂ in sintered porous alumina. This mismatch can be explained in terms of a complexity of the pore network and differences in reaction kinetics. Model calculations reveal that there is a scattering for the predicted residual porosity as a function of the Thiele modulus for TiN. Moreover, this Thiele modulus can not fully account for the changes in densification at different temperatures. Given these uncertainties it is likely that a residual porosity of less than one percent can be obtained if the Thiele modulus is smaller than 1 × 10^–4. However, a CVI process with such a small Thiele modulus will not be practical, because of the concomitant long process times. Therefore, more precise conditions for the individual process parameters, i.e. concentration, reactor pressure, and temperature are deduced from the model calculations.Nomenclature a, b, c reaction order constants - C _i(x, t) concentration of species i at axial position x and time t (mole m^–3) - C _i ^o bulk concentration of species i (mole m^–3) - C _i ^* (x, t) dimensionless concentration of species i at axial position x and time t - D _e(x, t) effective diffusion coefficient at axial position x and time t (m²s^–1) - D _ij(x, t) binary diffusion coefficient (m²s^–1) - D _K(x, t) Knudsen diffusion coefficient at position x and time t (m²s^–1) - F correction factor for effective diffusion coefficient - k growth rate constant (ms^–1(m³mole^–1)^a+b-1) - K _i adsorption-desorption equilibrium constant (m³mole^–1) - L length of a pore (m) - M _i molecular weight of species i (g mole^–1) - M _ij harmonic mean of the molecular weights of species i andj (g mole^–1) - M _s molecular weight of deposit (g mole^–1) - m _t measured mass increase (g) - n _i stoichiometric number - P reactor pressure (Pa) - R(C _i) growth rate (mole(m^–2s^–1)) - r(x, t) pore radius at position x and time t (m) - r ^o initial pore radius (m) - r ^* dimensionless pore radius - S geometrical surface area (m²) - s ^t fraction of free titanium sites at the surface of TiN - s ⁿ fraction of free nitrogen sites at the surface of TiN - T temperature (K) - t time (s) - t _p process time (s) - U K _HCl/(K _{H 2} C _{H 2})^1/2 (m³ mole^–1) - V volume of alumina substrate (m³) - W K _TiCl3(m³ mole^–1) - X volume of infiltrated deposit relative to initial pore volume - x axial distance (m) - x ^* dimensionless axial distance - z number of time steps - dummy variable for integration - porosity of sintered porous alumina substrate - ratio of the volume over the surface area perpendicular to the flux (m) - density deposit (kg m^–3) - _ij a characteristic length (Å) - tortuosity factor of substrate - Thiele modulus - _D collision integral     

Systematic hardness measurements on mixed and doped crystals of rubidium halides

Efforts are made to improve the hardness of rubidium halide crystals by (i) solid solution hardening and (ii) impurity hardening. Systematic microhardness measurements have been made on rubidium halide mixed crystals (RbBr-RbI and KI-RbI) and rubidium halide crystals doped with Sr²⁺ ions. The composition dependence of the hardness of mixed crystals follows the law ΔH _v =K x (1− x),where ΔH _v is the enhancement in hardness,K a constant andx and (1 −x) the concentrations of the first and second component of the mixed crystals, respectively. The hardness of doped crystals increases with the concentrationC of the dopant according to the law, ΔH _v+6 =k C ^m ,wherek andm are constants. The relative efficacy of the two methods of hardening is discussed.     

Mechanical wave interaction at a diffuse interface

An investigation is made of the reflection coefficientR of an ultrasonic wave impinging on a boundary at which there is a gradual transition of material properties from one material to another. The study is confined to the one-dimensional case of compression waves incident at 90° on the boundary. Both analytical and numerical solutions forR are presented for various interfacial profiles, and are seen in general to be strongly dependent on wave frequency. Measurements are made on diffusion bonded samples of nickel to copper, and the results compared with theory. Significant scatter is found in the experimental results due primarily to the migration of oxygen into the diffusion zone. A second set of measurements on diffusion bonded samples of nickel to carbon steel show general agreement with theory.Nomenclature A _i ,B _i ,C _i ,i=1, 2 coefficients in the expression forX (x) - c _i concentration of speciesi - c _L compression wave velocity - D _i diffusion coefficient for speciesi - diffusion coefficient for a binary mixture - E(x) Young's modulus - stiffness for the case of one-dimensional strain, =(1–v)E(x)/(1+v)(1–2v) - one-dimensional stiffness within bluk materiali - J _i ,i=1, 2 flux of speciesi - k(x) wave number at spatial co-ordinatex, equal to 2/ - k _i ,i=1, 2 wave number within bulk materiali - K(x) pseudo-wavenumber, defined by Eq. (8). - k _B Boltzmann's constant - L thickness of interfacial zone - f wave frequency - m, N material profile parameters, used to specifyK(x) - Q activation energy for diffusion - R reflection coefficient - t time - T transmission coefficient - u(x, t) displacement of molecules due to wave action,u(x, t)=X(x) (t) - X(x) spatial portion of functionu(x, t) - x spatial variable - Y(x, t) (x, t)/ - Z _i ,i=1, 2 mechanical impedance of bulk materiali - strain - wavelength of sound - Poisson's ratio - material density - (x, t) stress - (t) temporal portion of functionu(x, t) - _i (x),i=1, 2 functional form ofX(x) within interfacial region - angular frequency of wave     

Slow Dynamics of Nonequilibrium Density Fluctuations in Concentrated Hard-Sphere Suspensions

The coupled diffusion equations recently proposed for concentrated hard-sphere suspensions of interacting Brownian particles, the nonlinear deterministic diffusion equation with the self-diffusion coefficient D _S((x, t)) for the average local volume fraction (x, t), and the linear stochastic diffusion equation with D _S((x, t)) for the density fluctuations n(x, t) are numerically solved under a spatially inhomogeneous, nonequilibrium initial state. Thus, in a supercooled region where < _g, the slow evolution of the cluster-like glassy domains with (x, t) _g and the slow relaxation of the nonequilibrium density fluctuations are shown to be caused by the dynamic singularity of the self-diffusion coefficient, D _S((x, t)) (1–(x, t)/_g)², where is a particle volume fraction, _g = (4/3)³/(7 ln 3 – 8 ln 2 + 2) is the colloidal glass transition volume fraction, and is the crossover volume fraction.     

Convergence,accuracy and stability of finite element approximations of a class of non-linear hyperbolic equations

Numerical stability criteria and rates of convergence are derived for finite element approximations of the non- linear wave equation u_tt?F(u_x) = f(x, t), where F(u_x) possesses properties generally encountered in non-linear elasticity. Piecewise linear finite element approximations in x and central difference approximations in t are studied.