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Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1990"


Journal ArticleDOI
Zhigang Suo1
TL;DR: In this article, the Lekhnitskii and Stroh formalisms for interfacial fracture mechanics for anisotropic solids are formalized and a simple rule is formulated that allows one to construct the complete solutions from mode III solutions in an isotropic, homogeneous medium.
Abstract: 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.

673 citations


Journal ArticleDOI
TL;DR: In this article, the spectral variation of the optical properties of flame soot particles is determined by combining classical and dynamic light scattering measurements with the Kramers-Kronig relations.
Abstract: The spectral variation of the optical properties of soot particles is determined by combining classical and dynamic light scattering measurements with the Kramers-Kronig relations. Particle size and number densities are determined from scattering/extinction and autocorrelation measurements at the wavelength of 0.488 μm. This information is then combined with the spectral extinction measurements in the wavelength range 0.2 to 6.4 μm to determine the spectral variation of the refractive indices of flame soot. Results are presented for a premixed propane-oxygen flame with a fuel equivalence ratio ϕ = 1.8. The sensitivity of the technique and its advantage over the previous methods are discussed.

574 citations


Journal ArticleDOI
TL;DR: In this article, the H-means are introduced for studying oscillations and concentration effects in partial differential equations, and applications to transport properties and homogenisation are given as an example of the new results which can be obtained by this approach.
Abstract: New mathematical objects, called H-measures, are introduced for studying oscillations and concentration effects in partial differential equations. Applications to transport properties and to homogenisation are given as an example of the new results which can be obtained by this approach.

465 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the topological properties of the symplectic diffeomorphism groups are related to the topology of the phase space geometry of the manifold.
Abstract: In this paper we show that symplectic maps have surprising topological properties. In particular, we construct an interesting metric for the symplectic diffeomorphism groups, which is related, but not obviously, to the topological properties of symplectic maps and phase space geometry. We also prove a certain number of generalised symplectic fixed point theorems and give an application to a Hamiltonian system.

412 citations


Journal ArticleDOI
TL;DR: In this article, a laminar flamelet model of pre-mixed turbulent combustion is described, in which a characteristic length scale is used to control the flamelet surface-to-volume ratio.
Abstract: A laminar flamelet model of pre-mixed turbulent combustion is described in which a characteristic length scale $\hat{L}\_{y}$ controls the flamelet surface-to-volume ratio. An analysis, based on the Bray-Moss-Libby model of turbulent combustion, leads to the conclusion that $\hat{L}\_{y}$/l is proportional to the ratio of the laminar burning velocity to the turbulence velocity u$^{\prime}$, where l is the integral length scale of the turbulence. A fractal flame model and an analysis of experimental time series data both support this conclusion. Several different theories for the turbulent burning velocity are shown to be equivalent to each other and to be generalizations of the classical theory of Kolmogorov, Petrovski & Piskonov. A method of characteristics analysis confirms the resulting expression. This expression, containing only one disposable constant which must be of order unity, is compared with a published correlation of a large amount of experimental data. This leads to an experimental determination of the ratio of effective to true laminar burning velocities, as a function of Karlovitz number, which shows satisfactory agreement with results of strained laminar flame calculations.

400 citations


Journal ArticleDOI
TL;DR: In this article, the authors describe the basic procedures and the development of a general purpose computer program (SWANDYNE-X) for solving deformations in geomaterials.
Abstract: The behaviour of all geomaterials, and in particular of soils, is governed by their interaction with the pore fluid. The mechanical model of this interaction when combined with suitable constitutive discription of the solid phase and with efficient, discrete, computation procedures, allows most transient and static problems involving deformations to be solved. This paper describes the basic procedures and the development of a general purpose computer program (SWANDYNE-X). The results of the computations are validated by comparison with experimental results obtained on physical models tested in the Cambridge Centrifuge.

374 citations


Journal ArticleDOI
TL;DR: In this article, a detailed study of the methods of analysing the experimental data obtained from fracture mechanics tests using double-cantilever beam, end loaded split and end notched flexure specimens.
Abstract: One of the most important mechanical properties of a fibre-polymer composite is its resistance to delamination. The presence of delaminations may lead not only to complete fracture but even partial delaminations will lead to a loss of stiffness, which can be a very important design consideration. Because delamination may be regarded as crack propa­gation then an obvious scheme for characterizing this phenomenon has been via a fracture mechanics approach. There is, therefore, an extensive literature on the use of fracture mechanics to ascertain the interlaminar fracture energies, G c , for various fibre-polymer composites using different test geometries to yield mode I, mode II and mixed mode I/II values of G c . Nevertheless, problems of consistency and discussions on the accuracy of such results abound. This paper describes a detailed study of the methods of analysing the experimental data obtained from fracture mechanics tests using double-cantilever beam, end loaded split and end notched flexure specimens. It is shown that to get consistent and accurate values of G c it is necessary to consider aspects of the tests such as the end rotation and deflection of the crack tip, the effective shortening of the beam due to large displacements of the arms, and the stiffening of the beam due to the presence of the end blocks bonded to the specimens. Analytical methods for ascertaining the various correction constants and factors are described and are successfully applied to the results obtained from three different fibre-polymer composites. These composites exhibit different types of fracture behaviour and illustrate the wide range of effects that must be considered when values of the interlaminar fracture energies, free from artefacts from the test method and the analysis method, are required.

373 citations


Journal ArticleDOI
TL;DR: In this paper, the second order Hamiltonian system (HS) has a homoclinic orbit q emanating from 0, where q ∊ ℝn and V ∊ C1 (ℝ ×ℽn ℽ) is T periodic in t. The orbit q is obtained as the limit as k → ∞ of 2kT periodic solutions (i.e. subharmonics) qk of (HS).
Abstract: Consider the second order Hamiltonian system:where q ∊ ℝn and V ∊ C1 (ℝ ×ℝn ℝ) is T periodic in t. Suppose Vq (t, 0) = 0, 0 is a local maximum for V(t,.) and V(t, x) | x| → ∞ Under these and some additional technical assumptions we prove that (HS) has a homoclinic orbit q emanating from 0. The orbit q is obtained as the limit as k → ∞ of 2kT periodic solutions (i.e. subharmonics) qk of (HS). The subharmonics qk are obtained in turn via the Mountain Pass Theorem.

364 citations


Journal ArticleDOI
TL;DR: In this paper, a power-law continuum model for small-strain nonlinear elasticity of granular media near states of zero stress, as it relates to the pressure-dependent incremental linear elasticity and wave speeds is presented.
Abstract: Following is an analysis of the small-strain nonlinear elasticity of granular media near states of zero stress, as it relates to the pressure-dependent incremental linear elasticity and wave speeds. The main object is elucidation of the p ½ dependence of incremental elastic moduli on pressure p , a dependence observed in numerous experiments but found to be at odds with the p ½ scaling predicted by various micromechanical models based on hertzian contact. After presenting a power-law continuum model for small-strain nonlinear elasticity, the present work develops micromechanical models based on two alternative mechanisms for the anomalous pressure scaling, namely: (1) departures at the single-contact level from the hertzian contact, due to point-like or conical asphericity; (2) variation in the number density of hertzian contacts, due to buckling of particle chains. Both mechanisms result in p ½ pressure scaling at low pressure and both exhibit a high-pressure transition to p ½ scaling at a characteristic transition pressure p *. For assemblages of nearly equal spheres, a non-hertzian contact model for mechanism (1) and percolation-type model for (2) yield estimates of p * of the form p * = c μ ˆ ∝ 3 . Here c is a non-dimensional coefficient depending only on granular-contact geometry, while α ≪ 1 is a small parameter representing spherical imperfections and μ ˆ is an appropriate elastic modulus of the particles. Then, with R representing particle radius and h a characteristic spherical tolerance or asperity height, it is found that α = ( h / R ) ½ for mechanism (1) as opposed to α = h / R for (2). Limited data from the classic experiments of Duffy & Mindlin on sphere assemblages tend to support mechanism (1), but more exhaustive experiments are called for. In addition to the above analysis of reversible elastic effects, a percolation model of inelastic ‘shake-down’ or consolidation is given. It serves to describe how prolonged mechanical vibration, leading to the replacement of point-like or inactive contacts by stiffer Hertz contacts may change the pressure-scaling behaviour of particulate media. The present analysis suggests that pressure-dependence of elasticity may provide a useful means of characterizing the state of consolidation and stability of dense particulate media.

334 citations


Journal ArticleDOI
TL;DR: In this paper, a general theory of the tension field is developed for application to the analysis of wrinkling in isotropic elastic membranes undergoing finite deformations, where the principal contribution is a partial differential equation describing a geometrical property of tension trajectories.
Abstract: A general theory of the tension field is developed for application to the analysis of wrinkling in isotropic elastic membranes undergoing finite deformations. The principal contribution is a partial differential equation describing a geometrical property of tension trajectories. This is one of a system of two equations which describes the state of stress independently of the deformation. This system is strongly elliptic at any stable solution, whereas the deformation is described by a system of parabolic type. Controllable solutions, i.e. those states that can be maintained in any isotropic elastic material by application of edge tractions and lateral pressure alone, are obtained. The general axisymmetric problem is solved implicitly and the theory is applied to the solution of two representative examples. Existing small strain theories are shown to correspond to a singular limit of the general theory, at which the underlying system changes from elliptic to parabolic type.

301 citations


Journal ArticleDOI
TL;DR: In this paper, the mathematical ideas of differentiable dynamics have had a profound impact on physics and some success stories are discussed, while attention is also directed to the excessive optimism of some currently attempted applications.
Abstract: Under the name of chaos the mathematical ideas of differentiable dynamics have had a profound impact on physics. Some success stories are discussed. Attention is also directed to the excessive optimism of some currently attempted applications. It is shown how and why they should fail.

Journal ArticleDOI
TL;DR: In this article, it is shown that the characteristic initial value problem for the Einstein equations in vacuum or with perfect fluid source is well posed when data are given on two transversely intersecting null hypersurfaces.
Abstract: A method is described by means of which the characteristic initial value problem can be reduced to the Cauchy problem and examples are given of how it can be used in practice. As an application it is shown that the characteristic initial value problem for the Einstein equations in vacuum or with perfect fluid source is well posed when data are given on two transversely intersecting null hypersurfaces. A new discussion is given of the freely specifiable data for this problem.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the critical probability of a percolation process on a lattice L equals the limit of a slice of L as the thickness of the slice tends to infinity.
Abstract: We prove a general result concerning the critical probabilities of subsets of a lattice L . It is a consequence of this result that the critical probability of a percolation process on L equals the limit of the critical probability of a slice of L as the thickness of the slice tends to infinity. This verification of one of the standard hypotheses of the subject settles many questions concerning supercritical percolation.

Journal ArticleDOI
TL;DR: In this article, an energetically consistent theory is presented for dynamics of partly elastic collisions between somewhat rough rigid bodies with friction that opposes slip, which is based on separately accounting for frictional and non-frictional sources of dissipation.
Abstract: An energetically consistent theory is presented for dynamics of partly elastic collisions between somewhat rough rigid bodies with friction that opposes slip. This theory is based on separately accounting for frictional and non-frictional sources of dissipation. Alternative theories derived from Newton’s impact law or Poisson’s impact hypothesis are shown to be valid only for central (collinear) or non-frictional collisions; generally the latter theories yield erroneous energy dissipation if small initial slip stops during collision between eccentric bodies. Collision processes are complex when small slip is stopped by friction; then either the direction of slip reverses or contact points roll without slip. An inconsistent theory based on Newton’s impact law can yield erroneous energy increases when slip stops during collision; the consistent theory always dissipates energy. The impact law that specifies a simple proportionality between normal components of contact velocity for incidence and rebound is not applicable in any range of incident velocities with small slip if the collision is non-collinear with friction. In Percussion the force or Impetus whereby one body is moved may cause another body against which it strikes to be put in motion, and withal lose some of its strength or swiftness. (J. Wallis, 1668)

Journal ArticleDOI
TL;DR: In this paper, a local existence and regularity theory is given for nonlinear parabolic initial value problems (x′(t) = f(x(t))), and quasilinear early value problems.
Abstract: In this paper a local existence and regularity theory is given for nonlinear parabolic initial value problems (x′(t) = f(x(t))), and quasilinear initial value problems (x′(t)=A(x(t))x(t) + f(x(t))). This theory extends the theory of DaPrato and Grisvard of 1979, and shows how various properties, like analyticity of solutions, can be derived as a direct corollary of the existence theorem.

Journal ArticleDOI
TL;DR: In this paper, a theory of attractors of parabolic reaction-diffusion equations in bounded domains is built, and the properties of semigroups corresponding to equations with solutions in spaces of growing and decreasing functions essentially differ.
Abstract: There is a large number of papers in which attractors of parabolic reaction-diffusion equations in bounded domains are investigated. In this paper, these equations are considered in the whole unbounded space, and a theory of attractors of such equations is built. While investigating these equations, specific difficulties arise connected with the noncompactness of operators, with the continuity of their spectra, etc. Therefore some new conditions on nonlinear terms arise. In this paper weighted spaces are widely applied. An important feature of this problem is worth mentioning: namely, properties of semigroups corresponding to equations with solutions in spaces of growing and of decreasing functions essentially differ.

Journal ArticleDOI
Z. Bilicki1, J. Kestin1
TL;DR: In this paper, the authors explore the potential of the homogeneous relaxation model (HRM) as a basis for the description of adiabatic, one-dimensional, two-phase flows.
Abstract: The paper explores the potential of the homogeneous relaxation model (HRM) as a basis for the description of adiabatic, one-dimensional, two-phase flows. To this end, a rigorous mathematical analysis highlights the similarities and differences between this and the homogeneous equilibrium model (HEM) emphasizing the physical and qualitative aspects of the problem. Special attention is placed on a study of dispersion, characteristics, choking and shock waves. The most essential features are discovered with reference to the appropriate and convenient phase space Ω for HRM, which consists of pressure P , enthalpy h , dryness fraction x , velocity w , and length coordinate z . The geometric properties of the phase space Ω enable us to sketch the topological pattern of all solutions of the model. The study of choking is intimately connected with the occurrence of singular points of the set of simultaneous first-order differential equations of the model. The very powerful centre manifold theorem allows us to reduce the study of singular points to a two-dimensional plane Π , which is tangent to the solutions at a singular point, and so to demonstrate that only three singular-point patterns can appear (excepting degenerate cases), namely saddle points, nodal points and spiral points. The analysis reveals the existence of two limiting velocities of wave propagation, the frozen velocity a f and the equilibrium velocity a e . The critical velocity of choking is the frozen speed of sound. The analysis proves unequivocally that transition from ω a f to w > a f can take place only via a singular point. Such a condition can also be attained at the end of a channel. The paper concludes with a short discussion of normal, fully dispersed and partly dispersed shock waves.

Journal ArticleDOI
TL;DR: In this article, a function space approach is employed to obtain bifurcation functions for which the zeros correspond to the occurrence of periodic or aperiodic solutions near heteroclinic cycles.
Abstract: A function space approach is employed to obtain bifurcation functions for which the zeros correspond to the occurrence of periodic or aperiodic solutions near heteroclinic or homoclinic cycles. The bifurcation function for the existence of homoclinic solutions is the limiting case where the period is infinite. Examples include generalisations of Silnikov's main theorems and a retreatment of a singularly perturbed delay differential equation.

Journal ArticleDOI
TL;DR: In this paper, the authors studied the way in which the transition amplitude to an initially unoccupied state increases to its exponentially small final value in the adiabatic approximation, for a 2-state quantum system.
Abstract: The way in which the transition amplitude to an initially unoccupied state increases to its exponentially small final value is studied in detail in the adiabatic approximation, for a 2-state quantum system. By transforming to a series of superadiabatic bases, clinging ever closer to the exact evolving state, it is shown that transition histories renormalize onto a universal one, in which the amplitude grows to its final value as an error function (rather than via large oscillations as in the ordinary adiabatic basis). The time for the universal transition is of order $\surd (\hslash /\delta)$ where $\delta $ is the small adiabatic (slowness) parameter. In perturbation theory the pre-exponential factor of the final amplitude renormalizes superadiabatically from the incorrect value ${\textstyle\frac{1}{3}}\pi $ (for the ordinary adiabatic basis) to the correct value unity. The various histories could be observed in spin experiments.

Journal ArticleDOI
TL;DR: Uniform elastic strain fields are found in two-phase fibrous media of arbitrary transverse geometry, and in any phase geometry. as discussed by the authors exploited the existence of such fields to establish exact connections between transformation and mechanical influence functions and concentration factors.
Abstract: Uniform elastic strain fields are found in two-phase fibrous media of arbitrary transverse geometry, and in two-phase media of any phase geometry. In initially stress-free fibrous solids, a single uniform field can be created by certain proportionally changing tractions derived from a uniform overall stress. In the presence of phase eigenstrains, many overall stress states can be superimposed to create uniform strain fields in fibrous media. The existence of such fields is exploited to establish a number of exact results for two-phase fibre systems. These include universal connections between phase and overall moduli, and between components of phase stress and strain fields; expressions for new transformation influence functions and concentration factors in terms of their mechanical counterparts; and also expressions for the overall stresses and strains caused by phase eigenstrains. Examples are presented for macroscopically monoclinic fibrous composites with transversely isotropic phases. In two-phase media of arbitrary phase geometry there is only a single uniform stress and strain field for each non-vanishing eigenstrain state. The existence of this field is utilized in derivation of exact connections between transformation and mechanical influence functions and concentration factors.

Journal ArticleDOI
Hermann Bondi1
TL;DR: In this article, the difficulties of conservation laws in general relativity are discussed, with special reference to the non-tangible nature of gravitational energy and its transformation into tangible forms of energy.
Abstract: The difficulties of conservation laws in general relativity are discussed, with special reference to the non-tangible nature of gravitational energy and its transformation into tangible forms of energy. Inductive transfer of energy is marked out as wholly distinct from wave transfer. Slow (adiabatic) changes are utilized to make clear, in the axi-symmetric case, that the mass of an isolated body is conserved irrespective of any local changes (e.g. of shape) and that in inductive transfer the movement of energy between two bodies can readily be traced by the changes in their masses.

Journal ArticleDOI
TL;DR: In this article, the three-dimensional structures of thirteen MX and MX$2}$ molten salts (M, metal; X, halide) have been modelled from neutron diffraction data by using the Reverse Monte Carlo method.
Abstract: The three-dimensional structures of thirteen MX and MX$\_{2}$ molten salts (M, metal; X, halide) have been modelled from neutron diffraction data by using the Reverse Monte Carlo method. Although the structures are highly disordered the dominant structural symmetries of the short-range order can be determined by using bond angle correlations and spherical harmonic invariants. It is found that cations in the alkali chlorides tend to be octahedrally coordinated by anions (and vice versa), although the large number of \`vacancies' leads to coordination numbers lower than six. CuCl has a tetrahedral coordination of anions about cations. In MX$\_{2}$ melts small cations are octahedrally coordinated except in ZnCl$\_{2}$ where the coordination is tetrahedral. As the cation size increases the coordination tends to cubic. In all cases the local structural symmetries in the melt are similar to those in the corresponding crystals. It is shown that the \`pre-peak' which occurs in the partial structure factor A$\_{\text{MM}}$(Q) for MX$_{2}$ melts with small cations, and which has been associated with intermediate range ordering, arises from a local density fluctuation due to cation `clustering'. This is the first time that a visual picture of such intermediate-range order has been obtained.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the idempotent rank of K(n, r) is the Stirling number of the second kind, defined as the cardinality of a minimal generating set of idempots.
Abstract: The subsemigroup Sing n of singular elements of the full transformation semigroup on a finite set is generated by n(n − l)/2 idempotents of defect one. In this paper we extend this result to the subsemigroup K(n, r) consisting of all elements of rank r or less. We prove that the idempotent rank, defined as the cardinality of a minimal generating set of idempotents, of K(n, r) is S(n, r) , the Stirling number of the second kind.

Journal ArticleDOI
TL;DR: In this paper, the stability and instability properties of solitary wave solutions φ(x − ct) of a general class of evolution equations of the form Muttf(u)x=0, which support weakly nonlinear dispersive waves were studied.
Abstract: This paper studies the stability and instability properties of solitary wave solutions φ(x – ct) of a general class of evolution equations of the form Muttf(u)x=0, which support weakly nonlinear dispersive waves. It turns out that, depending on their speed c and the relation between the dispersion (i.e. the order of the pseudodifferential operator) and the nonlinearity, travelling waves maybe stable or unstable. Sharp conditions to that effect are given.

Journal ArticleDOI
TL;DR: In this paper, invariant manifold theory is applied to singular perturbation problems and a specific condition is given to ensure the existence of heteroclinic connections between normally hyperbolic invariant manifolds.
Abstract: Based on Fenichel's geometric idea, invariant manifold theory is applied to singular perturbation problems. This approach clarifies the nature of outer and inner solutions. A specific condition is given to ensure the existence of heteroclinic connections between normally hyperbolic invariant manifolds. A method to approximate the connections is also presented.

Journal ArticleDOI
TL;DR: In this paper, a model for the large-scale structures in a turbulent shear layer is introduced, which is applicable to most turbulent flows, but with certain reservations, the model is only applicable to the case of turbulent mixing in a two-stream compressible shear.
Abstract: For over a quarter of a century it has been recognized that turbulent shear flows are dominated by large-scale structures. Yet the majority of models for turbulent mixing fail to include the properties of the structures either explicitly or implicitly. The results obtained using these models may appear to be satisfactory, when compared with experimental observations, but in general these models require the inclusion of empirical constants, which render the predictions only as good as the empirical database used in the determination of such constants. Existing models of turbulence also fail to provide, apart from its stochastic properties, a description of the time-dependent properties of a turbulent shear flow and its development. In this paper we introduce a model for the large-scale structures in a turbulent shear layer. Although, with certain reservations, the model is applicable to most turbulent shear flows, we restrict ourselves here to the consideration of turbulent mixing in a two-stream compressible shear layer. Two models are developed for this case that describe the influence of the large-scale motions on the turbulent mixing process. The first model simulates the average behaviour by calculating the development of the part of the turbulence spectrum related to the large-scale structures in the flow. The second model simulates the passage of a single train of large-scale structures, thereby modelling a significant part of the time-dependent features of the turbulent flow. In both these treatments the large-scale structures are described by a superposition of instability waves. The local properties of these waves are determined from linear, inviscid, stability analysis. The streamwise development of the mean flow, which includes the amplitude distribution of these instability waves, is determined from an energy integral analysis. The models contain no empirical constants. Predictions are made for the effects of freestream velocity and density ratio as well as freestream Mach number on the growth of the mixing layer. The predictions agree very well with experimental observations. Calculations are also made for the time-dependent motion of the turbulent shear layer in the form of streaklines that agree qualitatively with observation. For some other turbulent shear flows the dominant structure of the large eddies can be obtained similarly using linear stability analysis and a partial justification for this procedure is given in the Appendix. In wall-bounded flows a preliminary analysis indicates that a linear, viscous, stability analysis must be extended to second order to derive the most unstable waves and their subsequent development. The extension of the present model to such cases and the inclusion of the effects of chemical reactions in the models are also discussed.

Journal ArticleDOI
TL;DR: In this paper, the authors measured the high strain rate response of polycarbonate and polymethyl methacrylate (PMMA) and found that the fracture stress for both materials obeys a thermal activation rate theory of Eyring type.
Abstract: The high strain rate response of polycarbonate (PC) and polymethyl methacrylate (PMMA) are measured using a split Hopkinson torsion bar for shear strain rates Ẏ from 500 s -1 to 2200 s -1 , and temperatures in the range —100°C to 200°C. The yield and fracture behaviours are compared with previous data and existing theories for Ẏ -1 . We find that PC yields in accordance with the Eyring theory of viscous flow, for temperatures between the beta transition temperature T β ≈ — 100°C and the glass transition temperature T g = 147°C. At lower temperatures, T T β , backbone chain motion becomes frozen and the shear yield stress is greater than the Eyring prediction. Strain softening is an essential feature of yield of PC at all strain rates employed. Poly methyl methacrylate fractures before yield in the high strain rate tests for T T g 120°C. It is found that the fracture stress for both materials obeys a thermal activation rate theory of Eyring type. Fracture is thought to be nucleation controlled, and is due to the initiation and break down of a craze at the fracture stress τ f . Examination of the fracture surfaces reveals that failure is by the nucleation and propagation of inclined mode I microcracks which link to form a stepped fracture surface. This reveals that failure is by tensile cracking and not by a thermal instability in the material. The process of shear localization is fundamentally different from that shown by steel and titanium alloys.

Journal ArticleDOI
TL;DR: In this article, a general method is developed to predict the effective conductivity of an infinite, statistically homogeneous suspension of particles in an arbitrary (ordered or somewhat disordered) configuration.
Abstract: A general method is developed to predict the effective conductivity of an infinite, statistically homogeneous suspension of particles in an arbitrary (ordered or disordered) configuration. The method follows closely that of 'stokesian dynamics', and captures both far-field and near-field particle interactions accurately with no convergence difficulties. This is accomplished by forming a capacitance matrix, the electrostatic analogue of the low-Reynolds-number resistance matrix, which relates the monopole (charge), dipole and quadrupole of the particles to the potential field of the system. A far-field approximation to the capacitance matrix is formed via a moment expansion of the integral equation for the potential. The capacitance matrix of the infinite system is limited to a finite number of equations by using periodic boundary conditions, and the Ewald method is used to form rapidly converging lattice sums of particle interactions. To include near-field effects, exact two-body interactions are added to the far-field approximation of the capacitance matrix. The particle dipoles are then calculated directly to determine the effective conductivity of the system. The Madelung constant of cohesive energy of ionic crystals is calculated for simple and body-centred cubic lattices as a check on the method formulation. The results are found to be in excellent agreement with the accepted values. Also, the effective conductivities of spherical particles in cubic arrays are calculated for particle to matrix conductivity ratios of infinity, 10 and 0.01.

Journal ArticleDOI
TL;DR: In this article, it was shown that there exists a constant ρ ρ such that the electrical resistance of the Sierpinski carpet is at most 4π 4π ρ 4 ρ 2 ρ + ρ 1/4 ρ 0.
Abstract: Let F$\_{\text{n}}$ be the nth stage in the construction of the Sierpinski carpet. Let R$\_{\text{n}}$ be the electrical resistance of F$\_{\text{n}}$ when the left and right sides are each short-circuited, and a voltage is applied between them. We prove that there exists a constant $\rho $ such that ${\textstyle\frac{1}{4}}\rho ^{n}\leq $ R$\_{n}\leq $ 4$\rho ^{n}$. The motivation for this result came from the problem of establishing (a) the existence and (b) the value of the `spectral dimension' of the Sierpinski carpet. In this and a subsequent paper, we settle (a) and give bounds for (b).

Journal ArticleDOI
TL;DR: In this paper, the exponential small probability of transition between two quantum states, induced by the slow change over infinite time of an analytic hamiltonian, was shown to have a geometric origin.
Abstract: The exponentially small probability of transition between two quantum states, induced by the slow change over infinite time of an analytic hamiltonian $\hat{H}$ = H($\delta $t)$\cdot \hat{\boldsymbol{S}}$ (where $\delta $ is a small adiabatic parameter and $\hat{\boldsymbol{S}}$ is the vector spin-$\frac{1}{2}$ operator), contains an additional factor exp {$\Gamma \_{\text{g}}$} of purely geometric origin (that is, independent of $\delta $ and $\hslash $). For $\Gamma \_{\text{g}}$ to be non-zero, $\hat{H}$ must be complex hermitian rather than real symmetric, and the hamiltonian curve H($\tau $) must not lie in a plane through the origin nor be a helix identical (up to rigid motion) with its time reverse. An expression is given for $\Gamma _{\text{g}}$, as an integral from the real t axis to the complex time of degeneracy of the two states. Explicit examples are given. The geometric effect could be observed in experiments with spinning particles.