Stress singularity due to traction discontinuity on an anisotropic elastic half-plane

In a half-plane problem with x₁ paralleling with the straight boundary and x₂ pointing into the medium, the stress components on the boundary whose acting plane is perpendicular to x₁ direction may be denoted by t₁ = [σ₁₁, σ₁₂, σ₁₃]^T. Stress components σ₁₁ and σ₁₃ are of more interests since σ₁₂ is completely determined by the boundary conditions. For isotropic materials, it is known that under uniform normal loading σ₁₁ is constant in the loaded region and vanishes in the unloaded part. Under uniform shear loading, σ₁₁ will have a logarithmic singularity at the end points of shear loading. In this paper, the behavior of the stress components σ₁₁ and σ₁₃ induced by traction-discontinuity on general anisotropic elastic surfaces is studied. By analyzing the problem of uniform tractions applied on the half-plane boundary over a finite loaded region, exact expressions of the stress components σ₁₁ and σ₁₃ are obtained which reveal that these components consist of in general a constant term and a logarithmic term in the loaded region, while only a logarithmic term exists in unloaded region. Whether the constant term or the logarithmic term will appear or not completely depends on what values of the elements of matrices Ω and Γ will take for a material under consideration. Elements for both matrices are expressed explicitly in terms of elastic stiffness. Results for monoclinic and orthotropic materials are all deduced. The isotropic material is a special case of the present results.     

Simple constitutive modelling of nonlinear viscoelasticity under general extension

We study the behaviour of a single integral constitutive equation, capable of providing analytic expressions for the viscoelastic stress in extensional flows of a variety of deformation histories and geometries, ranging from uniaxial to equibiaxial. It is based on the use of a stress damping function, with a power-law dependence on the elongation, λ: h(λ) = 1/λⁿ. The parameter n (0 ≤ n ≤ 2) signifies the nonlinear viscoelastic character of the material and, therefore, is an inverse measure of network connectivity strength of the underlying microstructure. This renders the constitutive approach applicable to incompressible polymers of a variable degree of branching, strain hardening and stress thinning behavior. Methods of connecting n with the macromolecular architecture and the alignment strength of the flow are also explored.     

Singular,hypersingular and singular free electromagnetic fields at wedge tips in metamaterials

The engineering response of metamaterials has a dramatic impact on the physics, optics and engineering communities, because they offer electromagnetic properties that are difficult or impossible to achieve with conventional materials. In this paper, an asymptotic analysis of the electromagnetic fields at multi-material wedges composed of metamaterials is proposed. This is made possible by removing the assumption of positive electric permittivities and magnetic permeabilities, an hypothesis which usually applies to conventional materials. Exploring the whole range of variability of these electromagnetic properties, it is shown that, in addition to the classical real eigenvalues 0 ? λ < 1 leading to power-law singularities of the type O(r^λ?1) as r → 0, it is also possible to find imaginary eigenvalues leading to hypersingular solutions, as well as nonsingular configurations for a suitable choice of the negative electric permittivities and magnetic permeabilities of the media. Moreover, to fully characterize the asymptotic fields, the analysis is not only restricted to the determination of the lowest real and complex eigenvalues, but is also extended to the evaluation of the higher-order nonsingular ones. The obtained analytical results collected in synthetic diagrams are expected to have impact on the design of micro- and nano-electro-mechanical systems.     

A generalized stress intensity factor to be applied to rounded V-shaped notches

In the presence of sharp (zero radius) V-shaped notches the notch stress intensity factors (N-SIFs) quantify the intensities of the asymptotic linear elastic stress distributions. They are proportional to the limit of the mode I or II stress components multiplied by the distance powered 1 − λ_i from the notch tip, λ_i being Williams’ eigenvalues. When the notch tip radius is different from zero, the definition is no longer valid from a theoretical point of view and the characteristic, singular, sharp-notch field diverges from the rounded-notch solution very next to the notch. Nevertheless, N-SIFs continue to be used as parameters governing fracture if the notch root radius is sufficiently small with respect to the notch depth.Taking advantage of a recent analytical formulation able to describe stress distributions ahead of rounded V-notches, the paper gives a generalized form for the notch stress intensity factors, in which not only the opening angle but also the tip radius dimension is explicitly involved. Such parameters quantify the stress redistribution due to the root radius with respect to the sharp notch case.     

Coalescence of sessile droplets of varying viscosities for line printing

Coalescence of sessile droplets is studied experimentally with water–glycerin mixtures of different viscosities. Effects of viscosity on the dimensionless spreading length (Ψ) and the center-to-center distance (L) are investigated for two droplets; the first droplet (D_s) is stationary on a substrate and the second droplet (D₀) landing at a center-to-center distance L from the first droplet. For a low viscosity fluid, Ψ is maximum when L approaches zero (or λ → 1, where λ = 1 − L/D_s), which represents a head-on collision. For a high viscosity fluid, Ψ is minimum when λ → 0.6. The effect of λ on line printing for various viscosities is also examined by printing multiple droplets. We found that the larger the viscosity, the less the breakup between droplets; viscosities smaller than 60 wt% glycerin yielded line breakup. The overlap ratio of λ > 0.3 produced not a line, but a bigger droplet or puddle because of coalescence. Data obtained in this work can provide insights for the fabrication of conductive microtracks or microinterconnects in printed-electronics applications where a line breakup between droplets would lead to an electrical circuit short.     

Revisiting the perforation of ductile plates by sharp-nosed rigid projectiles

The process of ductile plate perforation by sharp-nosed rigid projectiles is further examined in this work through 2D numerical simulations. We highlight various features concerning the effective resisting stress (σ_r) which a finite thickness plate, with a flow stress of Y_t, exerts on the projectile during perforation. In particular, we show that the normalized resisting stress (σ_r/Y_t) can be represented as a unique function of the normalized thickness of the plate (H/D, where H is plate thickness and D is projectile diameter), for a large range of normalized thicknesses. Our simulations for very thin target plates show that the penetration process is achieved through the well-known dishing mechanism, where the target material is pushed forward by the projectile’s nose. An important observation, which emerges from our simulations, is that the transition between the dishing and the hole enlargement mechanisms takes place at a normalized thickness of about H/D = 1/3. We also find that the normalized resistive stress for intermediate plate thicknesses, 1/3 < H/D < 1.0, is relatively constant at a value of σ_r/Y_t = 2.0. This range of thicknesses conforms to a state of quasi plane stress in the plates. For thicker plates (H/D > 1) the σ_r/Y_t ratio increases monotonically to values which represent the resistance to penetration of semi-infinite targets, where the stress state is characterized by plane strain conditions. Using a simple model, which is based on energy conservation, we can predict the values of the ballistic limit velocities for many projectile/target combinations, provided the perforation is done through the ductile hole enlargement mechanism. Good agreement is demonstrated between predictions from our model and experimental data from different sources, strongly enhancing the confidence in both the validity and usefulness of our model.     

Some exact solutions for rotating flows of a generalized Burgers’ fluid in cylindrical domains

The velocity field and the adequate shear stress corresponding to the flow of a generalized Burgers’ fluid model, between two infinite co-axial cylinders, are determined by means of Laplace and finite Hankel transforms. The motion is due to the inner cylinder that applies a time dependent torsional shear to the fluid. The solutions that have been obtained, presented in series form in terms of usual Bessel functions J₁( ? ), J₂( ? ), Y₁( ? ) and Y₂( ? ), satisfy all imposed initial and boundary conditions. Moreover, the corresponding solutions for Burgers’, Oldroyd-B, Maxwell, second grade, Newtonian fluids and large-time transient solutions for generalized Burgers’ fluid are also obtained as special cases of the present general solutions. The effect of various parameters on large-time and transient solutions of generalized Burgers’ fluid is also discussed. Furthermore, for small values of the material parameters, λ₂ and λ₄ or λ₁, λ₂, λ₃ and λ₄, the general solutions corresponding to generalized Burgers’ fluids are going to those for Oldroyd-B and Newtonian fluids, respectively. Finally, the influence of the pertinent parameters on the fluid motion, as well as a comparison between models, is shown by graphical illustrations.     

Logarithmic singularity of an elastic composite wedge subjected to out-of-the-plane extensional strain

When an elastic composite wedge is not under a plane strain deformation, an out-of-the-plane extensional strain exists. The singularity analysis for the stresses at the apex of the composite wedge reduces to a system of non-homogeneous linear equations. When the composite wedge consists of two anisotropic elastic materials, it is shown that the stresses have the (ln r) term for all combinations of wedge angles with few exceptions. The same is true when the materials are isotropic except that the (ln r) term may appear in the form of r(ln r) in the displacements only. For these isotropic composite wedges therefore the stresses are bounded, though not continuous, at the apex. However, there are isotropic composite wedges for which the stress singularity is logarithmic. Conditions are given for isotropic composite wedges for which the stresses are (a) uniform, (b) non-uniform but bounded and (c) logarithmic. Unlike the r^−λ singularity, the existence of the (ln r) term does not depend on the complete boundary conditions.     

三维双材料结构的应力奇异性分析

应用有限单元法子模型技术，对具有不同界面角的三维双材料结构的应力奇异性进行了分析。结果表明，应用子模型技术估算三维双材料结构的应力奇异性指数是有效的。然后分析了界面端线和界面端点处附近奇异性指数，得到了一些重要而有趣的结果。最后对消除三维双材料结构应力奇异性的几何条件进行了讨论。     

Failure in notched tension bars due to high-temperature creep: Interaction between nucleation controlled cavity growth and continuum cavity growth

For axi-symmetrically notched tension bars [Dyson, B.F., Loveday, M.S., 1981, Creep Fracture in Nimonic 80A under Tri-axial Tensile Stressing, In: Ponter A.R.S., Hayhurst, D.R. (Eds.), Creep in Structures, Springer-Verlag, Berlin, pp. 406–421] show two types of damage propagation are shown: for low stress, failure propagates from the outside notch surface to the centre-line; and for high stress, failure propagates from the centre-line to the outside notch surface. The objectives of the paper are to: identify the physics of the processes controlling global failure modes; and, describe the global behaviour using physics-based constitutive equations.Two sets of constitutive equations are used to model the softening which takes place in tertiary creep of Nimonic 80A at 750 °C. Softening by multiplication of mobile dislocations is firstly combined, for low stress, with softening due to nucleation controlled creep constrained cavity growth; and secondly combined, for high stress, with softening due to continuum void growth. The Continuum Damage Mechanics, CDM, Finite Element Solver DAMAGE XX has been used to study notch creep fracture. Low stress notch behaviour is accurately predicted provided that the constitutive equations take account of the effect of stress level on creep ductility. High stress notch behaviour is accurately predicted from a normalized inverse cavity spacing d/2? = 6, and an initial normalized cavity radius r_hi/? = 3.16 × 10^?3, where 2? is the cavity spacing, and d is the grain size; however, the constants in the strain rate equation required recalibration against high stress notch data. A void nucleation mechanism is postulated for high stress behaviour which involves decohesion where slip bands intersect second phase grain boundary particles. Both equation sets accurately predict experimentally observed global failure modes.     

Asymptotic fields around an interfacial crack with a cohesive zone ahead of the crack tip

This paper considers an interfacial crack with a cohesive zone ahead of the crack tip in a linearly elastic isotropic bi-material and derives the mixed-mode asymptotic stress and displacement fields around the crack and cohesive zone under plane deformation conditions (plane stress or plane strain). The field solution is obtained using elliptic coordinates and complex functions and can be represented in terms of a complete set of complex eigenfunction terms. The imaginary portion of the eigenvalues is characterized by a bi-material mismatch parameter ε = arctanh(β)/π, where β is a Dundurs parameter, and the resulting fields do not contain stress singularity. The behaviors of “Mode I” type and “Mode II” type fields based on dominant eigenfunction terms are discussed in detail. For completeness, the counterpart for the Mode III solution is included in an appendix.     

Spatiotemporal variations of PM2.5 and PM10 concentrations between 31 Chinese cities and their relationships with SO2, NO2, CO and O3

The variations of mass concentrations of PM_2.5, PM₁₀, SO₂, NO₂, CO, and O₃ in 31 Chinese provincial capital cities were analyzed based on data from 286 monitoring sites obtained between March 22, 2013 and March 31, 2014. By comparing the pollutant concentrations over this length of time, the characteristics of the monthly variations of mass concentrations of air pollutants were determined. We used the Pearson correlation coefficient to establish the relationship between PM_2.5, PM₁₀, and the gas pollutants. The results revealed significant differences in the concentration levels of air pollutants and in the variations between the different cities. The Pearson correlation coefficients between PMs and NO₂ and SO₂ were either high or moderate (PM_2.5 with NO₂: r = 0.256–0.688, mean r = 0.498; PM₁₀ with NO₂: r = 0.169–0.713, mean r = 0.493; PM_2.5 with SO₂: r = 0.232–0.693, mean r = 0.449; PM₁₀ with SO₂: r = 0.131–0.669, mean r = 0.403). The correlation between PMs and CO was diverse (PM_2.5: r = 0.156–0.721, mean r = 0.437; PM₁₀: r = 0.06–0.67, mean r = 0.380). The correlation between PMs and O₃ was either weak or uncorrelated (PM_2.5: r = −0.35 to 0.089, mean r = −0.164; PM₁₀: r = −0.279 to 0.078, mean r = −0.127), except in Haikou (PM_2.5: r = 0.500; PM₁₀: r = 0.509).     

Periodic group crack problems in an infinite plate

This paper investigates periodic group crack problems in an infinite plate. The periodic group crack is composed of infinite groups with numbering from j = −∞, …, −2, −1, 0, 1, 2, …, to j = ∞, and the groups are placed periodically. The same loading condition and the same geometry are assumed for cracks in all groups. A singular integral equation is used to solve the problems. The singular integral equation is formulated on cracks of the 0th group (or the central group) with the collection of influences from the infinite groups. The influences of many neighboring groups to the central group are evaluated exactly. Meantime, the influences of many remote groups to the central group can be summed up into one term approximately. The stress intensity factors at crack tips can be evaluated from the solution of the singular integral equation. It is found from some sample problems that the obtained results are very accurate. Finally, several numerical examples are presented and interaction among the group cracks is addressed.     

A unified definition for stress intensity factors of interface corners and cracks

Based upon linear fracture mechanics, it is well known that the singular order of stresses near the crack tip in homogeneous materials is a constant value −1/2, which is nothing to do with the material properties. For the interface cracks between two dissimilar materials, the near tip stresses are oscillatory due to the order of singularity being −1/2 ± iε and −1/2. The oscillation index ε is a constant related to the elastic properties of both materials. While for the general interface corners, their singular orders depend on the corner angle as well as the elastic properties of the materials. Owing to the difference of the singular orders of homogeneous cracks, interface cracks and interface corners, their associated stress intensity factors are usually defined separately and even not compatibly. Since homogenous cracks and interface cracks are just special cases of interface corners, in order to build a direct connection among them a unified definition for their stress intensity factors is proposed in this paper. Based upon the analytical solutions obtained previously for the multibonded anisotropic wedges, the near tip solutions for the general interface corners have been divided into five different categories depending on whether the singular order is distinct or repeated, real or complex. To provide a stable and efficient computing approach for the general mixed-mode stress intensity factors, the path-independent H-integral based on reciprocal theorem of Betti and Rayleigh is established in this paper. The complementary solutions needed for calculation of H-integral are also provided in this paper. To illustrate our results, several different kinds of examples are shown such as cracks in homogenous isotropic or anisotropic materials, central or edge notches in isotropic materials, interface cracks and interface corners between two dissimilar materials.     

Theoretical stress analysis of intersecting cylindrical shells subjected to external loads transmitted through branch pipes

A thin shell theoretical solution of two normally intersecting cylindrical shells subjected to thrust-out force and three kinds of moments transmitted through branch pipes is presented in this paper. The solutions of modified Morley equation, which can be applicable up to ρ₀ = d/D ⩽ 0.8 and λ = d/(DT)^1/2 ⩽ 8 and the order of accuracy is raised to O(T/D), for the four loading cases are given. The accurate continuity conditions of generalized forces and displacements at the intersecting curve of two cylindrical shells for the four loading cases and the condition of the uniqueness of displacements are derived in this paper. The presented results are verified by experimental and numerical results successfully. They are in agreement with WRC Bulletin 297 when d/D is small.     

Cracked infinite cylinder with two rigid inclusions under axisymmetric tension

This paper considers the problem of an axisymmetric infinite cylinder with a ring shaped crack at z = 0 and two ring-shaped rigid inclusions with negligible thickness at z = ±L. The cylinder is under the action of uniformly distributed axial tension applied at infinity and its lateral surface is free of traction. It is assumed that the material of the cylinder is linearly elastic and isotropic. Crack surfaces are free and the constant displacements are continuous along the rigid inclusions while the stresses have jumps. Formulation of the mixed boundary value problem under consideration is reduced to three singular integral equations in terms of the derivative of the crack surface displacement and the stress jumps on the rigid inclusions. These equations, together with the single-valuedness condition for the displacements around the crack and the equilibrium equations along the inclusions, are converted to a system of linear algebraic equations, which is solved numerically. Stress intensity factors are calculated and presented in graphical form.     

Bow shocks formed by plasma collisions in laser irradiated semi-cylindrical cavities

The formation of shocks in plasmas created by short pulse laser irradiation (λ = 800 nm, I ≈ 1 × 10¹² W cm^?2) of semi-cylindrical cavities of different materials was studied combining visible and soft X-ray laser interferometry with simulations. The plasma rapidly converges near the axis to form a dense bright plasma focus. Later in time a long lasting bow shock is observed to develop outside the cavity, that is shown to arise from the collision of plasmas originating from within the cavity and the surrounding flat walls of the target. The shock is sustained for tens of nanoseconds by the continuous arrival of plasma ablated from the target walls. The plasmas created from the heavier target materials evolve more slowly, resulting in increased shock lifetimes.     

Flow structures around an equilateral triangle arrangement of three spheres

This paper represents the results of an experimental study on the flow structure around a single sphere and three spheres in an equilateral-triangular arrangement. Flow field measurements were performed using a Particle Image Velocimetry (PIV) technique and dye visualization in an open water channel for a Reynolds number of Re = 5 × 10³ based on the sphere diameter. The distributions and flow features at the critical locations of the contours of the velocity fluctuations, the patterns of sectional streamlines, the vorticity contours, the turbulent kinetic energy, the Reynolds stress correlations and shedding frequency are discussed. The gap ratios (G/D) of the three spheres were varied in the range of 1.0 ⩽ G/D ⩽ 2.5 where G was the distance between the sphere centers, and D was the sphere diameter which was taken as 30 mm. Due to the interference of the shedding shear layers and the wakes, more complex features of the flow patterns can be found in the wake region of the two downstream spheres behind the leading sphere. For G/D = 1.25, a jet-like flow around the leading sphere through the gap between the two downstream spheres occurred, which significantly enhanced the wake region. It was observed that a continuous flow development involving shearing phenomena and the interactions of shedding vortices caused a high rate of fluctuations over the whole flow field although most of the time-averaged flow patterns were almost symmetric about the two downstream spheres.