Symmetric Representation of Stress and Strain in the Stroh Formalism and Physical Meaning of the Tensors L, S, L(θ) and S(θ)

The Stroh formalism for two-dimensional deformation of an anisotropic elastic material does not give the stress _ij explicitly in a symmetric form. It does not give an explicit expression for the strain _ij at al. Mantic and Paris [1] have recently derived an explicit symmetric representation of stress. We present here a new and elementary derivation that is more straight forward and transparent. The derivation does not require consideration of the surface traction or the normalization of the Stroh eigenvectors. The new derivation also provides an explicit symmetric representation of strain. Moreover, it allows us to deduce two of the three Barnett–Lothe tensors L, S [2] and the associated tensors L ( ), S ( ) [3], resulting in a physical interpretation of these tensors and the component ( L S )₂₁.     

A new secular equation for slip waves along the interface of two dissimilar anisotropic elastic half-spaces in sliding contact

We consider wave propagation along the interface of two dissimilar anisotropic elastic half-spaces that are in sliding contact. A new secular equation is obtained that covers all special cases in one equation. One special case is when a Rayleigh wave (called the R$R$-wave) can propagate in both half-spaces with the same wave speed. Another special case is when a slip wave (called the S$S$-wave) can propagate in each of the half-spaces with the same wave speed. If a Rayleigh wave and a slip wave can propagate in one of the half-spaces it is called the RS-wave. In this case an interfacial slip wave exists in which the other half-space is at rest unless an RS-wave can also propagate in the other half-space. The results for general anisotropic elastic materials are applied to orthotropic materials.     

Nonlinear interfacial waves in a circular cylindrical container subjected to a vertical excitation

Singular perturbation theory of two-time scale expansions was developed in inviscid fluids to investigate the motion of single interface standing wave in a two-layer liquid-filled circular cylindrical vessel, which is subjected to a vertical periodical oscillation. It is assumed that the fluid in the circular cylindrical vessel is inviscid, incompressible and the motion is irrotational, a nonlinear amplitude equation including cubic nonlinear and vertically forced terms, was derived by the method of expansion of two-time scales without taking the influence of surface tension into account. By numerical computation, it is shown that different patterns of interface standing wave can be excited for different driving frequency and amplitude. We found that the interface wave mode become more and more complex as increasing of upper to lower layer density ratio γ$\gamma$. The traits of the standing interface wave were proved theoretically. In addition, the dispersion relation and nonlinear amplitude equation obtained in this article can reduce to the known results for a single fluid when γ=0,_h2→_h1$\gamma =0,h_2\to h_1$.     

Generalized Darboux transformation and localized waves in coupled Hirota equations

In this paper, we construct a generalized Darboux transformation to the coupled Hirota equations with high-order nonlinear effects like the third dispersion, self-steepening and inelastic Raman scattering terms. As application, an N$N$th-order localized wave solution on the plane backgrounds with the same spectral parameter is derived through the direct iterative rule. In particular, some semi-rational, multi-parametric localized wave solutions are obtained: (1) vector generalization of the first- and the second-order rogue wave solutions; (2) interactional solutions between a dark–bright soliton and a rogue wave, two dark–bright solitons and a second-order rogue wave; (3) interactional solutions between a breather and a rogue wave, two breathers and a second-order rogue wave. The results further reveal the striking dynamic structures of localized waves in complex coupled systems.     

On dispersive propagation of surface waves in patchy saturated porous media

Frequency-dependent velocity and attenuation for Rayleigh-wave propagation along a vacuum/patchy saturated porous medium interface are investigated in the low frequency band (0.1–1000 Hz). Conventional patchy saturation models for compressional waves are extended to account for Rayleigh wave propagation along a free surface. The mesoscopic interaction of fluid and solid phases, as a dominant loss mechanism in patchy saturated media, significantly affects Rayleigh-wave propagation and attenuation. Researches on the dispersion characteristics at low frequencies with different gas fractions in patchy saturated media also demonstrate a strong correlation between the Rayleigh-wave mode and the fast compressional wave. Especially, the strongest attenuation with the maximum value of 1/Q$1/Q$ for Rayleigh waves are obtained in the frequency range of 1–200 Hz. Numerical results show that the significant dependence of velocity and attenuation on frequencies and gas fractions presents a distinctive dynamical response of Rayleigh waves in the time domain.     

Spatiotemporal breather in diffraction decreasing media

The similarity transformation between the (3+1$3+1$)-dimensional nonlinear Schrödinger equation with different distributed transverse diffraction and the standard nonlinear Schrödinger equation is found, and a spatiotemporal breather solution is given based on this transformation. The control for the evolutional behaviors of a spatiotemporal breather is discussed. Our results manifest that the relation between the maximum accumulated time _Tm$T_m$ and the accumulated time, _T0$T_0$, with the maximum amplitude, is the basis to realize the control and manipulation of propagation behaviors of breathers, such as fast and slow excitations, sustainment and restraint. These results are potentially useful for future experiments in the optical communications and Bose–Einstein condensations.     

A scaling analysis for point–particle approaches to turbulent multiphase flows

Simple dimensional arguments are used in establishing three different regimes of particle time scale, where explicit expression for particle Reynolds number and Stokes number are obtained as a function of nondimensional particle size (d/η)$(d/\eta )$ and density ratio. From a comparative analysis of the different computational approaches available for turbulent multiphase flows it is argued that the point–particle approach is uniquely suited to address turbulent multiphase flows where the Stokes number, defined as the ratio of particle time scale to Kolmogorov time scale (_τp/_τk)$({\tau }_p/{\tau }_k)$, is greater than 1. The Stokes number estimate has been used to establish parameter range where point–particle approach is ideally suited. The point–particle approach can be extended to handle “finite-sized” particles whose diameter approach that of the smallest resolved eddies. However, new challenges arise in the implementation of Lagrangian–Eulerian coupling between the particles and the carrier phase. An approach where the inter-phase momentum and energy coupling can be separated into a deterministic and a stochastic contribution has been suggested.     

Chaining of weakly interacting particles suspended in viscoelastic fluids

An asymptotic theory based on multipole expansions is presented for multiparticle interactions in unbounded, weakly viscoelastic, creeping flows. The theory accounts for non-Newtonian sphere–sphere interactions that are of order O(De(a/R)²)$\text{O}(De{(a/R)}^2)$, where De is the Deborah number, a the sphere radius and R is the sphere–sphere separation. Analytic expressions are derived for the non-Newtonian correction to the multisphere mobility matrix for non-neutrally buoyant sedimenting spheres, and for neutrally buoyant spheres suspended in a shear flow. It is shown that these expressions give rise to particle chaining in simulations of interacting spherical particles.     

Green’s formula and singularity at a triple contact line. Example of finite-displacement solution

The various equations at the surfaces and triple contact lines of a deformable body are obtained from a variational condition, by applying Green’s formula in the whole space and on the Riemannian surfaces. The surface equations are similar to the Cauchy’s equations for the volume, but involve a special definition of the ‘divergence’ (tensorial product of the covariant derivatives on the surface and the whole space). The normal component of the divergence equation generalizes the Laplace’s equation for a fluid–fluid interface. Assuming that Green’s formula remains valid at the contact line (despite the singularity), two equations are obtained at this line. The first one expresses that the fluid–fluid surface tension is equilibrated by the two surface stresses (and not by the volume stresses of the body) and suggests a finite displacement at this line (contrary to the infinite-displacement solution of classical elasticity, in which the surface properties are not taken into account). The second equation represents a strong modification of Young’s capillary equation. The validity of Green’s formula and the existence of a finite-displacement solution are justified with an explicit example of finite-displacement solution in the simple case of a half-space elastic solid bounded by a plane. The solution satisfies the contact line equations and its elastic energy is finite (whereas it is infinite for the classical elastic solution). The strain tensor components generally have different limits when approaching the contact line under different directions. Although Green’s formula cannot be directly applied, because the stress tensor components do not belong to the Sobolev space H¹(V)$H^1(\text{V})$, it is shown that this formula remains valid. As a consequence, there is no contribution of the volume stresses at the contact line. The validity of Green’s formula plays a central role in the theory.     

Acoustic wave propagation in inhomogeneous,layered waveguides based on modal expansions and hp-FEM

A new model is presented for harmonic wave propagation and scattering problems in non-uniform, stratified waveguides, governed by the Helmholtz equation. The method is based on a modal expansion, obtained by utilizing cross-section basis defined through the solution of vertical eigenvalue problems along the waveguide. The latter local basis is enhanced by including additional modes accounting for the effects of inhomogeneous boundaries and/or interfaces. The additional modes provide implicit summation of the slowly convergent part of the local-mode series, rendering the remaining part to be fast convergent, increasing the efficiency of the method, especially in long-range propagation applications. Using the enhanced representation, in conjunction with an energy-type variational principle, a coupled-mode system of equations is derived for the determination of the unknown modal-amplitude functions. In the case of multilayered environments, h$h$- and p$p$-FEM have been applied for the solution of both the local vertical eigenvalue problems and the resulting coupled mode system, exhibiting robustness and good rates of convergence. Numerical examples are presented in simple acoustic propagation problems, illustrating the role and significance of the additional mode(s) and the efficiency of the present model, that can be naturally extended to treat propagation and scattering problems in more complex 3D waveguides.     

Adhesive contact on power-law graded elastic solids: The JKR–DMT transition using a double-Hertz model

A cohesive zone model of axisymmetric adhesive contact between a rigid sphere and a power-law graded elastic half-space is established by extending the double-Hertz model of Greenwood and Johnson (1998). Closed-form solutions are obtained analytically for the surface stress, deformation fields and equilibrium relations among applied load, indentation depth, inner and outer radii of the cohesive zone, which include the corresponding solutions for homogeneous isotropic materials and the Gibson solid as special cases. These solutions provide a continuous transition between JKR and DMT type contact models through a generalized Tabor parameter μ$\mu$. Our analysis reveals that the magnitude of the pull-off force ranges from (3+k)πRΔγ/2$(3+k)\pi R\mathrm{\Delta }\gamma /2$ to 2πRΔγ$2\pi R\mathrm{\Delta }\gamma$, where k$k$, R$R$ and Δγ$\mathrm{\Delta }\gamma$ denote the gradient exponent of the elastic modulus for the half-space, the radius of the sphere and the work of adhesion, respectively. Interestingly, the pull-off force for the Gibson solid is found to be identically equal to 2πRΔγ,$2\pi R\mathrm{\Delta }\gamma ,$ independent of the corresponding Tabor parameter. The obtained analytical solutions are validated with finite element simulations.     

Rayleigh waves with impedance boundary conditions in anisotropic solids

The paper is concerned with the propagation of Rayleigh waves in an elastic half-space with impedance boundary conditions. The half-space is assumed to be orthotropic and monoclinic with the symmetry plane _x3=0$x_3=0$. The main aim of the paper is to derive explicit secular equations of the wave. For the orthotropic case, the secular equation is obtained by employing the traditional approach. It is an irrational equation. From this equation, a new version of the secular equation for isotropic materials is derived. For the monoclinic case, the method of polarization vector is used for deriving the secular equation and it is an algebraic equation of eighth-order. When the impedance parameters vanish, this equation coincides with the secular equation of Rayleigh waves with traction-free boundary conditions.     

Molecular modeling of cracks at interfaces in nanoceramic composites

Toughness in Ceramic Matrix Composites (CMCs) is achieved if crack deflection can occur at the fiber/matrix interface, preventing crack penetration into the fiber and enabling energy-dissipating fiber pullout. To investigate toughening in nanoscale CMCs, direct atomistic models are used to study how matrix cracks behave as a function of the degree of interfacial bonding/sliding, as controlled by the density of C interstitial atoms, at the interface between carbon nanotubes (CNTs) and a diamond matrix. Under all interface conditions studied, incident matrix cracks do not penetrate into the nanotube. Under increased loading, weaker interfaces fail in shear while stronger interfaces do not fail and, instead, the CNT fails once the stress on the CNT reaches its tensile strength. An analytic shear lag model captures all of the micromechanical details as a function of loading and material parameters. Interface deflection versus fiber penetration is found to depend on the relative bond strengths of the interface and the CNT, with CNT failure occurring well below the prediction of the toughness-based continuum He–Hutchinson model. The shear lag model, in contrast, predicts the CNT failure point and shows that the nanoscale embrittlement transition occurs at an interface shear strength scaling as _τs~_εf,CNTσCNT${\tau }_s~{\epsilon }_{f,CNT}{\sigma }_{CNT}$ rather than _τs~_σCNT${\tau }_s~{\sigma }_{CNT}$ typically prevailing for micron scale composites, where _εf,CNT${\epsilon }_{f,CNT}$ and _σCNT${\sigma }_{CNT}$ are the CNT failure strain and stress, respectively. Interface bonding also lowers the effective fracture strength in SWCNTs, due to formation of defects, but does not play a role in DWCNTs having interwall coupling, which are weaker than SWCNTs but less prone to damage in the outerwall.     

Study of the rotational diffusivity coefficient of fibres in planar contracting flows with varying turbulence levels

The Fokker–Planck equation is solved by describing the evolution of a 3D fibre orientation state along a planar contraction. A constant value of the effective rotational diffusion coefficient was determined for four different turbulent flow cases in planar contractions, reported experimentally in the literature. Two hypotheses for the non-dimensional rotational diffusivity are presented, each based on two different turbulent time scales, i.e. the Kolmogorov time scales and the time scale associated with large energy bearing eddies. These hypotheses are dependent on either the Reynolds number, based on the Taylor micro-scale, and/or a non-dimensional fibre length. The hypothesis, based on the assumption of long fibres, _Lf/η?25$L_{\text{f}}/\eta ?25$, compared to the Kolmogorov scale and in the limit of large _Reλ${\mathit{Re}}_{\lambda }$ seems to capture the basic trends presented in the literature. This hypothesis has also the feature of predicting effects of varying fibre length within certain limits. Accordingly, by modeling the variation of turbulent quantities along the contraction in a CFD analysis, local values of rotational diffusivity can be evaluated with the mentioned hypothesis, based on either Kolmogorov time scale or Eulerian integral time scale.