Cosserat interphase models for elasticity with application to the interphase bonding a spherical inclusion to an infinite matrix

Interphases are often modeled as interfaces with zero thickness using jump conditions that can be developed based on approximate shell or membrane models which are valid for specific limited ranges of the elastic material parameters. For a two-dimensional problem it has been shown (Rubin and Benveniste, 2004) that the Cosserat model of a finite thickness interphase is a unified model that is accurate over the full range of elastic parameters. In contrast, many other interphase models are valid for only limited ranges of the elastic parameters. In this paper, the accuracy of different Cosserat models of a finite thickness interphase that connects a spherical inclusion to an infinite matrix is examined. Specifically, four Cosserat interphase models are considered: a general shell (GS)$(\mathit{GS})$, a membrane-like shell (MS)$(\mathit{MS})$, a simple shell (SS)$(\mathit{SS})$ and a generalized membrane (GM)$(\mathit{GM})$. The models (GS)$(\mathit{GS})$ and (MS)$(\mathit{MS})$ both satisfy restrictions on the strain energy function of the interphase that ensure exact solutions for all homogeneous three-dimensional deformations, while the other models (SS)$(\mathit{SS})$ and (GM)$(\mathit{GM})$ do not satisfy these restrictions. The importance of these restrictions is examined for the three-dimensional inhomogeneous inclusion problem being considered. This is the first test of the accuracy of an elastic interphase model for a spherical interphase.     

Lateral migration of a capsule in a plane Poiseuille flow in a channel

Three-dimensional numerical simulation is presented on the motion of a deformable capsule undergoing large deformation in a plane Poiseuille flow in a channel at small inertia. The capsule is modeled as a liquid drop surrounded by an elastic membrane which follows neo-Hookean law. The numerical methodology is based on a mixed finite-difference/Fourier transform method for the flow solver and a front-tracking method for the deformable interface. The methodology can address large deformation of a capsule over a wide range of capsule-to-medium viscosity ratio. An extensive validation of the methodology is presented on capsule deformation in linear shear flow and compared with the boundary-element/integral simulations. Motion of a capsule in wall-bounded parabolic flow is simulated over an extended period of time to consider both transient and steady-state motion. Lateral migration of the capsule towards the centerline of the channel is observed. Results are presented over a range of capillary number, viscosity ratio, capsule-to-channel size ratio, and lateral location. After an initial transient phase during which the capsule deforms very quickly, the flow of the capsule is observed to be a quasi-steady process irrespective of capillary number (Ca)$(\mathit{Ca})$, capsule-to-channel size ratio (a/H)$(a/H)$, and viscosity ratio (λ)$(\lambda )$. Migration velocity and capsule deformation are observed to increase with increasing Ca$\mathit{Ca}$ and a/H$a/H$, but decrease with increasing λ$\lambda$, and increasing distance from the wall. Numerical results on the capsule migration are compared with the analytical results for liquid drops, and capsules with Hookean membrane which are valid in the limit of small deformation. Unlike the prediction for liquid drops, capsules are observed to migrate toward the centerline for 0.2?λ?5$0.2?\lambda ?5$ range considered here. The migration velocity is observed to depend linearly on (a/H)³$(a/H{)}^3$, in agreement with the small-deformation theory, but non-linearly on Ca$\mathit{Ca}$ and the distance from the wall, in violation of the theory. Using the present numerical results and the analytical results, we present a correlation that can reasonably predict migration velocity of a capsule for moderate values of a/H$a/H$ and Ca$\mathit{Ca}$.     

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.     

A Riemannian approach to strain measures in nonlinear elasticity

The isotropic Hencky strain energy appears naturally as a distance measure of the deformation gradient to the set SO(n)$\mathrm{SO}(n)$ of rigid rotations in the canonical left-invariant Riemannian metric on the general linear group GL(n)$\mathrm{GL}(n)$. Objectivity requires the Riemannian metric to be left-GL(n)$\mathrm{GL}(n)$-invariant, isotropy requires the Riemannian metric to be right-O(n)$\mathrm{O}(n)$-invariant. The latter two conditions are only satisfied for a three-parameter family of Riemannian metrics on the tangent space of GL(n)$\mathrm{GL}(n)$. Surprisingly, the final result is basically independent of the chosen parameters.     

Relative periodic orbits in plane Poiseuille flow

A branch of relative periodic orbits is found in plane Poiseuille flow in a periodic domain at Reynolds numbers ranging from Re=3000$\mathit{Re}=3000$ to Re=5000$\mathit{Re}=5000$. These solutions consist in sinuous quasi-streamwise streaks periodically forced by quasi-streamwise vortices in a self-sustained process. The streaks and the vortices are located in the bulk of the flow. Only the amplitude, but not the shape, of the averaged velocity components does change as the Reynolds number is increased from 3000 to 5000. We conjecture that these solutions could therefore be related to large- and very large-scale structures observed in the bulk of fully developed turbulent channel flows.     

Effect of cylinder proximity to the wall on channel flow heat transfer enhancement

Heat-transfer enhancement in a uniformly heated slot mini-channel due to vortices shed from an adiabatic circular cylinder is numerically investigated. The effects of gap spacing between the cylinder and bottom wall on wall heat transfer and pressure drop are systemically studied. Numerical simulations are performed at Re=100$\mathit{Re}=100$, 0.1?Pr?10$0.1?\mathrm{Pr}?10$ and a blockage ratio of D/H=1/3$D/H=1/3$. Results within the thermally developing flow region show heat transfer augmentation compared to the plane channel. It was found that when the obstacle is placed in the middle of the duct, maximum heat transfer enhancement from channel walls is achieved. Displacement of circular cylinder towards the bottom wall leads to the suppression of the vortex shedding, the establishment of a steady flow and a reduction of both wall heat transfer and pressure drop. Performance analysis indicates that the proposed heat transfer enhancement mechanism is beneficial for low-Prandtl-number fluids.     

Temporal behavior of laser induced elastic plate resonances

This paper investigates the dependence on Poisson’s ratio of local plate resonances in low attenuating materials. In our experiments, these resonances are generated by a pulse laser source and detected with a heterodyne interferometer measuring surface displacement normal to the plate. The laser impact induces a set of resonances that are dominated by Zero Group Velocity (ZGV) Lamb modes. For some Poisson’s ratio, thickness-shear resonances are also detected. These experiments confirm that the temporal decay of ZGV modes follows a t^−0.5$t^{-0.5}$ law and show that the temporal decay of the thickness resonances is much faster. Similar decays are obtained by numerical simulations achieved with a finite difference code. A simple model is proposed to describe the thickness resonances. It predicts that a thickness mode decays as t^−1.5$t^{-1.5}$ for large times and that the resonance amplitude is proportional to D^−1.5$D^{-1.5}$ where D$D$ is the curvature of the dispersion curve ω(k)$\omega \left(k\right)$ at k=0$k=0$. This curvature depends on the order of the mode and on the Poisson’s ratio, and it explains why some thickness resonances are well detected while others are not.     

Explicit expression of polarization vector for surface waves,slip waves,Stoneley waves and interfacial slip waves in anisotropic elastic materials

We present explicit expression of the polarization vector for surface waves and slip waves in an anisotropic elastic half-space, and Stoneley waves and interfacial slip waves in two dissimilar anisotropic elastic half-spaces. An unexpected result is that, in the case of interfacial slip waves, the polarization vector for the material in the half-space _x2≥0$x_2\ge 0$ does not depend explicitly on the material property in the half-space _x2≤0$x_2\le 0$. It depends on the material property in the half-space _x2≤0$x_2\le 0$ implicitly through the interfacial slip wave speed υ$\upsilon$. The same is true for the polarization vector for the material in the half-space _x2≤0$x_2\le 0$.     

Lamb wave mode decomposition for structural health monitoring

Lamb waves propagate over large distances in plate-like thin structures and they have received great attention in the structural health monitoring (SHM) field as an efficient means to inspect a large area of a structure by using only a small number of sensors. The times-of-flight of the Lamb wave modes are useful for detecting damage generated in a structure. However, due to the dispersive and multi-mode nature of Lamb waves, it is very challenging to decompose Lamb wave modes into symmetric and anti-symmetric modes for potential applications to structural health monitoring. Thus, we propose an efficient Lamb wave mode decomposition method based on two fundamental rules: the group velocity ratio rule and the mode amplitude ratio rule. The group velocity ratio rule means that the ratio of the group velocities of _A0$A_0$ and _S0$S_0$ modes must be constant. The mode amplitude ratio rule means that the ratio of the magnitudes of _A0$A_0$ and _S0$S_0$ modes in a measured response signal must be always greater than one once the center frequency of the input signal is determined, such that the magnitude of the _A0$A_0$ mode in the excited signal is larger than that of the _S0$S_0$ mode, and vice versa. The proposed method is verified through experiments conducted for a plate specimen.     

Wake response of a stationary finite-sized particle in a turbulent channel flow

Here we consider the effect of a finite-sized stationary particle in a channel flow of modest turbulence at _Reτ=178.12${\mathit{Re}}_{\tau }=178.12$. The size of particle is varied such that the particle Reynolds number ranges from about 40 to 450. The location of the particle is chosen to be either in the buffer layer (_yp+=17.81)$(y_{p+}=17.81)$ or at the channel center. Fully resolved direct numerical simulations of the turbulent channel flow around the particles is performed. Here the ambient turbulence intensity relative to the mean velocity seen by the particle is large (I=23.16%)$(I=23.16\%)$ in the buffer region, while it is substantially lower (I=4.09%)$(I=4.09\%)$ at the channel center. We present results on turbulence modulation due to the particle in terms of wake dynamics and vortex shedding.     

The two-dimensional free-space Green’s function and its derivatives in a tangent-normal system

The two-dimensional free-space Green’s function, G⁽²⁾$G^{\left(2\right)}$, and its derivatives, are used extensively in the formulation of scattering and diffraction problems through its presence in single- and double-layer potentials, and their use in integral equations. The vast majority of the results from elementary classical mathematical physics for G⁽²⁾$G^{\left(2\right)}$ is based on Cartesian coordinate-space, either directly as a Hankel function in coordinate-space or through a transform, such as the Weyl transform, also based on Cartesian coordinate-space. However, if the geometry of the problem is not Cartesian, for example in scattering from a rough surface, there are difficulties in using a transform representation for G⁽²⁾$G^{\left(2\right)}$ which depends on Cartesian geometry, as the standard Weyl transform does. Here we formulate transform-space representations using a tangent-normal coordinate system. The result for G⁽²⁾$G^{\left(2\right)}$ is a new Weyl-type tangent-normal transform representation from which the results for the vector derivatives of the single-layer potential, the double-layer potential, and the vector derivatives of the double-layer potential follow quite simply. The latter three results can be expressed in terms of two new spectral functions in tangent-normal space, _S1$S_1$ and _S2$S_2$. The overall results are new representations for G⁽²⁾$G^{\left(2\right)}$ and its derivatives which may be useful in integral equation formulations of scattering problems for non-Cartesian geometries.     

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 non-homogeneous constitutive model for human blood: Part II. Asymptotic solution for large Péclet numbers

In tube flow of healthy human blood the formed elements typically migrate away from vessel walls, leaving a plasma-rich, cell-depleted region there. In larger tubes (corresponding in size to arteries, for example) and at physiologically realistic flow rates, very thin wall boundary layers may develop which, nonetheless, have an impact upon the bulk flow properties. In this paper the non-homogeneous blood model of Moyers-Gonzalez et al. [M. Moyers-Gonzalez, R.G. Owens and J. Fang, A non-homogeneous constitutive model for human blood. Part I. Model derivation and steady flow, submitted for publication] is used in combination with a novel matched asymptotic method, to study the boundary layer behaviour of the steady tube flow of blood at high Péclet numbers Pe$Pe$ and in vessels of diameters corresponding to those of small arteries. A boundary layer thickness of O(Pe^−1/2)$O(P{\text{e}}^{-1/2})$ is predicted. In the absence of stress diffusion (the homogeneous case, with Pe=∞$Pe=\infty$) no cell migration takes place and the size and number density of red cell aggregates along the axis of symmetry remains constant at all flow rates. In the non-homogeneous case, however, even at very high values of Pe$Pe$, particles migrate, introducing a thin apparent slip layer next to the wall and affecting the aggregate distribution throughout the flow, even on the axis of symmetry.     

Inversion for weak triclinic anisotropy from acoustic axes

Acoustic axes are directions in anisotropic elastic media, in which phase velocities of two or three plane waves (P$P$, S1$S1$ or S2$S2$ waves) coincide. Acoustic axes are important, because they can cause singularities in the field of polarization vectors and anomalies in the shape of the slowness surface. The maximum number of acoustic axes in triclinic anisotropy is 16, and their directions depend on anisotropy parameters in a complicate way. Under weak anisotropy approximation this dependence simplifies and the directions of acoustic axes can be used for the inversion for anisotropy parameters. The maximum acoustic axes under weak anisotropy is 16, the minimum number of acoustic axes is zero. In the inversion, we can retrieve 13 combinations of anisotropy parameters provided we use directions of 7 acoustic axes at least. Under weak anisotropy approximation, the directions of acoustic axes are insensitive to strength of anisotropy; hence we cannot invert for absolute values of weak anisotropy parameters, but only for their relative values. Numerical tests have shown that the inversion is applicable only to very weak anisotropy with strength of less than 5%, provided that the acoustic axes used in the inversion are determined with an accuracy of 0.1^°$0.1^{°}$ or better. In this case the inversion yields an average error for elastic parameters of less than 10%. In order to invert for the total set of 21 anisotropy parameters it is necessary to combine the measurements of the directions of the acoustic axes with measurements of other attributes of elastic waves in anisotropic media.     

The stress concentration factor for slightly roughened random surfaces: Analytical solution

We derive an analytical solution to the stress concentration factor (_kt)$(k_t)$ for slightly roughened random surfaces. Topology is assumed to possess Gaussian distribution of heights and auto correlation length, ACL  . For our development, we combine Gao’s first-order perturbation method, the Hilbert transform, and an energy conservation principal related to the Parseval theorem.The root-mean-square (RMS) value of _kt$k_t$ results in a function of the ratio RMS-roughness to ACL. The derived formula agrees with experimental results previously reported. The results provide insight for more efficient design.     

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.