On Spherical Inhomogeneity With Steigmann–Ogden Interface

2018 ◽  
Vol 85 (12) ◽  
Author(s):  
Anna Y. Zemlyanova ◽  
Sofia G. Mogilevskaya
Keyword(s):  
Surface Tension ◽  
Analytical Solutions ◽  
Effective Properties ◽  
Spherical Surface ◽  
Far Field ◽  
Elastic Space ◽  
Equilibrium Equations ◽  
Hydrostatic Loading ◽  
Ogden Model ◽  
Special Case

The problem of an infinite isotropic elastic space subjected to uniform far-field load and containing an isotropic elastic spherical inhomogeneity with Steigmann–Ogden interface is considered. The interface is treated as a shell of vanishing thickness possessing surface tension as well as membrane and bending stiffnesses. The constitutive and equilibrium equations of the Steigmann–Ogden theory for a spherical surface are written in explicit forms. Closed-form analytical solutions are derived for two cases of loading conditions—the hydrostatic loading and deviatoric loading with vanishing surface tension. The single inhomogeneity-based estimates of the effective properties of macroscopically isotropic materials containing spherical inhomogeneities with Steigmann–Ogden interfaces are presented. It is demonstrated that, in the case of vanishing surface tension, the Steigmann–Ogden model describes a special case of thin and stiff uniform interphase layer.

scholarly journals Optimal Shape and First Integrals for Inverted Compressed Column

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 334
Author(s):  
Enes Kacapor ◽  
Teodor M. Atanackovic ◽  
Cemal Dolicanin
Keyword(s):  
Variational Principle ◽  
First Integrals ◽  
Arbitrary Cross Section ◽  
Optimal Shape ◽  
Equilibrium Equations ◽  
Concentrated Force ◽  
Cross Sectional ◽  
Nonlinear Equilibrium ◽  
Elastic Column ◽  
Special Case

We study optimal shape of an inverted elastic column with concentrated force at the end and in the gravitational field. We generalize earlier results on this problem in two directions. First we prove a theorem on the bifurcation of nonlinear equilibrium equations for arbitrary cross-section column. Secondly we determine the cross-sectional area for the compressed column in the optimal way. Variational principle is constructed for the equations determining the optimal shape and two new first integrals are constructed that are used to check numerical integration. Next, we apply the Noether’s theorem and determine transformation groups that leave variational principle Gauge invariant. The classical Lagrange problem follows as a special case. Several numerical examples are presented.

scholarly journals Quadratic integrals of a multi-scalar cosmological model

2020 ◽  
Vol 35 (10) ◽  
pp. 2050068 ◽  
Author(s):  
Sameerah Jamal
Keyword(s):  
Scalar Field ◽  
Cosmological Model ◽  
Conservation Laws ◽  
Analytical Solutions ◽  
Physical System ◽  
Noether Symmetries ◽  
Spatial Curvature ◽  
Special Case ◽  
Friedmann Robertson Walker ◽  
Field Integral

In the context of Friedmann–Robertson–Walker (FRW) spacetime with zero spatial curvature, we consider a multi-scalar tensor cosmology model under the pretext of obtaining quadratic conservation laws. We propose two new interaction potentials of the scalar field. Integral to this task is the existence of dynamical Noether symmetries which are Lie–Bäcklund transformations of the physical system. Finally, analytical solutions of the field are found corresponding to each new model. In one of the models, we find that the scale factor mimics [Formula: see text]-cosmology in a special case.

Analytical solutions of a spherical nanoinhomogeneity under far-field unidirectional loading based on Steigmann–Ogden surface model

2020 ◽  
Vol 25 (10) ◽  
pp. 1904-1923
Author(s):  
Youxue Ban ◽  
Changwen Mi
Keyword(s):  
Flexural Rigidity ◽  
Elastic Stiffness ◽  
Surface Model ◽  
Far Field ◽  
Stress Concentrations ◽  
Equilibrium Equations ◽  
Legendre Series ◽  
Simple Method ◽  
Shear Moduli ◽  
Special Cases

For a solid surface or interface that is subjected to transverse loading, the influence of its flexural resistibility to bending deformation becomes significant. A spherical inhomogeneity or void embedded in an infinite elastic medium under the application of nonhydrostatic loads represents a typical example. In this work, we consider the most fundamental loading of a far-field unidirectional tension. Analytical displacements and stresses are developed by the coupling of a Steigmann–Ogden surface mechanical model, the simple method of Boussinesq displacement potentials, the semi-inverse method of elasticity, and Legendre series representations of spherical harmonics. The problem is then solved by converting the equilibrium equations of displacement into a linear system with respect to the Legendre series coefficients. The developed solutions are general in the sense that they may reduce to their classical or Gurtin–Murdoch counterparts as special cases. Analytical expressions reveal that the derived solution depends on four dimensionless ratios from among surface material parameters, shear moduli ratio, and inhomogeneity or void radius. In particular, instead of depending on both flexural parameters in the moment–curvature relation, one fixed combination is sufficient to represent the surface flexural rigidity. This is in contrast with the influence of the in-plane elastic stiffness, in which both surface Lamé parameters matter. Parametric studies further demonstrate that, for metallic inhomogeneities or voids with radii between 10 nm and 100 nm, the effects of surface flexural rigidity on stress distributions and stress concentrations are significant.

Capillary–gravity waves of solitary type and envelope solitons on deep water

1993 ◽  
Vol 252 ◽  
pp. 703-711 ◽  
Author(s):  
Michael S. Longuet-Higgins
Keyword(s):  
Surface Tension ◽  
Dispersion Relation ◽  
Gravity Waves ◽  
Deep Water ◽  
Small Amplitude ◽  
Finite Amplitude ◽  
Surface Slope ◽  
Carrier Wave ◽  
Envelope Solitons ◽  
Special Case

The existence of steady solitary waves on deep water was suggested on physical grounds by Longuet-Higgins (1988) and later confirmed by numerical computation (Longuet-Higgins 1989; Vanden-Broeck & Dias 1992). Their numerical methods are accurate only for waves of finite amplitude. In this paper we show that solitary capillary-gravity waves of small amplitude are in fact a special case of envelope solitons, namely those having a carrier wave of length 2π(T/ρg)1½2 (g = gravity, T = surface tension, ρ = density). The dispersion relation $c^2 = 2(1-\frac{11}{32}\alpha^2_{\max)$ between the speed c and the maximum surface slope αmax is derived from the nonlinear Schrödinger equation for deep-water solitons (Djordjevik & Redekopp 1977) and is found to provide a good asymptote for the numerical calculations.

scholarly journals A New Form of the General Solution of the Elastic Space Axisymmetric Problem in Pavement Mechanics

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Jia Liang ◽  
Guo Jian ◽  
He Shikai
Keyword(s):  
General Solution ◽  
Axisymmetric Problem ◽  
Hankel Transform ◽  
Elastic Half Space ◽  
Displacement Function ◽  
Winkler Foundation ◽  
Elastic Space ◽  
Equilibrium Equations ◽  
Present Solution ◽  
New Form

In order to analyze the stress and displacement of pavement, a new form of the general solution of the elastic space axisymmetric problem is proposed by the method of mathematics reasoning. Depending on the displacement function put forward by Southwell, displacement function is derived based on Hankel transform and inverse Hankel transform. A new form of the general solution of the elastic space axisymmetric problem has been set up according to a few basic equations as the geometric equations, constitutive equations, and equilibrium equations. The present solution applies to elastic half-space foundation and Winkler foundation; the stress and displacement of pavement are obtained by mathematical deduction. The example results show that the proposed method is practically feasible.

scholarly journals Analysis of Smart Piezo-Magneto-Thermo-Elastic Composite and Reinforced Plates: Part II – Applications

2014 ◽  
Vol 1 (1) ◽  
Author(s):  
D. A. Hadjiloizi ◽  
A.L. Kalamkarov ◽  
Ch. Metti ◽  
A. V. Georgiades
Keyword(s):  
Effective Properties ◽  
Electric Displacement ◽  
Micromechanical Model ◽  
Composite Plates ◽  
Local Fields ◽  
Orthotropic Materials ◽  
Constant Thickness ◽  
Equilibrium Equations ◽  
Static Approximation ◽  
Effective Coefficients

AbstractA comprehensive micromechanical model for the analysis of a smart composite piezo-magneto-thermoelastic thin plate with rapidly varying thickness is developed in Part I of thiswork. The asymptotichomogenization model is developed using static equilibrium equations and the quasi-static approximation of Maxwell’s equations. The work culminates in the derivation of general expressions for effective elastic, piezoelectric, piezomagnetic, dielectric permittivity and other coefficients. Among these coefficients, the so-called product coefficients are determined which are present in the behavior of the macroscopic composite as a result of the interactions between the various phases but can be absent from the constitutive behavior of some individual phases of the composite structure. The model is comprehensive enough to also allow for calculation of the local fields of mechanical stresses, electric displacement and magnetic induction. The present paper determines the effective properties of constant thickness laminates comprised of monoclinic materials or orthotropic materials which are rotated with respect to their principal material coordinate system. A further example illustrates the determination of the effective properties of wafer-type magnetoelectric composite plates reinforced with smart ribs or stiffeners oriented along the tangential directions of the plate. For generality, it is assumed that the ribs and the base plate are made of different orthotropic materials. It is shown in this work that for the purely elastic case the results of the derived model converge exactly to previously established models. However, in the more general case where some or all of the phases exhibit piezoelectric and/or piezomagnetic behavior, the expressions for the derived effective coefficients are shown to be dependent on not only the elastic properties but also on the piezoelectric and piezomagnetic parameters of the constituent materials. Thus, the results presented here represent a significant refinement of previously obtained results.

Sensitivity of horizontal flows to forcing geometry

2001 ◽  
Vol 432 ◽  
pp. 419-441
Author(s):  
ISAO KANDA ◽  
P. F. LINDEN
Keyword(s):  
Reynolds Number ◽  
Analytical Solutions ◽  
Stokes Flow ◽  
Large Reynolds Number ◽  
Two Dimensional ◽  
Vortex Pattern ◽  
Inverse Energy Cascade ◽  
Horizontal Flows ◽  
Source Sink ◽  
Special Case

We investigate the horizontal flow produced by source–sink forcing in a stably stratified fluid. The forcing jets are kept laminar and are placed along the boundary of a square domain. We find that the resultant flow patterns are extremely sensitive to the forcing geometry. The single dominant vortex pattern, interpreted as the result of inverse energy cascade of two-dimensional turbulence in our previous work (Boubnov, Dalziel & Linden 1994), turns out to be a special case. We show that some of the steady patterns resemble the eigenmodes of the Helmholtz equation as the inviscid vorticity equation. Although there are significant discrepancies in the streamfunction vs. vorticity relations between the observed flows and the analytical solutions, we identify the differences as a result of viscous diffusion of vorticity from the source flows. We also study the transition from forced to decaying flow. The flow assumes the properties of Stokes flow at quite large Reynolds number, indicating transformation into patterns with small advective acceleration.

The motion of a viscous drop sliding down a Hele-Shaw cell

1986 ◽  
Vol 163 ◽  
pp. 59-67 ◽  
Author(s):  
Kalvis M. Jansons
Keyword(s):  
Surface Tension ◽  
Viscous Fluid ◽  
Contact Angle Hysteresis ◽  
Leading Edge ◽  
Governing Equations ◽  
Drop Shape ◽  
Viscous Drop ◽  
Special Case ◽  
Hele Shaw Cell ◽  
Effect Of Surface

The motion of a viscous drop in a vertical Hele-Shaw cell is studied in a limit where the effect of surface tension through contact-angle hysteresis is significant. It is found that a rectangular drop shape is a possible steady solution of the governing equations, although this solution is unstable to perturbations on the leading edge. Even though the unstable edge is one where a viscous fluid is moving into a less viscous fluid, in this case air, this is shown to be a special case of the well-known Saffman—Taylor instability. An experiment is performed with an initially circular drop in which it is observed that the drop shape becomes approximately rectangular except at the leading edge, where it becomes rounded and sometimes has a ragged appearance.A drop sliding down a vertical Hele-Shaw cell is an example of a system where the action of surface tension is not always one of smoothing, since in this case it leads to the formation of right-angle corners at the back of the drop (rounded only slightly on the lengthscale of the gap thickness of the cell).

scholarly journals Mullins–Sekerka stability analysis for melting-freezing waves in helium

1994 ◽  
Vol 5 (1) ◽  
pp. 21-37
Author(s):  
Joseph D. Fehribach
Keyword(s):  
Surface Tension ◽  
Growth Rate ◽  
Stability Analysis ◽  
Original Problem ◽  
The Other ◽  
Field Conditions ◽  
System Of Equations ◽  
Far Field ◽  
Other Hand ◽  
The Stability

This paper considers the stability of melt-solid interfaces to eigenfunction perturbations for a system of equations which describe the melting and freezing of helium. The analysis is carried out in both planar and spherical geometries. The principal results are that when the melt is freezing, under certain far-field conditions, the interface is stable in the sense of Mullins and Sekerka. On the other hand, when the solid is melting (at least when the melting is sufficiently fast), the interface is unstable. In some circumstances these instabilities are oscillatory, with amplitude and growth rate increasing with surface tension and frequency. The last section considers the original problem of Mullins and Sekerka in the present notation.

Sign in / Sign up

Export Citation Format

Share Document