One-dimensional Elastic Continuum Research Articles

Распространение плоских волн в упругой полуплоскости, усиленной по своей границе накладкой бесконечной длины

The dynamic contact problem of the propagation of plane elastic waves incident from infinity onto the boundary of an elastic half-plane reinforced by an infinite plate of small thickness is studied. Questions related to the dynamic mutual influence of an elastic half-plane with an stringer attached to its boundary are studied. As a physical model for the stringer, a one-dimensional elastic continuum model is taken. The solution of the problem is constructed by direct application of the Fourier transform. Analytical expressions are obtained that represent the distribution of wave components in all parts of the half-plane.

Traveling Waves in a One-Dimensional Elastic Continuum Model of Cell Layer Migration with Stretch-Dependent Proliferation

Collective cell migration plays a substantial role in maintaining the cohesion of epithelial cell layers and in wound healing. A number of mathematical models of this process have been developed, all of which reduce to essentially a reaction-diffusion equation with diffusion and proliferation terms that depend on material assumptions about the cell layer. In this paper we extend a one-dimensional mathematical model of cell layer migration of Mi et al. (Biophys. J., 93 (2007), pp. 3745-3752) to incorporate stretch-dependent proliferation, and show that this formulation reduces to a generalized Stefan problem for the density of the layer. We solve numerically the resulting partial differential equation system using an adaptive finite difference method and show that the solutions converge to self-similar or traveling wave solutions. We analyze self-similar solutions for cases with no prolifera- tion, and necessary and sufficient conditions for existence and uniqueness of traveling solutions for a wide range of material assumptions about the cell layer.

Brillouin scattering from porous silicon-based optical Bragg mirrors

Brillouin light scattering experiments were done on porous silicon-based optical Bragg mirrors with modulation wavelengths of ∼100 nm. By using a combination of pseudo-reflection and backscattering geometries, phonon dispersion curves along the superlattice modulation axis were mapped. Excellent agreement is obtained with the bulk acoustic mode band structure calculated using a one-dimensional elastic continuum model. In addition to zone-folding of the bulk longitudinal mode dispersion curve, the samples are marked by a surface-localized acoustic mode at the superlattice-air interface. The frequency of this mode lies near the upper edge of a phononic band gap centered at ∼16 GHz. These results, along with optical reflectance data showing visible-range photonic band gaps, reveal that these samples are one-dimensional hypersonic phononic-photonic crystals.

Continuum Elastic Model of Epithelial Sheet Migration

A key component of a wound healing process is rapid migration of epithelial cells that covers the wound area and helps to protect the underlying tissue. In our recent paper∗ we have developed a one-dimensional continuum mechanical model of intestinal epithelial cell layer, that incorporates lamellipod forces at both the wound edge and in the interior of the layer, adhesion forces between the cell layer and the substrate, and elastic stress within the cell layer. Here, the model is extended to two-dimensions and solved numerically using a level set method, which tracks the moving wound edge on a fixed grid. The model is calibrated by comparing the position of wound edge with experimentally observed positions in scratch-wound assay experiments. These comparisons show good qualitative agreement between model results and experimental observations. The models supports experimental observations that the time to wound closure varies with initial wound shape and area, and that the closure is possible, albeit slower, if boundary lamellipod formation is inhibited.∗ Qi. et al., One-dimensional elastic continuum model of enterocyte layer migration, Biohys. J., 93, 3745-3752 (2007).

An atomistic-continuum Cosserat rod model of carbon nanotubes

The focus of the present work is an atomistic-continuum model of single-walled carbon nanotubes (CNTs) based on an elastic rod theory which can exhibit geometric as well as material nonlinearity [Healey, T.J., 2002. Material symmetry and chirality in nonlinearly elastic rods. Mathematics and Mechanics of Solids 7, 405–420]. In particular, the single-walled carbon nanotube (SWNT) is modeled as a one-dimensional elastic continuum with some finite thickness bounded by the lateral surface. Exploitation of certain symmetries in the underlying atomic structure leads to suitable representations of the continuum elastic strain energy density in terms of strain measures that capture extension, twist, bending, and shear deformations [Healey, T.J., 2002. Material symmetry and chirality in nonlinearly elastic rods. Mathematics and Mechanics of Solids 7, 405–420]. Bridging between the atomic scale and the effective continuum is carried out by parameterization of the continuum elastic energy and determination of the parameters using unit cell atomistic simulations over a range of deformation magnitudes and types. Specifically, the proposed model takes into account (a) bending, (b) twist, (c) shear, (d) extension, (e) coupled extension and twist, and (f) coupled bending and shear deformations. The extracted parameters reveal benefits of accounting for important anisotropic and large-strain effects as improvements over employing traditional, linearly elastic, isotropic, small-strain, continuum models to simulate deformations of atomic systems such as SWNTs. It is envisioned that the proposed approach and the extracted model parameters can serve as a useful input to simulations of SWNT deformations using existing nonlinearly elastic continuum codes based, for example, on the finite element method (FEM).

One-Dimensional Elastic Continuum Model of Enterocyte Layer Migration

Necrotizing enterocolitis is the leading cause of death from gastrointestinal disease in preterm infants. It results from an injury to the mucosal lining of the intestine, leading to translocation of bacteria and endotoxin into the circulation. Intestinal mucosal defects are repaired by the process of intestinal restitution, during which enterocytes migrate from healthy areas to sites of injury. In this article, we develop a mathematical model of migration of enterocytes during experimental necrotizing enterocolitis. The model is based on a novel assumption of elastic deformation of the cell layer and incorporates the following effects: i), mobility promoting force due to lamellipod formation, ii), mobility impeding adhesion to the cell matrix, and iii), enterocyte proliferation. Our model successfully reproduces the behavior observed for enterocyte migration on glass coverslips, namely the dependence of migration speed on the distance from the wound edge, and the finite propagation distance in the absence of proliferation that results in an occasional failure to close the wound. It also qualitatively reproduces the dependence of migration speed on integrin concentration. The model is applicable to the closure of a wound with a linear edge and, after calibration with experimental data, could be used to predict the effect of chemical agents on mobility, adhesion, and proliferation of enterocytes.

Dynamic Phase Transitions in a One-Dimensional Elastic Continuum

This paper summarizes the results of some recent investigations on the continuum mechanical modeling of thermo-elastic phase transitions in solids carried out jointly with J. K. Knowles. Our most recent efforts have been focussed on the role of a kinetic law and an initiation criterion in modeling dynamic phase transitions, and their relationship to theories that include viscosity and strain-gradient effects.

Effect of material constants on the orientation structure of ferroelectric liquid crystal cells

The orientation structure of ferroelectric liquid-crystal cells in layer-normal geometry was studied based on a one-dimensional elastic continuum model which included three nonequal elastic constants. By minimizing the distortion energy and the surface energies, the critical cell thickness for the uniform, splayed, and helical states has been obtained as functions of the cone angle, the helical pitch, the elastic constants, and the surface interaction coefficients. By a proper choice of the handedness of the liquid crystal helicoid and the material constants, the critical cell thickness for the uniform state can be made very large, and the constraint on small polar surface-interaction coefficient for the stabilization of uniform state can be relaxed to a great extent. It is also shown that bistable states can only be stabilized in cells with no polarity difference between the substrate surfaces.

Load transfer from a web to two parallel sheets

This paper deals with the elastostatic transfer of a tensile load from an infinite strip-shaped web of uniform rectangular cross section to two identical parallel sheets of infinite extent. The web is perpendicular to the sheets and is continuously bonded to the latter along finite portions of its “top” and “bottom” edges. The sheets are treated within the two-dimensional theory of generalized planes stress. On the other hand, two alternative models for the web are considered: (a) the web likewise is regarded as a two-dimensional continuum within the conventional theory of generalized plane stress and (b) the web is idealized as a one-dimensional elastic continuum. In either case, an integral equation with a Cauchy-type singular kernel is obtained for the density of the bond force between the web and the sheets. This equation, in turn, is reduced to a Fredholm integral equation of the second kind. Numerical results for the axial force in the web and the bond force are presented for several combinations of the governing material and geometrical parameters.

Certain contact problems for a half-space reinforced by elastic gussets: PMM vol. 36, n≗5, 1972, pp. 770–787

Contact problems for bodies with elastic reinforcements in the form of gussets (stringers) of slight thickness are directly connected with questions of load transmission from the gussets to elastic bodies, which are important for engineering practice. Various plane problems have been investigated in many papers. For example, two fundamental problems on the transmission of a load from a gusset infinite in both directions to a semi-infinite and infinite plate have been examined in [1]. The model of a one-dimensional elastic continuum of the gusset is taken as the fundamental physical model. A number of papers devoted to various extensions and modifications of the fundamental Melan problems was later executed within the scope of the physical assumptions in [1]. A sufficiently complete bibliography of these papers is contained in [2, 3], The papers [4–7] are devoted to giving a foundation for the model of the one-dimensional model of the elastic continuum of the gusset and to investigating some other contact problems for a half-plane with elastic gussets. While the domain of plane contact problems for bodies with elastic gussets of slight thickness has been developed sufficiently well, the domain of three-dimensional contact problems for bodies with elastic gussets of slight cross section has hardly ever been investigated, and the authors know of no papers in this area where a rigorous solution of such problems would be presented. In some sense, paper [8], referring to questions of determining the contact stresses on the lateral surface of a cylindrical rod imbedded in an elastic space or half-space, is an exception. Such a situation in the area of three-dimensional problems is explained by the fact that significant mathematical difficulties are encountered in their solution. Moreover, the model of one-dimensional elastic continuum for the gusset in combination with the model of contact along a line is not directly applicable in the formulation of three-dimensional contact problems for bodies with elastic gussets of small cross section. The model of contact along an area, when it is assumed that the stresses in the contact zone are distributed uniformly in the transverse direction, also does not correspond completely to reality. In contrast to the case of plane problems, a new approach is proposed herein to the formulation of three-dimensional contact problems for a half-space reinforced by elastic gussets of small cross section. Three kinds of contact problems are then examined, namely, problems when the elastic half-space is reinforced on some part of its boundary by an infinitely long gusset, a semi-infinite gusset. and a gusset of finite length. In the proposed formulation, the solution of these problems reduces to solving integro-differential equations with kernels expressible by complete elliptic integrals of the first and second kinds, under definite boundary conditions. An effective method of solving these equations is proposed.

Detailed Wave Equation and Dual Amplitude: Relativistic Quantum Dynamics of String Model of Hadrons

Relativistic quantum dynamics of a one-dimensional elastic continuum (string) is represented by a “detailed wave equation” which is defined at every point of the string and has local coupling to an external field at the end of string. The actual form of this wave equation is severely determined from its self-consistency, which is guaranteed by the existence of a closed algebra for the “invariant hamiltonian density operator”. The algabra in turn directly represents the “gauge invariance” of the theory. The invariant hamiltonian density operator consists of real and imaginary parts, of which the latter represents nonlocal interaction inside the string and exhibits a remarkable Hilbert transform relation. The detailed wave equation is brought to the set of the global wave equation corresponding to the wave equation in the usual sense and an infinite number of subsidiary conditions which work to suppress unphysical states. The unconventional definition of normal model operators {Cµr} enables a uniform treatment of external and internal motions and the factorization of all wave operators into convolutions. The theory leads to dual amplitude for a particular value of intercept. A completely equivalent formulation of theory with the introduction of a proper-time like parameter is also given, which supplies further insights. Finally speculations are made for generalizing the model to more realistic cases.

Comparison of Continuum and Discrete Lattice Results for Gas–Surface Interactions

Closed-form analytical results for trapping and gas–surface energy exchange have been derived for a simple model employing a one-dimensional elastic continuum representation of the solid with (1) a truncated harmonic potential and (2) a Morse potential for the interaction between the gas atom and the surface of the continuum. The results consist of formulas for the thermal accommodation coefficient (a.c.) and critical trapping energy, which are compared with corresponding results of McCarroll and Ehrlich and of Trilling for one-dimensional discrete lattices. With the truncated harmonic potential, the continuum gives practically the same values of a.c. and critical trapping energy as the discrete lattice, in accordance with Landau's 1935 prediction, for cases in which the collision frequency is considerably smaller than the cutoff frequency of the lattice. With a Morse potential, the variation of a.c. with temperature is qualitatively correct for the continuum solid model, but the a.c. values appear to be considerably larger than those predicted by Trilling for a discrete lattice. The analysis for the continuum brings out the fact that both the truncated harmonic and Morse potential formulas for a.c. can be expressed in the form of a similarity relation (Bμ)−1/2α = H(ρ), where α is the a.c., H is a function of ρ, B is the dimensionless potential-well depth, μ is the gas-atom–solid-atom mass ratio, ρ is the average energy per gas atom/potential-well depth. Experimental a.c. data for several gas–solid combinations appear to be correlated fairly well on a plot of α / (Bμ)1/2vsρ.

On the Diffusion of Load From a Transverse Tension Bar Into a Semi-Infinite Elastic Sheet

This paper deals with the load diffusion from a tension bar of finite length and uniform cross section into a semi-infinite sheet, the axis of the bar being perpendicular to the edge of the sheet. The bar is regarded as a one-dimensional elastic continuum, whereas the elastic sheet is treated within the two-dimensional theory of generalized plane stress. Three alternative models for the stringer-attachment are considered: (a) line-contact, (b) area-contact based on matching the axial stringer-strain and the corresponding average sheet-strain across the width of the strip of adhesion, and (c) area-contact based on matching the stringer strain and the corresponding sheet-strain along the center line of the strip of adhesion. It is shown that the line-contact model, in contrast to both area-contact models, does not admit the transmission of portions of the applied load through forces concentrated at the ends of the adhering bar segment. Further, asymptotic estimates are deduced for the end slopes of the load-diffusion curves appropriate to the three models under consideration. The integro-differential equation for the stringer-force in Case (b) and Case (c) is reduced to a standard Fredholm integral equation, which is solved numerically. The results thus obtained are compared with available experimental findings.