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2012

Volume 72, Issue 1 (partial)


Stabilization in a State-Dependent Model of Turning Processes

Qingwen Hu, Wieslaw Krawcewicz, and Janos Turi

SIAM J. Appl. Math. 72, pp. 1-24 (24 pages)

Online Publication Date: January 03, 2012

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We consider a two-degree-of-freedom model for turning processes which involves a system of differential equations with state-dependent delay. Depending on process parameters (e.g., spindle speed, depth of cut) the cutting tool can exhibit unwanted vibrations, resulting in a nonsmooth surface of the workpiece. In this paper we propose a feedback law to stabilize the turning process for a large range of system parameters. The feedback law introduces a generic nonhyperbolic stationary point into the model, which generates the main technical challenge of this work. We establish the stability equivalence between the differential equations with state-dependent delay and a corresponding nonlinear system with the delay fixed at its stationary value. Then we show the stability of that nonlinear system with constant delay by computing its normal form. Finally, we obtain conditions on system parameters which guarantee the stability of the state-dependent delay model at the nonhyperbolic stationary point.

Lyapunov Functions and Global Stability for Age-Structured HIV Infection Model

Gang Huang, Xianning Liu, and Yasuhiro Takeuchi

SIAM J. Appl. Math. 72, pp. 25-38 (14 pages)

Online Publication Date: January 03, 2012

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We study the basic age-structured population model describing the HIV infection process, which is defined by PDEs. The model allows the production rate of viral particles and the death rate of productively infected cells to vary and depend on the infection age. By using the direct Lyapunov method and constructing suitable Lyapunov functions, dynamical properties of the age-structured model without (or with) drug treatment are established. The results show that the global asymptotic stability of the infection-free steady state and the infected steady state depends only on the basic reproductive number determined by the burst size. Further, we establish mathematically that the typical ODE and DDE (delay differential equation) models of HIV infection are equivalent to two special cases of the above PDE models.

Geometry-Driven Charge Accumulation in Electrokinetic Flows between Thin, Closely Spaced Laminates

B. S. Tilley, B. Vernescu, and J. D. Plummer

SIAM J. Appl. Math. 72, pp. 39-60 (22 pages)

Online Publication Date: January 03, 2012

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Fluid flows through anisotropic media are found in a wide variety of geophysical and biological systems. The macroscale behavior of these systems depends on the microstructure, which in turn may depend on local and global physical processes. Classically, geometric restrictions are needed to model these systems on the largest length scale, and we are interested in developing effective models which relax these restrictions. To explore the development of these multiscale models, we consider an array of closely spaced, purely dielectric rigid laminates with nonuniform thickness. The laminate thickness and spacing varies on a length scale much longer than the characteristic thickness of the laminates. In the spacing between the laminates, an electrically conducting fluid is driven by an applied electric field through electroosmosis and electrophoresis along with an applied pressure gradient. Debye layers occur at the laminate-fluid interface, which are assumed to be much smaller than the laminate thickness. From a modification of the classical homogenization approach that relies on a physical microscale constraint in place of a geometric constraint, we derive an effective set of equations that describe the fluid pressure, the anion and cation concentrations in the fluid, and the electric potential. Anisotropic dispersion effects in the electric field are included, and electroneutrality in the fluid is not imposed. We find that gradients in the laminate spacing can lead to charge accumulation when electroosmosis and the electrophoresis induced from the anisotropic dispersion effects balance.

Sampling in Flat Detector Fan Beam Tomography

Steven H. Izen

SIAM J. Appl. Math. 72, pp. 61-84 (24 pages)

Online Publication Date: January 10, 2012

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In fan beam tomography, functions in $\mathbb{R}^2$ are reconstructed from integrals along rays which emanate from source points on a circle exterior to the object. The ray from each source can be parameterized either by the angle subtended with the ray through the origin or by the location of intersection with a line through the origin perpendicular to the central ray. The former models a curved detector and the latter a flat detector. Requirements for reconstructing an essentially bandlimited function from regular samples acquired with a curved detector are well known [F. Natterer, SIAM J. Appl. Math., 53 (1993), pp. 358–380]. These results are extended to the flat detector. The essential bandregion of flat fan beam transform of an essentially bandlimited function is computed and shown to be a superset of the scaled version of the corresponding bandregion of the curved fan beam transform of the function. Sampling conditions for the flat fan beam transform follow and differ from those appearing in the literature [F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, SIAM, Philadelphia, 2001].

A Semianalytical Solution for a Compressible Turbulent Axisymmetric Jet

Babak Emami, Markus Bussmann, and Honghi N. Tran

SIAM J. Appl. Math. 72, pp. 85-98 (14 pages)

Online Publication Date: January 10, 2012

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The change of variables (see [G. Kleinstein, Quart. J. Appl. Math., 20 (1962), pp. 49–54], and also see [G. Kleinstein, J. Spacecraft, 1 (1964), pp. 403–408]) applied to the mean flow equations for a free compressible axisymmetric shear layer is extended to the turbulence model equations. This reduces these nonlinear PDEs to the form of nonhomogeneous cylindrical heat transfer equations, which can be integrated using Green's functions. The integration results in a fixed point problem, the solution to which is obtained numerically. The method is validated against experimental data of free axisymmetric compressible turbulent jets, and good agreement is obtained.

The Generalized Graetz Problem in Finite Domains

Jérôme Fehrenbach, Frédéric de Gournay, Charles Pierre, and Franck Plouraboué

SIAM J. Appl. Math. 72, pp. 99-123 (25 pages)

Online Publication Date: January 13, 2012

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We consider the generalized Graetz problem associated with stationary convection-diffusion inside a domain having any regular three-dimensional translationally invariant section and finite or semi-infinite extent. Our framework encompasses any previous “extended” and “conjugated” Graetz generalizations and provides theoretical bases for computing the orthogonal set of generalized two-dimensional Graetz modes. The theoretical framework includes both heterogeneous and possibly anisotropic diffusion tensors. In the case of semi-infinite domains, the existence of a bounded solution is shown from the analysis of two-dimensional operator eigenvectors which form a basis of $L^2$. In the case of finite domains a similar basis can be exhibited, and the mode's amplitudes can be obtained from the inversion of newly defined finite domain operator. Our analysis includes both the theoretical and practical issues associated with this finite domain operator inversion as well as its interpretation as a multireflection image method. Error estimates are provided when numerically truncating the spectrum to a finite number of modes. Numerical examples are validated for reference configurations and provided in nontrivial cases. Our methodology shows how to map the solution of stationary convection-diffusion problems in finite three-dimensional domains into a two-dimensional operator spectrum, which leads to a drastic reduction in computational cost.

On the Self-similar Diffraction of a Weak Shock into an Expansion Wavefront

John K. Hunter and Allen M. Tesdall

SIAM J. Appl. Math. 72, pp. 124-143 (20 pages)

Online Publication Date: January 24, 2012

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We study an asymptotic problem that describes the diffraction of a weak, self-similar shock near a point where its shock strength approaches zero and the shock turns continuously into an expansion wavefront. An example arises in the reflection of a weak shock off a semi-infinite screen. The asymptotic problem consists of the unsteady transonic small disturbance equation with suitable matching conditions. We obtain numerical solutions of this problem, which show that the shock diffracts nonlinearly into the expansion region. We also solve numerically a related half-space problem with a “soft” boundary, which shows a complex reflection pattern similar to one that occurs in the Guderley Mach reflection of weak shocks.

The Reversing of Interfaces in Slow Diffusion Processes with Strong Absorption

J. M. Foster, C. P. Please, A. D. Fitt, and G. Richardson

SIAM J. Appl. Math. 72, pp. 144-162 (19 pages)

Online Publication Date: January 24, 2012

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This paper considers a family of one-dimensional nonlinear diffusion equations with absorption. In particular, the solutions that have interfaces that change their direction of propagation are examined. Although this phenomenon of reversing interfaces has been seen numerically, and some special exact solutions have been obtained, there was previously no analytical insight into how this occurs in the general case. The approach taken here is to seek self-similar solutions local to the interface and local to the reversing time. The analysis is split into two parts, one for the solution prior to the reversing time and the other for the solution after the reversing time. In each case the governing PDE is reduced to an ODE by introducing a self-similar coordinate system. These ODEs do not readily admit any nontrivial exact solutions and so the asymptotic behavior of solutions is studied. By doing this the adjustable parameters, or degrees of freedom, which may be used in a numerical shooting scheme are determined. A numerical algorithm is then proposed to furnish solutions to the ODEs and hence the PDE in the limit of interest. As examples of physical problems in which a PDE of this type may be used as a model the authors study the spreading of a viscous film under gravity and subject to evaporation, the dispersion of a population, and a nonlinear heat conduction problem. The numerical algorithm is demonstrated using these examples. Results are also given on the possible existence of self-similar solutions and types of reversing behavior that can be exhibited by PDEs in the family of interest.

Geographic Profiling from Kinetic Models of Criminal Behavior

George O. Mohler and Martin B. Short

SIAM J. Appl. Math. 72, pp. 163-180 (18 pages)

Online Publication Date: January 24, 2012

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We consider the problem of estimating the probability density of the “anchor point” (residence, place of work, etc.) of a criminal offender given a set of observed spatial locations of crimes committed by the offender. Starting from kinetic models of criminal motion and target selection, we derive the probability density of anchor points using the Fokker–Planck equation and Bayes' theorem. Here, geographic inhomogeneities such as housing densities and geographic barriers (bodies of water, parks, etc.) are naturally incorporated into the probability density estimate, as well as directional bias and distance to crime preferences in offender target selection. The resulting equations are steady state advection-diffusion-reaction PDEs. We test our methodology against crime data provided by the Los Angeles Police Department, and our results highlight the benefits of incorporating these elements of criminal behavior and geographic inhomogeneities into profiling estimates.

Elliptic-Spline Solutions for Large Localizations in a Circular Blatz–Ko Cylinder Due to Geometric Softening

Hui-Hui Dai and Xiaochun Peng

SIAM J. Appl. Math. 72, pp. 181-200 (20 pages)

Online Publication Date: January 24, 2012

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It has been known that geometric softening can induce strain localizations in solids. However, it is very difficult to analytically capture the localized deformation states within a three-dimensional framework, especially when the deformation is large. In this paper, we introduce a novel approach, which resembles the use of a spline to approximate a curve, to construct analytical (asymptotic) solutions for large localizations in a circular cylinder composed of a Blatz–Ko material due to geometric softening. The asymptotic normal form equation (in the form of an ODE) valid for the axial stretch in a small neighborhood is first derived and then a set of these equations, each valid in a small neighborhood, can be obtained. The union of these small neighborhoods can cover a large range of the axial stretch, and as a result this set of equations governs the deformation states for the axial stretch in a large interval. Through a phase-plane analysis on this set of ODEs we manage to obtain the analytical solutions (in the form of a spline of elliptic integrals) for the large strain localizations. Both a force-controlled problem and a displacement-controlled problem are solved and the analytical results capture well the nonuniqueness of the stress-displacement relation and the snap-through phenomenon, which are often observed in experiments when strain localizations happen. In addition, some insightful information on the bifurcation points is obtained. The important geometric size effect is also discussed through the analytical solutions.

An Asymptotic Theory for the Re-Equilibration of a Micellar Surfactant Solution

I. M. Griffiths, C. D. Bain, C. J. W. Breward, S. J. Chapman, P. D. Howell, and S. L. Waters

SIAM J. Appl. Math. 72, pp. 201-215 (15 pages)

Online Publication Date: January 24, 2012

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Micellar surfactant solutions are characterized by a distribution of aggregates made up predominantly of premicellar aggregates (monomers, dimers, trimers, etc.) and a region of proper micelles close to the peak aggregation number, connected by an intermediate region containing a very low concentration of aggregates. Such a distribution gives rise to a distinct two-timescale re-equilibration following a system dilution, known as the $\tau_1$ and $\tau_2$ processes, whose dynamics may be described by the Becker–Döring equations. We use a continuum version of these equations to develop a reduced asymptotic description that elucidates the behavior during each of these processes.

Total Resonant Transmission and Reflection by Periodic Structures

Stephen P. Shipman and Hairui Tu

SIAM J. Appl. Math. 72, pp. 216-239 (24 pages)

Online Publication Date: January 24, 2012

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Resonant scattering of plane waves by a periodic slab under conditions close to those that support a guided mode is accompanied by sharp transmission anomalies. For two-dimensional structures, we establish sufficient conditions, involving structural symmetry, under which these anomalies attain total transmission and total reflection at frequencies separated by an arbitrarily small amount. The loci of total reflection and total transmission are real-analytic curves in frequency-wavenumber space that intersect quadratically at a single point corresponding to the guided mode. A single anomaly or multiple anomalies can be excited by the interaction with a single guided mode.

Nonlinear Waves in Shallow Honeycomb Lattices

Mark J. Ablowitz and Yi Zhu

SIAM J. Appl. Math. 72, pp. 240-260 (21 pages)

Online Publication Date: January 26, 2012

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The linear spectrum and corresponding Bloch modes of shallow honeycomb lattices near Dirac points are investigated. Via perturbation theory, the dispersion relation is found to have threefold degeneracy at leading order with eigenvalue splitting at the following two orders; i.e., the threefold eigenvalue splits into single and double values. Multiscale perturbation methods are employed to describe the nonlinear dynamics of the associated wave envelopes. The dynamics of the envelope depends on different asymptotic balances whereupon a three-level nonlinear Dirac-type equation or a two-level nonlinear Dirac equation is derived. The analysis agrees well with direct numerical simulations.

Global Dynamics of a General Class of Multistage Models for Infectious Diseases

Hongbin Guo, Michael Y. Li, and Zhisheng Shuai

SIAM J. Appl. Math. 72, pp. 261-279 (19 pages)

Online Publication Date: February 02, 2012

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We propose a general class of multistage epidemiological models that allow possible deterioration and amelioration between any two infected stages. The models can describe disease progression through multiple latent or infectious stages as in the case of HIV and tuberculosis. Amelioration is incorporated into the models to account for the effects of antiretroviral or antibiotic treatment. The models also incorporate general nonlinear incidences and general nonlinear forms of population transfer among stages. Under biologically motivated assumptions, we derive the basic reproduction number $R_0$ and show that the global dynamics are completely determined by $R_0$: if $R_0\leq 1$, the disease-free equilibrium is globally asymptotically stable, and the disease dies out; if $R_0>1$, then the disease persists in all stages and a unique endemic equilibrium is globally asymptotically stable.

Single-spiral-vortex Model for a Cavitating Elastic Curvilinear Foil

A. Y. Zemlyanova and Y. A. Antipov

SIAM J. Appl. Math. 72, pp. 280-298 (19 pages)

Online Publication Date: February 09, 2012

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A two-dimensional nonlinear inverse fluid-structure interaction problem for a curvilinear elastic hydrofoil is considered. A cavity formed behind the foil is modeled according to the single-spiral-vortex model by Tulin. The fluid-structure problem is decoupled by the method of successive approximations. For the cavitation problem, the foil is modeled as a polygon. The method of conformal mappings and the Riemann–Hilbert problem are employed at this stage. The classical detachment mechanism for a smooth arc is satisfied for the polygon approximately. The deformation of the smooth foil is described by the governing equations of the thin shell theory with the clamped-clamped boundary conditions. The loading acting on the middle surface of the foil is prescribed as the difference between the fluid and vapor pressure computed in the fluid problem. Numerical results include those for the cavity profile, the drag coefficient, the pressure distribution, the speed, and the displacements of the elastic foil.

Uniqueness of Limit Cycles in a Rosenzweig–MacArthur Model with Prey Immigration

Jitsuro Sugie and Yasuhisa Saito

SIAM J. Appl. Math. 72, pp. 299-316 (18 pages)

Online Publication Date: February 09, 2012

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Many natural predator and prey populations persist while their densities show sustained oscillations. Hence these populations must be regulated in such a way that the densities are kept away from the values where extinction is likely to occur. On the other hand, nonspatial simple predator-prey models show vigorous oscillations that can bring the populations to the brink of extinction or beyond. Predator-prey systems that are kept in the laboratory also tend to show fluctuations in densities that are severe enough to drive them to extinction. Since the amount of space that laboratory populations live in is small compared to that of natural populations, one is readily led to the hypothesis that spatial interactions must contribute to the regulation of natural predator-prey systems. In this paper, we construct a simplest type of spatially interacting populations by taking into account constant immigration of prey for a predator-prey model with a Holling type II functional response and derive necessary and sufficient conditions for both the uniqueness of limit cycles and the global asymptotic stability of a positive equilibrium. From these results, it is fully suggested (mathematically) that the prey immigration dampens the large fluctuations emerging in the predator-prey model and also stabilizes a positive equilibrium globally.

Noise Source Localization in an Attenuating Medium

Habib Ammari, Elie Bretin, Josselin Garnier, and Abdul Wahab

SIAM J. Appl. Math. 72, pp. 317-336 (20 pages)

Online Publication Date: February 14, 2012

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In this paper we consider the problem of reconstructing the spatial support of noise sources from boundary measurements using cross correlation techniques. We consider media with and without attenuation and provide efficient imaging functionals in both cases. We also discuss the case where the noise sources are spatially correlated. We present numerical results to show the viability of the different proposed imaging techniques.

Strongly Nonlinear Beat Phenomena and Energy Exchanges in Weakly Coupled Granular Chains on Elastic Foundations

Yuli Starosvetsky, M. Arif Hasan, Alexander F. Vakakis, and Leonid I. Manevitch

SIAM J. Appl. Math. 72, pp. 337-361 (25 pages)

Online Publication Date: February 14, 2012

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We study the dynamics of weakly interacting, strongly nonlinear one-dimensional granular chains mounted on elastic foundations. These chains are composed of a number of identical linearly elastic beads interacting with each other through Hertzian contact. No dissipative effects, such as plasticity or dry friction effects, are taken into account in our analysis. Assuming zero precompression between beads, the dynamics of the system under consideration is strongly (essentially) nonlinear, having no linear component. The complete absence of linear structural acoustics in these chains led to their characterization as “sonic vacua.” The two sources of strong nonlinearity in the considered granular chains are (i) the nonlinearizable Hertzian law interaction between adjacent beads in compression, and (ii) the possible separations between beads leading to bead collisions in the absence of compressive forces. In the current study we demonstrate that the weakly coupled granular chains possess complex dynamics leading to strong energy exchanges between them. Three different types of nonlinear beat phenomena are analytically studied, based on spatially periodic traveling waves, stationary breathers, and propagating breathers, respectively. We employ a complexification—averaging methodology that leads to smooth slow flow reduced models of the dynamics despite the discontinuous nature of the bead interactions. Verification of the derived analytical approximations with direct numerical simulations is also performed.

Modeling, Asymptotic Analysis, and Simulation of an Energy Tower

Maria Bauer and Ingenuin Gasser

SIAM J. Appl. Math. 72, pp. 362-381 (20 pages)

Online Publication Date: February 14, 2012

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A new model for the description of an energy tower is presented. For the modeling of the transient flow in an energy tower and the related power production a one-dimensional approach is taken. Although many simplifying assumptions are imposed, this approach is thought to incorporate most of the main physical effects. The model is derived from the general one-dimensional Euler equations of gas dynamics for a compressible humid air gas mixture. Low Mach number asymptotics allow us to handle the related problems. Numerical simulations are performed. Optimal values for the spray rate and the height of the spraying region are identified. The model gives results which are in accordance with our physical understanding. For lack of an energy tower prototype, a quantitative comparison is not possible at the moment.

A Generalized Birkhoff–Rott Equation for Two-dimensional Active Scalar Problems

Hui Sun, David Uminsky, and Andrea L. Bertozzi

SIAM J. Appl. Math. 72, pp. 382-404 (23 pages)

Online Publication Date: February 16, 2012

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In this paper we derive evolution equations for the two-dimensional active scalar problem when the solution is supported on one-dimensional curves. These equations are a generalization of the Birkhoff–Rott equation when vorticity is the active scalar. The formulation is Lagrangian and it is valid for nonlocal kernels ${\bf K}$ that may include both a gradient and an incompressible term. We develop a numerical method for implementing the model which achieves second order convergence in space and fourth order in time. We verify the model by simulating classic active scalar problems such as the vortex sheet problem (in the case of inviscid, incompressible flow) and the collapse of delta ring solutions (in the case of pure aggregation), finding excellent agreement. We then study two examples with kernels of mixed type, i.e., kernels that contain both incompressible and gradient flows. The first example is a vortex density model which arises in superfluids. We analyze the effect of the added gradient component on the Kelvin–Helmholtz instability. In the second example, we examine a nonlocal biological swarming model and study the dynamics of density rings which exhibit complicated milling behavior.

Solving Initial Value Problem by Matching Asymptotic Expansions

Yuri Skrynnikov

SIAM J. Appl. Math. 72, pp. 405-416 (12 pages)

Online Publication Date: February 16, 2012

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An asymptotic expansion in powers of small dimensionless dispersion coefficient is sought to solve an initial value problem posed for an advection diffusion equation modeling orientation of pulp fibers in a steady fully turbulent flow. The regular expansion is shown to be nonuniform in a small neighborhood of $\phi=0$. Although the highest order derivative with respect to the orientation angle $\phi$ is multiplied by the small parameter, application of matched asymptotic expansions to obtain the inner solution in a small neighborhood of $\phi=0$ matchable with the regular expansion turned out to be unsuccessful. The multiple scales do not lead to the solution either. The problem is solved by matching two asymptotic expansions, one solving the initial value problem in a small neighborhood of the initial point, while another one solves the equation at large distances from the initial point. Thus, this is an example of using the method of matched asymptotic expansions to satisfy the given initial condition by the long-distance approximation of the solution to a nonsingular partial differential equation.

Global Dynamics of a Hyperbolic-Parabolic Model Arising from Chemotaxis

Tong Li, Ronghua Pan, and Kun Zhao

SIAM J. Appl. Math. 72, pp. 417-443 (27 pages)

Online Publication Date: February 21, 2012

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We prove global existence and qualitative behavior of classical solutions for a hyperbolic-parabolic system describing chemotaxis on bounded domains. It is shown that classical solutions to the initial-boundary value problem of the one-dimensional model exist globally in time for large initial data, and the solutions converge to constant equilibrium states exponentially in time, which rigorously demonstrates the collapsing of cell populations in chemotaxis. Moreover, similar results are established for the multidimensional model when the initial data are small.

Self-Similar Voiding Solutions of a Single Layered Model of Folding Rocks

T. J. Dodwell, M. A. Peletier, C. J. Budd, and G. W. Hunt

SIAM J. Appl. Math. 72, pp. 444-463 (20 pages)

Online Publication Date: February 21, 2012

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In this paper we derive an obstacle problem with a free boundary to describe the formation of voids at areas of intense geological folding. An elastic layer is forced by overburden pressure against a V-shaped rigid obstacle. Energy minimization leads to representation as a nonlinear fourth-order ordinary differential equation, for which we prove there exists a unique solution. Drawing parallels with the Kuhn–Tucker theory, virtual work, and ideas of duality, we highlight the physical significance of this differential equation. Finally, we show that this equation scales to a single parametric group, revealing a scaling law connecting the size of the void with the pressure/stiffness ratio. This paper is seen as the first step toward a full multilayered model with the possibility of voiding.
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