Application of displacement and traction boundary integral equations for fracture mechanics analysis

A general formulation by dual boundary integral equations and a computational solution algorithm for the general mixed-mode crack in a linearly elastic, isotropic medium is presented. Traction boundary integral equations are collocated at the points on one crack surface while displacement boundary integral equations are collocated at the opposite points on the other surface to ensure a unique solution. The hypersingular and strongly singular integrals in the traction and displacement boundary integral equations are regularized before the numerical implementation. The singular integration elements on both crack surfaces are replaced by smooth curved auxiliary surfaces to avoid the direct integration over the singular elements. Usage of these detoured auxiliary contours is justified by certain identities of the fundamental solutions. Convergence tests for integration order and subdivision of the auxiliary surface elements were performed. To demonstrate the accuracy and efficiency of the present technique, the deformation and the stress intensity factors for two- and three-dimensional embedded and edge crack problems are given

Towards the in vivo mechanical characterization of abdominal wall in animal model: application to hernia repair

El trabajo presentado en esta tesis se centra en el diseño e implementación de una metodología que permita caracterizar in vivo el comportamiento mecánico pasivo de la pared abdominal. Esta metodología permitiría a los cirujanos disponer de información mecánica relevante sobre un paciente especí co, lo que podría contribuir a mejorar el tratamiento quirúrgico de hernias mediante malla protésica. El tratamiento quirúrgico de hernias consiste en cerrar la debilidad creada en el músculo, ya sea directamente con puntos de sutura o mediante la implantación de una malla protésica. En el caso de la malla, ésta es la responsable de absorber las tensiones a las que el músculo se ve sometido durante el tiempo en el que se produce la regeneración de tejido. Para reducir el riego de aparición de dolor postoperatorio, rotura o rasgadura de tejido o incluso una recidiva, la malla debe mimetizar la respuesta mecánica de la zona de la pared donde vaya a ser colocada, que a su vez puede variar de un paciente a otro en función de su edad, género, índice de masa corporal u otras características físicas. Un mejor conocimiento de las propiedades mecánicas del abdomen en paciente especí co ayudaría al cirujano a determinar qué malla protésica se puede considerar la ideal, mecánicamente hablando. Por todo ello, el trabajo que aquí se presenta plantea una aproximación in vivo para caracterizar la pared abdominal sobre un modelo animal y su posterior implementación en casos de patologías herniarias. En un primer paso, se ha realizado un estudio biomecánico del cierre en línea alba, que ayudase a entender los aspectos mecánicos y biológicos que tienen lugar durante la curación de la herida a corto y largo plazo. A continuación, se han llevado a cabo ensayos mecánicos de in ado sobre la pared, que combinados con el uso de cámaras y técnicas de adquisición de imagen han permitido extraer la respuesta del tejido de una manera no invasiva. Este estudio experimental, se ha llevado a cabo sobre especímenes sanos y otro herniados y reparados con distintas mallas quirúrgicas, lo que ha permitido extrapolar el efecto in vivo que provocan estas mallas. A partir de los datos experimentales también se ha desarrollado un análisis numérico que permitiese caracterizar la respuesta mecánica especí ca de cada espécimen. A este efecto, dicha caracterización se ha tratado como un problema inverso y resuelto primeramente mediante un análisis de super cies de respuesta y después con un algoritmo propio aplicado a modelos hiperelásticos. Finalmente, también se ha reconstruido un modelo de elementos nitos de la cavidad abdominal que permite simular el efecto producido por distintas mallas protésicas así como su alteración respecto al tejido sano

Boundary element solution of Poisson\u27s equations in axisymmetric laminar flows

The primitive variable Navier-Stokes equations may be replaced by two equations using the derived variable of vorticity. These equations model separately the kinematic and kinetic parts of the problem. Two boundary element solutions for the kinematic equations were developed for axisymmetric flow geometries. The first was based on the fluid mechanics analogy of the Biot and Savart formula for the magnetic effects of a current. The second was the solution of the vector Poisson\u27s velocity equation using the direct boundary element equation. Numerical integration algorithms were developed which were used for all integrals;Integral solutions for Poisson\u27s pressure equation and Poisson\u27s vector potential equation were derived using the direct boundary element equation. The equations were integrated using the algorithms developed for the velocity solutions;The axisymmetric laminar Navier-Stokes solution was completed by solving the kinetic vorticity transport equation with finite difference methods. Two finite difference methods developed for the complete 2 dimensional non-linear Burger\u27s equation were modified for use on the axisymmetric form of the vorticity transport equation;This complete Navier-Stokes solution was then used to verify the form of the six boundary element equations and the accuracy of the integration algorithm developed. This was done by solving three steady state flow problems and one time dependent flow problem which were designed to simulate flow in power hydraulic components;Flow problems were encountered which produced ill-conditioned kinematic systems with attendant unstable solutions and large errors. Solution algorithms were developed which stabilized the associated matrix operator and improved solution performance. The method is based on the theory and numerical methods of Tikhonov regularization as it applies to linear algebraic systems of equations

Mathematical problems arising in interfacial electrohydrodynamics

In this work we consider the nonlinear stability of thin films in the presence of electric fields. We study a perfectly conducting thin film flow down an inclined plane in the presence of an electric field which is uniform in its undisturbed state, and normal to the plate at infinity. In addition, the effect of normal electric fields on films lying above, or hanging from, horizontal substrates is considered. Systematic asymptotic expansions are used to derive fully nonlinear long wave model equations for the scaled interface motion and corresponding flow fields. For the case of an inclined plane, higher order terms are need to be retained to regularize the problem in the sense that the long wave approximation remains valid for long times. For the case of a horizontal plane the fully nonlinear evolution equation which is derived at the leading order, is asymptotically correct and no regularization procedure is required. In both physical situations, the effect of the electric field is to introduce a non-local term which arises from the potential region above the liquid film, and enters through the electric Maxwell stresses at the interface. This term is always linearly destabilizing and produces growth rates proportional to the cubic power of the wavenumber - surface tension is included and provides a short wavelength cut-off, that is, all sufficiently short waves are linearly stable. For the case of film flow down an inclined plane, the fully nonlinear equation can produce singular solutions (for certain parameter values) after a finite time, even in the absence of an electric field. This difficulty is avoided at smaller amplitudes where the weakly nonlinear evolution is governed by an extension of the Kuramoto-Sivashinsky (KS) equation. Global existence and uniqueness results are proved, and refined estimates of the radius of the absorbing ball in L2 are obtained in terms of the parameters of the equations for a generalized class of modified KS equations. The established estimates are compared with numerical solutions of the equations which in turn suggest an optimal upper bound for the radius of the absorbing ball. A scaling argument is used to explain this, and a general conjecture is made based on extensive computations. We also carry out a complete study of the nonlinear behavior of competing physical mechanisms: long wave instability above a critical Reynolds number, short wave damping due to surface tension and intermediate growth due to the electric field. Through a combination of analysis and extensive numerical experiments, we elucidate parameter regimes that support non-uniform travelling waves, time-periodic travelling waves and complex nonlinear dynamics including chaotic interfacial oscillations. It is established that a sufficiently high electric field will drive the system to chaotic oscillations, even when the Reynolds number is smaller than the critical value below which the non-electrified problem is linearly stable. A particular case of this is Stokes flow, which is known to be stable for this class of problems (an analogous statement holds for horizontally supported films also). Our theoretical results indicate that such highly stable flows can be rendered unstable by using electric fields. This opens the way for possible heat and mass transfer applications which can benefit significantly from interfacial oscillations and interfacial turbulence. For the case of a horizontal plane, a weakly nonlinear theory is not possible due to the absence of the shear flow generated by the gravitational force along the plate when the latter is inclined. We study the fully nonlinear equation, which in this case is asymptotically correct and is obtained at the leading order. The model equation describes both overlying and hanging films - in the former case gravity is stabilizing while in the latter it is destabilizing. The numerical and theoretical analysis of the fully nonlinear evolution is complicated by the fact that the coefficients of the highest order terms (surface tension in this instance) are nonlinear. We implement a fully implicit two level numerical scheme and perform numerical experiments. We also prove global boundedness of positive periodic smooth solutions, using an appropriate energy functional. This global boundedness result is seenin all our numerical results. Through a combination of analysis and extensive numerical experiments we present evidence for global existence of positive smooth solutions. This means, in turn, that the film does not touch the wall in finite time but asymptotically at infinite time. Numerical solutions are presented to support such phenomena

Fully nonlinear interfacial waves in a bounded two-fluid system

We study the nonlinear flow which results when two immiscible inviscid incompressible fluids of different densities and separated by an interface which is free to move and which supports surface tension, are caused to flow in a straight infinite channel. Gravity is taken into consideration and the velocities of each phase can be different, thus giving rise to the Kelvin-Helmholtz instability. Our objective is to study the competing effects of the Kelvin-Helmholtz instability coupled with a stably or unstably stratified fluid system (Rayleigh-Taylor instability) when surface tension is present to regularize the dynamics. Our approach involves the derivation of two-and three-dimensional model evolution equations using long-wave asymptotics and the ensuing analysis and computation of these models. In addition, we derive the appropriate Birkhoff-Rott integro-differential equation for two-phase inviscid flows in channels of arbitrary aspect ratios. A long wave asymptotic analysis is undertaken to develop a theory for fully nonlinear interfacial waves allowing amplitudes as large as the channel thickness. The result is a set of evolution equations for the interfacial shape and the velocity jump across the interface. Linear stability analysis reveals that capillary forces stabilize short-wave disturbances in a dispersive manner and we study their effect on the fully nonlinear dynamics described by our models. In the case of two-dimensional interfacial deflections, traveling waves of permanent form are constructed and it is shown that solitary waves are possible for a range of physical parameters. All solitary waves are expressed implicitly in terms of incomplete elliptic integrals of the third kind. When the upper layer has zero density, two explicit solitary-wave solutions have been found whose amplitudes are equal to h/4 or h/9 where 2h is the channel thickness. In the absence of gravity, solitary waves are not possible but periodic ones are. Numerically constructed traveling and solitary waves are given for representative physical parameters. The initial value problem for the partial differential equations is also addressed numerically in periodic domains, and the regularizing effect of surface tension is investigated. In particular, when surface tension is absent it is shown that the system of governing evolution equations terminates in a singularity after a finite time. This is achieved by studying a 2 x 2 system of nonlinear conservation laws in the complex plane and by numerical solution of the evolution equations. The analysis shows that a sinusoidal perturbation of the flat interface and a cosine perturbation to the unit velocity jump across the interface, develop a singularity at time tc = ln 1/ε+0 (ln(ln 1/ε)) where ε is the initial amplitude of the disturbances. This result is asymptotic for small ε and is derived by studying the asymptotic form of the flow characteristics in the complex plane. We also derive the analogous three-dimensional evolution equations by assuming that the wavelengths in the principal horizontal directions are large compared to the channel thickness. Surface tension is again incorporated to regularize short-wave Kelvin-Helmholtz instabilities and the equations are solved numerically subject to periodic boundary conditions. Evidence of singularity formation is found. In particular, we observe that singularities occur at isolated points starting from general initial conditions. This finding is consistent with numerical studies of unbounded three-dimensional vortex sheets (see Introduction for a discussion and references). In the final part of this work we consider the vortex-sheet formulation of the exact nonlinear two-dimensional flow of a vortex sheet which is bounded in a channel. We derive a Birkhoff-Rott type integro-differential evolution equation for the velocity of the interface in terms of the vorticity as well as the evolution equation for the unnormalized vortex sheet strength. For the case of a spatially periodic vortex sheet, this Birkhoff-Rott type equation is written in terms of Jacobi\u27s functions. The equation is shown to recover the limits of unbounded and non-periodic flows which are known in the literature

Kerr electro-optic tomography for determination of nonuniform electric field distributions in dielectrics

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1999.Includes bibliographical references (p. 255-260).by Afşin Üstündağ.Ph.D