Boundary Value Problems In Mechanics Research Articles

On the solution of boundary value problems of mechanics for the Poisson equation and related equations

To a boundary value problem, we assign an auxiliary problem of determining the spectrum of eigenfunctions and eigenvalues (EFV). After the main problem has been reduced to a form with homogeneous boundary conditions, it becomes possible to prove theorems about the formulas for the solution of the boundary value problem with linear equations of elliptic type for multidimensional multiply connected domains by using the spectral expansion in the Fourier series. We find conditions under which the action of second-order differential operators on the obtained solutions in the Fourier series can be computed not only in the interior of the domain but also on its boundary. But if these conditions are not satisfied, then the series for second-order differential operators do not converge on the boundary. The proposed method for the expansion in the EFV can be used not only in plane but also in spatial problems if the domain of complicated shape can be represented as a combination of bounded domains with known EFV spectra. As one of the examples, we consider the problem of torsion of an elastic rod whose cross-section consists of a rectangle and a half-disk.

On evolving deformation microstructures in non-convex partially damaged solids

The paper outlines a relaxation method based on a particular isotropic microstructure evolution and applies it to the model problem of rate independent, partially damaged solids. The method uses an incremental variational formulation for standard dissipative materials. In an incremental setting at finite time steps, the formulation defines a quasi-hyperelastic stress potential. The existence of this potential allows a typical incremental boundary value problem of damage mechanics to be expressed in terms of a principle of minimum incremental work. Mathematical existence theorems of minimizers then induce a definition of the material stability in terms of the sequential weak lower semicontinuity of the incremental functional. As a consequence, the incremental material stability of standard dissipative solids may be defined in terms of weak convexity notions of the stress potential. Furthermore, the variational setting opens up the possibility to analyze the development of deformation microstructures in the post-critical range of unstable inelastic materials based on energy relaxation methods. In partially damaged solids, accumulated damage may yield non-convex stress potentials which indicate instability and formation of fine-scale microstructures. These microstructures can be resolved by use of relaxation techniques associated with the construction of convex hulls. We propose a particular relaxation method for partially damaged solids and investigate it in one- and multi-dimensional settings. To this end, we introduce a new isotropic microstructure which provides a simple approximation of the multi-dimensional rank-one convex hull. The development of those isotropic microstructures is investigated for homogeneous and inhomogeneous numerical simulations.

Mechanical analysis of fault slip data: Implications for paleostress analysis

Stress inversions are a useful and popular tool for structural geologist and seismologist alike. These methods were first introduced by Wallace (1951) and Bott (1959) and subsequent studies continue to be based on their assumptions: the remote stress tensor is spatially uniform for the rock mass containing the faults and temporally constant over the history of faulting in that region, and the slip on each fault surface has the same direction and sense as the maximum shear stress resolved on that surface from the remote stress tensor. Furthermore, successful implementation requires that slip accumulates on faults of diverse orientation. Many studies employ these methods on isolated faults or on fault systems with limited ranges of orientations, which can lead to erroneous results. We propose a new method that incorporates the effects of mechanical interaction of the entire fault or fault system, and solves the complete mechanical problem rather than employing empirical relationships between slip and stress or strain (or strain rate). The method requires knowledge of the fault geometry and information on at least one slip vector component along portions of the known fault geometry. For example, if throw is known, the strike-slip component can be solved for. We test the method using a single synthetic fault with anisotropic roughness similar to that measured at fault outcrops. While the orientation of remote stress may be determined precisely, the lack of diverse fault orientations introduces a systematic error in the remote stress ratio. We further test the effect of diversity of fault orientations and find that Wallace–Bott type inversions do not perform as well for limited ranges of orientations when compared to the proposed method. Finally, we use published data from the 1999 Chi-Chi, Taiwan, earthquake, and find that the method using surface data only, and surface data with subsurface focal mechanisms, produce similar results. The resulting stress orientations are in good agreement with results from Wallace–Bott inversions. Furthermore, the slip distribution is in general agreement with kinematic slip inversions using coseismic surface deformation. Stress inversion methods using fault slip data can thus be improved upon, significantly in some cases, by solving a mechanical boundary value problem that takes into account the geometry of faults or fault systems. As a bonus, the solution provides the stress, strain, and displacement fields throughout the region and the slip distributions on the faults.

Improving the differentiability of generalized fourier series solutions of boundary value problems of mechanics by using boundary functions

We prove a theorem on conditions for the differentiation of generalized Fourier series. We show that Fourier series solutions of boundary value problems can in general be differentiated term by term only once. To improve the differentiability properties of such series, we suggest to use pth-order boundary functions. We suggest an algorithm for constructing boundary functions for classical domains. This approach is illustrated by a new solution, with improved differentiability properties, of the problem on the torsion of an elastic rod of rectangular cross-section.

Solution of boundary value problems in mechanics of composite plates and shells

Solution of boundary value problems in mechanics of composite plates and shells

Numerical analysis of forth-order boundary value problems in fluid mechanics and mathematics

Numerical analysis of forth-order boundary value problems in fluid mechanics and mathematics

О решении низкоразмерных краевых задач квантовой механики методом канторовича - приведение к обыкновенным дифференциальным уравнениям

О решении низкоразмерных краевых задач квантовой механики методом канторовича - приведение к обыкновенным дифференциальным уравнениям

Modeling of a Galfenol transducer using the bidirectionally coupled magnetoelasticmodel

The bidirectionally coupled magnetoelastic model (BCMEM) developed by Mudivarthi et al (2008 Smart Mater. Struct. 17 035005) has been extended to include electriccurrents in its magnetic finite element formulation. This enables the model tocapture the magnetoelastic behavior of magnetostrictive materials subjected toelastic stresses and magnetic fields applied not only using permanent magnetsbut also the current carrying coils often used in transducer applications. Thismodel was implemented by combining finite element solutions of mechanical andmagnetic boundary value problems using COMSOL Multiphysics 3.4 (finiteelement modeling software) with an energy-based nonlinear magnetomechanicalconstitutive model. The coupling variables are magnetostriction and magneticpermeability, which are dependent on both the magnetic (magnetic flux density) andthe mechanical (stress) states of a magnetostrictive material. In this research,the BCMEM was used to simulate actuator load lines for a magnetostrictiveFe84Ga16 alloy, which were then compared with experimental data (Datta and Flatau 2008Proc. SPIE 6929 69291Z). Also, the ability of the BCMEM to capture theΔE effect in Galfenol was demonstrated. Finally, the use of the BCMEM as a tool fortransducer design optimization is demonstrated by using the model to visualizethe influence of different magnetic circuit designs on transducer performance.

Finite Element Analysis of General Magnetomechanical Coupling Phenomena

AbstractIn this paper we consider the derivation of a finite element model for fully–coupled magneto–mechanical boundary value problems based on a vector–valued magnetic potential. The formulation allows for the incorporation of nonlinear, anisotropic and hysteretic constitutive response. A piezomagnetic inclusion problem is computed to test the implementation. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the Singularities in Fracture and Contact Mechanics

Generally, the mixed boundary value problems in fracture and contact mechanics may be formulated in terms of integral equations. Through a careful asymptotic analysis of the kernels and by separating nonintegrable singular parts, the unique features of the unknown functions can then be recovered. In mechanics and potential theory, a characteristic feature of these singular kernels is the Cauchy singularity. In the absence of other nonintegrable kernels, Cauchy kernel would give a square-root or conventional singularity. On the other hand, if the kernels contain, in addition to a Cauchy singularity, other nonintegrable singular terms, the application of the complex function theory would show that the solution has a non-square-root or unconventional singularity. In this article, some typical examples from crack and contact mechanics demonstrating unique applications of such integral equations will be described. After some remarks on three-dimensional singularities, the key examples considered will include the generalized Cauchy kernels, membrane and sliding contact mechanics, coupled crack-contact problems, and crack and contact problems in graded materials.

An Approximate Solution for Boundary Value Problems in Structural Engineering and Fluid Mechanics

Variational iteration method (VIM) is applied to solve linear and nonlinear boundary value problems with particular significance in structural engineering and fluid mechanics. These problems are used as mathematical models in viscoelastic and inelastic flows, deformation of beams, and plate deflection theory. Comparison is made between the exact solutions and the results of the variational iteration method (VIM). The results reveal that this method is very effective and simple, and that it yields the exact solutions. It was shown that this method can be used effectively for solving linear and nonlinear boundary value problems.

Discrete gradient method in solid mechanics

AbstractA discrete method to boundary value problems in solid mechanics is presented. In this method, the unknown variable and its derivative are defined only at nodes. A discrete gradient operator is constructed with the aid of a tensorial identity on the Voronoi diagram. This operator is utilized in a weak form to derive a discrete Galerkin formulation for the boundary value problem. The theoretical underpins of the methodology are discussed, and the details of computational implementation in two‐dimensional elasticity, both small strain and finite strain, are provided. Several benchmark tests are presented to demonstrate the accuracy, convergence, and other properties of the method. Copyright © 2007 John Wiley & Sons, Ltd.

An approach to solving mechanics problems for materials with multiscale self-similar microstructure

This article proposes an efficient method for solving mechanics boundary value problems formulated for domains with multiscale self-similar microstructure. In particular, composite materials for which one of the phases has a fractal-like structure with scale cut-offs are considered. The boundary value problems are solved using a finite element procedure with enriched shape functions that incorporate information about the geometric complexity. The use of these shape functions makes possible the definition of a unique, parametrically defined model from which the solution for configurations with an arbitrary number of scales can be derived. The proposed method is primarily useful for structures with a large number of self-similar scales for which using the usual finite element method would be too expensive. In order to exemplify the method, a 2D composite with fractal microstructure is considered and several boundary value problems are solved.

Integral-projection method and singularities of its application in mating shells with inconsistent parameters of approximation

We describe a numerical integral-projection method used by the authors for the approximate solution of systems of interrelated two-dimensional linear boundary-value problems in mechanics of composite shell systems. The method is based on discretization in each shell substructure of a two-dimensional problem along one of coordinates using a projection-grid variant of the Galerkin-Petrov method and its subsequent transformation to a system of ordinary differential equations; by integration and introduction of sought functions as unknown derivatives, the system is reduced to a system of integral equations being solved by the method of mechanical quadratures. The method is characterized by the fact that its application requires no additional conditions of conformity with discretization parameters of substructures being mated.

Finite element analyses of cracks in piezoelectric structures: a survey

Piezoelectric materials have widespread applications in modern technical areas such as mechatron- ics, smart structures or microsystem technology, where they serve as sensors or actuators. For the assessment of strength and reliability of piezoelectric structures under combined electrical and mechanical loading, the existence of cracklike defects plays an important role. Meanwhile, piezoelectric fracture mechanics has been established quite well, but its application to realistic crack configurations and loading situations in piezoelec- tric structures requires the use of numerical techniques as finite element methods (FEM) or boundary element methods (BEM). The aim of this paper is to review the state of the art of FEM to compute the coupled electro- mechanical boundary value problem of cracks in 2D and 3D piezoelectric structures under static and dynamic loading. In order to calculate the relevant fracture parameters very precisely and efficiently, the numerical treatment must account for the singularity of the mechanical and electrical fields at crack tips. The follow- ing specialized techniques are presented in detail: (1) special singular crack tip elements, (2) determination of intensity factors KI-KIV from near tip fields, (3) modified crack closure integral, (4) computation of the electromechanical J -integral, and (5) exploitation of interaction integrals. Special emphasis is devoted to a realistic modeling of the dielectric medium inside the crack, leading to specific electric crack face boundary conditions. The accuracy, efficiency, and applicability of these techniques are examined by various example problems and discussed with respect to their advantages and drawbacks for practical applications.

Comment on “Exact analytic solutions of nonlinear boundary value problems in fluid mechanics (Blasius equations)” [J. Math. Phys. 46, 033101 (2005)

A simple procedure is given to transform the Blasius equation into an Abel equation of the second kind.

A multi-step linearization technique for a class of boundary value problems in non-linear mechanics

Non-linear finite element analyses of structures (such as beams) involve construction of weak solutions for the governing equations. While a weak approach weakens the differentiability requirements of the so-called shape functions, the governing equations are only satisfied in an integral sense and not point-wise, or, even path-wise. Moreover, use of a finite mesh leads to a stiffening of the numerical model. While strong solutions obtained through some of the existing mesh-free collocation methods overcomes some of these lacunae to an extent, the quality of the numerical solutions would be considerably improved if the computational algorithm were able to faithfully reproduce (or approximate or preserve) certain geometrical features of the response surfaces or manifolds. This paper takes the first step towards realizing this objective and proposes a multi-step transversal linearization (MTL) technique for a class of non-linear boundary value problems, which are treated as conditionally dynamical systems. Numerical explorations are performed, to a limited extent, through applications to large deflection analyses of planar beams with or without plastic deformations.

Exact analytic solutions of nonlinear boundary value problems in fluid mechanics (Blasius equations)

In this work it is shown that, by a series of admissible functional transformations, the generalized Blasius equation in fluids can be exactly reduced to a three-term generalized Emden–Fowler equation. Furthermore, the restricted in axisymmetric flows and simplified forms of this equation can be reduced to (i) two-term generalized Emden–Fowler equations; (ii) generalized Emden–Fowler equations; (iii) Emden–Fowler equations of the normal form; and (iv) Abel equations of the second kind. By means of a recently developed mathematical solution methodology (Panayotounakos, Fifth Greek Congress on Mechanics, Xania, Crete, 22–25 June 2004, Hellas, Greece), we provide exact analytic solutions for the simplified as well as for the restricted forms of the above-mentioned Blasius equations. Thus, it is proved that important, unsolvable in exact form problems in nonlinear fluid dynamics now can be analytically solved.

Investigation of coupled boundary value problems of continuum mechanics and mathematical physics academician

Investigation of coupled boundary value problems of continuum mechanics and mathematical physics academician

Finite Element Modeling in Musculoskeletal Biomechanics

Numerical approximation of the solutions to continuum mechanics boundary value problems, by means of finite element analysis, has proven to be of incalculable benefit to the field of musculoskeletal biomechanics. This article briefly outlines the conceptual basis of finite element analysis and discusses a number of the key technical considerations involved, specifically from the standpoint of effective modeling of musculoskeletal structures. The process of conceiving, developing, validating, parametrically exercising, and interpreting the results of musculoskeletal finite element models is described. Pertinent case study examples are presented from two series of finite element models, one involving total hip implant dislocation and the other involving femoral head osteonecrosis.