Elasticity With Mathematica

This book is intended for researchers, engineers and students in solid mechanics, materials science and physics who are interested in using the power of modern computing to solve a wide variety of problems of both practical and fundamental significance in elasticity. Extensive use of Mathematica in the book makes available to the reader a range of recipes that can be readily adjusted to match particular tastes or requirements, to visualize solutions, and to carry out symbolic and numerical analysis and optimization.

This book, first published in 2007, introduces key ideas and principles in the theory of elasticity with the help of symbolic computation. Differential and integral operators on vector and tensor fields of displacements, strains and stresses are considered on a consistent and rigorous basis with respect to curvilinear orthogonal coordinate systems. As a consequence, vector and tensor objects can be manipulated readily, and fundamental concepts can be illustrated and problems solved with ease. The method is illustrated using a variety of plane and three-dimensional elastic problems. General theorems, fundamental solutions, displacements and stress potentials are presented and discussed. The Rayleigh-Ritz method for obtaining approximate solutions is introduced for elastostatic and spectral analysis problems. Containing more than 60 exercises and solutions in the form of Mathematica notebooks that accompany every chapter, the reader can learn and master the techniques while applying them to a large range of practical and fundamental problems.

Elasticity: Theory, Applications, and Numerics, Third Edition, continues its market-leading tradition of concisely presenting and developing the linear theory of elasticity, moving from solution methodologies, formulations, and strategies into applications of contemporary interest, such as fracture mechanics, anisotropic and composite materials, micromechanics, nonhomogeneous graded materials, and computational methods. Developed for a one- or two-semester graduate elasticity course, this new edition has been revised with new worked examples and exercises, and new or expanded coverage of areas such as spherical anisotropy, stress contours, isochromatics, isoclinics, and stress trajectories. Using MATLAB software, numerical activities in the text are integrated with analytical problem solutions. These numerics aid in particular calculations, graphically present stress and displacement solutions to problems of interest, and conduct simple finite element calculations, enabling comparisons with previously studied analytical solutions. Online ancillary support materials for instructors include a solutions manual, image bank, and a set of PowerPoint lecture slides. Thorough yet concise introduction to linear elasticity theory and applications Only text providing detailed solutions to problems of nonhomogeneous/graded materials New material on stress contours/lines, contact stresses, curvilinear anisotropy applications Further and new integration of MATLAB software Addition of many new exercises Comparison of elasticity solutions with elementary theory, experimental data, and numerical simulations Online solutions manual and downloadable MATLAB code

This textbook's methodological approach familiarizes readers with the mathematical tools required to correctly define and solve problems in continuum mechanics. Covering essential principles and fundamental applications, this second edition of Continuum Mechanics using Mathematica® provides a solid basis for a deeper study of more challenging and specialized problems related to nonlinear elasticity, polar continua, mixtures, piezoelectricity, ferroelectricity, magneto-fluid mechanics and state changes (see A. Romano, A. Marasco, Continuum Mechanics: Advanced Topics and Research Trends, Springer (Birkhäuser), 2010, ISBN 978-0-8176-4869-5). Key topics and features: * Concise presentation strikes a balance between fundamentals and applications * Requisite mathematical background carefully collected in two introductory chapters and one appendix * Recent developments highlighted through coverage of more significant applications to areas such as wave propagation, fluid mechanics, porous media, linear elasticity. This second edition expands the key topics and features to include: * Two new applications of fluid dynamics: meteorology and navigation * New exercises at the end of the existing chapters * The packages are rewritten for Mathematica 9 Continuum Mechanics using Mathematica®: Fundamentals, Applications and Scientific Computing is aimed at advanced undergraduates, graduate students and researchers in applied mathematics, mathematical physics and engineering. It may serve as a course textbook or self-study reference for anyone seeking a solid foundation in continuum mechanics.

Thermoelasticity, Second Edition reviews advances in thermoelasticity and covers topics ranging from stationary problems of thermoelasticity to variational theorems of stationary thermoelasticity; stresses due to the action of a discontinuous temperature field in an infinite elastic body; the action of heat sources in the elastic space; and thermal inclusions in an infinite disc and semi-infinite disc. Three different sets of differential equations describing the fields of strain and temperature are presented. This book is comprised of 12 chapters and begins with a discussion on basic relations and equations of thermoelasticity. Thermoelasticity is treated as a synthesis of the theory of elasticity and the theory of heat conduction. Some particular cases of thermoelasticity are then investigated, including stationary problems, the theory of thermal stresses, and classical dynamic elasticity. Dynamic effects due to the action of a non-stationary temperature field are examined, along with plane harmonic waves in an elastic space and thermal stresses in plates, shells, and viscoelastic bodies. The final chapter focuses on micropolar thermoelasticity, magnetothermoelasticity, and thermopiezoelectricity. This monograph will be of interest to physicists and mechanical engineers.

Since the first edition of this book was published, there have been major improve- TM TM ments in symbolic mathematical languages such as Maple and Mathematica and this has opened up the possibility of solving considerably more complex and hence interesting and realistic elasticity problems as classroomexamples. It also enables the student to focus on the formulation of the problem (e. g. the appropriate governing equations and boundary conditions) rather than on the algebraic manipulations, with a consequent improvement in insight into the subject and in motivation. During the past 10 years I have developed files in Maple and Mathematica to facilitate this p- cess, notably electronic versions of the Tables in the present Chapters 19 and 20 and of the recurrence relations for generating spherical harmonics. One purpose of this new edition is to make this electronic material available to the reader through the Kluwer website www. elasticity. org. I hope that readers will make use of this resource and report back to me any aspects of the electronic material that could benefit from improvement or extension. Some hints about the use of this material are contained in Appendix A. Those who have never used Maple or Mathematica will find that it takes only a few hours of trial and error to learn how to write programs to solve boundary value problems in elasticity.

This text presents a highly original treatment of the fundamentals of FEM, developed using computer algebra, based on undergraduate-level engineering mathematics and the mechanics of solids. The book is divided into two distinct parts of nine chapters and seven appendices. The first chapter reviews the energy concepts in structural mechanics with bar problems, which is continued in the next chapter for truss analysis using Mathematica programs. The Courant and Clough triangular elements for scalar potentials and linear elasticity are covered in chapters three and four, followed by four-node elements. Chapters five and six describe Taig’s isoparametric interpolants and Iron’s patch test. Rayleigh vector modes, which satisfy point-wise equilibrium, are elaborated on in chapter seven along with successful patch tests in the physical (x,y) Cartesian frame. Chapter eight explains point-wise incompressibility and employs (Moore-Penrose) inversion of rectangular matrices. The final chapter analyzes patch-tests in all directions and introduces five-node elements for linear stresses. Curved boundaries and higher order stresses are addressed in closed algebraic form. Appendices give a short introduction to Mathematica, followed by truss analysis using symbolic codes that could be used in all FEM problems to assemble element matrices and solve for all unknowns. All Mathematica codes for theoretical formulations and graphics are included with extensive numerical examples.