Problems Of Nonlinear Deformation

Interest in nonlinear problems in mechanics has been revived and intensified by the capacity of digital computers. Consequently, a question offundamental importance is the development of solution procedures which can be applied to a large class of problems. Nonlinear problems with a parameter constitute one such class. An important aspect of these problems is, as a rule, a question of the variation of the solution when the parameter is varied. Hence, the method of continuing the solution with respect to a parameter is a natural and, to a certain degree, universal tool for analysis. This book includes details of practical problems and the results of applying this method to a certain class of nonlinear problems in the field of deformable solid mechanics. In the Introduction, two forms of the method are presented, namely continu ous continuation, based on the integration of a Cauchy problem with respect to a parameter using explicit schemes, and discrete continuation, implementing step wise processes with respect to a parameter with the iterative improvement of the solution at each step. Difficulties which arise in continuing the solution in the neighbourhood of singular points are discussed and the problem of choosing the continuation parameter is formulated.

Complicated problems in nonlinear mechanics pose a challenge - many cannot be solved with existing closed-form methods. You would probably like easier methods for obtaining analytical and numerically exact solutions for finite elements, updated or total Lagrangian formulation, and arc-length methods of nonlinear elastic problem solving. Nonlinear Mechanics, Second Edition gives you what you want - convenient methods of analysis and valuable data for comparison. This is the only book to offer a comprehensive treatment of structural components with variable thickness and a variable modulus of elasticity. It is also the only one to cover closed-form solutions for the dynamic and inelastic analysis of members and plates that are subjected to small and large deformations by including axial and vertical restraints. The author uses exact and approximate solutions for static, dynamic, and inelastic analysis. It also discusses aspects of nonlinear vibration of elastically supported beams, nonlinear response of nonuniform rotor blades, and a new concept of airfoil design. With more than 30% updated and new material, this edition is revised and reorganized to meet the needs of both academia and industry. Easy-to-follow equation derivations, example problems, step-by-step procedures, and iterative approaches create a thorough reference that fills present needs and equips you for the challenges of the future.

The book provides a rigorous axiomatic approach to continuum mechanics under large deformation. In addition to the classical nonlinear continuum mechanics – kinematics, fundamental laws, the theory of functions having jump discontinuities across singular surfaces, etc. - the book presents the theory of co-rotational derivatives, dynamic deformation compatibility equations, and the principles of material indifference and symmetry, all in systematized form. The focus of the book is a new approach to the formulation of the constitutive equations for elastic and inelastic continua under large deformation. This new approach is based on using energetic and quasi-energetic couples of stress and deformation tensors. This approach leads to a unified treatment of large, anisotropic elastic, viscoelastic, and plastic deformations. The author analyses classical problems, including some involving nonlinear wave propagation, using different models for continua under large deformation, and shows how different models lead to different results. The analysis is accompanied by experimental data and detailed numerical results for rubber, the ground, alloys, etc. The book will be an invaluable text for graduate students and researchers in solid mechanics, mechanical engineering, applied mathematics, physics and crystallography, as also for scientists developing advanced materials.

Modern machine design challenges engineers with a myriad of nonlinear problems, among them fatigue, friction, plasticity, and excessive deformation. Today's advanced numerical computer programs bring optimal solutions to these complex problems within reach, but not without a trained and experienced overseer. Nonlinear Problems in Machine Des

From the reviews: "A unique feature of this book is the nice blend of engineering vividness and mathematical rigour. [...] The authors are to be congratulated for their valuable contribution to the literature in the area of theoretical thermoelasticity and vibration of plates." Journal of Sound and Vibration

Classic in the field covers application of theory of finite elasticity to solution of boundary-value problems, analysis of mechanical properties of solid materials capable of large elastic deformations. Problems. References.

The scientists of the seventeenth and eighteenth centuries, led by Jas. Bernoulli and Euler, created a coherent theory of the mechanics of strings and rods undergoing planar deformations. They introduced the basic con cepts of strain, both extensional and flexural, of contact force with its com ponents of tension and shear force, and of contact couple. They extended Newton's Law of Motion for a mass point to a law valid for any deformable body. Euler formulated its independent and much subtler complement, the Angular Momentum Principle. (Euler also gave effective variational characterizations of the governing equations. ) These scientists breathed life into the theory by proposing, formulating, and solving the problems of the suspension bridge, the catenary, the velaria, the elastica, and the small transverse vibrations of an elastic string. (The level of difficulty of some of these problems is such that even today their descriptions are sel dom vouchsafed to undergraduates. The realization that such profound and beautiful results could be deduced by mathematical reasoning from fundamental physical principles furnished a significant contribution to the intellectual climate of the Age of Reason. ) At first, those who solved these problems did not distinguish between linear and nonlinear equations, and so were not intimidated by the latter. By the middle of the nineteenth century, Cauchy had constructed the basic framework of three-dimensional continuum mechanics on the founda tions built by his eighteenth-century predecessors.

This book presents the fundamentals of nonlinear mechanics within a modern computational approach based mainly on finite element methods. Both material and geometric nonlinearities are treated. The topics build up from the mechanics of finite deformation of solid bodies through to nonlinear structural behaviour including buckling, bifurcation and snap-through. The principles are illustrated with a series of solved problems. This book serves as a text book for a second year graduate course and as a reference for practitioners using nonlinear analysis in engineering and design.