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Presenting original results from both theoretical and numerical viewpoints, this text offers a detailed discussion of the variational approach to brittle fracture. This approach views crack growth as the result of a competition between bulk and surface energy, treating crack evolution from its initiation all the way to the failure of a sample. The authors model crack initiation, crack path, and crack extension for arbitrary geometries and loads.
This book exposes a number of mathematical models for fracture of growing difficulty. All models are treated in a unified way, based on incremental energy minimization. They differ from each other by the assumptions made on the inelastic part of the total energy, here called the "cohesive energy". Each model describes a specific aspect of material response, and particular care is devoted to underline the correspondence of each model to the experiments. The content of the book is a re-elaboration of the lectures delivered at the First Sperlonga Summer School on Mechanics and Engineering Sciences in September 2011. In the year and a half elapsed after the course, the material has been revised and enriched with new and partially unpublished results. Significant additions have been introduced in the occasion of the course "The variational approach to fracture and other inelastic phenomena", delivered at SISSA, Trieste, in March 2013. The Notes reflect a research line carried on by the writer over the years, addressed to a comprehensive description of the many aspects of the phenomenon of fracture, and to its relations with other phenomena, such as the formation of microstructure and the changes in the material’s strength induced by plasticity and damage. Reprinted from the Journal of Elasticity, volume 112, issue 1, 2013.
This book addresses new questions related to the asymptotic description of converging energies from the standpoint of local minimization and variational evolution. It explores the links between Gamma-limits, quasistatic evolution, gradient flows and stable points, raising new questions and proposing new techniques. These include the definition of effective energies that maintain the pattern of local minima, the introduction of notions of convergence of energies compatible with stable points, the computation of homogenized motions at critical time-scales through the definition of minimizing movement along a sequence of energies, the use of scaled energies to study long-term behavior or backward motion for variational evolutions. The notions explored in the book are linked to existing findings for gradient flows, energetic solutions and local minimizers, for which some generalizations are also proposed.
By the detailed analysis of the modern development of the mechanics of deformable media can be found the deep internal contradiction. From the one hand it is declared that the deformation and fracture are the hierarchical processes which are linked and unite several structural and scale levels. From the other hand the sequential investigation of the hierarchy of the deformation and destruction is not carried out. The book’s aim is filling this mentioned gap and investigates the hot topic of the fracture of non-ideal media. From the microscopic point of view in the book we study the hierarchy of the processes in fractured solid in the whole diapason of practically used scales. According the multilevel hierarchical system ideology under “microscopic we understand taking into account the processes on the level lower than relative present strata. From hierarchical point of view the conception of “microscopic fracture can be soundly applied to the traditionally macroscopic area, namely geomechanics or main crack propagation. At the same time microscopic fracture of the nanomaterials can be well-grounded too. This ground demands the investigation on the level of inter-atomic interaction and quantum mechanical description. The important feature of the book is the application of fibred manifolds and non-Euclidean spaces to the description of the processes of deformation and fracture in inhomogeneous and defected continua. The non-Euclidean spaces for the dislocations’ description were introduced by J.F. Nye, B.A. Bilby, E. Kröner, K. Kondo in fiftieth. In last decades this necessity was shown in geomechanics and theory of seismic signal propagation. The applications of non-Euclidean spaces to the plasticity allow us to construct the mathematically satisfying description of the processes. Taking into account this space expansion the media with microstructure are understood as Finsler space media. The bundle space technique is used for the description of the influence of microstructure on the continuum metrics. The crack propagation is studied as a process of movement in Finsler space. Reduction of the general description to the variational principle in engineering case is investigated and a new result for the crack trajectory in inhomogeneous media is obtained. Stability and stochastization of crack trajectory in layered composites is investigated. The gauge field is introduced on the basis of the structure representation of Lie group generated by defects without any additional assumption. Effective elastic and non-elastic media for nanomaterials and their geometrical description are discussed. The monograph provides the basis for more detailed and exact description of real processes in the material. The monograph will be interesting for the researchers in the field of fracture mechanics, solid state physics and geomechanics. It can be used as well by the last year students wishing to become more familiar with some modern approaches to the physics of fracture and continual theory of dislocations. In Supplement, written by V.V.Barkaline, quantum mechanical concept of physical body wholeness according to H. Primas is discussed with relation to fracture. Role of electronic subsystem in fracture dynamics in adiabatic and non-adiabatic approximations is clarified. Potential energy surface of ion subsystem accounting electron contribution is interpreted as master parameter of fracture dynamics. Its features and relation to non-euclidean metrics of defected solid body is discussed. Quantum mechanical criteria of fracture arising are proposed.
This book contains contributions presented at the IUTAM Symposium "Fracture Phenomena in Nature and Technology" held in Brescia, Italy, 1-5 July, 2012.The objective of the Symposium was fracture research, interpreted broadly to include new engineering and structural mechanics treatments of damage development and crack growth and also large-scale failure processes as exemplified by earthquake or landslide failures, ice shelf break-up and hydraulic fracturing (natural or for resource extraction or CO2 sequestration), as well as small-scale rupture phenomena in materials physics including, e.g. inception of shear banding, void growth, adhesion and decohesion in contact and friction, crystal dislocation processes and atomic/electronic scale treatment of brittle crack tips and fundamental cohesive properties. Special emphasis was given to multiscale fracture description and new scale-bridging formulations capable to substantiate recent experiments and tailored to become the basis for innovative computational algorithms.
Novel techniques for modeling 3D cracks and their evolution in solids are presented. Cracks are modeled in terms of signed distance functions (level sets). Stress, strain and displacement field are determined using the extended finite elements method (X-FEM). Non-linear constitutive behavior for the crack tip region are developed within this framework to account for non-linear effect in crack propagation. Applications for static or dynamics case are provided.
Variational Methods in the Mechanics of Solids contains the proceedings of the International Union of Theoretical and Applied Mechanics Symposium on Variational Methods in the Mechanics of Solids, held at Northwestern University in Evanston, Illinois, on September 11-13, 1978. The papers focus on advances in the application of variational methods to a variety of mathematically and technically significant problems in solid mechanics. The discussions are organized around three themes: thermomechanical behavior of composites, elastic and inelastic boundary value problems, and elastic and inelastic dynamic problems. This book is comprised of 58 chapters and opens by addressing some questions of asymptotic expansions connected with composite and with perforated materials. The following chapters explore mathematical and computational methods in plasticity; variational irreversible thermodynamics of open physical-chemical continua; macroscopic behavior of elastic material with periodically spaced rigid inclusions; and application of the Lanczos method to structural vibration. Finite deformation of elastic beams and complementary theorems of solid mechanics are also considered, along with numerical contact elastostatics; periodic solutions in plasticity and viscoplasticity; and the convergence of the mixed finite element method in linear elasticity. This monograph will appeal to practitioners of mathematicians as well as theoretical and applied mechanics.
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