Riassunto
Mechanical Engineering, an engineering discipline borne of the needs of the in dustrial revolution, is once again asked to do its substantial share in the call for industrial renewal. The general call is urgent as we face profound issues of pro ductivity and competitiveness that require engineering solutions, among others. The Mechanical Engineering Series features graduate texts and research mono graphs intended to address the need for information in contemporary areas ofme chanical engineering. The series is conceived as a comprehensive one that covers a broad range of concentrations important to mechanical engineering graduate education and re search. We are fortunate to have a distinguished roster ofconsulting editors on the advisory board, each an expert in one ofthe areas ofconcentration. The names of the consulting editors are listed on the next page of this volume. The areas of concentration are applied mechanics, biomechanics, computational mechanics, dynamic systems and control, energetics, mechanics ofmaterials, processing, ther mal science, and tribology. Frederick A. Leckie,the series editor for applied mechanics, and I are pleased to presentthis volume in the Series: Nonlinear Computational Structural Mechan ics: New Approaches and Non-Incremental Methods of Calculation, by Pierre Ladeveze. The selection of this volume underscores again the interest of the Me chanical Engineering series to provide our readers with topical monographs as well as graduate texts in a wide variety of fields.
Contenuti
1 The Reference Problem for Small Disturbances.- 1.1. Notation.- 1.2. The reference problem.- 1.3. Sufficient conditions assuring uniqueness.- 1.4. Analogy with the basic problem of fluid mechanics.- 2 Material Models.- 2.1. Formulation with internal variables.- 2.2. Examples of material models.- 2.3. Formulation of the constitutive relation.- 2.4. Normal formulation of a constitutive model.- 2.5. Error as measured by the constitutive relation (error in CR).- 2.5.1. Some classical properties of constitutive models.- 2.5.2. Error measures for a standard normal formulation.- 2.5.3. Illustration of the notion of admissibility as regards internal variables.- 2.5.4. Error in the sense of Drucker—functional formulation.- 2.5.5. Extensions.- 3 Solution Methods for Nonlinear Evolution Problems.- 3.1. The principle of incremental methods.- 3.2. Differential equation formulation of the reference problem.- 3.3. A general presentation of some classical methods for solving nonlinear problems.- 3.3.1. The geometric scheme associated with the problem.- 3.3.2. Algorithms for two search directions.- 3.3.3. Description of the different stages.- 3.3.4. Examples of directions of descent and ascent.- 3.3.5. Errors and error indicators.- 3.3.6. A convergence result.- 3.4. Other approaches to nonlinear evolution problems.- 4 Principles of the Method of Large Time Increments.- 4.1. Mechanics framework for the method of large time increments.- 4.2. Algorithms for two search directions.- 4.3. The local step.- 4.3.1. General case.- 4.3.2. Examples: plastic and viscoplastic materials with isotropic hardening.- 4.4. The global linear step.- 4.4.1. Quasi-static linear global step.- 4.4.2. The linear global step in dynamics.- 4.5. Convergence.- 4.5.1. Principal hypotheses.- 4.5.2. Basic identities.- 4.5.3. Convergence results.- 4.6. A posteriori error estimates.- 4.6.1. A first set of error indicators.- 4.6.2. Analysis of the caseH+=H-=L(Lsymmetric and positive).- 4.6.3. Other error indicators.- 4.7. Remarks.- 5 A Preliminary Example: A Beam in Traction.- 5.1. Quasi-static analysis for a viscoplastic material.- 5.2. Static analysis for a hyperelastic material.- 6 A “Mechanics Approximation” and Numerical Implementation.- 6.1. Discretization in time and space.- 6.2. Numerical treatment of the local step.- 6.3. Treatment of the linear global step in statics.- 6.3.1. Approximation on ? × [0T] (Principle P3).- 6.3.2. Iterative method for solving the linear global step.- 6.3.3. Remarks.- 6.4. Decomposition and approximation of the “radial loading” type for a function defined on ? × [0T].- 6.4.1. Approximation of order 1.- 6.4.2. Properties of the associated eigenvalue problem.- 6.4.3. Approximation of ordermand convergence properties.- 6.4.4. Remarks.- 6.5. Applications and analysis of performance.- 6.5.1. Example 1.- 6.5.2. Example 2.- 6.5.3. Example 3.- 7 Modeling and Calculation for Structures under Cyclic Loads.- 7.3. Treatment of the linear global step.- 7.4. A one-dimensional example.- 7.5. Example: viscoplastic disk with a loading of 1,000 cycles.- 8 Formulation and “Parallel” Strategies in Mechanics.- 8.1. Remarks on the degree of parallelism in the equations of reference.- 8.2. Partioning of the body into sub-structures and interfaces.- 8.2.1. Principles of the partioning method.- 8.2.2. Examples of interfaces.- 8.2.3. Modeling of an interface.- 8.2.4. New formulation with partitioning of the reference problem.- 8.3. Treatment of a static assemblage of elastic structures.- 8.3.1. Formulation of the problem.- 8.3.2. The local step: $${{s}_{n}} \to {{\hat{s}}_{{n + \frac{\hbox{$\scriptstyle 1$}}{\hbox{$\scriptstyle 2$}} }}}$$.- 8.3.3. The semi-global linear step: $${{\hat{s}}_{{n + \frac{\hbox{$\scriptstyle 1$}}{\hbox{$\scriptstyle 2$}} }}} \to {{s}_{{n + 1}}}$$.- 8.3.4. Example.- 8.4. Convergence for a static assemblage of elastic structures.- 8.4.1. Principal hypotheses.- 8.4.2. A preliminary convergence result (µ a positive constant).- 8.4.3. Convergence results for µ = 0.- 8.5. Dynamic and static treatment of an assemblage of structures with nonlinear behavior.- 8.5.1. A new formulation with partitioning of the reference problem.- 8.5.2. The local step $${{s}_{n}} \to {{\hat{s}}_{{n + \frac{\hbox{$\scriptstyle 1$}}{\hbox{$\scriptstyle 2$}} }}}$$.- 8.5.3. The linear step (semi-global) $${{\hat{s}}_{{n + \frac{\hbox{$\scriptstyle 1$}}{\hbox{$\scriptstyle 2$}} }}} \to {{s}_{{n + 1}}}$$.- 8.5.4. Convergence of the method.- 9 Modeling and Computation for Large Deformations.- 9.1. Material quantities and modeling of their behavior.- 9.2. Pure material formulation of large deformations—bases.- 9.3. Kinematic and other properties.- 9.3.1. Calculation of A as a function of $$\dot{\Sigma }$$.- 9.3.2. Calculation of Q as a function of $$\dot{\Sigma }$$.- 9.3.3. Calculation of ? and R as functions ofV.- 9.3.4. Other properties.- 9.4. Purely material formulation of the equilibrium of the body—properties and approximations.- 9.4.1. Reference formulation.- 9.4.2. The approximation A~1.- 9.4.3. The notion of radial loading.- 9.4.4. The problem in velocity.- 9.5. Two different representations of the modeling and computation of large deformations.- 9.5.1. Presentation with “linear” equilibrium equations.- 9.5.2. Presentation with “nonlinear” equilibrium equations.- 9.6. Approaches to large time increments.- 9.6.1. A first approach.- 9.6.2. Large time increment approaches to constitutive models with internal variables.- 9.6.2.1. Presentation with “linear” equilibrium equations.- 9.6.2.2. Presentation with “nonlinear” equilibrium equations.- 9.7. Remarks and an example.
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