Nonlinear Computational Structural Mechanics: New Approaches and Non-Incremental Methods of Calculation

ISBN-13: 9781461271413 / Angielski / Miękka / 2012 / 222 str.

Pierre Ladeveze; J. G. Simmonds

Opis

This book treats computational modeling of structures in which strong nonlinearities are present. It is therefore a work in mechanics and engineering, although the discussion centers on methods that are considered parts of applied mathematics. The task is to simulate numerically the behavior of a structure under various imposed excitations, forces, and displacements, and then to determine the resulting damage to the structure, and ultimately to optimize it so as to minimize the damage, subject to various constraints. The method used is iterative: at each stage an approximation to the displacements, strains, and stresses throughout the structure is computated and over all times in the interval of interest. This method leads to a general approach for understanding structural models and the necessary approximations.

Spis treści

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: $$ \to {{\hat}_{{n + \frac{\hbox{$\scriptstyle 1$}}{\hbox{$\scriptstyle 2$}} }}}$$.- 8.3.3. The semi-global linear step: $${{\hat}_{{n + \frac{\hbox{$\scriptstyle 1$}}{\hbox{$\scriptstyle 2$}} }}} \to {_{{n + 1}}}$$.- 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 $$ \to {{\hat}_{{n + \frac{\hbox{$\scriptstyle 1$}}{\hbox{$\scriptstyle 2$}} }}}$$.- 8.5.3. The linear step (semi-global) $${{\hat}_{{n + \frac{\hbox{$\scriptstyle 1$}}{\hbox{$\scriptstyle 2$}} }}} \to {_{{n + 1}}}$$.- 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.