Nonlinear Computational Structural Mechanics

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.

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.