معلومات المقرر

Course Contents
Part I: One-Dimensional Plasticity: formulation and numerical implementation
1. Derivation of one-dimensional constitutive equations (strain decomposition, flow rule, yield condition, loading/unloading conditions, and consistency conditions) for perfect plasticity building on the phenomenological interpretation using a one-dimensional friction device
2. Extension to isotropic and kinematic hardening
3. The strong and weak form of the initial boundary value problem (IBVP)
4. Discretized and linearized version of IBVP and algorithmic procedure emphasizing the place of constitutive equations (stress-strain relation, algorithmic tangent modulus)
5. Integration algorithms (return map algorithms) for the constitutive equations, the incremental forms of rate-independent plasticity models through general midpoint rules, the elastic predictor-plastic corrector return map algorithms for incremental equations of perfect plasticity and isotropic hardening models, algorithmic tangent modulus

Part II: Three-dimensional classical rate-independent plasticity
1. Stress space governing equations parallel to the one-dimensional constitutive equations
2. Geometric interpretation of elastic unloading, plastic loading, neutral loading, associativity of flow rule
3. Classical J2 flow theory (von Mises yield criterion) for plain strain and 3D problems with the derivation of constitutive equations and algorithmic tangent moduli
4. Generalization to the general quadratic form of classical plasticity and emphasizing the special cases of von Mises isotropic criterion and general anisotropic criterion of Hill

Part III: Maximum plastic dissipation and convex optimization perspective
1. 2nd law of thermodynamics based interpretation of plasticity models with the introduction of essential definitions
2. Statement of maximum plastic dissipation principle and its interpretation as constraint convex optimization problem
3. Mathematical preliminaries of convex optimization (definition of a convex set, convex functions, the method of Lagrange multipliers for convex constraint optimization, KKT optimality conditions)
4. Derivation of constitutive equations (perfect plasticity and isotropic hardening) from convex optimization principles and the interpretation of loading/unloading conditions as KKT optimality conditions

Part IV: Integration algorithms for plasticity
1. Incremental form of constitutive equations, geometric interpretation as closest point projection, and strain-driven algorithmic procedure
2. Algorithmic treatment of constraint convex optimization problems and transferring these concepts for solving incremental equations
3. Radial return map algorithm for J2 plasticity
4. General return map algorithms (closest point projection algorithms, cutting plain algorithms)

Part V: Further topics
1. Subdifferential interpretation for non-smooth failure surfaces from an optimization perspective
2. Algorithmic treatment of non-smooth plasticity with an example of Tresca criterion
3. Direct extension to rate-dependent viscoelasticity with 1D example
4. Extension to finite strain plasticity

Literature
• Simo, J.C. and Hughes, T.J., 2006. Computational Inelasticity. Springer Science & Business Media.
• de Souza Neto, E.A., Peric, D. and Owen, D.R., 2011. Computational Methods for Plasticity: Theory and Applications. John Wiley & Sons.

Preconditions
A Background in continuum mechanics and linear finite element analysis is expected.

Further Grading Information
[b]Learning Outcomes: [/b]After the completion of the course, the students would be capable of developing a rigorous understanding of integration algorithms for elastoplastic constitutive problems and their mathematical foundations from the convex optimization perspective, implementing multidimensional problems for inelastic solids focusing on the return map algorithms for rate-independent plasticity models, linearization of nonlinear global governing equations, its discretization and solution within the context of the finite element method. The course will serve as a strong foundation for the structural, geotechnical, aerospace, and automobile applications in both academic and industry settings.

Official Course Description
This course covers the theoretical and implementation foundations of numerical methods used in small strain analysis of elastoplastic solids within the finite element method framework (FEM). The course follows the natural progression from numerical methods for one-dimensional rate-independent elastoplastic governing equations to generalized multidimensional problems in inelastic solids. In particular, the course attempts to approach the mathematical foundations of these numerical methods from the perspective of convex optimization theory. Special attention is dedicated to the computer implementation of the taught methods for benchmark elastoplastic problems through exercises and a final project.