Computational Plasticity

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