## Modeling of Localized Inelastic Deformation

taught by Milan Jirásek

in Prague, Czech Republic, on 17-21 September 2012

### Theme

This course provides an overview of modeling approaches used in the mechanics of inelastic materials and structures, with special attention to the objective description of highly localized deformation modes such as cracks or shear bands. In 2012 it attracted 30 participants from 13 European countries and Canada. It is included among the RILEM educational courses.

### Main topics

• Introduction: notation, fundamentals of tensor algebra, basic types of inelastic material behavior, principles of incremental-iterative nonlinear analysis.
• Elastoplasticity: physical motivation, basic equations in one dimension, extension to multiaxial stress, postulate of maximum plastic dissipation, associated and nonassociated plastic flow, hardening and softening, tangent stiffness.
• Fracture mechanics: stress concentration around defects, asymptotic fields in the vicinity of a crack tip, local and global criteria for crack propagation, fracture toughness and fracture energy, nonlinear process zone, cohesive crack models.
• Damage mechanics: physical motivation, basic equations in one dimension, isotropic damage models, smeared crack models, anisotropic damage models based on principles of strain equivalence and of energy equivalence, damage deactivation due to crack closure, combination of damage and plasticity.
• Strain localization: physical aspects, structural size effect, conditions of stability and uniqueness, discontinuous bifurcation, incipient weak discontinuity, localization analysis based on acoustic tensor, loss of ellipticity and its mathematical and numerical consequences, classification of models for localized inelastic behavior, mesh-adjusted softening modulus (crack band approach).
• Regularized continuum models: classification of enriched continuum theories, nonlocal formulations of the integral type, explicit and implicit gradient formulations, continua with microstructure, localization analysis, implementation aspects, application examples.
• Strong discontinuity models: cohesive crack and cohesive zone models, finite elements with incorporated discontinuities (embedded crack models, extended finite elements), implementation aspects and examples. A pdf file with the presentation can be downloaded from here.

### Level

• The course is designed for graduate students at the doctoral level, but it can be equally useful to motivated master students, post-doctoral researchers, or senior researchers who are not specialists in this field. Similar courses were given by the lecturer at the Swiss Federal Institute of Technology in Lausanne (1998), Czech Technical University in Prague (1998), Universität Stuttgart (1998), Rheinisch-Westfälische Technische Hochschule in Aachen (1999), Universität der Bundeswehr in Munich (2000), and Universitat Politècnica de Catalunya in Barcelona (2002). In its current format, the course has been taught in Prague every year since 2004.
• Prerequisites: fundamentals of elasticity, plasticity and finite element methods.
• sample chapter of the lecture notes is available for free downloading. Note that this excerpt is taken from an old version of the lecture notes. The material distributed at the course has been updated and extended. The complete set of lecture notes has about 300 pages in a dense format.

Last update: 30 September 2012