After the course the student is able to
- Derive strain and stress measures in tensor format based on the definition of large deformation kinematics.
- Describe the elastic behavior of rubberlike materials with appropriate hyperelastic material models.
- Describe the plastic behavior of metals with a yield function and with isotropic and kinematic hardening.
- Implement a numerical algorithm for plasticity in a finite element program.
- Describe failure with a continuum damage model.
- Perform numerical simulations of forming processes with a finite element program and analyze and evaluate the results
- Understand the plastic instability criterion for sheet metal
The course Nonlinear Solid Mechanics prepares students to perform nonlinear finite element analysis with large deformation elastic and plastic material models. Nonlinear behavior can be triggered by nonlinear stress-strain relations of the material, like in rubber or plastic deformation of metals or by nonlinear displacement-strain relations when deformations or rotations are large. Although most structures are designed to operate in the linear regime, nonlinear behavior is essential in a number of applications. This is the case e.g. in forming processes, where material is permanently deformed into a desired shape or in the case where large deformations of components determine the performance, as in rubber seals. Another application of nonlinear analysis is to determine the safety margin when structures are overloaded. Will a linearly designed structure catastrophically collapse, or can an overload be sustained at a minor loss of functionality?|
In the mechanics of large deformations, new definitions for stress and strain will be introduced (the classical strain definition is not valid for large deformations). Large deformations almost always lead to nonlinear stress–strain relations. Attention will be given to mathematically correct kinematical relations. The transformation from the undeformed to the deformed state is described with the deformation gradient tensor. Strain measures are defined in a Eulerian and a Lagrangian frame. Strain-rate definitions are based on the velocity gradient and relations with the total strain measures are derived. Special attention is given to objectivity of these relations.
The stress in a material is defined based on the total strain or based on strain-rate relations. For the purpose of this course, a short overview of applicable vector and tensor equations is provided. For rubber-like materials, hyperelastic models are introduced with special attention to nearly-incompressible models. The focus will be on models based on invariants of the stretch tensor, like the neo-Hookean model and the Mooney–Rivlin model.For metal plasticity, elasto-(visco) plastic models are introduced, as well as the numerical algorithms to implement them in a finite element code. Isotropic and anisotropic yield functions are introduced for use in bulk forming and in sheet metal forming simulations (Tresca, Von Mises and Hill). Work hardening models based on isotropic and kinematic hardening theories are introduced with commonly used hardening relations (Swift and Voce). An instability criterion and fracture criterion is introduced for use in simulations of sheet forming processes. Finally, continuum damage models are considered to model failure of the material.
The course comes with a practical assignment to design and simulate forming of areal product using a dedicated commercial software program (Abaqus). This assignment will provide further insight in the plastic behavior of metal during a specific forming process (deep drawing).