Strain hardening

The attached notes are written for a course on plasticity.  I will update new posts on my twitter account:  https://twitter.com/zhigangsuo.  Rheology is the science of deformation. This science poses a question for every material: given a history of stress, how do we find the history of strain?

We can certainly apply the history of stress to the material and record the history of strain. We can also use computers to simulate the movements of electrons and atoms and molecules. These brute-force approaches, however, do not work by themselves. The stress has six components and can each undergo countless histories. There are other factors and their histories to consider: temperature, electric field, humidity, pH, etc. There are so many materials: metals, glasses, rubbers, liquids, toothpastes, and chewing gums. They all behave differently. How many experiments and simulations do we have to run?

In various courses we have been learning a hybrid approach. Given a material, we paly with it and watch it deform. We relate histories of stress and histories of strain by constructing a phenomenological model. We determine parameters in the model by running experiments. We try to understand the model by thinking about electrons and atoms and molecules, sometimes aided by computers. The hybrid approach dates back at least to Hooke’s law of elasticity and Newton’s law of viscosity. More recent examples include various models of thermoelasticity, viscoelasticity, large-strain elasticity, plasticity, poroelasticity, electrorehology, and chemorheology.

This course focuses on plasticity, but we also compare plasticity with other types of rheological behavior. Furthermore, we examine how the rheology of materials affects phenomena of inhomogeneous deformation, such as necking, cavitation, creasing, and shear localization. In a separate course, we have examined how the rheology of materials affects fracture.

In this lecture we will construct the elastic, isotropic hardening model for a metal. The model achieves something extraordinary: it uses the experimental record of a single history (i.e., the tensile stress-strain curve) to describe all histories of loading and unloading, tension and compression. Here we will consider a rod of metal under axial force, but later in the course we will generalize the model to multiaxial loading.