# Introduction

After extracting metal from their ores and adding different elements to obtain a precise alloy optimised for a given end usage, the alloy often starts as a relatively large billet from which objects have to be fabricated. Hence, the large billets have to be reduced by mechanical deformation processes such as forging, rolling and extrusion, to reduce and change their shapes.

These processes are energy intensive and require expensive machinery. It would be inappropriate to over-design such machinery since that would be unnecessarily expensive, while under-designing would prevent the alloy from being deformed. Therefore, it is important to know the required loads needed to achieve the necessary deformations for different alloys.

This TLP examines various approaches that can be used to provide estimates of the loads (forces or stresses) required when deforming metallic objects. Some of the approaches are two-dimensional, and this introduces the concepts of plane strain and plane stress. In addition, the precise nature of the alloy will affect its mechanical behaviour and so the idea of homogeneous deformation is also included.

The approaches included in this TLP to estimate deformation loads are:

- Levy-Mises equations leading to plane stress and plane strain
- Slip line field theory
- Work formula
- Limit analysis and hodographs
- Finite element analysis

This module addresses how materials deform (change shape) when subjected to applied forces (stresses). Clearly, the nature of such deformations will depend on the class of material (metal, polymer, glass, etc.) as well as the precise microstructure of the specific material. In the following, the deformation is assumed to be homogeneous, i.e. it is independent of microstructural features such as grain size, dislocation density and defects. Thus the properties of the material are assumed to be isotropic.