Applied Mathematical Modelling of Industrial Metal Forming

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Metal forming is the shaping of metal; examples from manufacturing include rolling metal to create thin sheets and stamping flat sheets of metal to form car body panels. There is currently an industrial need for smarter metal forming in order to create new products, to reduce scrap, to compensate for more variable materials (e.g. recycled metals), to reduce costs, and to reduce energy usage. In the 21st century, one might expect real-time computer control of metal forming processes, which would monitor the metal workpiece during the forming process and adapt the process to correct any problems and consistently obtain the desired end result. However, the computer controller needs a theoretical model to predict what would happen if it were to make a change, in order to find the right changes to make, and such theoretical models are currently unavailable; computer simulations (using finite elements for example) are too slow for use in real-time. The current state of the art is to use computer finite element simulations during process development or to diagnose problems, and then to use simple controllers (such as PID controllers) to blindly follow the pre-prescribed forming procedure. Clearly new modelling techniques could be expected to give a substantial improvement.The ambitious aim of this project is to investigate techniques for mathematical modelling in continuum solid mechanics and plasticity, the outcome of which could be used to provide predictive theoretical models for industrial metal forming. Unlike existing computer simulations (such as finite elements) which work in all situations but which are slow, the aim here is to take advantage of properties of particular metal forming processes (such as symmetry, or small parameters such as thin sheets, small deformations, etc), and create bespoke simplified models specific to each of these processes. By accounting for these properties in a rigorous way, and using best practice mathematical techniques (such as asymptotics and stability theory), quick-to-compute models with a guaranteeable accuracy could be produced. Such models would be eminently suitable for use in online control of the metal forming process. The aim of this project is to work out how to do this in specific detail, and to produce a number of industrially relevant examples.There are a number of related areas of research which, while they will certainly inform this work, they will not be directly worked on in this project. These include research into atomic scale modelling of metals and alloys, and research into the correctness of the governing equations of plastic continuum mechanics (a branch of pure mathematical analysis). In this project, we will take the governing equations that are accepted and used in finite element computations of metal forming used in industry, and instead of solving them on a computer, we will investigate the theoretical development of modelling techniques and simplified models based on these equations.It is hoped the results of this project will be a number of techniques for creating bespoke mathematical models of particular metal forming processes, together with a few such models specific to particular metal forming processes of industrial relevance. The specific models have been chosen to be relevant to UK industry, in particular to Tata Steel and Primetals Technologies. These few models will be validated against either numerical results or practical experiments. The project would also result in a skilled team of researchers and a validated body of new applied mathematical knowledge, which would give industry the confidence to partner with the PI and invest in subsequent projects applying this knowledge in industrial practice.