Friction plays a central role in life; in transport, in manufacturing, in process engineering, in medical devices and in everyday human activities. Friction has commanded the attention of Amontons, Coulomb and Da Vinci and their simplistic, empirical laws have been the cornerstone of friction theory. At the conceptual and theoretical levels the vast modern day friction literature has revealed the enormous complexity of even the simplest processes and the limitations of the early friction laws. Friction is intimately linked to both adhesion, contact geometry and wear and all require an appreciation of the highly non-equilibrium and non-linear processes occurring over multiple length scales. The challenge presented is that friction in realistic engineering contacts cannot be predicted. Understanding the physical and chemical processes at contacting interfaces is the only route to cracking the tribological enigma. The research gap addressed in this Programme Grant is linked to the development of accurate experimental and numerical simulations of friction. We appreciate that the search for a unified model for friction prediction is futile because friction is system dependent. However, the goal to predict friction is achievable. We have identified 4 key areas where there are current challenges in understanding the origins of friction because of different complexities as outlined below: - Reactive surfaces; in many systems the frictional contact brings about chemical reactions that can only be described by non-equilibrium thermodynamics. We need accurate kinetic rate data for reactions which can only be provided by advanced in-situ chemical analysis - Extreme interfaces; these can be described as any interfaces that are inducing high strain rate material deformation and combined with electrochemical or chemical reactions. Simulation and sensing are key to improving the understanding. - Non-linear materials; in engineering and in biological systems we see the evolution of "soft" materials for tribological applications. Predicting friction in these systems relies on understanding the rheology/tribology interactions. - Particles and 2nd phase materials; for materials processing or for understanding the transport of wear particles in a contact we need to understand particle-particle friction in complex contact conditions where fracture/deformation are occurring.

This Centre for Doctoral training in Industrially Focused Mathematical Modelling will train the next generation of applied mathematicians to fill critical roles in industry and academia. Complex industrial problems can often be addressed, understood, and mitigated by applying modern quantitative methods. To effectively and efficiently apply these techniques requires talented mathematicians with well-practised problem-solving skills. They need to have a very strong grasp of the mathematical approaches that might need to be brought to bear, have a breadth of understanding of how to convert complex practical problems into relevant abstract mathematical forms, have knowledge and skills to solve the resulting mathematical problems efficiently and accurately, and have a wide experience of how to communicate and interact in a multidisciplinary environment. This CDT has been designed by academics in close collaboration with industrialists from many different sectors. Our 35 current CDT industrial partners cover the sectors of: consumer products (Sharp), defence (Selex, Thales), communications (BT, Vodafone), energy (Amec, BP, Camlin, Culham, DuPont, GE Energy, Infineum, Schlumberger x2, VerdErg), filtration (Pall Corp), finance (HSBC, Lloyds TSB), food and beverage (Nestle, Mondelez), healthcare (e-therapeutics, Lein Applied Diagnostics, Oxford Instruments, Siemens, Solitonik), manufacturing (Elkem, Saint Gobain), retail (dunnhumby), and software (Amazon, cd-adapco, IBM, NAG, NVIDIA), along with two consultancy companies (PA Consulting, Tessella) and we are in active discussion with other companies to grow our partner base. Our partners have five key roles: (i) they help guide and steer the centre by participating in an Industrial Engagement Committee, (ii) they deliver a substantial elements of the training and provide a broad exposure for the cohorts, (iii) they provide current challenges for our students to tackle for their doctoral research, iv) they give a very wide experience and perspective of possible applications and sectors thereby making the students highly flexible and extremely attractive to employers, and v) they provide significant funding for the CDT activities. Each cohort will learn how to apply appropriate mathematical techniques to a wide range of industrial problems in a highly interactive environment. In year one, the students will be trained in mathematical skills spanning continuum and discrete modelling, and scientific computing, closely integrated with practical applications and problem solving. The experience of addressing industrial problems and understanding their context will be further enhanced by periods where our partners will deliver a broad range of relevant material. Students will undertake two industrially focused mini-projects, one from an academic perspective and the other immersed in a partner organisation. Each student will then embark on their doctoral research project which will allow them to hone their skills and techniques while tackling a practical industrial challenge. The resulting doctoral students will be highly sought after; by industry for their flexible and quantitative abilities that will help them gain a competitive edge, and by universities to allow cutting-edge mathematical research to be motivated by practical problems and be readily exploitable.