This book contains thirteen chapters. After the introduction in Chapter 1, Chapter 2 recalls the pertinent spatial and material continuum kinematics in bulk volumes. Chapter 3 reviews the corresponding continuum kinematics on dimensionally reduced smooth manifolds. Chapter 4 revisits the relevant continuum kinematics at singular sets elaborating on the jumps in the non-linear deformation maps and their associated tangent, cotangent and measure maps. Chapter 5 represents the formulation of generic balances for generic volume extensive quantities. Chapter 6 applies the formats of the generic balances to the spatial and material tangent, cotangent and measure maps to formulate kinematical 'balances'. Chapter 7 details the generic balances for the case of mechanical balances of mass, spatial momentum and its vector moment, respectively. Chapter 8 explores the consequences of the mechanical balances by elaborating on local and global formats of the balance of kinetic energy and the balance of material momentum. Chapter 9 capitalises on the referential setting when introducing the notions of spatial and material virtual displacements and discussing the accompanying spatial and material virtual work principles. Chapter 10 expands on the related variational setting in terms of extended Hamilton and Dirichlet principles for conservative elasto-dynamic and elasto-static cases. Chapter 11 specifies the generic balances for the case of thermo-dynamical balances of energy and entropy. Chapter 12 exploits the consequences of the thermo-dynamical balances and the resulting formats of the dissipation power inequalities. Chapter 13 sketches the consequences for computational mechanics by outlining the material force method based on finite element discretisation of the material virtual work principle and highlights its applicability to geometrically non-linear fracture mechanics by some computational examples.

Paul Steinmann is a full professor at FAU Erlangen-Nuernberg, where he is active since 2007, and co-director of the GCEC in Glasgow. He is a member of GACM, GAMM and EUROMECH and was member of the managing board of ECCOMAS, member of the IACM general council, and president of the DEKOMECH (German Committee for Mechanics), the adhering organisation to IUTAM. He supervises several doctoral and postdoctoral researchers in the fields of material modelling, multi-scale methods, multi-physics problems, non-standard continua, configurational-failure-fracture mechanics, biomechanics as well as general developments in finite element and discretisation methods.

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