Linear Momentum Balance in Continuum Mechanics

This lecture focuses on the evaluation of linear momentum balance in continua. It begins with the definition of the stress vector and the stress tensor, emphasizing their properties, such as objectivity and independence from the coordinate system. The instructor introduces the concept of second-order tensors, explaining how they map normal vectors to stress vectors. The lecture progresses to the tensor transformation law, detailing how stress components relate across different coordinate systems. The discussion includes the transport of linear momentum across boundaries, incorporating contributions from mass flux and stress. The instructor highlights the importance of signs in these expressions, ensuring consistency in momentum transport. The lecture culminates in deriving a global expression for linear momentum balance, applying the divergence theorem to transform surface integrals into volume integrals. The final governing equation for linear momentum conservation is presented, laying the foundation for future studies in fluid and solid mechanics. The instructor emphasizes the complexity of the closure problem, indicating the need for further exploration in kinematics.