ABOUT THE COURSE:This course develops the mathematical and physical foundations of continuum mechanics, essential to understanding both solid and fluid mechanics. It emphasizes tensor algebra, stress and strain measures, conservation laws, and constitutive modelling, enabling a deep understanding of key continuum concepts. Designed as a foundational course, it equips students with the core concepts required for advanced courses such as elasticity, plasticity, advanced fluid mechanics, turbulence, and FEM. It also builds essential skills for computational mechanics, CFD, and FEA—key areas in aerospace, automotive, civil and materials industries. The focus is on conceptual clarity, mathematical rigor, problem-solving, and broad applicability across engineering disciplines.INTENDED AUDIENCE: Undergraduate and Early Postgraduate students of Mechanical Engineering / Civil Engineering / Biotechnology / Engineering PhysicsPREREQUISITES: Undergraduate Mechanics (Mechanics 101), Undergraduate Mathematics (Calculus and linear algebra).INDUSTRY SUPPORT: The following industry will find the course very useful: Aerospace, Automotive & Transportation, Oil & Gas, Mining, Energy, Construction, Semiconductors and Advanced Materials, Computational Mechanics & Software Companies, Biomechanics & Biomedical Engineering, Civil Engineering, Infrastructure, Manufacturing, and Research Institutes.

Syllabus

Week 1:Introduction and mathematical foundations (Module 1):Cartesian tensors, indicial notation, summation rule, Kronecker delta.
Week 2:Mathematical foundations: permutation symbol, epsilon-delta identity, vector and tensor products.
Week 3:Mathematical foundations: matrices and determinants, tensor transformations, isotropy and invariance.
Week 4:Mathematical foundations: principal values and directions, tensor calculus, integral theorems.
Week 5:Stress principles (Module 2): body and surface forces, definition of the Cauchy stress tensor, equilibrium equations.
Week 6:Stress principles: stress transformation laws, principal stresses and directions, Stress maxima and minima, Mohr’s circle, plane stress, spherical and deviatoric stress components.
Week 7:Kinematics (Module 3): configurations, deformation and motion, material and spatial coordinates, Lagrangian and Eulerian descriptions, material derivative.
Week 8:Kinematics: deformation gradient tensor, Lagrangian and Eulerian finite strain tensors, infinitesimal deformation theory and the infinitesimal strain tensor, normal and shear strain tensors, dilatation, and plane strain.
Week 9:Kinematics: differential displacement vector, infinitesimal rotation tensor, velocity gradient tensor, rate of deformation tensor, vorticity tensor, material derivatives of elements.
Week 10:Conservation laws (Module 4): (Reynolds) transport theorem, equation of the conservation of mass in Eulerian and Lagrangian forms, linear momentum principle, Piola-Kirchoff stress tensors, angular momentum principle.
Week 11:Constitutive modelling (Module 5): introduction and closure problem, 4th-order constitutive tensor, linear isotropic and anisotropic models for solids and fluids; non-linear constitutive models including hyperelasticity, plasticity, non-Newtonian fluid behaviour and viscoplasticity; time-dependent models such as viscoelasticity, creep, stress relaxation, thixotropy and rheopexy.
Week 12:Application of continuum mechanics: derivation of Navier-Stokes equation for linear and non-linear fluids; special cases; modelling and solving specific fluid dynamics problems using continuum mechanics principles.