Come not, as some, without preparation,
But con his paragraphs o'er and o'er,
To be able to say, when you hear his oration,
That he gives you his book, and nothing more;
Yet not the less take down his words in writing,
As if the Holy Spirit were inditing!
Mephistopheles, Goethe: Faust, Part I, lines 1910–1911
Module 1 — Mathematical Toolkit
Tensor algebra: scalars, vectors, and tensors; tensor operations; tensor product and contraction; symmetric and skew-symmetric tensors; tensor invariants; eigenvalues and eigenvectors; orthogonal tensors; change of basis and tensor transformations.
Tensor calculus: scalar, vector, and tensor fields; gradient, divergence, and curl operators; material derivative; integral theorems of vector calculus; compact tensor notation.
Module 2 — Kinematics of Deformable Bodies
General theory of deformation: motion and deformation; reference and current configurations; displacement field; deformation gradient tensor; changes in length, area, and volume; Jacobian of deformation; homogeneous deformations; rigid body motions; strain measures; right and left Cauchy–Green tensors; Green–Lagrange strain tensor; polar decomposition theorem; velocity field; velocity gradient tensor; stretching tensor; spin tensor.
Infinitesimal theory: small-deformation approximation; infinitesimal strain tensor; infinitesimal rotation tensor; geometrical interpretation of infinitesimal strain.
Module 3 — Basic Mechanical Principles
Mass balance: conservation of mass; continuity equation in integral and local form.
Force and moment balance: body and surface forces; traction vector; Cauchy theorem; Cauchy stress tensor; balance of linear momentum; balance of angular momentum; symmetry of the stress tensor; external and internal power; kinetic energy; inertial power.
Referential formulation: Piola transformation; first and second Piola–Kirchhoff stress tensors; referential balance equations; power expenditure in the reference configuration.
Frame indifference: change of observer; objectivity; principle of material frame indifference; consequences for constitutive equations.
Module 4 — Constitutive Theory
Constitutive equations: constitutive variables; determinism; locality; material symmetry; thermodynamic admissibility; frame indifference.
Elastic materials: strain-energy density; constitutive equations for elastic solids; consequences of frame indifference; infinitesimal elasticity; isotropic linear elasticity; Hooke’s law in tensor form.
Module 5 — Atomistic-to-Continuum Bridging
Failure of classical continuum mechanics at the nanoscale: continuum hypothesis and characteristic length scales; surface-to-volume ratio effects; size-dependent elastic behavior; Representative Volume Element (RVE); breakdown of continuum assumptions at the nanoscale.
Cauchy–Born rule: Bravais lattices; unit cells and primitive vectors; affine lattice deformation; Cauchy–Born hypothesis; interatomic potentials; derivation of strain-energy density; recovery of elastic constants; limits of validity.
Statistical mechanics primer: microstates and macrostates; phase space; microcanonical, canonical, and grand canonical ensembles; ensemble averages and time averages; ergodic hypothesis; Liouville equation; conservation of phase-space volume.
Irving–Kirkwood–Noll procedure: continuum fields from discrete atomic systems; microscopic density and momentum fields; ensemble averaging; conservation equations from the Liouville equation; microscopic stress tensor; kinetic and virial contributions; non-uniqueness of pointwise stress definitions.
Hardy procedure: localization functions and smoothing kernels; regularization of Dirac delta distributions; density and momentum density fields; Hardy stress tensor; bond function; geometric interpretation of stress localization; spatial localization of atomistic quantities.
REFERENCES
Lecture notes
Other references
M.E. Gurtin, An Introduction to Continuum Mechanics} Academic Press, 1982.
M.E. Gurtin, E. Fried, L. Anand, The Mechanics and Thermodynamics of Continua, Cambridge University Press, 2010.
E.B. Tadmor, R.E. Miller, Modeling Materials, Cambridge, 2011.
C. Truesdell, A First Course in Rational Continuum Mechanics, Academic Press, 1977.