Elasticity and continuum mechanics
Stress and strain tensors, Hooke’s law in three dimensions, elastic wave speeds, strings and membranes and plates, viscoelasticity, and the energy principle.
An elastic solid is a continuum that stores energy when deformed and recovers its shape when released. Describing it takes two tensor fields — the stress that carries the internal forces and the strain that measures the deformation — linked by a constitutive law that is the three-dimensional form of Hooke’s ut tensio sic vis. From that law follow the elastic moduli, the two speeds of elastic waves, the mechanics of thin structures, and the energy principle behind computational solid mechanics.
- 7.1 Stress: force per unit area on a plane — the traction vector and the Cauchy stress tensor.
- 7.2 Strain: the deformation of a material element — the displacement field, why only its symmetric gradient is strain, and volumetric strain.
- 7.3 Hooke’s law in 3-D and the elastic moduli — the two Lamé constants and the engineering moduli , , , .
- 7.4 Elastic waves: P-waves and S-waves — the Navier–Cauchy equation and the two wave speeds, and why fluids carry only one.
- 7.5 Strings, membranes, and plates — tension versus bending stiffness, and linear versus quadratic dispersion.
- 7.6 Viscoelasticity and the energy principle — Maxwell and Kelvin–Voigt models, mechanical impedance, and the variational basis of finite elements.