Mechanics of continuously distributed matter — solid mechanics, elasticity, plasticity, fracture, viscoelasticity. The bridge between classical mechanics (point particles) and fluid dynamics (deformable continua), unified by stress-strain…
continuum-mechanics
Continuum hypothesis (mechanics)
Matter is modelled as a continuous medium so that fields like density, stress, and velocity are well-defined at every point. Valid when…
Cauchy stress tensor
Second-order tensor σ_{ij} that gives the force per unit area transmitted across an oriented surface element with normal n: t_i = σ_{ij}…
Infinitesimal strain tensor
Symmetric tensor ε_{ij} = ½(∂u_i/∂x_j + ∂u_j/∂x_i) capturing local deformation of a body relative to its reference configuration, valid for…
Constitutive equation
A material-specific relation tying stress to strain (and possibly strain rate, history, temperature). Closes the balance laws to yield…
Generalised Hooke's law
For linear elastic materials, stress is linear in strain: σ_{ij} = C_{ijkl} ε_{kl}, where the stiffness tensor C has up to 21 independent…
Linear elasticity
The infinitesimal-strain Hookean theory — closed-form Green's functions for isotropic media (Kelvin, Boussinesq), Saint-Venant principle,…
Plasticity (framework)
Theory of irreversible deformation: yield surface in stress space, flow rule, hardening law. Governs metal forming, geomaterials, granular…
von Mises yield criterion
Isotropic yield surface based on the second invariant of the deviatoric stress: yielding occurs when J₂ = ½ s_{ij} s_{ij} reaches a…
Fracture mechanics
Analysis of crack initiation, growth, and arrest. Linear elastic fracture mechanics (LEFM) introduces the stress-intensity factor K and the…
Balance of linear momentum (continuum form)
Local form of Newton's second law for a continuum: ∇·σ + b = ρ a. Closed by a constitutive equation, this is the master equation for solid…
Navier–Cauchy equations
Equation of motion for an isotropic linear elastic medium, obtained by combining Hooke's law with the momentum balance: (λ+μ)∇(∇·u) + μ∇²u…
Material (substantial) derivative
Time derivative following a fluid/material particle: D/Dt = ∂/∂t + (v·∇). Bridges Eulerian (field-fixed) and Lagrangian…
Lagrangian vs Eulerian description
Two complementary kinematic frames: Lagrangian tracks material particles by their reference position; Eulerian fixes spatial coordinates…
Viscoelasticity
Materials whose response combines elastic (energy-storing) and viscous (energy-dissipating) modes — polymers, biological tissue, glasses…