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…
Cauchy stress (Cauchy 1822)
A-L Cauchy 1822 stress tensor t_i = sigma_ij n_j; foundation of continuum-mechanics + finite-element-method + structural-analysis.
Mooney-Rivlin (1940)
Mooney-Rivlin 1940 hyperelastic strain-energy; modern Ogden 1972 + Yeoh 1990 incompressible-rubber models.
Prandtl-Reuss (1924)
Prandtl 1924 + Reuss 1930 von Mises J2 incremental-plasticity flow rule; modern crystal-plasticity finite-element-method.
Damage (Kachanov 1958)
L Kachanov 1958 + J Lemaitre 1985 continuum-damage-mechanics; modern Gurson + porous-plasticity for metal-fracture prediction.
Cauchy stress (1822)
A Cauchy 1822 stress-tensor σ_ij; foundational text in continuum-mechanics axioms; modern atomistic-continuum bridges.
Truesdell-Noll (1965)
C Truesdell-W Noll 1965 'Non-linear field theories of mechanics'; modern frame-indifference + objective-stress-rates.
Reynolds transport (1903)
O Reynolds 1903 transport-theorem d/dt∫_V φdV; foundation of conservation-of-mass + momentum + energy in continua.
Mooney-Rivlin (1940)
M Mooney 1940 + R Rivlin 1948 hyperelastic strain-energy W=C1(I1-3)+C2(I2-3); modern rubber + biological-tissue models.
Boundary layer (Prandtl 1904)
L Prandtl 1904 boundary-layer-theory; foundation of viscous-flow approximations + drag-reduction + aircraft design.
Eshelby tensor (Eshelby 1951)
J Eshelby 1951 elastic energy-momentum tensor; modern configurational-mechanics + crack-tip + dislocation-driving-force.