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0542-4221

Introduction to theory of elasticity

Also listed as: מבוא לתורת האלסטיות

The theory floor under every serious solid-mechanics result. Stress and strain as tensors, boundary-value problems formulated and solved, and the classical two-dimensional solutions.

Semester
Semester B
Weekly hours
4h
Counts as
Core: solids
Interest areas
Structures and computational mechanics · Aeronautics and space

What it covers

Formulation

  • Index and tensor notation, vectors and second-order tensors
  • Strain analysis: displacement gradient, infinitesimal strain tensor, compatibility
  • Stress: traction vector, Cauchy stress tensor, equilibrium, principal stresses, invariants
  • Generalised Hooke's law, isotropic elasticity, Lamé constants, E, ν, G, K
  • Navier displacement equations, Beltrami–Michell stress equations, Saint-Venant's principle

Two-dimensional elasticity

  • Plane stress and plane strain, and when each is the right idealisation
  • Airy stress function and the biharmonic equation
  • Cartesian solutions: beam bending
  • Polar solutions: thick cylinder (Lamé), stress concentration at a hole (Kirsch), the wedge and Flamant problems

Torsion and energy methods

  • Torsion of non-circular bars, Prandtl stress function, warping
  • The membrane analogy
  • Energy methods and an introduction to three-dimensional problems

Results worth carrying out

  • Infinitesimal strain tensor

    εij=12(ui,j+uj,i)\varepsilon_{ij} = \tfrac{1}{2}\left(u_{i,j} + u_{j,i}\right)
  • Equilibrium

    σij,j+fi=0\sigma_{ij,j} + f_i = 0
  • Hooke's law, isotropic

    σij=λεkkδij+2μεij\sigma_{ij} = \lambda\,\varepsilon_{kk}\,\delta_{ij} + 2\mu\,\varepsilon_{ij}
  • Airy stress function

    4ϕ=0\nabla^4\phi = 0
  • Kirsch, stress at a hole edge

    σθθ=σ(1+2cos2θ),Kt=3\sigma_{\theta\theta} = \sigma\left(1 + 2\cos 2\theta\right), \qquad K_t = 3

Figures worth knowing

  • Three-dimensional Mohr's circle
  • Stress-concentration contours around a hole
  • Airy-function stress fields
  • Membrane analogy for torsion
  • Photoelastic fringe patterns

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