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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

Comes after

  • Mechanics of solids 1

Opens up

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=λ εkk δij+2μ εij\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+2cos⁡2θ),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

Quick answers

When is Introduction to theory of elasticity taught?

Introduction to theory of elasticity (0542-4221) is taught in Semester B (spring), 4 weekly hours. It counts toward “Core: solids” in the Tel Aviv University Mechanical Engineering elective requirements.

What do I need before 0542-4221?

Before Introduction to theory of elasticity you need: Mechanics of solids 1.

What software is used in Introduction to theory of elasticity?

Introduction to theory of elasticity is normally taught with MATLAB, Mathematica, ABAQUS, COMSOL.

Which jobs does Introduction to theory of elasticity prepare you for?

Introduction to theory of elasticity is part of the preparation for these roles: Simulation and FEA engineer, Materials and failure engineer, Aerospace structures engineer.

Which interest areas include Introduction to theory of elasticity?

Introduction to theory of elasticity sits in these interest areas: Structures and computational mechanics, Aeronautics and space.

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