Voigt Notation Explained Visually | From σ = C:ε to a 6x6 Matrix

What is Voigt notation in continuum mechanics?

Voigt notation is a compact way to turn symmetric second-order tensors into 6-component vectors.

In small-strain linear elasticity, the tensor law is

σ = C : ε

or in index notation,

σij = Cijkl εkl

Because the Cauchy stress tensor and infinitesimal strain tensor are symmetric, each 3x3 tensor has only six independent components.

Voigt notation packs them as

σV = [σ11, σ22, σ33, σ12, σ23, σ13]

For the engineering strain vector, we use

εV = [ε11, ε22, ε33, γ12, γ23, γ13]

where

γij = 2εij

This shear convention is the critical point.

For isotropic linear elasticity, the engineering-Voigt stiffness matrix has:

λ plus 2G on the normal diagonal entries,

λ on the normal off-diagonal coupling entries,

and G on the shear diagonal entries.

Important warning:

In tensor form,

σ12 = 2Gε12

But engineering shear strain is

γ12 = 2ε12

Therefore,

σ12 = Gγ12

So the engineering-Voigt shear entry is G, not 2G.

Memory trick:

Normal strain stays ε.
Shear strain becomes γ = 2ε.
That is why the shear block uses G.

This video is part of the tensor notation and continuum mechanics shorts series.

#VoigtNotation #ContinuumMechanics #TensorNotation #HookesLaw #SolidMechanics #Shorts

Видео Voigt Notation Explained Visually | From σ = C:ε to a 6x6 Matrix канала DEM & Granular Mechanics Lab