Index notation for tensors and vectors
An introduction to index notation for vectors and tensors in continuum mechanics.
Blog
Technical articles on continuum mechanics, rock mechanics and scientific programming, with worked examples.
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Continuum mechanics
A series on stress analysis, best read in the order below.
Transformation of a tensor to a new coordinate system
Rotation of a tensor into a new coordinate system, with application to the Cauchy stress tensor.
Calculation of normal and shear stress on a plane
How to calculate the normal and shear stress on an arbitrary plane through a point of a body in equilibrium.
Principal stresses and stress invariants
Derivation of the principal stresses and the stress invariants from the Cauchy stress tensor.
Deviatoric stress and invariants
Definition of the hydrostatic and deviatoric parts of the stress tensor and calculation of the invariants of the deviatoric stress tensor.
Octahedral stresses
Definition of the octahedral planes and of the normal and shear stresses acting on them.
Plane stress and plane strain
Definition of plane stress and plane strain, the form of the stress and strain tensors, and the stress–strain relations in each case.
Mohr's circle
How to construct Mohr's circle for the two-dimensional analysis of the stress state.
Mohr's circle in three dimensions
How to construct Mohr's circle diagram in three dimensions.
Equilibrium equations
Derivation of the equilibrium equations in three dimensions and proof of the symmetry of the Cauchy stress tensor.
Rock mechanics
Mohr–Coulomb failure criterion
The Mohr–Coulomb failure criterion and its calibration on experimental data.
Programming
Digitize scientific plots with Python
A step-by-step method to digitize scientific plots and extract their data into arrays with Python.