2.3 Cauchy Stress¶
Let the plane with unit normal \(\mathbf{n}\) bisect a loaded material body, such that \(\mathbf{n}\) represents the outward normal (pointing away from the material) to one of the two halves. The traction vector \(\mathbf{t}\) at any point \(P\) on this cross-section represents the limit of the reaction force \(\Delta\mathbf{f}_{n}\) acting over an elemental area \(\Delta A_{n}\) in the neighborhood of \(P\), as \(\Delta A_{n}\to0\). For a non-polar medium, it is assumed that the reaction moment \(\Delta\mathbf{m}_{n}\) acting over \(\Delta A_{n}\) at \(P\) vanishes as \(\Delta A_{n}\to0\). Since there can be an infinite number of planes passing through the point \(P\), it follows that \(\mathbf{t}\) may assume an infinity of values for each different \(\mathbf{n}\). Therefore, we define a unique tensorial measure \(\boldsymbol{\sigma}\) at \(P\), which returns the traction \(\mathbf{t}\) for any unit normal \(\mathbf{n}\),
Here, \(\boldsymbol{\sigma}\) is the Cauchy stress tensor at any point \(P\) inside the material body. The Cauchy stress tensor, a spatial tensor, is the actual physical stress, since the corresponding traction vector is the force in the deformed configuration, per unit deformed area.