Stress Tensor

The stress vector on a plane passing through a given spatial point $x$ at a given time $t$ depends only on the unit normal vector $n$ to the plane. Thus, let $T$ be the transformation such that \[{{t}_{n}}=Tn.................................(1)\] Let's giving a small tetrahedron which have one normal angle as one of the vertices. This tetrahedron is a small portion from the whole body with $P$ the point on the normal vertrice. The size of the tetrahedron is approching zeros. At that limit, we can think of point $p$ passing incline plane.

 The Newton's second law written for this tetrahedron is \[\sum{F={{t}_{-{{e}_{1}}}}(\Delta {{A}_{1}})}+{{t}_{-{{e}_{2}}}}(\Delta {{A}_{2}})++{{t}_{-{{e}_{3}}}}(\Delta {{A}_{3}})+{{t}_{n}}\Delta {{A}_{n}}=ma.....................(2) \] since assumming the portion very small in size, the mass of the portion which is the product of three lengths will approching zero before the left ide of the equation. therefore, \[\sum{F={{t}_{-{{e}_{1}}}}(\Delta {{A}_{1}})}+{{t}_{-{{e}_{2}}}}(\Delta {{A}_{2}})++{{t}_{-{{e}_{3}}}}(\Delta {{A}_{3}})+{{t}_{n}}\Delta {{A}_{n}}=0.......................(3) \] Let the unit normal vector of the incline plane $ABC$ be \[n={{n}_{1}}{{e}_{1}}+{{n}_{2}}{{e}_{2}}+{{n}_{3}}{{e}_{3}}.....................(4) \] Therefore, the area $\Delta {{A}_{1}}$, $\Delta {{A}_{1}}$, and $\Delta {{A}_{1}}$ are \[\begin{align} & \Delta {{A}_{1}}={{n}_{1}}\Delta {{A}_{n}} \\ & \Delta {{A}_{2}}={{n}_{2}}\Delta {{A}_{n}} .....................(5) \\ & \Delta {{A}_{3}}={{n}_{3}}\Delta {{A}_{n}} \\ \end{align}\] using equation (3) and (5) \[{{t}_{-{{e}_{1}}}{{n}_{1}}}+{{t}_{-{{e}_{2}}}}{{n}_{2}}+{{t}_{-{{e}_{3}}}}{{n}_{3}}+{{t}_{n}}=0.......................(6)\] from the action reaction \[{{t}_{-{{e}_{1}}}}=-{{t}_{{{e}_{1}}}},{{t}_{-{{e}_{2}}}}=-{{t}_{{{e}_{2}}}},{{t}_{-{{e}_{3}}}}=-{{t}_{{{e}_{3}}}}\] therefore, equation (6) is \[{{t}_{{{e}_{1}}}}{{n}_{1}}+{{t}_{{{e}_{2}}}}{{n}_{2}}+{{t}_{{{e}_{3}}}}{{n}_{3}}={{t}_{n}}.......................(7)\] using equation (7) and (4), we can proove equation (1) \[\begin{align} & {{t}_{n}}=Tn=T({{n}_{1}}{{e}_{1}}+{{n}_{2}}{{e}_{2}}+{{n}_{3}}{{e}_{3}})\text{ =}{{n}_{1}}T{{e}_{1}}+{{n}_{2}}T{{e}_{2}}+{{n}_{3}}T{{e}_{3}}={{n}_{1}}{{t}_{1}}+{{n}_{2}}{{t}_{2}}+{{n}_{3}}{{t}_{3}} \\ & \\ & \\ \end{align}\] be a linear transformation. It is called the stress tensor or the Cauchy stress tensor resource: ISBN: 978-0-7506-8560-3