# POLYFLOW - How to identify the several components of a tensor ?

 What is a tensor? What is the meaning of the component Tij of a stress tensor? What is an invariant of a tensor? How many components does a tensor have? What is the meaning of T11, T12, T22 in a 2D planar flow? What is the meaning of T11, T12, T22, T33 in a 2D axisymmetric flow? What is the meaning of T11, T12, T22, T13, T23 and T33 in a 3D flow? What is a tensor? In physics, some quantities are scalar, such as the temperature or the pressure. Other quantities are vectors, such as the velocity or the force. Those quantities are attached to a point. Next to those, there exist other quantities, somewhat more complex, which depend on the location of the point considered AND on the orientation of an infinitesimal element attached to it. Those quantities are represented by means of tensors. What is the meaning of the component Tij of a stress tensor? It is the j-th component of the force applied on an infinitesimal element, whose orientation is along the i-direction. Note that individual components of a tensor are not really relevant, on the same way as individual components of a velocity vector are not really more relevant either... What is an invariant of a tensor? A quantity whose value does not depend on the reference frame selected for describing the corresponding tensor. How many components does a tensor have? Most tensors of interest are symmetric. The nature being 3D, tensors generally have six independent components. However... - for 2D planar flow, we have three non-zero components; - for 2D axisymmetric flows, we have four non-zero components; - for all other flows, we usually have six non-zero components. What is the meaning of T11, T12, T22 in a 2D planar flow? The reference frame is (x,y), both directions being referred to as 1 and 2. Hence, T11 refers to Txx, T22 refers to Tyy, and T12 refers to Txy. What is the meaning of T11, T12, T22, T33 in a 2D axisymmetric flow? The reference frame is (r,z), both directions being referred to as 1 and 2. They are respectively the radial and axial directions. A third direction (theta or t), referred to as 3, is also considered : the hoop, azimuthal or circumferential direction. Hence, T11 refers to Trr, T22 refers to Tzz, T12 refers to Trz, and T33 refers to as Ttt. What is the meaning of T11, T12, T22, T13, T23 and T33 in a 3D flow? The reference frame is (x,y,z), both directions being referred to as 1, 2 and 3. Hence, T11, T22 and T33 respectively refer to Txx, Tyy and Tzz ; while T12, T13 and T23 respectively refer to Txy, Txz and Tyz. REMARK: In case of a multi-modes viscoelastic model with the EVSS formulation, the total extra-stress tensor will be calculated following the formula T = S + *eta*D where D is the rate of deformation tensor and S is the sum of the elastic contribution of each modes (S = S1 + S2 + ... + Sn where n is the number of relaxation modes). In this case, S__211 will represent Sxx of the second relaxation mode; S__323, Syz of the third mode, ... Note that the individual contributions of the modes are of little interest, since the force balance results from the sum of all contributions.