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In continuum mechanics, the Cauchy stress tensor (symbol , named after Augustin-Louis Cauchy), also called true stress tensor[1] or simply stress tensor, completely defines the state of stress at a point inside a material in the deformed state, placement, or configuration. The second order tensor consists of nine components and relates a unit ...
Invariants of tensors. In mathematics, in the fields of multilinear algebra and representation theory, the principal invariants of the second rank tensor are the coefficients of the characteristic polynomial [1] where is the identity operator and are the roots of the polynomial and the eigenvalues of . More broadly,any scalar-valued function is ...
The incompressible momentum Navier–Stokes equation results from the following assumptions on the Cauchy stress tensor: [5] the stress is Galilean invariant: it does not depend directly on the flow velocity, but only on spatial derivatives of the flow velocity. So the stress variable is the tensor gradient .
Lode coordinates. Surfaces on which the invariants , , are constant. Plotted in principal stress space. The red plane represents a meridional plane and the yellow plane an octahedral plane. Lode coordinates or Haigh–Westergaard coordinates . [1] are a set of tensor invariants that span the space of real, symmetric, second-order, 3-dimensional ...
Thus the stress state of the material must be described by a tensor, called the (Cauchy) stress tensor; which is a linear function that relates the normal vector n of a surface S to the traction vector T across S. With respect to any chosen coordinate system, the Cauchy stress tensor can be represented as a symmetric matrix of 3×3
Objective stress rate. In continuum mechanics, objective stress rates are time derivatives of stress that do not depend on the frame of reference. [1] Many constitutive equations are designed in the form of a relation between a stress-rate and a strain-rate (or the rate of deformation tensor). The mechanical response of a material should not ...
Stress triaxiality. In continuum mechanics, stress triaxiality is the relative degree of hydrostatic stress in a given stress state. [1] It is often used as a triaxiality factor, T.F, which is the ratio of the hydrostatic stress, , to the Von Mises equivalent stress, . [2][3][4] Stress triaxiality has important applications in fracture ...
Alternative stress measures. In continuum mechanics, the most commonly used measure of stress is the Cauchy stress tensor, often called simply the stress tensor or "true stress". However, several alternative measures of stress can be defined: [1][2][3] The Kirchhoff stress (. τ {\displaystyle {\boldsymbol {\tau }}} ).