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With these assumptions, the stress and strain rate tensors here are symmetric and have a trace of zero, properties that allow their invariants and squares to be simplified from the general definitions. The deviatoric stress tensor is related to an effective stress by its second principal invariant: [3]
where is the first invariant of the stress tensor, is the second invariant of the deviatoric part of the stress tensor, is the yield stress in uniaxial compression, and is the Lode angle given by θ = 1 3 cos − 1 ( 3 3 2 J 3 J 2 3 / 2 ) . {\displaystyle \theta ={\tfrac {1}{3}}\cos ^{-1}\left({\cfrac {3{\sqrt {3}}}{2}}~{\cfrac {J_{3}}{J_{2 ...
A real tensor in 3D (i.e., one with a 3x3 component matrix) has as many as six independent invariants, three being the invariants of its symmetric part and three characterizing the orientation of the axial vector of the skew-symmetric part relative to the principal directions of the symmetric part.
In continuum mechanics, the maximum distortion energy criterion (also von Mises yield criterion [1]) states that yielding of a ductile material begins when the second invariant of deviatoric stress reaches a critical value. [2] It is a part of plasticity theory that mostly applies to ductile materials, such as some metals.
The initiation of crazing normally requires the presence of a dilative component of the stress tensor and can be inhibited by applying hydrostatic pressure. From a solid mechanics perspective this means that a necessary condition for craze nucleation is having a positive value of I 1 {\displaystyle I_{1}} , the first stress invariant that ...
The yield surface is usually expressed in terms of (and visualized in) a three-dimensional principal stress space (,,), a two- or three-dimensional space spanned by stress invariants (,,) or a version of the three-dimensional Haigh–Westergaard stress space. Thus we may write the equation of the yield surface (that is, the yield function) in ...
where is the first invariant of the Cauchy stress and is the second invariant of the deviatoric part of the Cauchy stress. The constants A , B {\displaystyle A,B} are determined from experiments. In terms of the equivalent stress (or von Mises stress ) and the hydrostatic (or mean) stress , the Drucker–Prager criterion can be expressed as
The right side of the equation is in effect a summation of hydrostatic effects, the divergence of deviatoric stress and body forces (such as gravity). All non-relativistic balance equations, such as the Navier–Stokes equations, can be derived by beginning with the Cauchy equations and specifying the stress tensor through a constitutive relation .