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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.
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 ...
The first index i indicates that the stress acts on a plane normal to the X i-axis, and the second index j denotes the direction in which the stress acts (For example, σ 12 implies that the stress is acting on the plane that is normal to the 1 st axis i.e.;X 1 and acts along the 2 nd axis i.e.;X 2). A stress component is positive if it acts in ...
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.
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 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 latter is the most used and states that yielding of a ductile material begins when the second invariant of deviatoric stress ... Example of a yield-safe region in ...
Other examples of sets of tensor invariants are the set of principal stresses (,,) or the set of kinematic invariants (,,). The Lode coordinate system can be described as a cylindrical coordinate system within principal stress space with a coincident origin and the z-axis parallel to the vector ( σ 1 , σ 2 , σ 3 ) = ( 1 , 1 , 1 ...