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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.
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.
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 ...
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
Surfaces on which the invariants , , are constant. Plotted in principal stress space. The first principal invariant of the Cauchy stress (), and the second and third principal invariants (,) of the deviatoric part of the Cauchy stress are defined as:
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]
[1] are a set of tensor invariants that span the space of real, symmetric, second-order, 3-dimensional tensors and are isomorphic with respect to principal stress space. This right-handed orthogonal coordinate system is named in honor of the German scientist Dr. Walter Lode because of his seminal paper written in 1926 describing the effect of ...