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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 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
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 ...
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]
As it is a second order tensor, the stress deviator tensor also has a set of invariants, which can be obtained using the same procedure used to calculate the invariants of the stress tensor. It can be shown that the principal directions of the stress deviator tensor s i j {\displaystyle s_{ij}} are the same as the principal directions of the ...
Considering that polymers have a viscoelastic behaviour an effect of loading rates and temperatures on shear yield stress and on crazing yield is observed. [ 59 ] [ 60 ] When the loading conditions ( T , ϵ ˙ {\displaystyle T,{\dot {\epsilon }}} ) are such that the tensile stress for shear yielding is lower than the crazing stress no crazing ...
where is the first invariant of the Cauchy stress, is the second invariant of the deviatoric part of the Cauchy stress, and ,, are material constants. Yield criteria of this form have also been used for polypropylene [2] and polymeric foams. [3]