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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 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 ...
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:
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
Elastic response of rubber-like materials are often modeled based on the Mooney–Rivlin model. The constants , are determined by fitting the predicted stress from the above equations to the experimental data. The recommended tests are uniaxial tension, equibiaxial compression, equibiaxial tension, uniaxial compression, and for shear, planar ...
A two-dimensional flow that, at the highlighted point, has only a strain rate component, with no mean velocity or rotational component. In continuum mechanics, the strain-rate tensor or rate-of-strain tensor is a physical quantity that describes the rate of change of the strain (i.e., the relative deformation) of a material in the neighborhood of a certain point, at a certain moment of time.