Von Mises yield criterion
The equivalent stress is a term from the strength of materials and describes a fictional uniaxial tension, which is the same material stress like a real, multi-axial stress state.
Thus, the real three-dimensional stress state can be compared from normal stresses and shear stresses in all three spatial directions, directly with the parameters from the uniaxial tensile test (material characteristics, such as yield strength or tensile strength) in the component, consisting.
Basics
For a full description of the stress state in a component specifying the stress tensor ( tensor of 2nd stage) is necessary in general. This includes the general case six different voltage values (since the shear stresses are pairwise equal). Due to the transformation of the stress tensor in an excellent coordinate system ( the principal axis system ), the shear stresses to zero and three excellent (normal ) stresses ( principal stresses ) describe the stress state of the system are equivalent.
The elements of the vector of the principal stresses and the stress tensor can then be converted to a scalar, that is to satisfy two conditions:
- The one it is intended to describe the stress state as fully as possible (equivalence here can not be reached: are emerging information loss during the transition from the vector of the principal stresses for comparison voltage on )
- Secondly, it should definitely be a failure-relevant information.
The calculation rule for forming these scalar equivalent stress are called equivalent stress hypothesis or as failure rule. As part of a sustainability analysis comparing the reference voltage with allowable stresses. By selecting the hypothesis it implicitly contains the failure mechanism and is thus a value which expresses the exposure of the component under the given load. The choice of the equivalent stress hypothesis therefore always depends on the strength behavior of the material to be detected and the load case from (static, oscillating, impact ).
There are a number of hypotheses for calculating the equivalent stress. They are often combined in engineering mechanics, the term strength theories. The application depends on the material behavior and to some extent on the application area ( such as when a standard that a particular hypothesis calls ) from.
Most often, the distortion energy hypothesis is applied according to von Mises in engineering and construction. Except as set forth herein, there are other hypotheses.
Change in shape hypothesis ( von Mises )
After the change of shape hypothesis, even distortion energy hypothesis (in short: GEH ) called or Mises stress, failure of the structure occurs when the distortion energy exceeds a threshold (see also distortion or deformation ). This hypothesis is used for ductile materials (eg steel) under static and cyclic loading. The Mises stress is most commonly used in engineering and construction - for the most common materials ( not too brittle) under normal load (alternating, not jerky ) is the GEH used. Important application areas are the calculations of waves, which are claimed to both bending and torsion as well as the steel construction. The GO is constructed so that when the hydrostatic stress conditions ( equal tension in all three directions ) shows a reference voltage of zero. Because plastic flow of metals is isochoric and even extreme hydrostatic pressures do not affect the flow initiation ( experiments of Bridgman ).
Description in the general stress state:
Different notation:
Description of the main stress state:
, And are the principal stresses.
Description plane stress:
Description plane strain condition with:
Description in the GOCE:
The second invariant of the stress deviator is:
Shear stress hypothesis ( Tresca, Coulomb, Saint- Venant, Guest)
It is assumed that the failure of the material, the major principal stress difference is responsible (name in some FE programs: intensity). These principal stress difference represents twice the value of the maximum shear stress - thereby it is in tough material under static load, which failed by flow ( shear fracture ) applied. In Mohr's circle is the critical size of the diameter of the largest circle. The shear stress hypothesis is very general terms and in engineering application because of the formula set is easier to handle compared to GEH and you lie with her compared to Von Mises (GEH ) on the safe side (there are in any doubt somewhat larger values for the comparison voltage and thus a little more safety margin out ).
Spatial stress state:
, And are the principal stresses.
Plane stress:
Principal stress hypothesis ( Rankine )
It is assumed that the component due to the largest standard power fails. In Mohr's circle, the critical point is the maximum principal stress. The hypothesis is used for materials which with brittle fracture without flow fail:
- Brittle materials (eg cast iron or welded seams ) with predominantly static tensile
- Brittle and ductile materials under shock loading.
Spatial stress state:
Plane stress:
Square rotationally symmetric model ( Burzyński - Yagn )
With the approach
Follow models:
- Cone of Drucker-Prager ( Mirolyubov ) with,
- Paraboloid of Balandin ( Burzyński -Torre ) with,
- Ellipsoid with Beltrami,
- Ellipsoid of Schleicher,
- Hyperboloid of Burzyński - Yagn with,
- Single-shell hyperboloid.
The quadratic models can be explicitly resolve after what their practical use promoted.
The Poisson's ratio at train can be combined with
Calculate. The use of rotationally symmetric models for brittle failure
Has not been studied enough.
Combined rotationally symmetric model ( Huber)
The model of Huber consists of the ellipsoid of Beltrami
And an output coupled to him in cross-section cylinder
With the parameter.
The transition in the interface is continuous differentiable. The Poisson's ratios at train and print result to be
The model was developed in 1904. However, it sat at first not because it was (for example ) understood by several scientists as a discontinuous model.
Unified Strength Theory (Mao - Hong Yu )
The model of the Unified Strength Theory (UST ) consists of two hexagonal pyramids of Sayir which are rotated 60 ° to each other:
With and.
With results in the model of Mohr- Coulomb ( single -shear theory of Yu ) and follows the twin -shear theory of Yu ( cf. Pyramid of Haythornthwaite ).
The Poisson's ratios of the train and the pressure follow as
Geometric - mechanical model ( Altenbach - Bolchoun - Kolupaev )
Often the strength of hypotheses on the basis of the voltage angle
Formulated. Several models are isotropic material behavior in the approach
Summarized.
And the parameters describing the geometry of the surface in the plane. The restrictions
Arising from the Konvexitätsanforderung.
The parameters and define the location of the hydrostatic nodes. For materials that do not fail under the uniform 3D printing load (steel, brass, etc. ) concerns. For materials that fail under the uniform 3D printing ( rigid foams, ceramics, sintered materials) follows.
The integer powers and describing the curvature of the meridian. The meridian is a straight line and a parabola.