Equivalent stress

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Equivalent stress
Tresca and Mises strength criterion in the stress area

The equivalent stress is a term from the strength of materials . This denotes a fictitious uniaxial stress which , based on a certain material-mechanical or mathematical criterion, represents a hypothetically equivalent material stress as a real, multi-axis stress state .

Based on the comparison stress, the real, generally three-dimensional stress state in the component in the strength or yield condition can be compared with the characteristic values ​​from the uniaxial tensile test (material characteristic values, e.g. yield point or tensile strength ).

Basics

To fully describe the stress state in a component, it is generally necessary to specify the stress tensor (symmetrical tensor, 2nd level). In the general case (equilibrium of forces and moments) this contains six different stress values (since the shear stresses assigned to one another are equal). By transforming the stress tensor into an excellent coordinate system (the principal axis system ), the shear stresses become zero and three exceptional (normal) stresses (the principal stresses ) describe the stress state of the system in an equivalent manner .

The elements of the vector of the principal stresses or the stress tensor can now be converted into a scalar that should meet two conditions:

  • On the one hand, it should describe the stress state as comprehensively as possible (equivalence can no longer be achieved here: information is always lost during the transition from the vector of the principal stresses to the equivalent stress)
  • on the other hand, it should definitely represent failure-relevant information.
Areas of application of strength hypotheses. SH: shear stress hypothesis, GEH: shape change hypothesis, NH: normal stress hypothesis.

The calculation rule for forming this scalar equivalent stress are called equivalent stress hypothesis or as failure rule . As part of a load-bearing capacity analysis, the equivalent stress is compared with the permissible stresses. Due to the choice of the hypothesis, it implicitly contains the failure mechanism and is therefore a value that expresses the risk to the component under the given load. The choice of the respective equivalent stress hypothesis therefore always depends on the strength behavior of the material to be verified and on the load case (static, oscillating, impact).

There are a number of hypotheses for calculating the equivalent stress. In technical mechanics, they are often summarized under the term strength hypotheses. The application depends on the material behavior and partly also on the area of ​​application (if a standard requires the use of a certain hypothesis).

The von Mises shape change energy hypothesis is most commonly used in mechanical engineering and construction . There are other hypotheses besides those mentioned here.

Shape change hypothesis (von Mises)

The Mises equivalent stress can be determined graphically using the
Mohr stress circles

According to the shape change hypothesis , also known as the shape change energy hypothesis (short: GEH ) or Mises equivalent stress according to Richard von Mises , failure of the component occurs when the shape change energy exceeds a limit value (see also distortion or deformation ). This hypothesis is used for tough materials (e.g. steel ) under static and changing loads. The Mises equivalent stress is most commonly used in mechanical engineering and construction - the GEH can be used for most common materials (not too brittle) under normal loads (alternating, not jerky). Important areas of application are the calculation of shafts that are subjected to both bending and torsion , as well as steel construction . The GEH is designed in such a way that hydrostatic stress states ( stresses of the same magnitude in all three spatial directions) result in a comparison stress of zero. The plastic flow of metals is isochoric and even extreme hydrostatic pressures have no influence on the start of flow (experiments by Bridgman).

Description in general stress state:

other notation:

Description in the main stress state:

, and are the principal stresses.

Description in the plane stress state:

Description in the plane state of distortion with:

Description in invariant representation:

where the second invariant of the stress deviator is:

The shape change hypothesis represents a special case of the Drucker-Prager flow criterion, in which the limit stresses for compression and tension are equal.

Shear stress hypothesis (Tresca, Coulomb, Saint-Venant, Guest)

It is assumed that the failure of the material is due to the greatest principal stress difference (designation in some FE programs : intensity). This main stress difference corresponds to twice the value of the maximum shear stress - it is therefore applied to tough material under static load, which fails due to flow (sliding fracture). In Mohr's circle of tension , the critical size is the diameter of the largest circle. The shear stress hypothesis is also used in general in mechanical engineering , since the formula apparatus is easier to use compared to GEH and you are on the safe side with it compared to Von Mises (GEH) (in case of doubt, slightly larger values ​​for the equivalent stress are used and thus a little more safety reserves).

Spatial state of tension:

, and are the principal stresses.

Level stress state (provided and have different signs):

Principal normal stress hypothesis (Rankine)

It is assumed that the component fails due to the greatest normal stress. In Mohr's stress circle , the critical point is the maximum principal stress. The hypothesis is used for materials that fail with separation fracture , without flow:

Spatial state of tension:

For

otherwise

For

Level stress state:

Quadratic rotationally symmetric criterion (Burzyński-Yagn)

With the approach

follow the criteria:

- Printer-Prager cone (Mirolyubov) with ,

- Balandin Paraboloid (Burzyński-Torre) with ,

- Beltrami ellipsoid with ,

- Schleicher ellipsoid with ,

- Burzyński-Yagn hyperboloid with ,

- single-shell hyperboloid.

The quadratic criteria can be resolved explicitly according to what promoted their practical use.

The Poisson's ratio in tension can be changed with

to calculate. The application of rotationally symmetric criteria for brittle failure

has not been adequately investigated.

Combined rotationally symmetrical criterion (Huber)

Huber's criterion consists of the Beltrami ellipsoid

For

and a cylinder coupled to it in section

For

with the parameter .

The transition in the cut is continuously differentiable. The transverse contraction numbers for tension and compression result from

The criterion was developed in 1904. However, it did not catch on initially, as it was understood by several scientists as a discontinuous model.

Unified Strength Theory (Mao-Hong Yu)

The Unified Strength Theory (UST) consists of two hexagonal pyramids of Sayir, which are rotated by 60 ° against each other:

with and .

With the criterion of Mohr-Coulomb (single-shear theory by Yu), with the Pisarenko-Lebedev criterion and with follows the twin-shear theory of Yu (cf. pyramid by Haythornthwaite).

The Poisson's contraction numbers when pulling and pushing follow as

Cosine approach (Altenbach-Bolchoun-Kolupaev)

Often the strength hypotheses are made using the stress angle

formulated. Several criteria of isotropic material behavior are being considered

summarized.

The parameters and describe the geometry of the surface in the plane. You need the terms

meet, which result from the convexity requirement. An improvement to the third condition is suggested in FIG.

The parameters and describe the position of the points of intersection of the flow surface with the hydrostatic axis (space diagonal in the main stress space). These intersections are called hydrostatic nodes. For the materials that do not fail under the even 3D printing load (steel, brass, etc.), results . For the materials that fail under uniform 3D printing (hard foams, ceramics, sintered materials), the following applies .

The integer powers and , describe the curvature of the meridian. The meridian is with a straight line and with a parabola.

literature

  • J. Sauter, N. Wingerter: New and old strength hypotheses . (= VDI progress reports. Series 1. Volume 191). VDI-Verlag, Düsseldorf 1990, ISBN 3-18-149101-2 .
  • S. Sähn, H. Göldner: Breakage and assessment criteria in strength theory. 2nd Edition. Fachbuchverlag, Leipzig 1993, ISBN 3-343-00854-0 .
  • H. Mertens: For the formulation of strength hypotheses for multi-axis, phase-shifted vibration loads. In: Z. angew. Math. And Mech. Volume 70, No. 4, 1990, pp. T327-T329.

Individual evidence

  1. Christian Hellmich and others: Script for exercises from strength theory . In: Strength teaching script for civil engineers . tape 2017/18 , no. 202.665 . Institute of Mechanics for Materials and Structures, Vienna University of Technology; Vienna, January 2018 ( tuwien.ac.at ).
  2. ^ Frank Faulstich: Drucker-Prager equivalent voltage. Retrieved April 2, 2020 .
  3. lecture on Plasizität the University of Siegen. University of Siegen, accessed April 2, 2020 .
  4. L. Issler, H. Ruoß, P. Häfele: Strength theory - basics . Springer, Berlin / Heidelberg 2003, ISBN 3-540-40705-7 , pp. 178 .
  5. ^ W. Burzyński: About the hypotheses of exertion. In: Schweizerische Bauzeitung. Volume 94, No. 21, 1929, pp. 259-262.
  6. NM Beljaev: Strength of materials. Mir Publ., Moscow 1979.
  7. MT Huber: The specific deformation work as a measure of effort. Czasopismo Techniczne, Lwow 1904.
  8. H. Ismar, O. Mahrenholz: Technical Plastomechanics. Vieweg, Braunschweig 1979.
  9. M.-H. Yu: Unified Strength Theory and its Applications. Springer, Berlin 2004.
  10. M. Sayir: On the flow condition of the plasticity theory. In: Ing. Arch. 39, 1970, pp. 414-432.
  11. H. Altenbach, A. Bolchoun, VA Kolupaev: Phenomenological Yield and Failure Criteria. In: H. Altenbach, A. Öchsner (Ed.): Plasticity of Pressure-Sensitive Materials. (= ASM series ). Springer, Heidelberg 2013, pp. 49–152.

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