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A yield surface is a five-dimensional surface in the six-dimensional space of stresses. The yield surface is usually convex and the state of stress of inside the yield surface is elastic. When the stress state lies on the surface the material is said to have reached its yield point and the material is said to have become plastic. Further deformation of the material causes the stress state to remain on the yield surface, even though the shape and size of the surface may change as the plastic deformation evolves. This is because stress states that lie outside the yield surface are non-permissible in rate-independent plasticity, though not in some models of viscoplasticity.
Contents
- Invariants used to describe yield surfaces
- Examples of yield surfaces
- Tresca yield surface
- von Mises yield surface
- Burzyski Yagn criterion
- MohrCoulomb yield surface
- DruckerPrager yield surface
- BreslerPister yield surface
- WillamWarnke yield surface
- BigoniPiccolroaz yield surface
- Cosine Ansatz Altenbach Bolchoun Kolupaev
- References
The yield surface is usually expressed in terms of (and visualized in) a three-dimensional principal stress space (
Invariants used to describe yield surfaces
The first principal invariant (
where (
where
A related set of quantities, (
where
Another related set of widely used invariants is (
The
The principal stresses and the Haigh–Westergaard coordinates are related by
A different definition of the Lode angle can also be found in the literature:
in which case the ordered principal stresses (where
Examples of yield surfaces
There are several different yield surfaces known in engineering, and those most popular are listed below.
Tresca yield surface
The Tresca yield criterion is taken to be the work of Henri Tresca. It is also known as the maximum shear stress theory (MSST) and the Tresca– (TG) criterion. In terms of the principal stresses the Tresca criterion is expressed as
Where
Figure 1 shows the Tresca–Guest yield surface in the three-dimensional space of principal stresses. It is a prism of six sides and having infinite length. This means that the material remains elastic when all three principal stresses are roughly equivalent (a hydrostatic pressure), no matter how much it is compressed or stretched. However, when one of the principal stresses becomes smaller (or larger) than the others the material is subject to shearing. In such situations, if the shear stress reaches the yield limit then the material enters the plastic domain. Figure 2 shows the Tresca–Guest yield surface in two-dimensional stress space, it is a cross section of the prism along the
von Mises yield surface
The von Mises yield criterion is expressed in the principal stresses as
where
Figure 3 shows the von Mises yield surface in the three-dimensional space of principal stresses. It is a circular cylinder of infinite length with its axis inclined at equal angles to the three principal stresses. Figure 4 shows the von Mises yield surface in two-dimensional space compared with Tresca–Guest criterion. A cross section of the von Mises cylinder on the plane of
Burzyński-Yagn criterion
This criterion
represents the general equation of a second order surface of revolution about the hydrostatic axis. Some special case are:
The relations compression-tension and torsion-tension can be computed to
The Poisson's ratios at tension and compression are obtained using
For ductile materials the restriction
is important. The application of rotationally symmetric criteria for brittle failure with
has not been studied sufficiently.
The Burzyński-Yagn criterion is well suited for academic purposes. For practical applications, the third invariant of the deviator should be introduced in the equation, e.g.
Mohr–Coulomb yield surface
The Mohr–Coulomb yield (failure) criterion is similar to the Tresca criterion, with additional provisions for materials with different tensile and compressive yield strengths. This model is often used to model concrete, soil or granular materials. The Mohr–Coulomb yield criterion may be expressed as:
where
and the parameters
Figure 5 shows Mohr–Coulomb yield surface in the three-dimensional space of principal stresses. It is a conical prism and
Drucker–Prager yield surface
The Drucker–Prager yield criterion is similar to the von Mises yield criterion, with provisions for handling materials with differing tensile and compressive yield strengths. This criterion is most often used for concrete where both normal and shear stresses can determine failure. The Drucker–Prager yield criterion may be expressed as
where
and
Figure 7 shows Drucker–Prager yield surface in the three-dimensional space of principal stresses. It is a regular cone. Figure 8 shows Drucker–Prager yield surface in two-dimensional space. The elliptical elastic domain is a cross section of the cone on the plane of
Bresler–Pister yield surface
The Bresler–Pister yield criterion is an extension of the Drucker Prager yield criterion that uses three parameters, and has additional terms for materials that yield under hydrostatic compression. In terms of the principal stresses, this yield criterion may be expressed as
where
Willam–Warnke yield surface
The Willam–Warnke yield criterion is a three-parameter smoothed version of the Mohr–Coulomb yield criterion that has similarities in form to the Drucker–Prager and Bresler–Pister yield criteria.
The yield criterion has the functional form
However, it is more commonly expressed in Haigh–Westergaard coordinates as
The cross-section of the surface when viewed along its axis is a smoothed triangle (unlike Mohr–Coulumb). The Willam–Warnke yield surface is convex and has unique and well defined first and second derivatives on every point of its surface. Therefore, the Willam–Warnke model is computationally robust and has been used for a variety of cohesive-frictional materials.
Bigoni–Piccolroaz yield surface
The Bigoni–Piccolroaz yield criterion is a seven-parameter surface defined by
where
describing the pressure-sensitivity and
describing the Lode-dependence of yielding. The seven, non-negative material parameters:
define the shape of the meridian and deviatoric sections.
This criterion represents a smooth and convex surface, which is closed both in hydrostatic tension and compression and has a drop-like shape, particularly suited to describe frictional and granular materials. This criterion has also been generalized to the case of surfaces with corners.
Cosine Ansatz (Altenbach-Bolchoun-Kolupaev)
For the formulation of the strength criteria the stress angle
can be used.
The following criterion of isotropic material behavior
contains a number of other well-known less general criteria, provided suitable parameter values are chosen.
Parameters
which follow from the convexity condition. A more precise formulation of the third constraints is proposed in.
Parameters
The integer powers