Wikipedia / Yield surface
Wikipedia / Yield surface
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.
The yield surface is usually expressed in terms of (and visualized in) a three-dimensional principal stress), a two- or three-dimensional space spanned by stress invariants) or a version of the three-dimensional Haigh–Westergaard stress space. Thus we may write the equation of the yield surface (that is, the yield function) in the forms:
are the principal stresses.
are the second and third principal invariants of the deviatoric part of the Cauchy stress.
.
is the stress angle or Lode angle
Invariants used to describe yield surfaces
) of the Cauchy stress) of the deviatoric) of the Cauchy stress are defined as:
, and
is the identity matrix.
), are usually used to describe yield surfaces for cohesive frictional materials such as rocks, soils, and ceramics. These are defined as
is the equivalent stress makes the use of these quantities problematic in practice.
) which describe a cylindrical coordinate system (the Haigh–Westergaard coordinates). These are defined as:
plane is also called the Rendulic plane is sometimes called the Lode parameter was first given by Novozhilov V.V. in 1951, see also
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:
) are related by
Note. It is incorrect (false) information that Chakrabarty did define shear stress mode angle. The first work in which rational arguments and discussion is delivered why it is wise and useful, where it is elucidated that adopting definition of shear stress mode angle with sin(.) function physically means accepting pure shears as comparison reference states, and it is explained why it is very beneficial.
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–Guest (TG) criterion. In terms of the principal stresses the Tresca criterion is expressed as
is the tensile yield strength.
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 plane.
von Mises yield surface
The von Mises yield criterion is expressed in the principal stresses as
is the yield strength in uniaxial tension.
Figure 3 shows the von Mises yield surface in the three-dimensional space of principal stresses. It is a circular cylinder produces the elliptical shape of the yield surface.
Burzyński-Yagn criterion
This criterion
represents the general equation of a second order surface of revolution about the hydrostatic axis. Some special case are:
(Maxwell (1865), Huber (1904), von Mises (1913), Hencky (1924)),
(Botkin (1940), Drucker-Prager (1952), Mirolyubov (1953)),
(Burzyński (1928), Balandin (1937), Torre (1947)),
(Beltrami (1885)),
(Schleicher (1926)),
(Burzynski (1928), Yagn (1931)),
(Kuhn (1980))
(Filonenko-Boroditsch (1960), Gol’denblat-Kopnov (1968), Filin (1975)).
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 in the odd and even power should be introduced in the equation, e.g.:
Huber criterion
The Huber criterion consists of the Beltrami ellipsoid and a scaled von Mises cylinder in the principal stress space, see also
is continuously differentiable. The criterion represents the "classical view" with respect to inelastic material behavior:
and
The Huber criterion can be used as a yield surface with an empirical restriction for Poisson's ratio.
The modified Huber criterion, see also, cf.
consists of the Schleicher ellipsoid with the restriction of Poisson's ratio at compression
follows with the compression / tension relation
.
should be considered in these criteria.
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
.
. In Figure 6 Rr and Rc are used for Syc and Syt, respectively, in the formula.
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
.
Figure 7 shows Drucker–Prager yield surface in the three-dimensional space of principal stresses. It is a regular cone line, but usually selected by convention to be those in the compression regime. (biaxial fit). The Drucker-Prager yield criterion is also commonly expressed in terms of the material cohesion and friction angle.
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
gives the yield surface an ellipsoidal is the yield stress in biaxial compression, the parameters can be expressed as
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–Coulomb). 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.
Podgórski and Rosendahl trigonometric yield surfaces
, the Podgórski criterion reads
-plane
), Ivlev ) and also the cubic criterion of Sayir (the Ottosen criterion and the isotoxal (equilateral) hexagons of the Capurso criterion. The von Mises - Tresca transition . The isogonal (equiangular) hexagons of the Haythornthwaite criterion containing the Schmidt-Ishlinsky criterion (regular hexagon) cannot be described with the Podgórski criterion.
The Rosendahl criterion reads
-plane
), and also the bicubic criterion and the isotoxal dodecagons of the unified yield criterion of Yu . The isogonal dodecagons of the multiplicative ansatz criterion of hexagonal symmetry containing the Ishlinsky-Ivlev criterion (regular dodecagon) cannot be described by the Rosendahl criterion.
is relevant for theoretical investigations.
-substitution
which is sufficient for many applications, e.g. metals, cast iron, alloys, concrete, unreinforced polymers, etc.
Bigoni–Piccolroaz yield surface
The Bigoni–Piccolroaz yield criterion is a seven-parameter surface defined by
is the "meridian" function
is the "deviatoric" function
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.
-plane. They are subject to the constraints
which follow from the convexity condition. A more precise formulation of the third constraints is proposed in.
.
– a parabola.
Barlat's Yield Surface
For the anisotropic materials, depending on the direction of the applied process (e.g., rolling) the mechanical properties vary and, therefore, using an anisotropic yield function is crucial. Since 1989 Frederic Barlat has developed a family of yield functions for constitutive modelling of plastic anisotropy. Among them, Yld2000-2D yield criteria has been applied for a wide range of sheet metals (e.g., aluminum alloys and advanced high-strength steels). The Yld2000-2D model is a non-quadratic type yield function based on two linear transformation of the stress tensor:
:
are the transformed matrices (by linear transformation C or L):
where s is the deviatoric stress tensor.
for principal values of X’ and X”, the model could be expressed as:
and:
are eight parameters of the Barlat's Yld2000-2D model to be identified with a set of experiments.
See also
References