- Yield surface
A yield surface is a five-dimensional surface in the six-dimensional space of stresses. 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 surface itself may change shape and size as the plastic deformation evolves.
The yield surface is usually expressed in terms of (and visualized in) a three-dimensional principal stress space (), a two- or three-dimensional space spanned by stress invariants () or a version of the three-dimensional Haigh-Westergaard space. Thus we may write the equation of the yield surface (that is, the yield function) in the forms:
* where are the principal stresses.
* where is the first invariant of the Cauchy stress and are the second and third invariants of the deviatoric part of the Cauchy stress.
* where are scaled versions of and and is a function of .
* where are scaled versions of and , and is the Lode angle.Invariants used to describe yield surfaces
The first invariant of the Cauchy stress (), and the second and third invariants of the deviatoric part of the Cauchy stress () are defined as:where is the Cauchy stress and are its principal values, is the deviatoric part of the Cauchy stress and are its principal values.
The quantities are usually used to describe yield surfaces for cohesive frictional materials such as rocks, soils, and ceramics. These quantities are defined as:where is the equivalent stress.
The quantities describe a cylindrical coordinate system (the Haigh-Westergaard coordinates) and are defined as:The plane is also called the Rendulic plane. The angle is called the Lode angle [Lode, W. (1926). Versuche ueber den Einfuss der mitt leren Hauptspannung auf das Fliessen der Metalle Eisen Kupfer und Nickel. Zeitung Phys., vol. 36, pp. 913-939.] and the relation between and was first given by Nayak and Zienkiewicz in 1972 [Nayak, G. C. and Zienkiewicz, O.C. (1972). Convenient forms of stress invariants for plasticity. Proceedings of the ASCE Journal of the Structural Division, vol. 98, no. ST4, pp. 949–954.]
The principal stresses and the Haigh-Westergaard coordinates are related by:
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 [Tresca, H. (1864). "Mémoire sur l'écoulement des corps solides soumis à de fortes pressions." C.R. Acad. Sci. Paris, vol. 59, p. 754.] or "maximum shear stress" yield criterion is taken to be the work of
Henri Tresca . It is also referred as the Tresca-Guest (TG) criterion. The functional form of this yield criterion is:In terms of the principal stresses the Tresca criterion is expressed as: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 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 plane.von Mises yield surface
The
von Mises yield criterion (also known as Prandtl-Reuss yield criterion) has the functional form:This yield criterion is often credited toMaximilian Huber andRichard von Mises (seevon Mises stress ). It is also referred to as the Huber-von Mises-Hencky (HMH) criterion.The von Mises yield criterion is expressed in the principal stresses as:where is the yield stress in uniaxial tension.
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 produces the elliptical shape of the yield surface.
Mohr-Coulomb yield surface
The Mohr-Coulomb yield (failure) criterion is a two-parameter yield criterion whichhas the functional form: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 and are the yield (failure) stresses of the material in uniaxial compression and tension, respectively. If then the Mohr-Coulomb criterion reduces to the Tresca-Guest criterion.
Figure 5 shows Mohr-Coulomb yield surface in the three-dimensional space of principal stresses. It is a conical prism and determines the inclination angle of conical surface. Figure 6 shows Mohr-Coulomb yield surface in two-dimensional stress space. It is a cross section of this conical prism on the plane of .
The following formula was used to plot the surface in Fig. 5 ::
Drucker-Prager yield surface
The Drucker-Prager yield criterion has the function form: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 are the uniaxial yield stresses in compression and tension respectively.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 ellipsoidal-shaped elastic domain is a cross section of the cone on the plane of and encloses the elastic domain for the Mohr-Coulomb yield criterion.
Bresler-Pister yield surface
The Bresler-Pister yield criterion is an extension of the Drucker-Prager yield criterion that uses three parameters.
The Bresler-Pister yield surface has the functional form:In terms of the principal stresses, this yield criterion may be expressed as:where are material constants. The additional parameter gives the yield surface a ellipsoidal cross section when viewed from a direction perpendicular to its axis. If is the yield stress in uniaxial compression, is the yield stress in uniaxial tension, and 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-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.
References
ee also
*
Yield (engineering)
*Plasticity (physics)
* Stress
*Henri Tresca
*von Mises stress
*Mohr-Coulomb theory
* Strain
*Strain tensor
*Stress-energy tensor
*Stress concentration
*3-D elasticity
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