Information about Yield Surface
Yield surface is described in three dimensional space of stresses, and encompasses the elastic region of material behavior. The states of stress of material inside the yield surface are elastic, when the stress reaches this surface it reaches the yield point. Then the material behaviour becomes plastic, because the stress cannot cross this surface.
Useful means of describing yield surface include expressing it in the terms of principal stresses (
), or using stress invariants (
).
There are several different yield surfaces known in engineering, and those most popular are listed below.
Figure 1 shows the TG criterion 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 the material is subject to shearing and one of principal stresses becomes smaller (or larger) than the others, then the yield surface is reached and material enters the plastic domain.
Figure 2 shows the Tresca-Guest criterion in two dimensional space, it is a cross section of the prism along the
plane.
In the non-principal stresses, it takes the form of
Figure 3 shows the HMH criterion in the three dimensional space of principal stresses. It is a circular cylinder of infinite length, with the same angle to all three axes.
Figure 4 shows the Huber-Mises-Hencky criterion in two dimensional space compared with Tresca-Guest criterion. HMH is a cross section of this cylinder on the plane of, which produces an ellipse.
and 
which are the maximum values for compression and tension for given material. This model is often used to model concrete, soil or granular materials. This model is the first one that takes shearing into account. It is expressaed as follows:
To plot this surface on Fig. 5 the following formula was used:
Figure 5 shows Mohr-Coulomb criterion in three dimensional space of principal stresses. It is a conical prism. If
then it becomes Tresca-Guest criterion, thus 
determines the inclination angle of conical surface.
Figure 6 shows Mohr-Coulomb criterion in two dimensional space, it is a cross section of this conical prism on the plane of, which produces a shape shown below.
Figure 7 shows Drucker-Prager criterion in three dimensional space of principal stresses. It is a regular cone.
Figure 8 shows Drucker-Prager criterion in two dimensional space, it is a cross section of this cone on the plane of, which produces an ellipsioidal shape. It is compared here with Mohr-Colulomb criterion.
function. Thus cylinder's section along its axis is no longer a rectangle (or rather two parallel lines, since the cylinder has infinite length) but a parabola.
Useful means of describing yield surface include expressing it in the terms of principal stresses (
), or using stress invariants ().
There are several different yield surfaces known in engineering, and those most popular are listed below.
List of symbols used in this article
Following symbols are used below:
- principal stress in direction along first axis
- principal stress in direction along second axis
- principal stress in direction along third axis
- yield stress for single parametric yield surface
- - yield strength during compression
- - yield strength during tension
- ratio of yield strengths
- material cohesion
- stress coefficient
- - material stiffness for two parametric yield surface (
)
Tresca - Guest yield surface
This is the most simple yield surface, and it is taken to be the work of Henri Tresca. It is also referred as the TG criterion. In terms of the principal stresses it is expressed as
Figure 1 shows the TG criterion 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 the material is subject to shearing and one of principal stresses becomes smaller (or larger) than the others, then the yield surface is reached and material enters the plastic domain.
)
Figure 1: View of Tresca-Guest criterion in 3D space of principal stresses
Figure 2 shows the Tresca-Guest criterion in two dimensional space, it is a cross section of the prism along the
plane.
)
Figure 2: Tresca-Guest criterion in 2D space (math:4/233C33FC10B3CA5038AFF398.gif)
Huber - Mises - Hencky, also known as Prandtl - Reuss yield surface
This is another simple yield surface, which perhaps explains why it is credited to so many authors. Who is the real author depends on the university, although often it is credited to Maximilian Huber and Richard von Mises (see von Mises stress). It is also referred to as the HMH criterion. It is expressed in the principal stresses as
In the non-principal stresses, it takes the form of
Figure 3 shows the HMH criterion in the three dimensional space of principal stresses. It is a circular cylinder of infinite length, with the same angle to all three axes.
)
Figure 3: View of Huber-Mises-Hencky criterion in 3D space of principal stresses
Figure 4 shows the Huber-Mises-Hencky criterion in two dimensional space compared with Tresca-Guest criterion. HMH is a cross section of this cylinder on the plane of
, which produces an ellipse.
)
Figure 4: Comparison of Tresca-Guest and Huber-Mises-Hencky criteria in 2D space (math:4/233C33FC10B3CA5038AFF398.gif)
Mohr - Coulomb yield surface
It is a first two-parametric yield surface, the parameters are and which are the maximum values for compression and tension for given material. This model is often used to model concrete, soil or granular materials. This model is the first one that takes shearing into account. It is expressaed as follows:
To plot this surface on Fig. 5 the following formula was used:
Figure 5 shows Mohr-Coulomb criterion in three dimensional space of principal stresses. It is a conical prism. If
then it becomes Tresca-Guest criterion, thus determines the inclination angle of conical surface.
)
Figure 5: View of Mohr-Coulomb criterion in 3D space of principal stresses
Figure 6 shows Mohr-Coulomb criterion in two dimensional space, it is a cross section of this conical prism on the plane of
, which produces a shape shown below.
)
Figure 6: Mohr-Coulomb criterion in 2D space (math:4/233C33FC10B3CA5038AFF398.gif)
Drucker - Prager yield surface
This criterion is most often used for concrete, both normal and shear stresses are taken into account.
Figure 7 shows Drucker-Prager criterion in three dimensional space of principal stresses. It is a regular cone.
)
Figure 7: View of Drucker-Prager criterion in 3D space of principal stresses
Figure 8 shows Drucker-Prager criterion in two dimensional space, it is a cross section of this cone on the plane of
, which produces an ellipsioidal shape. It is compared here with Mohr-Colulomb criterion.
)
Figure 8: Drucker-Prager and Mohr-Coluomb criterions in 2D space (math:4/233C33FC10B3CA5038AFF398.gif)
Brestler - Pister criterion
This criterion is a first criterion that uses three parameters. It is similar to HMH criterion but additional parameter affects the cylinder radius using an function. Thus cylinder's section along its axis is no longer a rectangle (or rather two parallel lines, since the cylinder has infinite length) but a parabola.
Willam - Warnke criterion
This is the most advanced yield surface, it takes the idea from Brestler - Pister a bit further and applies it to Mohr-Colulomb criterion. The resulting surface is smooth (unlike Mohr-Colulumb) and has first and second derivative fully defined on every point of its surface which is an important property. This smoothness allows optimisations during calculations when searching for a yield point on the surface (using gradient method for instance).See also
- Henri Tresca
- von Mises stress
- Mohr-Coulomb theory
- Yield (engineering)
- Stress
- Strain
- Strain tensor
- Stress-energy tensor
- Stress concentration
- 3-D elasticity
Stress is a measure of force per unit area within a body. It is a body's internal distribution of force per area that reacts to external applied loads. Stress is often broken down into its shear and normal components as these have unique physical significance.
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yield strength or yield point of a material is defined in engineering and materials science as the stress at which a material begins to plastically deform. Prior to the yield point the material will deform elastically and will return to its original shape when the applied
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Cohesion or cohesive attraction or cohesive force in chemistry is the intermolecular attraction between like-molecules. Cohesion explains phenomena such as surface tension.
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Henri Edouard Tresca (October 12, 1814 РJune 21, 1885) was a French mechanical engineer, and a professor at the Conservatoire National des Arts et M̩tiers in Paris.
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prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. Thus these joining faces are parallelograms. All cross-sections parallel to the base faces are the same. A prism is a subclass of the prismatoids.
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Tytus Maksymilian Huber (also known as Maksymilian Tytus Huber, January 4 1872 - 1950) was a world renowned Polish mechanical engineer, educator and scientist.
He was an important member of the pre-war Polish scientific foundation, Kasa im. Józefa Mianowskiego.
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He was an important member of the pre-war Polish scientific foundation, Kasa im. Józefa Mianowskiego.
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Richard Edler von Mises (Lemberg(now Lviv) 19 April 1883 - Boston, 14 July 1953) was a scientist who worked on fluid mechanics, aerodynamics, aeronautics, statistics and probability theory.
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Von Mises stress, , or simply Mises stress, is a scalar function of the deviatoric components of the stress tensor that gives an appreciation of the overall magnitude of the shear components of the tensor.
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cylinder is a quadric surface, with the following equation in Cartesian coordinates:
This equation is for an elliptic cylinder, a generalization of the ordinary, circular cylinder (a = b).
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This equation is for an elliptic cylinder, a generalization of the ordinary, circular cylinder (a = b).
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ellipse (from the Greek ἔλλειψις, literally absence) is the locus of points on a plane where the sum of the distances from any point on the curve to two fixed points is constant.
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Concrete is a construction material that consists of cement (commonly Portland cement) as well as other cementitious materials such as fly ash and slag cement, aggregate (generally a coarse aggregate such as gravel limestone or granite, plus a fine aggregate such as sand or
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SOiL is a five-piece Hard Rock band from Chicago, Illinois, United States. They formed in 1997 and are still active. They are signed to DRT Entertainment and have released four albums, their most recent being True Self which was released in March 27 2006.
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A granular material is a conglomeration of discrete solid, macroscopic particles characterized by a loss of energy whenever the particles interact (the most common example would be friction when grains collide).
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Concrete is a construction material that consists of cement (commonly Portland cement) as well as other cementitious materials such as fly ash and slag cement, aggregate (generally a coarse aggregate such as gravel limestone or granite, plus a fine aggregate such as sand or
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Cone (from the Greek κώνος, Latin conu) is a basic geometrical shape. It may also refer to:
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- Cone (software), a text-based e-mail client and news client for Unix-like operating systems.
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parabola (from the Greek: παραβολή) (IPA pronunciation: /pəˈrab(ə)lə/
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Henri Edouard Tresca (October 12, 1814 РJune 21, 1885) was a French mechanical engineer, and a professor at the Conservatoire National des Arts et M̩tiers in Paris.
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Von Mises stress, , or simply Mises stress, is a scalar function of the deviatoric components of the stress tensor that gives an appreciation of the overall magnitude of the shear components of the tensor.
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Mohr-Coulomb theory is a mathematical model (see yield surface) describing the response of a material such as rubble piles or concrete to shear stress as well as normal stress.
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yield strength or yield point of a material is defined in engineering and materials science as the stress at which a material begins to plastically deform. Prior to the yield point the material will deform elastically and will return to its original shape when the applied
..... Click the link for more information.
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Stress is a measure of force per unit area within a body. It is a body's internal distribution of force per area that reacts to external applied loads. Stress is often broken down into its shear and normal components as these have unique physical significance.
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strain is the geometrical expression of deformation caused by the action of stress on a physical body. Strain is calculated by first assuming a change between two body states: the beginning state and the final state.
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The strain tensor, ε, is a symmetric tensor used to quantify the strain of an object undergoing a small 3-dimensional deformation:
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- the diagonal coefficients εii are the relative change in length in the direction of the i
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The stress-energy tensor (sometimes stress-energy-momentum tensor) is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics.
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A stress concentration (often called stress raisers or stress risers) is a location in an object where stress is concentrated. An object is strongest when force is evenly distributed over its area, so a reduction in area, e.g.
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3-D elasticity is one of three methods of structural analysis. This method is used for analyzing structures that behave in a linearly elastic fashion. There are 15 partial differential equations that must be simultaneously solved to get the state of stress at any point in an
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