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## Mohr Coulomb criteria for ductile materials

The Mohr-Coulomb criteria is based on the internal friction that happens among the material particles.

This criteria has its origin from the Mohr break theory. This theory is based on a set of hypothesis and the is based on a set of observations carried out by Mohr at the beginning of the 20th century. It is also based on the Mohr circles and ideas related to them.

The main idea of this theory implies finding how to calculate the creep stress for a material, when the results for the creep tension, creep stress and creep shear stress trial is known. With this data, the Mohr circles are plotted, and a enfolding curve for the three circles is plotter. This enfolding curve can be a line, a parabola or a curve of any shape.

According to this theory, the collapse of the material takes places if the tangent stress and the normal stress in a point of the material reach a critical combination of stresses:

This equation represents two curves in the Mohr plane. The state will be safe if every Mohr circle, obtained from the main stresses are inside the curves determined by the latter equation. The state will not be safe if any Mohr circle is tangent to any curve.

The curves defined by the latter equation are called fail Mohr enfondings.

The latter expression can explicitly be expressed if the fail Mohr enfolding are lines:

Where,

c          the cohesion

μ          The internal frictional coefficient, μ = tg φ

It is convenient to express the Mohr circle parametrically:

Where,

σm        Mean stress

τm        Mean Shear stress

α          Angle between the fail plane and the axis with minor stress.

The yield condition will be

And taking into account,

The following expressions are obtained,

Consequently, the yield criteria will be,

And, expressed in terms of the main stresses,

Consequently, the yield criteria for compression is,

And the yield criteria for tension is,

The yield surfaces are surfaces that intersect forming an hexagonal pyramid.

This criteria is applicable to fragile materials. Its main characteristics are the following,

• It is a simplification of the more general Mohr theory
• It is widely used
If the cohesion coefficient and the internal frictional angle are kept constant, the latter equation is transformed into

Where