Columb Failure
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Question 1
With the aid of annotated diagrams and the appropriate formula, describe:
Coulomb failure criterion
The coulomb failure criterion assumes, based on experimental data that failure in a rock or soil takes place along a plane due to the shear stress acting along that plane, and motion is assumed to be resisted by a frictional type force whose magnitude equals the normal stress acting along the plane multiplied by the coefficient of internal friction specific to each material. In a welded surface sliding movement is also resisted by an internal coheasive force, which is the resistance of a material to shear under zero normal load and is a result of inter-molecular forces, frictional forces (ect), and must be overcome to initiate failure. Experimental data relating to the failure criterion is gathered for a material using a triaxial testing (fig 1) , where (sigma 2) ( ) = (sigma 3) ( ), therefore a two dimensional failure can be assumed.
Figure 1 ( triaxial test with stress tensors)
The criterion can be expressed mathematically as failure will occur if:
And failure will not occur if:
Where:
The coloumb failure criterion expressed in the equation above defines a straight line on the (sigma shear) – (sigma normal) plane and has a slope equal to the coefficient of internal friction (FIG 2)
Figure 2 (graph of mohr diagram)
Where ( coeeficient of internal friction (
) = tan x angle of internal friction ( )
The form of criterion means that mohrs circle construction will be useful in analysis where the unstable and stable fields are highlighted in the failure envelope above, where it is thus refered to as a mohr diagram.
Limitations
Experimental data shows that the coulomb failure criterion does not apply close to the point of zero normal stress and into the tensile part of the mohr diagram, it also relies on the angle of internal friction which is physically meaningless for tensile normal stress indicating mathematically that the criterion is inappropriate. Another problem is encountered with high normal stress vaules where material begins to behave in a ductile manner at a point referred to as the yield stress and the criterion no longer applies where von misses failure criterion applies. Finally experiments under true triaxial conditions REFERENCE? (sigma 1) >(sigma 2) > (sigma 3) have revealed that the intermediate stress value can have a pronounced effect on value of (sigma 1) at failure, limiting its applicability to brittle deformation within the Earths crust where this situation can occur.
Griffith failure criterion
As touched on in section A coulombs failure criterion is successful in empirically accounting for macroscopic brittle behaviour of most geological materials under compression however it does not explain the physical mechanism of fracturing on a microscopic level and does not predict deformation under tensile stresses, this is where the Griffith failure criterion comes into play.
Calculations of the tensile strength of materials based on strength of atomic bonds in the constituents of a solid are approximately 2 orders of magnitude higher than experimental data, Griffiths failure criterion is based on the idea that all materials contains numerous microscopic and sub-microscopic cracks such as cracks pores and crystal lattice defects, that reduce substantially the strength of a material. In other words there are no materials that are entirely homogeneous.
Expression mathematically
In contrast to coulomb, Griffith found a non linear relationship between the principal stresses for critically stressed rocks, the relationship which is called the Griffith failure criterion where failure will occur if:
And failure will not occur if:
Where: T= critical tensile stress which is calculated experimentally and varies from rock to rock due to variations in their heterogeneities
This equation can also be represented in a mohr diagram (figure 3), where it defines a parabola where the tensile strength x 2 (2T) is the intersection with the horizontal axis therefore the tensile strength of a rock is half its cohesive strength, which is close to experimental data.
Fig 3 Graph
Limitations
For non porous rocks the Griffith failure criterion is reasonably realistic approximation for the compressional regime, however the Griffith criterion predicts that the uniaxial compressive strength should be 8 times the uniaxial tensile strength, while experiments