8.2: Cutting Process and Failure Criteria
 Page ID
 29465
In granular materials a number of failure mechanisms can be distinguished. For clarity of definitions, the following definitions are used:

Flow Type. Failure is based on plastic shear failure. Nondestructive, continues. Both the stressstrain curve according to Figure 82 and the nondestructive plastic deformation show ductile behavior. This type of failure will only occur at very high pressures and/or temperatures. The flow of magma is an example of this.

Tear Type: UCS/BTS=large. Failure based on 100% tensile failure. This type of failure will occur when the UTSBTS absolute value is small compared to the UCS value. This is a discontinues mechanism.

Chip Type: UCS/BTS=medium. Failure based on a combination of shear failure and tensile failure, with a crushed zone near the tool tip. The fractions of shear failure and tensile failure depend on the UCS/BTS ratio. A large ratio results in more tensile failure, a small ratio in more shear failure. This is a discontinues mechanism.

Shear Type: UCS/BTS=small. Failure based on 100% shear failure. This type of failure occurs when the UTSBTS value is larger and the normal stresses in the shear plane are high, usually at larger blade angles.
This is a discontinues mechanism.

Crushed Type: Cataclastic failure based on shear, similar to the Flow Type and the Shear Type like in sand.
The Crushed Type is based on cataclastic failure, disintegration of the grain matrix. This mechanism will be identified as pseudoductile since it shows ductile behavior in the stressstrain curve of Figure 82, but it is destructive and not plastic.
When cutting in dredging practice, blade or pick point angles of about 60 degrees are used. With these blade angles often the Chip Type of cutting mechanism occurs. Smaller blade angles may show the Tear Type cutting mechanisms, while larger blade angles often show the Shear Type of cutting mechanism. The higher the normal stresses in the rock cut, the less likely the occurrence of tensile failure.
When the pick point starts penetrating the rock, usually very high normal stresses occur in front and below the tip of the pick point, resulting in crushing of the rock. Destroying the grain matrix. In a stressstrain diagram this behavior is ductile, but since its also destructive its named pseudoductile. Now if the layer thickness is very small, like in oil drilling, the crushed zone may reach the surface and the whole process is of the Crushed Type. If the layer cut is thicker, like in dredging, the Chip Type cutting mechanism may occur, a combination of mechanisms. In the crushed zone and the intact rock a shear plane can be identified based on the minimum deformation work principle. When the pick point progresses, the shear stress on this shear plane increases. When the shear stress exceeds the shear strength (cohesion) a brittle shear crack will occur. It is not necessary that the shear stress exceeds the shear strength over the full length of the shear plane, it only has to exceed the shear strength at the beginning of the shear crack as in the Nishimatsu (1972) approach. When the pick point progresses, the normal and shear stresses increase, resulting in a Mohr circle with increasing radius. Now if the radius increases faster than the normal stress at the center of the Mohr circle, the minimum principal stress decreases and may even become negative. When it becomes negative it may become smaller than the negative tensile strength, resulting in tensile failure.
So in time it starts with a crushed zone, then a shear plane with possibly shear failure and than possibly tensile failure. If the tensile strength is large, it is possible that only shear failure occurs. If the tensile strength is small, it is possible that only tensile failure occurs. Crushing will start if locally a certain criterion is exceeded. Often the Mogi (1966) criterion is applied, giving a certain ratio between the maximum principal stress and the minimum principal stress. Ratio’s used are 3.4 for sandstone and 4.2 for limestone, while Verhoef (1997) found 6 for limestone. Of course crushing does not start instantly, but gradually, based on the structure of the rock, especially the distribution of the microcracks and the skeleton. With the hypothesis that crushing starts where the rock is the weakest, one may assume that crushing starts at the scale of the microcracks, giving relatively large particles still consisting of many grains. With increasing normal stress these particles will also be fragmented into smaller particles. This process will go on until the smallest possible particles, the rock grains, result. Up to the Mogi (1966) criterion intact rock is assumed, however some fracturing or crushing may already have taken place.
From the perspective of the angle of internal friction, one may assume that the angle of internal friction is based on the internal structure of the rock, and as long as the rock is intact, the angle of internal friction may change slightly based on the stress situation, but not to much. However, when fracturing and crushing starts, the internal structure of the rock is changing and this will result in a decreasing angle of internal friction. Decreasing until the angle of internal friction of the smallest particles, the rock grains is reached at high confining pressures.
Verhoef (1997) shows a complete failure envelope of intact rock, including Mogi’s brittleductile transition. Vlasblom (20032007) refers to this failure envelope. Figure 84 shows this failure envelope, where the maximum normal stress is based on a hydrostatic compression test. So based on hydrostatic pressure, the material is crushed, without the presence of shear. This hydrostatic compressive strength (HCS) is a few times the UCS value of the rock. In the figure HCS+UTS=3.5·UCS. Not all rocks show this kind of behavior however. It is important to know that this envelope is based on triaxial tests on intact rock.
Verhoef (1997) also shows in figure D3 a different failure envelope beyond the brittleductile transition point, which is more related to the cutting process. Beyond this transition point the crushed rock still has a certain internal friction angle, which will be discussed later and is shown in Figure 814 and Figure 815.
It is thus very important to determine the failure criterion envelope based on tests where shear failure occurs.
Figure 85 shows how a failure envelope can be constructed by connecting failure points of different stress situations. The figure shows the UTS, BTS and UCS Mohr circles, the Mohr circle at the Mogi criterion, the Mohr circle of a hydrostatic compression test and three additional Mohr circles. Connecting the failure points gives the failure curve. Surrounding the Mohr circles gives the envelope where Mohr circles have to stay inside to prevent failure. At confining stresses exceeding the Mogi point the two envelopes are slightly different.
As mentioned, the apparent shear strength and the internal friction angle of the intact and the crushed rock may differ. In the case where the Mogi criterion describes the shear strength and the angle of internal friction of the crushed rock, the failure curve for higher normal stresses may be a straight line tangent to the Mogi criterion point. Figure 86 shows this type of behavior. The Zijsling (1987) experiments at very high confining pressures show this type of behavior for cutting loads in Mancos Shale. The experiments of Zijsling (1987) will be discussed in chapter 9.
It is however also possible that the shear strength and the internal friction angle of the crushed rock decrease to a certain minimum with increasing normal stresses larger than the Mogi point to a point A or B in Figure 87. For higher normal stresses the failure curve will follow a straight line as is shown in the figure. The Zijsling (1987) experiments at very high confining pressures show this type of behavior for cutting loads in Pierre Shale.
When increasing the bottomhole pressure (confining pressure) from 0 MPa to 50 MPa, first the cutting forces and thus the normal stresses and shear stresses increase up to a maximum, after which the cutting forces decrease, but at a certain bottomhole pressure this decrease stops and the cutting forces increase slightly with further increasing bottomhole pressure. So there was still an internal friction angle, but very small. The bottomhole pressure is a good indication of the confining pressure. The Zijsling (1987) experiments did show that the material was crushed.
It should be mentioned that the layer thickness was very small in these experiments, resulting in a crushed zone reaching to the surface. In other words, the rock was crushed completely.
8.2.1. Some Relations
The relation between shear strength (cohesion) c, internal friction angle φ and the minimum and maximum principal stresses can be derived according to, using the basic MohrCoulomb relations:
\[\ \tau=\mathrm{c}+\sigma \cdot \tan (\varphi)\tag{81}\]
And:
\[\ \tau=\frac{\sigma_{\max }\sigma_{\min }}{2} \cdot \cos (\varphi) \quad\quad\quad\sigma=\frac{\sigma_{\max }+\sigma_{\min }}{2}\frac{\sigma_{\max }\sigma_{\min }}{2} \cdot \sin (\varphi)\tag{82}\]
This gives:
\[\ \frac{\sigma_{\max }\sigma_{\min }}{2} \cdot \cos (\varphi)=\mathrm{c}+\left(\frac{\sigma_{\max }+\sigma_{\min }}{2}\frac{\sigma_{\max }\sigma_{\min }}{2} \cdot \sin (\varphi)\right) \cdot \tan (\varphi)\tag{83}\]
Multiplying with cos(φ) and reorganizing gives:
\[\ \begin{array}{left}\frac{\sigma_{\max }\sigma_{\min }}{2} \cdot \sin ^{2}(\varphi)+\frac{\sigma_{\max }\sigma_{\min }}{2} \cdot \cos ^{2}(\varphi)\\
=\mathrm{c} \cdot \cos (\varphi)+\frac{\sigma_{\max }+\sigma_{\min }}{2} \cdot \sin (\varphi)\\
\frac{\sigma_{\max }\sigma_{\min }}{2}=\mathrm{c} \cdot \cos (\varphi)+\frac{\sigma_{\max }+\sigma_{\min }}{2} \cdot \sin (\varphi)\\
\sigma_{\max } \cdot(1\sin (\varphi))=2 \cdot \mathrm{c} \cdot \cos (\varphi)+\sigma_{\min } \cdot(1+\sin (\varphi))\\
\sigma_{\max }=\frac{2 \cdot \mathrm{c} \cdot \cos (\varphi)+\sigma_{\min } \cdot(1+\sin (\varphi))}{(1\sin (\varphi))}\end{array}\tag{84}\]
This equation can also be written as:
\[\ \sigma_{\max }=\sigma_{\min } \cdot \tan ^{2}\left(\frac{\pi}{4}+\frac{\varphi}{2}\right)+2 \cdot \mathrm{c} \cdot \tan \left(\frac{\pi}{4}+\frac{\varphi}{2}\right)\tag{85}\]
This relation is valid for all linear failure criteria with a cohesion c and an internal friction angle φ. Now if two Mohr circles are found with index 1 and 2. Index 1 for the smallest circle and index 2 for the largest circle, the following relation is valid in relation to the failure curve and internal friction angle:
\[\ \frac{1+\sin (\varphi)}{1\sin (\varphi)}=\frac{\sigma_{\max , 2}\sigma_{\max , 1}}{\sigma_{\min , 2}\sigma_{\min , 1}}=\mathrm{r}\tag{86}\]
This gives:
\[\ \sin (\varphi)=\frac{\mathrm{r}1}{\mathrm{r}+1} \quad\text{ and }\quad \cos (\varphi)=\frac{2 \cdot \sqrt{\mathrm{r}}}{\mathrm{r}+1} \quad\text{ and }\quad \tan (\varphi)=\frac{\mathrm{r}1}{2 \cdot \sqrt{\mathrm{r}}}\tag{87}\]
Once the internal friction angle is found, the cohesion can be determined as:
\[\ \mathrm{c}=\frac{\mathrm{U} \mathrm{C S}}{2} \cdot\left(\frac{1\sin (\varphi)}{\cos (\varphi)}\right)=\frac{\mathrm{U} \mathrm{C S}}{2 \cdot \sqrt{\mathrm{r}}}\tag{88}\]
So the MohrCoulomb relation is:
\[\ \tau=\frac{\mathrm{U C S}}{2 \cdot \sqrt{\mathrm{r}}}+\sigma \cdot \frac{\mathrm{r}1}{2 \cdot \sqrt{\mathrm{r}}}=\frac{\mathrm{U} \mathrm{C S}+\sigma \cdot(\mathrm{r}1)}{2 \cdot \sqrt{\mathrm{r}}}\tag{89}\]
8.2.2. Brittle versus Ductile
The terms ductile failure and brittle failure are often used in literature for the failure of materials with shear strength and tensile strength, but what do the words ductile and brittle mean?
In materials science, ductility is a solid material's ability to deform under tensile stress; this is often characterized by the material's ability to be stretched into a wire. Malleability, a similar property, is a material's ability to deform under compressive stress; this is often characterized by the material's ability to form a thin sheet by hammering or rolling. Both of these mechanical properties are aspects of plasticity, the extent to which a solid material can be plastically deformed without fracture. Ductility and malleability are not always coextensive – for instance, while gold has high ductility and malleability, lead has low ductility but high malleability. The word ductility is sometimes used to embrace both types of plasticity.
A material is brittle if, when subjected to stress, it breaks without significant deformation (strain). Brittle materials absorb relatively little energy prior to fracture, even those of high strength. Breaking is often accompanied by a snapping sound. Brittle materials include most ceramics and glasses (which do not deform plastically) and some polymers, such as PMMA and polystyrene. Many steels become brittle at low temperatures (see ductilebrittle transition temperature), depending on their composition and processing. When used in materials science, it is generally applied to materials that fail when there is little or no evidence of plastic deformation before failure. One proof is to match the broken halves, which should fit exactly since no plastic deformation has occurred. Generally, the brittle strength of a material can be increased by pressure. This happens as an example in the brittleductile transition zone at an approximate depth of 10 kilometers in the Earth's crust, at which rock becomes less likely to fracture, and more likely to deform ductile.” (Source Wikipedia).
Rock has both shear strength and tensile strength and normally behaves brittle. If the tensile strength is high the failure is based on brittle shear, but if the tensile strength is low the failure is brittle tensile. In both cases chips break out giving it the name Chip Type. So rock has true brittle behavior. Under hyperbaric conditions however, the pore under pressures will be significant, helping the tensile strength to keep cracks closed. The result is a much thicker crushed zone that may even reach the surface. Crushing the rock is called cataclastic behavior. Since the whole cutting process is dominated by the crushed zone, this is named the Crushed Type. Due to the high pore under pressures the crushed material sticks together and visually looks like a ductile material. That’s the reason why people talk about ductile behavior of hyperbaric rock. In reality it is cataclastic behavior, which could also be named pseudoductile behavior.
Now whether the high confining pressure result from a high hyperbaric pressure or from the cutting process itself is not important, in both cases the pseudoductile behavior may occur. Figure 82 shows the stressstrain behavior typical for brittle and ductile behavior. Based on this stressstrain behavior the term ductile is often used for rock, but as mentioned before this is the result of cataclastic failure.
Gehking (1987) stated that pseudoductile behavior will occur when the ratio UCS/BTS<9. Brittle behavior will occur when the ratio UCS/BTS>15. For 9<UCS/BTS<15 there is a transition between brittle and pseudoductile. The geometry of the cutting equipment and the operational conditions are not mentioned by Gehking (1987).
Mogi (1966) found a linear relation between the minimum and maximum principal stress at the transition brittle to pseudoductile failure. For sandstone he found σ_{max}=3.4·σ_{min}, and for limestone σ_{max}=4.2·σ_{min}. Those values give an indication, since other researchers found σ_{max}>6·σ_{min} (Verhoef, 1997). Now assuming σ_{max}=α·σ_{min} and combining this with Hoek & Brown (1988), gives:
\[\ \begin{array}{left}\sigma_{\min }=\mathrm{UCS} \cdot \frac{\mathrm{m}+\sqrt{\mathrm{m}^{2}+4 \cdot(\alpha1)^{2}}}{2 \cdot(\alpha1)^{2}}\\
\sigma_{\max }=\alpha \cdot \mathrm{UCS} \cdot \frac{\mathrm{m}+\sqrt{\mathrm{m}^{2}+4 \cdot(\alpha1)^{2}}}{2 \cdot(\alpha1)^{2}}\end{array}\tag{810}\]
This gives for the center of the Mohr circle:
\[\ \begin{array}{left}\sigma_{\text {center }}=(\alpha+1) \cdot \frac{\text { UCS }}{2} \cdot \frac{\mathrm{m}+\sqrt{\mathrm{m}^{2}+4 \cdot(\alpha1)^{2}}}{2 \cdot(\alpha1)^{2}}\\
\tau_{\max }=(\alpha1) \cdot \frac{\mathrm{UCS}}{2} \cdot \frac{\mathrm{m}+\sqrt{\mathrm{m}^{2}+4 \cdot(\alpha1)^{2}}}{2 \cdot(\alpha1)^{2}}\end{array}\tag{811}\]
Figure 817 shows the Mogi criterion both for the top of the Mohr circle curve and the failure curve. Left of the Mogi criterion point there will be brittle failure, on the right there will be pseudoductile failure. When the coefficient α increases, the Mogi points move to the left.
In the case of a straight failure plane this gives for the normal and shear stress:
\[\ \sigma=\frac{\frac{\mathrm{c}}{\cos (\varphi)} \cdot\left(\frac{\alpha+1}{\alpha1}\sin (\varphi)\right)}{1\frac{\tan (\varphi)}{\cos (\varphi)} \cdot\left(\frac{\alpha+1}{\alpha1}\sin (\varphi)\right)} \quad\text{ and }\quad \tau=\mathrm{c + \sigma \cdot \operatorname { tan } ( \varphi )}\tag{812}\]
Which is also shown in Figure 817. If the angle of internal friction is to high, the brittleductile transition will never be reached. The criterion for this is:
\[\ \frac{\alpha1}{\alpha+1}>\sin (\varphi)\tag{813}\]
8.2.3. Based on UTS and UCS
Here a linear envelope tangent to the UTS and the UCS Mohr circles is assumed, based on the assumption that the failure curve always has to be tangent to at least two Mohr Coulomb circles. This gives for the principal stresses:
\[\ \begin{array}{left}\sigma_{\min , 1}=\text{UTS}\\
\sigma_{\max , 1}=0\\
\sigma_{\min , 2}=0\\
\sigma_{\max , 2}=\text{UCS}\\
\mathrm{r=\frac{U C S0}{0U T S}=\frac{U C S}{U T S}=m}\end{array}\tag{814}\]
This method results in a rather high value for the internal friction angle and consequently a rather low value for the shear strength (cohesion). To find a good estimate for the internal friction angle, there should be two Mohr circles based on shear failure. In this case one circle is based on shear failure, but the other circle is based on tensile failure. So this method is rejected.
Figure 89 shows the Mohr circles for UTS, BTS and UCS for UCS=100 MPa, UTS=BTS=15 MPa. The resulting angle of internal friction φ=47.7^{o}. The transition brittleductile according to Mogi (1966) does not exist, the angle of internal friction is too high.
8.2.4. Based on BTS and UCS
Here a linear envelope tangent to the BTS and the UCS Mohr circles is assumed, based on the assumption that the failure curve always has to be tangent to at least two Mohr Coulomb circles. This gives for the principal stresses:
\[\ \begin{array}{left}\sigma_{\min , 1}=\text{BTS}\\
\sigma_{\max , 1}=3 \cdot \text{BTS}\\
\sigma_{\min , 2}=0\\
\sigma_{\max , 2}=\text{UCS}\\
\mathrm{r=\frac{U C S3 \cdot B T S}{0B T S}=\frac{U C S3 \cdot B T S}{B T S}=\frac{U C S}{B T S}3=m3}\end{array}\tag{815}\]
This method results in a rather high value for the internal friction angle and consequently a rather low value for the shear strength (cohesion), although the internal friction angle will be lower than from the first method. To find a good estimate for the internal friction angle, there should be two Mohr circles based on shear failure. In this case one circle is based on shear failure, but the other circle is based on tensile failure. So this method is rejected. Figure 810 shows the Mohr circles for UTS, BTS and UCS for UCS=100 MPa, UTS=BTS=15 MPa. The resulting angle of internal friction φ=34.8^{o}. The transition brittleductile according to Mogi (1966) is at a normal stress of 316 MPa.
8.2.5. Hoek & Brown (1988)
Over the years Hoek & Brown (1988) developed a failure criterion for rock, based on the UCS and BTS values of the specific rock. The generalised criterion is empirical and yields:
\[\ \sigma_{\max }=\sigma_{\min }+\mathrm{U C S} \cdot\left(\mathrm{m} \cdot \frac{\sigma_{\min }}{\mathrm{U C S}}+\mathrm{s}\right)^{\mathrm{a}} \quad\text{ with }\quad\frac{\mathrm{a}=\mathrm{0 .5}}{\mathrm{s}=\mathrm{1 .0}}\text{ for intact rock}\tag{816}\]
The parameters m and s are material properties. The parameter m is related to the ratio of the UCS value to the BTS value according to:
\[\ \mathrm{m}=\frac{\mathrm{U C S}^{2}\mathrm{B T S}^{2}}{\mathrm{U C S} \cdot \mathrm{B T S}} \quad\text{ for }\frac{\mathrm{B T S}}{\mathrm{U C S}} \ll \mathrm{1} \quad \mathrm{m}=\frac{\mathrm{U C S}}{\mathrm{B T S}}\tag{817}\]
The parameter s is a measure for the amount of fractures in the rock and equals 1 for intact rock. The stresses σ_{min} and σ_{max} are the minimum and maximum principal stresses of the Mohr circle considered. The BTS value can also be represented as a function of m and s according to:
\[\ \mathrm{B T S}=\frac{\mathrm{U C S}}{2} \cdot\left(\mathrm{m}\sqrt{\mathrm{m}^{2}+4 \cdot \mathrm{s}}\right)\tag{818}\]
Based on:
\[\ \sigma_{\text {center }}=\frac{\sigma_{\max }+\sigma_{\min }}{2}\text{ and }\tau_{\max }=\frac{\sigma_{\max }\sigma_{\min }}{2}\tag{819}\]
An equation can be derived relating the maximum shear stress \(\ \tau_\text{max} \) (the top of the Mohr circle) to the normal stress at the center of the Mohr circle σ_{center}.
\[\ \tau_{\max }=\frac{1}{8} \cdot\left(\mathrm{m} \cdot \mathrm{UCS}+\sqrt{(\mathrm{m} \cdot \mathrm{U C S})^{2}+\mathrm{1 6} \cdot\left(\mathrm{m} \cdot \mathrm{U C S} \cdot \sigma_{\text {center }}+\mathrm{U C S}^{2}\right)}\right)\tag{820}\]
This equation results in a curve through the tops of the Mohr circles and is not yet a failure criterion. For the failure criterion Hoek & Brown (1988) give the following method; First determine a variable h according to:
\[\ \mathrm{h}=1+\frac{\mathrm{1 6} \cdot(\mathrm{m} \cdot \sigma+\mathrm{s} \cdot \mathrm{U C S})}{\mathrm{3} \cdot \mathrm{m}^{2} \cdot \mathrm{U C S}}\tag{821}\]
Now an angle θ can be determined:
\[\ \theta=\frac{1}{3} \cdot\left(\frac{\pi}{2}+\operatorname{atan}\left(\frac{1}{\sqrt{\mathrm{h}^{3}1}}\right)\right)\tag{822}\]
Based on the angle θ the instantaneous internal friction angle can be determined, which is also the tangent to the failure criterion:
\[\ \varphi=\operatorname{atan}\left(\frac{1}{\sqrt{4 \cdot \mathrm{h} \cdot \cos ^{2}(\theta)1}}\right)\tag{823}\]
Rock Type 
Class 
Group 
Texture 

Coarse 
Medium 
Fine 
Very Fine 

Sedimentary 
Clastic 
Conglo merates (21±3) Breccias (19±5) 
SandStones (17±4) 
SiltStones (7±2) Grey wackes (18±3) 
Clay Stones (4±2) Shales (6±2) Marls (7±2) 

Nonclastic 
Carbonates 
Crystalline Limestone (12±3) 
Sparitic Limestone (10±2) 
Micritic Limestone (9±2) 
Dolo mites (9±3) 

Evaporites 
Gypsum (8±2) 
Anhydrite (12±2) 

Organic 
Chalk (7±2) 

Meta morphic 
Non Foliated 
Marble (9±3) 
Hornfels (19±4) Meta Sandstone (19±3) 
Quartzite (20±3) 

Slightly Foliated 
Migmatites (29±3) 
Amphi bolites (26±6)  
Foliated 
Gneiss (28±5) 
Schists (12±3) 
Phyllites (7±3) 
Slates (7±4) 

Igneous 
Plutonic 
Light 
Granite (32±3) Granodiorite (29±3) 
Diorite (25±5) 

Dark 
Gabbro (27±3) Norite (20±5) 
Dolerite (16±5) 

Hypabyssal 
Porphyries (20±5) 
Diabase (15±5) 
Peridotite (25±5) 

Volcanic 
Lava 
Rhyolite (25±5) Andesite (25±5) 
Dacite (25±3) Basalt (25±5) 
Obsidian (19±3) 

Pyroclastic 
Agglomerate (19±3) 
Breccia (19±5) 
Tuff (13±5) 
Last but not least, the shear stress \(\ \tau\), matching the normal stress σ can be determined:
\[\ \tau=(\cot (\varphi)\cos (\varphi)) \cdot \frac{\mathrm{m} \cdot \mathrm{U C S}}{\mathrm{8}}\tag{824}\]
A second way of determining the failure criterion curve is with the following two equations, based on the minimum principal stress:
\[\ \sigma=\sigma_{\min }+\frac{\mathrm{U C S}}{2} \cdot \sqrt{\mathrm{m} \cdot \frac{\sigma_{\min }}{\mathrm{U C S}}+\mathrm{s}} \cdot{\left(1\frac{\mathrm{m}}{\mathrm{m}+4 \cdot \sqrt{\mathrm{m} \cdot \frac{\sigma_{\min }}{\mathrm{U C S}}+\mathrm{s}}}\right)}\tag{825}\]
\[\ \tau=\frac{\mathrm{U C S}}{2} \cdot \sqrt{\mathrm{m} \cdot \frac{\sigma_{\min }}{\mathrm{U C S}}+\mathrm{s}} \cdot \sqrt{ \mathrm{\left(1  \left( \frac { \mathrm { m } } { \mathrm { m } + 4 \cdot \sqrt { \mathrm { m } \cdot \frac { \sigma _ { \operatorname { min } } } { \mathrm { U C S } } + \mathrm { s } } } \right) ^ { 2 } \right)}}\tag{826}\]
Figure 811 and Figure 812 show the Hoek & Brown failure criterion for the top of the Mohr circles (A) and for the real failure condition (B). Although still based on UTS or BTS and UCS and not on two tests with shear failure, the resulting failure curve seems more realistic, which seems logic since it is based on many experiments. The Mohr circles for UTS, BTS and UCS are determined for UCS=100 MPa, UTS=BTS=15 MPa. The transition brittleductile according to Mogi (1966) is at a normal stress of 150 MPa.
Taking an average internal friction angle from a normal stress of zero to a normal stress of 240 MPa gives φ=27.1^{o}.
8.2.6. Parabolic Envelope UTS and UCS
Based on the UTS or BTS and the UCS a parabole can be constructed of the normal stress as a function of the shear stress, with boundary conditions that the parabole goes through the UTS or BTS point (shear stress equals zero, normal stress equals UTS or BTS and derivative d_{σ}/d_{\(\ \tau\)} equals zero) and touches the UCS Mohr circle as a tangent. With m=UCS/UTS or m=UCS/BTS the equation of this parabole is:
\[\ \mathrm{\sigma=\frac{1}{U T S \cdot(\sqrt{m+1}1)^{2}} \cdot \tau^{2}U T S}\tag{827}\]
It is more convenient to write this equation in the form where the shear stress is a function of the normal stress, giving:
\[\ \tau^{2}=\mathrm{UTS} \cdot(\sqrt{\mathrm{m}+1}1)^{2} \cdot(\sigma+\mathrm{UTS})\tag{828}\]
Figure 813 shows the resulting parabole. Although still based on UTS or BTS and UCS and not on two tests with shear failure, the resulting failure curve seems more realistic. The Mohr circles for UTS, BTS and UCS are determined for UCS=100 MPa, UTS=BTS=15 MPa. The transition brittleductile according to Mogi (1966) is at a normal stress of 104 MPa. Taking an average internal friction angle from a normal stress of zero to a normal stress of 240 MPa gives φ=18.6^{o}.
8.2.7. Ellipsoid Envelope UTS and UCS
Instead of a parabole, also an ellipse can be used. The advantage of an ellipse is that it gives more flexibility for the shape of the failure envelope. The general equation for an ellipse is:
\[\ \frac{(\sigma(\mathrm{a}\mathrm{U T S}))^{2}}{\mathrm{a}^{2}}+\frac{\tau^{2}}{\mathrm{b}^{2}}=1\tag{829}\]
In order to find an estimate for the radii a and b, it is assumed that the ellipse also touches the UCS Mohr circle in the same point as the parabole. With:
\[\ \mathrm{f}=\frac{\mathrm{1}}{\mathrm{U T S} \cdot(\sqrt{\mathrm{m}+\mathrm{1}}\mathrm{1})^{2}}\tag{830}\]
This gives for the normal stress of the parabole to Mohr circle tangent point:
\[\ \mathrm{\sigma_{p}=\frac{(1f \cdot U C S)+\sqrt{(1f \cdot U C S)^{2}+4 \cdot f \cdot U T S}}{2 \cdot f}}\tag{831}\]
And for the shear stress at the tangent point:
\[\ \tau_{\mathrm{p}}^{2}=\mathrm{U T S} \cdot(\sqrt{\mathrm{m}+1}1)^{2} \cdot\left(\sigma_{\mathrm{p}}+\mathrm{U T S}\right)\tag{832}\]
Comment: For sandstone a residual internal friction angle of 15 degrees and for limestone 25 degrees have been found at the brittleductile transition points.
For a given radius a this gives:
\[\ \mathrm{b}^{2}=\frac{\tau_{\mathrm{p}}^{2}}{\left(1\frac{\left(\sigma_{\mathrm{p}}(\mathrm{a}\mathrm{U T S})\right)^{2}}{\mathrm{a}^{2}}\right)}\tag{833}\]
Figure 814 shows both the parabolic and the ellipsoid failure envelopes. The ellipsoid failure envelope is determined for a=1.75·UCS. The Mohr circles for UTS, BTS and UCS are determined for UCS=100 MPa, UTS=BTS=15 MPa. At low normal stresses the parabolic and ellipsoid failure envelopes behave almost identical. Also the Mogi brittleductile transition points are very close. Chosing a>10·UCS gives about identical envelopes in the normal stress range considered.
8.2.8. Linear Failure Criterion
The best way to determine the angle of internal friction is to execute at least two tests with different confining pressures in the range of normal stresses the cutting process is expected to operate. Figure 816 shows this for a φ=20o internal friction angle. The Mohr circles for UTS, BTS and UCS are determined for UCS=100 MPa, UTS=BTS=15 MPa. The transition brittleductile according to Mogi (1966) is at a normal stress of 95 MPa.
8.2.9. The Griffith (Fairhurst, 1964) Criterion
Griffith (Fairhurst, 1964) has derived a criterion for brittle failure. His hypothesis aasumes that fracture occurs by rapid extension of submicroscopic. Preexisting flaws, randomly distributed throughout the material. He defined two criteria. The first criterion is:
\[\ \begin{array}{left}3 \cdot \sigma_{\min }+\sigma_{\max } \leq 0\\
3 \cdot \operatorname{UTS}+\sigma_{\max } \leq 0 \quad\text{ or }\quad \sigma_{\max } \leq 3 \cdot \mathrm{UTS}\\
3 \cdot \operatorname{BTS}+\sigma_{\max } \leq 0 \quad\text{ or }\quad \sigma_{\max } \leq 3 \cdot$ BTS\end{array}\tag{834}\]
Failure will occur when σ_{min}=UTS or σ_{min}=BTS, which is satisfied in the Brazilian split test. However when:
\[\ \mathrm{3} \cdot \sigma_{\min }+\sigma_{\max }>0\tag{835}\]
Failure will occur when:
\[\ \left(\sigma_{\max }\sigma_{\min }\right)^{2}\mathrm{8} \cdot \mathrm{U T S} \cdot\left(\sigma_{\max }+\sigma_{\min }\right)=\mathrm{0}\tag{836}\]
With:
\[\ \left(\frac{\sigma_{\max }\sigma_{\min }}{2}\right)^{2}=4 \cdot \mathrm{UTS} \cdot\left(\frac{\sigma_{\max }+\sigma_{\min }}{2}\right)\tag{837}\]
This can be written as a parabole for the center of the Mohr circles:
\[\ \tau_{\mathrm{max}}^{2}=4 \cdot \mathrm{UTS} \cdot \sigma_{\mathrm{center}}\tag{838}\]
For a UCS test this gives:
\[\ \left(\frac{\mathrm{U C S}}{2}\right)^{2}=4 \cdot \mathrm{U T S} \cdot \frac{\mathrm{U C S}}{2} \quad$ or $\quad \frac{\mathrm{U C S}}{\mathrm{U T S}}=\mathrm{8}\tag{839}\]
If the UCS/UTS or UCS/BTS ratio is larger than 8, brittle failure will occur.
The Griffith criterion as mentioned here is not the failure curve, but the curve connecting the tops of the Mohr circles.
In the original articles tensile is positive and compression negative, resulting in a sign change compared with the equations mentioned here. Als the minimum and maximum principal stresses were reversed.
8.2.10. Conclusions & Discussion
6 concepts for the angle of internal friction and the failure criteria have been discussed. Figure 814 and Figure 817 show these criteria. To find the best failure criterion curve, many tests should be carried out at different confining pressures, resulting in shear failure and a set of Mohr circles. Since this information is not always available, The Hoek & Brown, Parabole or Ellipse approximations can be used. The preference of the author is the Ellipse Envelope method or the Linear Failure Criterion method, where the internal friction angle is based on the average of the Parabole Envelope method or measured by experiments.
Above the brittleductile transition normal stress, the failure curve will decrease according to Verhoef (1997), based on research of van Kesteren (1995). As mentioned before, at higher stress situations there will be fracturing and crushing. This results in a decrease of the angle of internal friction. The higher the normal stresses, the stronger the fracturing and crushing, the smaller the angle of internal friction. When this starts there is a decrease of the angle of internal friction, while the failure curve is still increasing. However at a certain stress situation the failure curve may be at a maximum, since the angle of internal friction decreases to much. This maximum is often close to the Mogi (1966) criterion. Since intact rock and crushed rock are two different materials with different properties, one has to be very carefull with the interpretation of the resulting failure curve. In fact the material has continuously changing properties from the moment is starts fracturing and crushing. First larger particles are formed, consisting of many rock grains. When the stresses increase, these particles will also be fractured or crushed, resulting in smaller particles, until the rock grains are left.
When the angle of internal friction decreases faster than the increase of normal stresses, the failure curve decreases. This does however not mean that there is negative internal friction, normally the tangent to the failure curve. Just that the angle of internal friction decreases faster than the increase of normal stresses and most probably that the shear strength of the crushed rock decreases to zero. Verhoef (1997) and Vlasblom (20032007) show a failure curve reducing to zero for very high normal stresses. This seems to be unlikely to happen. It would imply that at very high normal stresses the shear stress equals zero, so no friction at al, which sounds like liquid behavior. It is more likely that the crushed rock, once completely crushed, will have a residual internal friction angle and possibly a residual shear strength. The latter is possible, for example when the particles are so small that van der Waals forces start playing a role. But this will depend completely on the type and composition of the rock.
Figure 814 shows a residual internal friction angle for both the ellipse and the parabole, tangent to the failure envelopes at the Mogi brittlecriterion.
For the models derived in this chapter, a constant internal friction angle is assumed, where this constant internal friction angle should match the stress state of the cutting process considered.