 Tenic rock, you have sedimentary rock, and it could be many different things. So you have basalt, that's what we use to protect the coastline of Holland. Everybody knows marble, we often use it either for tiles or for tables and that kind of stuff. So those are types of rock, you could also say stone. But usually with stone we already talk about pieces of stone that we use for building purposes. And rock is what we call the intact material. In dredging we use cutter heads to cut rock. And that's not exceptional, we often do that and we go to rocks of about 60 MPa. And I noticed some people didn't follow the introduction dredging course where I already looked into the soil mechanics. Not everybody had soil mechanics. So if you want to know a little bit more about the material properties and the behavior, you should download the soil mechanics lecture notes which I put in under week one. So there you can find, and that's just for information to give you a feeling what are we talking about. What kind of values do you have if you talk about compressive strength, what is it, and how big is it, what are we talking about. So those numbers you can find in those lecture notes. You could also go to some courses of petroleum engineering where they have specific courses about that, where they give a lot of information, but that means you have to look through complete lecture notes of a whole course and that could be a little bit too much. Well, those are two types of cutter heads. So you see the cutter head, you see the teeth on the cutter head, and in this picture I show a lot of different types of pickpoints, we call them pickpoints. And the pickpoints are in fact the tools that cut the rock. And those pickpoints are normally mounted on an adapter. You put a steel bar through it to fix it, and that means when it's broken or when you have too much wear, you can take it off and replace it. And on some projects this happens every hour. So after one hour the whole cutter head is either too much damaged or all the pickpoints are worn out and you have to replace all the pickpoints. So that could be very expensive, but for a dredging company if you know this is the kind of job you do, then in the offer you make you calculate how many pickpoints do we expect to use, and that's part of the deal you make. So for those people in the companies it's very important. They have some idea about the forces, the power, but also the wear that may occur. In fact right now we have one student who will investigate this for Boscalis. We know something about wear, but we don't have good models where you can actually predict the wear. There is something but not good enough. This student has to model the whole three-dimensional movements of each pickpoint, determine how much rock it is cutting, determine the forces in 3D, then once you know the forces on each pickpoint with some models you could try to determine the wear of each pickpoint. In fact you could say the wear for a specific type of rock, the wear depends on how much cutting energy you use. So if you know the specific energy of the type of rock and you know the volume that has been cut by that pickpoint that would give you a good indication of the wear of that pickpoint. Now the problem is that if a pickpoint like this is worn out it means usually the top is worn out. So if you are cutting like this the bottom has a lot of wear. That means you will also get a force at the bottom, a force which we normally don't have. And this force takes care that the swing wires of the cutter dredge, the cutter dredge is dredging like this with at the top the cutter head. Here you have a wire, we call it the swing wire, and that swing wire has to pull. But if there is too much wear on those pickpoints you get a force in this direction on the teeth and the swing force has to become too big. And usually when they reach the limit of the swing wire, the swing wire can generate a certain power and a certain torque. Well, when you are at the limit of what the swing wire can do usually they will say the cutter head is worn too much because if you are at the limit of that swing wire you have to reduce speed, your production goes down. And if I don't have maximum production I'm losing money. And then replacing the cutter head is cheaper than continuing too long with a worn cutter head because your production is really decreasing rapidly. So they would like to know that. Okay, this gives you an impression of what those things look like and it's also a picture that I used in the lecture notes. When you are cutting rock we have to distinguish between brittle cutting and ductile cutting. Brittle looks like this. So if I measure the force as a function of time I get an irregular sort of sawtooth. So I have big peaks and I have smaller peaks and that has to do with big pieces of rock that collapse and smaller pieces that collapse. So it's not just one piece that breaks out it could first be a number of smaller pieces and then one big piece. And that's why you have all those peaks and finally a big peak. What does it look like? Well brittle looks like this. So really pieces of material break out and that process of breaking out could be because of tensile failure when you exceed the tensile stress but it could also be shear failure. So in contrary with clay, in clay if you have shear failure it's plastic failure which means the clay stays intact but if you are cutting rock and you have shear failure you get a discrete shear plane and the material collapses and a piece of rock will break out. A chip is just a piece of material that breaks out. We call it a chip. Here you see ductile cutting so you are cutting and you get a nice continuous material so that's what you would have when you are cutting steel or cutting clay. Now what we will see in this lecture is that under atmospheric conditions rock fails like this in general. So brittle so we get chips, discrete chips but under very high pressures we call it hyperbaric cutting usually rock fails like this. We are still thinking about a new word for what happens under hyperbaric conditions because if you really talk about ductile cutting it means plastic deformation the material stays intact. So after the cutting you still have that type of material but if I am hyperbaric cutting it looks like this but as soon as I take away the pressure I have powder. So just imagine you are cutting a normal brick that you use for building houses you cut it under very high pressure then the bindings between the particles collapse but because of huge under pressures it stays together, it sticks together so in the beginning when you observe it through a window you think it's a nice ductile cutting process but if I take away that pressure then what's left is powder I made all the bindings between the particles collapse so it's just powder. So in fact it looks like ductile but it's not and there is not yet a good word for that so I challenged my PhDs to come up with some new term for this type of cutting process. What does it look like if you do measurements? Here you have the strain, epsilon is the strain so it's the amount of volume, deformation sigma is the pressure and then you can see if you are cutting brittle you have a buildup of stresses up to a certain maximum stress which we call the strength of the material so that would be the shear strength of the material then the material collapses and after the collapse the stresses go down very rapidly in some materials they will go down to zero if you really have a chip breaking out then after that there is no stress anymore so it would go down to zero but that depends on the process it doesn't have to go all the way to zero if you have ductile then in fact you get a sort of asymptotic behavior where the stresses increase and here they start increasing more slowly but at the end they will stay at a certain level so there is a maximum level and not like in this picture where it continues to increase there is a maximum level to that stress and that's also what we call the shear strength of such a material you have materials in between and then you also get a curve in between which we call brittle ductile so it's a sort of transition well what happens really and in fact what really happens is something we don't have an equation for that yet so the equations I will show you are simplified models which give a good indication of the cutting forces but for this real process we don't have an equation or a model a model does not always mean you have one equation it could be a whole algorithm or whatever but it doesn't exist yet so what happens here this is the pick point and you can see here this is the wear flat so this is a worn pick point and we study that because in reality all pick points are worn after they are used for 10 minutes so what happens well in front of this pick point so here you have the layer you are cutting in front of the pick point here you have an area with very high normal stresses those stresses are so high that the bindings between the particles collapse so you get particles it's not a material with strength anymore it's like sand so that's what happens over here we call it the crushed zone so the material is crushed it doesn't exist anymore as the original material so this is the crushed zone then this crushed zone will have a certain normal stress or stress distribution if you want so it's pushing in every direction to the outside of the crushed zone that will take care that you get stresses shear stresses and normal stresses in the intact rock normally that will result in some shear plane here so the material wants to start shearing over there it's a shear plane, number 2 but once it starts shearing you get a bending moment on the chip and the bending moment will take care that in this area you get a so-called tensile crack it will break because of tensile failure well that means in this process you could say you have 4 areas but this is the crushed zone this is the area with high normal stresses this is the shear plane zone and the tensile crack zone so you have 4 areas and each area has a different behavior so if we look in this graph basically it's the PQ graph which is derived from the tau sigma graph so from the Mohr circle graph but you could also draw this picture in a Mohr circle graph basically what you have here you have the tensile strength so in this area with the tensile crack you are in this area that's where you get tensile cracks so that's the tensile area the shear plane is this area because you have to exceed the shear strength of the material, the failure condition then area number 3 here in front of the crushed zone is at the top of this curve and area number 4 the crushed zone is at the end of the curve so basically in one cutting process you have all regions of the whole curve in fact if we would draw the curve in the normal Mohr-Colombe graph with the tau and the sigma it would look almost the same so you have four different areas and different types of failure that may occur at the same time and that makes it difficult to have one model for this one theory for this so I didn't find a theory where you can actually calculate forces based on this model so then what to do well you have to simplify the model and see ok maybe under certain conditions I can apply another theory and the first theory I want to discuss is the so called Evans theory what did Mr. Evans do Mr. Evans said if we take a cone a 2D cone with a certain top angle alpha or in fact it's 2 alpha and you push against a piece of rock and in fact they often use this theory in rock cutting when you are drilling tunnels in Switzerland or when you are drilling a coal that kind of material they often use this because they have this kind of a tool and they push or they hammer against the rock and they want it to collapse so what did Mr. Evans say he said suppose here you have a piece of a circle and at that whole circle we will have tensile failure so the material will fail because of tensile so if you push hard enough you split the rock in fact that's the idea behind this theory ok so how big is this circle well from the tip of the big point you can make a vertical line then you can define an angle beta and the top, the origin of the circle depends on the angle beta if I make my angle beta different I will get a different origin of that circle so what you do is you just define this picture you can determine the total force because everywhere perpendicular on that circle you have your tensile stress so you can integrate that and then you find an equation for the total force and that total force is this T then you can say ok what will nature do nature will make it collapse where the T is the smallest at the smallest force where it could collapse it will collapse so I can take the derivative of that T with respect to beta and determine for which beta do I have the smallest force and then I will say ok that will be the beta where the material will collapse by the way the picture is in the lecture notes with the complete derivation of those forces so you will also find it there well so what do we have we found a beta based on minimum energy we call it the minimum energy principle and then if you notice T you can also determine the forces required in both directions you have a force R the forces on the pick point required to generate this force T because they have to be in equilibrium the nice thing about this model is that you have the force R on both sides of the pick point because in fact the pick point is splitting the material so both sides of the pick point have to push against the material to make it fail so that's the Evans model I will show you the resulting equation after this here we have the HI again so the HI is always the thickness of the layer we are cutting and so on and here you have the delta which was the external friction angle so that's if we look at the equations the equation you find is not too complicated this is the cutting force this is the horizontal cutting force and this is the vertical cutting force and in this case because the pick point is horizontally pressing against the rock in this case the vertical force is zero because in both directions you have that R force and they compensate in the vertical direction so you don't have a vertical force in this model here you can see the cutting force is the tensile strength so if I know my tensile strength I can determine the force this is the layer thickness the width of the pick point and here you have a relation where the alpha was half the top angle of the pick point and delta is the external friction angle you can see that the internal friction angle does not play a role which makes sense because it's failing on tensile not on shear if I would have failure on shear I need my internal shear angle but it's failing on tensile but still I have the forces on my pick point and there I have my external friction angle so that's why the external friction angle plays a role but the internal does not if you divide it like here with the production then this is the specific energy in this case so if you know the alpha and the delta of your the alpha of your pick point and the delta of your rock you can say the specific energy is a constant times the tensile strength and in literature you can find values for that constant from practice but you can also use this theory to calculate the constant now you can imagine if you are dredging with such a cutter head with those 3D pick points then a pick point will never push against the rock horizontally it will have a certain angle so if I just rotate the whole picture and that's what I'm doing here I'm rotating this over an angle epsilon so the original picture from the previous slide is this white rectangle so I just rotated the whole picture but I have to compensate for that epsilon with a number of angles because now if you look vertically this R and this R are not compensating anymore completely in the vertical direction because I rotated the picture so I'm just rotating the previous picture and take the correct signs and cosines for the forces and then I get this so what do we have? here we have the total cutting force the cutting force is in the direction of the pick point not yet in horizontal direction so it's in the direction of the pick point to the axis of the pick point well the beginning is the same I see I miss an eye here so you have the tensile strength again the layer thickness, the width of the blade and now you see that in this sign an epsilon is added for the rest the equation is the same as the previous equation but to determine horizontal and vertical force for horizontal I have to take the cosine of that epsilon for the vertical force I have to take the sign because the direction is not horizontal anymore of this fc and also for the specific energy because specific energy is always based on the horizontal force because the velocity is in the horizontal direction so here you also get that cosine that you had here to determine your specific energy now what is interesting is could we also simulate a real pick point with a wear flat bottom with this method well we can because this angle epsilon we just make that angle epsilon equal to half the top angle of the pick point so here you have your alpha which was 2 times alpha was the top angle of the pick point if we make the epsilon exactly equal to alpha then the bottom of the pick point becomes horizontal and then we can say ok so now we have a pick point with a horizontal bottom and that horizontal bottom in fact is the wear of the pick point so we just rotate over half the top angle of the pick point and that way we can simulate a pick point with wear and that's nice because also in that study of the student who just started with wear the question is but how do we know the vertical force on the bottom of the pick point well with this theory it follows from the theory so you know the force here in fact it's this r times a certain cosine and then if you know the normal force you can also calculate the friction force because you know the friction angle the external friction angle so it's just a matter of rotating the original model over the right angle and you can simulate pick points with wear well what kind of equation do you get since we rotated with an epsilon that is equal to alpha that's what we did the epsilon can be deleted or can be replaced in the equation and you can see that because of that here we get a 2 alpha for the rest the equation is the same well we have to multiply with cosine alpha and sine alpha to get the horizontal and the vertical component and if you use this horizontal force in this specific energy equation you get this equation for the specific energy and it's again it's linear it's proportional to the tensile strength this is a nice model but it only works if you have tensile failure if I don't have tensile failure if I already know I will not get tensile failure for example because my tensile strength is too high that could be the case then I cannot use this model because it doesn't apply well if I do not have tensile failure and later I will show you a graph and with that graph you can determine whether you will have tensile failure or brittle failure if I don't have tensile failure then what do I have well the only other option is shear failure it's either shearing or it's breaking by tensile there's another model made by a person called Nishimatsu a Japanese and those models are already pretty old but in literature we didn't really find better models so Mr. Nishimatsu made a model and it looks very much like the clay cutting model where he said ok I have a pick point I have a shear plane on that shear plane I calculate a normal stress and a shear stress and if I take the equilibrium of forces horizontal and vertical I can determine my forces but Mr. Nishimatsu added something he said the shear stress in this shear plane does not have to be the same everywhere because what could happen is that near the tip of the blade I have a very high shear stress so the material will start to fail there but once it starts failing the next piece of material will also fail because where it has failed you don't have strength anymore so then the remaining part of the rock has to take care of the forces so he suggested to use a certain distribution over the length of the shear plane and that distribution he used a factor N for that distribution and here you can see this is the tip of the blade this is the free surface if I take an N of zero everywhere my shear stress is one is the maximum shear stress but if I use an N with another value for example I use let's say the one which is the center one then you get a linear distribution of your stress over the shear plane I can also use an N that is bigger for example the eight which gives me the lowest line and you can see you have a shear stress like this then he integrated the stress distribution and the result is that in your forces you get a factor one divided by N plus one so if you would use this method and you have for example an N equal to one a linear stress distribution you would get one divided by two in front of the equation distribution oh so if you are pushing with a pick point against the rock then it makes sense that just in front of the pick point you have your maximum normal stress and shear stress but maybe at the point where it reaches the surface the stress is still zero so you get a certain distribution of stress from the tip of the pick point to the surface in the clay cutting we acted like everywhere in the shear plane the shear stress was the same namely the shear strength of the material but it doesn't have to be like that because if I would have maximum shear stress at the point at the tip of the blade and still zero at the surface what will happen if I continue pushing my pick point the material will start to fail near the tip of the blade because material can have a shear crack just like a tensile crack it can also have a shear crack well if that shear crack starts occurring near the tip of the blade then that part of the shear plane cannot resist forces anymore because it already cracked so that means at the end of that crack you get a very high stress concentration resulting in the crack to move forward can you imagine? okay so you start with a certain stress distribution for example this one, the lowest one it means you have a high stress near the tip of the blade but near the surface the stress is still zero you can only know if you do experiments with that material and see if you can measure those stresses that's the only way to do it in fact what Mr. Nitshumasu did he created this theory and then from experiment he is not measuring the stresses he is just saying okay so if I measure a force because forces can be measured then the force would be like 50% of what would be expected based on this part of the equation he would say oh then N has to be one you understand? so he put a very nice theory behind it with the whole stress distribution but at the end nobody cares about this picture anymore everybody only cares about this one this term because with that term you can always have a correction to make it match and then you say oh my N was one or two or three so that's the way it works in reality some coefficients here I made some graphs they are also in the lecture notes of the coefficients so with the coefficients I mean this whole term and here it's called LAPDA HF and LAPDA VF H for horizontal V for vertical and the F is for the flow type because basically he is also working with shear failure so that's the flow type here you can see what those coefficients look like based on here the blade angle and here the angle of internal friction so you can see that if you have a very don't have internal friction but in rock you always have some internal friction you would have the lowest line and if you have a very high internal friction 45 which is too high for normal rock rock normally would be 20-25 degrees so you are in the middle and then from this graph you can actually read which coefficient you have to use in the previous equations so this one is for the horizontal force and this one is for the vertical force well then we go to the theory that I want to show you because the previous are some theories from literature and with the model you already had for sand and clay and so on you can also solve this problem in fact the result is almost the same as Nishimatsu except for that n factor because I don't believe in that n factor because if you don't really measure the stress distribution but you just use an n to make it match that's not really what I want I want a controllable theory so the definitions are the same as with sand and clay which forces do we have? well if you cut rock under atmospheric conditions then we have the sea I still call it cohesion but with rock everybody would say shear strength in clay they often use the word cohesion but in rock or in steel we don't use the word cohesion we say shear strength or maybe the UCS, the Unconfined Compressive Strength so the sea that's the shear force based on the shear strength because you are shearing the material later I will show you what happens for tensile failure but this is for shear failure ok that will result in a normal force and a shear force on the shear plane under an angle phi so under the angle of internal friction and then on the blade we only have this normal force and the shear force because we assume rock does not have adhesion it's not sticky normally if you put your hand on rock your hand will not stick to the rock so it doesn't have adhesion so those are the forces on this layer that we are cutting so in fact we start with a model where we act like it fails like clay and then later we will see where we can use it and where we can't use it on the blade so you only have that normal force and that shear force we also have the moments but for atmospheric rock we will not use them but for the moments you only have those two normal forces because all the other forces go through the tip of the blade so they are not resulting in a moment only the normal forces are resulting in a moment like I said for atmospheric rock we don't need the moment equation what is the resulting equation and if you would write the C as a force based on the stress you would see it's the same equation as Mr. Nishimatsu apart from that N vector so this is the resulting K2 and then you put them in the horizontal and vertical force equations and this is what you find so the big difference with Mr. Evans is that here you also have the internal friction in the equation and you don't have the tensile strength and Mr. Evans is completely based on tensile strength this equation is based on shear failure so you have the shear strength in your equation what does it look like if you make more circle of that process so you can take the stresses on the shear plane and on the blade and construct a more circle well if you do that then the top of the circle should touch the failure line under an angle phi so that's the phi of your rock here you have the so-called UCS value why? because I made a circle that on this side goes through the origin if it goes through the origin then this maximum stress here is what we call the UCS value you can determine that with a triaxial test on a sample of rock and then you will find this circle but you need a couple of circles to determine the phi from one circle you can never determine the phi because in this case the failure line is not going through the origin so you need at least two points to find the failure line and this failure line here you have the tensile strength you could determine the tensile strength for example with the BTS test the Brazilian tensile strength test and that way you get your tensile strength like I told you yesterday I think if you know the tensile strength here and the UCS you can construct the failure lines but it doesn't have to be so rectangular like here it could be a smooth curve but in fact we are not really interested in that all we are interested in is what is the tensile strength and where is this failure line that's what we are interested in ok we will stop for a break continue in 15 minutes we are busy with the moor circle of rock cutting this is the moor circle of a UCS test and yesterday I already I already showed you the moor circle in the case of tensile failure and tensile failure could occur like this in fact it's not exactly the same tensile failure that you saw in the first picture explaining the whole process of rock cutting but it's a model and the model gets close to measurements from reality and I will show you some of those measurements later well this is the situation that you get when you have tensile failure so you determine your moor circle and in fact once you know the forces on the shear plane and the forces on the blade you can determine the total moor circle in the rock and if it would be a moor circle like this then you can see here you exceed the tensile strength because the tensile strength is at this point well you can do many things to say okay how can I calculate the forces in such a case what I do is reduce the moor circle just make the radius of the moor circle smaller in such a way that this point just touches the tensile strength in the lecture notes you can find the whole derivation of how I do that but in fact I'm reducing the shear strength of the material in such a way that you get a circle just touching the tensile strength if you get such a circle you can see it touches the tensile strength here but the top does not touch the shear strength anymore so this is a circle where you get tensile failure but you don't have shear failure if you want shear failure the whole big circle should move to the right in such a way that it doesn't touch the tensile strength anymore that's very simple if somewhere the circle exceeds a failure criterion then that's where it will fail and in this case it exceeds the tensile strength so you get tensile failure this is a picture a graph I still have to change the graph one of my PhDs give me some tip he said this looks like one thousand and this looks like ten thousand but the dot is the decimal dot so this is just one and this is just ten so why not just leave all the zeros away and I will do that in the next issue so I will change the graphs to make them a little bit more clear in fact if you are below a line it means you have either ductile or shear failure if you are above a line you have brittle failure based on tensile failure and here vertically you see the UCF which is the compressive strength of the material divided by the BTS which is the tensile strength of the material so compressive strength divided by tensile strength if you get a very high value for this ratio it means either means the compressive strength is very high or the tensile strength is very low but in both cases it means the ratio so the tensile strength is very small compared to the compressive strength in that case it's easy to get brittle failure so here you can see above lines those are the lines for different angles of internal friction above a line I could have tensile failure below a line I cannot I put two lines in here this line is a line they often use in dredging it's a ratio value of 9 and in dredging they say if the ratio UCS, BTS is below 9 we will not have brittle failure we will have ductile failure and above this line they say we will have brittle failure and in between you have a certain transition ratio transition region now let's look at real values for rock a real value for the internal friction angle would be 20 degrees something near 20 degrees here we have the 20 degrees which is this curve now if you look at that curve you see it crosses the 9 at a blade angle of 50-55 degrees that's the normal angle of pick points in dredging so apparently if I use a cutter head 50-55 degrees blade angle and I look at a real rock of 20 degrees internal friction angle I'm exactly at this factor 9 that they use in dredging so this graph gives you a feeling could I have tensile failure or not what you can also see is if I go to very high large blade angles it's more difficult to get tensile failure and the reason for that is if I have like a 110 degree blade and I get so much compressive stress in front of the blade I will never have tensile stresses anymore so very high or big blade angles will not result in tensile failure in oil drilling you do you use 110 and that means in oil drilling when I use those stratopax the diamond blades, the small blades I will never have tensile failure it doesn't mean it cannot be brittle because brittle can also be a brittle shear failure if I get a discrete shear plane that's also brittle brittle just means pieces of material break out that's what brittle means so I have two types of brittle tensile brittle and shear brittle both can be brittle next picture in fact shows a little bit the same here you have the BTS the tensile strength divided by the cohesion and normally the cohesion is about 50% of the compressive strength not completely because we have an internal friction angle and here I put the BTS negative because tensile strength is negative and again also in this picture it means if you are below the line you have what we call ductile or shear if you are above a line you have brittle failure so you can use one of the two graphs and I'm planning to improve the readability of the graphs a little bit to add some title so in the graph you already can see what exactly does it mean here I have the brittle coefficients so in the equations you have coefficients and you can see in fact with the vertical coefficient it stops here and that means beyond that point you cannot have tensile failure these are graphs for tensile failure but if it can't exist that's where the line stops just read it in the lecture notes for some more explanation then we go to hyperbaric rock cutting and that's what that CEO of IHC was talking about first of all when we are cutting rock and this is a specific energy graph in literature you can find some data not much but you can find some data of people who did experiments cutting experiments on different types of rock this is a sort of marble and if you analyze those measurements what do you see if I don't have this by the way this pressure is the hydrostatic pressure of the water and this is in MPA 100 kPa is 10 meters of water so 1 MPA is 100 meters of water and that means this 10 is 1000 meters of water so that's what it means now what do you see if I have zero pressure that also means zero meters of water I find specific energies in this region now if I increase my water depth here so 1 MPA would be 100 meters I can see it increases rapidly so apparently the water depth influences the specific energy and also the cutting forces so they increase with increasing water depth but at a certain point those points follow a line with less steepness it's not as steep as in the beginning well this is brittle here is brittle cutting this is what we call ductile cutting so you can see that in the first 100 meters of water depth you get a sort of transition between brittle cutting and ductile cutting because the water plays a role and the water plays a role just like in sand cutting because of pore pressures you get pore pressures in your rock and they influence the specific energy this is the same kind of graph in a limestone and you can see the same type of behavior so this is brittle again it increases rapidly until it becomes ductile and there it follows this ductile line I have one more but this is forces and not energy but the principle is the same here you see brittle again and here you can see the ductile behavior and somewhere here at about one MPa those two lines would intersect well what is the difference between hyperbaric rock cutting and atmospheric rock cutting if you have hyperbaric rock cutting you have to add pore pressures to your equation so under atmospheric conditions those pore pressures hardly play a role they could play a role but it's neglectable and how to determine if pore pressures play a role very simple if you know the compressive strength tensile strength of your rock suppose it's 10 MPa and you have 10 meters of water depth which could give you a maximum under pressure of 2 bar which is 200 kPa then this 200 kPa is very small compared with the 10 MPa it's 2% so you can say ok if it's just 2% I can neglect it I would go to 1000 meters where I have 100 bar and 100 bar would be 10 MPa then the under pressure is of the same magnitude as the compressive strength of the material and suddenly those under pressures start playing a role so it's just about the ratio between the hydrostatic pressure at the location of your cutting device and the compressive strength of the material that also means that if you have a very high compressive strength like 100 MPa you have to go to much bigger water depth before you get the same influence on your cutting process so very strong rocks will not have so much influence of the hyperbaric cutting effect it's especially the weaker rocks that you have to deal with but if you do deep sea mining the mining tools in dredging we say we can go up to 60 MPa compressive strength with normal cutter heads so I assume if you put that kind of equipment at the sea floor 60 MPa would also roughly be the limit and then you can see what would be the effect so if I have 3000 meters of water depth I get 30 MPa hydrostatic pressure if I take that limit of 60 MPa then it's 50% of the compressive strength so it would influence the cutting process but later I will show you how it influences so the difference between atmospheric cutting and hyperbaric cutting is that now we have the W1 here and the W2 here and I'm not going to explain all the other forces because we already did that many times just to make the picture complete in this case I also added the A, the adhesion just to get equations that are more generic but normally we will say we don't have adhesion in rock but in the equations you can find that A so if you would have a rock with some adhesion you can take it into account well, forces on the blade also we added the W2 and those are the resulting equations so I took the generic equation from the beginning again I just deleted all the terms that are not to be taken into account and what is left is the W2 term, the W1 term and the C term and then you substitute that in the horizontal and vertical equations here I deleted in fact in this equation I deleted the A terms so I assume no adhesion but in the lecture notes you also have the adhesion but normally we do not take that into account well I will put this on blackboard so you don't have to copy all the equations now we have a problem we could have two problems the equations are again based on the flow type now the flow type is a nice type for ductile cutting and we said okay under hyperbaric conditions we will often have something that looks like ductile cutting so we can use those equations very well but there may still be a case of tensile failure or the curling type when did we have the curling type if you cut very thin layers of material now in deep sea mining we will not do that usually you will have centimeters maybe 5 centimeter thickness but in oil drilling you work you are more scraping the material and they work with a layer thickness less than 1 millimeter I have examples I will show you 0.15 millimeter and 0.3 millimeter so it's really very thin and in that case you will get the curling type in offshore drilling they call it the balling type so they don't call it curling but they call it balling so the fact that the chip is not following the blade but it starts curling they call it balling and it often happens when you are oil drilling so they have to take measures to make the material flow away because in oil drilling you want the material that has been cut to mix with the fluid so they can transport it out of the hole otherwise if it stays there it will block the whole hole to determine how the curling or balling will look like you need the moment equation well we have the normal force here the underwater force and here also so if you have equations for those forces and you know the radius in both cases you can determine the moment equilibrium now the point is if you get curling or balling the chip will not follow the whole blade and just like with clay you want to know how far will the chip be in touch with the blade so in fact you act like you have a shorter blade you don't have to write this down but I just wanted to show you the equations this in fact is this moment equilibrium so you can see the W2, W1 and the C this is the moment on the blade on the layer cut so in the shear plane this is the moment on the blade and what I did is replace the height of the blade with H-axons and H-axons would be the height of the blade required to get an equilibrium of moments so that's why I give it that action so H-axons is always smaller than the real height of the blade well if you write this down it will have the shape of a second degree function and you have to determine the roots of that function so you have an A, B and C here I put the A, B and C and you really have to put this in a computer program because it's too much to do by hand you could put it in an excel sheet or something and then this is the solution for your H-axons again all the equations are in the lecture notes so you can follow it there so that's how to find that H-axons well, is it true? that's the question so right now we are cooperating with Shell Oil in this research and a man of Shell, Mr. Zeisling did research in the 80s up to 100 MPa and 100 MPa is 1000 bar they had a test facility where they could actually do cutting tests up to 1000 bar which equals 10,000 meters of water depth but that's where they go with oil drilling because oil drilling doesn't stop where you hit the seafloor you can go 5 to 10 km deeper to find the oil and then you get very high hydrostatic pressures so this FN in fact in our way of thinking is the horizontal force and here you can see tests with this is 0.3 mm and this is 0.15 mm and at the time when they did those tests they were not exactly sure what it depended on because they said maybe it depends on the already existing pore pressures in the rock because if you are oil drilling the rock is at a certain distance under the surface so inside the rock you already have a certain pore pressure so they did tests that's what you see here with unconditioned means 0, 21 MPa, 30 and 35 MPa but that's the pore pressure they had in the rock samples but then they found it doesn't depend on those pore pressure it just depends on the hydrostatic pressure based on the outside pressure this pressure doesn't really play a role that was the conclusion of that research and that's where they stopped they didn't make any model so with the equations I just showed you I checked the measurements of Mr. Zeisling and you can see well it's not exact and there is some scatter in those measurements but they get close enough to say okay this is a good way of predicting the forces in oil drilling I must say so for that the theory works but in this verification I assumed full cavitation in the layer of material you are cutting so I'm just cutting 0.15 mm and the under pressure is 1,000 bar because the permeability of the rock is so low it starts cooking in the pores it actually happened and that was also one of the conclusions of Mr. Zeisling this gives the specific energy and what was interesting about the specific energy is if you have two layer thicknesses and the 0.15 and the 0.3 mm do they give the same specific energy? well you could maybe still see a difference between the two layer thicknesses I can't so we said okay so it's proportional with the layer thickness by the way here at zero you can still see it's below the line and this was tensile failure so under atmospheric conditions for the rest I think it follows the theoretical line pretty well so the blue line also in the previous slide I calculated theoretically with the theory I just showed you then what can we do with that? well if you have that theory so both atmospheric and hyperbaric based on the theory you can determine specific energy oh yeah I should tell one thing about those measurements when you are cutting such a thin layer of material the crushed zone that you saw in the first slide occupies the whole thickness of the layer because the layer is so thin so everything is crushed so it doesn't mean you can use it one to one in dredging because maybe in dredging not everything is crushed oh because the under pressure so you should realize that outside you have a hyperbaric pressure the hydrostatic pressure of maybe 1,000 bar but at least hundreds of bars between the particles the pressure is zero so you get the effect of vacuum packed coffee so because of this under pressure the particles stick together and it looks when you do a test it looks like it's a very nice continuous chip and that's also what you see with that balling and curling it's a continuous chip that is curling on the blade but as soon as they take away the pressure it falls apart and it's powder so it sticks together because of the under pressure this is a graph I made for the specific energy based on the Seisling experiment so based on that theory with full cavitation this is for a 60 degree blade so this one you could apply in dredging where you have a cutter head with 55 or 60 degree pickpoints so what do you see? first of all, brittle is at the bottom I have a number of lines here those straight lines are for atmospheric cutting and the black line is ductile cutting under atmospheric conditions the other lines are if the tensile strength is much less than the compressive strength in fact here you can see the ratio, the R, the ratio between the compressive strength and the tensile strength so if that ratio is high it means you have a very low tensile strength and that would give you the lowest curve here so that's fractured material with very low tensile strength now what happens if we are under water and we can start here with 10 bars that's this one and 10 bars would be 100 meter of water you get this red line so you can see that already 100 meters of water has some influence but like I said before especially with a softer rock so here you have the compressive strength so if I have a low compressive strength already 100 meters of water can increase the specific energy but if you go to the stronger rocks so this area is from 10 to 100 MPa so that's the stronger rock you can see it hardly influences the black line for ductile cutting but if I would go to very deep water here you have 1000 bar well 1000 bar is 10 kilometers so that would be very deep you can see that almost over the whole range it's almost a horizontal line and that means everything is determined by the pore pressures and not by the strength of the rock anymore but that would be at 10 kilometers of water depth well and you see lines for all the water depths in between so from this picture you can actually get your specific energy but it's based on completely crushing the material like in the Seisling experiment so if you have a situation like in deep sea mining where you use a pick point and chips break out the specific energy is probably less so this is based on completely crushing the material but if you know this and you know ok here I can find a specific energy which is an upper limit to what will happen in reality ok then at least I have an upper limit in reality if I have bigger chips the specific energy could be lower but probably it will never be higher so this is an upper limit to the specific energy here you get the graph for rock cutting 110 degrees that's those diamond cutting bits and one of the things you see is that here for atmospheric cutting you don't see the different lines anymore for tensile fracture because like I explained you don't have tensile fracture with a 110 degree bit you could have like in sand wedge forming theoretically but I calculated that the internal friction angle is relatively small and the shear angle is about 10 degrees and then you are still far below the 180 although your blade angle is 110 but if you have an internal friction of let's say 20 and an external of 16 and a shear angle of 10 you are far enough below the 180 to still use the theory that's because all the other angles are so small so this is the chart you can use for drilling oil drilling and I checked those values in this range with the experiments of Mr. Seisling in fact we wrote a paper together for this summer in Rio de Janeiro a conference where we published all those graphs and I must say in the 80s I already worked together with Mr. Seisling and I got my PhD on this research and he did those experiments but for some reason we never communicated about it so we never combined it to one theory and then 20 years later we met again and then suddenly we you do that and you do that we can combine it to a theory and then we also have measurements experiments to verify the results this is just a zoom-in of the previous graph so it's easier to read so although in this region hyperbaric cutting we call it ductile like I explained to you it's not really ductile because if you take away the pressure you have powder and the powder is not ductile I think this is the... oh yeah for deep sea mining we like to know exactly what is happening and I already talked a little bit about what the PhDs are doing well this is a picture of one of my PhDs but this is when he did his master's thesis what you can see here is a lot of particles and this is from EDEM EDEM is discrete element method software and what you do is you define particles discrete particles and you define how those particles are connected in fact you could say ok the particles do not have any tensile strength but they have internal friction sliding friction, rolling friction but you can also put springs and dampers between all the particles so each combination of particles of two particles has a spring and a damper in between them in every direction and this is like a shoebox with 20,000 something like 20,000 of those particles and then the software will actually calculate the interaction between each set of two particles and based on that it will calculate where particles go so particles the connection between particles can break can actually break I will show you in a video a little later but in the case of sand you don't have tensile strength but you have friction and in fact this shows you the cutting of sand we simulate it sand and if it has a different color than blue it means it has velocity so based on velocity the color of each particle will change and that way you can see exactly here where you have your shear plane well this shows when the blade has moved forward and the particles fall over the blade and you can see exactly what's happening this is without under pressure this is dry sand because we are not yet capable of doing this with pore pressures and that's the challenge of two of my PhDs in four years time they have to implement pore pressures into the method in order to simulate hyperbaric rock cutting now the videos I will show you they are made by one of the PhDs he started beginning of the year and he said yeah those commercial software EDEM and ETASCA and so on they are too closed which means if you want to add something it's difficult it's like you have a finite element program and it doesn't do exactly what you want and you want to change it usually that's not possible so he said you know what I'm going to develop everything myself and he started beginning of the year and so what you see now is already the result of what he did he did more but this is part of what he did in the first year and at the end we want to be capable that you have a big point going through the rock and you can see exactly how the rock will fail what happens and then if you adjust the hydrostatic pressure you can okay first we do it atmospheric and you can see the chips breaking out and then we add hydrostatic pressure and you can see how it will change so we will get some videos the first video is just to test the software so here you can see a silo that's why you thought it's from transportation engineering but it's not, it's just testing the methodology so you see all those particles at a certain point they open the silo and just by gravity all the particles they are falling out and you can imagine if you do this on a PC and each time step so you work with time steps so like one time step is one thousandth of a second and then each time step you have to calculate all the interactions between all the particles and then that's how the software gives you such a result the second one I will do the cutting of sand so in fact at the beginning you see that's what you always do with this software from the top you just drop all the particles, they fall then when all the particles are at the bottom you press a button connect them, activate the interactions so otherwise you would think in rock if you just drop the particles how can it be rock but that's because once the particles are not moving at the bottom anymore you turn on all the connections and from that moment on you can deal with it as real rock you can also understand that this way because you just drop the particles you get a sort of random distribution and what this guy did is the size of the particles is different so you have a distribution of sizes of particles you drop them then you get a random distribution at the bottom and that should match real material better the next one this is material so the previous one was sand without tensile strength between the particles this is material with tensile strength and you can really see those shocks it's not yet perfect so if you see something that's not real it's just a test of the software but you can see those shocks in the material and you can see chips breaking out and shooting away and well for being busy for one year I think it's already a good job so this promise is something for the next three years he also made some simulations of tests in fact the graph you see is the graph of the shear stresses in the material so this is also cutting of something with tensile strength and you can see if you see the green line moving you can see where pieces break out is the recording okay? good then one more or two more brittle then this is the simulation of a UCS test so in fact you have a cylinder of material and there you see the crack so this is material with tensile strength and internal friction and you see the green line approaching that curve and then at the top you can see you get a crack and then of course you have to calibrate this with real samples and you have to do a number of real tests and then see is my software predicting what happens in reality so one more which is also a UCS test and if it would be the real material then for example you could say oh it should shear I should enter like 45 degrees well those are the things he is testing the biggest problem with this kind of simulations is that normally we work with what we call macro parameters and what is a macro parameter well you know internal friction angle because how do you measure the internal friction angle you take a sample of sand or rock or whatever put it in a triaxial machine press until it breaks and based on the Mohr circle you say that's my angle of internal friction in this software you have to define the interactions between individual particles we call that the micro parameters and so if I define a tensile strength between two particles how is that related to the tensile strength of a block of material if I determine the sliding friction or the spring stiffness or the damper how is that related to the bulk we call it bulk parameters also nobody knows and in the application of this type of software that's one of the biggest issues how to connect those bulk parameters to the micro parameters between individual particles nobody knows yet so in fact from what I know on internet they are forming a sort of a database of all the people who work with this kind of software and in the database they explain what kind of experiment they did with the software and this is the bulk parameter and this is how I modeled the micro parameter and so you can see you are at the edge of science because for example like I told you the pore pressures are not yet implemented in this type of software nobody did that so if those PhDs do that as the first in the whole world that means we can do calculations nobody else can do it also means you can write publications for very good journals because they like this kind of development okay that's it for today next week I will do erosion processes so if you have questions about all the cutting processes don't wait until the last week if you can already ask questions next week or please do because then we still have time maybe I could make an extra picture or something to explain something but if you have questions please ask see you next week