- Email: [email protected]
Outside or inside
COMMENT
Outside or inside Where to put the skeleton?
David Taylor | Trinity College, Dublin | [email protected] Vertebrates like you and me have evolved with a skeleton on the inside: an endoskeleton. This may seem perfectly normal but in nature it’s very much the exception. Exoskeletons are more common, making up the great majority of species, notably the arthropods, which include insects and all those creatures which you normally only encounter as “seafood”, such as lobsters and crabs. It is believed that the weight of all the ants in the world is greater than the weight of all the people. Skeletons are essentially mechanical organs; they exist to provide stiffness for posture and movement, and skeletal material needs to have strength and toughness. So if we ask “Why do some animals have exoskeletons and some endoskeletons” it’s the sort of design question that would be familiar to any structural engineer and materials scientist. In a previous article1 I offered to replace your bones with new ones made from steel; here I’m asking
Typical exoskeleton (a lobster) and endoskeleton (the author).
62
whether it would be advantageous to give you an exoskeleton, but using the same bone material that you have now. This brings us into areas in which materials science overlaps with structural engineering, because both material properties and geometry need to be considered. And needless to say biology gets involved as well. Tubes are commonly used as structural elements because, weight for weight, they are good at resisting bending and torsion. By moving material away from the centre of the cross-section we can increase the second area moments I and J (commonly called the moments of inertia) without increasing the amount of material used. Bending is found in almost all structures and causes a lot of trouble for engineering designers. It’s present in our bones because body forces act off-axis, through joints, and muscles pull from the side. It may surprise you to learn that we still don’t have precise figures for the stresses that arise in bones: the analysis is complicated by dynamic loading factors and, primarily, by the fact that most bones are loaded by multiple sets of muscles, making the problem wildly indeterminate. However, as a reasonable approximation, we can consider a typical bone to be a tube with an approximately circular cross-section, loaded in bending but with significant levels of axial force, usually compression (e.g. body weight acting through the legs) but occasionally tension, as when a monkey hangs from a branch. In our bones the ratio between the radius r and the wall thickness t is typically about 2. Suppose we made this radius larger, giving us (for the same weight) a wider, thinner-walled tube which will have a bigger I value and so better resist bending forces? Such a bone would certainly be stiffer and stronger: the maximum stress for a given applied force would
MARCH 2011 | VOLUME 14 | NUMBER 3
be reduced, so failure by yielding or brittle fracture would be less likely. The problem is that two other failure modes start to become significant: buckling and splitting. Buckling is a complex phenomenon which few people really understand. The simplest type is Euler buckling, which is what happens when compression is applied to a long, thin object such as a drinking straw or a sheet of paper. It arises because, above a certain critical force, the object is unstable with respect to elastic deflections perpendicular to the applied force. This is a completely elastic phenomenon, not dependant on the material’s strength at all. It may sometimes occur in the bones of vertebrates2, but rarely. Since it depends on the ratio between the I value and the length, if we change our endoskeleton into an exoskeleton, Euler buckling will be less likely to occur, so we don’t have to worry about it. Sadly that’s not the end of the buckling story, because there are two other types. Ovalization buckling occurs because when you bend a circular tube it becomes oval, in such a way as to reduce I in the plane of bending. This causes an instability which sets in at a critical value of the bending moment3. Local buckling is what happens when you press hard on the ends of a drinks can. It suddenly “gives”, creating wrinkles in the surface. This time it’s not the whole structure which buckles, just some local regions. Both of these types of buckling are potential problems for exoskeletons because the critical stress is proportional to the ratio t/r, which of course is going to be small. Longitudinal splitting happens because when you bend a tube you create tensile strains in the circumferential direction3. These can cause failure in materials like bone which are anisotropic, being relatively weak in that direction. For bone the ratio
COMMENT
of strengths in the best and worst directions is a factor of 2 – 3, which is enough to make this failure mode a possibility. In considering all these different failure mechanisms we face a problem which is quite common in engineering design. We can work out the loading conditions under which each individual failure mode occurs and so find the most likely one for any given shape of tube and applied loading conditions. But interactions occur between the different loading modes. For example, in a region of behavior where yielding is the dominant mode, but close to a region where buckling dominates, the actual load at failure is less than predicted by either equation. It’s difficult to incorporate these effects into the theory, so in structural engineering they are usually dealt with using empirical equations. The crucial material properties involved are the ratio between Young’s modulus and longitudinal strength, and the ratio of the strengths in the longitudinal and circumferential directions. Working from these, I did some sums. I imagined a typical arm or leg bone turned into an exoskeleton, increasing its diameter from 32 mm to 110 mm. To keep constant bone weight the thickness would reduce from 8 mm to just 1.8 mm. I estimated that, for the typical combination of bending and axial loads, this new bone would be about twice as strong as the original one. Alternatively I could choose to keep the same strength, in which case it would be significantly lighter. This is a pretty big advantage for an animal in the wild. A stronger bone would be less likely to break in a fight or an accident; a lighter bone means you can move around more quickly, with less expenditure of energy. So you can either fight that lion, or run away from it: the choice is yours. This suggests that there would be a significant pressure to evolve exoskeletons. And yet, we didn’t do it. One potential problem is joints. If you feel your knee (or someone else’s) you’ll notice that the bone gets very much bigger near the joint, and this is true for all the articulating joints in our bodies, giving bones their typical shape with bulbous ends. Why is this? The answer seems obvious but it’s not – in fact insects do just the opposite: their bones get smaller at the joints. One might assume that the increased bone area is there to reduce stresses on the bones, but actually the cross section doesn’t change much (the bone in that region becomes thin and porous) and anyway the loads near the joints are not much different from
what they are at mid-shaft. You might imagine that it’s done to reduce wear, but any tribologist will tell you that wear depends on the applied force, not the stress, so increasing the contact area doesn’t help. In fact (and rather counter-intuitively) small joints are better than large ones because the amount of wear is proportional to the sliding distance, which is reduced if you have a smaller radius. The real reason why our joints are so big is to protect the cartilage. Cartilage is necessary to give us a low-friction surface but it’s a much weaker material than bone. It is really the weakest link in the chain of our musculoskeletal system, as anyone suffering from arthritis will tell you. Insects don’t use cartilage, they make their bones from a different material (based on chitin) which they harden locally at the joint surface, and many of their joints are designed as flexible hinges rather than articulations. So if we are going to design an exoskeleton from bone, we can’t just simply copy the designs used by arthropods. The other issue in designing joints for exoskeletons is that you have to put all the muscles inside the bones rather than outside. This creates some problems for the kinematics of the system: muscles are basically pieces of string which can get shorter, exerting forces only along their own axes; there are restrictions regarding the amount of shortening that they can do, and they can’t support any compressive load. To achieve movement you need to attach the muscles to the bones so as to create lever arms, and these always operate at a mechanical disadvantage. The resulting forces passing through the attachments can be several times your full body weight, contributing considerably to the stresses in the bones themselves. There are some other considerations that come into play in exoskeleton design, which I don’t have space to discuss here, such as resistance to impacts of the kind which pathologists like to call “blunt force trauma”, i.e. someone attacks you with a hammer or a karate chop. This can be a problem for a thin-walled structure. Taking all these factors into consideration, I have tried to design an exoskeleton using bone and I think I can make one which is better than the equivalent endoskeleton. I won’t show you my designs here because I won’t believe them myself until I test them experimentally, but on paper they look pretty good. The final problem for the exoskeleton is a biological one: how to grow it? As we grow to maturity, we
need to change the sizes and shapes of our bones, whether they are inside or outside our bodies. This is a big issue for arthropods. Their solution is to periodically shed their skeletons and grow new ones. This is normally done once a year and it’s a huge disadvantage: most crustaceans die during this moulting stage, when they are defenseless against predators. Bones are remodeled as we grow thanks to the actions of special cells: osteoclasts, which dissolve unwanted regions of bone, and osteoblasts, which make new bone. Through a remarkably subtle feedback system, the details of which are still unclear, these cells work together to allow bones to grow, and to change their geometry in response to changes in their stress environment. In order to take advantage of this remodeling system you need to have cells within the bone to direct operations and cells on the surfaces to resorb and deposit as required. Cells couldn’t live on an external surface: they need to be supplied with nutrients through the blood supply, and anyway they would just get rubbed off. Our bony exoskeleton would need to have at least a thin layer of skin on it to protect these cells. So it wouldn’t be a “true” exoskeleton but let’s not be pedantic, it would still be an exoskeleton in the mechanical sense, and it would be able to grow continuously, avoiding the need to moult. But wait a minute, we already have one of those – it’s called the skull. It seems that we have evolved an exoskeleton after all, to protect the brain; we just haven’t got around to extending it over the rest of the body. This design exercise may seem rather silly, since the fact is that exoskeletons made of bone don’t exist. In this sense it’s similar to the question I asked in my previous article1. As I said then, asking silly questions is important in science; here it has allowed us to think about why different types of skeletons have evolved in different species and to consider all the challenges that any skeleton has to face. The resulting analysis reminds us that mechanical performance is very important in nature, and emphasizes the role to be played by materials scientists and engineers in furthering our understanding of the living world.
REFERENCES 1. Taylor, D., Mater Today (2010) 13 (3), 6. 2. Currey, J. D., J Morphol (1967) 126, 1. 3. Wegst, U. G. K., and Ashby, M. F., J Mater Sci (2007) 42, 9005.
MARCH 2011 | VOLUME 14 | NUMBER 3
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Outside or inside Where to put the skeleton?
David Taylor | Trinity College, Dublin | [email protected] Vertebrates like you and me have evolved with a skeleton on the inside: an endoskeleton. This may seem perfectly normal but in nature it’s very much the exception. Exoskeletons are more common, making up the great majority of species, notably the arthropods, which include insects and all those creatures which you normally only encounter as “seafood”, such as lobsters and crabs. It is believed that the weight of all the ants in the world is greater than the weight of all the people. Skeletons are essentially mechanical organs; they exist to provide stiffness for posture and movement, and skeletal material needs to have strength and toughness. So if we ask “Why do some animals have exoskeletons and some endoskeletons” it’s the sort of design question that would be familiar to any structural engineer and materials scientist. In a previous article1 I offered to replace your bones with new ones made from steel; here I’m asking
Typical exoskeleton (a lobster) and endoskeleton (the author).
62
whether it would be advantageous to give you an exoskeleton, but using the same bone material that you have now. This brings us into areas in which materials science overlaps with structural engineering, because both material properties and geometry need to be considered. And needless to say biology gets involved as well. Tubes are commonly used as structural elements because, weight for weight, they are good at resisting bending and torsion. By moving material away from the centre of the cross-section we can increase the second area moments I and J (commonly called the moments of inertia) without increasing the amount of material used. Bending is found in almost all structures and causes a lot of trouble for engineering designers. It’s present in our bones because body forces act off-axis, through joints, and muscles pull from the side. It may surprise you to learn that we still don’t have precise figures for the stresses that arise in bones: the analysis is complicated by dynamic loading factors and, primarily, by the fact that most bones are loaded by multiple sets of muscles, making the problem wildly indeterminate. However, as a reasonable approximation, we can consider a typical bone to be a tube with an approximately circular cross-section, loaded in bending but with significant levels of axial force, usually compression (e.g. body weight acting through the legs) but occasionally tension, as when a monkey hangs from a branch. In our bones the ratio between the radius r and the wall thickness t is typically about 2. Suppose we made this radius larger, giving us (for the same weight) a wider, thinner-walled tube which will have a bigger I value and so better resist bending forces? Such a bone would certainly be stiffer and stronger: the maximum stress for a given applied force would
MARCH 2011 | VOLUME 14 | NUMBER 3
be reduced, so failure by yielding or brittle fracture would be less likely. The problem is that two other failure modes start to become significant: buckling and splitting. Buckling is a complex phenomenon which few people really understand. The simplest type is Euler buckling, which is what happens when compression is applied to a long, thin object such as a drinking straw or a sheet of paper. It arises because, above a certain critical force, the object is unstable with respect to elastic deflections perpendicular to the applied force. This is a completely elastic phenomenon, not dependant on the material’s strength at all. It may sometimes occur in the bones of vertebrates2, but rarely. Since it depends on the ratio between the I value and the length, if we change our endoskeleton into an exoskeleton, Euler buckling will be less likely to occur, so we don’t have to worry about it. Sadly that’s not the end of the buckling story, because there are two other types. Ovalization buckling occurs because when you bend a circular tube it becomes oval, in such a way as to reduce I in the plane of bending. This causes an instability which sets in at a critical value of the bending moment3. Local buckling is what happens when you press hard on the ends of a drinks can. It suddenly “gives”, creating wrinkles in the surface. This time it’s not the whole structure which buckles, just some local regions. Both of these types of buckling are potential problems for exoskeletons because the critical stress is proportional to the ratio t/r, which of course is going to be small. Longitudinal splitting happens because when you bend a tube you create tensile strains in the circumferential direction3. These can cause failure in materials like bone which are anisotropic, being relatively weak in that direction. For bone the ratio
COMMENT
of strengths in the best and worst directions is a factor of 2 – 3, which is enough to make this failure mode a possibility. In considering all these different failure mechanisms we face a problem which is quite common in engineering design. We can work out the loading conditions under which each individual failure mode occurs and so find the most likely one for any given shape of tube and applied loading conditions. But interactions occur between the different loading modes. For example, in a region of behavior where yielding is the dominant mode, but close to a region where buckling dominates, the actual load at failure is less than predicted by either equation. It’s difficult to incorporate these effects into the theory, so in structural engineering they are usually dealt with using empirical equations. The crucial material properties involved are the ratio between Young’s modulus and longitudinal strength, and the ratio of the strengths in the longitudinal and circumferential directions. Working from these, I did some sums. I imagined a typical arm or leg bone turned into an exoskeleton, increasing its diameter from 32 mm to 110 mm. To keep constant bone weight the thickness would reduce from 8 mm to just 1.8 mm. I estimated that, for the typical combination of bending and axial loads, this new bone would be about twice as strong as the original one. Alternatively I could choose to keep the same strength, in which case it would be significantly lighter. This is a pretty big advantage for an animal in the wild. A stronger bone would be less likely to break in a fight or an accident; a lighter bone means you can move around more quickly, with less expenditure of energy. So you can either fight that lion, or run away from it: the choice is yours. This suggests that there would be a significant pressure to evolve exoskeletons. And yet, we didn’t do it. One potential problem is joints. If you feel your knee (or someone else’s) you’ll notice that the bone gets very much bigger near the joint, and this is true for all the articulating joints in our bodies, giving bones their typical shape with bulbous ends. Why is this? The answer seems obvious but it’s not – in fact insects do just the opposite: their bones get smaller at the joints. One might assume that the increased bone area is there to reduce stresses on the bones, but actually the cross section doesn’t change much (the bone in that region becomes thin and porous) and anyway the loads near the joints are not much different from
what they are at mid-shaft. You might imagine that it’s done to reduce wear, but any tribologist will tell you that wear depends on the applied force, not the stress, so increasing the contact area doesn’t help. In fact (and rather counter-intuitively) small joints are better than large ones because the amount of wear is proportional to the sliding distance, which is reduced if you have a smaller radius. The real reason why our joints are so big is to protect the cartilage. Cartilage is necessary to give us a low-friction surface but it’s a much weaker material than bone. It is really the weakest link in the chain of our musculoskeletal system, as anyone suffering from arthritis will tell you. Insects don’t use cartilage, they make their bones from a different material (based on chitin) which they harden locally at the joint surface, and many of their joints are designed as flexible hinges rather than articulations. So if we are going to design an exoskeleton from bone, we can’t just simply copy the designs used by arthropods. The other issue in designing joints for exoskeletons is that you have to put all the muscles inside the bones rather than outside. This creates some problems for the kinematics of the system: muscles are basically pieces of string which can get shorter, exerting forces only along their own axes; there are restrictions regarding the amount of shortening that they can do, and they can’t support any compressive load. To achieve movement you need to attach the muscles to the bones so as to create lever arms, and these always operate at a mechanical disadvantage. The resulting forces passing through the attachments can be several times your full body weight, contributing considerably to the stresses in the bones themselves. There are some other considerations that come into play in exoskeleton design, which I don’t have space to discuss here, such as resistance to impacts of the kind which pathologists like to call “blunt force trauma”, i.e. someone attacks you with a hammer or a karate chop. This can be a problem for a thin-walled structure. Taking all these factors into consideration, I have tried to design an exoskeleton using bone and I think I can make one which is better than the equivalent endoskeleton. I won’t show you my designs here because I won’t believe them myself until I test them experimentally, but on paper they look pretty good. The final problem for the exoskeleton is a biological one: how to grow it? As we grow to maturity, we
need to change the sizes and shapes of our bones, whether they are inside or outside our bodies. This is a big issue for arthropods. Their solution is to periodically shed their skeletons and grow new ones. This is normally done once a year and it’s a huge disadvantage: most crustaceans die during this moulting stage, when they are defenseless against predators. Bones are remodeled as we grow thanks to the actions of special cells: osteoclasts, which dissolve unwanted regions of bone, and osteoblasts, which make new bone. Through a remarkably subtle feedback system, the details of which are still unclear, these cells work together to allow bones to grow, and to change their geometry in response to changes in their stress environment. In order to take advantage of this remodeling system you need to have cells within the bone to direct operations and cells on the surfaces to resorb and deposit as required. Cells couldn’t live on an external surface: they need to be supplied with nutrients through the blood supply, and anyway they would just get rubbed off. Our bony exoskeleton would need to have at least a thin layer of skin on it to protect these cells. So it wouldn’t be a “true” exoskeleton but let’s not be pedantic, it would still be an exoskeleton in the mechanical sense, and it would be able to grow continuously, avoiding the need to moult. But wait a minute, we already have one of those – it’s called the skull. It seems that we have evolved an exoskeleton after all, to protect the brain; we just haven’t got around to extending it over the rest of the body. This design exercise may seem rather silly, since the fact is that exoskeletons made of bone don’t exist. In this sense it’s similar to the question I asked in my previous article1. As I said then, asking silly questions is important in science; here it has allowed us to think about why different types of skeletons have evolved in different species and to consider all the challenges that any skeleton has to face. The resulting analysis reminds us that mechanical performance is very important in nature, and emphasizes the role to be played by materials scientists and engineers in furthering our understanding of the living world.
REFERENCES 1. Taylor, D., Mater Today (2010) 13 (3), 6. 2. Currey, J. D., J Morphol (1967) 126, 1. 3. Wegst, U. G. K., and Ashby, M. F., J Mater Sci (2007) 42, 9005.
MARCH 2011 | VOLUME 14 | NUMBER 3
63