 Gharials and alligators (Figure 1) belong to the crocodilian family, they share similar body plans, hunt in the water, and both rely on stealth and explosive speed to ambush their prey. Despite all their similarities, the shape of their skulls and jaws are different. These differences reflect their different diets: while gharials almost exclusively eat fish, alligators eat a variety of prey which range from fish, turtles, to small deer. Gharials snap at fish, alligators use the power of their jaws to grab prey and drag it under the water. As discussed in the exploration of levers the gharial jaw geometry provides it with a faster jaw motion while generating lower biting forces. The jaw forces were calculated under the assumption that the jaw muscle forces for the gharial and alligator are the same. We are now going to try to determine is this is a reasonable assumption.
 The jaws of the gharial seem very slender in comparison with alligator's. As such they do not look as strong as alligator jaws, but are they?

Figure1 Comparison of Gharial on the left and Alligator on the right

Solid Mechanics
 Intuitively we can expect the alligator's jaw to be stronger than the gharial's jaw, but can we show this?
 To understand how structures respond to forces, we use the theory of continuum mechanics.
 Continuum mechanics tells us that materials deform when we push or pull on them.
 Forces and material deformations are grouped into 3 types (Figure 2):
 Compression (a)
 Tension (b)
 Shear (c)
 When materials experience a load they deform, and the amount of deformation is a property of the material, it is called stiffness. Materials with a greater level of stiffness deform less when they experience a force.
 All things being equal, larger structures are stronger than smaller ones. For example, when comparing two similar ropes made from the same materials, the thicker rope can hold more weight. In order to take into account the differences in size we have the concept of stress, which is the force applied divided by the area through which it "flows". Continuing with the rope example, we can expect that if the cross sectional area of a rope were to double, the force would also have to double in order to maintain the stress level.
 In reality, shapes are rarely as simple as the shapes in Figure 2. Over time physicists have developed equations for estimating the stresses some commonly used structural shapes. These are shapes that can be readily described using mathematical expressions, such as circles, rectangles, and ellipses. Examples of these shapes can be seen on the Wikipedia page on moments of inertia
 What about complex irregular shapes like the jaw of an alligator?
 Can we draw a lesson from out study of the motion of thelizard jump?

Figure2 Types of loads: acompression, btension, cshear

Using computers to determine the stiffness of crocodilian jaws
 Our goal is to compare the strength of the gharial jaw to the alligator jaw.
 Like in the jumping lizard example we have a complex problem. Like before we are going to divide the problem into many simple problems which can be solved with the aid of computer.
 To have a good comparison of the jaws, we are going to compare the amount of deformation the skulls suffer when loaded with the same force.
 We are going to subject both skull to the same biting muscle force and then compare the amount of the deflection the skulls experience.
 Our goal is to compare, not to measure actual deflections.
 Calculating the deflection of simple shapes can be easily calculated, but the deflection of a shape as complex as an alligator jaw is very difficult to calculate.
 Like the lizard jump example where the flight of the lizard was divided into many small intervals where the motion was simplified. The crocodilian jaw can be approximated as many simple shapes stuck together in a mesh (Figures 3 and 4) The deflection of each section can then calculated and combined to get an estimate of the deflection of the complex structure.
 The formal method for analyzing complex structures, by dividing them into small and simple element is called Finite Element Method. The analysis of structures using the finite element method is called finite element analysis (FEA).
 The finite elements method is widely used in engineering to predict the behavior of complex structures.
 The steps needed to conduct FEA are the following:
 Create finite element model
 Model  Create a shape that represents the structure being analyzed.
 Mesh  Divide the complex shape into simple elements.
 Load  Define forces which simulate the forces which the structure might experience in use.
 Example: Figures 4 and 5
 Solve
 Using a solver, solve the finite element problem in order to get stresses and deflection for the elements in the model
 Process the results
 Combine the results such that the stresses and deflections can be visualized and understood.
 Example: Figures 5 and 6
Understanding the results
 Figures 5 and 6 show an exaggerated representation of the deflection which a the skulls of alligators and gharials might experience if they are subjected to the same biting force.
 What does this tell you about the strength of their jaws?
 The gharial jaw deflects or bends much more than the alligator jaw. Is this okay? Is the gharial at a disadvantage?
 Not necessarily, the gharial's jaw might not be exposed to the same forces as an alligator.
 gharials eat almost exclusively fish, which do not require the same amount of force to capture, kill, and eat as would be required by an alligator as it tries to catch an antelope.
Can you imagine other uses for FEA in biology?

Figure3 Alligator jaw divided into simple shapes, or meshed
Figure4 Gharial jaw divided into simple shapes, or meshed
Figure5 Meshed alligator jaw with applied forces and exaggerated deformation (colors represent the total deformation).
Figure6 Meshed gharial jaw with applied forces and exaggerated deformation (colors represent the total deformation).
