When Hooke's Law is no longer enough...

According to Hooke's law known from school textbooks, the elongation of a body should be directly proportional to the applied stress. However, many materials that are of great importance in modern technology and everyday life only approximately comply with this law or behave completely differently. Physicists and engineers say that such materials have rheological properties. The study of these properties will be the subject of some interesting experiments.

Rheology is the study of the properties of materials whose behavior goes beyond the theory of elasticity based on the aforementioned Hooke's law. This behavior is associated with many interesting phenomena. These include, in particular: the delay in the return of the material to its original state after a voltage drop, i.e., elastic hysteresis; increase in body elongation at constant stress, otherwise called flow; or a multiple increase in the resistance to deformation and hardness of an initially plastic body, up to the appearance of properties characteristic of brittle materials.

lazy ruler

One end of a plastic ruler 30 cm or more long is fixed in the vise jaws so that the ruler is vertical (Fig. 1). We reject the upper end of the ruler from the vertical by only a few millimeters and release it. Note that the free part of the ruler oscillates several times around the vertical equilibrium position and returns to its original state (Fig. 1a). The observed oscillations are harmonic, since at small deflections the magnitude of the elastic force acting as a guiding force is directly proportional to the deflection of the end of the ruler. This behavior of the ruler is described by the theory of elasticity. 

Rice. 1. Study of elastic hysteresis using a ruler

1 – ambulance,

2 - vise jaws, A - deviation of the end of the ruler from the vertical

In the second part of the experiment, we deflect the upper end of the ruler by a few centimeters, release it, and observe its behavior (Fig. 1b). Now this end is slowly returning to the equilibrium position. This is due to the excess of the elastic limit of the ruler material. This effect is called elastic hysteresis. It consists in the slow return of the deformed body to its original state. If we repeat this last experiment by tilting the top end of the ruler even more, we will find that its return will also be slower and may take up to several minutes. In addition, the ruler will not return exactly to the vertical position and will remain permanently bent. The effects described in the second part of the experiment are just one of rheology research subjects.

Returning bird or spider

For the next experience, we will use a cheap and easy to buy toy (sometimes even available in kiosks). It consists of a flat figurine in the form of a bird or other animal, such as a spider, connected by a long strap with a ring-shaped handle (Fig. 2a). The entire toy is made of a resilient, rubber-like material that is slightly sticky to the touch. The tape can be stretched very easily, increasing its length several times without tearing it. We conduct an experiment near a smooth surface, such as mirror glass or a furniture wall. With the fingers of one hand, hold the handle and make a wave, thereby tossing the toy onto a smooth surface. You will notice that the figurine sticks to the surface and the tape stays taut. We continue to hold the handle with our fingers for several tens of seconds or more.

Rice. 2. A vivid example of elastic hysteresis, shown using a return cross

1 - spider figurine, 2 - rubber band,

3 - handle, 4 - palm, 5 - surface

After some time, we notice that the figurine will abruptly come off the surface and, attracted by a heat shrink tape, will quickly return to our hand. In this case, as in the previous experiment, there is also a slow decay of voltage, i.e., elastic hysteresis. The elastic forces of the stretched tape overcome the forces of adhesion of the pattern to the surface, which weaken over time. As a result, the figure returns to the hand. The material of the toy used in this experiment is called by rheologists viscoelastic. This name is justified by the fact that it exhibits both sticky properties - when it sticks to a smooth surface, and elastic properties - due to which it breaks away from this surface and returns to its original state.

descending man

Photo 1. A figurine descending a vertical wall is also a great example of elastic hysteresis.

This experiment will also use a readily available toy made of viscoelastic material (photo 1). It is made in the form of a figure of a man or a spider. We throw this toy with deployed limbs and turned upside down on a flat vertical surface, preferably on a glass, mirror or furniture wall. A thrown object sticks to this surface. After some time, the duration of which depends, among other things, on the roughness of the surface and the speed of throwing, the top of the toy comes off. This happens as a result of what was discussed earlier. elastic hysteresis and the action of the weight of the figure, which replaces the elastic force of the belt, which was present in the previous experiment.

Under the influence of weight, the detached part of the toy bends down and breaks off further until the part again touches the vertical surface. After this touch, the next gluing of the figure to the surface begins. As a result, the figure will be glued again, but in a head-down position. The processes described below are repeated, with the figures alternately tearing off the legs and then the head. The effect is that the figure descends along a vertical surface, making spectacular flips.

Fluid plasticine

Rice. 3. Plasticine flow test

a) initial situation, b) final situation;

1 - palm, 2 - upper part of plasticine,

3 - indicator, 4 - constriction, 5 - torn piece of plasticine

In this and several subsequent experiments, we will use the plasticine available in toy stores, known as "magic clay" or "tricolin". We knead a piece of plasticine in a shape similar to a dumbbell, about 4 cm long and with a diameter of thicker parts within 1-2 cm and a narrowing diameter of about 5 mm (Fig. 3a). We grab the molding with our fingers by the upper end of the thicker part and hold it motionless or hang it vertically next to the installed marker indicating the location of the lower end of the thicker part.

Observing the position of the lower end of the plasticine, we note that it is slowly moving down. In this case, the middle part of the plasticine is compressed. This process is called the flow or creep of the material and consists in increasing its elongation under the action of constant stress. In our case, this stress is caused by the weight of the lower part of the plasticine dumbbell (Fig. 3b). From a microscopic point of view current this is the result of a change in the structure of the material subjected to loads for a sufficiently long time. At one point, the strength of the narrowed part is so small that it breaks under the weight of the lower part of the plasticine alone. The flow rate depends on many factors, including the type of material, the amount and method of applying stress to it.

The plasticine we use is extremely sensitive to flow, and we can see it with the naked eye in just a few tens of seconds. It is worth adding that magic clay was invented by accident in the United States, during World War II, when attempts were made to produce a synthetic material suitable for the production of tires for military vehicles. As a result of incomplete polymerization, a material was obtained in which a certain number of molecules were unbound, and the bonds between other molecules could easily change their position under the influence of external factors. These "bouncing" links contribute to the amazing properties of bouncing clay.

stray ball

Rice. 4. Set for testing plasticine for spreading and stress relaxation:

a) initial situation, b) final situation; 1 - steel ball,

2 - transparent vessel, 3 - plasticine, 4 - base

Now squeeze the magic plasticine into a small transparent vessel, open at the top, making sure that there are no air bubbles in it (Fig. 4a). The height and diameter of the vessel should be several centimeters. Place a steel ball about 1,5 cm in diameter in the center of the upper surface of the plasticine. We leave the vessel with the ball alone. Every few hours we observe the position of the ball. Note that it goes deeper and deeper into the plasticine, which, in turn, goes into the space above the surface of the ball.

After a sufficiently long time, which depends on: the weight of the ball, the type of plasticine used, the size of the ball and the pan, the ambient temperature, we notice that the ball reaches the bottom of the pan. The space above the ball will be completely filled with plasticine (Fig. 4b). This experiment shows that the material flows and relieve stress.

Jumping plasticine

Form a ball of magic playdough and quickly toss it onto a hard surface such as the floor or wall. We notice with surprise that the plasticine bounces off these surfaces like a bouncy rubber ball. Magic clay is a body that can exhibit both plastic and elastic properties. It depends on how quickly the load will act on it.

When stresses are applied slowly, as in the case of kneading, it exhibits plastic properties. On the other hand, with the rapid application of force, which occurs when colliding with a floor or wall, plasticine exhibits elastic properties. Magic clay can be briefly called a plastic-elastic body.

Tensile plasticine

Photo 2. The effect of slow stretching of magic clay (the length of the stretched fiber is about 60 cm)

This time, form a magic plasticine cylinder about 1 cm in diameter and a few centimeters long. Take both ends with the fingers of your right and left hands and set the roller horizontally. Then we slowly spread our arms to the sides in one straight line, thereby causing the cylinder to stretch in the axial direction. We feel that the plasticine offers almost no resistance, and we notice that it narrows in the middle.

The length of the plasticine cylinder can be increased to several tens of centimeters, until a thin thread is formed in its central part, which will break over time (photo 2). This experience shows that by slowly applying stress to a plastic-elastic body, one can cause a very large deformation without destroying it.

hard plasticine

We prepare the magic plasticine cylinder in the same way as in the previous experiment and wrap our fingers around its ends in the same way. Having concentrated our attention, we spread our arms to the sides as quickly as possible, wanting to sharply stretch the cylinder. It turns out that in this case we feel a very high resistance of plasticine, and the cylinder, surprisingly, does not elongate at all, but breaks in half its length, as if cut with a knife (photo 3). This experiment also shows that the nature of the deformation of a plastic-elastic body depends on the rate of stress application.

Plasticine is fragile like glass

Photo 3. The result of the rapid stretching of magic plasticine - you can see many times less elongation and a sharp edge, resembling a crack in a fragile material

This experiment shows even more clearly how the stress rate affects the properties of a plastic-elastic body. Form a ball with a diameter of about 1,5 cm from magic clay and place it on a solid, massive base, such as a heavy steel plate, anvil, or concrete floor. Slowly hit the ball with a hammer weighing at least 0,5 kg (Fig. 5a). It turns out that in this situation the ball behaves like a plastic body and flattens out after a hammer falls on it (Fig. 5b).

Form the flattened plasticine into a ball again and place it on the plate as before. Again we hit the ball with a hammer, but this time we try to do it as quickly as possible (Fig. 5c). It turns out that the plasticine ball in this case behaves as if it were made of a fragile material, such as glass or porcelain, and upon impact it shatters into pieces in all directions (Fig. 5d).

Thermal machine on pharmaceutical rubber bands

Stress in rheological materials can be reduced by raising their temperature. We will use this effect in a heat engine with a surprising principle of operation. To assemble it, you will need: a tin jar screw cap, a dozen or so short rubber bands, a large needle, a rectangular piece of thin sheet metal, and a lamp with a very hot bulb. The design of the motor is shown in Fig. 6. To assemble it, cut out the middle part from the cover so that a ring is obtained.

Rice. 5. Method for demonstrating plasticine and brittle properties of plasticine

a) slow hitting the ball b) slow hitting

c) a quick hit on the ball, d) the effect of a quick hit;

1 - plasticine ball, 2 - solid and massive plate, 3 - hammer,

v - hammer speed

In the center of this ring we put a needle, which is the axis, and put elastic bands on it so that in the middle of their length they rest against the ring and are strongly stretched. The elastic bands should be placed symmetrically on the ring, thus, a wheel with spokes formed from elastic bands is obtained. Bend a piece of sheet metal into a crampon shape with arms stretched out, allowing you to place the previously made circle between them and cover half of its surface. On one side of the cantilever, at both of its vertical edges, we make a cutout that allows us to place the wheel axle in it.

Place the wheel axle in the cutout of the support. We rotate the wheel with our fingers and check if it is balanced, i.e. does it stop in any position. If this is not the case, balance the wheel by slightly shifting the place where the rubber bands meet the ring. Put the bracket on the table and illuminate the part of the circle protruding from its arches with a very hot lamp. It turns out that after a while the wheel starts to rotate.

The reason for this movement is the constant change in the position of the center of mass of the wheel as a result of an effect called rheologists. thermal stress relaxation.

This relaxation is based on the fact that a highly stressed elastic material contracts when heated. In our engine, this material is wheel-side rubber bands protruding from the bracket bracket and heated by a light bulb. As a result, the center of mass of the wheel is shifted to the side covered by the support arms. As a result of the rotation of the wheel, the heated rubber bands fall between the shoulders of the support and cool down, since there they are hidden from the bulb. Cooled erasers lengthen again. The sequence of the described processes ensures the continuous rotation of the wheel.

Not only spectacular experiments

Rice. 6. The design of a heat engine made of pharmaceutical rubber bands

a) side view

b) section by an axial plane; 1 - ring, 2 - needle, 3 - pharmaceutical eraser,

4 - bracket, 5 - cutout in the bracket, 6 - bulb

Now rheology is a rapidly developing field of interest to both physicists and specialists in the field of technical sciences. Rheological phenomena in some situations can have an adverse effect on the environment in which they occur and must be taken into account, for example, when designing large steel structures that deform over time. They result from the spreading of the material under the action of acting loads and its own weight.

Accurate measurements of the thickness of the copper sheets covering steep roofs and stained glass windows in historic churches have shown that these elements are thicker at the bottom than at the top. This is the result currentboth copper and glass under their own weight for several hundred years. Rheological phenomena are also used in many modern and economical manufacturing technologies. An example is plastics recycling. Most products made from these materials are currently manufactured by extrusion, drawing and blow molding. This is done after heating the material and applying pressure to it at an appropriately selected rate. Thus, among other things, foils, rods, pipes, fibers, as well as toys and machine parts with complex shapes. Very important advantages of these methods are low cost and non-waste.