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Discontinuous deformation in an elastic material. Part 2. Energy dissipative and storage applications.

INTRODUCTION

Dissipative and Storage Applications

It has been established [1] that energy dissipation is possible from a perfectly elastic material. It has conventionally been believed that the maximum work required to stretch a material between two states of elongation is the area under the stress-strain curve for loading and that the work that is recoverable in the contraction process is the corresponding area under the unloading curve. However this holds good only for smooth continuously varying deformations. In a discontinuous process, the work required to stretch is more than the area under the loading curve and the work that is recoverable on contraction is much less than the area under the unloading curve. In a cyclic discontinuous process there is a difference between work performed and recovered, which is an energy change in the material that also gets dissipated as heat. This dissipation is not due to established internal friction causes such as viscoelasticity. This dissipation is because of the jump or discontinuous nature of the deformation process. This can be observed in any discontinuous deformation process, such as a belt mechanism of transmitting power. In this work the numerous applications that are possible for such dissipative devices will be presented. By modifying the deformation process, the energy losses associated with dissipation in a cyclic deformation process can be minimized. The device undergoing such a process can then be made to function as an energy storage device. The feasibility for such energy storage applications will also be established.

Dissipative Applications. Dissipative applications include nonfriction brakes, tension and torque controllers, and torque converters.

Brakes are generally friction brakes, either dry friction or viscous friction [2, 3]. In the use of dry friction brakes, such as those used on automobiles, the mechanical energy of the sliding components is converted to thermal energy that leads to very high temperatures, especially in the skin of the braking components, and wear. These high temperatures can lead to problems such as warping and loosening of the metal parts, as well as be a source of ignition causing fires. In an application for the dissipative device in this work, as a nonfriction brake, these problems can be avoided.

Another application is as a constant tension device. A constant tension device can be made to function as a tension controller in applications such as in the drawing of polymer sheets and fibers where a constant tension should be maintained. The stretch and contraction points in the discontinuous rubber deformation process occur by means of sudden jumps and not continuously. Hence, the stress states in the belt are constant for a given region and vary over the length of the belt just as the stretch varies. It takes a constant torque to turn such a device and so it can be applied as a tension controller or a constant torque controller.

A third application for the dissipative device will be as a torque converter. Torque converters can be used for the startup of massive systems such as conveyor belts, escalators and clutches in automobiles and trucks. This application permits apparent slippage to occur between the input and output speeds whenever the torque reaches a critical value. When this occurs the torque remains constant and the speeds are independent of each other with the excess work being dissipated as heat. Below this critical torque, the speeds are identical with no dissipation. Clutches or torque converters of this type are required in start up situations, e.g. when an electric motor is used to start an escalator from rest the torque will become so high the motor will overload and burn up if unprotected. In many such systems an automotive hydraulic transmission is used to isolate the electric motor from the escalator thereby limiting the torque. In the present application presented here, with the dissipation mechanism that occurs from the rubber belt, the need for an automatic hydraulic transmission is eliminated. Two such torque converters also have the potential to replace differential gear assemblies in automobiles, as will be discussed.

Storage Applications. Storage applications include regenerative brake devices and mechanical energy batteries.

By understanding the energy dissipation mechanism it is simple to understand that using several stretching steps, rather than a single step process, leads to less and less dissipation. In effect all of the energy is stored as strain energy and it all can be recovered. This is very attractive because good elastomers have tremendous strain energy potential. Simple calculations indicate that about 100 pounds of rubber can contain the kinetic energy of a large automobile moving at ~60 Mph. This kinetic energy would be converted into strain energy and could be used to accelerate the vehicle on demand. This is the working principle behind a regenerative brake device. The difficulty with this argument is that to uniformly stretch 100 pounds of rubber, huge forces of the order of 500,000 pounds will be required. In the present system explained in this work, by using spools of stretched elastomer, these forces could be contained on the spools and only a small amount of elastomer will be stretched at a time. This would solve the packaging problem and makes the process more practical.

Bardwich [4] describes an invention where he uses the kinetic energy from vehicles during their normal operation and not just during braking. Bardwich et al. [5] use inertial flywheels in their inventions, to store energy. This is a complex energy transfer mechanism and requires complex electro-mechanical coupling because of the high RPMs required for efficient flywheels. Elastomeric storage devices have since been used to store energy more efficiently. Hoppie [6] uses elastomers in regenerative brake systems. The above and his other inventions [7-9] relate to regenerative brake systems of automotives where he uses solid elastomeric rods. The storage assembly includes a plurality of cylindrical rubber bars that are torsionally stressed to store energy. In these rods, one end is rotated relative to the other, thereby torsionally stressing each rod. The rubbers are torsionally stressed during braking of the vehicle and then drained off during acceleration. The existence of separate transmission paths for energy delivery to and transfer from storage devices leads to high cost and energy losses. Gill [10] provides an energy recovery system for automobiles with a common energy transmission path for both energy delivery and removal with respect to storage device. He torsionally stresses a tubular elastomeric storage member, as opposed to solid elastomeric bodies used by Hoppie. Jayner [11] uses solid elastomeric bodies that are longitudinally stressed to store energy. He uses an elastomeric membrane which absorbs and stores the energy of a moving vehicle that is decelerating by stretching itself and which can later transmit the stored energy to enable the vehicle to be accelerated, by contracting.

With the use of the technique for a regenerative brake device, as proposed in this work, some of the problems inherent with the above systems can be avoided. The application and the advantage it brings over these existing systems will be discussed in detail in this work. One of the biggest advantages of using elastomers for regenerative brake devices is that they can deliver and accept energy at near explosive like rates. A typical automobile battery has a capacity of about 80 Amp-hr at 12 V. This is roughly a Kilo-watt hour of energy or well over 2,500,000 foot-pounds of energy. A 3200 pound automobile traveling at 100 Mph has less than half that amount of kinetic energy. Why there are no simple regenerative brakes using a single electric battery in an automobile? The reason is that batteries cannot accept or deliver energy at high rates. To have a practical device using batteries one needs a huge bank of batteries such that the rates of energy input and output are small when one considers the battery mass. One does not have this difficulty with elastomeric materials. Elastic energy storage devices can have numerous applications as energy batteries. Possible applications besides regenerative brakes include starting devices, e.g. it is difficult to start diesel engines because of heat loss due to slow cranking speeds. One could use an electric battery to charge an elastomeric battery and then crank the engine at a very high speed for easy starting, especially at low temperatures. Obvious extensions to this idea are starters for small gasoline powered devices, such as lawn mowers. This would also make an interesting device for powering tools and toys, especially in applications where electric energy presents hazards that are unacceptable.

Thermomechanical Engines

The concept of discontinuous deformation in an elastomer can be used to design thermomechanical heat engines using an elastomer as the working substance, in a manner not utilized previously. A thermomechanical engine, which is the same as a heat engine, will involve the transformation of thermal energy into mechanical work. Elastomers possess unique gas-like thermodynamic properties, as has been established. This gas-like or entropic thermodynamic behavior is unique to long chain polymeric materials and discrete particulate matter such as gases whose behavior is governed not by internal energy but by statistical or entropic considerations. Gas thermodynamic cyclic processes form the basis for heat engines and heat pumps [12]. In a similar manner, elastomers have also been used to make rubber-heat-engines.

[FIGURE 1 OMITTED]

Rubber-heat-engines were first extensively studied in the 1930s by Weigand and Snyder [13, 14]. Farris [15] then presented his work (1977) on the potential use of rubbery solids as the "working fluid" in heat engine and heat pump applications. Lyon et.al [16] investigated the possibility of using synthetic elastomers as working substances in a heat engine at the laboratory and pilot scale. They obtained work of the order of 1 J/g under optimum conditions which exceeded natural rubber by a factor of 10. In a related area, mechanochemistry is a field of research concerned with the direct isothermal transformation of chemical energy into mechanical work. Katchalsky et al. have also built engines based on the above principle using regenerated collagen as the working substance [17, 18]. In these engines a continuous collagen belt uses the difference in chemical potential between a concentrated salt solution and a dilute salt solution to run the system. Polymers have also been synthesized which convert electromagnetic energy into useful work [19, 20]. The specific mechanism by which polymers interact with these diverse energy sources to do work depends on the physical and chemical structure of the individual polymer material. However, it is the ability of polymers and gases alike to store and release mechanical energy as changes of entropy, which provides the thermodynamic route to power production using these materials.

Rubber heat engines utilize the property that a stretched rubber shows a positive coefficient of force with temperature. In an isothermal heat engine cycle, such as Farris' and Lyon's [16] the rubber is stretched from [[lambda].sub.min] to [[lambda].sub.max] at a low temperature, [T.sub.lo]. This stretch between [[lambda].sub.min] and [[lambda].sub.max] is a continuous process and not a shock or discontinuous stretch. At [[lambda].sub.max], the rubber is heated to a temperature [T.sub.hi.]. Because of a positive coefficient of force with temperature for the stretched rubber, the load on the rubber increases. The rubber is then contracted at [T.sub.hi] in a continuous manner from [[lambda].sub.max] to [[lambda].sub.min]. At [[lambda].sub.min], the temperature of the rubber is brought down to [T.sub.lo] and the load then decreases back to the starting point. This process then continues in a cyclic fashion (Fig. 1). Heat is provided as input to the system and work is obtained as output. The work obtained per cycle is the area enclosed under the stress--strain curve.

The current design of a thermomechanical heat engine as proposed in this work is based on the same principle, a positive coefficient of force with temperature for a stretched rubber. The new feature introduced is that it is based on the principle of a discontinuous stretch between [[lambda].sub.min] and [[lambda].sub.max]. Consequently, the work that is obtained from this heat engine will not be dependent on the path of the stress-strain curve, as in the existing models, but will depend only on the stress and strain values at these points of discontinuity.

THEORY OF ENERGY DISSIPATION, BASED ON DISCONTINUOUS STRETCH AND CONTRACTION

This concept has been established in detail in the previous work [1]. Consider the device shown in Fig. 2. The device performs a cyclic operation. Here 1 and 2 are concentric pulleys and a rubber strip or belt is wound over them moving from one to the other when the pulleys rotate. The rubber strip is stretched and contracted sequentially at the jump points P and Q, respectively, which are the points of discontinuous deformation. At P, the rubber strip leaves the smaller pulley and gets stretched instantly when it leaves the smaller pulley. At Q, the strip contracts from the higher to the lower elongation when it is unwound from the larger pulley. In this cyclic process the work that is consumed in the jump stretch is not what is recovered in the contraction. The difference in work is dissipated as change in energy and heat, as shown in Fig. 3. This is the dissipation per unit volume of rubber belt subjected to the cyclic process. In a 100% ideal case, the dissipated energy per unit volume of a rubber belt subjected to a jump deformation process between states 1 and 2 is given as ([[sigma].sub.2] - [[sigma].sub.1])([[lambda].sub.2] - [[lambda].sub.1]). This is the difference in work between what goes into stretching and what is recovered on contraction. This was established in the previous publication. The objectives of this study are to utilize this and related properties of elastomers towards energy dissipative, energy storage, and heat engine applications.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

EXTENDING THE PRINCIPLE TO APPLICATIONS

Dissipative Applications

Nonfriction Brakes. The basis behind this application is the utilization of energy dissipation from a purely elastic material. Consider the device as shown in Fig. 2. The kinetic energy of a moving vehicle can be transmitted to a system which by means of simply stretching and contracting a rubber belt ends up dissipating this energy as heat, thus consuming energy without the use of friction brakes. The energy is consumed in true thermodynamic free extensions and free contractions, much like the throttling of a gas. The amount of energy that gets dissipated can be controlled by controlling the volume of rubber used and its distribution between the two stretch states. Unlike a friction brake, which produces very high skin temperatures because the dissipated energy is confined to such a small volume of material, these thermodynamic brakes dissipate the energy over the entire volume of the material. Consequently, these brakes result in relatively low temperature rises and the coolest component is the surface. In friction brakes the temperatures are very high with the temperatures being the highest at the surface. Figure 2 is not the only model for dissipation. A number of alternatives are possible, such as in Fig. 4. In this case, the two pulleys are of the same size, but do not rotate at the same speed. As the belt moves from the slower to the faster pulley it is jump-stretched and as it moves from the faster to the slower pulley it is jump-contracted. Over a cyclic process this leads to dissipation from the rubber belt.

Tension Controller. The above devices can also be utilized as tension controllers. It takes a constant torque to turn a device such as in Fig. 2 or 4. This is because the stretch and contraction points in the discontinuous deformation process occur by means of sudden jumps and not continuously. So the stress states in the belt are constant for a given region and vary over the length of the belt just as the stretch varies. Applications such as drawing of polymer sheets require a uniform tension to be maintained during the drawing process. Since the above device requires a constant torque to turn, it can be used for such applications.

Torque Converters and Alternative to Differentials. Consider the device shown in Fig. 5a and b. In Fig. 5a, the two small identical pulleys are connected by means of a chain. This ensures that the two systems rotate at the same angular velocity. The two large pulleys of different radii are connected by a rubber belt. The pulleys are connected to shafts as shown in Fig. 5b. The shaft connected to pulley B is the driving shaft, which is powered by a motor system. The torque delivered by this is transmitted to A which then transmits torque to the driven shaft, as can be seen from Fig. 5b. The driven shaft is connected to a moving system.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Below a certain critical torque, the driving torque is transmitted from B to A and there are no losses of energy because the rubber belt does not rotate relative to the pulley. When the driving torque reaches this critical value, the torque still gets transmitted but energy is also dissipated from the rubber belt. As discussed in the previous sections, the movement of the rubber belt will involve dissipation, since it involves a process of jump stretches and contractions of this elastomeric member. So a certain amount of the input energy from the driving shaft will be dissipated while the torque is transmitted to the driven shaft. There is thus a torque conversion involved in the process. Depending on the amounts of dissipation, the angular velocity of the output shaft is determined. The velocities for the input and output shafts will now be different and 100% transmission is no longer possible. This torque converter can be used in starting large massive conveyers, such as escalators. These are systems that require huge forces or torques to get started from rest. Once started, they are more easily maintained. In the above torque converter, below a critical torque there is a 100% conversion of the energy, hence once the system is up it will be running and the drive would be very efficient. There is a maximum torque transmitted before the system appears to slip. Once started, the system requires much less energy to be maintained in motion. Interestingly, on many of these massive systems that are driven by electric motors, an automotive hydraulic transmission is used between the motor and the conveyor to prevent electrical overload. The same thing can be done with a few pulleys and rubber belts. This type of design can be used in place of friction clutches in current "stick shift" automobile transmissions.

Two of the above torque converters can be used to replace differential gears in automobiles. Differential gears in automobiles are used in drive axle assemblies and are the means that make it possible for vehicles to move the wheels that drive the automobile such that one wheel can turn faster than the other in going around a corner. The axles are made in two separate pieces and each piece is connected at the middle to a differential gear assembly. This permits the two rear wheels to turn at different speeds, as is necessary in driving along a curved road or in making a sharp turn. With a mechanical differential the torque transmitted to each wheel is the same while the speeds of the wheels can be independent, however, the sum of the speeds is equal to the input speed. Differentials are also used in all wheel-drive vehicles to permit the front and rear wheels to turn at different speeds, thereby eliminating excess tire wear. One of the features of using differential gears is that when one of the drive wheels is lifted from the ground and the engine started, this wheel will freely spin while the other wheel on the ground will carry no torque. This is also what happens when one rear wheel is in soft mud or ice and the other wheel is on firm ground. The former will spin in the mud or ice, but the latter will not move. In our invention this problem will be overcome. This problem is solved in some designs by using limited-slip sensors to brake a freely rotating wheel.

Using two of the above torque converters, consider the driven shaft as the rear axles in an automobile. Let the drive shaft be connected to these axles by means of the torque converter system rather than a differential gear. Then the torque from the drive shaft is transmitted to the axles. But not all the energy is always transmitted, as discussed above, since there is also a dissipation mechanism involved when the torque reaches a critical value. In both cases, whether in using a differential gear system or the above invention, the input torque equals the sum of all outputs. With differential gears, if one wheel was on slippery ground, it would spin completely while no torque would be transmitted to the other wheel. This is because the torque transmitted to the two wheels has to be the same because of the mathematics governing differentials. This can be avoided with the present invention. In the present case, the torque values need not be equal. The two wheels can rotate at different speeds, since for both the axles there is also the heat mechanism which acts as an output. Thus there is always a two-wheel drive, somewhat equivalent to a positive traction differential, which is much more complex than an ordinary differential. With two different sources for dissipation, the two wheels can turn at different speeds, independent of each other, which is needed when the automobile turns around corners.

The above torque conversion principle enables this invention to be used in place of differential gears in automobiles or other vehicles. This is very useful since it replaces complex and heavy machinery with a much simpler device that uses an elastomer. Since the rotations of the two axles are independent of each other, it is now possible to have a wheel move on firm ground, even when the other is stuck in ice or thin mud. This is a very useful advantage. It is also possible to have variations in the design of the device used as torque converter. The above is one example. In another case, two rubber belts could be used rather than one chain and one rubber belt. Yet another variation could be the use of the devices as shown in Figs. 2 or 4 where a constant torque is used to turn the device and heat is dissipated over the loading-unloading cycle.

Storage Applications

Regenerative Brake Device. On braking of a moving machine, say an automobile, if the kinetic energy which is being lost can be used to stretch a large amount of rubber, then the kinetic energy gets transformed into potential energy of the stretched rubber. If this stored energy can be recovered to restart the machine when required, then that is a regenerative brake device. By utilizing the discontinuous deformation principle for an elastomer and by introducing intermediate steps for stretching and contraction to improve efficiency, the feasibility for the present invention as a regenerative brake device will be discussed.

[FIGURE 6 OMITTED]

On braking, if the kinetic energy is used to stretch an elastomer instantly from [[lambda].sub.1] to [[lambda].sub.2], then the energy per unit volume of elastomer that is transformed is that indicated in Fig. 6. This is due to the discontinuity of the stretch. To utilize this energy which can now be used to accelerate the vehicle, the elastomer will have to be unloaded. On contracting instantly from [[lambda].sub.2] to [[lambda].sub.1], the work or energy obtained per unit volume of elastomer will be that indicated in Fig. 7. This is not very large and there are huge losses involved. However, if instead of contracting directly from [[lambda].sub.2] to [[lambda].sub.1], we introduce a number of pulleys with varying sizes, decreasing from R to r, then we can obtain much more work returned. Consider the introduction of a pulley of size a such that R > a > r. Let R/a = [[lambda].sub.2]/[[lambda].sub.3]. Then the work that will be returned on contracting the rubber from [[lambda].sub.2] to [[lambda].sub.3] will be as shown in Fig. 8. On contracting from the second pulley to the third, i.e. from [[lambda].sub.3] to [[lambda].sub.1], further work will be obtained. Thus, the total work that is now returned is the sum of the two areas in Fig. 9, which is more than just the area in Fig. 7. Likewise, if the intermediate pulley was not used for the stretching process, then the work that gets transformed in stretching between [[lambda].sub.1] and [[lambda].sub.2] will be that shown in Fig. 6. However, on use of the intermediate pulley, the work transformed now will be that shown in Fig. 10.

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

If even more intermediate pulleys could be used, then the amount of work recovered is enhanced. In other words, if the unloading process was a continuous one from the higher extension to the lower, rather than a direct jump, then more work or energy is obtained. Ideally if infinite number of pulleys could be used to gradually reduce the elongation from [[lambda].sub.2] to [[lambda].sub.1], then the work returned will be the sum of infinite small rectangles, which will sum up to the area under the unloading curve (Fig. 11). This is the ideal scenario where the work that is recovered is equal to the area under the unloading curve. If this also equals the area under the loading curve, then there will be no dissipation. This is not practical, but still by using an appropriate number of pulleys, as close as possible to the maximum energy can be obtained.

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

The above invention is a regenerative brake system where large amounts of work are used to stretch an elastomeric member and a part of this work is recovered by contraction in a specific manner. One of the disadvantages with the existing systems, as mentioned before, is that all of the material is stretched at the same time. This requires a huge force. It also requires a massive structured system to contain a large mass of elastomer. In the present invention this will not be a problem because only a small amount of material will be stretched at a given time. Hence the loads required are small by comparison. Another disadvantage with the existing devices is that they provide nonuniform braking and energy return depending on the position on the stress--strain curve. The present invention overcomes this disadvantage too since it requires a constant torque to turn. Hence there is a uniform braking device and energy source. The other advantage that these elastomeric energy devices offer is that unlike electric batteries they can deliver and accept energies at near explosive like rates. A rubber sling shot was first used to power vehicles in high speed crash tests and only small amounts were needed. Elastomers have also been used to launch aircraft and gliders. This ability to deliver and accept energies at very high rates leads to the next application, the elastomer as a mechanical energy battery.

Mechanical Energy Batteries. This property of high speed energy delivery enables functions as a motor and as a starting device for other equipment. Cyclic loading tests performed on Spandex, a thermoplastic polyurethane elastomer, showed that these fibers provide energy of 20 J/g on contracting them from extensions of 500% [21]. By sequentially stretching and contracting these fibers in stages, close to the maximum energy will be obtained on unloading. Equipment like golf carts and aircraft food and beverage carts can be powered using this energy. Less than a pound of Spandex would be needed to power a typical aircraft food cart during flight. Steel springs are currently used to power specialized radios by bending (winding the steel in a manner similar to that used in windup clocks) the steel and using the energy that is returned when they are unwound. For the same mass, the energy from stretched Spandex can power the device eight times as long as steel because the energy density potential is that much higher in rubber than in the best steels. These mechanical batteries can be charged by stretching the elastomer manually or using electric motors. This can be done for a golf cart before proceeding onto the course or in the case of the aircraft food/beverage carts during the time between flights when the trolleys are loaded with food.

[FIGURE 12 OMITTED]

The rapid energy delivery capacity of these rubber motors make them very useful for starting devices that require huge power loads. An example is in the starting of diesel engines. Another example is for starting small gasoline motors such as those used on lawnmowers. Only a very small fraction of a pound of Spandex would be required for this type of applications. The Spandex can be slowly wound to charge the system and then have the energy released rapidly.

Yet another application for the rubber batteries can be as replacement for impact tools. The high energy delivery can be used for impact hammers/drills and related applications. The mechanical batteries have no fire hazard associated with them which makes them an effective replacement in fire sensitive areas where electric devices are not favored.

Thermomechanical Engine. Current work involves the design of a thermomechanical heat engine using Spandex and the principle of discontinuous deformation. This design utilizes the properties that stretched elastomers have a positive coefficient of force with temperature. Consider the design shown in Fig. 12. This looks similar to the design in Fig. 2, designed to study dissipation from the elastic belts, but the design in Fig. 12 is for a heat engine that utilizes heat as the input and provides mechanical work as the output.

There are two states of stretch in the system. The discontinuous stretch from [[lambda].sub.1] to [[lambda].sub.2] is performed cold while the contraction from [[lambda].sub.2] to [[lambda].sub.1] is done hot. The ratio of [[lambda].sub.1] to [[lambda].sub.2] is determined by the ratio of the pulley radii, i.e. r to R, as has been established elsewhere [1]. Because of the cold the load on the stretched portion of the rubber strip will be smaller than normal and similarly due to heat, the load on the contracted portion of the strip will be greater than normal. This is because stretched elastomers have a positive coefficient of force with temperature. The corresponding values for the stress will be given as [[sigma].sub.c] and [[sigma].sub.h], respectively as shown in Figs. 12 and 13. From Fig. 12, in order to have the jump stretch performed under cold and the jump contraction performed under hot, the system must have a net clockwise moment. The net clockwise moment per unit volume of material for the above system is,

M = [[sigma].sub.h][[lambda].sub.2] + [[sigma].sub.c][[lambda].sub.1] - [[sigma].sub.h][[lambda].sub.1] - [[sigma].sub.c][[lambda].sub.2] = ([[sigma].sub.h] - [[sigma].sub.c])([[lambda].sub.2] - [[lambda].sub.1])

Hence, the requirement for the above system to work, in other words to obtain a positive work output, will be to have a sufficient temperature difference that leads to [[sigma].sub.h] being greater than [[sigma].sub.c] for the chosen values of [[lambda].sub.2] and [[lambda].sub.1] (i.e. R and r).

[FIGURE 13 OMITTED]

This can be observed from Fig. 13. In the above system heat is provided as the input, i.e. the temperature difference between the hot and the cold portions and mechanical work is obtained as the output, i.e. the shaded region in Fig. 13. To maximize the work output, the shaded area, i.e. ([[sigma].sub.h] - [[sigma].sub.c])([[lambda].sub.2] - [[lambda].sub.1]) has to be maximized. If the difference between [[lambda].sub.2] and [[lambda].sub.1] is large, then the temperature difference might not be sufficient to obtain [[sigma].sub.h] greater than [[sigma].sub.c]. In other words the drop in load at [[lambda].sub.2] might not be more than the increase in load at [[lambda].sub.1]. In that case, the net work output will not be positive and the heat engine will not work. So in order to obtain maximum work output, the optimal value for ([[lambda].sub.2] - [[lambda].sub.1]) must be obtained and by providing an appropriate temperature difference, the difference between [[sigma].sub.h] and [[sigma].sub.c] should be obtained as large as possible.

A design of a heat engine along similar lines is in progress. The difference between this heat engine and the existing engines that also utilize rubber as the working substance is that in this engine there are only two states of deformation, [[lambda].sub.1] and [[lambda].sub.2], and these are obtained by a discontinuous shock jump. In existing engines the deformation between [[lambda].sub.1] and [[lambda].sub.2] is continuous and does not require a condition that [[sigma].sub.h] should be greater than [[sigma].sub.c]. All that those require is that the area under the cyclic stress-strain loop is positive (Fig. 1). In our model, effective temperature control and transfer to the selective portions of the working fiber is important to obtain [[sigma].sub.h] and [[sigma].sub.c] accurately. This is currently being studied.

CONCLUSIONS

The dissipation from elastic rubber belts arising due to discontinuous deformation mechanisms can be used in applications such as nonfriction brakes, tension controllers, and torque converters. By modifying the deformation process to include many steps and thereby make it more continuous, efficient energy storage devices can be created. Energy storage applications include regenerative brake devices and mechanical energy batteries that can be used as replacements for electric batteries. Ongoing work includes utilizing the property that elastomers have a positive coefficient of force with temperature to design a heat engine that uses heat as the input and produces mechanical work as output, in the form of discontinuous deformation of the elastomer.

ACKNOWLEDGMENTS

Assistance from the National Environmental Technology Institute (NETI) and the Materials Research Science and Engineering Center (MRSEC), University of Massachusetts, Amherst for their help in developing this work is greatly appreciated.

NOMENCLATURE
M moment per unit volume of material
r radius of small pulley
R radius of large pulley
T temperature
[lambda] elongation
[sigma] engineering stress
[omega] angular velocity

Subscripts
1 lower state of stretch
2 higher state of stretch
c cold
h hot
hi high
lo low
max maximum
min minimum


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Arun Raman, Richard J. Farris

Polymer Science and Engineering Department, University of Massachusetts Amherst, Amherst, Massachusetts 01003

Correspondence to: Richard J. Farris; e-mail: rjfarris@polysci.umass.edu
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