University Of Glasgow. School of Engineering. Department of Mechanical Engineering. Final Year Project

Transcription

1 University Of Glasgow School of Engineering Department of Mechanical Engineering Final Year Project Manufacture and Characterisation of Magneto-Rheological Elastomers By Mr Bruce Miller Supervised by Dr Philip Harrison and Ms Gerlind Schubert 1 Abstract This report deals with the manufacturing, testing and characterisation of Magneto Rheological Elastomers (MRE s). MRE s are a set of smart composite materials consisting of an elastomeric matrix with magnetic particles dispersed in said matrix. When a magnetic field is applied the stiffness of the material changes instantaneously and reversibly. This is due to the interaction of the magnetic particles dispersed in the matrix. This report deals with the manufacturing processes and how to optimize them for the best mechanical properties and most time efficient methods. Furthermore the method by which these materials are compression tested while a uniform magnetic field is applied will be dealt with. Finally the results from experiments will be used to compare the mechanical properties of MRE s with different volume percentages of magnetic particles, samples with Isotropic and Anisotropic particle structures and finally samples with applied magnetic fields. These results will be used in future work to generate a general constitutive model for these materials. List of Objecticves: Design a test setup to create uniform magnetic field on samples while compression testing Compression tests on samples with different curing times to optimize manufacturing technique Compression Tests on anisotropic and isotropic samples with different Volume percentage of Carbonyl magnetic particles (10%,20%,30%) Compression tests on anisotropic and isotropic samples under the influence of a magnetic field Model material behaviour using Abaqus FEA 1

2 Table of Contents List of Figures... 3 List of Equations... 3 List of Tables... 4 Glossary of Nomenclature... 4 Acknoledgements... 5 Introduction... 5 Background Information... 5 Manufacturing Process... 6 Test Method... 9 Test Setup Test Results Pure Rubber, 10%, 20%, 30% Isotropic samples cured for 24 Hours at 25 o C Compression of Pure Rubber with different curing conditions Compression of 10% volume Isotropic samples with different curing conditions Compression of 10%, 20% and 30% Anisotropic samples Compression of 10%, 20% and 30%, Anisotropic and Isotropic Samples with applied magnetic fields % Isotropic Samples % Isotropic Samples % Isotropic Samples % Anisotrpic Samples % Anisotropic Samples % Anisotropic Samples Modelling of MREs Modelling of 10% Isotropic Samples Modelling of 20% Isotropic Samples Modelling of 30% Isotropic Samples Modelling of Pure Rubber Conclusions Bibliography Appendix A : Calculation for Solenoids Appendix B: Faulty test setup results

3 List of Figures Figure 1 Picture of Heating Plates... 8 Figure 2 Picture of Electromagnet with mould in between poles... 9 Figure 3 Picture of Electromagnet... 9 Figure 4 Original Magnetic Setup Figure 5 Revised Magnetic Setup Figure 6 Rendered Drawing of Test Setup Figure 7 Picture of Test Setup Figure Figure 9 Microscopic Picture of 10% Isotropic Sample Figure 10 Microscopic Picture of 20% Isotropic Sample Figure 11 Graph of all Isotropic Sample Configurations Figure 12 Graph of Pure Rubber with different Curing Conditions Figure 13 Graph of 10%v CIP samples with different curing conditions Figure 14 Microscopic Picture of 10% Anisotropic Sample Figure 15 Buckling Samples Figure 16 Graph of all volume percentages of CIP for Anisotropic and Isotropic structures Figure 17 Graph of 10% Isotropic CIP Samples with different applied magnetic fields Figure 18 Graph of 20% Isotropic CIP Samples with different applied magnetic fields Figure 19 Graph of 30% Isotropic Samples with different applied magnetic fields Figure 20 Graph of 30% Anisotropic Samples with different applied magnetic fields Figure 21Graph of 20% Anisotropic samples with different applied magnetic fields Figure 22 Graph of 30% Anisotropic samples with different applied magnetic fields Figure 23 10% Isotropic Samples with 400mT applied magnetic field Figure 24 10% Isotropic Samples with 270mT applied magnetic field Figure 25 10% Isotropic samples no applied magnetic field Figure 26 20% Isotropic Sample with 400mT applied magnetic field Figure 27 20% Isotropic Samples with 270mT applied magnetic field Figure 28 20% Isotropic Samples with no applied magnetic field Figure 29 30% Isotropic Samples with 400mT applied magnetic field Figure 30 30% Isotropic Samples with 270mT applied magnetic field Figure 31 30% Isotropic Samples with no applied field Figure 32 Pure Rubber Modelling Curves List of Equations Equation 1 Equation for Mass of CIP []... 8 Equation 2 Standard Ogden Model Equation 3 Standard Mooney Rivlin Model Equation 4 Standard Neo Hookean Model

4 List of Tables Table 1 Rubber Poperties... 6 Table 2 CIP properties... 7 Table 3 Calculation of Masses... 7 Table 4 Youngs Moduli for all Isotropic Sample Configurations Table 5 Youngs Moduli of Pure Rubber with different Curing Times Table 6 Youngs Moduli of 10%v CIP samples with different curing times Table 7 Youngs Moduli for Anisotropic Samples Table 8 Table of Youngs Moduli for 10% Isotropic Samples Table 9 Table of Youngs Moduli for 20% Isotropic Samples Table 10 Table of Youngs Moduli for 30% Isotropic Samples Table 11 Table of Youngs Moduli for 10% Anisotropic samples Table 12 Table of Young s Moduli for 20% Anisotropic Samples Table 13 Table of Youngs Moduli for 30% Anisotropic samples Table 14 Table of Coefficients for Mooney-Rivlin Model Table 15 Table of Coefficients for Neo-Hookean Model Table 16 Table of Coefficients for Neo-Hookean Model Table 17 Table of Coefficients for Mooney-Rivlin Model Table 18 Table of R2 values for 30% Isotropic Samples Table 19 Youngs Modulus for 30% Isotropic Samples for Neo-Hookean Model Table 20 Youngs Modulus for 30% Isotropic Samples for Mooney-Rivlin Model Table 21 Youngs Moduli for Pure Rubber for Mooney Rivlin Model Table 22 Youngs Moduli for Pure Rubber Neo-Hookean model Glossary of Nomenclature MR effect- Magneto-Rheological effect whereby the stiffness of material increases with an applied magnetic field CIP- Carbonyl Iron Powder Mullins Effect- The instantaneous softening of rubbers when the all time maximum applied stress is reached FEA- Finite Element Analysis Magnetic Flux Density- This is the amount of magnetic field passing through a surface measured in Teslas 4

5 Acknoledgements I would like to thank Dr Harrison for helping me out during this project.he has always kept me organisedand wa always giving me new suggestions and some really good ideas to work on. I would also like to thank Gerlind Schubert for helping with all backround information on this topic as well as the analysis parts of the project. I would also like to thank John Davidson the materials lab technician. He really helped with the testing and the test setup. Introduction Magneto Rheological Elastomers are a set of smart composites whose mechanical properties can be reversibly and instantaneously changed when a magnetic field is applied. These materials can be used in systems where the ability to vary the stiffness of a component is required, such as vibration control systems and variable suspension systems in automobiles. Currently there is no available general constitutive model for these materials and the aim of Ms Schubert s PhD research project is to develop a constitutive model for these materials. The objective of this project is to undertake the preliminary compression testing of these materials to determine their stress strain behaviour. Furthermore the manufacturing process and testing procedures will be developed and refined throughout. The objective is to test all the relevant specimens and all configurations possible in order to assist with the development of the constitutive model. Background Information 1 Magneto Rheological Elastomers were first studied in 1995 by Toyota Central Research and Development Laboratories. They tested silicone gels with the magnetic particles aligned through dynamic shear experiments with small deformations. They noted the change in moduli due to the effects of the applied external magnetic field. Further research was carried out by Jolly and Carlsen in 1996 at the Thomas Lord Research Centre where they also tested silicon gels under small deformation shear with and without magnetic field and noted the change in storage modulus due to the applied magnetic field. The first applications for this kind of material were brought forth by the Ford Motor Company. They suggested and implemented the material as a suspension bushing thats stiffness could be altered to change ride comfort or handling quality of their motor cars. Kallio 2 carried out many experiments in 2003 in order to discover which materials were best to create an MR effect. Kallio conducted mainly small strain experiments and found that the best materials are Silicone Rubber Matrix with CIP magnetic particles. Farshad 3 in 2003 conducted compression testing up to 30% strain, using anistropic samples and isotropic samples for testing. 1 All Backround information has been researched through Ms Gerlind Schuberts 1 st Year Literature Revue 2 M. Kallio Preliminary tests on an MRE device 3 M. Farshad Magnetoactive elastomer composites 5

6 Furthermore testing with an applied magnetic field were conducted, although moving magnets were used which means further steps are needed in calculations to allow for the attractive forces of the magnets to be removed from the force measurements. Varga 4 in 2005 conducted similar compression testing but chose to use a solenoid coil to apply magnet fields to the samples. Applications for this material mainly involve vibration control as the forced response can be controlled by changing the stiffness components in springs. Manufacturing Process For this project a silicone rubber has been chosen as the matrix material along with Carbonyl Iron powder acting as the magnetic particle filler. The silicone rubber was chosen since has been the most widely used matrix material for MRE s in past research. Furthermore it has good mechanical properties and chemical resistance over a large range of temperatures. The rubber which is being used is defined by the manufacturer as MM240TV. Like most silicon rubbers it consists of two parts. Part A and Part B where part B acts as the hardener. The components are mixed at a ratio of 10:1. MM240TV has the properties defined in Table 1. Table 1 Rubber Poperties Figures from data sheet provided by ACC silicones Ltd Viscosity mpa.s Tensile Strength 5.4MPa Elongation at Break 330% Youngs Modulus 1.88MPa Hardness 40 o Shore A The magntic particle filler that has been chosen is Carbonyl Iron powder. This is a typical magnetic filler as it has high magnetic permeability, low remnant magnetisation and high saturation. 5 The CIP powder being used has the properties shown below in table 2 4 Z. Varga Magnetic Field Sensitive functional elastomers with tunable modulus 5 Dr Philip Harrison - Magneto-Rheological Elastomers: Manufacture, characterisation and modelling Presentation 6

7 Table 2 CIP properties Parameter Unit Specification Test method Iron Content g/100g Min 99.5 Calculated Carbon Content g/100g Max 0.05 IRS (RCA/Q-C-296) Permeability (µ i )* % RCA/Q C 302 Q-Value * a 100kHz % RCA/Q C 302 Particle Size Distribution D 50 µm Microtrac X 100 These values are based on the information provided by the provider BASF Ltd *Values are based on BASF s SQ standard There is a third component which has been added to the MRE that is being used for this project. It is defined by the manufacturer as ACC34 thinner. The reason that this is being used is that the rubber on its own has a high viscosity and this leads to the agglomeration of particles in the matrix. This in turn produces a non uniform distribution of particles. This can makes modelling much harder. Therefore this solvent thinner has been added at 30% weight to reduce the viscosity. This in turn allows for an even distribution of particles and also allows the particles to align with the magnetic fields easily. However adding the thinner will reduce the modulus of samples while no magnetic field is applied, this may lead to a higher MR effect. The amount of CIP that is required for testing is 10, 20 and 30 percent volume. The equivalent mass was calculated using the Table 3 and Equation 1 shown below. Table 3 Calculation of Masses Using 50g of Rubber Component A Therefore: Mass of part B = 0.1 x 50 = 5g Using 30 w% of Solvent therefore: Mass of solvent = 0.3 x (50+5) = 16.5g 7

8 Equation 1 Equation for Mass of CIP [6] Where v = relative volume of CIP and x = required mass, 71.5g is the total mass of pure rubber 1.06g/cm 3 is density of rubber and g/cm 3 is density of CIP. The next stage of the process was to mix the mixture thoroughly using a hand mixer for a minimum time of three minutes. The mixture is then degassed in a vacuum chamber for ten minutes in order to avoid air bubbles when the rubber is cured. The mixture is then poured into moulds. These moulds are designed according to the standard sample size for test method A of BS ISO The sample size required is 29mm ± 0.5mm diameter and 12.5mm ± 0.5mm height. After the rubber has been moulded it requires time to cure, 24 hrs at 25 o C or 1 hrs at 100 o C as defined by the manufacturer. In order to fast cure the rubber at 100 o C the heating plates shown in Figure 2 are used. Figure 1 Picture of Heating Plates 6 Equations and Calculations are from Guide for rubber mixing process by Ms Gerlind Schubert 7 BS ISO 7742 Determination of Compression stress strain properties for Vulcanized Rubbers 8

9 One of the objectives for this report is to optimize the manufacturing process therefore samples will be fast and slow cured to find the optimal curing time. In order to create anisotropy within the samples they will be cured while under the influence of a magnetic field produced by an electromagnet. As shown in figures 3 and 4. Figure 3 Picture of Electromagnet Figure 2 Picture of Electromagnet with mould in between poles This will create chains of magnetic particles within the samples. For this project a magnetic field density of 400mT (milli Teslas) was used during curing. This value was measured using a gauss meter at the centre of the poles with no medium present. Unfortunately the amplifier tends to get hot and becomes overloaded after time, this meant that the higher magnetic field densities that were measured (max of 1 Tesla) could not be used. 400mT was found to be a safe value in order to cure the samples for the necessary time without overloading the amplifier. Furthermore other research projects have successfully used similar field densities for creating anisotropy for example Varga 2006 [4]. Test Method For this project the British Standard for Determination of compression stress-strain properties for Rubber, vulcanized or thermoplastic (BS ISO 7743:2008) is being used. Test method A from the standard has been selected and requires that the compression plates be lightly coated in a film of lubricant. Furthermore the test requires four cycles at a speed of 10mm/min. From the recorded force displacement data the stresses and strains can be calculated. The standard also requires a minimum of three samples for each sample configuration. 9

10 Test Setup In order to test these materials with an applied magnetic field it is necessary to create a device or alternative setup to allow the magnetic field to pass through the samples while they are compressed. A paper by Varga [4] indicated that a solenoid coil was used to implement a uniform magnetic field during testing. The aim of this project was to use quite a high field density of 400mT, and after some calculations (see appendix A) it was decided that the required coil would be expensive and impractical for use. Another paper by Farshad [3] indicated that a pair of permanent magnets could be used to generate a uniform field through the sample. The test setup used by Farshad [3] indicated that the plates be wedged between two aluminium plates while the compression took place, thereby the magnets would move up and down with the compression. It was decided that this method was inaccurate due to the fact that the attractive force between the magnets would increase as the magnets were brought closer together. This would add false readings to the load cell and would mean that additional steps would have to be taken to interpret the results correctly. Therefore a new test setup was designed and assembled using neodymium permanent magnets. The technical drawing is shown in figure 4. Unfortunately when this setup was used there were some problems with the test results (see appendix B). Therefore the setup was reworked again the revised setup is shown in figure 5. The magnetic field density can be altered simply by making the distance between the magnets larger or smaller. During testing a magnetic field density of 400mT was measured between the poles at a separation of 36mm and this decreased to around 270mT at a distance of 47mm. From the drawings provide you will see that the magnets are held in place as the compression plates are free to move. 10

11 Figure 4 Original Magnetic Setup Figure 5 Revised Magnetic Setup 11

12 Figure 7 Picture of Test Setup Wooden Blocks Magnets Clamps Figure 6 Rendered Drawing of Test Setup 12

13 Test Results The tests that have been completed are: Compression of Pure rubber, 10%, 20%, 30% volume CIP cured for 24 hours at 25 o C. Purely Isotropic samples. Compression of Pure Rubber cured for 1 hour and 1.5 hours at 100 o C. Compression of 10% volume cured for 1 hour and 1.5 hours at 100 o C. Purely Isotropic samples. Compression of 10%, 20% and 30% Anisotropic samples cured for 1 hour at 100 o C under a magnetic field of 400mT. Compression with applied magnetic fields using samples of 10%, 20%, 30% volume CIP cured for 1 hour at 100 o C. Purely Isotropic samples. Compression with applied magnetic field using samples of 10%, 20%, 30% volume CIP cured for 1 hour at 100 o C under field of 400mT magnetic field. Purely Anisotropic Samples During the testing it was noticed that the first cycle of loading showed larger force than the subsequent three cycles. This is effect is known as the Mullins effect and is typical for rubbers. This effect is defined as: The Mullins Effect can be idealized for many purposes as an instantaneous and irreversible softening of the stress-strain curve that occurs whenever the load increases beyond its prior all-time maximum value. At times when the load is less than a prior maximum, nonlinear elastic behaviour prevails. [8] The main theory for why this occurs says that it is caused by the breaking of cross links in the matrix materials, thereby reducing the overall cross-linking density and therefore the stiffness of the material. This effect is illustrated by the full compression cycle test shown in figure 8. As you can see the first cycle is higher than the others and all the others are equal. 8 Reference from ( 13

14 Figure 8 Cycle 1 Upload Cycle 2-4 Upload All Unload Parts Due to this effect for the comparisons of all tests the upload part of the third cycle will be used. A mean curve will then be calculated from the third cycles of the four samples tested. This ensures that there will be no softening after this loading cycle and therefore provides more accurate data for creating a constitutive model. This of course means that if these materials are ever used in a practical application there will need to be a degree of conditioning before entering service. This would require the material to be loaded to a higher stress than that which would be expected to encounter during service. Pure Rubber, 10%, 20%, 30% Isotropic samples cured for 24 Hours at 25 o C The samples mixed for this testing regime were cured for 24 hours at 25 o C as prescribed by the manufacturer. The samples produced the stress-strain data shown in figures 11 and table 4. Figure 8 below shows the structure of an isotropic sample loaded with 10% volume CIP at a magnification factor of x20 using a microscope, the white flecks are CIP clusters. This photograph shows that there is no order to the distribution of particles. Figure 9 shows a 20% volume isotropic sample of the same magnification. From figure 9 it can be seen that there is a higher density of white flecks and therefore CIP. 14

15 Figure 9 Microscopic Picture of 10% Isotropic Sample Figure 10 Microscopic Picture of 20% Isotropic Sample 15

16 Figure 11 Graph of all Isotropic Sample Configurations Table 4 Youngs Moduli for all Isotropic Sample Configurations Sample Composition Cure Time (hrs) Mean Youngs Modulus (Mpa)for 0-10% strain Percentage Increase compared to pure rubber Pure % % % Note that for comparing Young s moduli we are considering only the 0-10% strain region due to the fact that this region has very linear stress-strain behaviour and therefore a relatively constant gradient, i.e. young s modulus, unlike the higher strain regions where the behaviour is very much non-linear. As expected the higher volume percentage of CIP produces higher stiffness s. The choice of material for a specific application will depend on the required stiffness but also the limit of weight. 16

17 Compression of Pure Rubber with different curing conditions The samples for this set of tests were cured at 100 o C for 1 hour, 100 o C for 1.5 hours and 25 o C for 24 hours. This is to check to see if there is a significant difference in stiffness for different curing times in order to optimize the manufacturing process. Figure 12 Graph of Pure Rubber with different Curing Conditions Table 5 Youngs Moduli of Pure Rubber with different Curing Times Sample Composition Cure Time (hrs) Mean Youngs Modulus (Mpa)for 10% strain % increase compared to 24 hrs Increase Factor Pure Pure Pure From these results the fast curing process produces higher stiffness s. This may be due to additional energy being provided by the heating process, creating more intermolecular bonding and therefore producing higher stiffness. Furthermore the 1.5 hours cured samples have slightly higher stiffness than the 1 hour cured since the additional heat energy is provided for a longer period. 17

18 Compression of 10% volume Isotropic samples with different curing conditions This set of test has been done to see if the stiffness of filled samples is altered by the additional heat energy provided as shown in the pure rubber samples shown before. The samples as before have been cured for 1.5 hours, 1 hour and 24 hours. These results are very important to optimize the manufacturing process. Figure 13 Graph of 10%v CIP samples with different curing conditions Table 6 Youngs Moduli of 10%v CIP samples with different curing times Sample Cure Time Composition (hrs) Mean Youngs Modulus (Mpa)for 10% strain Percentage increase compared to 24 hrs Increase Factor 10% % %

19 From these results it can be seen that the addition of heat has very little affect on the stiffness of the samples, there is only a 5% increase in stiffness. Therefore unlike the pure rubber there is only a small amount of additional bonding caused by the additional energy provided. This means that the interaction between the filler and the matrix does not allow for strengthening of the matrix, it can therefore be assumed that most of the heat energy is absorbed by the CIP particles. This may be due to a higher thermal conductivity. From these results all future samples will be cured for 1 hour at 100 o C. Compression of 10%, 20% and 30% Anisotropic samples These samples have been cured for 1 hour at 100 o C under a magnetic field of 400mT. This field strength was used mainly because the amplifier being used overloads after time while high currents are used. The value of 400mT was a safe magnetic field density that could be achieved for the full curing time needed. Furthermore other studies (VARGA [4]) have shown that 400mT is adequate to create anisotropy. Figure 14 shows the structure created by applying the magnetic field during curing. As you can see there are distinct chains of particles as opposed to random orientation shown in figure 8 and 9. Figure 14 Microscopic Picture of 10% Anisotropic Sample Direction of Alignment 19

20 There have been some problems with the compression tests for this anisotropic structure. The rubber samples when compressed actually buckled which is not supposed to happen the rubber should remain at a constant volume and therefore the diameter of the sample should increase with decreasing height. Figure 15and 16 shows one of the samples as it buckles. Figure 15 Buckling Samples The result from this buckling is a very steep upload at the beginning of the cycle then it flattens out. Interestingly this behaviour is very consisitent for all the samples. At first it was suspected that because the moulds between the poles of the electromagnet were not completely covered by the pole and it was thought that the flux density would be weaker at the edge of the poles. To stop this only one mould at a time was cured between the magnetic poles. Unfortunately this did not solve the problem. It was then discovered that the moulds that were being used had deformed due to the heat and pressure of the curing process. New top plates were manufactured with additional screw holes for extra security. Unfortunately this still did not stop the buckling. It may be that the shear modulus is much less than the compressive modulus. In order to stop the buckling occurring, the film of oil on the sample during testing was removed. This did stop the buckling but unfortunately this causes additional friction on the sample during testing which means that the force displacement will not be comparable with the other tests. The test results are shown in figure 14 and table 7, but it must be stated that these results will not be an accurate representation of the anisotropic samples. 20

21 Figure 16 Graph of all volume percentages of CIP for Anisotropic and Isotropic structures Table 7 Youngs Moduli for Anisotropic Samples Sample Composition Particle Orientation Mean Youngs Modulus (Mpa)for 10% strain % increase compared to equivalent isotropic 30% Anisotropic % Anisotropic % Anisotropic % Isotropic % Isotropic % Isotropic

22 Interestingly it can be seen that the higher percentages of CIP produce larger increases in the Young s modulus. This will be due to denser particle chains created by the magnetic field. The higher concentration of ferromagnetic elements will create a higher magnetic flux density within the samples when curing, meaning that the individual particles will be magnetically attracted towards each other creating thicker and longer chains. Due to the fact that buckling is a factor in the testing of Anisotropic samples, it may be more poignant to perform tensile tests instead of compression to discover the mechanical properties that Anisotropy creates. Compression of 10%, 20% and 30%, Anisotropic and Isotropic Samples with applied magnetic fields This set of tests will deal with the effect of applying a magnetic field through the samples while being compressed. The magnetic field should increase the stiffness of the samples and the higher volume percentages should produce a bigger MR effect. Furthermore the Anisotropic samples should produce a bigger increase compared to the Isotropic samples. The increase in stiffness is caused by the interaction of the magnetic particles in the matrix. The magnetic flux causes a magnetic dipole interaction between every particle in the matrix. In the Anisotropic samples the magnetic particles will be closer together, thereby increasing the attractive force between them as attractive force is a function of distance. 22

23 10% Isotropic Samples Figure 17 Graph of 10% Isotropic CIP Samples with different applied magnetic fields Table 8 Table of Youngs Moduli for 10% Isotropic Samples Sample Composition 0-10% Youngs Flux Density Modulus 10% % % As we can see the results are not as expected. The applied magnetic field should increase stiffness not decrease it. The reason for this is currently unknown. It is a completely unexpected result. The curves indicate that the samples with the 400mT applied field have higher stress values than the 270mT but the gradient of these lines i.e. Young s Modulus indicate that the stiffness is lower for the higher Flux Density. This is contrary to work that has been carried out previously. Furthermore the zero field samples, especially at higher strains, have much higher stresses than those samples with an applied field. The only thing that changed between the magnetically tested samples and the no field samples was that for the no field samples a standard compression setup for the Zwick Z250 test machine was used. That particular setup involved less pinned joints and fittings. These fittings may have been slightly loose at certain points and may have resulted in a slightly decrease in the measured force. 23

24 20% Isotropic Samples Figure 18 Graph of 20% Isotropic CIP Samples with different applied magnetic fields Table 9 Table of Youngs Moduli for 20% Isotropic Samples Sample Composition Flux Density (mt) 0-10% Youngs Modulus (MPa) 20% % % From figure 18 and table 9, we can see a similar result from the 10% isotropic samples in that the samples with the applied magnetic fields have lower stiffness s than the no field samples. This would support the theory that the magnetic test setup is slightly flawed in some way. This sample however shows that the Youngs modulus did increase slightly for a higher magnetic field. From these results I would assume that the 10% samples have become magnetically saturated and this may have had a detrimental effect on the mechanical properties. The 20% samples, having a higher Ferro-Magnetic content can have higher flux densities applied to them without saturating and therefore maintain an increase in stiffness. 24

25 30% Isotropic Samples Figure 19 Graph of 30% Isotropic Samples with different applied magnetic fields Table 10 Table of Youngs Moduli for 30% Isotropic Samples Sample Composition Flux Density (mt) 0-10% Youngs Modulus (MPa) 30% % % From the curves and values shown in figure 19 and table 10 respectively, we can see that the problem of the zero field samples having a higher stiffness than those with the applied magnetic field is still partly true as the 270mT field samples still have lower stiffness. However we are starting to see a larger MR effect as the stiffness of the 400mT sample is higher than the other two configurations. This shows that for higher magnetic particle loading we are getting a larger MR effect. If the problem with the magnetic test setup had not been present, we would have seen that the 270mT field samples curve would fall inbetween the other two configurations. 25

26 10% Anisotrpic Samples Figure 20 Graph of 30% Anisotropic Samples with different applied magnetic fields Table 11 Table of Youngs Moduli for 10% Anisotropic samples Sample Composition Flux Density (mt) Youngs Mod for 0-10% Strain MR Effect 10% % % From the results in figure 20 and table 11 we can see that compared to the Isotropic samples we are seeing a larger MR effect, this is due, as stated previously, to the magnetic particles being closer together. Still present is the fact that the zero field samples are producing higher stresses than the 270mT applied field samples. From the curves we can also see that there is a steep gradient in the 0.05 to 0.1 strain region followed by a levelling off of the curve after this region. This may have been caused by buckling but it is unclear at this point. 26

27 20% Anisotropic Samples Figure 21Graph of 20% Anisotropic samples with different applied magnetic fields Table 12 Table of Young s Moduli for 20% Anisotropic Samples Sample Composition Flux Density (mt) Youngs Mod for 0-10% Strain (MPa) MR Effect 20% % % As with 10% Anisotropic Samples we are seeing that the zero field samples have a similair than the 270mT field samples. Again it is suspected that a loose fitting in the magnetic test setup is to blame. We are also seeing that at higher strains for the zero field samples have a slightly higher stresses than the 400mT samples. The upload curves however show that all samples have very similar upload behaviour up to around 30% strain. This is unexpected as we would have expected larger differences in all curves for this sample configuration. 27

28 30% Anisotropic Samples Figure 22 Graph of 30% Anisotropic samples with different applied magnetic fields Table 13 Table of Youngs Moduli for 30% Anisotropic samples Sample Composition Flux Density (mt) Youngs Mod for 0-10% Strain (MPa) MR Effect 30% % % The curves shown in figure 22 are like what was expected, we can see that the lowest curve is the zero field samples, with the red line representing the 270mT field samples and just above that the 400mT field samples. There is very little difference in the stresses off the two magnetic samples. This graph is a good indicator of the sort of MR effect we would hope to achieve in practice with these materials. The Young s moduli figures shown in table indicate a larger MR effect than previous samples configurations. 28

29 Modelling of MREs In this report the modelling of the Isotropic samples will be concentrated on. The main reason is the experimental problems with the anisotropic samples which need to be solved first. I am using for analysis the Abaqus 6.8 Student Edition. Already implemented models for rubber-like materials are the Neo-Hookean, the Mooney-Rivlin and the Ogden Model which are all models for isotropic materials. Modelling an anisotropic material would require a user defined constitutive equation using UMAT. This would go beyond the scope of this report. A further assumption of the said implemented models is that the materials are fully incompressible i.e. the volume does not change during deformation and the poisson ratio is 0.5. Furthermore G the shear modulus is related to the Elastic modulus E by the equation, region., but this is only valid on the linear theory,i.e. in the small strain The standard hyper elastic models which will be used are the, Neo-Hookean and Mooney Rivlin. All these models are derived from the standard Ogden model shown in Equation 2. This model is based on the relationship between the Strain Energy Function 9 and the principal stretch ratios. Equation 2 Standard Ogden Model Where is the strain energy and, are the principal stretch ratios for each direction. is defined as the shear modulus, and is a material constant. The Mooney Rivlin model is obtained by setting N=2, = 2, = -2. This produces Equation 3 shown below. Equation 3 Standard Mooney Rivlin Model Where c 10, c 01, and the shear modulus is µ = The Neo-Hookean model is derived from the standard Ogden model by setting N=1, = 2, this results in Equation 4. Equation 4 Standard Neo Hookean Model 9 Equations from Non-Linear Solid Mechanics by Gerhard A Holzapfel 29

30 Where c 1 and the shear modulus is The Young s modulus can be calculated from the coefficients calculated by Abaqus shown by these equations: These constitutive model need to be fitted to experimental data to calculate the model parameters. This fitting will be done by Abaqus using a least square method. To get an idea of the suitability of the fit a coefficient of determination R 2 will be calculated. This coefficient is often used in statistics. It approaches 1 the better the fit, a value of 0 would mean the fit is useless. res fit exp norm S tot R res1 res... resn ( 2 Stot i _ 2 exp, i exp) S norm tot 2 2 Where res are the residuals, norm is the 2-Norm also known as the Euclidian length of the residuals, S tot is the sum of squares and finally R 2 is the coefficient of determination. Modelling of 10% Isotropic Samples Figure 23 10% Isotropic Samples with 400mT applied magnetic field 30

31 Coefficients for 10% Isotropic samples with 400mT applied magnetic field HYPERELASTICITY - MOONEY-RIVLIN STRAIN ENERGY D1 C10 C HYPERELASTICITY - NEO-HOOKEAN STRAIN ENERGY D1 C10 C Figure 24 10% Isotropic Samples with 270mT applied magnetic field Coefficients for 10% Isotropic Samples with 270mT applied magnetic field HYPERELASTICITY - MOONEY-RIVLIN STRAIN ENERGY D1 C10 C HYPERELASTICITY - NEO-HOOKEAN STRAIN ENERGY D1 C10 C

32 Figure 25 10% Isotropic samples no applied magnetic field Coefficients for 10% Isotropic Samples with no applied magnetic field HYPERELASTICITY - MOONEY-RIVLIN STRAIN ENERGY D1 C10 C HYPERELASTICITY - NEO-HOOKEAN STRAIN ENERGY D1 C10 C

33 From Figure 23 we can see that the best model fit is the Mooney-Rivlin. This was confirmed by the R2 coefficient which was calculated as for the Mooney-Rivlin and for the Neo-Hookean. From Figure 24 we can see that both models are quite good for most strain levels. The R2 coefficient for the Mooney Rivlin model the R2 value was and for the Neo-Hookean model R2 is this shows that the Mooney-Rivlin is still the better fit for this material configuration with and without magnetic fields. From Figure 25 we can also see that the Mooney Rivlin model is still the best fit, and the R2 values confirm this as they are for the Mooney-Rivlin and for the Neo- Hookean. Finally the Mooney-Rivlin Model is the best fit for this material tested with and without magnetic field. Using the coefficients shown previously Table 14 and Table 15 were constructed to show the differences in the Young s moduli from test data and coefficients. Table 14 Table of Coefficients for Mooney-Rivlin Model Samples Composition Applied Field (mt) C10 Coefficient C01 Coefficient Shear Modulu s (MPa) Youngs Modulu s (MPa) Youngs Modulus form test data (MPa) 10% % % Table 15 Table of Coefficients for Neo-Hookean Model Samples Compositio n Applied Field (mt) C10 Coefficient Shear Modulus (MPa) Youngs Modulus (MPa) Youngs Modulus form test data (MPa) 10% % %

34 Modelling of 20% Isotropic Samples Figure 26 20% Isotropic Sample with 400mT applied magnetic field Coefficients for 20% Isotropic Samples with 400mT applied magnetic field HYPERELASTICITY - MOONEY-RIVLIN STRAIN ENERGY D1 C10 C HYPERELASTICITY - NEO-HOOKEAN STRAIN ENERGY D1 C10 C

35 Figure 27 20% Isotropic Samples with 270mT applied magnetic field Coefficients for 20% Isotropic Samples with 270mT applied magnetic field HYPERELASTICITY - MOONEY-RIVLIN STRAIN ENERGY D1 C10 C HYPERELASTICITY - NEO-HOOKEAN STRAIN ENERGY D1 C10 C

36 Figure 28 20% Isotropic Samples with no applied magnetic field Coefficients for 20% Isotropic Sample with no applied magnetic field HYPERELASTICITY - MOONEY-RIVLIN STRAIN ENERGY D1 C10 C UNIAXIAL TENSION: UNSTABLE AT A NOMINAL STRAIN LARGER THAN UNIAXIAL COMPRESSION: UNSTABLE AT A NOMINAL STRAIN LESS THAN BIAXIAL TENSION: UNSTABLE AT A NOMINAL STRAIN LARGER THAN BIAXIAL COMPRESSION: UNSTABLE AT A NOMINAL STRAIN LESS THAN PLANAR TENSION: UNSTABLE AT A NOMINAL STRAIN LARGER THAN PLANAR COMPRESSION: UNSTABLE AT A NOMINAL STRAIN LESS THAN VOLUMETRIC TENSION: STABLE FOR ALL VOLUME RATIOS VOLUMETRIC COMPRESSION: STABLE FOR ALL VOLUME RATIOS 36

37 HYPERELASTICITY - NEO-HOOKEAN STRAIN ENERGY D1 C10 C From Figure 26 we can see that again the Mooney-Rivlin Model is the best fit for this sample configuration. This is confirmed with the R2 coefficient being for the Mooney-Rivlin and for the Neo-Hookean. Figure 27 shows that for lower strains both models are quite accurate but as strain increases the Neo-Hookean curve starts to move away from the test data. The Mooney-Rivlin is still the best model for this material. Furthermore this is confirmed with the R2 coefficient, it is for the Mooney-Rivlin and for the Neo-Hookean. In Figure 28 we are starting to see a few discrepancies between the Mooney-Rivlin model and the test data especially at higher strains. Even so the Mooney-Rivlin is still the best fitting model for this data as shown again by the R2 coefficient. For the Mooney-Rivlin R2 is and for the Neo-Hookean R2 is Unusually the Abaqus has stated the Mooney-Rivlin Model as unstable. Table 16 Table of Coefficients for Neo-Hookean Model Samples Compositio n Applied Field (mt) C10 Coefficient Shear Modulus (MPa) Youngs Modulus (MPa) Youngs Modulus form test data (MPa) 20% % % Table 17 Table of Coefficients for Mooney-Rivlin Model Samples Compositi on 37 Applied Field (mt) 20% % 270 C10 Coeffici ent C01 Coefficien t Shear Modulus (MPa) Youngs Modulus (MPa) Youngs Modulus form test data (MPa) %

38 Modelling of 30% Isotropic Samples Figure 29 30% Isotropic Samples with 400mT applied magnetic field Coefficients for 30% Isotropic Samples with 400mT applied magnetic field HYPERELASTICITY - MOONEY-RIVLIN STRAIN ENERGY D1 C10 C HYPERELASTICITY - NEO-HOOKEAN STRAIN ENERGY D1 C10 C

39 Figure 30 30% Isotropic Samples with 270mT applied magnetic field Coefficients for 30% Isotropic Samples with 270mT Magnetic Field HYPERELASTICITY - MOONEY-RIVLIN STRAIN ENERGY D1 C10 C HYPERELASTICITY - NEO-HOOKEAN STRAIN ENERGY D1 C10 C

40 Figure 31 30% Isotropic Samples with no applied field Coefficients for 30% Isotropic Samples with no applied magnetic field HYPERELASTICITY - MOONEY-RIVLIN STRAIN ENERGY D1 C10 C ***WARNING: UNSTABLE HYPERELASTIC MATERIAL UNIAXIAL TENSION: UNSTABLE AT A NOMINAL STRAIN LARGER THAN UNIAXIAL COMPRESSION: UNSTABLE AT A NOMINAL STRAIN LESS THAN BIAXIAL TENSION: UNSTABLE AT A NOMINAL STRAIN LARGER THAN BIAXIAL COMPRESSION: UNSTABLE AT A NOMINAL STRAIN LESS THAN PLANAR TENSION: UNSTABLE AT A NOMINAL STRAIN LARGER THAN PLANAR COMPRESSION: UNSTABLE AT A NOMINAL STRAIN LESS THAN VOLUMETRIC TENSION: STABLE FOR ALL VOLUME RATIOS VOLUMETRIC COMPRESSION: STABLE FOR ALL VOLUME RATIOS 40

41 HYPERELASTICITY - NEO-HOOKEAN STRAIN ENERGY D1 C10 C From these simulations we can see that the Mooney-Rivlin Model is a good model for the samples where a magnetic field is present. However when there is no magnetic field present this model becomes unstable. The R2 values for all the cycles are shown in Table 18. This means that this model will have to be altered in order to be used for simulations. Table 19 and Table 20 show a comparison of the simulated young s moduli to the calculated Modulus from the test data. Table 18 Table of R2 values for 30% Isotropic Samples Samples Composition Applied Field Strength (mt) R2 for Mooney Rivlin R2 for Neo- Hookean 30% % % Samples Compositio n Table 19 Youngs Modulus for 30% Isotropic Samples for Neo-Hookean Model Applied Field (mt) C10 Coefficient Shear Modulus (MPa) Youngs Modulus (MPa) Youngs Modulus form test data (MPa) 30% % % Samples Compositi on Table 20 Youngs Modulus for 30% Isotropic Samples for Mooney-Rivlin Model Applied Field (mt) 30% % 270 C10 Coefficient C01 Coefficient Shear Modulus (MPa) Youngs Modulus (MPa) Youngs Modulus from test data (MPa) %

42 Modelling of Pure Rubber Figure 32 Pure Rubber Modelling Curves Coefficients for Pure Rubber Testing HYPERELASTICITY - MOONEY-RIVLIN STRAIN ENERGY D1 C10 C HYPERELASTICITY - NEO-HOOKEAN STRAIN ENERGY D1 C10 C

43 Table 21 Youngs Moduli for Pure Rubber for Mooney Rivlin Model Samples Compositi on Applie d Field (mt) C10 Coefficient C01 Coefficient Shear Modulu s (MPa) Youngs Modulus (MPa) Youngs Modulus form test data (MPa) Pure Samples Compositio n Applie d Field (mt) Table 22 Youngs Moduli for Pure Rubber Neo-Hookean model C10 Coefficient Shear Modulus (MPa) Youngs Modulus (MPa) Youngs Modulus form test data (MPa) Pure The pure rubber is like all the rest of the samples that have been modelled in that the best fit is with the Mooney-Rivlin model. The R2 values are for the Mooney-Rivlin model and for the Neo-Hookean. In conclusion it would appear that the best model for this material is the Mooney-Rivlin model, it has consistently close to 1 R2 value providing a good fit. Conclusions The main conclusions of this report are that the manufacturing process has been optimized and now all future samples will be cured for 1hrs at 100 o C. Furthermore it is now possible to successfully create Anisotropy within the MRE s. Another main point from these set of test is that a method whereby a magnetic field is passed through the sample while being compressed has been achieved. Some of the problems that have occurred unexpectedly during these tests were the buckling of anisotropic samples. This means that in future it would be best to test the anisotropic samples in tension rather than compression. Unfortunately applying a magnetic field through the sample will be difficult and a new test setup will have to be created. Future work on this material will be carried out to create a general constitutive model, and to achieve this task many more tests will have to be undertaken to fully define the MRE s mechanical properties. 43

44 Bibliography [1] Gerlind Schuberts 1 st Year Report [2] M. Kallio Preliminary tests on an MRE device [3] M. Farshad Magnetoactive elastomer composites [4] Z. Varga Magnetic Field Sensitive functional elastomers with tunable modulus [5]Dr Philip Harrison, MRE Presentation [6] Equations and Calculations are from Guide for rubber mixing process by Ms Gerlind Schubert [7 ] BS ISO 7742 Determination of Compression stress strain properties for Vulcanized Rubbers [8] Reference from ( [9] Equations from Non-Linear Solid Mechanics by Gerhard A Holzapfel [10]Picture from [11] Referenced from (AWG 23) 44

45 Appendix A : Calculation for Solenoids First a target Magnetic Flux Density of B = 400 mteslas was set. This is a relatively high magnetic field but will be suitable for all experiments that need to be done. Furthermore the flux density can be changed by altering the current supply. The max current that the power supply can safely produce is 5A. Figure A1 [10] : A typical solenoid field. Lines represent magnetic field lines. Dots and X s are coil cross-section. Figure A1 shows a wire coil in cross-section and the arrowed lines represent the magnetic field lines. This shows that the field through a solenoid is very uniform and will be ideal for use when compressing samples. The basic equation for calculating the magnetic flux density at a inside the coil at a point away from the ends is given as: Where: B = magnetic flux density Unit is Tesla µ = Kµ 0 where k = relative permeability of core material and µ 0 =magnetic constant and is the permeability of a vacuum µ 0 = 4π x 10-7 unit is Tesla Metre per Ampere N =number of turns L = length of solenoid I = Current Unit is Amperes 10 Picture from 45

46 Therefore what is required to build this solenoid is the turns density N/L. With this the appropriate coil can be designed and manufactured. The core material is going to be air which has K=1 B (Tesla) I (Amperes) K µ (T.m/Ampere) e-006 Therefore: =1.5915e+006 turns/metre As you can see this number is completely impractical. 1.5 million turns in one metre would require a long amount of time to wind. Furthermore the wire diameter would be required to be e-004mm diameter and this diameter would not support the necessary current required as specified by the American Wire Gauge. This gauge specifies that a wire will carry a maximum current of 4.7 Amperes with a wire diameter of mm [11]. For all the above reasons the idea of a solenoid to apply the magnetic field was rejected. 11 Referenced from (AWG 23) 46