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Specialization: Transport Engineering and Logistics Report number: 2017.TEL.8154
Title: A survey of indentation rolling resistance models for belt conveyors
Author: M.A. Lodder
Title (in Dutch) Een overzicht van vervormingsrolweerstand modellen in riemtransport systemen
Assignment: literature
Confidential: no
Supervisor: Dr. Ir. Y. Pang
Delft University of Technology
FACULTY MECHANICAL, MARITIME AND MATERIALS ENGINEERING
Department Marine and Transport Technology Mekelweg 2 2628 CD Delft the Netherlands Phone +31 (0)15-2782889 Fax +31 (0)15-2781397 www.mtt.tudelft.nl
Student: M.A. Lodder Assignment type: Literature
Supervisor (TUD): Dr. Ir. Y. Pang Report number: 2017.TEL.8154
Specialization: TEL Confidential: no
Creditpoints (EC): 10
Subject: A survey of indentation rolling resistance models for belt conveyors
Belt conveyors are a widely used mode of continuous transport in bulk solids handling. In a changing economy where the environment impact and energy consumption have an increasing influence the needs of energy efficient transport increase. The energy consumption of belt conveyor systems is based in the motional resistances during operation. Where the largest contributor to the resistance force is the indentation rolling resistance according to the DIN22101. To analyze the energy
consumption of belt conveyors the amount of indentation rolling resistance needs to be determined, with respect to the design and operation of belt conveying systems.
Since the 1950’s previous researches have presented different models of estimating and calculating the indentation rolling resistance. By surveying, reviewing and comparing these models, the accuracy and efficiency of different methods will be indicated. The main tasks for this literature assignments include:
• To review the theory of the motional resistances of belt conveyors • To interpret and explain the indentation rolling resistance
• To review the existing models of calculating the indentation resistance • To compare the existing models regarding accuracy and efficiency
The report should be arranged in such a way that all data is structurally presented in graphs, tables, and lists with belonging descriptions and explanation in text.
The report should comply with the guidelines of the section. Details can be found on the website. The mentor,
Summary
In this report a survey is done of the literature on the theory of indentation rolling resistance in belt conveyor systems. The goal of this report is to gather as much of the existing research as possible and comparing them on their accuracy and efficiency.
This form of motional resistance is viewed in the total of motional resistances found in belt conveyor systems. The underlying physics that cause the phenomenon are discussed followed by the analytical models created by the different researches and their respective accuracy and efficiency.
Firstly from looking at the overall motional resistance it is found that the indentation rolling resistance of belt conveyor systems plays a major role in the total amount of motional resistances experienced in operation. For long horizontal belt conveyor systems the indentation rolling resistance can be responsible for up to 60 % of the total motional resistance.
The cause of indentation rolling resistance is named as being the viscoelastic material properties of rubber belt covers. The viscoelastic behaviour of materials is well described in literature and used to understand the analytical models. Viscoelastic behaviour is most commonly described as a
combination of elastic and viscous behaviour where it has the rigidity of an elastic solid but also the ability to dissipate energy and flow like a viscous fluid.
The analytical models of indentation rolling resistance in belt conveyor systems have been developed since the 1950’s and been improved ever since. The main theory is as follows:
A hard cylinder indents the viscoelastic material of the belt cover. The belt cover material responds to the compression but once the pressure on the material is removed the viscoelastic behaviour causes a delay in the expansion back to the original shape. This delay of expansion causes an asymmetric contact between cylinder and belt and in turn this asymmetric contact causes an
asymmetric vertical stress distribution where the leading front half of the cylinder experiences larger stress than the trailing edge. This pressure imbalance creates a resultant horizontal force opposing the motion of the belt called indentation rolling resistance.
The analytic models of the resistance models differ from each other on some key properties. The most important being the way the belt cover material is modelled. The models can be divided into 2D models including shear stress and one dimensional models that don’t consider shear in the stress relations. The viscoelastic model used to describe the behaviour is another property of the models that strongly influences their performance.
This reports concludes that the two dimensional model by Qiu provides the most accurate model to describe the indentation rolling resistance. The model by Jonkers is deemed the most efficient way to model the resistance. The model provided by Yan Lu and Fuyan Lin combines the best of the two properties and is the best overall performing model.
List of Symbols
In this report a lot of symbols are used in the different formulas. Because not all researches have a consensus about what symbols to use for certain characteristics not all papers use the symbols the same way. In this literature study nearly all the formulas will be quoted from authors so in this single report certain characteristics will be shown with different symbols when described by different researchers. To be as clear as possible, the meaning of the symbols will be explained in the text accompanying the formula. The axis systems used in the papers also differs per research, so all the model explanations will be accompanied by a figure showing the orientation of the axis used. The following parameters appear in every model for the indentation rolling resistance and are described with the following symbols unless specified otherwise.
• F = force [N]
• σ = stress [Pa]
• ε = strain []
Table of Contents
Summary ... I List of Symbols ... II Table of Contents ... III
1. Introduction ... 1
1.1 Objective of the research ... 1
1.2 Structure of the report ... 1
2. Motional resistances ... 3 2.1 F1: Main resistances ... 3 2.1.1 CEMA ... 3 2.1.2 DIN22101 ... 3 2.2 F2: Point forces ... 4 2.2.1 CEMA ... 4 2.2.2 DIN22101 ... 4 2.3 F3: Gradient resistance ... 5 2.3.1 CEMA ... 5 2.3.2 DIN22101 ... 5 2.4 F4: Special resistance ... 5 2.5 Group distribution ... 5 3. Indentation Resistance ... 7 3.1 Viscoelasticity ... 7
3.1.1 Classical elastic solid ... 8
3.1.2 Classical viscous fluid ... 8
3.1.3 Viscoelastic material ... 9
3.2 Rolling contact ... 14
4. Review & comparison of existing models ... 17
4.1. H.P. Lachmann (Lachmann, 1954) ... 17
4.2. W.D. May, E.L. Morris and D. Atack (May, Morris, & Atack, 1959) ... 18
4.3. S.C. Hunter (Hunter, 1961) ... 22
4.4. L.W. Morland (Morland, 1962) ... 26
4.5 C. Spaans (Spaans, 1978) ... 29
4.6. C.O. Jonkers (Jonkers, 1980)... 33
4.7. K.L. Johnson (Johnson, 1985) ... 37
4.8. G. Lodewijks (Lodewijks, The Rolling Resistance of Conveyor Belts, 1995) ... 39
4.10. Yan Lu and Fuyan Lin (Lu & Lin, 2016) ... 45
4.11. Qiu Xiangjun (Qiu, Full two-dimensional model for rolling resistance: Hard Cylinder on viscoelastic foundation of finite thickness, 2006) ... 47
4.12. Comparison of models ... 51
4.12.1 Cover material model ... 51
4.12.2. Viscoelastic model ... 51
4.12.3. System parameters ... 52
4.12.4. Accuracy ... 53
4.12.5. Efficiency ... 55
4.12.6. Multi criteria analysis ... 56
5. Conclusion & Recommendations ... 57
5.1 Conclusion ... 57
5.2 Recommendations... 58
1. Introduction
Belt conveyor systems have had a great impact on the current world. These systems were first seen around 1800 and since then they have greatly aided the industrial revolution. The most famous example is the automotive industry where Ford introduced a belt conveyor system in their factory to increase production numbers (De evolutie van massaproductie, 2017). Nowadays belt conveyors are still a widely used mode of transportation for bulk solid materials. Bulk solid materials like grain, coal, ore and sand or general material like luggage and boxes are still transported by belt conveyors. One of the reasons is because belt conveyor systems of short to medium distance are more efficient than transport with trucks. (Lodewijks, General introduction 3, 2015) (Zhang & Xia, 2010) Innovative solutions for conveying have become available since the arrival of the belt conveyor, but for systems needing continuous transport the conveyor is still a very good option.
Because belt conveyors are widely used their energy consumption is important for cost and
environmental reasons. A growing knowledge about climate change and limited energy resources is influencing the industries. In the current economy to stay competitive, engineers must design
conveyor systems that delivers the same capacity with reduced energy consumption. The energy cost of a belt conveyor can be up to 40% of the total operating cost. (Hager & Hintz, 1993) By minimizing energy consumption the environmental impact of operating the belt conveyors and the operating cost can be reduced. To minimize the energy consumption multiple researches have been done into the power requirement of belt conveyors. Starting from the fifties in Germany multiple researches investigated the problem of the resistance to motion of belt conveyors (Lachmann, 1954) (Leyen, 1962).These researches found that there are multiple partial resistances that influence the energy requirement of belt conveyors. From additional studies it was found that in horizontal belt conveyors of significant length the indentation rolling resistance accounts for up to 60 % of the motional
resistances (Hager & Hintz, 1993). Over the years multiple researchers have investigated the indentation rolling resistance. These researches have tried to find a calculation model that predicts the indentation rolling resistance in belt conveyor system with the properties of the systems. Because the different researches had different objectives and methods, a broad range of results has been found. With this spectrum of indentation rolling resistance models it can be difficult to know if you have considered all prior researches, what research to use for the most accurate result or which research is best applicable to your own needs.
1.1 Objective of the research
In this paper the different calculation models for the indentation rolling resistance of conveyors belts are surveyed, reviewed and compared. By collecting as much of these researches as possible and bundling them it will be attempted to get a better overview of the existing research. By comparing the existing research the best model of the discussed models is found regarding accuracy and efficiency. So to summarise the research objective in two research questions that is:
• What are the existing calculation models for the indentation rolling resistance in belt conveyor systems?
• What is the best calculation model regarding accuracy and efficiency
1.2 Structure of the report
In order to get a good idea about the forces at play in a conveyor belt an overview of the motional resistances of belt conveyors is given with the DIN and CEMA norms as an example. In the next chapter indentation rolling resistance will be explained in a general sense with viscoelastic theory. In the final chapter the comparison of the different calculation models of researches is provided. In this
chapter most of the existing research is reviewed and compared to obtain the most accurate and efficient models. Finally a conclusion of the findings is given and recommendations for further research are provided.
2. Motional resistances
The indentation rolling resistance is one of many resistance forces in a running belt conveyor system. To place the indentation rolling resistance into perspective all motional resistances of belt conveyors will be discussed here. It is important to know what part the indentation rolling resistance plays in the total force and power calculations. The calculation and interaction of conveying forces for the design of belt conveyors can be approached from different angles. In this chapter the CEMA and DIN approach will be considered. These approaches contain the calculating norm for two large bulk handling countries. USA and Germany. The CEMA approach is oriented around the tension and tension change in the belt of the conveyor where the German DIN 22101 uses the forces (resistances) which oppose the motion of a belt. The tension changes in the CEMA approach are caused by the resistance forces acting on the belt so in fact the two norms are treating the problems similarly.
Both methods aim to find a total motional resistance which is equal to the force that must be transferred from the driving pulley to the belt to keep moving. This total motional resistance is divided into four groups of resistance forces and is described with the following function:
𝐹𝑡𝑜𝑡𝑎𝑙= 𝐹1+ 𝐹2+ 𝐹3+ 𝐹4 (2.1)
The power requirement of belt conveyors is expressed as the product of these resistances to motion and the belt speed. In this chapter these four groups of motional resistances will now be discussed separately for each calculation norm, which forces they contain and how to find them.
2.1 F
1: Main resistances
Both methods have a group of main resistances. This group of resistances consists of forces that have the same value regardless of the direction of belt travel but will always work in the opposite direction of motion. Another property of this group is that its value increases when head to tail distance increases, in contrast to the point forces and secondary resistances.
2.1.1 CEMA
The CEMA universal design method takes the following resistances into account for the main resistance of a conveyor belt:
• Belt sliding on skirtboard seal • Idler seal friction
• Idler load friction
• Viscoelastic deformation of belt (indentation rolling resistance) • Idler misalignment
• Drag due to impact cradles/slider beds • Bulk material sliding on skirtboard
• Bulk material moving between idlers (internal friction of bulk solid material)
Each of these separate resistance forces can be determined with an individual function. The sum of all these resistances forms the main resistance in the CEMA universal design method. (Conveyor equipment manufacturers association, 2014)
2.1.2 DIN22101
The DIN22101 method uses a simplified manner to determine the main resistance which is based upon the coulomb dry friction. A coefficient of friction is multiplied with a normal force to get the friction force between two surfaces.
𝐹𝐻 = 𝐿 ∗ 𝑓 ∗ 𝑔 ∗ [𝑚′𝑅+ 2 ∗ (𝑚′𝐺+ 𝑚′𝐿) ∗ cos 𝛿] (2.2)
FH is the main resistance, L is the belt conveyor length, δ is the mean inclination angle of the installation , 𝑚′𝑅, 𝑚′𝐺, 𝑚′𝐿 are the mass of the rotating rolls, belt and bulk solid material
respectively. 𝑓 is a hypothetical friction coefficient for the upper and lower strand. 𝑓 is mainly determined by the rolling resistance of idlers on the carrying side, indentation resistance and flexure resistance for relatively deep belt sag. But it is also influenced by:
• Internal friction of bulk solid material • Belt conveyor alignment
• Belt tension • Operating conditions • Idler diameter • Spacing of idlers • Belt speed • Trough angle
DIN 22101 uses measured values for 𝑓 for accuracy. These values are determined in a testing atmosphere and range from 0.012-0.035. A table in DIN22101 provides what value should be used for a specific calculation. As we will see in chapter 4 these values can also be approximated with calculations. The indentation rolling resistance calculations of different researches can be converted to a friction coefficient by dividing the frictional force by the normal force.
𝑓 =𝐹𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛
𝐹𝑛𝑜𝑟𝑚𝑎𝑙 (2.3)
Obviously the other influences on the main rolling resistance are then not yet accounted for but it could be an approximation.
2.2 F
2: Point forces
The next group of forces can be classified as forces that act over a finite length and that length is small compared to the overall length of the conveyor. These forces are independent of the length of the conveyor and can be analysed as discrete sources.
2.2.1 CEMA
The CEMA universal design method puts the following sources into this group: • Belt bending on the pulley
• Pulley bearings
• Belt cleaners and ploughs
Similar to the main resistances these tension changes are calculated individually and added together to form a total tension change. (Conveyor equipment manufacturers association, 2014)
2.2.2 DIN22101
The DIN22101 method uses an approximate method to calculate these secondary resistances. The total resistance force for the conveyor is multiplied with the coefficient C to account for the secondary resistances. This value of C is dependent on the total length of the conveyor. For shorter lengths the secondary resistances are far greater than for longer conveyor lengths.
𝐶 = 1 +𝐹𝑁 𝐹𝐻
In this formula FN is the total of secondary and FH is the total of main resistances. The value of C is taken from a table relating the value of C to the belt length. The value for the secondary resistances can also be calculated by adding the individual secondary resistances. The individual resistances that are taken into account for this are:
• Friction resistance between material conveyed and the belt in the feeding zone
• Friction resistance between conveyor belt and lateral chutes in the acceleration zone of a feeding point
• Friction resistance caused by belt cleaners.
There are more secondary resistances like: the bending resistance of the conveyor belt where it runs over a pulley and the resistance of the bearings of non-driven pulleys. In most cases however this norm neglects these forces when their influence is small compared to the original secondary forces.
2.3 F
3: Gradient resistance
2.3.1 CEMA
The CEMA universal design method has a group of components called Energy components. This group contains elements which cause a change and tension by the loading of bulk solid onto the belt and accelerating it to belt speed and lifting or lowering it. The loading of the bulk solid causes inertia forces and the lowering or lifting is aided or counteracted by gravity. The functions used to calculate these are standard procedures. (Conveyor equipment manufacturers association, 2014)
2.3.2 DIN22101
The DIN22101 method only considers the resistance caused by the gradient in the conveyor system and the energy that needs to be added to overcome this.
𝐹𝑠𝑡= 𝐻 ∗ 𝑔 ∗ 𝑚′𝐿 (2.5)
For this formula of the slope resistance H is the height that must be overcome, g the acceleration due to gravity and m’L is the mass of the bulk solid material that is being conveyed.
2.4 F
4: Special resistance
The German DIN22101 further specifies a special resistance which encompasses resistances which do not occur with all belt conveyors and do not fit in one of the three other groups. This resistance is made up of individual resistances of which the sum forms the special resistance.
2.5 Group distribution
The distribution of these resistance fractions is different for different belt conveyor systems. For long horizontal heavy conveyors the bulk of the resistance force comes from the indentation rolling resistance, up to 60 %. These conveyors systems have a length around 1000 meters. Other
resistances are not negligible but are of less influence on the total energy requirement for this type of conveyor. For short horizontal conveyor systems of about 80 meters there is not one resistance fraction of such influence as for the long horizontal belt conveyors. For these short horizontal conveyors the secondary resistance (F2) is the main source of energy loss. 47 % of the motional resistances comes from the secondary resistances, but still 33 % of the motional resistance is formed by the indentation rolling resistance. For inclined conveyor systems the motional resistances are dominated by the gradient resistance (F3) , 66%. The indentation rolling resistance however still forms 22 % of the total motional resistances. (Hager & Hintz, 1993)
In these three types of conveyor systems the indentation rolling resistance always creates an
the motional resistance like the gradient resistance for example which is absent in the horizontal conveyors. The gradient resistance does form the largest portion of the total of 66 % so reducing it would have a large impact. However because the gradient resistance in a belt conveyor system cannot be reduced in the system properties, the best point to start reduction of the total resistance forces is in the indentation rolling resistance which lies in the group of main resistances (F1). A reduction there has by far the greatest achievable influence on the total energy requirement. Because of this fact, the indentation rolling resistance is one of the more extensively investigated properties of belt conveyors, and the main subject for this research. (Hager & Hintz, 1993)
3. Indentation Resistance
Fig. 1 Idler rolling on rubber belt (Lodewijks, The Rolling Resistance of Conveyor Belts, 1995)
In this chapter the general concept of the indentation rolling resistance will be explained. To start a spatial model is established, next viscoelastic theory will be discussed followed by the rolling contact phenomenon. So at the end of the chapter it should be clear how viscoelastic indentation rolling resistance works.
The problem of rolling resistance is quite substantial and applicable to a multitude of systems involving rolling parts like tires on a road or a ball bearing system. For this paper the main focus lies on belt conveyors where a viscoelastic belt is being pulled along and rolling over the top of idler rolls. In this situation the belt is made of a viscoelastic material like rubber or PVC and the idler rolls are usually made of a hard material like aluminium or steel. (Lodewijks, The Rolling Resistance of
Conveyor Belts, 1995) The most understandable way to handle the rolling resistance of a belt over an idler roll is to project it as a two dimensional problem which can later be extended in the third direction to account for the width of a belt. The material is assumed uniform in that third direction. The second simplification which can be made is to let the idler cylinder roll over a stationary belt. This is purely for simplification and understanding of the problem and has no influence on the forces of the phenomenon, see Fig. 1
With this spatial model in mind there are a couple of systems which can be identified as being responsible for the energy dissipation occurring in the indentation process. The sources of energy loss can be classified into those which arise through micro-slip and friction, those which are due to inelastic properties of the material and those due to roughness of the rolling surface. The largest contribution to the rolling friction is made by the inelastic properties of the material. (Johnson, 1985) So the micro-slip and surface roughness will be ignored for now.
To know how the inelastic properties of the used materials cause the indentation resistance in the belt it is first important to treat the subject of viscoelasticity.
3.1 Viscoelasticity
As has been established, the subject of this paper is belt conveyor systems. This excludes plate conveyor systems and implies that the used belt is a viscoelastic material. Viscoelastic materials are materials that behave partly in an elastic manner and partly in a viscous manner. To understand this viscoelastic behaviour the separate response of elastic solids and viscous fluids will be discussed.
3.1.1 Classical elastic solid
When a constant stress is applied instantaneously to a classic elastic solid it will deform
instantaneously to a fixed strain. If that stress would then be removed instantaneously the strain would disappear in a similar instantaneous manner. So the elastic solid completely recovers its original shape. The other way round, if a step strain would be applied to the solid in question the stress would also respond in a similar instantaneous manner to a constant stress value. Overall for an elastic solid it can be said that for every strain ε there corresponds a unique value of stress σ
independent of strain rate or how the value of strain is reached (strain history) (Wineman & Rajagopal, 2000). The other way around this is also true and can be described by the formula:
𝜎(𝑡) = 𝐸𝜀(𝑡) (3.1)
Where E is the Young’s modulus. This is known as Hooke’s law.
In a dynamic mechanical test, materials are subject to an oscillating strain which would also be the case for a belt rolling over idler rolls. To determine the behaviour of that material under such a repeating deformation the resulting stress is measured. Materials that behave accordingly to Hooke’s law, as said before, have a stress function that is proportional to the strain and the stress and strain signals are also in phase with each other (Lodewijks, Determination of Rolling Resistance of Belt Conveyors using Rubber Data: Fact of Fiction?, 2010). An elastic solid that goes through an entire deformation cycle where it is deformed and then returned to its original shape does not dissipate any energy. The work that is done in total amounts to zero. A one dimensional mechanical response of a classical elastic solid is best represented as a linear spring, see Fig. 2.a. The response of such a spring is characterized by the force-deformation relation:
𝐹 = 𝑘∆ (3.2)
Where F is the force, Δ the elongation and k is the spring constant. This representation helps to understand the material response of such a material.
3.1.2 Classical viscous fluid
When a constant stress is applied instantaneously to a classic viscous fluid it will not reach a fixed deformed state. The material shall flow and there will be continuous straining in time. The strain rate however, is a constant for constant fixed stress. If that stress would then be removed
instantaneously the strain ε will not change and no strain will be recovered. The strain rate will be reduced to zero in the instant of removing the stress. Such a material has no tendency to return to its previous shape. The other way around, if a step strain is applied to the solid and held fixed. A very large stress will be needed to produce this sudden strain and when the strain is maintained the stress required to maintain it will reduce immediately to zero. Different strain histories also do not affect the required stress for viscous fluids. Generally said, there is only one value of strain rate that corresponds to a specific stress and that stress is independent of strain or strain history (Wineman & Rajagopal, 2000). The formula to describe this relation is:
𝜎(𝑡) = 𝜇𝜀̇(𝑡) (3.3)
Where μ is the viscosity and 𝜀̇(𝑡) the strain rate. This is known as Newton’s law.
When a fluid behaves ideally according to Newton’s law under an oscillating strain, the stress signal is 90 degrees out of phase. The stress signal will lead the strain signal (Lodewijks, Determination of Rolling Resistance of Belt Conveyors using Rubber Data: Fact of Fiction?, 2010). A viscous fluid that goes through an entire deformation cycle where it is deformed and then returned to its original shape will dissipate all the energy that is needed for such a cycle. The work is completely converted to heat. The mechanical representation that helps understand a classical viscous fluid is a viscous damper (a piston in an oil bath in a cylinder), see Fig. 2.b. The mechanical response of such a viscous damper is characterized by the relation:
𝐹 = 𝑐∆̇ (3.4)
Where c is the viscosity and ∆̇ is the deformation rate.
3.1.3 Viscoelastic material
When the properties are combined in a viscoelastic material the response to step stress becomes a combination of the two. So again if a constant stress is applied instantaneously to the material the response consists of an instantaneous increase in strain and a continued straining at a non-constant rate. There is a distinction between materials that are solid-like and materials that are fluid-like. For materials that are solid-like the strain asymptotically approaches a constant value, for materials that are fluid-like the strain rate asymptotically approaches a constant value. If the applied stress is then removed instantaneously the viscoelastic materials will show some instantaneous strain recovery and a part of delayed strain recovery. Again materials that are solid-like and fluid-like behave
differently to this. For solid-like materials, generally over time all the strain is recovered, for fluid-like materials not all the strain will be recovered. Because of creep, the constant continued straining under constant stress, there always remains some residual strain in fluid-like viscoelastic materials. The other way around when a constant strain is applied instantaneously to the material the stress responds by instantaneously jumping to a value. The stress required to maintain the constant strain slowly decreases. For solid-like materials the stress required to maintain the constant strain never reduces to zero, a small stress always stays required to maintain the deformation. For fluid-like materials stress asymptotically reduces to zero. So after a long enough period of time no stress is required to keep the material in its deformed state, this is called stress relaxation.
When a viscoelastic material is subjected to different stress histories where each history arrives at the same value of stress after the identical amount of time the strain will be different in each case. The strain is very dependent on the sequence of stress preceding that moment in time. Because the strain is so dependent on the stress history it becomes very interesting to find a relationship between the stress strain and time. With such a relationship it becomes possible to determine stress σ(t) knowing the preceding strain history, or determine strain ε(t) knowing the preceding stress history. (Wineman & Rajagopal, 2000)
When a viscoelastic material is subjected to a harmonic load, the strain will also be a harmonic motion which lags behind the stress. This is formulated mathematically as:
𝜎 = 𝜎0sin 𝜔𝑡 (3.5)
𝜀 = 𝜀0sin(𝜔𝑡 − 𝛿) (3.6)
Where σ0 and ε0 are the amplitudes of respectively stress and strain, ω is the circular frequency and δ the phase angle. By taking a complex modulus of elasticity the relation between stress and strain can be written as:
𝜎̅ = 𝐸̅𝜀̅ (3.8) The real part of the complex modulus E’ determines the stored energy during a load cycle and is called the dynamic modulus of elasticity. If the material would have no viscous losses it would be the young’s modulus. The real part is a measure of the magnitude of elastic stress which is in phase with the strain. It can be found as:
𝐸′ =𝜎0 𝜀0
cos 𝛿 (3.9)
The imaginary part E’’ is a measure of the magnitude of the viscous stresses which are 90 degrees out of phase with the strain. The imaginary modulus determines the loss energy during a loading cycle and is therefore called the loss modulus.
𝐸′′=𝜎0 𝜀0
sin 𝛿 (3.10)
The tangent of the phase angle δ can be found as the division of the imaginary and real part. It is known as the hysteresis factor of coefficient of internal damping.
tan 𝛿 =𝐸
′′
𝐸′ (3.11)
The real and imaginary part of the modulus and the hysteresis factor can be found for materials as properties in the correct literature. (Jonkers, 1980)
The energy dissipation in a complete loading cycle of a viscoelastic materials is only partly. Some the work done to deform the material is lost in heat and some will be returned when the material is restored.
Material structure
Some explanation for the differences in behaviour for the materials can be found in the molecular structure. When an external force is applied to a material its molecular structure will develop internal forces by distortion. Elastic materials like metal have an atomic crystalline structure with strong interatomic forces. By deforming this crystalline structure the atomic forces will resist and create the internal force. The internal connections stay intact so after pressure is released the old structure can be restored. Fluids such as water have no lattice structure and only weak attractive forces. When deforming a fluid continuous movement of particles with respect to each other builds up the internal force to resist the deformation. The relative positions of the fluid particles have now been
rearranged and once the pressure is released there is nothing to restore the material to its old form. When it comes to viscoelastic materials, once again, a combination of both these phenomena is the cause of build-up of internal forces. The rubber-like material of the belts is made of polymers. These are flexible long chain molecules and the rearrangement of these molecules causes the internal forces needed for the reaction force. These long macro molecules can be crosslinked which gives the material a sort of lattice structure. Deformation of this structure causes the internal forces. What differentiates the viscoelastic materials from the elastic materials is that the molecular structure is able to rearrange itself to relieve some of the built up forces. When the external force is removed the molecules that have not been rearranged cause the material to return to its original shape. The difference between solid-like and fluid-like materials depends on the amount of crosslinking between the molecules. Solid-like materials are more crosslinked than fluid-like materials. This explains why solid-like materials always return to their original shape and fluid like materials creep and don’t always return to their original shape. Fluid-like materials are not crosslinked so if an external
pressure is applied the molecular structure can rearrange and the molecules can move relative to each other. The internal forces will eventually reduce to zero and no external force is needed to maintain the new state of the material. Viscoelastic solids have a preferred reference state to which it will mostly return after external loads are removed. Viscoelastic fluids don’t have this behaviour, any state can become the new reference state (Wineman & Rajagopal, 2000).
Mechanical analogues
As could be seen for the elastic and viscous materials, a one dimensional mechanical representation can help understand the behaviour of these materials. The force deformation relations that apply to these mechanical representations are analogous to the constitutive equations for a one dimensional response of the materials. So in order to find a relations between stress, strain and time for
viscoelastic materials a mechanical representation is developed. These analogues are also widely used in the explanation of indentation rolling resistance as will be seen in chapter 4. Because of the composite nature of viscoelastic materials the mechanical analogue for it is also a composition of multiple basic structural elements: the linear spring and the viscous damper. Different analogues have been developed over time. A short explanation of a few examples follows.
• Maxwell model
In this model a linear spring and a viscous damper are connected in series. The stress-strain relation of this mechanical analogue is described by a first order linear differential equation:
𝑞1𝜀̇ = 𝑝1𝜎̇ + 𝑝0𝜎 (3.12)
In which the parameters q1, p0 and p1 are defined as: 𝑞1= 1, 𝑝0=
1 𝜇, 𝑝1=
1
𝐸 (3.13)
And μ is the viscosity and E is the young’s modulus. With this function the stress can be described from a known strain and vice versa. For this some additional operators need to be known. The stress relaxation function for this model is:
𝐺(𝑡) =𝑞1 𝑝1
𝑒−(𝑝0⁄𝑝1)𝑡 (3.14)
𝐺(𝑡) = 𝐸𝑒−(𝐸 𝜇)𝑡⁄ (3.15)
The creep compliance is given as:
𝐽(𝑡) =𝑝1 𝑞1 (1 +𝑝0 𝑝1 𝑡) (3.16) 𝐽(𝑡) =1 𝐸(1 + 𝑡 𝜇) (3.17)
The stress response for an arbitrary strain history is:
𝜎(𝑡) = 𝐺(𝑡)𝜀(0) + ∫ 𝐺(𝑡 − 𝑠)𝜀̇(𝑠)𝑑𝑠
𝑡
0
(3.18) The strain response for an arbitrary stress history:
𝜀(𝑡) = 𝐽(𝑡)𝜎(0) + ∫ 𝐽(𝑡 − 𝑠)𝜎̇(𝑠)𝑑𝑠
𝑡
0
(3.19) S is a variable of integration here. From the stress relaxation function it can be seen that the Maxwell model exhibits stress relaxation and that the stress relaxes completely to zero. This complete stress relaxation indicates that the Maxwell model represents fluid behaviour.
Because of the linear second term in the creep compliance it can be seen that this also indicates a fluid response. The model has a constant strain rate under constant stress. The model reaches a constant strain immediately, instead of gradually as is usually the case with viscoelastic materials (Wineman & Rajagopal, 2000).
• Kelvin-Voigt model
In this model a linear spring and a viscous damper are placed in parallel. The constitutive equation for this model is:
𝑝0𝜎 = 𝑞0𝜀 + 𝑞1𝜀̇ (3.20)
With the parameters q0, q1 and p0 defined as:
𝑝0= 1, 𝑞0= 𝐸, 𝑞1= 𝜇 (3.21)
μ and E are the same as in the Maxwell model. The stress relaxation function for this model is: 𝐺(𝑡) = 𝑞1 𝑝0 𝛿(𝑡) +𝑞0 𝑝0 (3.22) 𝐺(𝑡) = 𝜇𝛿(𝑡) + 𝐸 (3.23)
Where δ(t) is the dirac delta function. The creep compliance is described for this model as: 𝐽(𝑡) = 𝑝0 𝑞0 (1 − 𝑒−( 𝑞0⁄ 𝑞1)𝑡) (3.24) 𝐽(𝑡) = 1 𝐸(1 − 𝑒 −(𝐸 𝜇⁄ )𝑡) (3.25)
The stress response to an arbitrary strain history and vice versa have the same form as with the Maxwell model only the stress relaxation function and creep compliance function are different.
In this Kelvin-Voigt model the stress response for an instantaneous elongation becomes unbounded calling for the dirac delta function. But once the strain is applied to the viscous damper the stress needed to maintain the strain falls directly back to zero and the only stress remaining is that needed to keep the spring deformed. The stress relaxation is thus
instantaneous. The creep compliance for this model is a gradual one. The strain
asymptotically approaches a constant value which indicates solid-like behaviour, there is however no instantaneous strain response to an instantaneous stress appliance (Wineman & Rajagopal, 2000).
m
• Three parameter solid or standard linear solid
This model consists of either a Maxwell model and a linear spring in parallel or a Kelvin-Voigt model and a linear spring in series. The option with the Maxwell model provides a system with a bounded creep and the Kelvin-Voigt model provides a system where instantaneous elongation is possible. These properties can not be reversed. Both models lead to the same constitutive equation involving stress, strain and time.
𝑝0𝜎 + 𝑝1𝜎̇ = 𝑞0𝜀 + 𝑞1𝜀̇ (3.26)
With the parameters q0, q1 and p0, p1 defined as: 𝑝0= 1 𝜇, 𝑝1= 1 𝐸, 𝑞0= 𝐸1 𝜇 , 𝑞1= (1 + 𝐸1 𝐸) (3.27)
In these definitions μ is the viscosity of the damper, E the young’s modulus of the spring in parallel with the Maxwell model and E1 the young’s modulus of the spring in series with the damper. The stress relaxation function for this model is:
𝐺(𝑡) =𝑞0 𝑝0 + (𝑞1 𝑝1 −𝑞0 𝑝0 ) 𝑒−(𝑝0⁄𝑝1)𝑡 (3.28) 𝐺(𝑡) = 𝐸1+ 𝐸𝑒−(𝐸 𝜇)⁄ 𝑡 (3.29)
Where E1 is the Young’s modulus for the added spring in parallel and E is the Young’s modulus for the spring in the Maxwell element. The creep compliance is then:
𝐽(𝑡) =𝑝0 𝑞0 + (𝑝1 𝑞1 −𝑝0 𝑞0 ) 𝑒−(𝑞0⁄𝑞1)𝑡 (3.30) 𝐽(𝑡) = 1 𝐸1 + ( 𝐸 𝐸1(𝐸 + 𝐸1) ) 𝑒−( 𝐸1𝐸 𝜇(𝐸+𝐸1) ⁄ )𝑡 (3.31) The response for an arbitrary strain history and an arbitrary stress history for this model are also presented in the same way as for the single Maxwell model only with different functions for the stress relaxation function and the creep compliance.
The stress relaxation that this model displays is one of more solid-like behaviour. For an instantaneous applied strain the stress gradually reduces to a non-zero amount. Some stress remains to keep the springs in the model deformed. The creep response displays partly instantaneous behaviour for an instantaneous applied stress and partly a gradual asymptotical increase in strain to a fixed value. Which also indicates good solid-like behaviour (Wineman & Rajagopal, 2000).
• N Maxwell elements in parallel (generalized Maxwell model)
The name gives an accurate description. Every spring and damper in the model has a unique value. The corresponding constitutive equation for this model is:
(𝐷 𝐸1 + 1 𝜇1 ) (𝐷 𝐸2 + 1 𝜇2 ) … (𝐷 𝐸𝑀 + 1 𝜇𝑀 ) 𝜎 = (𝑞̅0+ 𝑞̅1𝐷 + ⋯ + 𝑞̅𝑀𝐷𝑀)𝜀 (3.32)
This model has M Maxwell elements and D is a differential operator. In this form the model would indicate fluid-like behaviour. To obtain a more solid like response from this model the viscous damper for the Nth Maxwell element should be taken as infinity. Practically
rendering the Maxwell element as a spring. For the stress relaxation function a Prony series is used.
𝐺(𝑡) = 𝐺0+ 𝐺1𝑒−(𝐸1⁄𝜇1)𝑡+ ⋯ + 𝐺𝑀𝑒−(𝐸𝑀⁄𝜇𝑀)𝑡 (3.33)
Where G0, G1, …, GM are constants. For the creep response a similar function can be established where the constant Ji are different from the constants Gi.
𝐽(𝑡) = 𝐽0+ 𝐽1𝑒−(𝐸1⁄𝜇1)𝑡+ ⋯ + 𝐽𝑀𝑒−(𝐸𝑀⁄𝜇𝑀)𝑡 (3.34)
With these function the response for arbitrary stress or strain histories can be accurately be described. The function for that is the same as in all models but with the appropriate
functions for stress response and creep response. If a step elongation is applied to this model the total force can be found by taking the sum of the force from every Maxwell element plus one spring. So the Maxwell element will show stress relief but the spring will continue under stress providing the solid-like behaviour of this model for stress relief. A similar analogy for a step stress is not possible because with an applied total force it is impossible to know all the individual forces per Maxwell element (Wineman & Rajagopal, 2000).
3.2 Rolling contact
Now that it is established how viscoelastic materials behave when a stress or strain is applied, the next part to determine is the rolling contact between the idler rolls and the belt.
In the last section a function was derived that provides the stress for a given strain on the material and vice versa. So the next step is to look at the stress distribution that is occurring in a rolling contact between the idler roll and the belt.
Johnson describes rolling as:” a relative angular motion between two bodies in contact about an axis parallel to their common tangent plane.” For this motion it is convenient to take a frame of reference which moves with the point of contact. That way the two touching surfaces flow through the contact zone with velocities V1 and V2. (Johnson, 1985) It is possible that the two velocities have a different value, the belt material will then either deform or there will exist a relative motion between the idler
and the belt. Because of the penetration of the idler rolls into the belt, the speed of the belt is not uniform in all points of the contact area. This means that when the rolls penetrate the belt there will always be slip or distortion of the belt cover. (Spaans, 1978) This stick slip phenomenon was already deemed to be of lesser influence on the total resistance force so it will not be treated further. Rolling can be divided into free rolling where no tangential force is transmitted and tractive rolling, where the tangential force is non-zero. The focus for this article lies on tractive rolling. The manner in which this tractive force arises shall now be explained.
If the idler roll and the belt would both be assumed rigid the contact area between the two would be a line of incremental breadth. Johnson describes this as non-conforming contact in one direction and conforming in the perpendicular direction. When the two dimensional representation that was adopted earlier is taken, the contact area becomes an incremental point. This situation would not be able to produce any resistance to the rolling because no tractive forces can be developed. If the belt material is a purely elastic material the belt surface will follow the curvature of the cylinder precisely. The indentation will resemble a part of the circle with the same radius as the cylinder symmetrical around the centre line of the cylinder. The cylinder would be compressing the belt on the leading edge which would require energy. However, past the centreline of the cylinder the idler roll would release the pressure on the roll letting the belt expand again. Because the belt material behaves purely elastic, once the pressure is released from the material the strain will instantly respond. The force that was then needed to compress the belt material is then directly returned to the rolling motion of the idler roll by the expanding belt material. So tractive forces do develop in this situation but they would be completely balanced out.
The situation where a viscoelastic belt material is indented by a hard cylinder creates a force imbalance. The leading edge of the cylinder still compresses the belt material. The applied strain creates a certain stress in the material. When the strain is removed after the centreline of the cylinder the material needs some time to recover to its original shape. Due to the crosslinking in viscoelastic solids the material will always return to its original shape. It is the creep that causes that this will take some time. Because of this time delay in the recovery of the belt material, the pressure distribution of the contact area between idler and belt shows a discontinuity, see Fig. 7. Also because of this slower expanding of the belt material, the belt and the idler roll separate at a point closer to the centreline then the point where they first contact each other, see Fig. 1. This asymmetric contact-phenomenon and the resulting stress distribution result in a resistance force. (Lodewijks, The Rolling Resistance of Conveyor Belts, 1995) The asymmetric stress distribution where the leading edge of the cylinder roll experiences higher stresses than the trailing edge causes a moment on the roll which can be interpreted as a resistance force. Several researches have attempted to describe this effect with a model and a mathematical description. The next chapter shall treat this.
4. Review & comparison of existing models
In this chapter the different models and mathematical descriptions of the indentation rolling
resistance shall be treated. The researches handled in here have tried to create models to determine the indentation rolling resistance separately. However, it is not possible to measure this resistance component individually for the verification of either a physical or a mathematical model. (Spaans, 1978) The effects of flexure resistance always influences the measurements on indentation rolling resistance in the model setup. Spaans found it is possible to verify the models, albeit with limitations, by investigating the influence of the idler roll diameter on the behaviour of the indentation
resistance. This is possible because the idler roll diameter has no influence on the other components of the main resistance group. So the researches have tried to describe the indentation rolling
resistance of a conveyor belt with the properties of a belt conveyor system. For example the idler diameter, belt thickness and travelling speed.
The chronological order of the researches shall be followed as much as possible. In the early years most of the research was done in Germany where they began by identifying the different resistances that work in a belt conveyor system. Later attempts were made where the measured resistances were explained with theories. The focus here shall be on original models and mathematical descriptions of indentation rolling resistance that predict the resistance with system properties. Lachmann was the first to try and give a theoretical explanation for the values that were found.
4.1. H.P. Lachmann (Lachmann, 1954)
Lachmann considered the work needed for indentation of the belt cover material and the work that was retrieved when relaxing the material. In a force deformation diagram of an indentation cycle of the belt cover material the hysteresis effect can clearly be recognized.
See Fig. 8. The amount of work retrieved from the release of pressure is less than the amount of work needed to deform the belt cover material. The difference between the two is called hysteresis work, Wh. The relation between the maximum deformation work and the hysteresis work is called the loss factor ψE:
𝜓𝐸 =
𝑊ℎ
𝑊𝑚𝑎𝑥 (4.1)
In his work Lachmann argued that the rolling resistance should be proportional to the loss factor and to the 3/2 power of the vertical indentation force. He later found from his own measurements that the value for the exponent should be lower. (Jonkers, 1980)
4.2. W.D. May, E.L. Morris and D. Atack (May, Morris, & Atack, 1959)
The article of May, Morris and Atack about the rolling friction on a viscoelastic materials provides two separate models. An approximate one that uses a single relaxation time represented by two individual models, the Maxwell and the standard linear, and a more rigorous one that uses a distribution of relaxation times.
In their paper they give the following compact description of the resistance force: The concept of the indentation rolling resistance is based on the fact that a viscoelastic material takes time to recover from an indentation. The stresses relax during passage of the cylinder giving an asymmetrical stress pattern where the forces at rear of the cylinder are smaller than those at the front of the cylinder. This results in a horizontal force known as the indentation resistance.
Maxwell model
In the approximate solution the shear stresses are ignored and only the compressive stresses are considered. The approximate model of the viscoelastic material is described as an infinite number of cylinders being compressed without affecting neighbouring cylinders. With this model the depth of penetration becomes dependant on the thickness of the material. Firstly the behaviour of the
viscoelastic material is described by the conventional Maxwell model. A spring dashpot arrangement with a single relaxation time τ.
𝜏 =𝜂
𝐸 (4.2)
Where η is the viscosity of the dashpot and E is the modulus of the spring. With these models in place the compression can be given as a function of time:
𝑧𝑡 =
𝑉
2𝑅(2𝑎0𝑡 − 𝑉𝑡
2) (4.3)
Where V is the velocity of the cylinder, R the radius, a0 half the width of the indentation, t the time which the material has been subject to compression and zt the compression depth as a function of t, see also Fig. 9. If l is the thickness of the viscoelastic belt material the stress (σt) at point P after time t is given by Boltzmann’s superposition principle:
𝜎𝑡 𝐸 𝑙 ⁄ = 𝑧 + ∫ 𝜓(𝑡 − 𝜃) 𝑑𝑧𝜃 𝑑𝜃 𝑑𝜃 𝑡 0 (4.4)
E is the modulus of the material, θ is the elapsed time and ψ(t-θ) is the relaxation function of the material. For the single Maxwell model it is written as:
𝜓(𝑡 − 𝜃) = 𝑒(𝜃−𝑡)⁄𝜏− 1 (4.5)
If you substitute θ for t in equation (4.3) you get: 𝑧𝜃=
𝑉
2𝑅(2𝑎0𝜃 − 𝑉𝜃
2) (4.6)
The derivative of this equation can be inserted into equation (4.4) for the stress. 𝜎𝑡 𝐸 𝑙 ⁄ = 𝑧𝑡+ ∫ (𝑒 (𝜃−𝑡) 𝜏 ⁄ − 1) (𝑎0− 𝑉𝜃) 𝑉 𝑅𝑑𝜃 𝑡 0 (4.7)
By substituting the formula for zt and integrating the formula gives the stress under the cylinder at a point P, at time t after P comes in contact with the cylinder:
𝜎𝑡 𝐸 𝑙 ⁄ = 𝑉𝜏 𝑅 [(1 − 𝑒 −𝑡 𝜏 ⁄ ) (𝑎 0+ 𝑉𝜏) − 𝑉𝑡] (4.8)
The point on the cylinder where the belt loses contact is dependent on the relaxation time τ. By expressing the relaxation time as a fraction of the time T taken for the cylinder to move a distance equal to the semilength of contact: 𝜏 =𝑘𝑎0
𝑉 ⁄ = 𝑘𝑇, the σt is 0 if 𝑒[− 1 𝑘( 𝑎0−𝑥 𝑎0 )]= 1 − 1 𝑘 + 1( 𝑎0− 𝑥 𝑎0 ) (4.9)
In this k is a coefficient and from this formula (𝑎0− 𝑥)/𝑎0 can be determined graphically for
different values of k.
To calculate the force being exerted on the cylinder a small strip is taken on point P of length y0 and width dx. When this is applied to the function for the stress (4.8) it becomes:
𝐹𝑡 = 𝐸𝑉𝜏 𝑙𝑅 [(1 − 𝑒 −𝑡 𝜏 ⁄) (𝑎 0+ 𝑉𝜏) − 𝑉𝑡] 𝑦0𝑑𝑥 (4.10)
By substituting 𝑡 = (𝑎0− 𝑥) 𝑉⁄ you get the force exerted on the cylinder as a function of the
position: 𝐹𝑥= 𝐸𝑉𝜏 𝑙𝑅 [(1 − 𝑒 [(𝑎0−𝑥) 𝑉𝜏 ⁄ ] ) (𝑎0+ 𝑉𝜏) − 𝑎0+ 𝑥] 𝑦0𝑑𝑥 (4.11)
The total moment about the centre is the next thing that must be determined. By taking the integral over the surface in x direction on the force described in x.
𝑀 =𝐸𝑉𝜏 𝑙𝑅 𝑦0∫ [(1 − 𝑒 [(𝑎0−𝑥) 𝑉𝜏 ⁄ ] ) (𝑎0+ 𝑉𝜏) − 𝑎0+ 𝑥] 𝑥𝑑𝑥 𝑎0 𝑥 (4.12) 𝑀 =𝐸𝑉𝜏 𝑙𝑅 𝑦0[𝑉 3𝜏3 − (𝑉𝜏 2⁄ )(𝑎02+ 𝑥2) + 1 3(𝑎0 3− 𝑥3) − 𝑉𝜏(𝑉𝜏 + 𝑎 0)(𝑉𝜏 − 𝑥)𝑒−[ (𝑎0−𝑥) 𝑉𝜏 ⁄ ] ] (4.13) In terms of k, this becomes:
𝑀 =𝐸𝑦0𝑎0 4𝑘 𝑙𝑅 [𝑘 3−𝑘 2(1 + ( 𝑥 𝑎0 ) 2 ) +1 3(1 + ( 𝑥 𝑎0 ) 3 ) − 𝑘(𝑘 + 1) (𝑘 − 𝑥 𝑎0 ) 𝑒[− 1 𝑘(1− 𝑥 𝑎0)]] (4.14)
Because 𝑎02 = 2𝑅𝑧0 the function for the moment about the centre can be written as a function of
the depth of penetration z0: 𝑀 =4𝐸𝑦0𝑧 2𝑘 𝑙 [𝑘 3−𝑘 2(1 + ( 𝑥 𝑎0 ) 2 ) +1 3(1 + ( 𝑥 𝑎0 ) 3 ) − 𝑘(𝑘 + 1) (𝑘 − 𝑥 𝑎0 ) 𝑒[− 1 𝑘(1− 𝑥 𝑎0)]] (4.15)
This is equivalent to the moment of a frictional force about the centre.
As the stress pattern in the material depends on the velocity of the cylinder and the relaxation time, the load required to maintain the cylinder at a fixed depth will also depend on these parameters. The function for the load is the integral over the indentation width.
𝑊 =𝐸𝑉𝜏𝑦0 𝑙𝑅 ∫ [(1 − 𝑒 [(𝑎0−𝑥) 𝑉𝜏 ⁄ ]) (𝑎 0+ 𝑉𝜏) − 𝑎0+ 𝑥] 𝑎0 𝑥 𝑑𝑥 (4.16)
With the function
𝑒[− 1 𝑘( 𝑎0−𝑥 𝑎0 )]= 1 − 1 𝑘 + 1( 𝑎0− 𝑥 𝑎0 ) (4.17)
In mind the load in the cylinder becomes:
𝑊 =√2𝐸𝑅 1 2 ⁄ 𝑧 0 3 2 ⁄ 𝑦 0𝑘 𝑙 [1 − ( 𝑥2 𝑎2)] (4.18)
Standard linear solid model
All viscoelastic materials which exhibit stress relaxation under a fixed strain also show creep under constant load and time-dependent recovery when load is removed. The Maxwell model used up till now only shows instantaneous recovery with no subsequent time-dependent recovery, after the load is removed. By adding a second spring in parallel this problem can be overcome. Now the model exhibits stress relaxation at a fixed strain, creep under constant load and both instantaneous and time dependent recovery when the load is removed. So this standard linear model is an all-round better one for viscoelastic materials.
The stress at point P for the new model is now defined as: 𝜎𝑡 𝐸 𝑙 ⁄ = 𝑉 2𝑅(2𝑎0𝑡 − 𝑉𝑡 2)(1 − 𝑎) +𝑎𝑉𝜏 𝑅 [(𝑎0+ 𝑉𝜏) (1 − 𝑒 −𝑡⁄𝜏 ) − 𝑉𝑡] (4.19)
Because the modulus is now built up of two springs it can be written as: 𝐸 = 𝐸1+ 𝐸2 and 𝑎 = 𝐸2/(𝐸1+ 𝐸2)
E1 and E2 generally relate to each other as: E1=3E2 . And with that the stress can be formulated as: 𝜎𝑡 = 𝐸1𝑉 2𝑅𝑙(2𝑎0𝑡 − 𝑉𝑡 2) +𝐸2𝑉𝜏 𝑅𝑙 [(𝑎0+ 𝑉𝜏)(1 − 𝑒 −(𝑡 𝜏⁄ )) − 𝑉𝑡] (4.20)
From this stress the friction force is found in a similar manner to the single Maxwell model. Moments are taken about the centre of the cylinder.
𝑀 =𝐸1𝑦0𝑉 2𝑅𝑙 ∫ (2𝑎0𝑡 − 𝑉𝑡 2)𝑥𝑑𝑥 +𝐸2𝑉𝜏𝑦0 𝑅𝑙 ∫ [(𝑎0+ 𝑉𝜏) (1 − 𝑒 −((𝑎0−𝑥)⁄𝑉𝜏)) − 𝑎 0+ 𝑥] 𝑥𝑑𝑥 𝑎0 𝑥0 𝑎0 𝑥0 (4.21)
This is then worked out as:
𝑀 =𝐸1𝑦0𝑎0 4 8𝑅𝑙 [1 − 2 ( 𝑥0 𝑎0 ) 2 + (𝑥0 𝑎0 ) 4 ] + 𝐸2𝑦0𝑎04𝑘 𝑅𝑙 {𝑘 3−𝑘 2[1 + ( 𝑥0 𝑎0 ) 2 ] +1 3[1 − ( 𝑥0 𝑎0 ) 3 ] − 𝑘(𝑘 + 1) (𝑘 −𝑥0 𝑎0 ) 𝑒[− 1 𝑘(1− 𝑥0 𝑎0)]} (4.22)
Writing this in terms of the indentation depth and substituting E1 you get the following function for the friction force:
𝐹 =4𝐸2𝑦0𝑧0 2 𝑙 ∗ (3 8− 3 4( 𝑥0 𝑎0 ) 2 +3 8( 𝑥0 𝑎0 ) 4 + 𝑘 {𝑘3−𝑘 2[1 + ( 𝑥0 𝑎0 ) 2 ] +1 3[1 − ( 𝑥0 𝑎0 ) 3 ] − 𝑘(𝑘 + 1) (𝑘 −𝑥0 𝑎0 ) 𝑒[− 1 𝑘(1− 𝑥0 𝑎0)]}) (4.23) Generalized Maxwell model
In the rigorous approach the shear stresses in the belt cover material are no longer ignored. The stress pattern for this new material behaviour is determined with a method thought of by Yoh Han Pao. The stress-strain relations are determined in a purely elastic deformation. The Laplace transform is then taken of the elastic solution. The elastic modulus in the resulting equation is then replaced. And inversion of that equation gives the viscoelastic solution.
The viscoelastic material now also must be described with a model with multiple relaxation times. May Morris and Atack do this by taking multiple Maxwell models in parallel. This is derived from the equation for a single Maxwell model of relaxation time τ and then integrated over the whole range of relaxation times. The rolling friction then becomes:
𝐹 = 2𝑎0 3𝑦 0 (1 − 𝜎)𝑟2 ∫ ∫ ℎ(1 − ℎ2 2 − ℎ2 { 1 2𝑒 −[𝑇(1−ℎ)𝜏 ] +𝜏 𝑇[(1 + 𝜏 𝑇) (1 − 𝑒 −[𝑇(1−ℎ)𝜏 ] ) − (1 − ℎ)]} 𝑑 ln 𝜏 𝑑ℎ ∞ −∞ ℎ0 𝑎0 (4.24)
Where h is a ratio of 𝑥/𝑎0 . May Morris and Atack found that the load needed to keep the belt
indented the same amount increases with velocity of the roll. The coefficient of friction also depends on the velocity but this differs per viscoelastic material.
4.3. S.C. Hunter (Hunter, 1961)
Small summary of how hunter handles the paper: “the problem of a rigid cylinder rolling on the surface of a viscoelastic solid is solved in an approximation in which inertial forces are neglected. With the introduction of viscoelastic effects, the symmetry associated with the corresponding elastic problem is destroyed, and in particular the cylinder motion is impeded by a resistive force. For a standard linear solid, the resulting coefficient of friction, a function of the rolling velocity tends to zero for small and large values of V, and attains a single maximum at an intermediate value.
The paper by Hunter is concerned with a theoretical analysis of the rolling contact of a rigid cylinder over the surface of a viscoelastic half space. For this analysis hunter makes a couple of assumptions: The inertial forces in the viscoelastic medium are neglected. The mechanical properties of the solid are idealized in two ways: It is assumed that the solid is characterized by a fixed value of Poisson’s ratio and the behaviour in shear is taken to be that of a standard linear solid, for which the creep strain response to a unit step function stress pulse is given by:
𝛾(𝑡) = 𝜇𝐷−1(1 + 𝑓(1 − 𝑒−𝑡 𝜏⁄ )) (4.25)
With μD as the dynamic shear modulus, τ as the retardation time and f as the strength of the single-line retardation spectrum. A more general viscoelastic solid has multiple relaxation times and its creep response is given by:
𝛾(𝑡) = 𝜇𝐷−1[1 + ∑ 𝑓𝑛(1 − 𝑒−𝑡 𝜏⁄ 𝑛) 𝑛
] (4.26)
Firstly a viscoelastic analogue for the Boussinesq formula is derived with which the normal surface displacement is expressed as an integral over the distribution of surface pressure P(x,y).
𝑢𝑧 = (1 − 𝜈) 2𝜋𝜇𝐷 ∫ ∫ 𝑃(𝑥 ′, 𝑦′)𝑑𝑥′𝑑𝑦′ [(𝑥 − 𝑥′)2+ (𝑦 − 𝑦′)2]1 2⁄ (4.27)
In this formula uz is the displacement vector in the z coordinate, ν is the poisson’s ratio and x and y are position variables. Later on in the last part a two-dimensional rolling contact problem is solved. So what is done in this first section is: derive the normal surface displacement of a viscoelastic half space subject to a pressure distribution moving at uniform velocity V across the surface. With a half space that is larger than zero and has no shear tractions on the surface at z=0, the normal surface displacement and stress for an elastic material are given as:
𝜇𝑢𝑧 = −(1 − 𝜈)(𝜑)𝑧=0 (4.28) 𝜎𝑧= − ( 𝜕𝜑 𝜕𝑧)𝑧=0 (4.29) ∇2𝜑 = 0 (4.30)
So φ, a potential function, satisfies the Laplace equation and μ and ν are elastic shear modulus and Poisson’s ratio respectively. The analogue of these equations for a viscoelastic material are:
𝑢̅𝑧= − 1 − 𝜈 𝜇(𝑝)(𝜑̅1)𝑧=0 (4.31) 𝜎̅𝑧 = − ( 𝜕𝜑̅1 𝜕𝑧 )𝑧=0 (4.32) Here φ1(x,y,z,p) is a harmonic function of the coordinates x, y, z while ν denotes the ratio
𝜈 = 𝜆(𝑝)
2[𝜆(𝑝) + 𝜇(𝑝)] (4.33)
These analogues follow from the correspondence of transformed viscoelastic equations with equations of classical elasticity. The transformed viscoelastic equations are:
𝜎̅𝑖,𝑗= 𝜆(𝑝)𝜖̅𝑘𝑘𝛿𝑖,𝑗+ 2𝜇(𝑝)𝜖̅𝑖,𝑗 (4.34) 𝜖̅𝑖,𝑗= 1 2( 𝜕𝑢̅𝑖 𝜕𝑥𝑗 +𝜕𝑢̅𝑗 𝜕𝑥𝑖 ) (4.35)
The bars denote a Laplace transform with respect to time and λ(p) and μ(p) are transform moduli, with p the parameter entering Laplace transform, defined by either creep or relaxation behaviour in both shear and dilation. 𝜖 and 𝜎 represent the strain and stress tensors in their respective plains. As mentioned above the ν is the Poisson’s ratio and it will be taken as a constant. By inverting the functions for normal surface displacement and stress the following functions are obtained:
𝑢𝑧 = −(1 − 𝜈)𝜓1(𝑥, 𝑦, 0, 𝑡), 𝜎𝑧= − (
𝜕𝜑1(𝑥, 𝑦, 𝑧, 𝑡)
𝜕𝑧 )
𝑧=0
(4.36) φ1 and ψ1 are related by the Volterra equation:
𝜓1(𝑥, 𝑦, 𝑧, 𝑡) = ∫ 𝛾(𝑡 − 𝑡′) 𝜕 𝜕𝑡′ 𝑡 −∞ 𝜑1(𝑥, 𝑦, 𝑧, 𝑡′)𝑑𝑡′ (4.37)
Here 𝛾(𝑡) is the creep function in shear. The analysis will be restricted to steady-state problems by letting the observer travel with a constant velocity V in the positive x-direction. That way the stress and deformation fields appear stationary to the observer. Stress strain and displacement fields depend on x and t through the composite independent variable 𝑥 − 𝑉𝑡. By measuring x in a coordinate system also traveling with velocity V, the variable t can be eliminated from the analysis and the equations for normal surface displacement and stress.
By expanding φ in a double Fourier integral and substituting it in the equation for normal surface displacement and stress the following equations are obtained.
𝑢𝑧= −(1 − 𝜈) ∫ ∫ 𝜇−1(𝑖𝑞1𝑉)𝐴(𝒒)𝑒−𝑖𝒒𝒓𝑑𝑞1𝑑𝑞2 ∞ −∞ ∞ −∞ (4.38) 𝜎𝑧 = ∫ ∫ |𝒒|𝐴(𝒒)𝑒−𝑖𝒒𝒓𝑑𝑞1𝑑𝑞2 ∞ −∞ ∞ −∞ (4.39)
Here q=(q1, q2, 0), so this vector is restrained to the x-y plane. 𝜇(𝑖𝑞1𝑉) is the complex shear modulus for circular frequency q1V. Choosing
𝐴(𝒒) = − 𝑃0
4𝜋2|𝒒| (4.40)
Yields the fundamental case of a point force P0 acting at the moving origin.
𝜎𝑧= −𝑃0𝛿(𝑥)𝛿(𝑦) (4.41)
Where δ is the Dirac delta function. The associated displacement for this stress after some algebra becomes: 𝑢𝑧= 1 − 𝜈𝑃0 2𝜋𝜇𝐷 [(𝑥2+ 𝑦2)−12+ ∑ 𝑓𝑛∫ 𝑑𝜉𝑒−𝜉{(𝑥 + 𝑉𝜏𝑛𝜉)2+ 𝑦2}− 1 2 ∞ 0 𝑛 ] (4.42)
In which ξ is a dimensionless variable. By superimposing point-force solutions gives the displacement for a distributed pressure P(x,y):
𝑢𝑧 = 1 − 𝜈 2𝜋𝜇𝐷 [∫ ∫ 𝑃(𝑥 ′, 𝑦′)𝑑𝑥′𝑑𝑦′ [(𝑥 − 𝑥′)2+ (𝑦 − 𝑦′)2]1 2⁄ + ∑ 𝑓𝑛∫ 𝑒−𝜉 𝑃(𝑥′, 𝑦′)𝑑𝑥′𝑑𝑦′ [(𝑥 − 𝑥′+ 𝑉𝜏 𝑛𝜉)2+ (𝑦 − 𝑦′)2]1 2⁄ ∞ 0 𝑛 ] (4.43)
With this a quasi static response to of a viscoelastic half space to a moving load is given. This is for a viscoelastic material characterized by multiple relaxation times and in three dimensions. Next this solution is converted to a two dimensional rolling problem. Next step is to convert the solution to a standard linear solid which is specified by a single pair of parameters. The total equation then reduces to: 𝑢𝑧(𝑥) = − 1 − 𝜈 𝜋𝜇𝐷 {∫ 𝑃(𝑥′) log|𝑥 − 𝑥′|𝑑𝑥′+ 𝑓 ∫ 𝑑𝜉𝑒−𝜉∫ 𝑃(𝑥′) log|𝑥 + 𝑉𝜏𝜉 − 𝑥′|𝑑𝑥′ ∞ 0 } (4.44)
From this function and a first derivative to x the following function can be derived which closely resembles the differential form of the constitutive equation for the standard linear solid.
𝑢𝑧− 𝑉𝜏 𝑑𝑢𝑧 𝑑𝑥 = − 1 − 𝜈 𝜋𝜇𝐷 ∫ [(1 + 𝑓)𝑃(𝑥′) − 𝑉𝜏𝑑𝑃 𝑑𝑥′] log|𝑥 − 𝑥′|𝑑𝑥′ (4.45) 𝜖̇ + 𝜏−1𝜖 = 𝜇 𝐷{𝜎̇ + (1 + 𝑓)𝜏−1𝜎} (4.46)
This function rewritten in terms of η, a and b which are the position variables related to the contact width, see Fig. 10.
𝑢𝑧− 𝑉𝜏 𝑑𝑢𝑧 𝑑𝜂 = − 1 − 𝜈 𝜋𝜇𝐷 ∫ [(1 + 𝑓)𝑃(𝜂′) − 𝑉𝜏𝑑𝑃 𝑑𝜂′] log|𝜂 − 𝜂′|𝑑𝜂′ 𝑎 −𝑎 (4.47)
Now by means of algebra the function for the indentation is worked to account for the shape of the indenter. So now the indentation can be described in the contact zone, outside it and on the edge of the contact zone.
The pressure distribution is:
𝑃 = 𝜇𝐷𝑎 2 (1 − 𝜈)𝑅𝑉𝜏𝑒 ℎ𝜂 𝑎⁄ ∫ 𝑒−ℎ𝑦{(1 − 𝑦2)12+ (𝛤2 𝑎2− 𝛤1𝑦 𝑎 )/((1 − 𝑦 2)12} 𝑑𝑦 1 𝜂 𝑎⁄ (4.48) Where ℎ =𝑎(1+𝑓)
𝑉𝜏 and Γ1, Γ2 are parameters entering the pressure function. The pressure is imposed
to be zero on the edges of the contact zone. The load per unit length of the cylinder is described as: 𝑊 = 𝜋𝜇𝐷𝑎
2
2(1 − 𝜈)(1 + 𝑓)𝑅(1 + 2𝛤2
𝑎2) (4.49)
With R as the radius of the cylinder. Mister Hunter describes the force impeding the motion of the cylinder as a reaction of the viscoelastic solid with two components directed through the centre of gravity. 𝐹𝑧 = ∫ 𝑃 cos 𝜃 𝑑𝑥, 𝐹𝑥 = ∫ 𝑃 sin 𝜃 𝑑𝑥 𝑥1 𝑥2 𝑥1 𝑥2 (4.50)
Where sin 𝜃 = 𝑥/𝑅. Fz is the load per unit length of the cylinder W. Fx is described by the integral:
𝐹𝑥 = 𝑏𝑊 𝑅 + 𝑅 −1 ∫ 𝜂𝑃(𝜂)𝑑𝜂 𝑎 −𝑎 (4.51)
Worked out this becomes:
𝐹𝑥 = 𝑊 𝑅 [𝑏 − 𝑉𝜏 1 + 𝑓− 𝛤1 𝑎2 𝑎02 ] (4.52)
The coefficient of friction:
χ =𝐹𝑥 𝐹𝑧 = 1 𝑅[𝑏 − 𝑉𝜏 1 + 𝑓− 𝛤1 𝑎2 𝑎02 ] (4.53)
