Characteristics of elastomer

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Characteristics of elastomer materials

Abstract: This chapter presents a brief introduction to the physical properties of elastomeric materials and the technique used in their manufacture. Typical elastomers suitable for rubber-pad forming process are introduced and related concepts such as shape factor and shore hardness are explained. Linear and non-linear mechanical properties of elastomers and different hyperelastic material models for representing their physical behavior are discussed. These models are frequently used in fi nite element simulation of rubber deformation and rubber-pad forming processes.

Key words: elastomer mechanics; hyperelastic model; rubber.

3.1 Introduction

Elastomers or rubber-like materials have been used as an engineering material for nearly 150 years. The term elastomer, which is derived from elastic polymer, is a polymer which has the property of viscoelasticity, with low Young’s modulus and high yield strain. It consists of monomers which are linked together to form the polymer and are usually made

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Rubber-pad forming processes

of carbon, hydrogen, oxygen and/or silicon. At temperatures above the glass transition temperature, the elastomers are amorphous polymers with the possibility of segmental motion.

Rubbers are widely used as seals, adhesives, tires, springs, shock isolators, noise and vibration absorbers, corrosion and abrasion protection, and electrical and thermal insulators. Rubber materials possess high damping and large extensibility, which are very useful in suspending resonant vibrations. Due to their large elastic deformability they are widely used as impact absorption in the marine industries and even for blast protection. Panels and reinforced concrete walls coated with rubber can provide some level of protection to the occupants when the walls are subjected to airblast or explosive loading. The low modulus of the rubber made it an ideal material for building isolator bearings for earthquake protection.

According to ASTM D 1566, an elastomer is a material that can recover from large deformations quickly without the need for applying external force and is capable of withstanding high per cent of elongation before fracture. Under tension, elastomers can generally stretch 300–500 per cent before breaking, behaving as hyperelastic materials. They also have low thermal conductivity and show signifi cant hysteresis under cyclic loading.

Elastomer parts are mostly fabricated using a molding process. With this process inexpensive molds can be used to mold large or intricate shaped pairs at relatively low costs and the material characteristics can be optimized to suit the requirement of the application. The rubber to be used in the molding process is mixed with other chemicals (known as additives) and then heated, melted and processed into a mold. The molding process is based on the controlled temperature–pressure–time cycle. Once the chemical process is completed (‘cured’) the rubber is vulcanized and cooled.

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In order to give rubber its shape, the polymer chains in rubber are tied together in a process known as ‘crosslinking’ or ‘vulcanization’. Vulcanization is the process of adding sulphur or other equivalent curing chemicals which will modify the polymer mechanical characteristics by forming crosslinks. Table 3.1 listed the mechanical properties of general elastomers in comparison with other engineering materials. What characterizes elastomers are the low elastic modulus, Poisson’s ratio of about 0.5 (incompressible) and high values of percentage of elongation to fracture. The deformation of elastomers are non-linear and as such the value of the Young’s modulus can only serve as a guideline with little use in design evaluation. Since the modulus is strain dependent, analysis of rubber components must use rubber material models such as Mooney-Rivlin, Ogden and Arruda-Boyce which are based on strain energy function, as discussed later.

Material Modulus of elasticity (GPa)

Poisson’s ratio

Ultimate tensile stress (MPa)

Percentage of elongation to fracture (%)

Thermal conductivity (W/m K)

Elastomers 0.0007–0.0004

0.47–0.5 7–20 100–800 0.13–0.16

Aluminum alloys

70–79 0.33 100–550 1–45 177–237

High strength steels

190–210 0.27–0.3 550–1200 5–25 35–60

Titanium alloys

100–120 0.33 900–1200 10 7–7.5

Nylon plastic

2.1–3.4 0.4 40–80 20–100 0.3

Comparison of properties of elastomers and some engineering materials

Table 3.1

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3.2 Elastomer types

There are two broad divisions of elastomers. The fi rst is thermoset elastomer (TSE) which cannot be reversed to its initial chemistry once cured. Curing can be achieved through heat (above 200 ºC), chemical reaction as in the two-part epoxy or by radiation using electron beam processing. They are not soluble in solvents and cannot be reprocessed using heat. Examples of thermoset elastomeric materials are vulcanized natural rubber and nitrile rubber (NBR). Reheating thermoset materials will not cause them to melt as thermoplastics. Since they are less prone to heat, thermoset elastomers are used for tires and mounts seals, for example NBR has a typical temperature rating of 40 ºC to 120 ºC.

Thermoplastic elastomers (TPE) can be dissolved in suitable solvents, and when heated become safe which enables them to be reprocessed. TPEs are a class of copolymers which are a mix of a plastic and a rubber and thus have the advantages of both types of materials. The difference between TSE and TPE is the type of crosslinking bond; TSE has a covalent bond while TPE has the weaker dipole or hydrogen bond. TPEs are potentially recyclable since they can be molded, extruded and reused due to their plastic components and hot recyclable due to the thermoset characteristics of rubber. In recent years the availability of delinking compound (patent held by the Petra Group) has enabled the vulcanized rubber to be devulcanized and recycled into what is known as ‘Green Rubber’. Another advantage of TPE is that it requires little or no compounding or additional reinforcing agents, stabilizers or cure systems leading to consistent raw materials that are easily dyed and good quality control features. The shortcomings of TPE are their high cost and raw material, which is diffi cult to use

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with fi llers and poor chemical and heat resistance. In terms of recyclability TPE is preferred over TSE.

3.3 Compounding

Rubber compounding or formulation refers to the addition of certain chemicals to raw rubber in order to obtain the desired properties. The well-known chemicals are crosslinking agents, reinforcements, anti degradants and colorants. The crosslinking agents are required for establishing the crosslinks to interconnect at molecular level thus improving the strength and elasticity. Unformulated elastomers behave like a high molecular weight with low elasticity and strength. Through formulation long-chain molecules are chemically linked together, forming networks and transforming the material from a viscous liquid to elastic solid. This is what happens during vulcanization or curing, which increases the strength and the modulus and decreases the hysteresis. Sulphur is widely used as the vulcanization agent.

A deformed elastomer stores the input energy in terms of elastic potential energy in the chains. This is released upon crack growth and acts as the driving force for fracture to propagate. The remaining energy is dissipated into heat through the molecular motions. Tight network of the molecular chain due to high crosslink levels restricted the chain motion preventing the network from dissipating much of the energy which causes brittle fracture at low elongation. Too high crosslink level is undesirable and can lead to brittle fracture and too little crosslinks may not be strong enough to resist viscous fl ow failure. Therefore, the optimum density range of the crosslinks for practical use needs to be determined.

Reinforcing agents of the rubber compound act as stress arrestor and are required to have high specifi c area. This

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means that the particle must be smaller than 1 μm in size. Typical fi llers are carbon black and silica. A primary particle size as small as 0.1 μm can be obtained which can give a specifi c area of a few hundred square meters per gram of fi ller.

3.4 Typical elastomers used in rubber-pad forming processes

Natural rubber (NR) is derived from the latex of the rubber tree Hevea brasiliensis, which is grown in the equatorial belt. The biggest user of natural rubber is for automobile tires. There are other plants that produce latex which include Gutta-Percha (Palaquium gutta), rubber fi g (Ficus elastica) and Guayule (Parthenium argentatum); however, these never reach the economic signifi cance of Hevea brasiliensis. Natural rubber is also known as gum rubber. The natural rubber, when vulcanized, will have long chain-like molecules interconnected with crosslinks forming a molecular network. To increase the stiffness and abrasion resistance, fi llers made of carbon black and silica are added during curing.

Styrene-Butadiene Rubber (SBR) is a synthetic rubber copolymer made of two compounds of styrene and butadiene. SBR is usually blended with natural rubber in car tires due to its good abrasion resistance and aging stability. The reaction of SBR is through free radical polymerization where two monomers, a free radical acid and a chain transfer agent are mixed inside a pressure reaction vessel. Rubber with high styrene content is generally hard since the transition temperature of butadiene is extremely low.

Silicone rubber (SR) is an inorganic synthetic elastomer made of silicone polymer containing silicon, carbon, hydrogen and oxygen. The manufacture of silicone rubber parts requires heat for vulcanization. A two-stage process is

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used: the parts are molded and this is followed by a prolonged post-cure process. SR is suitable for injection molding. The high-temperature properties of SR are excellent; it is non-reactive, stable and resistant to the temperature of 55 ºC to 300 ºC. It also has a wide hardness range of 10 to 80 shore hardness. A good example of high temperature application of SR is the turbocharger hose. Its inert characteristics of not importing taste or smell made it the best candidate for food-contact applications.

Polyurethane (PU) also known as polycarbonates or simply urethane are linear polymers containing carbamate groups (–NHCO2). The PU polymers are produced through a chemical reaction of diisocyanate and a polyol. Production of PU is done in a reaction vessel where a condensation-reaction is performed (known as step growth polymerization). Commercially important PU also contains other functional groups like esters, ethers, amides and urea. One widely used application of PU is in rigid and fl exible foams where organic compounds containing carboxyl groups are used, causing a reaction that produces carbon dioxide bubbles throughout the foam. There are many advantages of polyurethane. Firstly, the high abrasion resistance, which makes it suitable for applications such as castor wheels. It also has high load-bearing capacity compared to other rubber and is widely used in metal forming pads, shock pads and machine mounts. Due to its good damping properties, hard urethanes are used as gears to reduce noise and soft urethanes are used for sound dampening. Due to its good tear strength, it is also used for diaphragms, gaskets, and drive belts. In general there are three different forms of PU including foams, coatings and elastomer. Thermoplastic PU elastomers can be molded into parts. PU coatings have good impact resistance and abrasion resistance, and water-based PU coatings are used in painting aircraft and automobiles.

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The largest market for PU is fl exible foam used in furniture cushioning. Semi-fl exible PU foam is used in car dashboard and door lines. In general the useful temperature range of PU is 30 ºC to 70 ºC. However, for specifi c application, there are manufacturers that offer high temperature PU hose which can operate in the range of 70 ºC up to 260 ºC where high-quality ether-polyurethane are used instead of the more general ester-polyurethane. For general application, PU is limited to applications where the temperature is below 70 ºC since the bond between urethane and metal weakens signifi cantly above 70 ºC. One of the major advantages of PU is that the raw material is liquid, which allows it to be pumped, mixed and dispensed at precise temperatures and ratios. They enter molds as liquid at room temperature and are cured at the same elevated mixing temperature. This enables very large polyurethane parts to be molded with thick cross-sections. The development of polyurethane elastomer has extended the use of rubber-pad forming techniques in manufacturing industry, as they are generally harder than rubber materials, more resistant to attack by lubricant oil, and have better wear resistance.

Compressive stress–strain curves for silicon rubber, natural rubber and polyurethane with equivalent hardness are compared in Figure 3.1. As illustrated in the fi gure, the polyurethane can be loaded beyond the conventional limits for rubber and this higher bearing load capacity permits design of smaller parts with possible saving in weight and cost of materials. These stress–strain curves can be used in design of experiments; however, the effect of shape factor should be taken into account. Shape factor is defi ned as the ratio of the area of one loaded surface to the total area of the unloaded surfaces which are free to bulge. Parts made from the same compound and having the same shape factor behave identically in compression, regardless of actual size or shape.

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The shape factor is defi ned as:

(3.1)

(3.2)

where L is the length of the rectangular block, W is the width of the block, t is the thickness (height) of the block, d is the diameter of the disc or cylinder and h is the height. These relations are limited to the blocks which have parallel loading faces and their thicknesses are not more than twice the smallest lateral dimension. The loading surfaces of the block should also be restrained from lateral movement. Figures 3.2 to 3.5 depict the effect of shape factor and hardness on the stress–strain behavior of polyurethane under compression. The concept of the hardness of the elastomer is explained in

Comparison of stress–strain curves of different elastomers

Figure 3.1

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Compressive stress–strain curves of polyurethane shore A hardness 60º with different shape factors

Figure 3.2


Figure 3.3

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Figure 3.4


Figure 3.5

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the next section. As a general rule: the harder the elastomer, the greater its load-bearing capacity.

3.5 Mechanical properties of elastomers – linear elastic

A linear elastic material obeys Hooke’s law where strain is proportional to stress. An elastomer has the elastic property where it returns to its original shape upon the release of the applied force. Elastomers in general are exposed to large deformation; however, in the case where the deformation is relatively small (say 25 per cent strain) the stress can be approximated using linear elastic analysis. This enables the rubber design problems for small deformation to be solved if the modulus of elasticity, E is known.

Elastic materials can be represented by two constants. The fi rst is the bulk modulus, K which is defi ned by the following relation between the applied pressure, P and the volumetric strain:

(3.3)

where ΔV is the volume reduction and V0 the initial volume.The second constant is the shear modulus, G which

describes the resistance to a simple shearing stress τ and is defi ned by the relation G = τ /γ where γ is the amount of shear, defi ned as the ratio of the lateral displacement d to the height h of the sheared block.

Based on isotropic material model, the modulus of elasticity is the ratio of the stress σ to the strain, ε:

(3.4)

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The three constants are inter-related by the following equation:

(3.5)

The Poisson’s ratio, ν is defi ned as the ratio of the minor axis strain to the major axis strain obtained by a simple tensile test and is given by

(3.6)

Solid elastomers have relatively high value of bulk modulus, K but with low shear modulus, G in the range of 0.5 to 5 MPa. Since rubber is considered almost incompressible, the Poisson’s ratio is close to one half (approximated as 0.499). For a completely incompressible elastomer (ν = 0.5), the elastic behavior at small strain can be described by a single elastic constant of G. The determination of the modulus of elasticity of elastomers is generally done by measuring the elastic indentation caused by a rigid indentor of a standard geometry (a cone or a sphere) into the surface under standard loading condition.

Non-linear scales are used to derive the hardness value of the rubber. The hardness is obtained by the difference between a small initial force and a much larger fi nal force. The International Rubber Hardness Degrees (IRHD) scale has a range of 0 to 100 corresponding to zero to infi nitely high elastic modulus. The IRHD employs a ball as the indenter and provides rapid measurement of rubber stiffness. In this test a dead load system measures the indentation depth of spherical indenter in the soft elastomers. A minor load establishes the 100 IRHD position fi rst and a subsequently major load is used to establish the penetration which is electronically measured and converted to IRHD units.

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The shore A scale is another hardness scale used widely in the United States. The indenter used is a 35º truncated cone with test load of 9.81 N and holding time of 15 sec. The shore D test uses a sharp 30º cone with test load of 49.05 N and holding time of 15 sec. The readings range is 30–95 points. The results of shore A scale and the IRHD scale are approximately equal over similar resiliency range. Shore D scale is used for ‘harder’ elastomers. In general conversion between the scales is not advisable due to poor correlation.

The equations used for the determination of modulus E are based on the assumption that the block is thicker than the radius of the indentor and much thicker than the indentation. The following relations hold between indenting force F and amount d of displacement of the rubber surface under an indentor:

For a fl at-ended cylindrical punch of radius a (Timoshenko and Goodier, 1970),

(3.7)

For a spherical indentor of radius a,

(3.8)

For a conical indentor with semi-angle of θ (Sneddon, 1975),

(3.9)

In each of these relationships the rubber is assumed to be completely incompressible. For compressible materials, the right-hand sides are divided by the factor 2(1 – ν). The manner in which load-bearing properties of polyurethane changes with shore A hardness at various compressive strains is shown in Figures 3.6, 3.7 and 3.8.

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Compressive stress vs. hardness (with different shape factors) at 10 per cent strain

Figure 3.6


Figure 3.7

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3.6 Mechanical properties of elastomers – non-linear elastic

As discussed previously, linear elasticity is valid for small strain problems. For cases involving large deformation, which is typical of elastomers, non-linear hyperelastic models must be used to account for the non-linearity between stress and strain.

The development of the hyperelasticity begins with the assumption that the long chain molecules are randomly oriented which gives rise to isotropy. As the elastomer is stretched, the molecules orient themselves bringing about anisotropy. This anisotropy follows the direction of straining which brings the whole elastomer back to isotropy condition. To model this peculiar behavior, strain energy potential is formulated as a function of the strain invariants which are independent of the choice of the axes.


Figure 3.8

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(3.10)

(3.11)

(3.12)

where λ1, λ2, λ3 are the principal stretch ratios (stretched length/unstretched length). In the free unstrained state, λ1, = λ2, = λ3 = 1 and the values of Ji become zero. For incompressible materials J3 is zero and resulted in only two independent measures of strain of J1 and J2. This limits the strain energy W, making it dependent on J1 and J2 only. For small strain values the strain energy is a linear function of J1 and takes the form of:

(3.13)

where C1 and C2 are constants. This form of strain energy function is known as Mooney-Rivlin equation. Since J1 and J2 are of second order in the strains εi, this equation is valid for a greater range of strains than the linear elastic strain theory which neglected the higher order strain.

Strains larger than 25 per cent will cause the fourth order terms to be signifi cant and cannot be neglected. For the equation above the strain is less than unity. To include the higher order terms of J2 (i.e. all terms of order ε4

i), it is necessary to include nine terms in the equation for W for compressible elastic solid and for incompressible elastic solids these terms are reduced to fi ve. It is clear that fi ve elastic coeffi cients are required in order to obtain the strain energy function of rubber.

3.7 Hyperelastic models and elastomer mechanics

There are three approaches to study the elasticity of elastomers. The fi rst approach is the thermodynamic

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approach which focuses on the macroscopic behavior of the material. The second is the statistical approach which uses the statistical mechanics to describe the behavior of the molecular dynamics. The third is the phenomenological approach which describes the observed elasticity behavior for large deformation particularly by representing the experimental data with mathematical equations. The two major relationships are the strain and the applied forces acting on the body and secondly the elastomeric material properties as a function of elastic energy stored in the elastomers. The phenomenological approach does not describe the physical mechanism of the elastomeric behavior. The elastic energy stored in the elastomer when it is deformed can be described by the state of the strain defi ned by the three principal stretch ratios λ1, λ2, λ3. Strain energy density (strain energy stored per unit volume) function is usually written in terms of invariants of the deformation tensor. There are several polynomial models used to represent this which include the Arruda-Boyce, Mooney-Rivlin, Neo-Hookean, Ogden, polynomial, reduced polynomial, Yeoh and Van der Waals. Most of these material models are available as a special material model in fi nite element method software which enables the wider adoption of these models in the design and analysis of elastomeric components. The various models of the elastomers can generally be grouped under the polynomial and the non-polynomial models. The polynomial models include the Mooney-Rivlin and the reduced polynomial; subsequently the Yeoh and Neo-Hookean models are a subset of the reduced polynomial model.

The selection of which model to use in the analysis depended very much on the availability of the experimental data. For limited test data such as uniaxial tension or compression data, the models of Arruda-Boyce, Van der

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Waals or reduced polynomial forms give reasonably good correlation. With the availability of both uniaxial and biaxial test data, the models of Ogden and Van der Waals are more accurate.

The strain energy potential, W, of elastomers can be written in the polynomial form as below:

(3.14)

where Cij and Di are constants. The parameter N can be up to six but in general the value of N is less than two when both the fi rst and second invariants are used.

For N = 1, the Mooney-Rivlin form is produced (Mooney, 1940):

(3.15)

From the above equation, only the linear terms of the deviatoric strain energy is retained. The Mooney-Rivlin model is generally valid for strain up to 100 per cent.

The reduced polynomial form can be obtained by setting all Cij with J ≠ 0 to zero in equation (3.14):

(3.16)

The Yeoh form is a special form of the reduced polynomial with N = 3 (Yeoh, 1993):

(3.17)

The simplest form is when N = 1, which produces the Neo-Hookean form:

(3.18)

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The Neo-Hookean strain energy potential is the simplest hyperelastic model. The Ogden strain energy potential is given in the form below (Ogden, 1986):

(3.19)

If N = 2, α 1 = 2 and α 2 = –2, the Mooney-Rivlin model is obtained. If N = 1 and α1 = 2, Ogden model degenerates to the Neo-Hookean material model.

For the Arruda-Boyce model, the strain energy potential is given below (Arruda and Boyce, 1993):

(3.20)

The model above depends on the fi rst invariants only. The physical interpretation is that the eight chains are stretched equally under the action of a general deformation state. The fi rst invariant, I1 = λ1

2 + λ22 + λ3

2 , directly represents this elongation. The Arruda-Boyce model is good up to more than 325 per cent of uniaxial strain and such material model is generally available in most FEM software where the user only needs to type in the stress-elongation data obtained from a uniaxial test. The form of the Van der Waals strain energy potential is (Treloar, 1975):

(3.21)

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where I~ = (1 – β)( J1 + 3) + β ( J2 + 3) and .

Determination of which material model to use is highly dependent upon various factors like, type of application, strain range, availability of required test data, etc. The quality of the input as material constants will directly affect the accuracy of the design analysis when using these material models. The material constants for rubber material depend on several aspects such as amount of strain, type of deformation, loading rate and strain history. One of the challenges in obtaining reliable experimental data is the softening effect of the rubber in the fi rst few cycles, known as the Mullin effect. Most of the softening occurs in the fi rst deformation, and after a few deformation cycles the rubber approaches a steady state with a constant stress–strain curve. It is important to suffi ciently condition the sample in order to obtain steady-state characteristics of the rubber sample.

The above mentioned hyperelastic models are widely used for fi nite element simulations of rubber deformation and rubber-pad forming processes. The fi nite element method (FEM) is a robust numerical technique that can be used to obtain the behavior and deformation mechanism of workpiece and tools during forming process. Unlike analytical methods, this method permits the analysis of complex deformation and forming processes without the necessity of developing and solving complex equations. A large number of commercial FEM packages, such as ABAQUS, ANSYS, DYNA3D and NASTRAN, can be used for fi nite element simulation of rubber-pad forming processes and can predict possible process defects such as wrinkling and rupture. When developing a new forming process it is helpful to fi rst simulate the process using FEM software and try different process variables and schemes, and in this way optimized process

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parameters may be determined prior to physical try-outs, and by taking necessary technical measures the production costs and time consumption can be reduced, trial and error methods can be signifi cantly reduced, and the number of physical experiments can be minimized.

Before fi nishing this chapter, it should be mentioned that, for convenience, from the next chapter the term ‘rubber’ will be used throughout the book to refer to the viscoelastic pressure medium which may be rubber, polyurethane or any other elastomeric material.

3.8 BibliographyArruda, E.M., Boyce, M.C. (1993). A three-dimensional constitutive

model for the large stretch behavior of rubber elastic materials. Journal of the Mechanics and Physics of Solids 41 (2), pp. 389–412.

Mooney, M. (1940) A theory for large elastic deformation. Journal of Applied Physics 11, pp. 582–597.

Ogden, R.W. (1986) Recent advances in the phenomenological theory of rubber elasticity. Rubber Chemistry and Technology 59 (3), pp. 361–383.

Sneddon, I. (1975). Applications of Integral Transforms in the Theory of Elasticity. New York, NY: Springer-Verlag.

Timoshenko, S.P., Goodier, J.N. (1970). Mathematical Theory of Elasticity. 3rd edn. New York, NY: McGraw-Hill.

Treloar, L.R.G. (1975). The Physics of Rubber Elasticity, 3rd edn. Oxford: Clarendon Press.

Yeoh, O.H. (1993) Some forms of the strain energy function for rubber. Rubber Chemistry and Technology 66 (5), pp. 754–771.

Engineering

Characteristics of elastomer