introduction to materials science || tensile behaviour of materials

Chapter 11.

Tensile behaviour of materials

11.1. Objectives • To interpret tensile test results and describe the mechanical properties

of a material.

• To understand the typical tensile curves for different materials.

• To relate nominal and true stresses and strains.

• To introduce the concept of strain energy.

In chapter 6., the elastic behaviour of ideal solids subjected to mechanical stress was described. It was shown that the strength of materials was generally much lower than that calculated theoretically by only considering the interatomic bond strengths (§ 6.3.4.). This divergence between theory and practice is the result of defects (dislocations), which were described in chapter 7.

In this chapter, the tensile behaviour of real materials will be studied. In service, materials are subjected to loads or forces which cause deformations (strain). It is important to know how materials behave during deformation. The most important mechanical properties are stiffness (elastic modulus), yield strength and ductility measured by tensile testing and hardness determined by various types of test.

This chapter is limited to the macroscopic and phenomenological aspects of mechanical properties. The microscopic aspects of deformation mechanisms and the hardening processes, which influence the mechanical behaviour of materials, will be presented in chapter 12.

11.2. Tensile properties

11.2.1. Strength of materials and mechanical properties

Distinction should be made between mechanical properties describing the specific behaviour of materials, and what is called the strength of materials which analyses the behaviour of structural elements subjected to mechanical stresses: machine parts, structural elements in civil engineering, etc.

The strength of materials is an engineering discipline using the specific properties of materials, such as the modulus of elasticity (Young's modulus) or the yield stress, to calculate the service stresses and strains of structural elements and assure dimensional correctness. On the other hand, the study of mechanical properties, considered in this chapter, aims to establish the intrinsic characteristics of materials when they are deformed.

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Introduction to Materials Science

262 Introduction to Materials Science

The application of an external force to a solid provokes initially an elastic deformation. For a large number of materials (metals, certain polymers), this elastic deformation, which is reversible, is followed by a permanent, irreversible plastic deformation. There are two types of limiting stresses: the yield strength Re, giving the stress in the material at the end of the region of elastic deformation, and the ultimate tensile strength R m , which is the value of the maximum load before rupture, divided by the original area of the specimen. These two strength values are the same in brittle materials such as ceramics and a significant number of organic polymers which break without any prior plastic deformation,

Two other important characteristics are also measured in the tensile test: the ductility of a material, which is the amount of permanent deformation at rupture and the toughness, which is the energy absorbed by a material at rupture. Materials are often required with a high elastic modulus and high toughness, i.e. ductile, but rigid materials with a high yield strength. In practice it is difficult to obtain such a combination of mechanical properties.

As already mentioned in chapter 6, organic polymers at ambient temperature, and metals and ceramics at high temperatures can have viscoelastic behaviour. Here, the instantaneous elastic reaction is followed by viscous flow leading to a significant change in the mechanical properties as a function of the duration of the application of the stress (creep). To be able to use viscoelastic materials sensibly, it is necessary to use extrapolation techniques to determine the mechanical behaviour over long times. This point will be discussed in chapter 12.

The mechanical properties of materials are measured by standard test procedures using defined test specimens subjected to specific loading conditions. A test specimen of the material is made to specified dimensions. These standard tests make it possible to compare data from different laboratories. Various service conditions can also be simulated by appropriate test procedures (accelerated aging, wear, fatigue, etc.). However considerable care must be taken when translating the results of laboratory tests to 'in service' performance where real conditions are often much more complex.

11.2.2. The tensile test

The tensile test is the most common mechanical test. An increasing tensile force is applied to a bar of standard dimensions until it ruptures after loading at a constant deformation rate. A series of important mechanical characteristics can be measured by recording the force applied by the tensile machine to the specimen and its progressive elongation. In most instances, the variation of the cross-section of the specimen during testing is not known and, as a general rule, the force F and elongation Δ/ are measured relative to the initial dimensions of the specimen which gives the nominal stress σ:

where So is the initial cross-section of the tensile specimen. In the same way, the nominal deformation ε is defined as:

(11.1.)

(11.2.) Al

£=

h

F σ = —

So

Tensile behaviour of materials 263

where / 0 corresponds to the initial length the specimen. The value of ε is generally, given as a percentage. Plotting σ=σ(ε) gives the tensile curve.

The tensile curve of a ductile metal is shown in figure 11.1. The following four characteristic parameters can be determined from a tensile curve of this type.

• The modulus of elasticity Ε (or Young's modulus) is given by the slope of the elastic part of the stress - strain curve. As mentioned in chapter 6, this elastic modulus is a function of the bond energies between the atoms or molecules making up the material.

• The yield strength R e giving the value of the stress beyond which the material deforms plastically. As plastic deformation often appears gradually, the yield strength is difficult to determine precisely, and usually the 0.2 % offset yield R02IS used. A line parallel to the elastic portion of the stress-strain curve and intersecting the strain ordinate at 0.2 % is drawn (figure 11.1) and the point at which this line intercepts the stress-strain curve determines the 0.2 % offset yield Ro.2.

• The ultimate tensile strength R m is defined as the maximum load supported by the specimen divided by the original cross-sectional area.

• The rupture elongation e R measures the plastic deformation of the specimen after rupture under tension. This is a measure of ductility.

The yield strength R e is important since it defines the limiting stress which must not be exceeded in order to avoid the permanent deformation of a part in service. A large difference between the values of R m and Re, together with a high value of £R, provides a good degree of security in case the yield stress is locally exceeded in a loaded part.

For safety reasons machine components are designed so as to maintain stress at a lower level than yield. The mechanical behaviour thus depends only on the modulus of elasticity E. For example, when the top of the Eiffel Tower experiences oscillations of the order of 50 cm amplitude in a strong wind, this deflection is not catastrophic because it remains well within the elastic limit of the steel used in the construction. However, even when the applied stress is lower than yield, prolonged periodic deformations can lead to fracture due to the phenomenon offatigue, which will be considered in chapter 13. In practice, the surface degradation of materials, due to corrosion, often adds to the effect of static or cyclic stresses, thereby reducing the strength.

During the thermodynamic study of uniaxial elongation (section 6.3.), it was possible to differentiate between two major classes of material: materials with en-thalpic elasticity (metals, organic and mineral glasses, semi-crystalline polymers) and elastomers, which have entropic elasticity. The general characteristics of the tensile curves of enthalpic elastic materials, by far the most numerous, are reviewed below followed by a description of the behaviour of elastomers in tension.

11.2.3. Stress - Strain Curves of enthalpic elastic materials

When a ductile metal, such as mild steel, is subjected to tensile stresses, the following effects are observed (figure 11.1.).


Figure 11.1. Stress σ - Strain ε curve for a cylindrical bar of ductile metal subjected to uniaxial elongation.

The initial deformation is elastic. When the stress is removed the specimen regains its initial shape and dimensions with a total and almost immediate reversal of the deformation. In metals, elastic deformation is normally linear. In materials with a high modulus of elasticity (metals and ceramics), the elastic deformation is usually less than 0.1 %. In this region, Poisson's Ratio ν (chapter 6.) is close to 0.3 for metals.

If the yield stress is exceeded, the specimen length is increased permanently after the stress is removed. This is due to an irreversible deformation (plastic deformation).

When the material is subjected to loading and unloading in the region of plastic deformation, the yield stress increases with successive deformations. This is illustrated in figure 11.2. Two additional loadings of a 0.2 % carbon steel specimen were carried out after the initial loading. The conventional 0.2 % offset yield Ro2

(ε = 0.2 %) is of the order of 200 MPa. After an additional loading cycle, the conventional 0.2 % offset y ie lds2 r eaches 350 MPa. After another loading cycle, it is greater than 400 MPa.

Work hardening generally occurs during the plastic deformation of metals and is marked by an increase in hardness and yield stress. This is due to an increase of dislocation density during plastic deformation, which tends to block dislocation movement. This will be covered in more detail in the following chapter.


(MPa) 500

0 0 2 3 (%)

Strain ε —

Figure 11.2. Stress σ- Strain ε curve in nominal values for a steel specimen containing 0.2 % of carbon. Two additional loading cycles (1) and (2) were made after the initial tensile test.

The phenomenon of work hardening has been mainly studied for metals but similar phenomena occur in other materials such as ductile thermoplastics (figure 11.3.) although different mechanisms are involved.

A tensile curve can show an inflection during the elastic-plastic transition at the yield point. This is particularly so for soft steels (figure 11.2.), and results from weak anchoring of the dislocations by the interstitial carbon atoms.

During the initial elastic deformation and for small amounts of plastic deformation, the elongation of the specimen goes with a homogeneous contraction along the whole sample length. Beyond the deformation corresponding to the maximum load of the nominal stress - strain curve, the contraction of the section ceases to be homogeneous and becomes more significant at some place in the specimen. At this point, the localised cross-section of the specimen is reduced, i.e. necking (figure 11.1.) occurs. The strength of the specimen, which is proportional to the cross-section, also decreases. Therefore, necking appears when the elongation exceeds the maximum of the tensile curve in figure 11.1. This value corresponds to the maximum load supported by the test specimen deformed in a homogeneous manner.

The reduction in area As is the relative percentage decrease in cross-section measured after rupture and is given by the following relationship:

So is the initial cross-section, and Sr, the cross-section after rupture. The reduction in area is also a measure of ductility: As varies from zero (brittle materials) up to nearby 100 as a function of ductility. When rupture occurs, the specimen suddenly contracts, releasing stored elastic energy (figure 11.1.). It is accompanied by sound emission as a result of the transformation of potential into kinetic energy.

Figure 11.3. shows the tensile curve of a ductile polymer, using moderate deformation (strain) rates of (1 to 100 %min _ 1 ) . In the first part, (points (1) and (2)),

(11.3.) Α = - ^ - Α χ ΐ 0 0 So


100 Strain ε (%)

Figure 11.3. Nominal stress - strain curve for a ductile thermoplastic polymer. Up to (A), a pseudo-elastic deformation can be observed allowing the apparent modulus of elasticity to be determined. (B) Formation of the neck and (C) plastic deformation with orientation of chains. The necking propagates across the whole sample and (D) sample break. Curves of this type are encountered for some amorphous thermoplastics (PC) at Τ < Tg

and semi-crystalline thermoplastics (ΡΕ - PP - PA 6-6).

this curve is similar to that of metals, although stresses are lower and strains greater. The initial part of the tensile curve is almost linear, but it does not necessarily correspond to ideal elastic behaviour. As was mentioned in Chapter 6, polymers are usually viscoelastic.

Necking occurs in ductile semi-crystalline polymers (polyamide, polyethylene, etc.) or amorphous polymers (polycarbonate). The difference with respect to metals is that the neck gradually propagates through the entire specimen (figure 11.3.). Plastic deformation is accompanied by a significant increase in strength, as a result of the orientation of chains in the stress direction. This is why necking propagates throughout the sample and plastic deformation can reach up to 500 %. This phenomenon of consolidation by plastic deformation is used in the manufacture of textile fibres, which undergo a drawing step just after spinning.

The modulus of elasticity, determined from the slope of the tensile curve at low strains, is an apparent modulus of elasticity depending on the strain rate. The apparent modulus may also be strongly influenced by temperature depending on the proximity of the glass transition. For these materials, only tests carried out at the same temperature, at the same strain rate and, for moisture sensitive polymers (polyamide), at the same relative humidity, are comparable. The variation in mechanical behaviour with temperature, specific to polymers, is illustrated in figure 11.4 which shows the behaviour under tension of poly(methylmethacrylate) (PMMA). The transition between brittle and ductile behaviour occurs around 40 - 50 °C, i.e. about 60 °C below Tg (-105 °C). The slope at the origin gives the apparent modulus of elasticity, which decreases considerably with temperature.


20 30 Strain m % —»~

Figure 11.4. Stress - strain curves for poly (methylmethacrylate) PMMA (organic glass) as a function of temperature (after Andrew, 1968).

10%

(a)

200 1 1 1 "

(Γ—Ξ HDPE

(b) g loo

10%

(c)

/ A I 2 O 3

Α1 alloy

j / . Mineral glass

Organic glass

Deformation 0,2%

NR HDPE Organic glass

Vulcanised natural rubber High density polyethylene Poly(methylmethacrylate)

Figure 11.5. Stress - strain curves for different types of materials: (a) large deformation (£max>100%); (b) moderate deformation (ε™*-10%); (c) small deformation (ε™χ < 0,2 %).


To compare the elastoplastic behaviour of different materials, it is often necessary to use different levels of strain (£ m a x = 0.2 %, 10 %, 500 %, (figure 11.5.)). The plastic deformation of metals and alloys involves dislocation movement. In ceramics, the movement of dislocations is severely impeded at ambient temperature by the high strength of the bonds and by the presence of ions of opposing sign also making atomic movement difficult. Ceramics, like mineral glasses, are mainly brittle materials which break without yielding. A variety of behaviours is observed in polymers, depending on the molecular structure and the temperature. Most amorphous glassy polymers (polystyrene, poly(methylmethacrylate)) are brittle at Τ < Tg. The semi-crystalline polymers (polyethylene, polypropylene) are generally ductile at temperatures between Tg and Tm. Below Tg9 the semi-crystalline polymers become brittle, for example, polypropylene, (Tg ~ -15 °C) or poly(ethylene terephthalate) (Tg ~ 70 °C).

11.2.4. Tensile Behaviour of Rubber

As mentioned in chapter 6., elastomers are a special class of materials. The polymer chains are bound to each other by cross-links (figure 5.5.). The elastomers are three-dimensional networks (figure 6.7) of elastic polymer segments. The thermal motion of very mobile chain segments induces the retraction force. The elastic retraction force is almost exclusively of entropic origin.

The tensile curve at high strain is given in figure 11.6. Since the initial portion of the stress-strain curve is not linear, the modulus of elasticity Ε is calculated from the slope at the origin. This is extremely low. This observation obvious when

(MPa)

Figure 11.6. Tensile stress-strain curve for vulcanised natural rubber (non-linear elasticity): (a) nominal stress-strain curve, (b) theoretical stress-strain curve (after Treloar, 1975).


the tensile stress-strain curve for vulcanised natural rubber (NR) is compared with other tensile curves (figure 11.5. (a)). Elastomers rupture at extremely high levels of deformation (ε ~ 700 %) without preliminary plastic deformation. Therefore, paradoxically, they break in a brittle manner. Non-cross-linked elastomers are very viscous viscoelastic liquids.

The glass transition temperature of elastomers Tg is close - 7 0 °C. They are always used at temperatures higher than Tg when cohesive forces between the chains segments are extremely weak. These are able to move relative to each other without modification of the interatomic distances and without appreciable variation of the valence angles, i.e. practically without any variation in the internal energy. The tensile curve for an elastomer in uniaxial compression and at low strain elongation was shown in figure 6.13.

From the kinetic theory of rubbers, it has been possible to deduce the theoretical curve of figure 11.6. (b) from expression (6.46). This expression has been verified by experiment up to strains of the order of 60 % (figure 6.13.). As shown in figure 11.6., the difference between experimental results and theory is not significant up to about 500 %. The important discrepancy observed at very high strains (ε > 500 %) result of a rapid increase in the elastic force of retraction between the ends of the rubbery segments (figure 11.7.). When the molecular chains are highly oriented, the tensile force is progressively exerted on the covalent bonds of the chain. This considerably increases the retractive force.

11.2.5. True stress and strain

In general, the results of tensile tests are expressed as a function of nominal rather than true stress values. The true stress σ,, which is the ratio between the applied force F and actual cross-section of the sample at a given moment, can be easily calculated by using simplifying hypotheses. By assuming that the volume of the sample remains constant as in the case of plastic deformation of metal alloys (v = 0,5), the initial volume (SWo) and the volume after deformation (S-l) will be equal, from which it follows that:

0 1 Extension ratio •

Figure 11.7. Variation of the retractive force between the extremities of a chain as a function of chain extension ///max. In inserts, variation of the shape of the chain (conformation) as a function of extension.


So/S=I/l0=l+£ (11.4.)

The true stress Gt = F/S is related to the nominal stress σ= F/S0 by:

σ/ = σ(1 + e ) (11.5.)

In the same way, the true strain ε, is expressed by considering the infinitesimal increase of the specimen from / to / + al. The real increase is therefore equal to dl/l and not to d///0 and the total true strain et is given by:

, d / (H .6 . )

ε, = 1^ = 1η( / / / 0 )

The true strain ε, is related to nominal strain ε by:

et =1η(1 + ε) (11.7.)

In these calculations, strains are expressed as fractions and not as percentages. Table 11.8 compares nominal and true strains in uniaxial tension and compression. The deviation between nominal and true strains is minimal below a strain of 0.1 or 10 %. Above this value, the deviation becomes more important.

Table 11.8. Comparison between nominal ε and true values s of strains in uniaxial tension and compression.

Compression Tension

-2.0 -1.0 -0.5 -0.2 -0.1 -0.01 0.01 0.1 0.2 0.5 1.0 2.0 ε -0.86 -0.63 -0.39 -0.18 -0.095 -0.01 0.01 0.11 0.22 0.65 1.72 6.4

Figure 11.9. shows the tensile curve for polycrystalline ductile copper expressed as nominal and true values of stress and strain. The maximum in the nominal stress-strain curve does not correspond to the maximum value of the intrinsic strength of the

Figure 11.9. Nominal and true stress - strain curves for polycrystalline copper. The true curve σ, - ε, increases gradually to rupture while the true curve σ - ε passes through a maximum value due to necking.


elastic material. The intrinsic strength of the material increases continuously until rupture. The maximum of the tensile curve for the nominal stress is only the result of necking which decreases the cross-section of the specimen. The tensile force required to deform the specimen is therefore lower in spite of the increase in strength of the material by work hardening.

In reality, the value Rm (nominal) is a measure of the strength of the specimen and not that of the material. The deviation between the nominal and real strain curves, can also be seen in the curves for uniaxial compression. The processes of plastic deformation do not depend on the loading direction, and there should be no difference between the tensile and compression curves. When nominal stresses and strains are used, there is a marked difference (figure 11.10. (a)). This is because compression causes a lateral dilation which increases the cross-section of the sample and, consequently, the force required for compression increases. In true terms, the tensile and compression curves are identical in absolute values (figure 11.10. (b)).

(a) (b)

Figure 11.10 Stress - strain curves in uniaxial extension and compression for a ductile metal. Comparison between nominal (a) and true (b) curves (after Ashby, Jones, 1980).

11.2.6. Deformation energy and anelastic behaviour

The deformation energy of a material can be deduced from the tensile curve. Ductile materials, which break after plastic deformation, have a rupture deformation energy which is much higher than that of brittle materials. The volume deformation energy Ue is given by the area under the tensile curve (figure 11.11.):

e ϋ = \σάε (11.8.)

0

Figure 11.11 shows the difference between elastic deformation energy Ue and plastic deformation energy Up.

As the tensile curve for most materials is linear in the elastic region, the volume elastic deformation energy Ue is equal to the area of the triangle (0 - 1 - ε\). Since σ= Εε, the volume elastic deformation energy Uecm be calculated:

U)=0,5E{e\)2

( U 9 )

After plastic deformation, the elastic deformation energy can be calculated in a similar way from the triangle (2 - ερ - ε}). U} is considerably larger than U}.


(2)

ill ) 4

Strain e —

Figure 11.11. Calculation of the volume elastic (Ue) and plastic (Up) deformation energies from the stress - strain curve for a ductile material at two degrees of strain (1 and 2).

For a perfectly elastic material, the tensile curve is reversible and the energy absorbed on loading is completely recovered when the load is removed (spring behaviour). Most materials have do not have an ideal elastic behaviour, and a part of the absorbed energy is dissipated within the specimen by internal friction mechanisms (movement of dislocations in metals or molecular chains in polymers). In this case, there is a difference between energy supplied to the system and the energy recovered. The unloading curve is no longer equivalent to the loading curve, even though the specimen returns to its initial length. During a deformation cycle, a hysteresis loop is formed. The loop surface represents the amount of energy dissipated as heat on loading and unloading (figure 11.12.). This anelastic effect, is a form of viscoelastic behaviour.

Figure 11.12. Anelastic deformation of a grey cast iron with formation of a hysteresis loop after loading and unloading. The energy dissipated during a deformation cycle is proportional to the black area. Two loading cycles were made. In the second case, the elastic limit was exceeded and consolidation of the material by work hardening occurred, with some permanent deformation.


In most materials, the anelastic effect is weak but sometimes this effect can be relatively important. In grey cast iron, a significant anelastic effect is observed due to the movement of dislocations in the graphite lamellae in the iron matrix. Elastomers also form hysteresis loops. In this case, energy dissipation result from friction of the elastomer chains on one another during dynamic deformations. It is this anelastic effect, which produces heat-build-up in tyres during flexing. Because of this behaviour, grey cast irons and elastomers are excellent dampers of mechanical vibrations and sound.

11.2.7. Hardness measurements

Tensile tests are often performed together with hardness measurements. These measure the resistance to localised deformation. The methods of measuring hardness (Brinell hardness, Vickers hardness, etc.) are based on the penetration of a very hard indenting device, a steel ball or a diamond, into the surface of the material. A hardness value is obtained from the dimensions or the depth of the indentation made by the indenting device under given loading conditions. The hardness measurement is one of the commonest mechanical tests because it is quick, non-destructive and only requires a simple apparatus. It can usually be related empirically to the tensile strength. However, it is a complex test and not easy to interpret.

11.3. Summary and conclusions

The modulus of elasticity E, the yield strength Re, the ultimate tensile strength Rm and the rupture elongation ε#, are important mechanical properties of materials. The values of these parameters are generally obtained from the nominal stress - strain curve.

Ideally the true stress - strain curve should be taken but, in practice, this is difficult because of necking. However, the most important mechanical properties, the modulus Ε and the yield strength Re (or the ultimate tensile strength Rm in the case of brittle materials) are practically independent of this choice.

The thermodynamic study of the uniaxial elongation of materials (chapter 6.) demonstrates that they can be divided into two groups: materials with enthalpic elasticity, constituting the majority of materials (metals, ceramics, organic and mineral glasses etc.), and the elastomers, which are materials with entropic elasticity. The analysis of the tensile curves gives the most important mechanical properties of these two main categories of materials.

In materials with enthalpic elasticity, there are fragile materials breaking with little or no plastic deformation, and ductile materials, which show a more or less important degree of plastic deformation before rupture. Ceramics, a large number of glassy thermoplastics, as well as thermoset polymers and elastomers, are brittle materials. Metals and some amorphous and semi-crystalline thermoplastics have a ductile behaviour.

Isotropic polymers have much lower mechanical strength than metals or ceramics. They can be significantly oriented and stretched, and then show a significant improvement in properties. This effect will be considered in more detail in Chapter 12. This capacity for plastic deformation, which is shown by some thermoplastic


polymers should not be confused with the high degree of elastic defor mobility of the elastomers which break without any plastic deformation in a brittle manner. The deformation energy of materials is measured by the area under the stress-strain curve.

11.4. Illustrative example: cables for cable cars

The movement of a modern, high-capacity cable car is by a wheel-train pulled by a traction cable, moving on two carrying cables (figure 11.13.).

Generally, a cable is manufactured by the cabling of twisted or parallel metal wires. This spiroid construction is known as 'stranded'. Several strands, with one or more layers of wires, can be cabled around a central core of natural or synthetic fibres. This cabling technique plays a key role in determining the properties of the cable.

Figure 11.13. Wheel-train of a cable car resting on two carrying cables, pulled by a traction cable (photo Von Roll).

The traction cable of a cable car system must be flexible because it has to be wound on to the driving drums. This requires that the strands are twisted in the same sense. The sense and the pitch of the cabling of the various layers of wires are chosen in such a way that the resultant torsion acting on the whole cable is very weak. The carrying cable has a smooth surface which facilitates movement of the wheel train. Carrying cables with a closed construction are produced with profiled wires (e.g. Ζ shaped cross-section) for the outer layer (figure 11.14.). The cabling of the outer layer of the profiled wires is produced by assembling the wires like a zip. Before the intrinsic properties of these metallic wires are defined, the operating conditions must be considered and the stresses acting on the cables calculated.


Figure 11.14. Section of a carrying cable with, respectively, one layer and one and a half external layers of profiled wires (after Huber 1980).

Table 11.15. summarises the main characteristics of the cable car system at the Klein Matterhorn, Zermatt, Switzerland. It can be seen that the actual weight of the cable is important. For this installation, the maximum free length of the two carrying cables is 2885 m corresponding to 66 tons weight which is more than three times the weight of a cabin transporting 100 passengers. Any improvement in the tensile strength of the cable wires would be beneficial because the cable weight could be reduced.

Table 11.15. Characteristics of the cable car Steg-Klein Matterhorn at Zermatt.

Overall length 3835 m Difference in altitude 891 m Average slope 25% Maximum slope 90.6 % Number of intermediate pillars 3 Maximum distance between two pillars 2885 m Transport 2 cabins Number of people per cabin 100 Speed 10 ms"1

Duration of journey 500 s Carrying cables per cabin 2

• diameter 45.2 mm • weight per unit length 11.47 kg/m • rupture force 219000 kg

Traction cable per cabin 1 • diameter 40 mm • weight per unit length 6.21 kg/m • rupture force 102000 kg

The wires are made of carbon steel with an ultimate tensile strength Rm of between 1700 and 2200 MPa and an elongation at rupture eR of 2.5 to 5 %. The safety factor of the carrying cable is greater than 3.5. This factor means that the maximum stresses experienced in service are 3.5 times lower than the ultimate tensile strength of the cable. This is the lowest safety factor used in this type of transport. The traction cable of cable car systems and the carrying cables of chairlifts have a safety factor of 5 while a elevator cable has a safety factor of 12.

The average lifetime of a carrying cable is 17 years and not only depends on the intrinsic strength of the steel wire and the cable construction, but also on the mainte-


nance. Generally, it is not rupture occurring inside cables due to fatigue or excessive loads which limit the lifetime, but internal damage resulting from corrosion. That is why the cable is greased during manufacture and this is regularly renewed every year.

The carbon steels used in the manufacture of cables are of eutectoid composition (0.8 % C). The high mechanical performance is mainly a result of the heat- treatment, which refines the perlitic structure (chapter 9., figure 9.16.). To obtain significant hardening, the wires are quenched and transformed in an isothermal bath (at about 500 °C) to form very fine perlite. The wires are then drawn, i.e. worked to further refine the microstructure and to introduce a large number of dislocations. For many years, this process called patenting has enabled relatively cheap high-strength steel wires to be produced with a tensile strength of about 2000 MPa (200 kg m m - 2 ) , which are used for cable car system cables and piano wires.

11.5. Exercises

11.5.1. What are the deformation and elongation of a steel wire, of 2.5 mm diameter and 3 m length, supporting a mass of 500 kg, given that the modulus of elasticity Ε of the steel is 210 GPa?

11.5.2. A bar of initial length L0 is stretched in uniaxial plastic elongation to length L\= 2 L0. This bar is then stretched by a second deformation to length L2 = 3 L0. Calculate the nominal and true elongation after each stage.

11.5.3. Calculate the elastic energy Ue stored in an aluminium alloy wire with a modulus of elasticity Ε = 70 Gpa, after an elastic deformation of 0.01 %.

11.5.4. An aluminium alloy specimen has a tensile strength R m = 300 MPa and a reduction in area, As of 77 %. Calculate the true tensile stress ar.

11.5.5. A steel bar 10 cm in diameter is subjected to an alternating axial load of 500 kN. Knowing that the modulus of elasticity is 200 GPa and Poisson's ratio ν is 0.3, calculate the maximum and minimum diameters of the bar during service.

11.5.6. Which type of polymers could be proposed to obtain: • greater tensile strength, • increased rigidity, • greater ductility?

In each case, explain the differences in behaviour of these various materials.

11.5.7. Is it possible to plastically deform in compression, an aluminium bar, 50 mm in diameter with a yield strength, Re of 150 MPa, using a press with a maximum capacity of 50 tons?

11.5.8. A standard tensile test is carried out on a copper-nickel alloy sample of initial diameter 12.5 mm and initial length 50 mm. Using the values in table 11.16, draw the initial part of the stress a - strain ε curve and calculate:

• the modulus of elasticity E\ • the 0.2 % offset yield stress RQX, • the ultimate tensile strength R m and the rupture elongation £ R


Table 11.16 Data of the tensile test

Load Elongation [kN] [mm]

5 0.015 15 0.045 26 0.500 35 1.300 48.5 (a) 7.000 39.5 (b) 18,700

(a) maximum load; (b) Breaking load.

11.5.9. A magnesium alloy wire of 1 mm diameter has a modulus of elasticity of 45 GPa. Plastic deformation occurs when the load reaches 10 kg. At a load of 12 kg, the total deformation of the thread is 1 %. Calculate the permanent deformation of the wire after this loading.

11.5.10. The tensile curve of a material is generally recorded at low strain rates, e.g. ael dt= 10 3 s"1. What difference would be observed in the modulus of elasticity Ε if the tensile test were carried out at a strain rate 100 times higher, for:

• a metal? • a polymer?

Explain your answers.

11.6. References and complementary readings

E.H. ANDREWS, Fracture in Polymers, Oliver and Boyd, London, 1968. M.F. ASHBY, D.R.H. JONES, Engineering Materials 2, An Introduction to Microstructures, Processing and Design, Pergamon, Oxford, 1986 and 2 n d ed., Butterword-Heinemann, Oxford, 1996. Τ. F. COUTNEY, Mechanical Behavior of Materials, McGraw-Hill, New York, 1990. R.W. DAVIDGE, Mechanical Behaviour of Ceramics, Cambridge University Press, Cambridge, 1979. R.W. HERTZBERG, Deformation and Fracture, Mechanics of Engineering Materials, 4th ed., John

Wiley, New York, 1996. K.J. PASCOE, An Introduction to the Properties of Engineering Materials, 3rd ed., Van Nostrand-Rein-

hold, Wokingham, Berkshire, U.K., 1978. D. ROSENTHAL, R.M. ASIMOW, Introduction to Properties of Materials, 2nd ed., Van Nostrand-Re in

hold, Wokingham, Berkshire, U.K., 1971. J. SCHULZ, Polymer Materials Science, Prentice-Hall, Englewood Cliffs, New Jersey, 1974. E. HUBER, Experiences et Progres lors de la Fabrication de Cables Porteurs en Construction Close, Int.

Seilbahn, Rundschau 7, 303 (1980).

introduction to materials science || tensile behaviour of materials

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