Tài liệu Friction, Wear, Lubrication P2 docx

VISCO-ELASTICITY, CREEP, AND STRESS RELAXATION Polymers are visco-elastic, i.e., mechanically they appear to be elastic under high strain rates and viscous under low strain rates.. When

The principal stresses can be thought of as being imposed upon the surfaces

of a new cube rotated relative to the original cube by an angle θ/2, as shown in

Figure 2.10

Now that one circle is found, two more can be found by looking into the “1”

and “3” faces Ifσz is a tensile stress state of smaller magnitude than σy then it

Figure 2.9 The Mohr circle for nonprincipal orthogonal stresses.

Figure 2.10 Resolving of nonprincipal stress state to a principal stress state (where there

is no shear stress in the “2,” i.e., z face).

1

2 2

3

2 2

′

′

x

xy

y

xy

lies between σ1 andσ3 and is designated σ2 By looking into the 1 face, σ2 and

σ3 are seen, the circle for which is shown in Figure 2.11 as circle 1

Circle 2 is drawn in the same way (Recall that in Figures 2.5, 2.7, and 2.8

only principal stresses were imposed.) The inner cube in Figure 2.10 has only principal stresses on it In Figure 2.11 only those principal stresses connected with the largest circle contribute to yielding The von Mises equation, Equation

1, suggests otherwise (The Mohr circle embodies the Tresca yield criterion, incidentally.) Equation 1 for principal stresses only is:

(σ1 – σ2)2 + (σ2 –σ3)2 + (σ3 – σ1)2 = 2Y2 (5)

which can be used to show that the Tresca and von Mises yield criteria are identical when σ2 = either σ1 or σ3, and farthest apart (≈15%) when σ2 lies half way between Experiments in yield criteria often show data lying between the Tresca and von Mises yield criteria

VISCO-ELASTICITY, CREEP, AND STRESS RELAXATION

Polymers are visco-elastic, i.e., mechanically they appear to be elastic under high strain rates and viscous under low strain rates This behavior is sometimes modeled by arrays of springs and dashpots, though no one has ever seen them

in real polymers Two simple tests show visco-elastic behavior, and a particular mechanical model is usually associated with each test, as shown in Figure 2.12 From these data of ε and σ versus time, it can be seen that the Young’s Modulus,

E (=σ/ε), decreases with time

The decrease in E of polymers over time of loading is very different from the behavior of metals When testing metals, the loading rate or the strain rates

Figure 2.11 The three Mohr circles for a cube with only principal stresses applied.

are usually not carefully controlled, and accurate data are often taken by stopping

the test for a moment to take measurements That would be equivalent to a stress

relaxation test, though very little relaxation occurs in the metal in a short time

(a few hours)

For polymers which relax with time, one must choose a time after quick

loading and stopping, at which the measurements will be taken Typically these

times are 10 seconds or 30 seconds The 10-second values for E for four polymers

are given in Table 2.1

Dynamic test data are more interesting and more common than data from

creep or stress relaxation tests The measured mechanical properties are Young’s

Modulus in tension, E, or in shear, G, (strictly, the tangent moduli E′ and G′) and

the damping loss (fraction of energy lost per cycle of straining), Δ, of the material

(Some authors define damping loss in terms of tan δ, which is the ratio E″/E′

where E″ is the loss modulus.) Both are strain rate (frequency, f, for a constant

amplitude) and temperature (T) dependent, as shown in Figure 2.13 The range

of effective modulus for linear polymers (plastics) is about 100 to 1 over ≈ 12

orders of strain rate, and that for common rubbers is about 1000 to 1 over ≈ 8

orders of strain rate

The location of the curves on the temperature axis varies with strain rate, and

vice versa as shown in Figure 2.13 The temperature–strain rate interdependence,

i.e., the amount, aT, that the curves for E and Δ are translated due to temperature,

can be expressed by either of two equations (with varying degrees of accuracy):

Figure 2.12 Spring/dashpot models in a creep test and a stress relaxation test.

Table 2.1 Young’s Modulus for Various Materials

Solid E Young’s Modulus

polyethylene ≈ 34,285 psi (10s modulus)

polystyrene ≈ 485,700 psi (10s modulus)

polymethyl-methacrylate ≈ 529,000 psi (10s modulus)

lime-soda glass ≈ 10 × 10 6 psi (69.5 GPa)

aluminum ≈ 10 × 10 6 psi (69.5 GPa)

where ΔH is the (chemical) activation energy of the behavior in question, R is the gas constant, T is the temperature of the test, and Tois the “characteristic temperature” of the material; or

where Ts = Tg + 50°C and Tg is the glass transition temperature of the polymer.1

The glass transition temperature, Tg, is the most widely known “characteristic temperature” of polymers It is most accurately determined while measuring the coefficient of thermal expansion upon heating and cooling very slowly The value

of the coefficient of thermal expansion is greater above Tg than below (Polymers

do not become transparent at Tg; rather they become brittle like glassy solids, which have short range order Crystalline solids have long range order; whereas super-cooled liquids have no order, i.e., are totally random.)

An approximate value of Tg may also be marked on curves of damping loss (energy loss during strain cycling) versus temperature The damping loss peaks are caused by morphologic transitions in the polymer Most solid (non “rubbery”) polymers have 2 or 3 transitions in simple cyclic straining For example, PVC shows three peaks over a range of temperature The large (or α) peak is the most significant, and the glass transition is shown in Figure 2.14 This transition is thought to be the point at which the free volume within the polymer becomes greater than 2.5% where the molecular backbone has room to move freely The

Figure 2.13 Dependence of elastic modulus and damping loss on strain rate and

temper-ature (Adapted from Ferry, J D., Visco-Elastic Properties of Polymers, John

Wiley & Sons, New York, 1961.)

Arrhenius a H

R T T T

o

: log( )= ⎛ −

⎝⎜

⎞

⎠⎟

T T T

s s

: log( ) . ( )

+ −

8 86

101 6

secondary (or β) peak is thought to be due to transitions in the side chains These take place at lower temperature and therefore at smaller free volume since the side chains require less free volume to move The third (or γ) peak is thought to

be due to adjacent hydrogen bonds switching positions upon straining

The glass–rubber transition is significant in separating rubbers from plastics: that for rubber is below “room” temperature, e.g., –40°C for the tire rubber, and that for plastics is often above The glass transition temperature for polymers roughly correlates with the melting point of the crystalline phase of the polymer

The laboratory data for rubber have their counterpart in practice For a rubber sphere the coefficient of restitution was found to vary with temperature, as shown

in Figure 2.15 The sphere is a golf ball.2

An example of visco-elastic transforms of friction data by the WLF equation can be shown with friction data from Grosch (see Chapter 6 on polymer friction) Data for the friction of rubber over a range of sliding speed are very similar in shape to the curve of Δ versus strain rate shown in Figure 2.13 The data for µ versus sliding speed for acrylonitrilebutadiene at 20°C, 30°C, 40°C, and 50°C

Figure 2.14 Damping loss curve for polyvinyl chloride.

Figure 2.15 Bounce properties of a golf ball.

are shown in Figure 2.16, and the shift distance for each, to shift them to Ts is

calculated

i.e., the 50°C curve must be shifted by 1.51 order of 10, or by a factor of 13.2

to the left (negative log aT) as shown The 40°C curve moves left, i.e., 100.87, the

30°C curve remains virtually where it is, and the 20°C curve moves to the right

an amount corresponding to 100.86

When all curves are so shifted then a “master curve” has been constructed

which would have been the data taken at 29°C, over, perhaps 10 orders of 10 in

sliding speed range

(See Problem Set question 2 f.)

DAMPING LOSS, ANELASTICITY, AND IRREVERSIBILITY

Most materials are nonlinearly elastic and irreversible to some extent in their

stress–strain behavior, though not to the same extent as soft polymers In the

polymers this behavior is attributed to dashpot-like behavior In metals the reason

is related to the motion of dislocations even at very low strains, i.e., some

dislocations fail to return to their original positions when external loading is

removed Thus there is some energy lost with each cycle of straining These losses

Figure 2.16 Example of WLF shift of data.

For this rubber, T thus T

To transform the 50 C data, log(a

T

+ −

×

101 6

8 86 50 29

101 6 50 29

8 86 21

101 6 21 1 51

T T T

s s

) . ( – )

are variously described (by the various disciplines) as hysteresis losses, damping losses, cyclic energy loss, anelasticity, etc Some typical numbers for materials are given in Table 2.2 in terms of

HARDNESS

The hardness of materials is most often defined as the resistance to penetration

of a material by an indenter Hardness indenters should be at least three times

harder than the surfaces being indented in order to retain the shape of the indenter

Indenters for the harder materials are made of diamonds of various configurations, such as cones, pyramids, and other sharp shapes Indenters for softer materials are often hardened steel spheres Loads are applied to the indenters such that there is considerable plastic strain in ductile metals and significant amounts of plastic strain in ceramic materials Hardness numbers are somewhat convertible

to the strength of some materials, for example, the Bhn3000 (Brinell hardness number using a 3000 Kg load) multiplied by 500 provides a fair estimate of the tensile strength of steel in psi (or use Bhn × 3.45 ≈TS, in MPa)

The size of indenter and load applied to an indenter are adjusted to achieve

a compromise between measuring properties in small homogeneous regions (e.g., single grains which are in the size range from 0.5 to 25 µm diameter) or average properties over large and heterogeneous regions The Brinell system produces an indentation that is clearly visible (≈3 – 4 mm); the Rockwell system produces indentations that may require a low power microscope to see; and the indentations

in the nano-indentation systems require high magnification microscopy to see For ceramic materials and metals, most hardness tests are static tests, though tests have also been developed to measure hardness at high strain rates (referred to as dynamic hardness) Table 2.3 is a list of corresponding or equivalent hardness numbers for the most common systems of static hardness measurement

Polymers and other visco-elastic materials require separate consideration because they do not have “static” mechanical properties Hardness testing of these materials is done with a spring-loaded indenter (the Shore systems, for example)

An integral dial indicator provides a measure of the depth of penetration of the

Table 2.2 Values of Damping Loss,

ΔΔΔΔ for Various Materials

steel (most metals) ≈ 0.02 (2%)

Δ = energy loss per cycle

strain energy input in applying the load

indenter in the form of a hardness number This value changes with time so that

it is necessary to report the time after first contact at which a hardness reading

is taken Typical times are 10 seconds, 30 seconds, etc., and the time should be

reported with the hardness number Automobile tire rubbers have hardness of

about 68 Shore D (10 s)

Notice the stress states applied in a hardness test With the sphere the substrate

is mostly in compression, but the surface layer of the flat test specimen is stretched

and has tension in it Thus one sees ring cracks around circular indentations in

brittle material The substrate of that brittle material, however, usually plastically

deforms, often more than would be expected in brittle materials In the case of

the prismatic shape indenters, the faces of the indenters push materials apart as

the indenter penetrates Brittle material will crack at the apex of the polygonal

indentation This crack length is taken by some to indicate the brittleness, i.e.,

Table 2.3 Approximate Comparison of Hardness Values

as Measured by the Most Widely Used Systems (applicable to steel mostly)

10mm 1/16” ball cone 1/8” ball pyramid ball 100 kg f 150 kg f 60 kg f 1–120 g

Comparisons will vary according to the work hardening properties of

mate-rials being tested Note that each system offers several combinations of

indenter shapes and applied loads.

the fracture toughness, or stress intensity factor, Kc (See the section on Fracture

Toughness later in this chapter.)

Hardness of minerals is measured in terms of relative scratch resistance rather than resistance to indentation The Mohs Scale is the most prominent scratch

hardness scale, and the hardnesses for several minerals are listed in Table 2.4

(See Problem Set question 2 g.)

RESIDUAL STRESS

Many materials contain stresses in them even though no external load is applied Strictly, these stresses are not material properties, but they may influence apparent properties Bars of heat-treated steel often contain tensile residual stresses just under the surface and compressive residual stress in the core When such a bar is placed in a tensile tester, the applied tensile stresses add to the tensile residual stresses, causing fracture at a lower load than may be expected

Compressive residual stresses are formed in a surface that has been shot peened, rolled, or burnished to shallow depths or milled off with a dull cutter Tensile residual stresses are formed in a surface that has been heated above the recrystallization temperature and then cooled (while the substrate remains unheated) Residual stresses imposed by any means will cause distortion of the entire part and have a significant effect on the fatigue life of solids

(See Problem Set question 2 h.)

FATIGUE

Most material will fracture when a small load is applied repeatedly Generally, stresses less than the yield point of the material are sufficient to cause fatigue fracture, but it may require between 105 and 107 cycles of strain to do so Gear teeth, rolling element bearings, screws in artificial hip joints, and many other mechanical components fail by elastic fatigue If the applied cycling stress exceeds the yield point, as few as 10 cycles will cause fracture, as when a wire coat hanger is bent back and forth a few times More cycles are required if the strains per cycle are small Failure due to cycling at stresses and strains above the yield point is often referred to as low-cycle fatigue or plastic fatigue

There is actually no sharp discontinuity between elastic behavior and plastic behavior of ductile materials (dislocations move in both regimes) though in high cycle

or elastic fatigue, crack nucleation occurs late in the life of the part, whereas in

low-cycle fatigue, cracks initiate quickly and propagation occupies a large fraction of part life Wöhler (in reference number 3) showed that the entire behavior of metal in fatigue could be drawn as a single curve, from a low stress at which fatigue failure will never occur, to the stress at which a metal will fail in a quarter cycle fatigue test, i.e., in a tensile test A Wöhler curve for constant strain amplitude cycling is shown in Figure

2.17 (few results are available for the more difficult constant stress amplitude cycling)

There are several relationships between fatigue life and strain amplitude available

in the literature A convenient relationship is due to Manson (in reference number 3)

who suggested putting both high-cycle fatigue and low-cycle fatigue into one equation:

Table 2.4 Mohs Scale of Scratch Hardness

O E (Equiv Knoop) Reference Minerals

silicate

Mg3Si4O10 (OH) 2 carbon, soft

grade

1.5

finger nail >2

sulfate

CaSO4

calcium

fluoride

fluorophosphate

Ca3P2O8 -CaF 2

glass, window >5

magnesium

oxide

aluminum silicate

KAlSi3O8

steel,

hardened

nickel,

electroless

8 sodium

chloride

fluorosilicate

Al2F2SiO4

aluminum

nitride

silicon nitride ≈ 9

boron nitride

(cubic)

O signifies original Mohs scale with basic values underlined and bold; E signifies the newer

extended range Mohs scale The original Mohs number ≈ 0.1 Rc in midrange, and the new

Mohs numbers ≈ 0.7(Vickers hardness number) 1,3