(4) 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. If σ z is a tensile stress state of smaller magnitude than σ y 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). σσ σσ σσ τ σσ σσ σσ τ 1 2 2 3 2 2 24 24 == + + − + == + − − + ′ ′ x xy xy xy y xy xy xy ()() ()() ©1996 CRC Press LLC lies between σ 1 and σ 3 and is designated σ 2 . By looking into the 1 face, σ 2 and σ 3 are seen, the circle for which is sho wn 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: (σ 1 – σ 2 ) 2 + ( σ 2 – σ 3 ) 2 + ( σ 3 – σ 1 ) 2 = 2Y 2 (5) which can be used to show that the Tresca and von Mises yield criteria are identical when σ 2 = either σ 1 or σ 3 , and farthest apart (≈15%) when σ 2 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. ©1996 CRC Press LLC 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, a T , 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) Nylon 6-6 ≈ 285,700 psi (10s modulus) steel ≈ 30 × 10 6 psi (207 GPa) brass ≈ 18 × 10 6 psi (126 GPa) lime-soda glass ≈ 10 × 10 6 psi (69.5 GPa) aluminum ≈ 10 × 10 6 psi (69.5 GPa) ©1996 CRC Press LLC 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 T o is the “characteristic temperature” of the material; or where T s = T g + 50°C and T g is the glass transition temperature of the polymer . 1 The glass transition temperature, T g , 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 T g than belo w. (Polymers do not become transparent at T g ; 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 T g may also be mark ed 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 RTT T o : log( ) =− ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Δ 11 WLF a TT TT T s s : log( ) .( ) (. ) = −− +− 886 101 6 ©1996 CRC Press LLC 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. ©1996 CRC Press LLC are shown in Figure 2.16, and the shift distance for each, to shift them to T s 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 a T ) as shown. The 40°C curve moves left, i.e., 10 0.87 , the 30°C curve remains virtually where it is, and the 20°C curve moves to the right an amount corresponding to 10 0.86 . 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 gs T =− ° =+ = −− +− °= − +− = −× + =− 21 29 886 101 6 88650 29 101 6 50 29 886 21 101 6 21 151 C and a TT TT T s s , , log( ) .( ) . ) .( – ) .( ) . . . ©1996 CRC Press LLC 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 Bhn 3000 (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, ΔΔ ΔΔ for Various Materials steel (most metals) ≈0.02 (2%) cast iron ≈0.08 wood ≈0.03–0.08 concrete ≈0.09 tire rubber ≈0.20 Δ= energy loss per cycle strain energy input in applying the load ©1996 CRC Press LLC 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) Brinell Rockwell Vickers 3000 kg, b c e diamond 10mm 1/16” ball cone 1/8” ball pyramid ball 100 kg f 150 kg f 60 kg f 1–120 g 10 62 ↑ 20 68 Έ 30 75 Έ 40 81 Έ 50 87 same as 100 60 93 Brinell 125 71 100 Έ 150 81 Έ 175 88 7 Έ 200 94 15 Έ 225 97 20 Έ 250 102 24 ↓ 275 104 28 276 300 31 304 325 34 331 350 36 363 375 38 390 ⎧ 400 41 420 Έ 450 46 480 requires Έ 500 51 540 carbide Έ 550 55 630 ball Έ 600 58 765 Έ 650 62 810 Έ 675 63 850 Έ 700 65 940 ⎩ 750 68 1025 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. ©1996 CRC Press LLC the fracture toughness, or stress intensity factor, K c . (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 10 5 and 10 7 c ycles 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: ©1996 CRC Press LLC Table 2.4 Mohs Scale of Scratch Hardness O E (Equiv. Knoop) Reference Minerals talc 1 hydrous mag. silicate Mg 3 Si 4 O 10 (OH) 2 carbon, soft grade 1.5 boron nitride ≈2 (hexagonal form) finger nail >2 gypsum 2 32 hydrated calcium sulfate CaSO 4 ⋅2H 2 O aluminum ≈2.5 ivory 2.5 calcite 3 135 calcium carbonate CaCO 3 calcium fluoride 4 163 fluorite 4 calcium fluoride CaF 2 zinc oxide 4.5 apatite 5 430 calcium fluorophosphate Ca 3 P 2 O 8 - CaF 2 germanium ≈5 glass, window >5 iron oxide 5.5 to 6.5 rouge magnesium oxide ≈6 periclase orthoclase 6 560 potassium aluminum silicate KAlSi 3 O 8 rutile >6 titanium dioxide TiO 2 tin oxide 6 to 7 putty powder ferrites 7 to 8 quartz 7 8 820 silicon dioxide, SiO 2 silicon ≈7 steel, hardened ≈7 chromium 7.5 nickel, electroless 8 sodium chloride >8 NaCl topaz 8 9 1340 aluminum fluorosilicate Al 2 F 2 SiO 4 garnet 10 fused zirconia 11 aluminum nitride ≈9 alumina 9 12 alpha, corundum, Al 2 O 3 ruby/sapphire 9 1800 silicon carbide >9 13 alpha, carborundum silicon nitride ≈9 boron carbide 14 4700 boron nitride (cubic) ≈14.5 diamond 10 15 7000 carbon 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 R c in midrange, and the new Mohs numbers ≈0.7(Vickers hardness number) 1,3 ©1996 CRC Press LLC [...]... called the surface free energy.) If the crack can be made to propagate quasistatically, Pδ=2Aγ: much mathematics of fracture is based on the principle of this energy balance The equation, d(δ/P)/dA = 2R /P2 is used, where the value of R at the start of cracking is called the critical strain energy release rate, i.e., the rate at which A increases Another part of fracture mechanics consists of calculating