Mechanical Behaviour of Engineering Materials - Metals, Ceramics, Polymers and Composites 2010 Part 11 pptx

392 11 Creep applied shear stress, l is the distance between the obstacles, and V ∗ is the activation volume (see section 6.3.2). The strain rate ˙ε is proportional to the vacancy current density j and to the dislocation density . Thus, ˙ε ∝ j holds. In the stationary state, the dislocation density  is usually proportional to the square of the stress [51]. 6 If we insert the equation for the current density (and use τ ∝ σ), we finally find ˙ε = Aσ 3 kT D 0 exp  − Q V + Q ex kT  = Aσ 3 kT D V (T ) . (11.13) A is a material parameter that has to be determined experimentally. The strain rate thus depends exponentially on the temperature and with a power law on the stress. Equation (11.13) can only be used to describe secondary creep because it does not take into account the evolution of the dislocation density during primary creep or the damage processes occurring in tertiary creep. The exponential dependence of the strain rate on the temperature has been confirmed exp erimentally. The relation between strain rate and stress is found to follow a power law, as predicted by the equation (see also section 11.1), but in reality the creep exponent typically takes values between 3 and 8. Due to the large variations of the creep exponent in different materials, the value of the factor A can differ by several orders of magnitude. The activation energy in equation (11.13) is frequently stated per mole in the units kJ/mol. In this case, Boltzmann’s constant k has to be replaced by the gas constant R in the equation as explained in appendix C.1. In deriving equation (11.13), we assumed that the vacancy diffusion oc- curs through the undistorted crystal (volume diffusion). However, vacancy transport can also occur mainly along lattice defects like dislocation lines (dis- location pipe diffusion). In this case, the activation energy for site exchange is smaller d ue to the lattice distortion. Because of this difference, vacancy trans- port along dislocation lines dominates at low temperatures, whereas volume diffusion is the faster mechanism at higher temperature. Large stresses also favour transport along dislocations because the dislocation density increases due to the formation of new dislocations (work hardening). We also required in deriving equation (11.13) that the product of external stress and activation volume is small compared to the thermal energy kT . This is not the case at high deformation speeds and stresses. In this case, the relation between strain rate and stress is exponential and the power law is not valid anymore (power-law breakdown) [51]. 6 A similar relation was already discussed in the context of work hardening, see equation (6.20). 11.2 Creep mechanisms 393 ¾ ¾¾ ¾ vacancy current 1 2 d Fig. 11.6. Movement of vacancies in diffusion creep 11.2.3 Diffusion creep At high temperatures, dislocation creep – i. e., dislocation movement aided by vacancy diffusion – is not the only mechanism contributing to deformation. Vacancy diffusion alone can cause a deformation without any dislocations being involved. In this process, called diffusion creep, grain boundaries are sources and sinks of vacancies. It is mainly this mechanism that determines the creep behaviour of ceramics. As illustrated in figure 11.6, vacancies are formed at grain boundaries with a normal vector oriented in the direction of the tensile stress. These vacancies move to grain boundaries with compressive stresses or lower values of the te nsile stress. The material itself moves in the opposite direction from regions with compressive to those with tensile stresses. The derivation of the strain rate in diffusion creep is analogous to that of the previous section. Again, the vacancy current density is calculated and related to the strain rate. As in section 11.2.2, we start by calculation the vacancy concentration. In equilibrium and without externally applied stress, the vacancy con- centration is n = exp(−Q V /kT), where Q V is again the energy for the formation of a vacancy. If we now consider a grain boundary whose normal direction is in the direction of the tensile stress (figure 11.6), the material can elongate in the loading direction if an atom from the crystal lattice is added at the grain boundary, creating an additional vacancy in the crystal. The external stress does some work because the material lengthens (on average) by the quotient of the volume Ω of the vacancy and the cross section of the grain boundary considered. The force equals the stress multiplied with the cross section of the grain 394 11 Creep b oundary. The work done is thus σΩ, independent of the size of the grain boundary. On the other hand, work σΩ must be done to create a vacancy in a region that is under compressive stress σ. The vacancy concentration in the regions 1 ○ and 2 ○ from figure 11.6 is thus n 1 = exp „ − Q V + σΩ kT « , n 2 = exp „ − Q V − σΩ kT « . The vacancy concentration gradient can be estimated, according to equation (11.10), as (n 1 − n 2 )/d, with the grain size d replacing the dislocation distance. Thus, a vacancy current density j = −D 0 exp „ − Q ex kT « n 1 − n 2 d = 2σΩ dkT D 0 exp „ − Q V + Q ex kT « (11.14) results, with d being the mean diffusion length, approximately equal to the grain size. The creep rate ˙ε is proportional to the vacancy current density divided by the size of the grain. Thus, we find ˙ε = A NH σΩ kT D 0 d 2 exp  − Q V + Q ex kT  = A NH σΩ kT D V d 2 . (11.15) Here, A NH is a material parameter, σ the external stress, Ω the volume of a vacancy, d the grain size, and D V the diffusion coefficient for self-diffusion through the bulk material. This process is called Nabarro-Herring creep. Since the stress dependence is linear, Nabarro-Herring creep is most important at low stresses, whereas dislocation creep is more important at high stresses. Since the creep rate is inversely proportional to the square of the grain size, creep is favoured if the grains are small. In contrast to time-independent plas- tic deformation, where small grains are preferred (grain boundary strengthen- ing, see section 6.4.2), large grains are advantageous in materials that creep. Vacancy diffusion needs not to occur through the bulk material in diffusion creep. Instead, vacancies may move directly along the grain boundaries (see figure 11.7). The activation energy of vacancy diffusion along a grain boundary is smaller than in the bulk because the lattice is distorted. As before, the vacancy current density j is inversely proportional to the grain size. The derivation of the current density in the bulk material can thus be copied exactly, simply replacing the activation energy with 11.2 Creep mechanisms 395 ¾ ¾ ¾¾ vacancy current d d Fig. 11.7. Movement of vacancies along grain boundaries in diffusion creep the smaller activation energy along the grain boundary. The number of vacancies moving through the grain per unit time is given by the vacancy current density multiplied by the cross section of the region the vacancies are moving through. This is dδ, where δ denotes the thickness of the grain boundary. In total, the rate of vacancy diffusion is thus jdδ. The growth rate equals this rate, normalised by the cross section of the grain i. e., jdδ/d 2 = jδ/d. To get the strain rate, we again need to divide by the grain size, yielding the final result ˙ε ∝ jδ/d 2 ∝ 1/d 3 . The strain rate for grain boundary diffusion creep is thus ˙ε = A C σΩ kT δD GB d 3 . (11.16) δ is the thickness of the grain boundary, D GB is the diffusion c oefficient of self-diffusion along the grain boundary, and A C is another material parameter. This process is called Coble creep. Due to the strong dep e nde nce on the grain size, Coble creep is most important if the grains are small. Since the activa- tion energy of self-diffusion along th e grain bound aries is smaller than in the volume, Coble creep is also dominant compared to Nabarro-Herring creep at low temperatures. Because the shape of the grains changes in diffusion creep, neighbouring grains have to deform in a compatible manner, analogous to the compatibility of deformation in grain boundary strengthening as discussed in section 6.4.2. This is one cause of grain boundary sliding, described in the next section. Diffusion creep is also important in fibre composites. It was shown in sec- tion 9.3.2 that the load transfer to a fibre is determined only by the aspect ratio, the quotient of length and diameter. That the length is considered to be important in technical applications is only due to the fact that the fibre diameter cannot be made arbitrarily small, whereas the length can be as large 396 11 Creep as desired. However, at high temperatures, the fibres can be unloaded by diffu- sion processes. Atoms of the matrix can move along the fibre-matrix interface and relax the stress between fibre and matrix. Similar to equation (11.16), the absolute size of the fibre now becomes important and only sufficiently long fibres can have a strengthening effect. 11.2.4 Grain boundary sliding At high temperatures, grains in metals and ceramics can move against each other. This process is called grain boundary sliding. The strain rate of grain boundary sliding cannot be estimated as simply as for the other processes. It is [26] ˙ε = A GBS δσ n D GB d . (11.17) As usual, A GBS is a material parameter, δ is the thickness of the grain bound- ary, σ the externally applied stress, D GB the diffusion coefficient of grain boundary diffusion, and d the diameter of the grain. The creep exponent n of grain boundary sliding usually takes values between 2 and 3. In m etals, grain boundary sliding usually contributes only slightly to the overall deformation, but it is nevertheless important for two reasons: First, in diffusion creep, grain boundary sliding ensures the compatibility of the grains during the deformation (see also section 6.4.2 and the end of the previous sec- tion) as sketched in figure 11.8. Second, at points where three grain boundaries meet (triple points), movement of the grain boundaries by sliding can cause a large concentration in local stresses and thus induce damage by rupture of the grain boundaries (see also section 11.3). It is thus doubly advantageous to increase the resistance of the grain boundary against sliding: deformation by diffusion creep is impeded, and the danger of early damage is reduced. This will be discussed further in section 11.4. In ceramics, the strength at high temperatures is often limited by grain boundary sliding. The reason for this is the presence of a glassy phase at the grain boundaries (see also section 7.1). These amorphous regions have a much lower softening temperature than the grains themselves. This ‘lubricating film’ eases sliding of the grains, without dislocation movement inside the grains be- ing necessary. One important goal in manufacturing ceramic high-temperature materials is thus to reduce the amount of glassy phase as much as possible. 11.2.5 Deformation mechanism maps The various creep mechanisms discussed so far differ in their temperature de- pendence because the activation energy of the mechanisms is different. Further- more, they differ in their stress-dependence. The creep exponent takes values 11.2 Creep mechanisms 397 ¾ ¾¾ ¾ d Fig. 11.8. Grain boundary sliding ensures the compatibility of grains which would b e violated if only diffusion creep would occur 10 –5 10 –4 10 –3 10 –2 10 –1 10 –0 0.0 0.2 0.4 0.6 0.8 1.0 T/T m ¿/G theoretic strength dislocation glide (time-independent plastic deformation) elastic deformation yield strength Coble creep volume diffusion Nabarro- Herring creep dislocation creep diffusion creep disl. pipe diffusion Fig. 11.9. Idealised deformation mechanism map (after [26]) between 1 in diffusion creep and 3 in dislo c ation creep, with even higher val- ues occurring in reality. Thus, depending on the external conditions, different creep mechanisms dominate the behaviour. So-called deformation mechanism maps allow to read off the dominant mechanism under different conditions. Figure 11.9 shows a schematic defor- mation mechanism map. In the diagram, the tempe rature and the external 398 11 Creep T/T m T/°C 10 –5 10 –4 10 –3 10 –6 10 –7 10 –8 10 –2 10 –1 0.0 0.2 0.4 0.6 0.8 1.0 4002000–200 600 ¾/G 10 3 10 2 10 1 10 0 10 –1 10 –2 10 –3 ¾/MPa theoretical strength dislocation glide elastic deformation yield strength Coble creep dislocation creep Nabarro- Herring creep (a) Aluminium Coble creep T/T m T/°C 10 –5 10 –4 10 –3 10 –6 10 –7 10 –8 10 –2 10 –1 0.0 0.2 0.4 0.6 0.8 1.0 200010000 3000 ¾/G 10 3 10 2 10 1 10 0 10 –1 10 –2 10 –3 ¾/MPa theoretic strength dislocation glide elastic deformation yield strength dislocation creep Nabarro- Herring creep (b) Tungsten Fig. 11.10. Deformation mechanism maps (after [35]). The grain size is 32µm in both cases stress, normalised by the relevant material parameters (melting temperature and shear modulus), are used as axes so that the dominant deformation mech- anism can be read off. At low external stresses and low temperatures, the material deforms elas- tically. At higher temperatures, diffusion creep starts, being stronger at small stresses than dislocation creep because of its lower creep exponent. Because of the lower activation energy for grain boundary diffusion, this mechanism is more important than bulk diffusion at low temperatures. Since the creep exponent is the same in both cases, the two regions are separated by a vertical line. If we move on to larger stresses, dislocation creep with its larger creep exponent becomes dominant. Vacancy diffusion along dislocation lines is more important than diffusion through the bulk material at lower temperatures since its activation energy is smaller. Because the creep exponent is larger for diffusion along the dislocations than through the bulk, the separating line is inclined. 11.2 Creep mechanisms 399 T/T m T/°C 10 –5 10 –4 10 –3 10 –6 10 –7 10 –8 10 –2 10 –1 0.0 0.2 0.4 0.6 0.8 1.0 600200–200 4000 800 ¾/G 10 3 10 2 10 1 10 0 10 –1 10 –2 10 –3 ¾/MPa theoretical strength dislocation glide elastic deformation yield strength dislocation creep d = 32 µm d = 100 µm d = 10 µm Fig. 11.11. Deformation mechanism map of silver as function of the grain size (after [35]) 10 –5 10 –4 10 –3 10 –2 10 –1 10 –0 0.0 0.2 0.4 0.6 0.8 1.0 T/T m ¿/G dislocation glide elastic derformation " 2 . " 1 . " 3 . " 4 . " 4 > . " 2 > . " 1 . " 3 > . diffusion creep dislocation creep Fig. 11.12. Idealised deformation mechanism map at different strain rates (af- ter [26]) At even higher stresses, time-independent plastic deformation begins . If the stress level reaches about one tenth of the shear modulus, the theoretical strength of the material is reached. Diagrams like this can be compiled for different materials and material states. Figure 11.10 shows the deformation mechanism maps of aluminium and tungsten at a grain size of 32 µm. Although both maps have the same overall structure as the schematic map from figure 11.9, they nevertheless differ in the size and exact shape of the different regions. Figure 11.11 shows how the grain size changes the deformation mecha- nisms, using the example of silver. The regions are sh own for three different 400 11 Creep ¾ ¾ ¾ ¾ ¾ ¾ grain 1 grain 2 former grain boundary ) (a) Accumulation of cavities at the grain boundaries (b) Mechanism of formation Fig. 11.13. Schematic illustration of cavern-type pores at grain boundaries grain sizes, making it easy to see that creep processes are more important at small grain size and start at lower temperatures. This is due to the grain size dependence of diffusion creep. Since creep processes are time-dependent, the dominant mechanism also depends on the strain rate. This can also be represented in the diagrams as shown in figure 11.12. At high strain rates i. e., high stresses, diffusion creep becomes less important in comparison to dislocation creep. 11.3 Creep fracture After sufficiently long loading times, creeping materials fail by creep fracture. The strain rate, which attained its minimum value during secondary creep, increases again, and tertiary creep starts, ending with the final fracture, so- called creep rupture. In most cases, creep fracture is distinguished by material failure at the grain boundaries, not inside the grains. In contrast to ductile fracture, creep fracture is thus usually intercrystalline. Transcrystalline frac- ture usually occurs only at high stresses [40]. That fracture occu rs at the grain boundaries indicates a damage process there. It is due to the formation of pores and microcracks. Microscopically, two different types of damage are distinguished in a creeping material. On the one hand, oval cavern-type pores can be formed at grain boundaries which are loaded under tension, on the other hand, wedge-type pores may be induced at triple points where three grain boundaries meet. Cavern-type pores form by diffusion processes in which the material in- creases its length in the direction of the tensile stress by moving atoms from 11.4 Increasing the creep resistance 401 ¾ ¾ ¾ ¾ grain 1 grain 2 grain 3 ) ¾ ¾ (a) Wedge-type pores at triple-p oints of the grains (b) Mechanism of formation Fig. 11.14. Schematic illustration of wedge-type pores at triple-points the region of the forming pore to neighbouring zones (figure 11.13). Completely analogous to the precipitation of particles (see section 6.4.4), a nucleation bar- rier has to be overcome to form a cavern-type pore, for the energy required to form an inner surface is proportional to the square of the pore diameter, whereas the energy gain depends cubically on the diameter. For small pores, the surface energy dominates. Pores are therefore not formed initially, but only after a long time. Their formation is favoured at high temperatures and large testing times because this raises the probability to overcome the nucle- ation barrier by thermal activation. Local stress concentrations, for example due to precipitates on the grain boundary or dislocation pile-up, increase the energy gain and thus favour pore formation. Wedge-type pores are formed by grain boundary sliding at triple points (see figure 11.14). In these regions, there is a large stress concentration (see section 11.2.4) that c an cause failure of the grain boundaries if their cleavage strength is exceeded. Accordingly, this type of damage usually occurs at high stresses. Both damage mechanisms cause a decrease of the effective cross section of the specimen and stress concentrations by notch effects. This is the reason for the rapid increase in the strain rate observed in tertiary creep. 11.4 Increasing the creep resistance Materials heavily loaded under creep conditions must meet particular require- ments. [...]... CrMo 5 10 CrMo 9-1 0 13 CrMo 4-4 21 CrMoV 5-1 1 108 136 308 410 69 85 226 221 285 349 34 41 118 135 137 212 36 68 49 92 austenitic steels X 5 CrNi 1 8-1 0 X 10 CrNiNb 1 8-9 X 5 CrNiMo 1 7-1 2-2 300 127 205 74 131 145 30 55 52 18 23 nickel-base superalloys IN 738 SC 16 360 155 240 110 increased further by using austenitic steels like X 5 CrNi 1 8-1 0 This is not only due to the higher content of alloying elements,... at a stress σ2 A creep experiment at this stress and a temperature T3 = 940℃ results in a time to failure of t3 = 173 hours What is the maximum service temperature to meet the life time demands? Exercise 33: Creep deformation In section 8.2.1, the time-dependent behaviour of polymers was described using spring -and- dashpot models a) Sketch a spring -and- dashpot model suitable to describe creep deformation!... to the time-dependence of elastic deformation, stress and strain are out of phase because the strain follows the current stress only with some delay The following time-dependence is assumed for stress and strain: σ(t) = σ0 sin(ωt + δ) , ε(t) = ε0 sin ωt (12.2) a) Sketch the time-dependence of stress and strain and explain the meaning of the parameter δ! b) Write the stress as a function of the strain!... forming highly elongated dislocation loops (figure 11. 17) This impedes dislocation movement and causes the high strength of these alloys The dislocations can cut the particles only at rather high stresses Thus, stresses of 100 MPa can be applied at temperatures of 1000℃ and service times of several thousand hours without material failure Dispersion-strengthened materials (see section 6.4.4) also have a high... as a function of time in a retardation experiment with prescribed stress σ! b) What is the result if you perform a relaxation experiment instead? c) In real-world polymers, part of the elastic deformation is time-independent Use a three-parameter model according to figure 8.7(b), assuming am infinite viscosity of the dashpot element in series, to describe the behaviour in a relaxation and retardation... yield strength to a value of 100 MPa? Exercise 21: Precipitation hardening The yield strength of an aluminium-copper alloy is to be increased by precipitation hardening by ∆Rp0.2 = 600 MPa a) Calculate the required particle spacing of incoherent particles! b) Calculate the particle radius, assuming a copper content of 4 vol-%! To simplify the calculation, neglect the solubility of copper in the aluminium... in chemical engineering Exercise 23: Design of a fluid tank You are the design engineer of a company working in chemical engineering Your task is to design the bearing plate of a tank used to store acids The tank has a capacity of 200 L with dimensions of L × B × H = 1000 mm × 400 mm × 510 mm The bearing plate must bear a total weight of 250 kg The bearing plate is to be made of a ceramic and is fixed... A and the value of the applied force cancel from the equation as expected e) According to equations (2.35) and (2.39), Young’s modulus is E 100 = (C11 − C12 )(C11 + 2C12 ) 1 = = 43.5 GPa , S11 C11 + C12 The simple estimate has roughly the right order of magnitude, but it is still way too large This is mainly due to the fact that the repulsion of diagonally neighbouring atoms eases the deformation and. .. viscoelastic polymers As explained in section 8.2, the time-dependent behaviour of polymers can be described using spring and dashpot elements The behaviour of a spring 418 12 Exercises element can be described with the equation σ = Eε, with σ being the stress, ε the strain, and E Young’s modulus A dashpot element behaves according to ε = σ/η, with strain rate ε and viscosity η ˙ ˙ a) Start considering the Kelvin... are linear-elastic Young’s modulus of the rubber band of booth B is twice as large as that at booth A: EB = 2EA At both booths, the take-off velocity of the candies is disappointingly small We want to find a way to increase the velocity without using additional material or changing the construction of the catapults a) Start by deriving equations for the stored elastic energy W (el) when the bands are . 2 grain 3 ) ¾ ¾ (a) Wedge-type pores at triple-p oints of the grains (b) Mechanism of formation Fig. 11. 14. Schematic illustration of wedge-type pores at triple-points the region of the forming pore. stresses. Thus, stresses of 100 MPa can be applied at tempera- tures of 1000℃ and service times of several thousand hours without material failure. Dispersion-strengthened materials (see section. 49 21 CrMoV 5-1 1 410 349 212 92 austenitic steels X 5 CrNi 1 8-1 0 127 74 30 X 10 CrNiNb 1 8-9 300 205 131 55 18 X 5 CrNiMo 1 7-1 2-2 145 52 23 nickel-base superalloys IN 738 360 155 SC 16 240 110 increased