High Temperature Strain of Metals and Alloys Part 6 doc

72 5 Simulation of the Parameters Evolution Fig. 5.2 Dependence of strain and structural parameters on time for nickel. The computer simulation uses the set of 12 ordinary differential equations. Two curves in each of six graphs correspond to two intersecting systems of parallel slip planes. T = 1073K, σ 1 =17.5MPa, σ 2 =16.5MPa. time t =0 deformation γ 1 = γ 2 =0 dislocation density ρ 1 = ρ 2 =2× 10 8 m −2 5.3 Results of Simulation 73 dislocation spacing in sub-boundaries λ 1 = λ 2 =50nm subgrain size D 1 = D 2 =3µm coefficients of the dislocation multiplication and emission, respectively, δ =2×10 4 m −1 ; δ s =4× 10 4 m −1 The test time is 5h = 1.8 × 10 4 s. Comparison of the obtained results (Fig. 5.2) with the experimental data shows remarkable overall agreement. Further analysis of data from the model leads to some interesting con- clusions: There are some differences in how the processes in both plane sets proceed. One can see an increase in strain in Fig. 5.2. The steady-state stage of creep occurs earlier under the lower stress. The strain value of 2% is observed in 1.5 × 10 4 s after the load has been applied. For comparison with the model data the experimental results are presented in Fig. 5.3 and Fig. 5.4. The inter-dislocation spacings in Fig. 5.4 were deter- mined from X-ray measurement data, as described in Chapter 2. There is an obvious fit of model and experimental data which is evidence that the physical model is adequate. Fig. 5.3 Strain versus time for nickel tested at 1073K. To be compared with the first graph in Fig. 5.2. Two specimens. B, σ 1 =10MPa; C, σ 2 =14MPa. 74 5 Simulation of the Parameters Evolution Fig. 5.4 The sub-boundary dislocation spacing versus time for nickel tested at 1073K. To be compared with the third graph in Fig. 5.2. Experimental data for the same two specimens as in Fig. 5.3. The density of dislocations increases during the high-temperature strain from 2 ×10 8 m −2 to (4.0–4.5) ×10 11 m −2 . The dislocation density increases very quickly after loading, within 75–100 s. At the steady-state stage the values ρ 1 and ρ 2 are almost constant, hence the rates of the mobile dislocation gen- eration and of the dislocation annihilation (immobilization) are equal. The sum of the positive terms on the right-hand side of Eq. (5.3) is equal to the sum of the absolute values of the negative terms. The velocity,V , of themobiledislocationsdecreases gradually.At the steady- state it is of the order of 20 nm s −1 , i.e. 80 interatomic spacings per second. V is less in the planes where the applied stress is less. In contrast, the climb velocity Q increases with time. Q is two orders less than V . The spacing between jogs in mobile dislocations 1) , λ, decreases from 50 to 35 nm in exactly the same way as in reality, see Fig. 5.4. The decrease in λ = l correlate with the decrease in the V value. The change in the subgrain size D (the last graph) differs somewhat from the observed one. The calculated values drop too quickly. The proposed model enables one to examine the influence of structural parameters on strain and on the strain rate. It is convenient to study the evolution of the investigated values. The strain decreases when the initial value λ is decreased or the initial value D is increased by means of suitable treatment. For example, if the initial average subgrain size D is 12µm instead of 3µm then the strain drops from 0.02 to 0.005. As regards λ this value is in the exponents in Eqs. (4.8), 1) In Fig. 5.2 and 5.5 the sub-boundary dislocation spacing is denoted as l. 5.3 Results of Simulation 75 (4.10), (5.4). Hence, the strain rate is strongly affected by the sub-boundary dislocations spacing. A seeming paradoxical result is of interest. When the initial value of the mobile dislocation density, ρ, increases sharply then the annihilation of dis- locations progresses. As a result the strain of the specimen decreases. In con- trast, when the initial value ρ is decreased, e.g. from 2 × 10 8 to 5 ×10 7 m −2 , the strain increases from 0.02 to 0.10. It might seem that the coefficients of the dislocation multiplications are chosen somewhat arbitrarily. Undoubtedly, the real values δ and δ s are un- known. We are forced to consider them as fitting coefficients. However, it Fig. 5.5 Dependence of strain and structural parameters on time for niobium. The computer simulation uses the set of 12 ordinary differential equations. Two curves in each of the six graphs correspond to two intersecting systems of parallel slip planes. T = 1370K, σ 1 =17.0MPa, σ 2 =16.5MPa. 76 5 Simulation of the Parameters Evolution Fig. 5.6 Strain versus time for niobium tested at 1370K, σ =44.1MPa. Fig. 5.7 The sub-boundary dislocations spacing versus time for niobium tested at 1370K. B and C are two crystallites of the same specimen. To be compared with the third graph in Fig. 5.5. turned out that changes in these values, even if by orders of magnitude, have only a small effect on the results of the calculations. For example, varying δ from 2 × 10 4 to 3.2 × 10 5 m −1 does not affect the deformation curve or the dislocation density. The increase in δ s from 4 × 10 4 to 1.6 × 10 5 leads to an increase in ρ at the steady-state stage up to 7 × 10 11 m −2 . In Fig. 5.5 the model data are presented for niobium. The typical experi- mental curves for niobium tested at 1370K are shown in Figs. 5.6–5.8. These curves show the parameters ε, λ = l, and D, respectively, during steady-state strain. One can see that the physical model fits the experimental data well. 5.4 Density of Dislocations during Stationary Creep 77 Fig. 5.8 The average subgrain size versus time for niobium tested at 1370K. B and C are two crystallites of the same specimen. 5.4 Density of Dislocations during Stationary Creep At the steady-state deformation the rates of the dislocation generation and annihilation seem to be equal and the parameters λ and D are constant. Thus, we should solve the system of equations to calculate the density of deforming dislocations during the constant strain rate (see Section 4.3): N + N a = ρ (5.12) N a /N =0.5D 3 ρ 1.5 (5.13) where ρ is the real root of the following equation: δρV + δ s ρ s V −0.5D 2 Vρ 2.5 − ρ V D =0 (5.14) These equations were solved numerically (the Newton method was used). The dislocation densities were computed for different values of the structural parameters D and λ. The results are presented in Fig. 5.9. The obtained results seem to be quite reasonable. The density of deforming dislocations is of the order of 10 11 m −2 . This density is strongly affected by the subgrain size as well as by the distance between sub-boundary dislocations. The larger the subgrains the smaller the density of dislocations that contribute to the deformation process. It is obvious that sub-boundaries of relatively large misorientation are sources for moving dislocations. 78 5 Simulation of the Parameters Evolution Fig. 5.9 The computed density of deforming dislocations N versus the sub-boundary dislocation spacing. B, the subgrain size D =60µm; C, D =30µm; D, D =10µm. The experimentally measured data (Table 3.5) are of the same order, 10 11 m −2 . For examplethemeasured density is (1.3–9.5) × 10 11 m −2 in nickel, (1.6–5.3) × 10 11 m −2 in niobium and so on. From Figs. 5.9 and 5.10 it can be seen that the total dislocation density, ρ, is one order greater than the deforming dislocation density, N . Since the dislocation density is affected by structural parameters the minimal strain rate depends on their values, too. The different values of D and λ can be obtained by a preliminary treatment of the metal. Fig. 5.10 The computed total dislocation density ρ versus the sub-boundary dislocation spacing. B, the subgrain size D =60µm; C, D =30µm; D, D =10µm. 5.4 Density of Dislocations during Stationary Creep 79 Tab. 5.1 Structural parameters in nickel after preliminary deformation and annealing. Deformation D, µm η, mrad λ, nm 0 1.83 3.40 73.2 0.03 1.18 5.40 46.1 0.07 1.00 7.07 35.2 In order to test the influence of the D and λ parameters, we deformed specimens of nickel by 3% and 7% at room temperature and then annealed them at 873K. As a result we obtained the various average subgrain sizes and misorientations (Table 5.1). The specimens revealed after treatment an improved creep resistance (Fig. 5.11). The difference in the creep rate for specimens with 7% deforma- tion is one order less than for specimens without preliminary deformation. However, at relatively high stress the values of the steady-state strain rates become equal. The sub-boundary dislocation distance decreases if the stress increases (Fig. 5.12). Fig. 5.11 The steady-state strain rate of the preliminary de- formed nickel specimens versus applied stress. Temperature 873K. B, without deformation; C, deformation 3%; D, deformation 7%. 80 5 Simulation of the Parameters Evolution Fig. 5.12 The distance between sub-boundary dislocations in the preliminary deformed nickel specimens versus applied stress. Temperature 873K. B, without deformation; C, deformation 3%; D, deformation 7%. 5.5 Summary A systemofdifferential equations has beenproposedto simulate the processes of high-temperature deformation in metals. Two intersecting crystalline sys- tems of parallel slip planes are considered. The dislocation slip velocity in each system is controlled by vacancy-producing jogs and depends on the dis- tances between the sub-boundary dislocations in the parallel planes of another system. The evolution of six values in each system was studied: the shear strain; the total dislocation density; the slip velocity of the dislocations; the climb veloc- ity of the dislocations to the parallel slip planes; the mean spacing between parallel dislocations in sub-boundaries; the mean subgrain size. The formulas that describe the changes in each parameter as a function of time and of the other parameters have been derived; a system of 12 ordinary differential equations was obtained. The Runge-Kutta methods were used for integration of the system. The quantitative model results show a satisfactory fit with experiments. The processes in each plane set happen somewhat differently. The density of mobile dislocations increases during the high-temperature strain from 2 ×10 8 m −2 to (4.0–4.5) ×10 11 m −2 . Experimental data have the same order of 10 11 m −2 . The total dislocation density, ρ, is one order greater than the deforming dislocation density, N. The coefficient of multiplication of mobile dislocations is found to be of the order of 2 × 10 4 m −1 . 5.5 Summary 81 The velocity of the mobile dislocations, V , decreases gradually. At the steady-state stage it is of the order of 20 nm s −1 . The value of V is less in the plane set where less applied stress operates. In contrast, the climb velocity, Q, increases with time. The value of Q is two orders less than that of V . The jog spacing in mobile dislocations decreases when the strain increases, as in reality. A decrease in λ correlates with a decrease in the V value. The strain rate of thespecimenisstrongly affected by the sub-boundary dislocation spacing. A preliminary decreased value of λ and increased value of D lead to the strain rate decreasing when the applied stress is relatively low. [...]... Strain of Metals and Alloys, Valim Levitin (Author) Copyright c 20 06 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim ISBN: 3-527-313389-9 84 6 High- temperature Deformation of Superalloys Fig 6. 1 Crystal structure of the γ phase The face-centered cubic cell contains 3 atoms of B-type (6 atoms/2 adjacent cells) and 1 atom of A-type (8 atoms/8 cells) The chemical formula is B3 A The crystal lattice parameter of. .. oxidation and an acceptable density Such alloys – superalloys – have been developed The largest applications of superalloys are in aircraft and industrial gas turbines, rocket engines, space vehicles, submarines, nuclear reactors and landing apparatus The structure of the majority of nickel-based superalloys consists of a matrix i.e of the γ phase and particles of the hardening γ phase The γ phase is... The re- 6. 1 γ Phase in Superalloys Fig 6. 2 Dependence of the solubility of the γ phase on temperature sults are shown in Fig 6. 2 Solution of the γ phase in the matrix, i.e in the γ solid solution, is observed above 1173K (900 ◦ C) The usual heat treatment of superalloys consists of heating above the temperature of the γ -solution and subsequent hardening The γ phase has remarkable properties, in particular,... dependence of strength on temperature The γ phase first hardens, up to about 1073K, and then softens This peculiarity is reflected in a similar dependence of the yield strength upon temperature in superalloys This is shown in Fig 6. 3 The yield stress increases as the temperature increases from 573 to 1073K Fig 6. 3 Dependence of the yield strength of a superalloy on temperature Experimental data (symbols) and. ..83 6 High- temperature Deformation of Superalloys 6. 1 γ Phase in Superalloys The high- temperature strength requirements of materials have increased with new developments in engine design The continual need for better fuel efficiency has resulted in faster-spinning, hotter-running gas turbine engines One of the most important requirements is resistance to high- temperature deformation... Publishing 85 86 6 High- temperature Deformation of Superalloys Fig 6. 4 Creep curves for EI437B superalloy The steady-state and the tertiary, accelerating stages of creep are observed T = 973K Stress: 1, 410; 2, 450; 3, 490; 4, 530MPa The symbols × show when specimens were taken for transmission electron microscopy study The creep curves of polycrystalline superalloys differ from the curves of pure metals After... to the parameter of the solid solution, so the misfit between two lattices is relatively small The misfit, δ, between precipitates and matrix is defined as δ = (aγ − aγ )/ aγ + aγ 2 (6. 1) The value of δ is negative for current commercial superalloys The magnitude and sign of the misfit also influence the development of microstructure under the operating conditions of stress and high temperature Furthermore,... excess of vacancies in one of the sublattices, which leads to deviations from stoichiometry Sublattices A and B of the γ phase can dissolve a considerable amount of other elements Many of the industrial nickel-based superalloys contain, in addition to chromium, aluminum, and titanium, also molybdenum, tungsten, niobium, tantalum and cobalt These elements are dissolved in the γ phase High Temperature Strain. .. reaches a minimum value of strain rate The steady-state stage creep is sometimes short This minimum creep rate stage is followed by a relatively long tertiary deformation, which results in rupture The representative curves of high- temperature tests for two superalloys are shown in Figs 6. 4 and 6. 5 For superalloys it is often preferred to measure the time tr until a rupture occurs instead of determining the... solid solution with a face-centered crystal lattice and randomly distributed different species of atoms By contrast, the γ phase has an ordered crystalline lattice of type L12 (Fig 6. 1) In pure Ni3 Al phase atoms of aluminum are placed at the vertices of the cubic cell and form the sublattice A Atoms of nickel are located at the centers of the faces and form the sublattice B In fact the phase is not . preliminary decreased value of λ and increased value of D lead to the strain rate decreasing when the applied stress is relatively low. High Temperature Strain of Metals and Alloys, Valim Levitin (Author) Copyright c . 20 06 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-313389-9 83 6 High- temperature Deformation of Superalloys 6. 1 γ  Phase in Superalloys The high- temperature strengthrequirementsof. representative curves of high- temperature tests for two superalloys are shown in Figs. 6. 4 and 6. 5. For superalloys it is often preferred to measure the time t r until a rupture occurs instead of determining