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3.8 Summary type for the cubic body-centered crystal lattice The fifth distinctive structural feature is the presence of jogs in mobile dislocations Moreover, the distances between jogs are very close to the distances between immobile dislocations in small-angle sub-boundaries The following conclusion can be drawn: the mobile dislocations arise from the sub-boundary dislocations It is as if the former bear a “stamp”, an imprint of the latter The generation of vacancies during the process under consideration is the last feature Loops and helicoids are formed when vacancies collapse It is logical conclude that sources of vacancies are activated during the hightemperature deformation process It should be noted that some features described above, especially, the formation of small-angle boundaries were observed in many studies However, we should to take into account all the structural peculiarities for an adequate understanding of the phenomenon under consideration Now we can proceed to describe the physical mechanism of high-temperature deformation of pure metals and single-phase alloys Our aim is to relate the microstructural observations to the measured strain rates 3.8 Summary Typical structural features are observed in pure metals and solid solutions, which are loaded at high temperatures These features are caused by certain physical mechanisms of deformation The average subgrain sizes, D and the average subgrain misorientations, η, have been systematically directly measured during high-temperature strain of the metals and alloys under investigation The values of D are of the order of 0.7–2.0 µm, the values of η are of the order of 2.9–6.0 mrad The substructure is formed inside crystallites during the primary stage of creep The value of D decreases and η increases during the primary stage The origin of the steady-state stage coincides with the end of substructure formation The steady-state creep occurs at almost constant mean values of both parameters It is the process of substructure formation in the primary stage that causes the decrease in the strain rate and the beginning of the steady-state stage The values of D and η are strongly dependent upon stress The greater the applied stress the greater the misorientation angles and the smaller the subgrains Investigations of single-phase two-component alloys not reveal any qualitative differences in the structure evolution from the processes occurring in 41 42 Structural Parameters in High-Temperature Deformed Metals pure metals Larger values of η are observed in solid solutions as compared with metals Most of the dislocations in specimens are associated in sub-boundaries The parallel sub-boundary dislocations are situated at equal distances from each other It follows from results of the Burgers vector determinations and from the repeating structural patterns that the parallel sub-boundary dislocations are of the same sign Two intersecting dislocation systems are often found inside sub-boundaries The sub-boundaries that have been formed are the sources of slipping dislocations At the same time the sub-boundaries act as obstacles to the movement of deforming dislocations Kinks and bends are observed in dislocations inside subgrains They are caused by jogs in the mobile screw components A great number of vacancy loops and helicoids are present in the structure The slipping dislocations with equidistant one-sign jogs generate vacancies It has been found that the average distances between sub-boundary dislocations, λ, and the mean spacings between jogs in mobile dislocations, z0 are close in value The conclusion is drawn that the sub-boundary dislocations generate mobile dislocations with vacancy-producing jogs in their screw components 43 Physical Mechanism and Structural Model of Strain at High Temperatures 4.1 Physical Model and Theory It follows from the obtained data that complex processes occur in the crystal lattice at high-temperature strain These processes are inherent to pure metals and to solid solutions at certain temperatures and under the applied stresses The quantitative theory, which we are going to develop, will be based on the experimental results presented in Chapter The dislocation density increases at the beginning of the plastic strain In the primary stage of deformation some of the generated dislocations form discrete distributions They enter into low-angle sub-boundaries The interaction of dislocations having the same sign is facilitated by the high-temperature conditions and applied stress These conditions make it easy for dislocations to move and for the edge components of dislocations to climb The edge dislocations can change their slip planes Why the dislocations form ordered sub-boundaries spontaneously? The immediate cause of the formation of the dislocation walls is the interaction between dislocations of the same sign that results in a decrease in the internal energy of the system The elastic energy of dislocations that are associated in subgrains, Es , is less, than the energy of dislocations distributed chaotically in the whole volume of the material, Ev One can compare these energies The values of Ev and Es are expressed as [18] Ev ≈ Es ≈ ρµb2 4π ρµb2 8π ln ln L b (4.1) F ρb2 (4.2) High Temperature Strain of Metals and Alloys, Valim Levitin (Author) Copyright c 2006 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim ISBN: 3-527-313389-9 44 Physical Mechanism of Strain at High Temperatures where ρ is the dislocation density, µ is the shear modulus, b is the Burgers vector, L is the size of a crystal, and F is the fraction of the crystal volume that is occupied by sub-boundaries Assuming reasonable values of ρ = 1012 m−2 , L = 0.3cm, F = 0.05, one can obtain Ev = 2.4 Es It follows from this ratio that an appreciable decrease in internal energy takes place due to the formation of sub-boundaries Well-formed sub-boundaries were observed in the experiments described in Chapter In fact the dislocation subgrains are two-dimensional defects They are sources, emitters of mobile dislocations, which contribute to strain In Section 4.6 and in Supplement one can find evidence that emission of mobile dislocations from sub-boundaries leads to the formation of jogs in them It is the dislocation sub-boundary that generate jogs in mobile dislocations The screw components of emitted dislocations “keep their origin in their memory” They contain equidistant one-signed jogs, although this is only in the physical model, actually, the distances between jogs, z0 , are distributed values A jog is a segment of dislocation, which does not lie in the slip plane The jog cannot move without generation of point defects, i.e vacancies The jogged dislocation can slip if there is a steady diffusion of vacancies from it The nonconservative slipping of jogged dislocations is dependent on the material redistribution The shorter the distance between jogs, the lower the dislocation velocity Hence, it is the diffusion process that controls the velocity of the slip of deforming dislocations Note that we consider vacancy-producing jogs only The interstitial-producing jogs are practically immobile because the energy of formation of the interstitial atoms is several orders greater than that of the vacancies That is why we have observed so many vacancy loops and helicoids in the structure of tested metals (Section 3.6) It appears that a subsequent coalescence of vacancies leads to cavity formation and rupture This seems to be the essence of the tertiary stage of creep The distance between dislocations in sub-boundaries decreases during the primary stage of deformation and the role of sub-boundaries as obstacles for slipping dislocations increases At the end of the substructure formation the dislocation arrangement is ordered and a steady-state stage begins During this stage a dislocation emission from sub-boundaries takes place The emitted dislocations are replaced in sub-boundaries with new dislocations, which move under the effect of applied stress Having entered a sub-boundary a new dislocation is absorbed by 4.2 Velocity of Dislocations it Jogs of the same sign appear along the screw components of the emitted dislocations The distance between jogs at mobile dislocations is equal to the distance between the immobile sub-boundary dislocations, see Eq (3.2) Figure 4.1 demonstrates the described model of steady-state strain (creep) It can be seen that atoms are required in order to complete the extra-planes during the motion of the jogs Vacancies are generated in the vicinity of jogs since atoms are consumed in completing the extra-planes Consequently, the relay-like motion of the vacancy-emitted jogged dislocations from one subboundary to another is the distinguishing feature of the high-temperature strain Fig 4.1 The physical model of the steady-state strain at high temperature The sub-boundaries are built of two systems: the pure screw and the 60◦ -dislocation systems Emission of mobile dislocations from sub-boundaries is shown The jogs at the mobile screw components have the same sign and generate vacancies 4.2 Velocity of Dislocations Consider a screw dislocation that is situated along the Oz axis of the coordinate system A dislocation with jogs moves in the direction Ox (see Fig 4.2) The material parameters are different in the volume and inside the dislocation “tube” Let us denote the coefficients of the diffusion of the vacancies by Dv and Dd , respectively In Figs.4.2 and 4.3 the energy of vacancy generation is denoted by E The energy of vacancy diffusion is denoted by U Subscripts j, d, and v refer to the jog, dislocation and volume, respectively For example, Ujd is the energy of 45 46 Physical Mechanism of Strain at High Temperatures Fig 4.2 Scheme of energetic barriers for the motion of the screw dislocation with vacancy-producing jogs The dislocation is slipping along Ox from left to right (see text for further explanation) the vacancy displacement from the jog in the dislocation “tube”; Udv is the energy of the vacancy displacement from the dislocation in the crystal volume and so on The following elementary events determine the process of the dislocation slip: the generation of vacancies near jogs (energy of activation Ed ); the diffusion of vacancies along the dislocations (energy of activation Ud ); transition of vacancies to the volume (Udv ); diffusion of vacancies in the volume (Uv ) The external applied stress performs work, dA, and facilitates the generation Fig 4.3 Scheme of energetic barriers for the motion of the screw dislocation with vacancy-absorbing jogs 4.2 Velocity of Dislocations of vacancies: Ujd = Ed + Udj − dA (4.3) The vacancy concentration in the vicinity of the jogs, cjp , increases under the effect of stress: cjp = c0 exp dA + ε0 kT = b3 − Ev kT exp dA + ε0 kT (4.4) where c0 is the equilibrium vacancy concentration in the volume at a given temperature, ε0 is the energy of bonding a dislocation and a vacancy, ε0 = Udv − Uvd In the case of vacancy-absorbing jogs the sequence of events is the inverse, i.e., the generation of vacancies in the volume (Ev ), diffusion of vacancies in the volume (Uv ), transition of vacancies in the dislocation “tube” Uvd , diffusion of vacancies in the “tube” Ud , joining the vacancy in the jog Udj The applied external stress facilitates joining of vacancies: Ujd = Ed + Udj + dA (4.5) In both cases the work of applied stress is given by dA = σzz b3 + σyz z0 b a (4.6) where σzz and σyz are the components of the stress tensor; a is the height of the jogs As super-jogs are unstable we may assume that a ≈ b and b z0 Hence dA = σyz b2 z0 (4.7) Now we will consider the velocity of the dislocations The velocity of the screw components with vacancy-producing jogs is given by the following expression [19] Vp = Ev + Uv + ε0 πνr0 z0 exp − bF (α) kT exp σyz b2 z0 + ε0 kT −1 (4.8) where ν is the Debye frequency, r0 is the radius of the dislocation “tube”, α = Vp r0 /2Dv , F (α) is a weak function The vacancy-absorbing jogs have velocity Va = Ev + Uv + ε0 πνr0 z0 exp − bF (α) kT − exp − σyz b2 z0 + ε0 kT (4.9) 47 48 Physical Mechanism of Strain at High Temperatures In practice we always have σyz b2 z0 kT Therefore Vp Va The velocity of the dislocations is exponentially dependent on stress One can see that the exponent (4.8) contains the sum Ev + Uv This implies that the effective energy of the jogged dislocation motion is close to the activation energy of diffusion The activation volume in Eq (4.8) is equal to b2 z0 The computed values of Vp vary in the range 10−11 to 10−2 cm s−1 As is shown in Fig 4.4 the velocity of dislocations with vacancy-absorbing jogs, Va , is less by many orders These dislocation components are immobile and not control the strain rate In Fig 4.5 the velocity of jogged dislocations in α-iron is shown The distance between jogs strongly influences the velocity of the dislocations Fig 4.4 Velocity of screw dislocations in nickel The distance between jogs is 36nm 1, 2, 3: jogs generate vacancies; , , : jogs absorb vacancies and : 673; and : 873; and : 1073K 4.3 Dislocation Density Fig 4.5 The effect of temperature, stress and distance between jogs on the velocity of screw dislocations in α-iron 1: 773K, z0 = 35nm; 2: 813K, z0 = 57nm; : 813K, z0 = 75nm; 3: 973K, z0 = 52nm; : 973K z0 = 75 nm 4.3 Dislocation Density The slip strain rate is given by [20] γ = bN V ˙ (4.10) where N is the density of deforming dislocations and V is their average velocity The total mobile dislocation density is assumed to be equal to the sum ρ = N + Na (4.11) where Na is the density of annihilating dislocations Unlike N the density Na does not contribute to the macroscopic strain Processes of the dislocation multiplication, annihilation, sub-boundary emission and immobilization, occur in metals during the high temperature strain The balance equation, which characterizes the change in the mobile dislocation density, can be written as ρ = ρ m + ρm + ρa + ρa , ˙ ˙d ˙s ˙d ˙s (4.12) 49 50 Physical Mechanism of Strain at High Temperatures where ρm is the rate of the density increase due to the dislocation multipli˙d cation, ρm is the rate of the density change on account of the sub-boundary ˙s emission, ρa is the annihilation rate, and ρa is the rate of the immobilization of ˙d ˙s dislocation by sub-boundaries (Subscript d refers to dislocations, subscript s to sub-boundaries, superscript m to multiplication and emission, superscript a to annihilation and immobilization) Therefore, two terms of Eq (4.12) are determined by the interactions of dislocations and the other two are related to the effect of sub-boundaries Consider each term of Eq (4.12) separately The number of newly generated dislocation loops is directly proportional to the dislocation density and to the dislocation velocity Hence the rate of multiplication of the mobile slip dislocations is given by ρm = δρV ˙d (4.13) where δ is a coefficient of multiplication of the mobile dislocations The coefficient has the unit of inverse length Similarly, the rate of emission of the mobile dislocations out of sub-boundaries is directly proportional to the sub-boundary dislocation density and to the dislocation velocity: ρ m = δ s ρs V ˙s (4.14) where ρs = η/bD = 1/λD is the density of dislocations in sub-boundaries; δs is a coefficient of dislocation emission Let us assume there are n+ positive dislocations and n− negative ones inside a subgrain The annihilation rate of dislocations of opposite signs is given by n+ n− V (4.15) na = −2 ˙d l where l = ρ−0.5 is the mean distance between dislocations inside subgrains Since ρ = n/D2 , ρ+ = ρ− = ρ/2, one can obtain ρa = −0.5D2 V ρ2.5 ˙d (4.16) Immobilization of the mobile dislocations occurs when they are captured by sub-boundaries The rate of immobilization is assumed to be directly proportional to the dislocation density and velocity and inversely proportional to the subgrain size: V ρa = −ρ ˙s (4.17) D Only deforming dislocations of density N come out of sub-boundaries; annihilating dislocations of density Na are eliminated inside subgrains 4.4 Rate of the Steady-State Creep The ratio of the densities of annihilating and deforming dislocations is given by Na ρa ˙s = a = 0.5D3 ρ1.5 (4.18) N ρd ˙ Taking into account the obtained results we may rewrite the differential equation for the dislocation density evolution during high-temperature deformation of metals as follows: ρ = δρV + δs ρs V − 0.5D2 V ρ2.5 − ρ ˙ V D (4.19) It will be noted that Eq (4.19) does not contain any arbitrary parameters, only values which have physical meaning determine the rate of the dislocation density evolution It is obvious from the derived formulas that the structural parameters λ and D both influence the dislocation density Consequently, the strain rate γ, ˙ Eq (4.10) is dependent on structural parameters not only due to the dislocation velocity V but also due to the dislocation density N Equation (4.19) can be solved for the steady-state stage (see Section 5.4) We shall also make use of Eq (4.19) in Chapter to develop a computer model of processes that take place during the high-temperature deformation of metals 4.4 Rate of the Steady-State Creep We should take into account that the creep tests are realized as a one-axis tension However, Eqs (4.7) and (4.8) contain the stress component σyz In accordance with the rules for tensor component conversions we have σy z = γy y γz y σyy (4.20) where the primes relate to the new coordinate system; γi i are cosines of angles between the respective axes Averaging the product of the cosines over all orientations of the new coordinate system and finding the average value of the function of two variables we arrive at σy z = σyy (4.21) π The conversion from γ to ε is similar to Eq (4.21) ˙ ˙ 51 52 Physical Mechanism of Strain at High Temperatures Now we can calculate the rate of the steady-state high-temperature deformation ε Three groups of physical parameters are needed: ˙ • • • External parameters: temperature T and stress σ Diffusion parameters: the energy of vacancy generation Ev and the energy of vacancy diffusion Uv Structural parameters: the subgrain size D and the distance between dislocations in the sub-boundaries λ [the last value depends on the subgrain misorientation angle η, see Eq (3.1)] These parameters and constants are the input data for calculations Values of η and D are listed in Tables 3.1 and 3.2 The diffusion constants can be easily found in the literature, e.g [21, 22] For example, Ev = 2.56 × 10−19 Jat.−1 , Uv = 1.92 × 10−19 Jat.−1 for α-iron, and 2.88 × 10−19 Jat.−1 and 1.68 × 10−19 Jat.−1 , respectively, for nickel The dislocation density has been measured by us, see Table 3.5 Thus, both multipliers in Eq (4.10) are known In other words we have gone from a microscopic level of thinking to a macroscopic one The ability of the physical model to represent the macrocreep behavior of metals correctly and quantitatively is shown in Figs 4.6–4.9 The theoretical curves are in very close agreement with the experimentally observed values of the steady-state creep rate in α-iron at 813, 873, 923 and 973K However, the calculated data lie somewhat lower The difference between computed and experimental curves corresponds to a coefficient of about 1.5, which one should insert into the right-hand part of Eq (4.10) It can be seen that the theory does not represent the experimental results at 773K The data obtained for nickel are illustrated in Figs 4.8 and 4.9 The solid lines that present the calculated data fit satisfactorily the experimental steady- Fig 4.6 The steady-state creep rate of α-iron as a function of stress Comparison of the experimental results (symbols and dotted lines) and computed data (solid curves); 1, 813K; 2, 923K 4.4 Rate of the Steady-State Creep state strain rates at the test temperatures 673, 873, and 1023K When the temperature is increased to 1173K the theoretical model fails Fig 4.7 The steady-state creep rate of α-iron as a function of stress Comparison of the experimental results (symbols and dotted lines) and computed data (solid curves): 1, 773K; 2, 873K; 3, 973K Fig 4.8 The steady-state creep rate of nickel as a function of stress Comparison of the experimental results (symbols and dotted lines) and computed data (solid curves) for nickel: 1, 673; 2, 873K Fig 4.9 The steady-state creep rate of nickel as a function of stress Comparison of the experimental results (symbols and dotted lines) and computed data (solid curves) 1, 1023K; 2, 1173K 53 54 Physical Mechanism of Strain at High Temperatures 4.5 Effect of Alloying: Relationship between Creep Rate and Mean-Square Atomic Amplitudes The alloying of metals within the solubility limit was found not to change the physical mechanism of strain, however, the strain rate is noticeably affected by the diffusion parameters For example, the activation energy of self-diffusion in nickel is 4.64 × 10−19 Jat.−1 [22]; in alloy Ni+9.5%W this value increases to 4.88 × 10−19 Jat.−1 [23] In Table 4.1 the data for nickel-based alloys are presented The calculated quantities of ε are in accordance with the measured ˙ ones Tab 4.1 Comparison of computed and observed values for the nickel-based solid solutions Alloy, at.% T, K σ, MPa Measured λ, nm Measured ε, 10−7 s−1 ˙ Calculated V, nm s−1 Calculated ε, 10−7 s−1 ˙ Ni+9.9Al 723 125 202 52 34 1.4 10.1 4.3 7.6 2.2 11.5 873 70 136 54 33 0.9 13.9 2.8 8.2 1.0 13.0 873 138 170 41 33 5.6 17.6 12.0 8.6 7.8 26.0 1023 50 90 46 32 1.6 7.6 6.0 17.0 1.8 10.8 Ni+9.5W Comparison of the temperature dependence of the mean-squared atomic displacements, u2 , and the steady-state creep rates, ε, for nickel and for its solid ˙ solutions clearly shows a connection between the two quantities (Fig 4.10) The amplitudes of atomic vibrations correlate with one of the main characteristics of the high-temperature strength – the rate of the stationary creep One can see in Fig 4.10 that the creep rate ε depends almost exponen˙ tially on the mean-squared atomic displacements over the temperature range 850–1050K, where the deformation is controlled by diffusion of vacancies in the solid solution The effect of alloying on ε increases with increasing ˙ temperature The decrease in the vibration amplitude under the influence of solution atoms of Al, Cr, and W in nickel results in an increase in the values of Ev and Uv This causes a decrease in the values of ε The most noticeable ˙ is the decrease of the creep rate under the influence of the tungsten additions (by three or four orders of magnitude at 850–1050K) 4.6 Formation of Jogs Fig 4.10 Comparison of the temperature dependence of the creep rates ε under stress of 80 MPa (a) and of the ˙ mean-square atomic amplitudes (b) for nickel-based alloys The composition of alloys (at.%): 1, nickel; 2, Ni+9.9Al; 3, Ni+9.5Cr; 4, Ni+9.6W 4.6 Formation of Jogs Low-Angle Sub-boundaries in f.c.c and b.c.c Crystal Lattices The distance between the sub-boundary dislocations, λ, is related to the angle between adjacent subgrains If we assume that the low-angle sub-boundary is constructed by two crossing dislocation systems, then a node N belongs to both intersecting dislocations (Fig 4.11) A dislocation node is simply a point where three or more dislocations meet At a dislocation node the sum of all Burgers vectors is equal to zero, bi = Vectors b2 and b4 enter the node, vectors b1 and b3 leave the node, Fig 4.11(a) In Fig 4.11(b) the same situation is presented after changing the signs of the vectors ξ3 and ξ4 During parallel slip of dislocations ξ1 and ξ2 the displacement of the crystal part takes place in the plane which is determined by the vectors ξ1 × V1 and ξ2 × V2 Thus, after an emission from the sub-boundary a jog of length b2 must be created in the mobile dislocation ξ1 , Fig 4.11(c) We prove in Supplement that dislocations of at least one sub-boundary systems contain the screw component Hence, the slipping jog has to generate lattice vacancies or interstitial atoms 55 ... (4. 1) F ρb2 (4. 2) High Temperature Strain of Metals and Alloys, Valim Levitin (Author) Copyright c 2006 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim ISBN: 3-527-313389-9 44 Physical Mechanism of. .. distinguishing feature of the high- temperature strain Fig 4. 1 The physical model of the steady-state strain at high temperature The sub-boundaries are built of two systems: the pure screw and the 60◦ -dislocation... Subscripts j, d, and v refer to the jog, dislocation and volume, respectively For example, Ujd is the energy of 45 46 Physical Mechanism of Strain at High Temperatures Fig 4. 2 Scheme of energetic