56 4 Physical Mechanism of Strain at High Temperatures Fig. 4.11 The jog formation in a dislocation emitted by a low-angle sub-boundary. (a) The initial position,  b i are the Burgers vectors;  ξ i are the unit dislocation vectors. (b) The same as (a) after changing the signs of the vectors  ξ 3 and  ξ 4 . P 1 and P 2 are the slip planes;  V 1 and  V 2 are the velocity vectors. (c) Dislocations with jogs after emission of the dislocation  ξ 1 . The typical Burgers vectors, slip planes and unit dislocation vectors have been selected for examination of sub-boundaries. The results are presented in Table 4.2. The angles <  b 1  ξ 1 and <  b 2  ξ 2 are not equal to 90 ◦ . This means that the dislocations of both systems contain screw components. Tab. 4.2 The crystallography of low-angle sub-boundaries. Lattice Slip <ξ 1 ξ 2  b 1  ξ 1 <b 1 ξ 1  b 2  ξ 2 <b 2 ξ 2 plane f.c.c. {111} 90 ◦ a 2 [110] √ 2 2 [110] 0 ◦ a 2 [ ¯ 110] √ 2 2 [ ¯ 110] 0 ◦ 60 ◦ a 2 [110] √ 2 2 [110] 0 ◦ a 2 [10 ¯ 1] √ 2 2 [10 ¯ 1] 0 ◦ 60 ◦ a 2 [011] √ 2 2 [ ¯ 1 ¯ 10] 120 ◦ a 2 [ ¯ 110] √ 2 2 [0 ¯ 11] 120 ◦ 60 ◦ a 2 [110] √ 2 2 [110] 0 ◦ a 2 [ ¯ 110] √ 2 2 [011] 60 ◦ b.c.c. {110} 90 ◦ a 2 [111] √ 2 2 [110] 35.3 ◦ a 2 [1 ¯ 11] √ 2 2 [1 ¯ 10] 35.3 ◦ 73.2 ◦ a 2 [111] √ 2 2 [ ¯ 1 ¯ 10] 144.7 ◦ a 2 [1 ¯ 11] √ 6 6 [1 ¯ 21] 19.5 ◦ 4.7 Significance of the Stacking Faults Energy 57 4.7 Significance of the Stacking Faults Energy The processes of high-temperature strain are dependent upon the nature of a metal, especially, upon peculiarities of dislocations in its crystal lattice. Metals have different values of the stacking fault energy which results in a different ability to change the slip plain, i.e. to climb into parallel slip plains. This difference leads to various types of macroscopic behavior at high temperature. In Ref. [24] four crept metals with face-centered crystal lattices: aluminum, nickel, copper and silver were investigated. The subgrain misorientations were measured with the X-ray rocking method at discrete time moments. Tests were carried out at the tensile rate of 0.5MPah −1 . The total dislocation density was calculated from the misorientation angles. All four metals reveal linear dependences of the misorientation angle on strain at room temperature. In Fig. 4.12 the data of tests at 0.45 T m are shown. The linear dependence remains only for silver. At T =0.68 T m all depen- dences η(ε) have a certain curvature (Fig. 4.13). The curves are ordered in the order of the stacking fault energies: Al, Ni, Cu, Ag, 290, 150, 70, and 25mJ m −2 , respectively. Pishchak [24] considers the dependence of deformation ε on stress σ.At room temperature there is a linear dependence, ε ∼ σ. At high temperatures he assumes the empirical equation ε = Aσ m to be the most appropriate. The exponent of the power function, m, turns out not to be a constant value but to increase with temperature from m =1to m =2. The temperature of the m change is equal to 0.30, 0.35, 0.40 and 0.60 T m for Al, Ni, Cu, Ag, respectively. Fig. 4.12 The average subgrain misorientation versus strain in four metals with face-centered crystal lattice. Temperature is equal to 0.45 T m . B, aluminum; C, nickel; D, copper; E, silver. The results were calculated from the data of Ref. [24]. 58 4 Physical Mechanism of Strain at High Temperatures Fig. 4.13 The same as in Fig. 4.12, but at temperature 0.68 T m . From our point of view, in metals with little stacking fault energy, the climb of the dislocation edge components is hindered and dislocations cannot change their slip plane. A higher temperature is needed in order for regular sub-boundaries to be formed. 4.8 Stability of the Dislocation Sub-boundaries As has been noted above, the sub-boundaries are both the sources of and the obstacles for deforming dislocations. Let us consider the effect of external stress and temperature on the sub- boundary dislocation emission. By dislocation emission we mean a thermally activated release of a dislocation from an immobile sub-boundary and its sub- sequent transformation into a mobile deforming dislocation. Our aim is to de- termine a threshold stress, above which the sub-boundaries are unstable and can be destroyed without the thermal activation. We shall analyze the effect of applied stress and temperature on the sub-boundary dislocation emission. Consider the boundary built by two perpendicular systems of equidistant parallel screw dislocations (Fig. 4.14). Assume first that there is no dislocation 2 in a slip plane P 1 . The components of stress affecting a sub-boundary dislocation 1 in the slip plane (y =0) are given by [18] σ yz =  (µb sinh 2πX)(1 − cos 2πZ) 2λ(cosh 2πX − 1)(cosh 2πX −cos 2πZ)  −  µb 2πλX  (4.22) 4.8 Stability of Dislocation Sub-boundaries 59 Fig. 4.14 A sub-boundary formed in the yOz-plane by two systems of screw dislocations. λ is the distance between adjacent dislocations, D is the subgrain size, P 1 (xOz) is the slip plane; a boundary dislocation under consideration is denoted by 1, another dislocation in the slip plane outside the boundary is denoted by 2. σ xz = µb 2πλX ; σ xy = µb sin 2πZ 2λ(cosh 2πX − cos 2πZ) (4.23) where µ is the shear modulus, X = x/λ, Y = y/λ, Z = z/λ. When the dislocation 1 deviates from the boundary the shear stress com- ponent σ yz acts on it. (For a screw dislocation the stress component is parallel to the dislocation line.) The value of this component depends upon the co- ordinates. The results of the calculations of the shear stress are shown in Fig. 4.15. Fig. 4.15 The stress component σ yz in units of µb/2λ as a function of distance. On the left: z/λ = const, on the right: x/λ = const. 60 4 Physical Mechanism of Strain at High Temperatures The curves have singularities at x =0. Within the sub-boundary (in the initial position) the stress components are therefore equal to zero. Thus, the dislocation inside the boundary is affected by the force F (0) = 0. The force reaches its maximum value near the node at a distance equal to the dislocation core radius r 0 . It is reasonable to assume that F (r) is a linear function within the range 0 <r<r 0 . Further F (r)=−bσ yz if r 0 ≤ r<r 1 , where r 1 is a distance at which the interaction force between the dislocation and the boundary is close to zero. The calculated dependences of force and energy on the distance from the deviated dislocation 1 to sub-boundary are shown in Fig. 4.16. One can see that the maximum returning force is achieved at a distance of the order of the dislocation core. This force acts in the opposite direction. Fig. 4.16 The force at which the sub-boundary acts on the emitted dislocation 1, and the activation energy versus the distance. r 0 =2b is assumed. Assume that the applied external stress is σ. The energy to be consumed by the emission is expressed as U = −  r 0 0 F (r)dr −  r 1 r 0 F (r)dr (4.24) The stress field of the sub-boundary tends to return the dislocation 1 to the sub-boundary. Thus, the dislocation is pinned with pinning point density 1/λ and is emitted by means of thermal activation. According to the theory of the rates of reactions [25] the dislocation can be regarded as a linear crystal with D/b degrees of freedom. The number of thermal activations per unit of time 4.8 Stability of Dislocation Sub-boundaries 61 can be represented by an expression of the form Γ = ν eff exp  ∆U kT  (4.25) where ν eff is a pre-exponential factor; ∆U is the activation energy and kT has its usual meaning. From Eqs. (4.22), (4.23) and (4.24) we obtain for one degree of freedom (z =0) U = µb 3 2π ln αe α r 1 b (4.26) where α = b/r 0 . Taking into account the work of the external stress we obtain ∆U = U − σb 2 λ (4.27) The activation energy is essentially less if there are n ≥ 2 slipping dislo- cations in the same slip plane. One can show that in this case the factor n appears before the second term on the right-hand side of Eq. (4.27). In Fig. 4.17 the calculated curves of the influence of temperature and stress on the Γ value are shown for two metals. The probability of dislocation emis- sion from the sub-boundary is strongly affected by the temperature and the number of dislocations in the slip plane. Fig. 4.17 The number of thermal activations per unit time as a function of stress and temperature. Solid lines, one dislocation in slip plane, dashed lines, two dislocations in slip plane. (a) Nickel, (b) vanadium. r 0 =2b and r 1 = λ is assumed. 62 4 Physical Mechanism of Strain at High Temperatures The condition of the stability of the boundary during strain is 1 Γ >τ creep . Here Γ −1 is the time interval before the emission begins and τ creep is the time interval during which the creep deformation occurs; e.g. for a creep time of 10 5 s then Γ < 10 −5 s. The results in Fig. 4.17 show the temperature and stress intervals where the sub-boundaries are observed. From Eqs. (4.26) and (4.27) we obtain the condition of the inactivated emis- sion of dislocations from the sub-boundaries: σ ≥ µb 2πnλ ln αe α λ b (4.28) Assuming λ =50nm, n =2, α =0.5 we obtain σ ≥ 2 ×10 −3 µ for nickel. When the external stress is higher than this value then the sub-boundaries are unstable and are destroyed. 4.9 Scope of Application of the Theory A well-read reader may ask: what is the distinction between this theory and the model published by Barrett and Nix [11]? This excellent article was the first to examine deeply the motion of jogged dislocations as a process which controls the strain rate. However, the authors conceived the jogs as being a result of thermal activation. The equations pro- posed by them take into account only a thermodynamic equilibrium number of jogs in the dislocations. In their opinion, the screw components therefore contain equidistant alternating jogs of opposite signs. They wrote: “The av- erage spacing between jogs, λ, has never been measured directly”, so they assumed a parameter λ which could not be measured. The quantitative eval- uation of the strain rate was out of the question at that time, of course. As a matter of fact, the adjacent jogs of opposite signs slip along the dis- location line easily and would simply annihilate each other. The equilibrium values of λ can affect neither the dislocation velocity nor the creep rate. According to our experimental results the sources of jogs of the same sign in mobile dislocations are the immobile sub-boundary dislocations and we be- lieve that the substructure formation plays a key role during high-temperature strain, being the process that affects the strain rate. The present theory is understood to be valid within certain limitations. When the temperature is relatively low, the dislocation climb is depressed 4.9 Scope of the Theory 63 and hence regular sub-boundaries cannot be formed. The lower limit to give a sufficient climb rate is about 0.40 or 0.45 T m . The low-temperature defor- mation is controlled by other processes, e.g. the overcoming of the Peierls stress in the crystal lattice. The stable sub-boundaries are of major significance in the process of high- temperature strain for pure metals and solid solutions. The upper stress limit of the sub-boundary stability depends upon the metal properties and temperature. The lower the shear modulus µ and the higher the tempera- ture, the lower the limit. An estimation, for instance, shows that in nickel at 0.6 T m sub-boundaries are destroyed by a stress of 2 × 10 −4 µ in 30h. The analysis shows that when the applied external stress is higher than about 2 × 10 −3 µ inactivated emission of dislocations from sub-boundaries occurs and the sub-boundaries break up. The upper limit of temperature is (0.70– 0.75)T m . Diffusion creep takes place (the mechanism of Herring-Nabarro) at higher temperatures and relatively lower stresses. It is necessary to empha- size that an adequate understanding of dislocation processes in these ranges of temperature and stress is of great practical importance. Most heat-resistant metals, steels and alloys operate at temperatures between 0.40 and 0.75 T m . The area of temperature and stress where the proposed mechanism of high- temperature deformation takes place, is shown in normalized coordinates in Fig. 4.18. Construction diagrams (maps) of this type were proposed by Ashby, e.g. in Ref. [26]. Fig. 4.18 The deformation map of nickel. The shaded area represents the interval of temperature and stress where the physical mechanism under consideration takes place. The numbers on the curves denote strain rates in s − 1 . 64 4 Physical Mechanism of Strain at High Temperatures Fig. 4.19 The same as in Fig. 4.18 but for iron. It isknownthatin iron allotropictransformationoccurs at 0.65 T m (Fig. 4.19). The mutual arrangement of the deformation areas is in other respects similar to the previous one, however, there is a quantitative difference. The strain rate of iron, which has a body-centered crystal lattice is considerably greater. For example, at 0.5 T m under a stress of 6 × 10 −4 µ strain rates for Ni and Fe are equal to 10 −7 and 10 −3 s −1 , respectively. 4.10 Summary The dislocation density increases at the beginning of plastic strain. In the pri- mary stage of deformation a part of the generated dislocations form discrete distributions. Dislocations penetrate low-angle sub-boundaries. The interac- tion of dislocations having the same sign is facilitated by high-temperature and applied stress. These conditions make it easier for edge components of dislocations to climb. The immediate cause of the formation of dislocation walls is the interaction between dislocations of the same sign resulting in a decrease in the internal energy of the system. At the end of the substructure formation the dislocation arrangements are ordered. Then a steady-state stage of strain begins. During this stage a dislocation emission from sub-boundaries takes place. In metals with small stacking fault energy the climb of the dislocation edge components is hindered. The ordered dislocation sub-boundaries require a higher temperature in order to form. The low-angle sub-boundaries are built up of parallel equidistant dislo- cations that contain screw components. The sub-boundary dislocations are 4.10 Summary 65 sources, emitters for mobile dislocations, which contribute to the specimen strain. Emission of mobile dislocations from sub-boundaries leads to the for- mation of the equidistant one-signed jogs. The distance between jogs at mo- bile dislocations is close to the distance between the immobile sub-boundary dislocations. The jogged dislocations can slip when there is a steady diffusion flux of generated vacancies from jogs. 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 it. The relay-like motion of the vacancy-emitted jogged dislocations from one sub- boundary to another one is the distinguishing feature of the high-temperature strain of single-phase metals and solid solutions. The velocity ofdislocationsdepends exponentially on the appliedstress.The exponent contains the sum of the activation energies of vacancy generation and vacancy migration. Processes of dislocation multiplication, annihilation, sub-boundary emis- sion and immobilization occur in metals during the high-temperature strain. The balance equation, which characterizes the change in the mobile disloca- tion density, has been derived. Three groups of physical parameters are needed for estimation of the steady-state strain rate ˙ε: • External parameters: temperature T and stress σ. • Diffusion parameters: the energy of the vacancy generation E v and the energy of the vacancy diffusion U v . • Structural parameters: the average subgrain size ¯ D and the mean distance between dislocations in sub-boundaries ¯ λ. The computed values of ˙ε fit the experimental data satisfactorily at certain temperature and stress conditions. The rate of the stationary creep correlates with the amplitude of atomic vibrations at high temperatures. The developed theory is valid within certain limits of the temperature and applied stress. When the temperature is relatively low, the dislocation climb is depressed and hence the regular sub-boundaries cannot be formed. The lower limit for sufficient climb rate is about 0.40 T m or 0.45 T m , the upper limit is 0.70 T m or 0.75 T m . The upper stress limit of the sub-boundary stability depends upon the metal properties and temperature. The lower the shear modulus µ and the higher the temperature, the lower the limit. An inactivated emission of dislocations from sub-boundaries occurs when the applied external stress is higher than about 2 × 10 −3 µ, the sub-boundaries then break up. [...]... 1.81 (5. 10) dt ρ dt 5. 2.7 System of Differential Equations Equations (5. 2), (5. 3), (5. 5), (5. 7), (5. 8), (5. 10) constitute half the equations of the required system The second half of the system is obtained by replacing subscript 1 with subscript 2 in these equations We have obtained a system of 12 ordinary differential equations This is the system to be used for computer simulation The general form of. .. 3.4 and Eq (3.2)] Let us consider equations, which relate to each of the enumerated parameters 5. 2 Equations 5. 2.1 Strain Rate Combining Eqs (4.10), (4.11) and (4.18) we arrive at bρ1 V1 dγ1 = 3 dt 0.5D1 ρ1 .5 + 1 1 (5. 2) for the first system of planes One can see from this equation that the strain rate depends on all structural parameters via the dislocation density and the dislocation velocity 5. 2.2... physical mechanism under consideration: High Temperature Strain of Metals and Alloys, Valim Levitin (Author) Copyright c 2006 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim ISBN: 3 -52 7-313389-9 68 5 Simulation of the Parameters Evolution • the relative shear strain γ • the total dislocation density ρ • the slip velocity of dislocations in their slip planes V • the climb velocity of dislocations to the parallel... permeability of the wall If χ equals to zero the dislocation cannot enter the wall The number of onesigned dislocations in the band of width λ is equal to ρλD/2, hence the rate of change of λ inside the first system of planes is dλ1 1 + χ1 = −ρ1 λ1 D1 Q1 χ1 dt 1 − χ1 (5. 8) The author [27] has obtained a semi-empirical formula: χ1 = 0. 45 1 σ1 − 0.23 (5. 9) where λ is measured in meters and σ in megapascals Fig 5. 1... with vacancy-producing jogs, Dv is the coefficient of vacancy diffusion, c0 is the equilibrium concentration of vacancies, σ is stress, and the values k and T have their usual meaning After differentiating Eq (5. 4) and substituting the value of λ2 for (z0 )1 we arrive at 4π(Dv )b4 c0 σ1 dV1 dλ2 σb2 λ2 = exp (5. 5) dt kT kT dt One can see from Eq (5. 5) that the distance between immobile dislocations... one The velocity of slip decreases after loading since λ2 decreases, dλ2 /dt < 0 5. 2.4 The Dislocation Climb Velocity The velocity of climb of the edge dislocation components is given by [21] Q= Ev + Uv 11νb2 exp − λ kT exp σb2 λ kT (5. 6) Taking the derivative of Eq (5. 6) one obtains dQ1 11νb2 =− 2 dt λ2 σ 1 b2 λ 2 Ev + Uv − 1 exp − kT kT exp σ1 b2 λ2 dλ2 (5. 7) kT dt 69 70 5 Simulation of the Parameters... (4.19) Hence dρ1 V1 2 = δρ1 V1 + δs ρs1 V1 − 0.5D1 V1 ρ2 .5 − ρ1 1 dt D1 (5. 3) 5. 2 Equations It has been noted that the first term of the right-hand side describes the multiplication rate of mobile dislocations The second term is related to the emission of mobile dislocations from sub-boundaries The third term corresponds to annihilation of the mobile dislocations of opposite sign Finally, the fourth term...67 5 Simulation of the Evolution of Parameters during Deformation 5. 1 Parameters of the Physical Model In the previous chapter the physical model of the high- temperature dislocation deformation in metals was worked out Recall that the model and the equations deal exclusively with “natural” parameters, which have well-defined... evaluation of the right-hand side of equations) and then using the information obtained to match a Taylor series expansion up to some higher order Program MATLAB enables one to solve the system and achieve a specified precision of the fourth order We use the so-called ODE 45 RungeKutta method with a variable step size The step size is continually adjusted to achieve a specified precision 5. 3 Results of Simulation:... The processes progress in time The approach is to make a system of ordinary differential equations and to solve the system numerically The results of the simulation are used to validate the correctness of the model as well as to study further the processes under consideration The forming of subgrains occurs during the high- temperature strain This phenomenon was described in Chapter 4 The model under . temperature and stress is of great practical importance. Most heat-resistant metals, steels and alloys operate at temperatures between 0.40 and 0. 75 T m . The area of temperature and stress where. up. High Temperature Strain of Metals and Alloys, Valim Levitin (Author) Copyright c  2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3 -52 7-313389-9 67 5 Simulation of the Evolution of. − 3.17 ρ 1.81 dρ 1 dt (5. 10) 5. 2.7 System of Differential Equations Equations (5. 2), (5. 3), (5. 5), (5. 7), (5. 8), (5. 10) constitute half the equations of the required system. The second half of the system is obtained