Philosophical Magazine, Vol. 85, No. 18, 21 June 2005, 1967–1987
Development of dislocation-based unified material model for simulating microstructure evolution in multipass hot rolling J. LIN*y, Y. LIUy, D. C. J. FARRUGIAz and M. ZHOUz ySchool of Manufacturing and Mechanical Engineering, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK zSwinden Technology Centre, Corus UK Ltd, Moorgate, Rotherham, S60 3AR, UK (Received 21 May 2004 and accepted in revised form 5 August 2004) Dynamic recrystallization and recovery are two competing processes. Both may continue after hot deformation, such as during passes in multipass hot rolling processes, reducing dislocation density of materials and allowing larger plastic deformation to be achieved. The main objective of this research is to develop a set of mechanism-based unified viscoplastic constitutive equations which model the evolution of dislocation density, recrystallization and grain size during and after hot plastic deformation. This set of constitutive equations are determined for a C-Mn steel using an evolutionary programming (EP) optimization technique and implemented into the commercial finite element (FE) solver ABAQUS for process simulations. Numerical procedures to simulate multipass rolling are developed. FE analysis is carried out to simulate the evolution of grain size, dynamic/static recrystallization and recovery, and to rationalize their effects on the viscoplastic flow of the material in a two-pass hot rolling process.
1. Introduction During industrial thermo-mechanical processing operations such as multipass hot rolling, deformation takes place in a series of passes separated by intervals of time. The dynamic microstructural changes, which occur during deformation, are dependent on strain rates, temperature and initial microstructure of materials [1]. These determine the flow stress, and hence the working forces, and also control the stored energy present at the end of deformation. The energy drives the static microstructural changes of recovery and recrystallization thereby influencing the kinetics of these thermally activated processes [2]. Grain growth may follow recrystallization and static microstructure changes between passes determine the initial microstructure for entry to the next pass. The microstructural evolution in multipass rolling processes has an effect on the final mechanical properties of processed materials. Quality assurance demands
*Corresponding author. Email:
[email protected] Philosophical Magazine ISSN 1478–6435 print/ISSN 1478–6443 online # 2005 Taylor & Francis Group Ltd http://www.tandf.co.uk/journals DOI: 10.1080/14786430412331305285
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require materials to be processed with known microstructures. Thus, the understanding and modelling of microstructural evolution during hot rolling and its effect on viscoplastic flow of materials provide fundamental information to optimize processing conditions of various carbon steels and to improve final product properties. Microstructural evolution under hot working conditions generally involves strain hardening, recovery, recrystallization and grain evolution, all of which are highly temperature and deformation rate-dependent [3, 4]. The hardening due to accumulation of dislocations, and the softening due to recrystallization and recovery are two competing processes [5, 6]. The occurrence of a dynamic balance between hardening and softening leads to a steady state flow of materials [7]. Much research has been carried out to model dynamic recrystallization during hot rolling [8, 9]. The condition of dynamic recrystallization taking place is normally controlled by a critical strain, "c [10], which is used to model the flow stress behaviour. In this case, it is difficult to take into account the dynamic and static recovery. There has also been much research carried out on static recrystallization after hot working [11–13] and the relationships between recovery, recrystallization and grain growth [14]. The mechanisms and driving forces for microstructure evolution are fairly well understood. Many types of equations have been developed to model the individual processes and stress–strain responses. The current research concentrates on the modelling of viscoplastic flow of materials and microstructure evolution during hot forming conditions. There are no constitutive equations available to model the effects of dislocation accumulation on the microstructure evolution, and interactive relationships between dislocations, microstructure evolution and viscoplastic deformation of materials. Thus unified viscoplastic constitutive equations are required to predict those interrelationships in hot metal forming. The research reported in this paper concentrates on the identification of mechanisms and driving forces for recovery, recrystallization, grain size evolution and their effects on the viscoplastic flow of metallic materials. Based on identified mechanisms, a set of unified viscoplastic constitutive equations is proposed. This equation set is determined from experimental data of a typical C–Mn steel at a temperature of 1373 K. Flow stress, dynamic/static recrystallization, recovery and grain size evolution are modelled. The developed unified material model is implemented into the finite element (FE) solver ABAQUS/Standard and multi-step FE procedures are developed to simulate microstructure evolution in a multipass rolling process.
2. Microstructure evolution in high temperature deformation 2.1. Dislocation and grain boundary sliding The dislocation structure developed during plastic deformation constitutes a driving force for microstructural evolution, such as recrystallization and grain growth, during and after deformation at high temperature. Dislocations mainly concentrate on subgrain boundaries, and the average dislocation density is expressed as . Taking the strain hardening and the recovery of dislocations into account in the same way as is frequently used in connection with creep type processes but neglecting recrystallization, the dislocation density rate can be described as [12, 13]: _ ¼ "_p =ðbl Þ 2M2
ð1Þ
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where "_p is the true plastic strain rate, b Burger’s vector, l the dislocation mean free path, M the mobility of grain boundaries and the average energy per unit length of a dislocation. The first term in equation (1) describes the development of dislocation and the second represents the (static) recovery due to annealing at high temperature. Once the dislocation density is accumulated to a critical value c at high temperature, recrystallization may take place [11]. A dynamic recovery based dislocation density evolution equation is expressed as [15]: pffiffiffi d=d"p ¼ K1 K2 ð2Þ where the coefficients K1 and K2 characterize the processes of dislocation storage and concurrent dislocation annihilation by recovery, respectively. Particularly the coefficient K2 represents a thermally activated process of dynamic recovery by dislocation cross-slip (at low temperature) or dislocation climb (at high temperature). Adding another term in equation (2), static recovery can then be taken into account. Thus, the evolution equation for the dislocation density is given as [15]: pffiffiffi _ ¼ K1 K2 "_p r ð3Þ where
pffiffiffi r ¼ r0 exp½U0 =ðKB TÞ sinh =ðKB TÞ
ð4Þ
and KB denotes the Boltzmann constant, U0 the activation energy, and r0 are constants, T is temperature. This constitutive equation enables the dislocation density evolution to be well modelled for a microstructure before dynamic/static recrystallization takes place. According to high temperature deformation mechanisms, grain size and strain rate play very important roles in dislocation density evolution. The dominant deformation mechanism for fine equiaxial grain structures under low strain rate deformation is grain boundary sliding. At a grain boundary, which defines the boundary separating two neighbouring grains having different crystallographic orientations, not all the atoms are properly bonded, giving rise to the increased boundary energy. In addition, impurity atoms often preferentially segregate along boundaries because of the higher energy state. It is therefore feasible for deformation to occur through grain boundary sliding. In this case, the dislocation density could not increase proportionally with plastic strain. For example, the dislocation density does not reach the critical value c for many cases of superplastic deformation, thus dynamic recrystallization does not occur. However, for large grain structures (for example, 200 mm) under high strain rates (for example, 10 s1, much greater than that in metal creep), there is little time to let grain boundary diffusion take place and the large grains are difficult to rotate during the deformation. The dominant deformation mechanism under this condition is due to slip of dislocations. Thus, dislocation density increases quickly with plastic strain, which may result in dynamic recrystallization, depending on the processing conditions. The recrystallization process refines grains, which may facilitate grain boundary sliding taking place. 2.2. Recrystallization When crystalline materials are deformed at high temperatures, the accumulated dislocations are destroyed by two separate processes. The one discussed above
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is dynamic recovery, which leads to the annihilation of pairs of dislocations, as well as to the formation of sub-grains. In high stacking fault energy materials (such as in Al alloys), such recovery processes completely balance the effects of strain- and work-hardening, leading to the established steady state flow stress. In materials of moderate to low stacking fault energy, dislocation density increases to appreciably high levels; eventually the local differences in density are high enough to permit the nucleation of recrystallization during deformation. Such dynamic recrystallization leads to the elimination of a large number of dislocations, and creates dislocationfree grains. The critical value of dislocations for recrystallization is expressed as [12, 13]: c ¼ 4surf =ðd Þ
ð5Þ
where surf is the grain boundary energy per unit area, and d* is the diameter of the recrystallized nucleus. During recrystallization, the fraction of the grain boundary area is mobile. This fraction varies slightly during the process. It has been shown [12, 13] that for static recrystallization this fraction increases with time in the beginning and decreases towards the end of the recrystallization. The velocity v() of a moving grain boundary is approximately given by the following expression: vðÞ ¼ M
ð6Þ
During dynamic recrystallization, it is likely that the time variation is smaller since some boundaries may be mobile over several cycles of recrystallization. Although many models have been proposed for modelling grain boundary movement and the growth of recrystallized grains [8, 9, 13, 14], the modelling of recrystallized volume fraction, S, normally uses empirical expressions, such as [8]: S ¼ 1 exp½ðK=D0 Þtn
ð7Þ
where K and n are constants, D0 is the initial grain size.
2.3. Grain size evolution During recrystallization, new grains are nucleated and the total number increases. Consequently, the average grain diameter, d, decreases. At the same time, normal grain growth takes place working in the opposite direction. Taking only recrystallization into account, the evolution of the average grain diameter, d, can be written as [12]: d_ ¼ d ðdf =dtÞ ln N
ð8Þ
where N is the number of new grains per old grain after one cycle of recrystallization, which may be grain size dependent; f is the number of recrystallization cycles, which can be a non-integer number. Static and dynamic grain growth works independently of each other, which is especially important for low strain rate viscoplastic deformation, such as superplastic deformation. The grain growth rate can be expressed as: d_ ¼ Msurf d r0 þ "_p d r1
ð9Þ
where r0, r1 and are constants. The first term of the equation represents the static grain growth, which is directly related to grain boundary mobility, M, and grain
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boundary energy density, surf. The second term describes plastic strain induced grain growth, which has been discussed by Cheong et al. [16]. Compared with grain refinement due to recrystallization, dynamic grain growth plays less important roles during the deformation process due to dynamic recrystallization, which takes place at the early stage of deformation in hot metal forming conditions and reduces the average grain size. However, during passes of hot forming processes, static grain growth becomes more important after recrystallization. 3. Development of unified viscoplastic constitutive equations High temperature deformation processes taking place predominantly by diffusion generally occur at a low stress level for small grain size structures over comparatively long periods. Under these conditions, a linear relation exists between the strain rate and applied stress. At a high stress level, coupled with high strain rates, dislocation controlled deformation predominates, for which the stress–strain rate relation becomes non-linear. Compared with power-laws, sinh-law equations are suitable for a wide range of strain rates and stress levels [17]. Thus the following viscoplastic constitutive equation is used, "_p ¼ A1 sinh½A2 ð R kÞd 4
ð10Þ
where R is an internal variable representing isotropic hardening of materials, which is directly related to dislocation density and will be discussed later. k is the yield stress, A1, A2 and 4 are temperature dependent material constants. The average grain size d is a function of time, plastic strain and recrystallization, which will be detailed later. The parameter 4 characterizes the effects of grain size on viscoplastic flow of materials. 3.1. Modelling of dislocation density Models describing dislocation evolution under plastic/viscoplastic deformation are reviewed in section 2.1. Equations (1)–(3) enable dislocation accumulation and both dynamic and static recovery to be modelled. The effect of recrystallization on dislocation density evolution is not considered. In this work, a normalized dislocation density concept is introduced by defining ¼ 1 i =, where i is the initial dislocation density and the dislocation density in deformed materials. The normalized dislocation density varies from 0 (the initial state) to about 1 (the saturated state of a dislocation network). In consideration of high temperature deformation mechanisms, recrystallization, static and dynamic recovery, a constitutive equation for normalized dislocation density evolution is proposed: _ ¼ ðd=d0 Þd ð1 Þ"_p c1 c2 ½c3 =ð1 S ÞS_ ð11Þ where d0, d, c1, c2 and c3 are material constants. The first term in equation (11) represents the development of dislocation density due to plastic strain and the dynamic recovery of the dislocation density. This term is similar to equation (2) apart from that the effect of grain size on normalized dislocation evolution is modelled using (d/d0)d in equation (11). If a material has a large average grain size, dislocation density increases quickly under viscoplastic deformation since less grain rotation and grain boundary sliding take place. The effect of the deformation mechanism of grain rotation and grain boundary sliding on the accumulation rate
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of dislocations is modelled by (d/d0)d. The dynamic recovery term enables the normalized dislocation density to be limited to the saturated state of a dislocation network being 1.0. The second term in equation (11) models the effect of static recovery on the evolution of dislocation density. This expression is similar to the second term in equation (1). For simplicity, the constant c1 is introduced to represent the combined effects of M (the mobility of grain boundary) and (the average energy per unit length of a dislocation). The constant, c2, replacing the power 2 in equation (1), provides higher flexibility of the equation, so that it is suitable for a wide range of materials and deformation conditions. The third term expresses the effect of recrystallization on the evolution of dislocation density. 3.2. Modelling of dynamic/static recrystallization As discussed before, recrystallization is directly related to dislocation density. When the dislocation density reaches a critical value c under a high temperature, giving sufficient time, recrystallization takes place. Although the critical value of dislocations for recrystallization, equation (5), has been recognized as an important parameter for modelling the start of recrystallization, this has not been used in constitutive modelling [see equation (7)]. In addition, a rate equation is required to represent the evolutionary nature of recrystallization. To improve this, the following equation is proposed to describe the evolution of recrystallized volume fraction S: S_ ¼ Q0 ½x c ð1 S Þð1 S ÞNq
ð12Þ
where Q0 and Nq are constants; c is the critical value of normalized dislocation density, below which recrystallization would not take place. It has been experimentally evident that there is a need of an incubation time for onset of recrystallization and the incubation time varies with the change in values of dislocation density while the dislocation density must exceed the critical value c [11]. Parameter x describes this phenomenon and is known as incubation fraction. The incubation fraction for the onset of recrystallization is given as: x_ ¼ A0 ð1 xÞ
ð13Þ
where A0 is a materiel constant. The recrystallized volume fraction S varies from 0 to 1 and its variation is cyclic, depending on the evolution of dislocation density. 3.3. Modelling of grain size evolution In consideration of static grain growth, plastic strain induced grain growth and the grain refinement due to recrystallization, a new evolution equation of average grain size is proposed in the form: d_ ¼ 0 d 0 þ 1 "_p d 1 a2 S_ 3 d 2 ð14Þ where 0, 0, 1, 1, 2, 2 and 3 are material constants. The first term is important to model grain growth during passes in multipass forming processes following a recrystallization cycle. The constant 0 is introduced to represent the combined effect of the parameters M (grain boundary mobility) and surf (grain boundary energy density) in equation (9). Thus the first two terms in equation (14) are almost identical to equation (9). In many cases of superplastic forming processes, recrystallization does not take place, thus the last term can be ignored. Compared with static grain
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growth between forming passes and grain refinement due to recrystallization, the dynamic grain growth, the second term in equation (14), is less important in multipass metal forming processes.
3.4. Formulation of uniaxial unified viscoplastic constitutive equations Constitutive equations for viscoplasticity have been developed for many metal materials [18, 19]. The equations enable a wide range of time dependent phenomena to be modelled, such as strain hardening, stress relaxation and ratchetting [20], and in addition enable important time dependent effects, such as strain rates, recovery and creep to be modelled. This work intends to develop a set of unified viscoplastic constitutive equations to model the evolution of recrystallization, dislocation density, hardening and grain size, to rationalize their inter-relationships and effects on viscoplastic flow of materials. The mechanism-based unified viscoplastic equations for hot metal forming may take the form: "_p ¼ A1 sinh½A2 ð R kÞd 4 S_ ¼ Q0 ½x c ð1 S Þð1 S ÞNq x_ ¼ A0 ð1 xÞ _ ¼ ðd=d0 Þd ð1 Þ"_p c1 c2 ½c3 =ð1 SÞS_
ð15Þ
R_ ¼ B_ d_ ¼ 0 d 0 2 S_ 3 d 2 ¼ E "T "p where B is material constant, E Young’s modulus (100 GPa for the C–Mn steel at 1373 K). The isotropic hardening parameter, R, is directly related to the dislocation density, which varies with plastic strain and recrystallization. The effects of recrystallization on flow stresses are modelled by the isotropic hardening and grain size variables.
3.5. Multiaxial large deformation model for finite element (FE) analysis Multiaxial constitutive equations for high temperature viscoplastic materials are obtained by assuming von-Mises behaviour and by defining an energy dissipation rate potential. ¼ ðA1 =A2 Þ cosh½A2 ðe kÞ
ð16Þ
1/2
where e ¼ (3SijSij/2) is the effective stress and Sij ¼ ij ij ij/3 the stress deviators. Assuming normality and the associated flow rule, the multiaxial relationship is given as: d"pij =dt ¼ _ o =oij ¼ 3A1 Sij =ð2e Þ sinh½A2 ðe kÞ ð17Þ On the reintroduction of the hardening and grain size evolution variables, the effective plastic strain rate "_pe can be written as: "_pe ¼ A1 ½sinh A2 ðe R kÞ=d 4
ð18Þ
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and then the set of multiaxial viscoplastic constitutive equations, implemented within a large strain formulation, may be written as: Dpij ¼ 3Sij =ð2e Þ"_pe S_ ¼ Q0 ½x c ð1 S Þ ð1 S ÞNq _ ¼ ðd=d0 Þd ð1 Þ"_pe c1 c2 ðc3 =ð1 SÞÞS_ x_ ¼ A0 ð1 xÞ
ð19Þ
R_ ¼ B_ d_ ¼ 0 d 0 2 S_ 3 d 2 ^ ij ¼ GDeij þ 2Dekk where Deij is the rate of elastic deformation, Deij ¼ DTij Dpij and DTij is the rate of total deformation, Dpij is the rate of plastic deformation, ^ij is the Jaumann rate of Cauchy stress, G and are the Lame´ elasticity constants. These multiaxial constitutive equations have been implemented into the large strain finite element solver ABAQUS through the user defined subroutine CREEP and used to simulate a multipass isothermal hot rolling process. The constants within the set of viscoplastic constitutive equations are determined using an evolutionary programming (EP)-based optimization system [21] for a C–Mn steel [22, 23] (symbols in figure 1). The formulation of the fitness functions for the optimization is detailed below. 4. Determination of the constitutive equations 4.1. Formulation of objective functions Optimization techniques for obtaining the material constants arising in the constitutive equations are based on minimizing the sum of the squares of the errors between the experimental and computed data. For the set of unified viscoplastic constitutive equations, three sub-objective functions are defined in terms of the square of the difference of experimental and computed data for stress–strain, recrystallization– strain and average grain size evolution–strain via f1 ðxÞ ¼ f2 ðxÞ ¼ f3 ðxÞ ¼
mj n1 X X j¼1 i¼1 ml n2 X X
2 e ðic Þj ðie Þj = Werr ðmax Þj ,
ð20Þ
2 ðSkc Þl ðSke Þl = Werr ðSmax eÞl and
ð21Þ
l¼1 k¼1 mp h n3 X X
i c e e dq p dq p = Werr ðdmax Þp 2 ,
ð22Þ
p¼1 q¼1
where f1(x), f2(x) and f3(x) are residuals for stress, recrystallization and average grain size, respectively. x(x ¼ [x1, x2, . . . , xs]) represents the material constants and S is the number of constants to be determined. ðic Þj and ðie Þj are computational and experie mental stresses for the same strain level i and strain rate j, ðmax Þj is the maximum experimental stress at strain rate j, Werr is the weighting factor for all the differences between experimental and computational data, mj is the number of stress–strain data
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Figure 1. Comparison of experimental data (symbols) with computed (solid curves) and predicted (dash curves) results for different strain rates for the variation of (a) stress, (b) recrystallized fraction, (c) grain size and (d) predicted normalised dislocation density with equivalent true strain, respectively.
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for the strain rate j, n1 is the number of strain rates considered for stress–strain curves. Similarly, ðSkc Þl and ðSke Þl are computational and experimental recrystallized e volume fraction for the same strain level k and strain rate l; ðSmax Þl is the maximum experimental recrystallized fraction at strain rate l, ml is the number of experimental recrystallized volume fraction–strain data for the strain rate l; n2 is the number of recrystallization–strain curves considered. ðdqc Þp and ðdqe Þp are computational and experimental average grain sizes for the same strain level q and strain rate p; e ðdmax Þp is the maximum experimental grain size at strain rate p; mp is the number of experimental average grain size–strain data for the strain rate p; n3 is the number of strain rates considered for grain size evolution. Three groups of experimental data, i.e. the relationships of effective stress, recrystallization and grain size against strain for different strain rates, are involved in the optimization. As the three groups of data have different units, it is difficult to choose weighting factors so that those three objective functions could play the same important roles in the optimization process. By referencing the objective functions proposed by Lin and Yang [21] and Li et al. [24], the formulated objective functions, equations (20), (21) and (22), are normalized by dividing the maximum values e e e of individual experimental curves, ðmax Þj , ðSmax Þl and ðdmax Þp . Thus the objective functions become dimensionless and can be simply added together to form a single global objective function, which is much easier for the optimization. To increase the sensitivity of the objective functions Werr ¼ 0.1 is selected, which magnifies the errors of the fitting 100 times for each pair of data points assessed. The calculated stress, ðic Þj , recrystallized volume fraction, ðSkc Þl , and grain size, c ðdq Þp , are not available directly and have to be determined from the constitutive equations (15) by means of a numerical integration method. The objective functions, f1(x), f2(x) and f3(x), require the global minimization subject to determining the value of each constant within a region defined by the lower and upper bounds, that is, xi ¼ {ai, bi} for i ¼ 1, 2, . . . , s. The global objective function is given as: f ðxÞ ¼
3 X
fh ðxÞ
ð23Þ
h¼1
The determination of material constants within the unified viscoplastic constitutive equations is to minimize the above global objective function. 4.2. Determination of material constants EP-based optimization techniques are developed and programmed for determining the material constants within the constitutive equations. The EP method is detailed elsewhere by Li et al. [24] and Lin et al. [25]. The constitutive equation set, equation (15), together with the formulated objective functions, equations (20)–(23), are implemented into the optimization package through a user defined subroutine. The experimental data reported by Medina and Hernandez [22, 23] used here for the optimization is a C–Mn steel with an average initial grain size of 189 mm. The determined material constants are listed in table 1 and the experimental data and computational results are shown in figure 1. From the objective functions, equations (20), (21) and (22), it can be seen that the three groups of experimental data, effective stress, recrystallization and grain size against strain for different strain rates, are involved in the optimization. The hardening variation of the material during viscoplastic deformation is modelled through the evolution of normalized dislocation
Dislocation-based model for microstructure evolution in hot rolling Table 1. A1 (s1) 1.81 106 c1 (-) 16.00 B (MPa) 75.59
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Determined constants for the set of unified viscoplastic constitutive equations. A2 (MPa1) 3.14 101
4 (-) 1.00
Q0(-) 30.00
c (-) 1.84 101
Nq(-) 1.02
c2 (-) 1.43
c3 (-) 8.00 102
d0 (mm) 36.38
d (-) 1.02
A0 (-) 40.96
0 (mm) 1.44
0 (-) 3.07
2 (mm) 78.68
2 (-) 1.20 101
3 (-) 1.06
density. Thus the material constants within the equations _ and R_ in equation set (15) are optimized by achieving the best fit to the experimental stress–strain curves for different strain rates (figure 1a). There is no experimental dislocation density data available in the optimization, since this data is very difficult to obtain in hot mechanical tests. The dash curves in figure 1d show the predicted variation of normalized dislocation density with equivalent true strain. In this paper, the computed curves refer to those, which have best fitting to the corresponding experimental data in the optimization process, and the others are known as predicted results using the equations and the determined constants. This optimization uses fast evolution programming (FEP) algorithm [24] and the following EP parameters are selected in this work: population size 500, tournament size 250, the number of generations 5000 and the minimization function. The experimental data [22, 23], symbols in figure 1, for equivalent stress, recrystallization and grain size evolution for two strain rates "_ ¼ 0:544 and 5.224 s1 are used for the optimization. The corresponding solid curves, which are plotted using the constitutive equations with the optimized constants listed in table 1, are able to approximate the experimental data. Close agreements are obtained for all the cases. In addition, two other strain rates: "_ ¼ 1:451 and 3.628 s1 are used to predict the flow stress, recrystallization and grain size evolution, which are denoted by the dash curves in figure 1. Two experimental peak stress data are available for the two strain rates, respectively. This indicates that the constitutive equation set enables the mechanical and physical behaviour of the material to be well modelled.
5. Microstructure evolution during hot deformation Computations are carried out for uniaxial loading cases with strain rates of 0.544, 1.451, 3.628, and 5.224 s1, and an initial grain size of 189 mm (typical of a reheated grain size at furnace discharge) at a temperature of 1373 K. The predicted/computed results for flow stress, volume fraction of recrystallization, grain size and normalized dislocation density are shown in figure 1. It can be seen that, for a low strain rate deformation, dynamic recrystallization takes place at a low plastic strain. In physics, it indicates that the material exposes a longer time under high temperature deformation conditions and the combined thermo-mechanical effects encourage the recrystallization taking place at a low value of dislocation density (figure 1d). Mathematically, the introduction of the onset variable, x, enables this thermomechanical behaviour to be modelled properly. The low strain rate deformation results in the recrystallization completing at a low strain.
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Grain growth is not significant during high strain rate hot deformation as the time span is short and the strain, before recrystallization taking place, is relatively low. However, once recrystallization takes place, new grains are nucleated and this results in the average grain size decreasing quickly. The flow stress increases quickly with the increment of strains. In addition to the strain rate effects, the hardening of the material is directly related to the increment of the dislocation density, as discussed by Sandstrom and Lagneborg [12, 13], where it is argued that: pffiffiffi ¼ b ð24Þ where is a constant of about 0.5–1.0 and the shear modulus. For a high strain rate deformation, dislocations are accumulated more quickly. Since little time is given to allow recovery taking place (figure 1d) this contributes to the high flow stress shown in figure 1a. Once the dislocation density reaches a critical level for a deformation rate, recrystallization takes place due to the thermal mechanical effect, as shown in (figure 1b). This reduces the dislocation density, as shown in figure 1d, and grain size, figure 1c, subsequently, decreases the flow stress, figure 1a. The steady state flow of the material is observed once the cycle of the dynamic recrystallization is completed. This also results in a balanced state of material hardening (dislocation density figure 1d), grain growth and refinement, figure 1c. The relationship between the flow stress and grain size for a steady state basically follows the empirical equation [15]: ¼ kd m
ð25Þ
where k is a parameter related to materials, Burgers vector and shear modulus. This flow behaviour of the material can be observed in figure 1. During hot deformation, if the dislocation density reaches a certain level, dynamic recrystallization may take place provided that an incubation time for the onset of recrystallization is sufficient. This critical value of dislocation density, c , corresponds to a critical strain, "c, for the dynamic recrystallization, which varies with strain rates, initial grain size and temperature. As a certain volume fraction of recrystallization is reached, the flow stress drops due to the grain refinement and the reduction of dislocation density. Calculations are carried out using the determined constitutive equations with different initial grain sizes and the variation of critical strains and peak stresses with strain rates are determined and shown in figure 2. High values of critical strains for recrystallization (figure 2a) and lower peak stresses (figure 2b) are obtained for the material with smaller initial grain sizes. This is due to that the deformation mechanism of grain boundary sliding and grain rotation is more likely to be dominant for a material with smaller grain size. This results in the lower rate of accumulation of dislocations and thus a lower flow stress. If a low strain rate (e.g. 0.3 s1) is applied to the material with a given initial grain size, the accumulation rate of dislocation is low due to static recovery (annealing). Thus a high value of critical strain is required to enable the dislocation density to be accumulated to the critical value c (figure 2a). If the deformation rate is high (e.g. 10 s1), the critical strain is also high (figure 2a). This is due to an incubation time required for recrystallization to take place although the incubation time is shorter for higher dislocation density. However, for the material under a relatively low deformation rate with a small initial grain size, grain boundary sliding and grain rotation may be the dominant deformation mechanism. This can slow down the
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0.5
(a)
0.45
dini = 120 µm
Critical Strain
0.4 0.35 0.3 0.25 0.2
dini = 189 µm
0.15 dini = 280 µm
0.1 0.05 0 0.1
1
10
100
-1
Strain Rate (s ) 150
(b)
Peak Stress (MPa)
140 130
dini = 280 µm
120 110 100
dini = 120 µm
90
dini = 189 µm
80 70 60 0.1
1
10
100
-1
Strain Rate (s ) Figure 2. Variations of (a) critical strain and (b) peak stress with strain rate for different initial grain sizes, dini.
accumulation rate of dislocations. Thus for a material with a given initial grain size, an optimum strain rate exists for the lowest critical strain occurring. In addition, a material may be able to demonstrate its own ‘superplastic phenomenon’ if a proper combination of initial grain size and deformation rate at a given high temperature is applied. Furthermore, an investigation illustrates a physical insight to the effect of initial grain size on thermo-mechanical behaviour during deformation. The predicted stress–strain relationships and dislocation density evolution for three initial grain sizes at "_ ¼ 0:544 s1 are shown in figure 3. At early stages of deformation, both dislocation density and flow stress increase rapidly for all cases. With a finer initial
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(a)
Equivalent True Stress (MPa)
100
dini = 280 µm dini = 189 µm
80 60
dini = 80 µm
40 20 0 0
0.2
0.4
0.6
0.8
1
0.8
1
Equivalent True Stain
Dislocation Density
0.4
(b) dini = 280 µm
0.3
dini = 189 µm 0.2
0.1
0
dini = 80 µm
0
0.2
0.4
0.6
Equivalent True Strain Figure 3. Effect of initial grain size on (a) stress–strain relationships and (b) dislocation density evolution at "_ ¼ 0:544 s1 .
grain size, dini ¼ 80 mm, the increase in dislocation density and flow stress is low. This indicates that more strains are carried by grain boundary sliding and grain rotation for the material with finer grains and the accumulation rate of dislocations is relatively low. Meanwhile, the accumulation of dislocations can eventually be balanced by the annihilation of dislocations due to dynamic recovery. For the materials with coarser initial grain sizes, dini ¼ 180 and 280 mm, in contrast both dislocation density and flow stress are higher than that with a finer initial grain size at the early stage of deformation. This is due to the smaller contribution of grain boundary sliding and grain rotation to deformation. While dislocation density exceeds a critical value c
Dislocation-based model for microstructure evolution in hot rolling
.B
Figure 4.
1981
.
A
FE model for a two-pass rolling process (dimension in mm).
during deformation, dynamic recrystallization takes place. Then, dislocation density and flow stress fall quickly. The grain sizes reduce significantly due to the recrystallization, which enhances grain boundary sliding and grain rotation. This results in the lower dislocation density. The accumulation of dislocations due to the following deformation and annihilation of dislocations due to dynamic recovery dynamically balance each other. Therefore, both dislocation density and flow stress reach a steady state. 6. Numerical procedures for the simulation of multipass hot rolling Simulation of a hot rolling process is carried out at a constant temperature of 1100 C. The FE model for a two-pass hot rolling is shown in figure 4. The initial thickness of the stock decreases to 30 mm after the first pass and reaches its thickness of 20 mm after the second. The viscoplastic deformation of the material is governed by the multiaxial constitutive equation set (18), which is implemented into the FE solver ABAQUS/standard through the user defined subroutine CREEP. To use an implicit numerical integration method, the gradients, o"pe =o"pe and o"pe =oe , are calculated and implemented into the system for the step time control. Due to its symmetry, only a half of the rolling system is modelled. Five hundred and forty eight-noded quadrilateral plane strain elements are used to mesh the workpiece. Friction, as a surface interaction property, is related to the contact pair, rollers and the top and left-hand side surfaces of the workpiece, by specifying a sticking friction coefficient. The detailed FE procedures for the multipass rolling simulation are given below: Stage 1: The workpiece is moved in contact with roller 1. Stage 2: The workpiece is rolled with a rolling velocity of 6.58 rad.s. It takes 0.92 seconds to let the material pass entirely through roller 1. Stage 3: The interpass time applied to workpiece is 20 seconds at a constant temperature 1100 C. The recovery, recrystallization and grain size evolution continue to occur. Then forward it to roller 2. Stage 4: The workpiece is rolled with a rolling velocity of 3.74 rad/s. 7. Results and discussion The FE analysis is carried out using the FE model and numerical procedures described above for a two-pass rolling process. Figure 5 shows field plots of effective
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Figure 5. Predicted distributions of (a) effective stress, (b) normalized dislocation density, (c) recrystallization fraction and (d) grain size for the first pass.
Dislocation-based model for microstructure evolution in hot rolling
1983
stress, dislocation density, recrystallization and grain size for the first pass. The left end of the workpiece almost keeps its initial microstructure. Stress and strain levels change dramatically around the rolling region, after which the stress reduces to a very low level (residual stress) and keeps a constant pattern except the leading end, where stress states are more complicated due to the contact conditions between the roller and the material stock. The microstructural evolution exhibits a different behaviour. The normalized dislocation density increases (figure 5b) immediately from the original state once the material is fed into the rollers. The field pattern of the normalized dislocation density for the initial deformed part is very much similar to that of the effective stress (figure 5a), since the increment of dislocation density is directly related to the plastic strain rates as described in the equation. The dislocation density reaches its maximum value at almost the same time and location as the maximum effective stress occurs (figure 5a). Once the dislocation density reaches the critical value, dynamic recrystallization takes place. It can be seen clearly from the field plot, shown in figure 5c, that the recrystallization does not start immediately when the material enters the roller. There is an obvious delay due to the onset parameter control and the critical dislocation density accumulated from the deformation. The recrystallization (static) continues after the material passes the roller (figure 5c) due to the high dislocation density (0.4, figure 5b). This results in the recrystallized volume fraction of the material increasing from about 0.4 just after the material has passed the roller to about 0.8 over a very short period. The continuous recrystallization and static recovery cause the dislocation density reduction after rolling, which can be seen from figure 5b. The grain refinement only takes place when the dynamic recrystallization begins. The average grain size (figure 5d) reduces to about 100 mm from 189 mm just after the first pass due to dynamic recrystallization and continues to decrease to about 60 mm due to the following static recrystallization. Figure 6 shows field plots of the same parameters around the rolling area for the second pass with a lower rolling speed of 3.74 rad/s. It can be seen that after 20 s relaxation, the normalized dislocation density (figure 6b) reduces to about zero due to recrystallization and recovery. This reduces the hardening due to plastic deformation at the first pass and increases the formability of the material. The stress level (figure 6a) is lower at the second pass due to a low rolling velocity and smaller grain size. The average grain size (figure 6d) reduces to about 45 mm after the second pass due to the continuous recrystallization (figure 6c). Thus there is a lower stress level in the second pass partially due to the lower rolling speed and partially due to the further refinement of grains, which enables grain boundary sliding to take place more easily. Figure 7 shows the variation of effective stress, dislocation density, recrystallization and grain size at locations A and B, indicated in figure 4 for the two pass rolling process. Location A is passed the second roller, but B is not. The recrystallized fraction increases quickly after the first pass. The recrystallization stops when the dislocation density vanishes due to recovery. This results in the material being partially recrystallized. This recrystallization continues at the second pass until the material is fully recrystallized, which is the starting point for the second cycle of recrystallization. The developed mechanism based unified viscoplastic constitutive equations, implemented into an FE solver, enable the detailed microstructure evolution, such as recrystallization and grain size, of the material during and after hot rolling
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Figure 6. Predicted distributions of (a) effective stress, (b) normalised dislocation density, (c) recrystallization fraction and (d) grain size for the second pass.
1985
Dislocation-based model for microstructure evolution in hot rolling
Effective Stress (MPa)
150
A
Roller 1
B
A
Roller 2
100
B B
A
A
B 50
Dislocation Density
0 0.8 0.6
A A
0.4
B 0.2
B
Recrystallisation Fraction
0 0.8
A
A
0.6
B
B
0.4 0.2
Grain Size (µm)
0 190 160
A
130
B
100
A
B
70 40 0.1
1
Times (s)
10 21
21.5
22
22.5
23
23.5
Times (s)
Figure 7. Variation of (a) effective stress, (b) dislocation density, (c) recrystallized fraction and (d) grain size at locations A and B, indicated in figure 4, for the two-passes rolling process. The computation is carried out with the rolling speeds of 6.58 and 3.74 rad/s for the two rollers.
to be modelled. For example, according to the evolution of recrystallization, dislocation density recovery and grain growth between rolling passes, the interpass time can be determined. If the interpass time is too short, the recovery of dislocations may not complete. This would result in high rolling forces and low formability of the material in the subsequent passes. If the interpass time is too long, the excess static grain growth after recrystallization may result in poor mechanical properties of the material produced. Thus by the use of this equation set within an FE package,
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it is possible to control the microstructure of the material processed and to optimize hot rolling processes. 8. Conclusions According to the analysis of deformation mechanisms and driving forces, which lead to microstructure changes in hot deformation, mechanism-based unified viscoplastic constitutive equations are developed. The equation set is determined from the experimental data for a C–Mn steel using the developed EP-based optimization software. The dimensionless fitness functions formulated are suitable for the multiple objective optimization problems and enable the material constants within the equations to be accurately determined. The set of mechanism-based unified viscoplastic constitutive equations determined here describes the kinetics of grain size, dynamic/static recrystallization and dislocation density evolution and enables the prediction of microstructure evolution of the material during and after hot deformation. The microstructure evolution and the effects on viscoplastic flow of the material in multipass hot rolling can be predicted with the use of the developed material model and FE analysis procedures. Acknowledgements The authors at Birmingham gratefully acknowledge that Corus UK Ltd provides financial support to Mr Y. Liu. The constructive suggestion and advice from Professor J Beynon at the University of Sheffield are greatly acknowledged. Nomenclature "p, "_p ^ij e ij0 Deij , D_ eij DTij , D_ Tij Dpij , D_ pij "c c c R d dini x S surf
Stress in uniaxial case Plastic strain and rate in uniaxial case Jaumann rate of Cauchy stress von-Mises stress Stress deviators Elastic strains and rates Total strains and rates Plastic strains and rates Critical strain for recrystallization Dislocation density Critical dislocation density for recrystallization Normalized dislocation density Normalized critical dislocation density for recrystallization Isotropic hardening Average grain size – average diameter of grains Initial average grain size Incubation parameter for onset of recrystallization Volume fraction of recrystallization Energy dissipation rate potential Grain boundary energy per unit area
Dislocation-based model for microstructure evolution in hot rolling
M b G KB
1987
Average energy per unit length of a dislocation Mobility of grain boundary Burger vector Shear modulus Lame´ elasticity constant Boltzman constant
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