flow stress and microstructure models of alloys
TRANSCRIPT
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8nd International Conference on Physical and Numerical Simulation of Materials Processing, ICPNS’16 Seattle Marriott Waterfront, Seattle, Washington, USA, October 14-17, 2016
Flow Stress and Microstructure Models of Alloys Lars-Erik Lindgren1
1Luleå University of Technology, 97187 Luleå, Sweden
ABSTRACT It is necessary to have sufficient accurate description of the response of a material in order to obtain relevant results from manufacturing simulations. This is one of the larger challenges when simulating manufacturing processes. The response may be a complex function of loading conditions and the current microstructure of the material. This is particular true for the plastic properties of the material. Further complications arise when thermo-mechanical manufacturing processes trigger phase transformations. It is then necessary to combine flow stress and microstructure models. The paper presents an approach combining a mechanism-based flow stress model with microstructure models when needed.
Keywords: Plasticity, Microstructure, Steel, Aerospace alloys
1. INTRODUCTION Computer simulation of manufacturing processes poses several challenges (Lindgren, To appear; Lindgren, Lundbäck, Edberg, & Svoboda, 2013; Lindgren, Lundbäck, & Fisk, 2013) and the modeling has reached various levels of maturity depending on the manufacturing process and the scope of the model. There are two main issues when modeling the mechanical behavior that often require considerable efforts. They are the modeling of surface and bulk material responses. The plastic behavior of the bulk material is in focus in this paper. The flow stress depends on the loading conditions and microstructural features like dislocation cell formation, recrystallization, precipitate evolution as well as phase changes. The microstructural evolution depends on the thermo-mechanical history. The phase transformation models illustrated below are based on the assumption that the changes are thermal driven.
The paper presents an approach based on mechanism-based flow stress model combined with microstructure models when needed. The described models are applicable to large-scale simulations. The flow stress models are demonstrated for two stainless steels, (AISI 316 and AISI 420), one aluminum alloy (AA5083) as well as Ti-6Al-4V and Alloy 718. Phase change models for low alloy steels, Ti-6Al-4V and a precipitate model for Alloy 718 are also shown. References with detailed description of the microstructure models are given in the text. The use of mixture rules for computing macroscopic properties in presence of phase changes are
excluded due to space limitations, see (Lindgren, 2007, To appear) for details about this.
2. FLOW STRESS MODELS A mechanism based plasticity based model is summarized in section 2.1. It is often called the Estrin-Mecking model (Mecking & Estrin, 1980) and sometimes the Bergström model (Bergström, 1983) as he formulated the model set-up quite early (Bergström, 1969/70).
2.1. Mechanism based plasticity model The dislocation density based plasticity model accounts for plasticity due to dislocation motion. The rate-dependent flow stress, σ y , is in the current model assumed to be an additive contribution written as σ y =σG +σ HP +σ SR . (1)
The two first terms are long-range contributions and the last denotes short-range contributions due to different mechanisms (Frost & Ashby, 1982). The books (Caillard & Martin, 2003; Lindgren, To appear; Messerschmidt, 2010; Shetty, 2013) and the papers by Kocks et al. (1981; 1975; 2003) give extensive descriptions about the approach and the underlying concepts.
The long-range distortion of a lattice is such that thermal vibrations cannot assist a moving dislocation when passing this region. Thus the opposite holds for a short-range disturbance of the lattice. The long-range stress is sometimes called an athermal
2
contribution (Conrad, 1970). The first term in Eq. (1) requires tracking the evolution of the density of immobile dislocations, ρi , as shown below. This will be an internal state variable (ISV) of the described model. The density of mobile dislocations, ρm , need not be tracked in the current model. They are related to the effective plastic strain rate, !ε p , as
!ε p =!γ p
M=ρmvbM
(2)
according to the Orowan equation (Orowan, 1940). Thus the plastic strain rate together with the density of immobile dislocations is an important variable. The derivation of the Orowan equation is based on the relative motion of two slip planes when a dislocation moves. This gives a plastic shear rate that, when averaging over possible slip planes, can be related to the effective plastic strain rate by the Taylor factor M (Mecking, Kocks, & Hartig, 1996; Taylor, 1938). b is the magnitude of the Burgers vector and is the average velocity of the dislocations. The stress needed to sustain a certain plastic strain rate depends on the interaction of these moving dislocations with features of different sizes in the material (Arzt, 1998).
2.1.1 Stress, plasticity and creep
The basic assumption underlying Eq. (1) is that it is stress that exceeds the long-range stress field in the lattice, which causes plastic straining. The equation can be rewritten as σ SR =σ
y − σG +σ HP( ) . (3) The von Mises effective stress, σ , is equal to the flow stress, σ y , during plastic deformation. This leads to σ SR =σ − σG +σ HP( ) . (4)
Thus the applied stress, σ , must first exceed the sum of the long-range contributions described in section 2.1.2. This overstress, or excess stress, is the short-range stress, σ SR , described in section 2.1.3. It drives the mobile dislocations and is therefore related to the plastic strain rate in Eq. (2).
The model can be extended with creep strains due to diffusional flow, as in the case of the model for AA 5083 described in section 2.2. The creep due to diffusion of matter has no or a very low threshold stress in contrast to plastic strain due for dislocation motion. The creep strains are included by the additive decomposition of total effective plastic strain rate, !ε , into !ε = !ε e + !ε p + !ε c . (5)
The superscripts denote e=elastic strain, p=plastic strain due to dislocation motion and c=creep strain due to diffusional flow.
2.1.2 Long-range contributions
The interaction between a moving dislocation segment and an immobile dislocation segment is a long-range interaction when they are parallel. This is the physical basis for what is called for strain hardening in a metal and accounted for by σG in Eq. (1). This long-range interaction is written as (Arzt, 1998) σG =αMGb ρi , (6)
where α is a proportionality factor to be calibrated, and G is the temperature dependent shear modulus. This term requires a set of evolution equations for the immobile dislocation density. They consist of hardening and softening contributions !ρi = !ρi
+( ) − !ρi−( ) . (7)
Mobile dislocation may become immobile when meeting obstacles. The probability of this immobilization is proportional to the distance between the obstacles, their strength and the density of mobile dislocations. The latter is proportional to the effective plastic strain rate according to Eq. (2). Thus the immobilization due to various obstacles can be written as
!ρi+( ) =
MbΛ!ε p . (8)
Λ is the mean free path of a moving dislocation. The immobilization is taken as additive contributions from various obstacles; i.e. dislocation cells, grain boundaries, and precipitates. Then the equation for the mean free path becomes 1Λ=Ks
s+Kg
g+Kp
lp. (9)
The calibration factors Ks,g,p represent the strength
of the obstacles and s is the size of dislocation cells, g is average grain size and lp is the distance between precipitates. The cell size is taken as proportional to 1 ρi in the current approach.
Different processes may contribute to the recovery term in Eq. (7). Various models for recovery are described in (Engberg, 1976) as well as flow stress reduction due to recrystallization in (Engberg & Lissel, 2008). The recovery is separated into static and dynamic recovery as !ρi−( ) = !ρsr
−( ) + !ρdr−( ) . (10)
Diffusion processes may cause immobile dislocation segments to climb and annihilate each other. The probability of this is therefore proportional to the
!γ p
v
3
density of immobile dislocations in square and the self-diffusivity of the material. The static recovery is written as
!ρsr−( ) = csrDv
Gb3
kBTρi2 − ρgrwn
2( ) . (11)
kB is Boltzmann’s constant, T is the absolute temperature and is a calibration parameter. The
grown in dislocation density is taken as ρgrwn =1010
m-2. It is introduced to prevent the dislocation density to become zero when hold for very long times. Dv is the self-diffusivity of the material and is written as Dv = Dv0e
−Qv kBT . (12) Dynamic recovery implies that moving and immobile dislocations annihilate each other. The former are proportional to the effective plastic strain rate, Eq. (2), and therefore recovery due to dislocation glide is written as !ρdr−( ) =Ωρi !ε
p . (13) The function Ω is
Ω =Ω0 + ʹΩr0D!ε p
⎛
⎝⎜
⎞
⎠⎟1 n
. (14)
The first term is a constant and accounts for the possibility that a dislocation moves towards an immobile with opposite sign and they cancel each other. The second term accounts for the possibility that diffusion can assist in this process. Bergström (1983) derived the rate dependent part assuming that the recovery occurs mainly for dislocations in cell walls and that this process is due to climb controlled by diffusion of excess vacancies created during the deformation. He derived n=3 and D was taken as the diffusivity for moving vacancies. Kocks and Mecking (2003; 1986) and later Estrin (1998) estimated n to be in the range 3-5 and the activation energy is due to climb. Their variant of the model is based on the assumption that the glide process brings dislocations near each other and the annihilation process is completed by climb.
The other long-range stress in Eq. (1) is the Hall-Petch effect. It is due to pile-up of dislocations at grain boundaries leading to the activation of other slip systems in the neighboring grains (Johnston & Feltner, 1970). The contribution is typically expressed as
σ HP =kHPg
. (15)
Temperature dependency is accounted for by scaling it versus the temperature dependent shear modulus. The conversion is obtained by
σ HP =kHP
GRT bG b
g= ʹkHPG
bg
, (16)
, where GRT denotes shear modulus at room temperature. Thus a non-dimensional Hall-Petch coefficient ʹkHP is obtained.
2.1.3 Short-range contributions
The σ SR term in Eq. (1) is the material resistance to plastic deformation due to short-range interactions where thermal activated mechanisms assist the applied stress in moving dislocations (Caillard & Martin, 2003). It is the stress required to drive the mobile dislocations as described in section 2.1.1. The book by Frost and Ashby (1982) gives a very good summary of the various mechanisms that are dominating in different temperature and strain rate regions. Short-range obstacles is a general classification of any disturbance of the lattice that is so small that thermal vibrations can, together with the stress, move the affected part of a dislocation through the disturbed region of the lattice. Solutes or precipitates are examples of short-range obstacles. It can also be the interaction between a moving dislocation segment and an immobile dislocation segment that are orthogonal to each other.
The general expression for short-range obstacles is
σ glide = τ0G 1− kBTΔf0Gb
3 ln!εref!ε p
⎛
⎝⎜
⎞
⎠⎟
⎛
⎝⎜⎜
⎞
⎠⎟⎟
1q
1p
. (17)
τ0G can be interpreted as the stress required to
bypass the obstacle at 0 K. Δf0Gb3 is the energy
barrier of the obstacle. The four calibration parameters are τ0 , Δf0 , p, and q. The model can be extended to account for dynamic strain ageing (Lindgren, Domkin, & Hansson, 2008). The relation is simplified to
σ sol =σ solref 1− T
T0sol
⎛
⎝⎜
⎞
⎠⎟
2 3⎛
⎝⎜⎜
⎞
⎠⎟⎟
3 2
. (18)
for solute contribution and thus the rate dependency is ignored in this case. The contribution is neglected above the temperature T0
sol . The coefficient σ solref
depends on the solutes and their distortion of the lattice, (Sieurin, Zander, & Sandström, 2006).
The two contributions above are added linearly in the current approach as they have quite different magnitudes (Nembach & Neite, 1985).
csr
4
2.2. Stainless steels The results of applying the plasticity model on two stainless steels, AISI 316L and AISI 420, are shown below. The first material model has been used in welding (Hedblom, Lindgren, Nordgren, & Gillander, 1999) and extrusion (Hansson & Domkin, 2005) simulations. The material model described below is an improvement of the model presented in (Lindgren et al., 2008). Figure 1 shows some of the calibration curves and Figure 2 the validation set.
Figure 1 Calibration set of flow stress curves for AISI 316L.
Figure 2 Validation set of flow stress curves for AISI 316L.
AISI 420 is a ferritic steel that will be formed in ductile ferritic state and then heat treated to create a hard martensitic structure. Gleeble tests were performed for temperatures up to 1000 °C and strain rates up to 0.1 s-1. Some samples were first austenized and tested during cooling and thereby the properties of the martensitic phase could be obtained. The flow stress curves for the ferritic state are shown in Figure 3 and for varying martensitic states in Figure 4. The flow stress model need be combined with austenization and martensite formation models.
Figure 3 Calibration set of flow stress curves for AISI 420 for the ferritic state.
Figure 4 Calibration set of flow stress curves for AISI 420 for varying fractions of martensite.
2.2. Al 5083 The model in (Liu et al., 2013) for simulating warm forming of the aluminum alloy AA 5083 has been simplified and improved by addition of creep due to grain boundary sliding that becomes important at high temperatures and low rates. Some data in Figure 5 comes from (Toros & Ozturk, 2010).
2.3. Ti-6Al-4V A flow stress model for Ti-6Al-4V has been developed for α- and β- phases and applied to additive manufacturing (Lundbäck & Lindgren, 2011). The flow stress curves shown in Figure 6 shows softening due to globularization. This is grain growth that reduces the dislocation density and is included in the flow stress model (B. Babu & Lindgren, 2013).
0 0.1 0.2 0.3 0.4 0.5 0.6True strain [-]
0
200
400
600
800
1000
1200
True
stre
ss [M
Pa]
Measured 20°C and 0.01 s-1
Computed 20°C and 0.01 s-1
Measured 20°C and 1 s-1
Computed 20°C and 1 s-1
Measured 200°C and 1 s-1
Computed 200°C and 1 s-1
Measured 400°C and 1 s-1
Computed 400°C and 1 s-1
Measured 900°C and 0.01 s-1
Computed 900°C and 0.01 s-1
Measured 900°C and 1 s-1
Computed 900°C and 1 s-1
Measured 900°C and 10 s-1
Computed 900°C and 10 s-1
Measured 1100°C and 0.01 s-1
Computed 1100°C and 0.01 s-1
Measured 1100°C and 1 s-1
Computed 1100°C and 1 s-1
Measured 1100°C and 10 s-1
Computed 1100°C and 10 s-1
20°C and 1 s-1
200°C and 1 s-1
400°C and 1 s-1
900°C and 0.01 s-1
900°C and 1 s-1
1100°C and 1 s-1
900°C and 10 s-11100°C and 10 s-1
1100°C and 0.01 s-1
20°C and 0.01 s-1
0 0.1 0.2 0.3 0.4 0.5 0.6True strain [-]
0
20
40
60
80
100
120
140
160
180
200
True
stre
ss [M
Pa]
Measured 1100°C and 0.01 s-1
Computed 1100°C and 0.01 s-1
Measured 1100°C and 1 s-1
Computed 1100°C and 1 s-1
Measured 1100°C and 10 s-1
Computed 1100°C and 10 s-1
Measured Hold test at 1100°CComputed Hold test at 1100°CMeasured ratejump test 1100°CComputed ratejump test 1100°C
Measured 1300°C and 0.01 s-1
Computed 1300°C and 0.01 s-1
Measured 1300°C and 1 s-1
Computed 1300°C and 1 s-1
Measured 1300°C and 10 s-1
Computed 1300°C and 10 s-1
1300°C and 10 s-1
1100°C and 1 s-1
1100°Chold test
1300°C and 0.01 s-1
1100°C and 0.01 s-1
1300°C and 1 s-1
1100°C strainrate jump test
1100°C and 10 s-1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2True strain [-]
0
200
400
600
800
1000
1200
True
stre
ss [M
Pa]
Heat to 25°C, rate 0.01s-1
Computed 25°C, rate 0.01s-1
Heat to 25°C, rate 1s-1
Computed 25°C, rate 1s-1
Heat to 200°C, rate 0.01s-1
Computed 200°C, rate 0.01s-1
Heat to 200°C, rate 1s-1
Computed 200°C, rate 1s-1
Heat to 400°C, rate 0.01s-1
Computed 400°C, rate 0.01s-1
Heat to 400°C, rate 1s-1
Computed 400°C, rate 1s-1
Heat to 600°C, rate 0.01s-1
Computed 600°C, rate 0.01s-1
Heat to 600°C, rate 1s-1
Computed 600°C, rate 1s-1
Heat to 800°C, rate 0.01s-1
Computed 800°C, rate 0.01s-1
Heat to 800°C, rate 1s-1
Computed 800°C, rate 1s-1
25 °C, 0.01 and 1.0 s-1
200 °C, 1.0 s-1 and 0.01 s-1
400 °C, 1.0 s-1 and 0.01 s-1
600 °C, 1.0 s-1 and 0.01 s-1
800 °C, 1.0 s-1 and 0.01 s-1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2Strain
0
500
1000
1500
2000
2500St
ress
Cool to 200°C, rate 0.01s-1
Comp--Cool to 200°C, rate 0.01s-1
Cool to 25°C, rate 0.01s-1
Comp--Cool to 25°C, rate 0.01s-1
Cool to 300°C, rate 0.01s-1
Comp--Cool to 300°C, rate 0.01s-1
Heat and cool to 400°C, rate 0.01s-1
Comp--Heat and cool to 400°C, rate 0.01s-1
5
Figure 5 Calibration flow stress curves for AA 5083.
Figure 6 Calibration flow stress curves for Ti-6Al-4V for different temperatures and a strain rate of 1 s-1.
2.4. Alloy 718 The flow stress model is aimed for simulations of welding and subsequent ageing of Alloy 718 and is described in (Fisk, Ion, & Lindgren, 2014). The alloy requires an ageing treatment in order to restore the precipitate hardening. The model is combined with a model for precipitate growth and includes their effect on the flow stress by an additional short-range term. Some flow stress curves are shown in Figure 7.
Figure 7 Calibration flow stress curves for Alloy 718 for different ageing states and strain rates at 400°C.
3. MICROSTRUCTURE MODELS 3.1 Phase changes in carbon steels The model for phase changes in hypo-eutectoid steels is described in (Lindgren, 2007; Oddy, McDill, & Karlsson, 1996). The current material will be subjected to induction hardening. The model is typically calibrated versus TTT-diagrams. The model can be applied to austenization for varying heating rates as in Figure 8. An example of a CCT-diagram for validation is shown in Figure 9.
Figure 8 Austenization of AISI 4150 steel lines with symbols for measurements from (Macedo, Cota, & Araújo, 2011).
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4Strain
0
50
100
150
200
250
300
350
Stre
ss
Measured 25 °C, 0.1 s-1
Computed 25 °C, 0.1 s-1
Measured 100 °C, 0.1 s-1
Computed 100 °C, 0.1 s-1
Measured 150 °C, 0.1 s-1
Computed 150 °C, 0.1 s-1
Measured 200 °C, 0.1 s-1
Computed 200 °C, 0.1 s-1
Measured 250 °C, 0.1 s-1
Computed 250 °C, 0.1 s-1
Measured Al Toros 300°C, 0.16 s-1
Computed Al Toros 300°C, 0.16 s-1
Measured Al Toros 300°C, 0.042 s-1
Computed Al Toros 300°C, 0.042 s-1
Measured Al Toros 300°C, 0.0083 s-1
Computed Al Toros 300°C, 0.0083 s-1
Measured 350 °C, 0.1s-1Computed 350 °C, 0.1s-1Measured 350 °C, 0.01s-1Computed 350 °C, 0.1s-1Measured 400 °C, 0.01s-1Computed 400 °C, 0.1s-1
True strain0 0.1 0.2 0.3 0.4 0.5 0.6
True
stre
ss [M
Pa]
0
200
400
600
800
1000
1200
1400
1600
1800Aged 0.001s-1
Aged 0.1s-1
Half-aged 0.1s-1
Half-aged 0.001s-1
Annealed 0.001s-1
Annealed 0.1s-1
0 100 200 300 400 500 600 700 800 900 1000Heating rate °Cs-1
800
850
900
950
1000
1050
1100
1150
1200
1250
Ac3
ModelData
6
Figure 9 CCT-diagram of AISI 4150 steel lines with symbols are measurements from Atlas Wärmebehandlung Der Stähle.
3.2 Phase changes in Ti-6Al-4V The phase change model is described in (Charles Murgau, Pederson, & Lindgren, 2012). The model was calibrated versus various data and a validation set, relevant for the additive manufacturing is shown in Figure 10. It is combined with the flow stress model in section 2.3.
Figure 10 α-phase during cyclic heating of Ti-6Al-4V. Data from (S. S. Babu, Kelly, Specht, Palmer, & Elmer, 2005).
3.3. Precipitate model for Alloy 718 Alloy 718 obtains its creep resistance during ageing where precipitates are formed. Simulation of this process combines the flow stress model in section 2.4 with a precipitate growth model. The validation of
the model versus measured sizes of precipitates for ageing is shown in Figure 10.
Figure 11 Growth of smallest and largest axes of γ’’-precipitates in Alloy 718 aged at 760°C, from (Fisk et al., 2014).
4. DISCUSSIONS Simulations of manufacturing processes may require the use of coupled flow stress and microstructure models. Examples of such models, applicable for large-scale simulations have been illustrated.
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10-2 100 102 104 106
Time [log(sec)]
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Ageing time [s]
Pa
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le s
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[nm
]
Mean particle diameter, L
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L − [2]
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