flow stress and microstructure models of alloys

1

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

















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 Hold test at 1100°CComputed Hold test at 1100°CMeasured ratejump test 1100°CComputed ratejump test 1100°C







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

















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





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 Al Toros 300°C, 0.16 s-1

Computed Al Toros 300°C, 0.16 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.

REFERENCES Arzt, E. (1998). Size effects in materials due to

microstructural and dimensional constraints: a comparative review. Acta Materialia, 46(16), 5611-5626. doi: 10.1016/s1359-6454(98)00231-6

Babu, B, & Lindgren, L-E. (2013). Dislocation density based model for plastic deformation and globularisation of Ti-6Al-4V. International Journal of Plasticity, 50, 94-108.

Babu, S S, Kelly, S M, Specht, E D, Palmer, T A, & Elmer, J W. (2005). Measurements of phase transformation kinetics during repeated cycling of Ti-6Al-4V using time-resolved x-ray diffraction. Paper presented at the Int. Conf. on Solid-Solid Phase Transformation in Inorganic Materials, Warrendale, USA.

Bergström, Y. (1969/70). A dislocation model for the stress-strain behaviour of polycrystalline [alpha]-Fe with special emphasis on the variation of the densities of mobile and immobile dislocations. Materials Science and Engineering, 5(4), 193-200.

Bergström, Y. (1983). The plastic deformation of metals - A dislocation model and its applicability. Reviews on powder metallurgy and physical ceramics, 2/3, 79-265.

10-2 100 102 104 106

Time [log(sec)]

0

100

200

300

400

500

600

700

800

900Te

mpe

ratu

re [°

C]

Ferrite 5%

Pearlite 5%

Bainite 5%

Martensite 5%

(F+P)

100

102

104

106

0

20

40

60

80

100

120

140

160

180

200

Ageing time [s]

Pa

rtic

le s

ize

[nm

]

Mean particle diameter, L

Mean particle height, h

L − [25]

h − [25]

L − [3]

h − [3]

L − [2]

h − [2]

L − [26]

h − [26]

L − [27]

h − [27]

L − [24]

h − [24]

L − [24]

h − [24]

7

Caillard, D., & Martin, J.L. (2003). Thermally Activated Mechanisms in Crystal Plasticity (Vol. 8). Oxford: Pergamon.

Charles Murgau, C, Pederson, R, & Lindgren, L-E. (2012). A model for Ti-ì6Al-ì4V microstructure evolution for arbitrary temperature changes. Modelling and Simulation in Materials Science and Engineering, 20(5), 055006.

Conrad, H. (1970). The athermal component of the flow stress in crystalline solids. Materials Science and Engineering, 6(4), 265-273. doi: http://dx.doi.org/10.1016/0025-5416(70)90054-6

Engberg, G. (1976). Recovery in a titanium stabilized 15% Cr - 15¤ Ni austenitic stainless steel (pp. 50). Stockholm: Royal Institute of Technology.

Engberg, G, & Lissel, L. (2008). A physically based microstructure model for predicting the microstructural evolution of a C-Mn steel during and after hot deformation. Steel Res. Int., 79(7), 47-58.

Estrin, Y. (1998). Dislocation theory based constitutive modelling: foundations and applications. Journal of Materials Processing Technology, 80-81, 33-39.

Fisk, M., Ion, J. C., & Lindgren, L. E. (2014). Flow stress model for IN718 accounting for evolution of strengthening precipitates during thermal treatment. Computational Materials Science, 82, 531-539. doi: http://dx.doi.org/10.1016/j.commatsci.2013.10.007

Frost, HJ, & Ashby, MF. (1982). Deformation-Mechanism Maps - The Plasticity and Creep of Metals and Ceramics. Oxford: Pergamon Press.

Hansson, S, & Domkin, K. (2005). Physical based material model in finite element simulation of extrusion of stainless steel tubes. Paper presented at the Conference Proceedings of the 8th International Conference on Technology of Plasticity ICTP, Verona.

Hedblom, E., Lindgren, L-E., Nordgren, G., & Gillander, L. (1999, August). Welding procedures with reduced residual stresses for components in nuclear power plants. Paper presented at the 15th Int Conf. on Structural Mechanics in Reactor Technology (SMiRT-15), Seoul, South Korea.

Johnston, T., & Feltner, C. (1970). Grain size effects in the strain hardening of polycrystals. Metallurgical and Materials Transactions B, 1(5), 1161-1167. doi: 10.1007/bf02900226

Kocks, U F. (1981). Kinetics of nonuniform deformation. In J. Christian, P. Haasen & T. Massalski (Eds.), Progress in Materials Science (pp. 185-241): Pergamon Press.

Kocks, U F, Argon, A S, & Ashby, M F. (1975). Thermodynamics and kintetics of slip (Vol. 19). Oxford: Pergamon Press Ltd.

Kocks, U F, & Mecking, H. (2003). Physics and phenomenology of strain hardening: the FCC case. Progress in Materials Science, 48(3), 171-273.

Lindgren, L-E. (2007). Computational welding mechanics. Thermomechanical and microstructural simulations: Woodhead Publishing.

Lindgren, L-E. (To appear). Finite Element Simulation of Manufacturing. Plasticity and microstructure models for metals and alloys.: Springer.

Lindgren, L-E, Lundbäck, A, Edberg, J, & Svoboda, A. (2013). Challenges in finite element simulations of chain of manufacturing processes. Paper presented at the Physical and numerical simulation of materials processing VII, Oulu, Finland.

Lindgren, Lars-Erik, Domkin, Konstantin, & Hansson, Sofia. (2008). Dislocations, vacancies and solute diffusion in physical based plasticity model for AISI 316L. Mechanics of Materials, 40(11), 907-919.

Lindgren, Lars-Erik, Lundbäck, Andreas, & Fisk, Martin. (2013). Thermo-Mechanics and Microstructure Evolution in Manufacturing Simulations. Journal of Thermal Stresses, 36(6), 564-588. doi: 10.1080/01495739.2013.784121

Liu, J., Edberg, J., Tan, M. J., Lindgren, L. E., Castagne, S., & Jarfors, A. E. W. (2013). Finite element modelling of superplastic-like forming using a dislocation density-based model for AA5083. Modelling and Simulation in Materials Science and Engineering, 21(2), 025006.

Lundbäck, A, & Lindgren, L-E. (2011). Modelling of metal deposition. Finite Element in Analysis and Design, 47, 1169-1177.

Macedo, Marciano Quites, Cota, André Barros, & Araújo, Fernando Gabriel da Silva. (2011). The kinetics of austenite formation at high heating rates. Rem: Revista Escola de Minas, 64, 163-167.

Mecking, H, & Estrin, Y. (1980). The effect of vacancy generation on plastic deformation. Scripta Metallurgica, 14, 815-819.

Mecking, H, Kocks, U F, & Hartig, Ch. (1996). Taylor factors in materials with many deformation modes. Scripta Materialia, 35(4), 465-471.

Mecking, H, Nicklas, B, Zarubova, N, & Kocks, UF. (1986). A "universal" temperature scale for plastic flow. Acta Metallurgica, 34(3), 527-535.

Messerschmidt, U. (2010). Dislocation Dynamics During Plastic Deformation. Berling: Springer.

Nembach, Eckhard, & Neite, Gunter. (1985). Precipitation hardening of superalloys by ordered [gamma]'-particles. Progress in Materials Science, 29(3), 177-319.

Oddy, A.S., McDill, J.M.J., & Karlsson, L. (1996). Microstructural predictions including arbitrary thermal histories, reaustenization and carbon

8

segregation effects. Canadian Metallurgical Quarterly, 35(3), 275-283.

Orowan, E. (1940). Problems of plastic gliding. Proceedings of the Physical Society, 52(1), 8.

Shetty, MN. (2013). Dislocations and Mechanical Behaviour of Materials. Delhi: PHI Learning.

Sieurin, Henrik, Zander, Johan, & Sandström, Rolf. (2006). Modelling solid solution hardening in stainless steels. Materials Science and Engineering: A, 415(1-2), 66-71.

Taylor, GI. (1938). Plastic strain in metals. Journal of the Institute of Metals, 62, 307-324.

Toros, S., & Ozturk, F. (2010). Modeling uniaxial, temperature and strain rate dependent behavior of Al–Mg alloys. Computational Materials Science, 49(2), 333-339. doi: http://dx.doi.org/10.1016/j.commatsci.2010.05.019

flow stress and microstructure models of alloys

Documents