The onset of twinning in metals: a constitutive description

doi:10.1016/S1359-6454(01)00300-7

Acta Materialia

Volume 49, Issue 19, 14 November 2001, Pages 4025-4039

https://doi.org/10.1016/S1359-6454(01)00300-7 Get rights and content

Abstract

A constitutive approach is developed that predicts the critical stress for twinning as a function of external (temperature, strain rate) and internal (grain size, stacking-fault energy) parameters. Plastic deformation by slip and twinning are considered as competitive mechanisms. The twinning stress is equated to the slip stress based on the plastic flow by thermally assisted movement of dislocations over obstacles, which leads to successful prediction of the slip-twinning transition. The model is applied to body centered cubic, face centered cubic, and hexagonal metals and alloys: Fe, Cu, brasses, and Ti, respectively. A constitutive expression for the twinning stress in BCC metals is developed using dislocation emission from a source and the formation of pile-ups, as rate-controlling mechanism. Employing an Eshelby-type analysis, the critical size of twin nucleus and twinning stress are correlated to the twin-boundary energy, which is directly related to the stacking-fault energy (SFE) for FCC metals. The effects of grain size and SFE are examined and the results indicate that the grain-scale pile-ups are not the source of the stress concentrations giving rise to twinning in FCC metals. The constitutive description of the slip-twinning transition are incorporated into the Weertman–Ashby deformation mechanism maps, thereby enabling the introduction of a twinning domain. This is illustrated for titanium with a grain size of 100 μm.

Introduction

The response of metals and ceramics to mechanical stresses can produce the following structural changes: slip (by dislocation motion); twinning (which also requires dislocation activity); phase transformations; and fracture [1]. Slip and fracture have received the greatest amount of attention from both theoretical and experimental researchers during the past 60 years. Mechanical twinning and displacive (martensitic) transformations also constitute a significant modes of deformation and can dominate under specific deformation conditions. Whereas dislocation motion is highly sensitive to strain rate and temperature (e.g. Becker [2] and Seeger [3], [4], [5]), twinning has a much lower sensitivity to these parameters. Nevertheless, it is well known that dislocation activity is intimately connected with twinning nucleation and growth. A comprehensive review of mechanical twinning has been recently provided by Christian and Mahajan [6], in addition to several other overview treatments [7], [8], [9], [10], [11], [12]. However, the classical deformation-mechanism maps, also called Weertman–Ashby maps [13], [14], do not have a twinning domain. This is most likely due to the absence of well-tested constitutive equations. Mechanical twinning can have two effects on the evolution of plastic deformation:

1.
It subdivides the grains and therefore increases the barriers to slip, and the work-hardening rate. This has been demonstrated by Mulford and Kocks [15], and successfully modeled by Asgari et al. [16], El-Danaf et al. [17], Kalidindi [18], [19], Staroselsky and Anand [20] and Karaman et al. [21].
2.
It contributes to plastic deformation due to twinning shear, which induces a decrease in the work hardening rate. This has been found in copper alloys by Vöhringer [22].

There has been in recent years a considerable effort devoted to the development of constitutive equations describing plastic deformation of metals and based on the fundamental aspects of dislocation motion, impeded by a variety of barriers. Cottrell [23] and Seeger [24] made important early contributions. Ono [25], Vöhringer [26], and Kocks et al. [27] varied the barrier shape and configuration to arrive at very satisfactory descriptions of the constitutive response. These ideas were incorporated into equations used in large-scale computational codes; prominent representatives are the Zerilli–Armstrong [28], [29] and the MTS [30] constitutive equations. A notable effort toward the incorporation of mechanical twinning into constitutive models was made in the constitutive equation developed by Armstrong and Worthington [31]. The computational and experimental studies by Zerilli and Armstrong [32] show that twinning can play a significant role. Karaman et al. [21] developed a constitutive equation for the combined slip/twinning deformation in Hadfield (FCC) steel; Tomé et al. [33] extended this to HCP metals. The research effort whose results are presented in this paper had as a primary objective the application of a constitutive description for the onset of twinning in conjunction with a constitutive equation for slip to obtain maps for the two regions (slip and twinning) for a variety of FCC, BCC, and HCP metals. A second objective was to use the constitutive description to obtain a linkage between fundamental nucleation parameters, grain size, and stacking-fault energy (SFE).

Section snippets

The twinning stress

There are excellent overviews, such as one by Christian and Mahajan [6], on the effects of internal (material) and external parameters on the twinning stress. Four of these aspects, relevant to the constitutive description implemented here, are discussed next. The critical event in twinning is, for most cases, nucleation. Growth can occur at stresses that are a fraction of the nucleating stress [8], [26]. It has been known for a long time that the local stress required to nucleate twinning is

An analytical description of the twinning stress

In general, the tendency for the occurrence of mechanical twinning in BCC and HCP metals [74], [75], [76] is quite strong at high strain rates and low temperatures, because the flow stress can be effectively raised up to the level required for twin formation. This is a direct result of their high strain-rate and thermal sensitivity. In BCC metals, twinning usually occurs prior to macro-yielding, and in many cases it is inhibited by significant plastic deformation. In FCC metals, which have a

Constitutive description of the slip-twinning transition

The rationale to be used in this section is that the onset of twinning occurs when the slip stress τ_S becomes equal to the twinning stress τ_T, that is $τ_{S} =τ_{T} .$ It will be assumed that there is a CRSS for twinning that is independent of the stress state. For an untextured polycrystalline aggregate we then write $σ_{S} M_{S} = σ_{T} M_{T} .$ If the orientation factors M_S and M_T are assumed to be equal to each other, there follows $σ_{S} =σ_{T} .$

The described rationale will be applied to typical metals representative of the three

Effect of stacking-fault energy

Figure 3 shows the significant effect of the SFE γ_SF on the twinning stress for FCC metals. As an illustration of the effect of SFE on the incidence of twinning, the Cu–Zn system is analyzed. Gallagher [89] and Vöhringer [90] correlated the SFE to the electron/atom (e/a) ratio in copper alloys and arrived at the following expression $ln γ_{SF} γ_{Cu} =K_{1} C C+C_{max}^{2},$ where γ_Cu is the stacking fault energy for copper, and C is the concentration of solute atoms. The maximum concentration of the solute is denoted

Analytical prediction of the critical nucleus size

In this section, an expression for the twin-stress dependence on SFE will be derived using the classical nucleation theory. It is recognized that this approach has a number of simplifying assumptions and that the complex dislocation interactions involved in twinning are not incorporated. The induced stress acting on the barrier will generate an elastic distortion, which has to supply the energy required to create a twin-matrix interface (γ_TB), and the energy needed for the formation of the twin

Grain-size effects and the size of pile-ups

It is simple to relate the local stress σ⁰₁₃, driving the twin formation, and the globally applied stress τ₁₃=τ by considering the number of dislocations at the pile-up and by using the equation developed by Eshelby et al. [95]. The shear stress σ⁰₁₃ is created by the pile-up equivalent dislocation with the Burgers vector nb and the shear stress nτ at a distance l/4 from the tip of the pile-up of length l. The externally applied stress τ is related to the number of dislocations n in the pile-up

Conclusions

An analytical treatment that describes the initiation of mechanical twinning is developed and presented in graphical form as strain rate–temperature plots. This constitutive description is applied to metals representative of these principal crystal systems: BCC (iron); FCC (copper and Cu–Zn brass); and HCP (titanium). For BCC metals, an equation for the twinning stress is derived from the consideration of the activation of Frank-Read sources. This provides a temperature and strain-rate

Acknowledgements

The presented research was funded by the US Army Research Office through the Multidisciplinary University Research Institute (MURI), Contract No. DAAH 04-96-1-0376, and by the Humboldt Foundation of Germany through a Senior Scientist Award to the first author. Mrs Q. Xue and D. Khieng were helpful in preparation of the manuscript. The assistance of Drs T. Dummer and M. Ehlers during the visit of M.A.M. to the University of Karlsruhe is appreciated. Discussions with Professors R. W. Armstrong,

References (98)

J.W. Christian et al.
Prog. Mater. Sci.
(1995)
R.A. Mulford et al.
Acta metall.
(1979)
S.R. Kalidindi
J. Mech. Phys. Sol.
(1998)
A. Staroselsky et al.
J. Mech. Phys. Sol.
(1998)
I. Karaman et al.
Acta mater.
(2000)
A.H. Cottrell
J. Mech. Phys. Solids
(1952)
P.S. Follansbee et al.
Acta metall. mater.
(1988)
H. Suzuki et al.
Acta metall.
(1958)
S. Mahajan et al.
Acta metall.
(1973)
D.R. Chichili et al.
Acta mater.
(1998)

D. Hull

Acta metall.

(1961)

T.C. Lindley et al.

Acta metall.

(1963)

D. Lahaie et al.

Scripta Met.

(1992)

A. Serra et al.

Acta metall. mater.

(1991)

J.R. Patel et al.

Acta metall.

(1953)

N.N. Thadhani et al.

Prog. Mater. Sci.

(1986)

K. Okazaki et al.

Acta metall.

(1973)

M. Kleintges et al.

Scripta Met.

(1980)

K. Okazaki et al.

Acta metall.

(1973)

Thomas, G., Private communication,...

R. Becker

Z. Phys.

(1925)

A. Seeger

Z. Naturf.

(1954)

A. Seeger

Z. Naturf.

(1954)

A. Seeger

Z. Naturf.

(1954)

E.O. Hall

Twinning

(1954)

N.V. Klasen-Neklyudova

Mechanical Twinning of Crystals

(1964)

S. Mahajan et al.

Int. Metal. Rev.

(1973)

N. Narita et al.

G.T. Gray

J. Weertman

Trans. AIME

(1963)

H.J. Frost et al.

Deformation-mechanism Maps

(1982)

S. Asgari et al.

Metall. Trans. A

(1970)

E. El-Danaf et al.

Metall. Mater. Trans.

(1999)

S.R. Kalidindi

O. Vöhringer

Z. Metall.

(1970)

A. Seeger

Phil. Mag.

(1954)

K. Ono

J. Appl Phys.

(1968)

Vöhringer, O. Die Strukturmechanischen Grundlagen der Plastischen Verformung von Vielkristallinen-Kupfer-Legierungen....

U.F. Kocks et al.

Prog. Mater. Sci.

(1975)

F.J. Zerilli et al.

J. Appl. Phys.

(1987)

F.J. Zerilli et al.

J. Appl. Phys.

(1990)

R.W. Armstrong et al.

F.J. Zerilli et al.

J. Phys. IV

(1997)

C.N. Tomé

Symposium on HCP Metals

(2001)

R.L. Bell et al.

Proc. R. Soc.

(1957)

P.B. Price

Proc. R. Soc.

(1961)

P. Haasen et al.

Z. Metallk.

(1960)

S. Miura et al.

Trans. Jap. Met.

(1968)

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The onset of twinning in metals: a constitutive description

Abstract

Introduction

Section snippets

The twinning stress

An analytical description of the twinning stress

Constitutive description of the slip-twinning transition

Effect of stacking-fault energy

Analytical prediction of the critical nucleus size

Grain-size effects and the size of pile-ups

Conclusions

Acknowledgements

Prog. Mater. Sci.

Acta metall.

J. Mech. Phys. Sol.

J. Mech. Phys. Sol.

Acta mater.

J. Mech. Phys. Solids

Acta metall. mater.

Acta metall.

Acta metall.

Acta mater.

Acta metall.

Acta metall.

Scripta Met.

Acta metall. mater.

Acta metall.

Prog. Mater. Sci.

Acta metall.

Scripta Met.

Acta metall.

Z. Phys.

Z. Naturf.

Z. Naturf.

Z. Naturf.

Twinning

Mechanical Twinning of Crystals

Int. Metal. Rev.

Trans. AIME

Deformation-mechanism Maps

Metall. Trans. A

Metall. Mater. Trans.

Z. Metall.

Phil. Mag.

J. Appl Phys.

Prog. Mater. Sci.

J. Appl. Phys.

J. Appl. Phys.

J. Phys. IV

Symposium on HCP Metals

Proc. R. Soc.

Proc. R. Soc.

Z. Metallk.

Trans. Jap. Met.