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Journal ArticleDOI

The onset of twinning in metals: a constitutive description

TL;DR: In this article, 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.
About: This article is published in Acta Materialia.The article was published on 2001-11-14 and is currently open access. It has received 1366 citations till now. The article focuses on the topics: Crystal twinning & Deformation mechanism.

Summary (4 min read)

1. 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.
  • To whom all correspondence should be addressed.
  • 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:.

2. THE TWINNING STRESS

  • There are excellent overviews, such as one by Christian and Mahajan [6], on the effects of internal and external parameters on the twinning stress.
  • Four of these aspects, relevant to the constitutive description implemented here, are discussed next.
  • Growth can occur at stresses that are a fraction of the nucleating stress [8, 26].
  • Their results, however, can also be interpreted as twinning being normally initiated by some defect configuration, because of the requirement of much higher stress for the homogeneous nucleation.
  • An interesting alternative to the above mechanisms, all based on dislocation reactions, is the proposal by Orowan [48] that twins nucleate homogeneously.

2.1. Effect of temperature and strain rate

  • Figure 1 shows a compilation of twinning stress vs temperature for a number of metals (both mono and polycrystals).
  • Bell and Cahn [34] observed a large scatter in single crystals.
  • There are also reports of gradual decrease in the twinning stress with increasing temperature for FCC metals, by Bolling and Richman [44], and Koester and Speidel [49].
  • Christian and Mahajan [6] discuss this topic in detail.
  • Figure 2(a) shows the results by Chichili et al. [53] which illustrate eloquently the effect of stress on twinning.

2.2. Effect of grain size

  • Another highly unique characteristic of twinning, first pointed out by Armstrong and Worthington [31], is the larger grain size dependence of the twinning stress, as compared with the slip stress.
  • The reason for the difference is not fully understood, but Armstrong and Worthington [31] suggest that twinning is associated with microplasticity, that is, dislocation activity occurring before the onset of generalized plastic deformation, whereas the yield stress is associated with generalized plastic deformation.
  • It is very plausible that microplasticity and overall deformation are controlled by different mechanisms, that is, elastic anisotropy, incompatibility stresses, and barriers to slip.
  • Recent results by El-Danaf et al. [17] reconfirm the significant effect of grain size on the propensity for twinning.
  • Meyers et al. [66] performed shock compression experiments on copper at 35 GPa and obtained profuse twinning for grain sizes of 117 and 315 µm, but virtually no twinning for a grain size of 9 µm.

2.3. Effect of stacking-fault energy

  • Suzuki and Barrett [36] and Venables [40, 41] proposed relationships between the SFE gSF and the twinning stress tT.
  • This is true mostly for FCC metals, and the classic plot by Venables [40, 41] shows this effect very clearly.
  • Figure 3 shows a compilation of results by Venables [40] and Vöhringer [68].
  • The twinning stress for a number of copper alloys is shown to vary with the square root of the SFE.
  • This effect is critically discussed in Section 5, where a new relationship is proposed.

2.4. Effect of texture

  • Gray et al. [63] have shown that texture has especially important effect on twinning in low-symmetry metals.
  • A dislocation moves in opposite sense along the same direction when the applied stress is reversed, while the critical resolved shear stress (CRSS) is independent of the sense of dislocation motion.
  • A twin, on the other hand, has a definite sense along which it shears.
  • In the presence of texture, however, the twinning stresses in compression and tension are different.
  • The analysis presented in this paper applies to untextured polycrystalline aggregates only.

2.5. Effect of stress state

  • The simulations by Serra and Bacon [69] and Serra et al. [70] predict an increase in the lattice spacing between the twin plane and adjacent planes, with the dilatation of ca 0.004.
  • Lebensohn and Tomé [71] discuss the effect of stress state on the critical stress for twinning and point out that this strain can affect the stress dependence.
  • In martensitic transformations, when the product phase has a lower density than the parent phase, the effect is quite pronounced [72,73].
  • Since there is no intrinsic difference in density between the twinned and untwinned regions, the effect is most probably of second-order for twinning.
  • It will not be incorporated in the computations presented herein.

3. AN ANALYTICAL DESCRIPTION OF THE TWINNING STRESS

  • In general, the tendency for the occurrence of mechanical twinning in BCC and HCP metals [74– 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.
  • In FCC metals, which have a much lower strain-rate sensitivity, but higher work hardening ability, twinning often occurs after significant plastic deformation, which raises the corresponding stress level.
  • The number of dislocations piled up is determined by the distance l between the source and barrier, and the applied stress.
  • This equation applies to the lowvelocity regime, before viscous drag and relativistic effects come into play, and breaks down at temperatures close to 0 K.
  • The parameter K is obtained by fitting equation (8) to experimental results reported for the twinning stress by Harding [51, 52].

4. CONSTITUTIVE DESCRIPTION OF THE SLIPTWINNING TRANSITION

  • The rationale to be used in this section is that the onset of twinning occurs when the slip stress tS becomes equal to the twinning stress tT, that is tS tT. (10).
  • It will be assumed that there is a CRSS for twinning that is independent of the stress state.
  • If the orientation factors MS and MT are assumed to be equal to each other, there follows sS sT. (12).
  • The described rationale will be applied to typical metals representative of the three crystalline systems of greatest importance for metals: Fe (BCC); Cu (FCC); and Ti (HCP).
  • No attempt was made at the present stage to compare the calculated slip-twinning transitions with experimental results on the initiation.

4.1. Iron (BCC)

  • The constitutive equations (8) and (9) from Section 3 are used in equation (12), with the addition of the Hall–Petch terms for slip and twinning, kS and kT, respectively.
  • The intersections of these curves are given by the solution of equation (13).
  • Figure 6(b) shows the slip-twinning transition for different grain sizes.
  • The values for kT and kS are given in Table 1.
  • The twinning domain for monocrystalline iron is much larger than for polycrystalline iron.

4.2. Copper (FCC)

  • It was not possible to apply the constitutive equation for twinning given in Section 3 to copper.
  • Attempts were made at obtaining the activation energy and dislocation velocity exponent m from Jassby and Vreeland [80], Greenman et al. [81], Kleintges and Haasen [82], and Suzuki and Ishi [83].
  • The constitutive responses are shown in Fig. 7, at two levels of plastic strain: 0.2 and 0.8.
  • It is seen that no twinning is obtained at 0.2, but that at plastic strain of 0.8 the twinning occurs for all strain rates, for the grain size of 10 µm.
  • These calculations were made for a constant grain size of 10 µm.

4.3. Titanium (HCP)

  • Zerilli and Armstrong [84, 85] demonstrated that the constitutive response of BCC metals can represent the behavior of titanium, with a few modifications to incorporate the decrease in work hardening rate as the temperature is increased.
  • Conrad et al. [87] observed similar effects.
  • The Hall–Petch slope for slip was obtained by Okazaki and Conrad [88] and was found to be relatively insensitive to interstitial content.
  • It should be noted that the calculations were carried out for Marz titanium, with 0.1%Oeq. and not with the material given by Zerilli and Armstrong [84], which has Oeq. 1% and a yield stress at ambient temperature and the strain rate 10 3 s 1 of 400 MPa.
  • The rise in the twinning stress with interstitial content is more significant than the slip stress; this explains why the tendency for twinning decreases with interstitial increase.

5. EFFECT OF STACKING-FAULT ENERGY

  • Figure 3 shows the significant effect of the SFE gSF on the twinning stress for FCC metals.
  • The maximum concentration of the solute is denoted by Cmax. (20) The effect of solid solution (Zn, Ag, Al, Sn, Ge) atoms on the mechanical response of Cu has been established quite carefully, as well as the effects of these solutes on the Hall–Petch relationship.
  • Equation (21) is based on the overcoming of short-range obstacles that have the shape dictated by the parameters p and q. Since Cu–Zn is FCC, twinning can occur after significant plastic deformation.

6. ANALYTICAL PREDICTION OF THE CRITICAL NUCLEUS SIZE

  • An expression for the twin-stress dependence on SFE will be derived using the classical nucleation theory.
  • The total change in free energy can thus be written as W gTBA.
  • Furthermore, suppose that the shear modulus of the parent material is G, while the shear modulus of the embryo (due to its possible crystalline reorientation relative to the surrounding matrix) is G∗.
  • An expression for rc similar to that given by equation (42) was originally derived for FCC metals by Venables [41].
  • The twin-boundary energy can be considered to be the sum of the coherent twin-boundary energy (gCTB) and the energy of the twinning dislocations.

7. GRAIN-SIZE EFFECTS AND THE SIZE OF PILEUPS

  • It is simple to relate the local stress s013, driving the twin formation, and the globally applied stress t13 = t by considering the number of dislocations at the pile-up and by using the equation developed by Eshelby et al. [95].
  • The externally applied stress t is related to the number of dislocations n in the pile-up by n aplt Gb , (49) where a is parameter that depends on the dislocation character (a = 1 for edge dislocations).
  • For the grain size of 104 nm, the maximum length of the pile-up is about 40 times smaller than the grain size d.
  • An even smaller ratio l/d is found for Fe–3%Si.
  • An alternative explanation would be that other mechanisms are responsible for the stress concentrations necessary to initiate twinning.

8. CONCLUSIONS

  • An analytical treatment that describes the initiation of mechanical twinning is developed and presented in graphical form as strain rate–temperature plots.
  • For FCC and HCP metals, the authors are not aware of any experimental results on the effect and the twinning stress is assumed to be constant.
  • This domain is here divided into “twinning” and “slip” domains.
  • The same procedure can be applied to any deformation-mechanism map, and it is suggested that this will complement the maps and enhance their usefulness.
  • The analysis requires the use of the Swegle–Grady [97] equation and the procedure is delineated in Ref. [98].

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TL;DR: In this paper, it is shown that to answer several questions of physical or engineering interest, it is necessary to know only the relatively simple elastic field inside the ellipsoid.
Abstract: It is supposed that a region within an isotropic elastic solid undergoes a spontaneous change of form which, if the surrounding material were absent, would be some prescribed homogeneous deformation. Because of the presence of the surrounding material stresses will be present both inside and outside the region. The resulting elastic field may be found very simply with the help of a sequence of imaginary cutting, straining and welding operations. In particular, if the region is an ellipsoid the strain inside it is uniform and may be expressed in terms of tabu­lated elliptic integrals. In this case a further problem may be solved. An ellipsoidal region in an infinite medium has elastic constants different from those of the rest of the material; how does the presence of this inhomogeneity disturb an applied stress-field uniform at large distances? It is shown that to answer several questions of physical or engineering interest it is necessary to know only the relatively simple elastic field inside the ellipsoid.

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TL;DR: A balanced mechanics-materials approach and coverage of the latest developments in biomaterials and electronic materials, the new edition of this popular text is the most thorough and modern book available for upper-level undergraduate courses on the mechanical behavior of materials as discussed by the authors.
Abstract: A balanced mechanics-materials approach and coverage of the latest developments in biomaterials and electronic materials, the new edition of this popular text is the most thorough and modern book available for upper-level undergraduate courses on the mechanical behavior of materials To ensure that the student gains a thorough understanding the authors present the fundamental mechanisms that operate at micro- and nano-meter level across a wide-range of materials, in a way that is mathematically simple and requires no extensive knowledge of materials This integrated approach provides a conceptual presentation that shows how the microstructure of a material controls its mechanical behavior, and this is reinforced through extensive use of micrographs and illustrations New worked examples and exercises help the student test their understanding Further resources for this title, including lecture slides of select illustrations and solutions for exercises, are available online at wwwcambridgeorg/97800521866758

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"The onset of twinning in metals: a ..." refers background in this paper

  • ...Weertman–Ashby map for titanium (from Frost and Ashby [14], figure 17....

    [...]

  • ...An immediate application of the constitutive description presented here is in the Weertman–Ashby deformation mechanism maps....

    [...]

  • ...However, the classical deformation-mechanism maps, also called Weertman–Ashby maps [13, 14], do not have a twinning domain....

    [...]

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TL;DR: An improved description of copper and ironcylinder impact (Taylor) test results has been obtained through the use of dislocation-mechanics-based constitutive relations in the Lagrangian material dynamics computer program EPIC•2.
Abstract: An improved description of copper‐ and iron‐cylinder impact (Taylor) test results has been obtained through the use of dislocation‐mechanics‐based constitutive relations in the Lagrangian material dynamics computer program EPIC‐2. The effects of strain hardening, strain‐rate hardening, and thermal softening based on thermal activation analysis have been incorporated into a reasonably accurate constitutive relation for copper. The relation has a relatively simple expression and should be applicable to a wide range of fcc materials. The effect of grain size is included. A relation for iron is also presented. It also has a simple expression and is applicable to other bcc materials but is presently incomplete, since the important effect of deformation twinning in bcc materials is not included. A possible method of acounting for twinning is discussed and will be reported on more fully in future work. A main point made here is that each material structure type (fcc, bcc, hcp) will have its own constitutive beha...

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Frequently Asked Questions (11)
Q1. What are the contributions in "The onset of twinning in metals: a constitutive description" ?

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 strong decrease in the twinning stress, observed when Mo is alloyed with Rh, has also been attributed to a SFE decrease [62]. 

Taking a value of 800 MPa for the normal stress, the slip-twinning transition was estimated for grain sizes of 3, 10, and 100 µm. 

The effect of work hardening can be incorporated into equation (21) by adding the term C2 n to the thermal component of stress; in FCC metals work hardening increases the density of forest dislocations, which constitute short-term barriers. 

In general, the tendency for the occurrence of mechanical twinning in BCC and HCP metals [74– 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. 

It is seen that no twinning is obtained at 0.2, but that at plastic strain of 0.8 the twinning occurs for all strain rates, for the grain size of 10 µm. 

Another highly unique characteristic of twinning, first pointed out by Armstrong and Worthington [31], is the larger grain size dependence of the twinning stress, as compared with the slip stress. 

The effect of solid solution atoms is manifested (both in the thermal and athermal components of stress) through the C2/3 relationship and the Labusch parameter L, which has different values for different solid solution atoms. 

The possibility of homogeneous nucleation of twins in near perfect HCP crystals was reported by Bell and Cahn [34] and Price [35]. 

The rise in the twinning stress with interstitial content is more significant than the slip stress; this explains why the tendency for twinning decreases with interstitial increase. 

Discussions with Professors R. W. Armstrong, G. Thomas and X. Markenscoff were most helpful in the development of the concepts presented in this paper.