Microscale modelling of multiple and higher-order deformation twinning

Acta Mech 226:371–384 (2015) DOI 10.1007/s00707-014-1172-7

Shyamal Roy · Rainer Glüge · Albrecht Bertram

Microscale modelling of multiple and higher-order deformation twinning

Received: 18 December 2013 / Revised: 13 April 2014 / Published online: 28 June 2014 © Springer-Verlag Wien 2014

Abstract We present a framework for the microscale modelling of higher-order deformation twinning. For this purpose, it is important to track the evolution of the lattice orientation as well as the deformation at each ¯ ¯ ¯ ¯ material point. The model is applied to a titanium single crystal with the 1011{ 1012} and 1¯ 123{11 22} twinning modes.

1 Introduction Many materials undergo solid-to-solid phase transformations upon thermal or mechanical loading, which take part in the shape memory effect in shape memory alloys (SMA), the transformation-induced plasticity (TRIP) effect, or the twinning-induced plasticity (TWIP) effect, etc., where deformation twinning is an intriguing and unavoidable factor. The deformation twinning [6] plays an important role in plastic deformation mechanism along with the crystallographic slip for many crystalline solids. Therefore, the underlying physics behind the twin creation and its propagation in the grain need to be well understood. Twinning of an already pre-twinned material, which is called higher-order twinning [2,7,24], is a frequent phenomenon in crystals of lower symmetry, such as magnesium and titanium. Experimentally, the highest twinning order is 3, as reported in the literature [24]. In this work, the modelling of higher-order twinning on the grain scale is emphasized. Twinning can be considered as a homogeneous shearing of a crystal lattice, or of a sublattice with an additional shuffling movement of the atoms. The shuffling/shear deformation positions the atoms in such a way that a rotated replica of the parent lattice is generated (see Fig. 1). Twinning not only effects the stress–strain response, but has also a strong influence on the microstructural evolution. In cubic metals, which have a large magnitude of twinning shear, the deformation twins are generally thin [14]. However, in crystals of lower symmetry, most twinning modes have a small shear number, and twins are considerably thicker than the twins in cubic crystals. The twinning can happen after a certain slip occurrence or even before that in the elastic region, resulting in stress drops in the absence of slip. This type of twinning is sensitive to temperature and strain rates. However, in this work, the focus is on the isothermal and mechanically induced deformation twinning, where a strain-rate effect has not been implemented explicitly. This kind of solid-to-solid phase change is not recognized from a chemical point of view, although certain characteristic features of phase changes are displayed. Twinning gives rise to sharp interfaces, at which the material properties that depend on the crystal orientation undergo a jump. The twins form plates inside the grains and can significantly alter the morphological and the crystallographic texture, thus influencing the effective material properties. Due to its polarity, twinning can cause a pronounced differential effect on the S. Roy (B) · R. Glüge · A. Bertram Institute of Mechanics, Otto-von-Guericke Universität Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany E-mail: [email protected]

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η2

S

η2

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. . .. .. . . .. . . . η2

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α

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Fig. 1 Schematic representation of distorted and undistorted planes. The twin and conjugate twin planes are K 1 and K 2 , respectively. k1 and k2 are the normal vectors to K 1 and K 2 , respectively. The twin and conjugate twin directions are η1 and η2 , respectively. 2α is the angle between η2 and the direction η2 to which η2 is rotated by a simple shear in twin system 1. A 3D representation of twinning is shown in the right figure

strength of the material and the forming limit. For many materials, these effects are not negligible and need to be incorporated in the material model. In particular, ductile TWIP steels and lightweight hcp metals such as magnesium and titanium, which are interesting for engineering applications, show extensive twin formation at room temperature. For such materials, the proper prediction of forming processes or deformation mechanisms requires a material model which includes mechanical twinning. 1.1 Notation We employ a symbolic notation that is compliant with most textbooks on continuum mechanics of solids, such as Liu [19], Itskov [16], Bertram [3]. Vectors are denoted as bold minuscules, and second-order tensors as bold majuscules. The composition of vectors and tensors has no special symbol, e.g. a = ABb, C = AB. A deformation is denoted by its displacement field u(x 0 ), where x 0 locates the material points in the undeformed placement. The coordinates in the current placement are x = u(x 0 ) + x 0 . The deformation gradient is F = x ⊗ ∇0 , where the index ‘0’ indicates the derivative with respect to x 0 . From F, we can calculate the right Cauchy-Green tensors C = F T F, where the upper T indicates the transpose. Green’s strain tensor is defined as E = 21 (C − I), with the identity tensor I. The spatial velocity gradient is given through ˙ F −1 . The superimposed dot indicates the time derivative with x 0 held constant. The Cauchy stress tensor L=F is denoted by σ , the second Piola–Kirchhoff-stress tensor by T = J F −1 σ F −T , where J is the determinant of F. The linear mapping between second-order tensors is denoted by a fourth-order tensor, e.g. T = C[E], where C is the stiffness tetrad. The Rayleigh product is defined between a second order and an arbitrary order tensor, but we need it only for the orders 2 and 4: R ∗ A = R AR T , (R ∗ C)[ A] = RC[R T AR]R T , where R is an orthogonal tensor (RT R = I) having positive determinant. ‘:’ denotes the double scalar contraction between tensors. ‘·’ refers to the scalar product between two vectors. 1.2 Geometric description of twinning modes A schematic representation of a simple shear is illustrated geometrically in Fig. 1 [17]. The plane containing η1 and k1 (the normal to K 1 ) is called plane of shear (S), and K 1 is called the twin plane or the shear plane to twin system 1. K 2 is the second undistorted plane to twin system 1, which is just rotated due to the shear deformation. The shear direction η1 inside K 1 is common to S. η2 lies inside K 2 and S, and is rotated into η2 by 2α. η1 , η2 , k1 and k2 (the normal to K 2 ) are all contained in the plane of shear S. Here, the conjugate + twinning modes are k1 , η1 and k2 , η2 . They satisfy S−1 1 S2 ∈ Orth , where Si denote the shear deformations S i = I + γ0 η i ⊗ k i . The latter notation is usually used by material scientists, while in continuum mechanics, the shear direction and shear plane normal are denoted by d and n, respectively. We will employ the latter denomination.

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2 Modelling of higher-order twinning 2.1 Outline of the modelling approach Before going into the details of the model, we give a brief overview on the modelling strategy and its peculiarities. One approach, proposed by Ericksen [10], is to incorporate phase changes through an elastic strain energy w(F) that is not quasiconvex in the deformation gradient F. The lack of quasiconvexity leads to multiple solutions, involving a decomposition of the bulk material into laminates [1]. However, in the quasistatic, rateindependent setting, the non-quasiconvexity of the strain energy renders the boundary value problem ill-posed. This ill-posedness needs to be overcome without eliminating the predictability of the decomposition into laminates. In order to make the mathematical problem well posed without loosing the predictability of lamination, two model enhancements are commonly employed. These are the incorporation of a time-dependency through kinetics or rate-dependence, and the introduction of a convex strain-gradient contribution to the strain energy. The kinetisation strategy introduces a time-dependency by taking into account inertia effects, which shifts the problem from global energy minimization to evolution tracking. This approach is laborious, since it compels us to deal with waves and shocks. A more convenient approach to shift the ill-posed energy minimization to evolution tracking is the incorporation of a viscous contribution in the material law. This approach enjoys some popularity. It is often applied to single out solutions when strain softening, crack propagation, strain-rate softening, damage, or phase changes occur [4,5,9,13,18]. It is also used to overcome the Taylor problem in crystal plasticity [15] of the choice of the active set of slip systems, and to transform the algebraic differential equations, governing ideal plastic material behaviour, to ordinary differential equations [22]. It is also simpler to implement, and thermodynamically consistent. In the second approach, a convex strain-gradient contribution is added to the strain energy. This prevents an infinite fine lamination, i.e. unphysical solutions are excluded. However, it is more demanding from a practical point of view, since one has to deal with higher-order gradients of the displacement field. This requires an enhancement of the finite element formulations which are usually employed [20]. Both methods (viscosity and strain gradient) have their physical interpretation. The mobility of interfaces is accounted for by a corresponding kinetic relation, while the strain-gradient contribution can be seen as a way of incorporating the interface energy (capillarity), penalizing very thin laminates energetically. Here, we use a viscous regularization, i.e. we introduce a strain-rate effect in the bulk material. Turteltaub [25] demonstrated that the interface kinetic relation can be obtained by a limit process from a bulk viscosity and a strain-gradient dependence. Thus, the trouble of explicit interface modelling can be avoided, and still, one can prescribe a kinetic relation by appropriate viscosity and capillarity, without introducing a phase field for each phase. However, in the present context, we only consider a viscous contribution, and the constitutive law for the stresses is assumed as ∂w 1 + ηC −1 C˙ C −1 ∂E 2 ∂w 1 −1 ˙ −1 + ηC C C , =2 ∂C 2

T=

(1) (2)

where the second Piola-Kirchhoff stresses T are expressed by an elastic law plus a viscous contribution, where η is the viscosity parameter. The elastic contribution is given through the derivative of a volume-specific strain energy w with respect to Green’s strain tensor E. By assuming that the elastic energy only depends on the material stretch tensor U from the polar decomposition F = RU, we account for the invariance of the material law under rigid body rotations. The second summand in Eq. (2) is the Newtonian strain-rate sensitivity σ = η D, written with respect to the reference placement, which assumes an isotropic and linear dependence of the Cauchy stresses σ on the symmetric part D of the velocity gradient L. The key problem is to find a proper function w(C). Ball and James [1] suggest the strain energy as w = min(w1 , w2 . . . wn ),

(3)

where w1 . . . wn stand for the elastic strain energies of the individual phases or variants that are considered for the material modelling. This approach assumes that the material always transforms into the phase that minimizes the elastic strain energy for a given deformation state. It does not reflect whether this phase is at that particular instant accessible, nor does it account for a strain path-dependence. Further, the phase transformation is not triggered by any stress criterion for twin nucleation or the like, but solely by the current state of deformation. In

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conclusion, the simplicity of the visco-elastic modelling comes along with certain assumptions on the material behaviour. However, we will demonstrate that these issues can be relieved to a certain extent. Higher-order twins are only accessible through preliminary lower-order twinning. By updating the list of individual strain energies that constitute w depending on the strain path, a strain path-dependency is introduced. Then, the sequence of activated twin systems becomes part of the material state. A similar kind of treatment to the ’successive updating mechanism’ has been used in [8] to model repeated martensitic transformations. This can also, in principle, account for an infinite number of transformations, including the possibility of returning to the parent phase (similar to ’detwinning’ in the present article), in the large strain settings. The article concisely explains the re-occurrence of phase transformations by constructing energy landscapes depending on the selected variants. In the present work, we do not consider the martensitic transformations, but the higherorder twinning in the same phase. However, the overall approach of deformation path tracking is similar. To include the stresses into the twin criterion, one can translate twinning criteria given in terms of stresses to the strain space. Thus, the strain energies can be designed such that quasiconvexity is lost at the strain states that correspond to the critical stress states.

2.2 Framework for the change of the stress-free placements We deal with multiple stress-free placements that are related to each other by twinning operations. In this section, we investigate how the elastic laws of the potential twins are defined with respect to a common reference placement. For this purpose, we need the notion of placements, the stress-free state, plastic directors, and material directors. We assume a material with evolving stress-free configurations. The reference placement is chosen to be stress-free. This enables us to express the stresses initially in terms of a deviation of the current placement x(x 0 , t) from the reference placement, namely by the elastic energy w(F). Let us now assume a change of the stress-free placement. We denote the change of placement from the new stress-free placement to the reference placement by the transformation P, where the letter P indicates that this deformation may well be considered as plastic. Since rigid body rotations do not affect the stresses,1 the intermediate placement is not uniquely determined unless further constraints are introduced. One way of doing this is to define plastic directors that are common to both stress-free placements (see Fig. 2). This is often done implicitly. For example, in the case of crystallographic slip, it is included by making the intermediate placement isoclinic. In case of crystallographic slip and twinning, the plastic deformation is a simple shear. Thus, the plastic directors that suffice to describe the deformation are the shear direction d and the shear plane normal n. Other directors may or may not change their orientation with respect to the plastic directors during the change of stress-free placement, depending on the underlying deformation mechanism. In the case of crystallographic slip, the crystal lattice does not change its orientation with respect to the plastic directors. In the case of twinning, the crystal lattice is rotated with respect to the plastic directors, namely by Rπ n = −I + 2n ⊗ n (type 1 twinning) or Rπ d = −I + 2n ⊗ n (type 2 twinning) or both hold (compound twinning). For other materials, the transformation of the plastic directors is not necessarily orthogonal, for example, when fibres in an elastoplastic matrix material deform with the material, which has been coined ’Material Plasticity’ by Forest and Parisot [11]. To formulate this mathematically, let us consider two stress-free configurations a and b (see Fig. 2), and an elastic behaviour described by the St.-Venant-Kirchhoff law. The two stress-free placements differ by the transformation P, and by a transformation of the material directors R. The plastic directors are denoted by p1 and p2 . In the case of slip and twinning, they coincide with the shear direction and shear plane normal. The stress-free placements a and b are treated as reference placements for the individual elastic laws, with respect to which we determine the Cauchy stresses in the current placement,    1 1 T σa = F a Ka F a F a − I F aT , JFa 2    1 1 F b Kb σb = F bT F b − I F bT . JFb 2 1

See[23] for a discussion of the corresponding principles.

(4) (5)

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current placement

Fb

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P −1 m2 m1 p2 R mp p1 stress-free placement a

m2 m1

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Fig. 2 The stress-free placements a and b differ by P, and the material director orientation differs by R. The plastic directors pi are the same in both stress-free placements. The material directors mi change their orientation with respect to the plastic directors from R m p to R m p during the change of stress-free placement

We take the stress-free placement a as the constant reference placement. Then, we can claim that F b = F a P, Kb = R ∗ Ka

(6) (7)

hold. Equation (7) implies that the stiffness tetrad is attached to the material directors. In the case of a crystalline material, R can only be a rotation. Using Eqs. (4), (5), (6), and (7), the elastic law in b can be written with respect to the reference placement,    1 1 T T σb = F a P(R ∗ Ka ) (8) P F a F a − P −T P −1 P P T F aT JFa J P 2    1 1 F a ( P R ∗ Ka ) (9) F aT F a − P −T P −1 F aT . = JFa J P 2 Comparing coefficients with the elastic law in a shows how the process alters the elastic law if written with respect to the constant reference placement. The term P −T P −1 stands for the change of the stressfree placement in terms of a right Cauchy-Green tensor, while the Rayleigh product with P R gives the transformation of the stiffness tetrad Ka . Note that it is possible to determine R and P experimentally. It is −T −1 P˜ , also worth noting that in case of R ∈ Orth+ , we can always expand P −T P −1 = P −T R R T P −1 = P˜ ˜ i.e. P = P R, which allows us to write    1 1 −T −1 F aT P˜ F a ( P˜ ∗ Ka ) (10) F aT F a − P˜ σb = JFa J P 2 ˜ Let us next specify the latter quantities for crystallographic slip and twinning. We with only one variable P. can drop the indices a and b now.

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slip m2

n

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n m2

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Fig. 3 Sketch of the change of the stress-free placement and the crystal lattice orientation for slip (a) and twinning (b). In both cases, the change of stress-free placement is given by P = I − d ⊗ n, and the stress-free configurations have the plastic directors d and n in common. In the first case, the crystal orientation is unaffected by the change of stress-free placement, i.e. R = I. In the second case, the crystal lattice is subjected to a reorientation, where the lattice basis is rotated by Rπ n = −I + 2n ⊗ n in this sketch

Crystallographic slip As sketched in Fig. 3a, the lattice orientation does not vary when crystal planes glide along each other, i.e. R = I. Here, the two placements can be prescribed to be separated by an isoclinic change of the stress-free placement, since they have the same lattice orientation. This is due to our premise that the plastic directors in both placements coincide. The glide manifests itself as a simple shear deformation, leading to P = I − γ d ⊗ n, with J P = 1 and γ changing continuously. The new elastic law is   1 σ = (11) F( P ∗ K) F T F − P −T P −1 F T . 2J Then, the Cauchy stresses are given by   1 F e P −1 ( P ∗ K) P −T F eT F e P −1 − P −T P −1 P −T F eT 2J   1 F e K F eT F e − I F eT , = 2J

σ =

(12) (13)

with F e = F P. Thus, in case of crystallographic slip, the knowledge of F e or P is sufficient to study the material behaviour. This is due to the coincidence of the plastic directors with the material directors before and after slip. Deformation twinning Unlike crystallographic slip, the lattice orientation changes with respect to the plastic directors when twinning occurs (Fig. 3b). The lattice orientations before and after twinning differ by Rπ n or Rπ d , while the deformation itself is again a simple shear deformation. The change of stress-free placement is discrete in case of twinning, i.e. γ0 is a constant. Here, after employing F e = F P, it remains σ =

1 F e (R ∗ K)[F eT F e − I]F eT . 2J

(14)

Unlike in crystallographic slip, R = I, R ∈ Orth+ remains in Eq. (14). We have seen that with R ∈ Orth+ , we can further use the substitution F e =  F e R T to simplify,  1   T  T FeK Fe Fe − I  Fe . (15) σ = 2J We then have the overall expression  F e = F P R.

(16)

For first-order twinning, one can give the elastic laws of the potential twins by only one transformation  Fe, calculated from d and n. However, for second-order twinning, the lattice orientation of the first-order twin must be known. Therefore, knowing only  F e is not sufficient, but also R is needed. Thus, to keep track of the

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deformation and the lattice orientation of a material point that undergoes higher-order twinning, one needs to keep track of at least two transformations. One choice is to track the lattice orientation R and the plastic transformation P. This is a remarkable difference to crystallographic slip, where tracking R is not at all required, since the lattice orientation does not change with respect to the plastic directors. 2.3 Elastic strain energy invariance We have seen that we need only d and n to determine the elastic law of the twin, namely by calculating  F e in Eq. (16). However, due to lattice symmetries, it is not invertible: Given d 1 and n1 , two orthonormal vectors d 2 and n2 inside the plane spanned by d 1 and n1 can be determined that give  F e Q using the same γ0 , where Q is an element of the symmetry group of K, such that the elastic energy and the Cauchy stresses do not differ at all. Thus, the change of the elastic law is invariant with respect to the conjugate twin systems d 1 , n1 and d 2 , n2 [12,27,28]. This holds for all compound twinning modes, for which both orientation relations ( Rπ n and Rπ d ) hold. However, this is not a purely mathematical issue, since most of the observed twins fall into this class. The non-invertibility has some consequences for the modelling approach. Firstly, the equivalence between conjugate twins applies also to the elastic energy. This renders the Ball and James approach (Eq. (3)) as insufficient to decide which twin system is activated. Secondly, a careful distinction between crystallographically equivalent and distinct twins is required. For ¯ ¯ 101 ¯ 1{ ¯ 1012} ¯ example, the family of 101¯ 1{10 12}, twin systems2 in hcp crystals, which constitutes 6 different twin systems, has to be accounted for by only three transformations of the elastic law, each corresponding to a pair of conjugate twin systems. The energy invariance can also occur between crystallographically non¯ ¯ and the six 2243{11 ¯ ¯ twin systems3 in hcp crystals. The equivalent twin systems like the six 112¯ 3{11 22} 2¯ 4} purely elastic approach does not allow for a distinction, although the twins display a very different behaviour.4 Both issues are resolved by incorporating a purely elastic model with a twinning criterion that does not rely on the strain energy. Such a twinning criterion could be used to inhibit twinning in ’unwanted’ twin systems of crystallographically distinct twin systems that are generated automatically due to the elastic energy invariance, as well as to decide which one out of two crystallographically equivalent conjugate twin systems has been activated. This information is needed to track the change of the stress-free placement and the lattice orientation. Therefore, an additional criterion (explained in 2.4) is needed. Otherwise, the minimum elastic strain energy approach limits the modelling to first-order twinning of crystallographically equivalent twins as done in [13]. 2.4 Twinning criterion As first ingredient, an effective elastic deformation gradient (F eff ), which maps from the present stress-free parent configuration to the current placement (Fig. 4), is determined. To estimate the amount of shear in each twin system, we employ γi = d i · F eff ni , ∀i ∈ (i, . . . , n),

(17)

where d i and ni are the shear directions and shear plane normals of the next potential twin variant. Since one can consider this as the projection of F eff into different Schmid tensors d i ⊗ ni , we refer to Eq. (17) as the projection criterion. The largest γi indicates the activated twin variant. The applicability of the criterion is validated by performing several computational experiments, with well-known preferred twinning modes. It works quite well for single crystal computational tests. The projection criterion is applicable only for small rotations, since it is not objective. Unfortunately, all attempts to exclude the polar part of F eff , or transforming d and n with F eff to the current placement, resulted in practically indistinguishable shear numbers in conjugate twin systems. This is due to the fact that the conjugate twins exhibit d ⊗ n ≈ (d  ⊗ n )T , and hence, γ ≈ γ  holds when the projection is carried out with the symmetric stretch tensor. At the final twin deformations, the projections coincide for both variants. ¯ ¯ are the tensile twins (appearing under c-axis-extension of an hcp crystal) for crystals having the axial ratio ( c ) 101¯ 1{10 12} a √ smaller than 3. 3 112 ¯ 3{11 ¯ ¯ ¯ ¯¯ √22} and 2243{1124} are the compression twins (appearing under c-axis-compression of an hcp crystal) for crystals c having a < 3. 4 {1122} ¯ twins are observed quite often, while {112¯ 4} ¯ twins are reported only under special conditions, such as very low temperatures or very fast strain rates [26]. 2

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Fig. 4 Schematic diagram of the modelling approach

Only at intermediate deformations, there is a difference between γ and γ  . However, although mathematically not identical, the difference in the projections of conjugate twin systems is so small that in simulations, small perturbations trigger different twinning modes. This results in an unrealistic element-wise mixing of conjugate twins. 2.5 Update rules As discussed in (2.2), two transformations (R, S) are defined, where R tracks the crystal orientation (rotation) and S the plastic deformation (shear deformation) that builds up by successive twinning shears. A schematic representation of the material model is shown in Fig. 4. We presume the reference placement as stress free, i.e. S0 = I. The initial orientation is given by R0 , which maps the reference crystal to its actual orientation in the reference placement. Assuming that n potential twin variants are possible for a particular type of material, these are defined with respect to the stress-free configuration of the parent. At the occurrence of twinning, an updating of R and S is performed according to the twin variant that is activated. To model second-order twinning, again n twin variants are defined with respect to the new lattice orientation and the stress-free parent configuration, including the possibility of detwinning, i.e. twinning back to the former parent configuration. The successive updating allows to model, at least in principle, infinite-order twinning. Twinning is accompanied by a discrete change of the stress-free placement and of the crystal lattice orientation. We define a constant reference crystal with respect to a constant basis ei , for which we denote the , and the twinning shear directions and shear plane normals  stiffness tetrad by K d 1...n and  n1...n . We will denote by bracketed indices the number of successive twinning operations, while non-bracketed indices indicate which twin system has been activated. In the case of several non-bracketed indices, from left to right, they indicate the sequence of activated twin systems. In the case of twinning of the order n on the ith twin system, the lattice orientation needs to be updated by R(n) = (−I + 2n(n−1),i ⊗ n(n−1),i )R(n−1) ,

(18)

where the vector n(n−1),i is obtained by applying the orientation R(n−1) of the parent crystal to the reference crystal, n(n−1),i = R(n−1) ni . The recurrence can be rewritten by inserting Eq. (19) into Eq. (18),   T R(n−1) ni R(n−1) R(n) = −I + 2 R(n−1) ni ⊗    T = R(n−1) −R(n−1) R(n−1) + 2 ni ⊗  ni = R(n−1) (−I + 2 ni ⊗  ni ).

(19)

(20) (21) (22)

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With the abbreviation  Ri = −I + 2 ni ⊗  ni , we can give the lattice orientation explicitly, e.g. after four successive twinning operations in the systems i, j, k, l as Ri jkl = R0  R  R  R  R.  i j k l R(1)  R(2)

(23)

Similarly, the plastic shear S(n) is obtained from S(n) = (I + γ0 d (n−1),i ⊗ n(n−1),i )Sn−1 , d (n−1),i = R(n−1) di , ni . n(n−1),i = R(n−1)

(24) (25) (26)

Again, one can obtain an explicit expression for S(n) in terms of the indices of the successively activated twin systems. For example, after third-order twinning in the systems i, j, k we have Si jk = Ri j (I + γ0 nk )RiTj Ri (I + γ0 n j )RiT R0 (I + γ0 ni )R0T , dk ⊗  d j ⊗ di ⊗ 

 S(1)

 S(2)

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(27)

T which can be summarized by inserting the explicit expression for Ri j..kl , the simplification RiTjk Ri j =  Rk and the abbreviation  Si = I + γ0 ni to di ⊗  T T Si jkl =R0  Ri  R j Rj  Ri  Sk  Sj  Si R0T .

(28)

We may drop the transpose everywhere except for R0 , since the  Ri are symmetric. One can show that  Ri as T  well as Ri Si are tensors of period two, i.e. taken to the power 2 they result in I. This property reflects the fact that successive twinning in the same twin system corresponds to a recovery of the parent, which is called detwinning. 3 FE model set-up A simple finite element (FE) simulation is performed by the commercial FE software Abaqus to examine the predictability of the model. We carry out a tensile test on a titanium single crystal, as shown in Fig. 5. A small perturbation is planted in the sample to produce a high-stress concentration to initiate twinning. A displacement boundary condition is applied at the top plane along the sample axis (z-axis in Fig. 5), while the bottom plane is fixed. The distance between the two planes is increased proportional to time during the deformation. Inside the top and bottom planes, the lateral straining is not restricted in order to avoid high-stress concentrations (and hence, twin nucleation sites) in these planes. Out of four different classes of twins, only two are frequently observed in the experiment for titanium. ¯ ¯ and compression twins 112¯ 3{11 ¯ ¯ In the tensile test, we expect to see These are tensile twins 101¯ 1{10 12} 22}. primarily tensile twins. The c-axis orientation inside the first-order twins is approximately 90° off the tension axis, i.e. as the sample is extended further, the first-order twins are compressed along their c-axis. Thus, the second-order twins should be compression twins. ¯ ¯ tensile twins and the 112¯ 3{11 ¯ ¯ For titanium, 12 twin systems are defined in total. The 101¯ 1{10 12} 22} compression twins are indexed by 1 to 6 and 7 to 12, respectively. The c/a ratio of titanium is approximately 1.587. The shear numbers are √ c/a 3 γ1...6 = (29) − √ ≈ 0.175, c/a 3

  c/a + arctan(c/a) − π/2 ≈ 0.218. (30) γ7...12 = 2 tan arctan 2

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Z=0

X

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Fig. 5 Initial stage of FE model before deformation

The 5 independent elastic constants for titanium are C11 = 123.1 GPa, C12 = 99.6 GPa, C13 = 68.8 GPa, C33 = 152.9 GPa, and C66 = 61.4 GPa, with respect to the normalized Cowin basis for the stiffness tetrad, ⎡ ⎤ 0 0 0 C11 C12 C13 ⎢ C11 C13 0 0 0 ⎥ ⎢ ⎥ C 0 0 0 ⎥ ⎢ 33 C=⎢ (31) Bi ⊗ B j , C11 − C12 0 0 ⎥ ⎢ ⎥ ⎣ ⎦ C66 0 sym C66 B 1 = e1 ⊗ e1 , B 2 = e2 ⊗ e2 , B 3 = e3 ⊗ e3 ,

(32)

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(33)

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(34)

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(e2 ⊗ e3 + e3 ⊗ e2 ),

(35)

and have been taken from Simmons and Wang [21]. The axis of transverse isotropy is the e3 axis. 3.1 Meshing Although the mesh dependency is relaxed by the viscous contribution of the material model, a proper choice of the meshing is challenging for the present model. For FE models with localization, the Abaqus user manual recommends the use of low-order elements and states that, in particular, incompatible mode elements are appropriate. Further, the meshing should be irregular, since twinning is facilitated along the mesh interfaces. Thus, a regular meshing, in conjunction with a specific crystal orientation, induces an anisotropy by facilitating twinning in specific twin systems. A further observation we made was that elements with only one integration point performed much better than elements with more integration points. The problem with the latter type of elements is that different twin systems may be activated at different integration points, making the twin interface cut through the element (see Fig. 6). It is observed that convergence, as well as the quality (apparent mixture of variants in single element) of the results, suffer from these multi-phase elements (for example, the interfaces in Fig. 6). In conclusion, there is only one element type that meets all requirements, namely the linear tetrahedral element (C3D4 in the Abaqus denomination). Due to the modest capabilities of the linear tetrahedral elements, the mesh must be as fine as possible. However, too small element sizes lead to very expensive computations. Thus, we refined the mesh close to the notch, where the strongest inhomogeneity is expected. A balance is made between the computation and the mesh size by studying the minimum elastic strain energy of the system over a range of different mesh sizes. The window of mesh sizes, in which the minimum energy of the simulated system does not change considerably, is recognized as useful and sufficient for the present study.

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Fig. 6 Cross-sectional view parallel to the sample axis. Apparent mixture of twin variants at the twin interface because of multiple integration points in an element. The legends are the same as in Fig. 7

3.2 Results As expected, multiple first-order twins are nucleated near the notch and propagate steadily into the sample (Fig. 7a, b). The second-order twinning follows in the recently generated first-order twinned region (Fig. 7c). The sequence is as expected, namely first the tensile and then the compression twinning inside the tensile twins (Figs. 7d, 8). As the simulation progresses, the first-order twins propagate into the whole sample, producing a grain structure, which is similar to one that is observed in experiments (see Figure 3 in [24]). Only after the sample underwent completely first-order twinning, second-order twins proceed to grow considerably. There are several reasons for this behaviour. Firstly, the nucleation requires a higher force than twin propagation, which is reproduced by this kind of modelling approach [13]. Thus, the established twins grow when a specific loading is reached, limiting thus effectively the maximum load. Therefore, the second-order twinning happens only close to the notch, where locally a sufficiently high load is reached for twin nucleation. Secondly, the issue is numerical in nature. The relative grain discretization for second-order twinning is somewhat more coarse than for first-order twins, making the material much stiffer. Thus, the nucleation load for second-order twinning is higher than for first-order twinning. This shows that the application of the present material model requires more elaborate numerical methods, like adaptive mesh refinement. In order to examine the effect of the perturbation (imperfection) size, a simulation with different notch sizes in the same sample has been performed (Fig. 9). It is noticed that the first nucleation happens at the biggest notch. If the viscosity is small, we observed the same behaviour. Moreover, the qualitative nature of twin formation or the activation of twin variants remain the same if a small notch size is used at the same position.

4 Summary In this work, a simple, mechanistic model for higher-order twinning is implemented. The focus is on the implementation of the transformation pathway branching [8], an issue that is often not appreciated. The accessibility of higher-order twin variants needs special attention, since a simultaneous implementation of all potential higher-order twin variants in one strain energy leads to unphysical behaviour [27]. The model rests on the minimum elastic strain energy approach of a multi-well potential, in conjunction with a viscous regularization. The higher-order twinning is accounted for by a successive updating of the list of individual strain energies (wells) that constitute a set of directly available crystal configurations. The elastic energy invariance between conjugate, compound twins prevents an identification of the activated twin variant from the minimum elastic strain energy approach alone. Therefore, a projection criterion is introduced in order to overcome the indecisiveness of the activated twinning mode. The model is kept simple. The material parameters of the model are only the elastic constants and the geometric information of the twin systems. Additionally, we have a regularizing viscosity parameter that is chosen sufficiently small in order to have a negligible influence on the stresses. The twin system information that is needed is purely geometric in nature and determined by the crystal lattice parameters.

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(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

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Fig. 7 Snapshots of higher-order twinning phenomena in single crystal under deformation controlled test; cross-sectional view parallel to the sample axis. The figures from top to bottom are with increasing time. The left figures are for the twin order, and the right are for the twin variants. a, b The first-order tensile twins. c, d The second-order compression twins in first-order twins. e, f The first-, second-, and the third-order twins

The simplicity of the model restricts its applicability. The model is able to predict the microstructural evolution qualitatively, but for realistic reproduction of the dynamic behaviour, e.g. the stresses or the interface kinetics, a model refinement is required. The same holds when one wants to account for the energy that is stored through the formation of interfaces. It depends on the material under consideration whether this contribution to the stored energy can be neglected or not. We applied the model to titanium, which develops thick twins and hence less interfaces. For example, it is not applicable without further modifications to TWIP steels, which develop very fine twins [14]. The main results are as follows: 1. The elastic law of arbitrary order of twinning is isomorphic to the elastic law of the parent crystal, and thus, connected by a single second-order tensor P. However, one needs two transformations to track the evolution (or twin transformation path), namely one for the deformation, and one for the reorientation. The reason therefore is that the decomposition of P must be tracked, such that the lattice orientation of the twin is known, which is needed for the prediction of the next generation of twins. For first-order twinning, one transformation is sufficient. 2. It is quite challenging to find practical twinning criteria that overcome the indecisiveness induced by the elastic energy invariance of the material model and maintain invariance under Euclidean transformations at the same time. If the polar part of the deformation is neglected, a peculiarity of the conjugate twin systems,

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Fig. 8 Snapshots of higher-order twinning phenomena in single crystal under deformation controlled test; cross-sectional view perpendicular to sample axis. The figures from top to bottom are with increasing time. The left and right figures are for twin order and twin variants, respectively. The legends are the same as in Fig. 7. a, b The first-order tensile twins. c, d The second-order compression twins in already first-order twinned region

Fig. 9 Multiple twinning phenomena in single crystal having 7 perturbations of different sizes under deformation controlled test. The legends are the same as in Fig. 7

namely the fact that d ⊗ n ≈ (d  ⊗ n )T , gives rise to numerically very similar results for both conjugate twin systems. In simulations, this results is an unrealistic element-wise mixing of conjugate twins. This behaviour does not improve when Schmid stresses are considered, although specific elastic laws may have a positive effect. In case of isotropy, the problem is identical. We tried, however, to find a criterion that is independent of the elastic law. 3. To predict the microstructural evolution, a simple, mechanistic model works quite well for materials with relatively thick twins. The simulations of a tensile test of a titanium single crystal show a grain refinement by higher-order twinning, where the sequence of activated twin systems (first tensile, then compression twins) is reproduced correctly. Of course, the simulations require locally a very fine discretization, making the model hard to exploit without an adaptive meshing.

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