A study of dislocation junctions in FCC metals by an orientation dependent line tension model

Acta Materialia 50 (2002) 4873–4885 www.actamat-journals.com

A study of dislocation junctions in FCC metals by an orientation dependent line tension model L. Dupuy a, M.C. Fivel b,∗ b

a Division of Engineering and Applied Science, California Institute of Technology, CA, USA GPM2/CNRS, Institut National Polytechnique de Grenoble, 101 Rue de la Physique BP 46, 38402 St Martin d’Heres Cedex, France

Received 27 March 2002; received in revised form 25 July 2002; accepted 26 July 2002

Abstract The formation and strength of dislocation junctions in FCC crystals have been calculated using an orientationdependent line tension model. The structure of the different types of junctions existing in FCC metals in the absence of an applied stress is examined with particular emphasis on the Lomer–Cottrell lock, the Hirth lock and the glissile junction. We have determined the ‘yield surface’ in stress space corresponding to the dissolution of junctions. Although this model represents a huge simplification of the physics of dislocations, the comparison with more sophisticated models shows that it is able to satisfactorily reproduce both the structure of junctions as well as their response to an applied stress. Moreover, it is in qualitative agreement with available experimental data. It is claimed that this simple model can provide useful parameters related to junction strength in higher level models of single crystal plasticity. Published by Elsevier Science Ltd on behalf of Acta Materialia Inc. Keywords: Modelling; Dislocation theory; Dislocation junctions

1. Introduction The mechanical behavior of FCC metals is controlled by the motion of dislocations which is strongly affected by the various types of obstacles that they may encounter. Among all these obstacles, forest dislocations are a particularly important source of hardening. From a theoretical perspective, the reactions between each of the 12 existing slip systems have been considered by sev-

∗

Corresponding author. Fax: +33(0)476826382. E-mail address: [email protected] (M.C. Fivel).

eral authors [1,2] in terms of long-range interaction, short-range interaction and screw-edge character of the dislocations. It was clearly demonstrated that these interactions strongly depend on the type of each dislocation and that the various resulting junctions must behave in different manners under applied stress. This analysis was subsequently confirmed by meticulous latent hardening tests performed on copper and aluminium single crystals [3,4], culminating in the ranking of the strength of the different junctions [4,5]. Despite several efforts [6–8] to link the strength of individual junctions to macroscopic stress and macroscopic constitutive equations, a large theor-

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etical gap still remains between the behavior of two intersecting dislocations and the behavior of a crystal which involves a statistical ensemble of dislocations. This observation led to the development of numerical models [9] aimed at simulating a large population of dislocations. Preliminary attempts to determine the so-called ‘hardening matrix’ have been performed recently using similar dislocation modelling [10,11]. However, the inherent prohibitive computation time imposes a rough integration of the behavior of individual junctions [12], which in turn, gives rise to large uncertainties over the obtained numerical values. These facts were a driving force for recent work performed by different groups to study individual dislocation junctions on very small scales [13–17]. Indeed, although it is still beyond reach to derive the hardening matrix from the individual behavior of the different dislocation junctions, such studies may be used as an input for higher level modelling, such as dislocation dynamics simulation. Among the different models of individual junctions conducted recently we note the use of atomistic simulations [13–15] and nodal dislocation dynamics simulations [16,17]. These models were able to describe the structure of the junctions as observed in transmission electron microscopy (TEM) [18] and to give the strength of these junctions. A compelling conclusion of these studies was the good agreement between these highly comprehensive models and the more simple model based on line tension approximation [6–8], resulting in a resurgence of interest in this model. Indeed, it can provide a unique and tractable description of junction strength. However, some features of the junction cannot be properly described by the line tension model when the edgescrew character of the dislocation is not taken into into account. For instance, the existence of the Hirth lock cannot be predicted by this model and no distinction between the Lomer–Cottrell junction and the glissile junction is possible. As pointed out by Hirth [1] these problems can be addressed using a line tension model which includes the orientation-dependence of the line tension. In the present study, the formation and strength of dislocation junctions is calculated using a line tension model. By contrast with previous studies

[6,7] the concept of orientation-dependent line energy is explicitly considered. In the first part, the main features of this model are described. Then, the structure of the different types of junctions existing in FCC metals in the absence of an applied stress is examined, namely, the Lomer–Cottrell lock, the Hirth lock and the glissile junction. Finally, the strength of these various junctions is calculated in the stress space giving a picture of their dissolution. Special attention is given to the analytic description of the junctions which could be implemented in higher level models of single crystal plasticity.

2. Orientation-dependent line tension The line tension of a dislocation is a convenient model which allows for the calculation of dislocation motion under an applied shear stress, the most famous application being the Frank–Read source. This model was refined by many authors [19–21] and has been recently reviewed by Mughrabi [22]. In this study, we restrict the discussion to the classical model of de Wit and Koehler [19] which is analytically tractable. Despite the severity of the approximations, the discrepancy between this model and more recent ones is generally rather small [23,24]. The main features of this model are briefly described below. It is commonly accepted that the energy associated with a dislocation is mainly associated with the long-range elastic displacement field induced in the crystal [2], while other contributions such as core energy are generally neglected. The cornerstone of the line tension model is the assumption that the total energy of a dislocation is the sum of the energy of each infinitesimal part, the latter being calculated as if it was a straight dislocation. The total energy of a dislocation may be written as

冕

Etot ⫽ E(q)ds

(1)

C

where q is the the angle between the Burgers vec→

→

tor b and the local tangent vector T , and E(q) is the energy per unit length associated with an infi-

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nite straight dislocation of this orientation. Since E(q) is p-periodic, the energy per unit length associated with a mixed dislocation can be given as a Fourier series, whose coefficients can account for elastic anisotropy [25]. However, we restrict our study to isotropic linear elastic solids characterized by their shear modulus m and their Poisson coefficient n. Thus, E(q) is given by

冉冊

R mb2 (1⫺ncos2q)ln E(q) ⫽ 4p(1⫺n) r0

冕

冕冕

C

sb dxdy

where C is the curve of the dislocation and A is the area swept out by the dislocation. The characteristics of the dislocation are now calculated using the usual variational procedure in which we find the energy minimizing configurations. Let → r 0(s) be the equilibrium parametrization of the curve, → u (s) be an arbitrary smooth →

(2)

where b is the Burgers vector magnitude, R and r0 are, respectively, the outer and inner cut-off radii. As junctions involve several entangled dislocations, it is first important to know the behavior of a single dislocation pinned at both ends. The main interesting features are its shape under an applied stress, the forces exerted on both pinning points and the critical stress required to activate the dislocation as a Frank–Read source. The latter is of importance since we will only discuss equilibrium configurations of junctions. Hence, it requires each participating dislocation to be stable and not to act as a Frank–Read source. We consider the geometry given in Fig. 1. The potential energy of a dislocation under a resolved shear stress s can be written as Ep ⫽ E(q) ds⫺

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(3)

function that satisfies → u ⫽ 0 at both pinning points. Let → r (s,t) be given by →

r (s,t) ⫽ → r 0(s) ⫹ t → u (s)

(4)

where t苸R. → r (s,t) defines an evolving curve with an interface C(t) and an area A(t). Using standard transport identities (e.g. [26]), the derivative of the potential energy with respect to t reduces to the form dEp ⫽ dt

冕冉 冉

⫺␬ E(q) ⫹

冊

冊

d2E → → → (q) N⫺sbN ·u ds 2 dq

C(t)

(5) →

where ␬ is the curvature, T the tangent to the curve →

and N the corresponding normal. From variational arguments, it follows that

冉

⫺␬ E(q) ⫹

A

冊

d 2E → → → (q) N⫺sbN ⫽ 0 . dq2

(6)

Eq. (6) can be interpreted as the balance of forces on each infinitesimal part ds of the dislocation (Fig. 2a). Several conclusions can now be drawn. First, the effective line tension T(q) can be deduced in the celebrated form [19] T(q) ⫽ E(q) ⫹

d 2E (q). dq2

(7)

Eq. (6) can also be written as ⫺ Fig. 1. Geometry of a Frank–Read source. The geometry is characterized by the distance L between the two pinning points as well as the angle f0 between the x-axis and the Burgers vector.

冉

冊

dE → d → → E(q)T ⫹ (q)N ⫽ sbN. ds dq

(8)

The force exerted by each part of the dislocation on adjacent parts (see Fig. 2b) can be identified as (e.g. [27])

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dE E(a⫺f0)cos(a)⫺ (a⫺f0)sin(a) ⫽ ⫹ sby ⫹ A2 dq (13)

Fig. 2. Balance of an infinitesimal part ds of the dislocation. (a) The applied stress field acts on the dislocation as given by the Peach-Koehler force sbds. This force is balanced by the line tension resulting from the curvature of the dislocation. (b) The line tension can be seen equivalently as the gradient of the forces exerted by the surrounding part of the dislocation on the considered segment ds.

where A1 and A2 are two constants. These two constants, as well as aO and aP (Fig. 1), can be calculated directly from the boundary conditions, namely the coordinates of the pinning points O and P. Following de Wit and Koehler [19], Eqs. (12) and (13) can now be used as angle-parametrizations of the dislocation. Rather than using these equations as complex differential equations, they can be used as parametric equations, the parameter being a. This trick enables a rapid and easy plot of the shape of the dislocation. Using Eq. (9), it is worth noticing that the constants A1 and A2 can be interpreted as the components, respectively of the y- and x-axes, of the →

force F O exerted by the dislocation on the pinning point O since dE A2 ⫽ E(aO⫺f0)cos(aO)⫺ (aO⫺f0)sin(aO) dq →

→

→

F ⫽ E(q)T ⫹

⫽ F O·→ e x ⫽ FxO

dE → (q)N. dq

(9)

An interesting feature resulting from anisotropic line tension is that the force is not tangent to the dislocation. Indeed, the second term of this equation tends to rotate the dislocation towards the screw orientation since it has the lowest energy. Eq. (6) can also be written in a Cartesian frame. We recall in fact the differential equations established by de Wit and Koehler [19]

冉 冉

冊 冊

d 2E da cos(a) E(q) ⫹ 2 (q) ⫽ ⫺sb dx dq

(10)

da d2E sin(a) E(q) ⫹ 2 (q) ⫽ ⫺sb dx dq

(11)

where a is the angle between the x-axis and the tan→

gent vector T . Integration of these equations leads to E(a⫺f0)sin(a) ⫹

dE (a⫺f0)cos(a) ⫽ ⫺sbx ⫹ A1 dq (12)

(14) A1 ⫽ E(aO⫺f0)sin(aO) ⫹

dE (a ⫺f )cos(aO) dq O 0

→

⫽ F O·→ e y ⫽ FyO. (15) Since our model only deals with static configurations, it is also important to insure the stability of each dislocation segment. Clearly, we must check whether the shear stress on each dislocation is below the critical shear stress sc needed for the operation of the dislocation segment as a Frank– Read source. This is actually equivalent to determining whether the set of Eqs. (12) and (13) admit solutions for A1, A2, aO and aP. From simple calculations, it follows that the dislocation is at equilibrium until the tangents at the pinning point are parallel. The critical stress sc can now easily be calculated using Eqs. (12) and (13) with the additional equation aP ⫽ aO⫺p.

(16)

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The exact critical stress can be calculated analytically in the case of an isotropic elastic solid and is displayed in Fig. 3 in terms of an adimensional stress sc. The effective critical stress sc is given by

Hirth junctions which have been experimentally determined as the strongest and significant junctions existing in face centered cubic single crystals [3,5].

mb R . sc ⫽ sc ln L r0

3.1. Model

冉冊

(17)

It can be seen that it is more difficult to bend a screw than an edge dislocation between two pinning points [28] which is in good agreement with more complex and realistic models [23]. Moreover, the basic dependence on 1 / L is found. This analysis will also allow us to make a direct comparison between the stress required to nucleate a dislocation and the stress required to break a junction.

We use the model of a junction proposed by Saada [6] as displayed in Fig. 4a. The two dislo→

→

cations with Burgers vectors b 1 and b 2 are pinned at both ends. Their initial lengths are L1 and L2 and the angles between the line of intersection of the two slip planes and these dislocations are y1 and y2. For sake of simplicity, we impose the intersection between the dislocations in the middle of each

3. Junctions In this study, we focus on interactions between dislocations which form attractive junctions. As pointed out earlier [2,6] the stress required for allowing two attractive dislocations to ‘unzip’ is higher than the stress required for allowing two repulsive dislocations to cross. We restrict this study to attractive Lomer–Cottrell, glissile and

Fig. 3. The normalized activation stress of a Frank–Read source as a function of the angle f0 between the x-axis and the Burgers vector. Case of an isotropic linear elastic solid with n ⫽ 0.347.

Fig. 4. Geometry of the junction. (a) Initial configuration. The geometry of the junction is characterized by the angle that the initial segment makes with the line of intersection of the two slip planes (y1 and y2) as well as the lengths of the segments (L1 and L2). (b) Equilibrium configuration. The junction segment [PQ] is characterized by its length L3.

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dislocation. The equilibrium configuration will be modelled as in Fig. 4b, the junction segment hav→

→

ing a length L3 and a Burgers vector b 3 ⫽ b 1 ⫹ →

b 2. The analysis of the junction amounts to finding the energy minimizing configuration for a system made of five interacting dislocation segments on the basis of the line tension model. This is basically equivalent to calculating the equilibrium of the two triple nodes P and Q. Since geometry requires these nodes to belong by construction to both planes (i.e. to the x-axis), the equilibrium of each node requires the sum of the x-components of the three forces induced by the three intersecting segments to be equal to zero (Fig. 5). Whatever the junction type and the applied stress, this can be written as

absence of Peierls stress, each segment is straight and the equilibrium of the junction can be computed easily. Indeed, a junction will form if it is energetically favorable. Using forces, it requires the following equation to be satisfied x0 x0 Fx0 3 ⱕF1 ⫹ F2

(19)

where Fx0 i denotes the x-components of the forces exerted by each dislocation on the intersection node in the initial unbalanced configuration (Fig. 4a).

冦

’ Fx0 1 ⫽ E1(⌽1 ⫹ y1)cosy1 ⫹ E1(⌽1 ⫹ y1)siny1 ’ Fx0 2 ⫽ E2(⌽2 ⫹ y2)cosy2 ⫹ E2(⌽2 ⫹ y2)siny2.

Fx0 3 ⫽ E3(⌽3)

(18)

(20)

where Fx1, Fx2 and Fx3 are the x-components of the forces exerted by each segment on the nodal point

E1, E2 and E3 are, respectively, the line energies of the dislocations L1, L2 and the junction segment. ⌽1, ⌽2 and ⌽3 are the angles between the Burgers vector of each segment and the direction of the junction (Fig. 4a). Eqs. (19) and (20) are an extension of earlier studies [1,7] in which the orientation-dependent line energy was only partially taken into account. We demonstrate our results by first considering the equilibrium configuration of a Lomer–Cottrell lock in A1 as shown in Fig. 6. This configuration has been chosen so as to make a direct comparison of our results with atomistic simulations [15] and nodal dislocation dynamics calculations [17]. Prior to junction formation, the dislocations have a length L1 ⫽ L2 ⫽ 300°A and lie at an angle y1 ⫽ y2 ⫽ 60° with the line of intersection of the two slip planes. The material parameters can be restricted to the Poisson coefficient (n ⫽ 0.347 [2]). No specific values for the Burgers vector, the shear modulus and the ratio of the cutoff radii are required: since each dislocation is straight, it can easily be seen from Eqs. (18) and (19) that mb2ln(R / r0) vanishes in the balance of each triple node. Despite the crudeness of the line tension model, the agreement with the other models is satisfactory since the length of the junction is well rendered. However, asymmetry of triple nodes

Fx3 ⫽ Fx1 ⫹ Fx2

→

in question. The forces F i(i ⫽ 1,2,3) are given by Eq. (9) on the basis of the line energy model of isotropic solids. 3.2. Junctions without applied stress We consider first the case where no shear stress is applied on the relevant slip systems. In the

Fig. 5.

Equilibrium of a triple node (P or Q).

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Fig. 7. Equilibrium configuration of two symmetric junctions. The critical configuration is displayed on the right: y ⫽ yc. Hence, the triple nodes are balanced and superimposed. On the left, equilibrium of P and Q requires g ⫽ yc.

Fig. 6. Structure of a Lomer–Cottrell lock in Al. L ⫽ 300°A and y ⫽ 60°. The length of the junction segment L3 is equal to 97°A.

[15,17] and local curvature cannot emerge from our model. The most compelling common feature of the different models is the equilibrium length of the junction segment as a function of the initial angle between dislocation arms and the line of intersection of the two slip planes. We now consider the symmetric case L ⫽ L1 ⫽ L2 and y ⫽ y1 ⫽ y2. In that case, the length of the junction segment L3 can be easily calculated as a function of L and y with the line tension model. Let yc be the angle y which satisfies Eqs. (20) and (19) as a strict equality. Hence, every dislocation pair intersecting at an angle below yc will form a junction. Above this critical angle, the junction will not be energetically favorable. More precisely, we define y⫺ c (ⱕ0) and yc⫹ (ⱖ0) such that the range of existence of a sym⫹ metric junction is [y⫺ c ;yc ]. It should be noticed ⫺ ⫹ first that yc and yc are independent of the length L of the two parent dislocations. The role of yc can be extended, as shown in Fig. 7. Indeed, as given by Eq. (18), the equilibrium of each triple point (P and Q) only depends on the angle g ( ⫽ g1 ⫽ g2) between the line of intersection of the two slip planes and the dislocation arms. It is therefore independent of the distance between the two

triple nodes, namely the length of the junction segment L3. It follows that

再

g ⫽ y⫺ c for y ⬍ 0

g ⫽ y+c for yⱖ0

.

(21)

A straightforward geometrical calculation gives the length of the junction segment as

冦

L3 ⫽ L

sin(y⫺ c ⫺y) for y ⬍ 0 siny⫺ c

sin(y+c⫺y) L3 ⫽ L for yⱖ0 siny+c

.

(22)

Several conclusions can be drawn in the light of this argument. First, the length of the junction segment simply scales with the length of the parent dislocations. Moreover, Eq. (22) is valid within the context of the line tension model whatever the line energy E(q). Hence, it applies to any anisotropic elastic solid whereas the features of each model are included in the critical angles. Values of y⫺ c and yc⫹ are reported in Table 1 for different types of junction in the case of an isotropic linear elastic solid with different Poisson coefficients n. In this particular case, it follows from Eqs. (9) and (18) ⫹ depend neither on the initial that y⫺ c and yc length of the dislocations, nor on m, b and R / r0. The results obtained with the line tension method are compared with other models in Fig. 8. As reported earlier [17] the agreement between these models is fairly good. It is suggested that the accuracy of this model is linked to the independence

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Table 1 Existence range of different types of symmetric junctions in the case of an isotropic linear elastic solid Poisson coefficient

n ⫽ 0a

Junction type Lomer–Cottrell lock Glissile junction Hirth lock

y⫺ c ⫺60.0° ⫺60.0° ⫺0°

a b

n ⫽ 0.347b yc⫹ +60.0° +60.0° +0°

y⫺ c ⫺51.0° ⫺46.0° ⫺30.3°

yc⫹ +78.0° +60.0° +30.3°

The case n ⫽ 0 corresponds to the non-orientation dependent line tension model as considered by Saada [6]. The case n ⫽ 0.347 corresponds to aluminium.

Fig. 8. Junction length of the Lomer–Cottrell lock in Al as a function of the initial line directions of the participating dislocations. L ⫽ 300°A. Comparison with a nodal model [17] and a quasicontinuum model [15] shows a good agreement.

of the final result on parameters which are not well characterized, such as R / r0. In light of these promising results for the Lomer–Cottrell lock, the glissile junction and the Hirth lock were also considered. The results are displayed in Fig. 9 in the case of symmetric junctions. The junction length, normalized by the initial dislocation length, is shown as a function of the intersection angle. Comparison with the earlier model of Saada [6], in which the line energy of a dislocation was taken as amb2, indicate several noticeable features. First, as pointed out earlier in the case of the Lomer–Cottrell lock [12,17], the

Fig. 9. Normalized junction length of the Lomer–Cottrell lock, the glissile junction and the Hirth lock in Al as a function of the initial line directions of the participating dislocations.

agreement between the line tension model and atomistic and nodal simulations is improved when orientation-dependence of the line energy is considered. Indeed, the Hirth lock exists in the context of the present model (Table 1) whereas this lock cannot be described with the isotropic line tension model (n ⫽ 0) since the force exerted by the junction (Fx3 ⫽ amb23 ⫽ 2amb2) is the double of the upper-bound of the force of each arm. Thus, the equilibrium of the triple nodes as given by Eq. (18), can only be attained in the degenerate case y1 ⫽ y2 ⫽ 0°. This limitation can be overcome using the orientation-dependent line energy model, as shown in Fig. 9. To the authors knowledge,

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neither atomistic nor nodal simulation results are available for a Hirth lock junction: the comparison with these models could undoubtedly be a more discriminating and definitive test for the accuracy of the line tension model than the Lomer–Cottrell lock, since the Hirth lock pushes this model to its limits. Finally, a distinction between the Lomer– Cottrell lock and the glissile junction can be made on the basis of the orientation dependent line tension model (Fig. 9) by way of contrast with the earlier model. Despite the fact that the Burgers vector of both junctions is of ⬍ 110 ⬎ type, the parent dislocations do not have the same orientation with respect to their Burgers vectors. As an illustration, the junction of the Lomer–Cottrell lock is an edge dislocation while the segment PQ of the glissile junction has a mixed edge-screw character. A direct comparison between the various junctions can also be made in light of Fig. 9. The Lomer–Cottrell lock forms the longest junctions while the Hirth lock forms the shortest junctions. This is again in good agreement with earlier work [29] which pointed out that the length of a junction should increase with increasing energy gain of the reaction as calculated by Hirth [1]. In addition, this ranking is in accord with experimental results. Indeed Lomer–Cottrell locks are frequently observed in FCC solids with the transmission electron microscope, while Hirth locks are very rare (e.g. [18]). Lomer–Cottrell locks are likely to be formed whenever two appropriate dislocations intersect each other since the junction reaction is energetically favorable on a wide angular domain ([⫺51.0°; 78.4°] in a symmetric configuration— Fig. 9). On the other hand, Hirth locks are seldom observed experimentally since it requires that two dislocations intersect with a small angle ([⫺30.3°; 30.3°] in a symmetric configuration—Fig. 9). In summary, the orientation-dependent line energy model is in good agreement with both experimental results and more elaborate junction simulations, such as atomistic and nodal models. It confirms that line tension is the main parameter governing the equilibrium of junctions when no stress is applied.

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3.3. Strength of junctions The simulations of junctions under applied stress are carried out through the line tension model in isotropic linear elastic solids described earlier. The material parameters are the same as in the previous section. For the sake of simplicity, we only consider the symmetric junctions with L1 ⫽ L2 ⫽ L and y1 ⫽ y2 ⫽ y. We start our simulations with the equilibrium configuration of the junction in the absence of applied stress. The resolved shear stresses on each dislocation are linearly increased with an appropriate increment until the critical breaking stress is reached. The equilibrium configuration is tracked at each step by adjusting the position of P and Q of the junction in order to satisfy the equilibrium of each triple point [Eq. (18)]. Thus we do not consider dynamic movements of the dislocations, and the critical stress is decreed to be attained when no global equilibrium position can be found. In addition, our model allows for insight into the mechanism of the breaking of the junction. In fact, two possible mechanisms are considered. First, one of the two triple points may have no equilibrium position, leading to the breaking of the junction by the so-called ‘unzipping’ process as proposed by Saada [6]. Alternatively, the junction can also be destroyed when the equilibrium position of the triple points is the same, leading to a junction length equal to zero. Furthermore we considered the stress space corresponding to all resolved shear stresses acting on the junction. The ‘yield surface’ corresponding to the dissolution of the junction is plotted in this space. In particular, this stress space corresponds to s1 and s2, the resolved shear stress on the slip system of each initial dislocation, in the case of Lomer–Cottrell and Hirth locks. The glissile junction requires an additional dimension s3 corresponding to the resolved shear stress on the segment PQ which is a glissile dislocation. However this third dimension was not considered to avoid more intricate configurations where segment PQ intersects the arms of the initial dislocations. Therefore s3 was chosen equal to zero. A significant motivation of this study was to establish convenient formulas to describe the yield surface Y of junctions as a function of their

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geometry and the elastic parameters of the medium: Y ⫽ Y(m, n, b, R, r0, L, y). The resolved shear stresses s1 and s2 can be first normalized in terms of s1 and s2 as follows s i ⫽ si

冉冊

mb R ln . L r0

(23)

The dependence of the yield surface on y can be further added owing to the results of extensive number of simulations. Indeed, it was shown that junctions usually break because one of the two triple nodes eventually becomes unstable as the applied stresses increase. As an example, simulations on Lomer–Cottrell locks in aluminium revealed that junctions undergo this ‘unzipping’ process over a wide angular range (y苸[⫺40°; ⫹ 63°]) whatever the loading direction. Comparison with the total existence range of the Lomer– Cottrell lock ([⫺51°; ⫹ 78.4°]) indicate that it can be considered as the mean breaking process. The alternative mechanism and its influence on the yield surface will be discussed subsequently. The angular dependence can now be demonstrated as follows. Let’s consider a junction which yields because of this unzipping mechanism. We can restrict our analysis to the triple node which will eventually attain an unstable configuration as displayed in Fig. 10. The equilibrium position of this triple point depends on the forces Fx1, Fx2 and Fx3 exerted by the two dislocations and the junction segment. However, Fx3 does not depend on the length L3 since geometry constraints the junction segment to lie in the x-direction. Thus, the behavior of Q will be identical for every junction with the same distance h ⫽ (L / 2)siny. In particular, these

Fig. 10. The equilibrium of the triple node Q is independent of the length L3 of the junction segment.

junctions will yield under the same loading conditions. Thus, stresses can further be normalized as

冉冊

mb R . s i ⫽ si ln Lsiny r0

(24)

The yield surface Y(m, n, b, R, r0, L, y) for each type of junction can now be synthesized as a reference yield surface y(n) using Eq. (24). Actually, two normalized yield surfaces must be defined per junction, depending on whether y is positive or negative. The argument is similar to the distinction between positive and negative angle y in the case of junction without applied stress. Several interesting features can be deduced from Eq. (24). An important prediction of our model is the scaling of the breaking stresses of symmetric junctions with the distance L between the pinning points: this scaling law was obtained from several more sophisticated junction models [13,15,17] and was already demonstrated with a uniform line energy by Saada [6]. Furthermore, it allows the direct comparison between the strength of a junction and the stress required to multiply dislocations, namely the activation stress of a Frank– Read source. Assuming similar characteristic length L, Figs. 3 and 11 can directly be compared independently of m, b, ln(R / r0) and L for different angles y. It results that Lomer–Cottrell locks and glissile junctions are noticeably stronger than Frank-Read sources over a wide range of angles. Unsurprisingly, this underlines the importance of these junctions in the work-hardening of metals. On the other hand, weaker junctions, such as Hirth locks, exhibit limited strength with regards to the stress required to multiply dislocations and therefore may be neglected in first approximation. This comparison may provide useful inputs for high order models involving a large number of dislocations (e.g. the edge-screw model [11]) since it points out the relevant junctions. The normalized yield surfaces y(n) of the three junctions are presented in Fig. 11 in the case of n ⫽ 0.347. It first emphasizes the necessity to take into account both resolved shear stresses s1 and s2. Owing to calculation complexity, earlier studies on line tension models only considered particular stress fields [6–8], one of the resolved shear stresses being generally fixed to zero. In order to

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destruction. The symmetry property of the yield surfaces results from the symmetry of the two triples nodes in our model. It was indeed pointed out by Shenoy [17] that this symmetry is lost when considering partial dislocations. It is interesting to compare the strength of the three different junctions considered in this study in light of available experimental data. A quantitative comparison is obviously difficult since latent hardening tests involve a very large number of junctions and different geometries. However, several qualitative observations can be made. Let’s first consider symmetric junctions with a given angle y ⫽ y1 ⫽ y2. Fig. 11 clearly demonstrates that the stress required to break a junction depends on the type of junction, culminating in the following ranking crit crit scrit Lomer-Cottrell ⬎ sglissile ⬎ sHirth.

Fig. 11. 1. Reference yield surface y of Lomer–Cottrell lock, glissile junction and Hirth lock in aluminium. The stresses are normalized using Eq. (24). (a) y ⬍ 0. (b) yⱖ0.

establish the link between the breaking of a single junction and latent hardening tests (e.g. [3]), it is necessary to consider the complete stress field. On the other hand, no attempt was made in this study in averaging the breaking stress conditions to predict quantitative hardening parameters. In addition, the yield surfaces can be divided into two antisymmetric parts about the line s2 ⫽ s1 (or s1 ⫽ s2). The part on the right (resp. left) corresponds to the disequilibrium of P (resp. Q) leading to junction

(25)

Simulations with different intersecting angles were performed with our model and confirmed this proposed ranking. This result is actually in good agreement with experimental results on aluminium and copper [4,5], since the measured hardening parameters are in the same order for these three junctions. Indeed, both the probability to form a junction and stress to break it are higher for a Lomer–Cottrell lock than for a Hirth lock. Moreoever, our model overcomes some of the flaws of the orientation-independent line energy model. First, as pointed out earlier, the Hirth lock is stable within the present model and its strength can be calculated. Furthermore, a ‘natural’ distinction between a glissile junction and Lomer–Cottrell lock can be made. Indeed it was generally suggested [3,7] that their main strength difference is linked to the glissile or cessile character of the reactant junction. In light of our model, this is not the only possible explanation and further simulations and experiments should be done to balance these possible effects. For completeness of this analysis, the yield surfaces were also studied in the angular range where the junctions may break owing to a mechanism other than unzipping. In this domain, the junction segment may fade before one triple point becomes unstable. Hence, the yield surfaces are necessarily smaller than predicted by Eq. (24). Fig. 12 displays

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4. Conclusion

Fig. 12. Normalized yield surface of Lomer–Cottrell locks in aluminium for different angles y over +63°. In this range, the angular dependence is no longer captured by Eq. (24).

the yield surfaces of Lomer-Cottrell locks in this critical range [ ⫹ 63°; ⫹ 78.4°]. It can be seen that the shrinkage of the yield surfaces is limited until y reaches values very close to the critical angle yc⫹ ⫽ 78.4°. As a consequence, Eq. (24) may be used on the full existence domain of each junction without significant bias. A comparison of the predictions of the line tension model, with more sophisticated simulations involving atomistic, nodal or edge-screw models, would be attractive. Indeed, many features are missed in our model, such as elastic anisotropy, jogs, core structure, dissociation or long range interactions. The ratio R / r0 has to be given first, since it is a specific ‘defect’ of the line tension. Following classical assumptions (e.g. [30]), the outer cut-off radius R may be taken as the typical size of the studied object which would be here the length of the parent dislocations L. Full comparison with other models is now possible. These results are indeed in agreement with an earlier nodal model [17] indicating that the strength of a junction is mainly controlled by the line energy of the reacting dislocations. However, it should be noticed that this point is currently debated [31], and this conclusion may not be a general rule.

The formation and strength of dislocation junctions in FCC crystals have been calculated using a line tension model. The concept of orientationdependent line energy is explicitly considered. It allows for a distinction to be made between Lomer–Cottrell locks and glissile junctions and to consider the Hirth lock. On that basis, a good agreement between this simple model and experimental results is found. It has also been demonstrated that the strength of a symmetric junction simply scales as (mb / L siny) ln(R / r0). Finally, comparison with more detailed simulations indicates that, as for Frank–Read sources, the governing feature is line tension. It suggests that other contributions, such as jogs, dissociation and long range interaction, may have smaller effects on junctions. The gap between the strength of a single junction and the work-hardening of a polycrystal is still huge and cannot be bridged by this model. However, it implies a limited number of parameters and provides results with a sensible precision. Therefore this simple and tractable model can provide useful parameters related to junction strength in higher level models of single crystal plasticity. In addition, this model might demonstrate a similar accuracy in other types of materials such as bodycentered metals.

Acknowledgements We are grateful to K. Bhattacharya, D. Rodney, R. Phillips, V. Shenoy and D.M. Weygand for illuminating discussions. Part of this work was supported by the Caltech ASCI program.

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