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A model of dislocation multiplication at a crack tip: influence on the brittle to ductile transition
Materials Science and Engineering A272 (1999) 83 – 89 www.elsevier.com/locate/msea
A model of dislocation multiplication at a crack tip: influence on the brittle to ductile transition Ge´rard Michot a,*, M.A. Loyola de Oliveira b,1, G. Champier a,2 b
a Ecole des Mines de Nancy, Parc de Saurupt, 54042 Nancy, France Uni6ersidade Federal de Espirito Santo, C.T., A6. Fernando Ferrari, Goiabeiras, Vitoria 29000 E.S., Brazil
Abstract It has been shown long ago experimentally that primary nucleation of dislocations in silicon takes place heterogeneously on defects along the crack tip. More recently, it has been observed that a source is easily activated at the intersection point of the crack front and of an attracted dislocation. The authors offer and discuss a source multiplication mechanism based on this stimulated emission process. One of the dislocations emitted at a primary source on the plane of maximum resolved shear stress cross-slips to a plane where it is attracted by the crack: the intersection event gives rise to a secondary source. Because shielding is very low at this point, there occurs an emission of a new bundle of dislocations. The process then starts again giving rise to an ‘avalanche multiplication’ of dislocations which strongly shield the crack. Soft/sharp brittle to ductile transitions (BDT) observed in semi-brittle materials result from such a high shielding rate coupled with a low/high threshold stress intensity factor for the activation of the primary sources. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Brittle to ductile transition; Crack; Dislocation source; Shielding; Silicon
1. Introduction For a cracked body without bulk sources, the emission of dislocations is restricted to the crack tip. Such a generation has been successfully investigated experimentally in silicon through in-situ X-ray topography [1]. It has been shown that the crack front’s ability to generate dislocations is decreased by improving the crack quality. This means that nucleation takes place heterogeneously on defects along the tip, such as the intersections of cleavage ledges with the front. This statement, which has been claimed long ago by one of
This paper is dedicated to Professor Herbert Herman on the occasion of his 65th birthday. * Corresponding author. Tel.: + 33-3-8358-4158; fax: + 33-3-83579794. E-mail addresses: [email protected] (G. Michot), [email protected] (M.A. Loyola de Oliveira) 1 Tel.: +55-27-335-2851; fax: +55-27-335-2650. 2 Tel.: +33-3-8358-4158; fax: +33-3-8357-9794.
the authors, is contrary to the usual assumption of homogeneous nucleation [2–4]. For this reason, introduction of defects in the emission models is relatively recent and still incomplete [5,6]. This is probably due to some reluctance to accept the fact that a material’s property, as important as the brittle to ductile transition (BDT), could be extrinsic and, as a consequence, could not be modelled in a one-to-one way. With regards to sources, only a few experimental results are available and most of them have been obtained on silicon by the authors. They have derived in particular that the number of dislocation sources increases with time during the test [7]. In this paper, the main features of crack/dislocation interactions (Section 2) will be recalled first before presenting outstanding results about the BDT as is reported in the literature (Section 3). Then, emphasis will be put on sharpness of the transition (Section 4) and its link with dislocation source multiplication at the crack tip (Section 5). Finally, conclusions will be given (Section 6).
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2. Crack and plasticity In a loaded cracked body the stress intensity factor K measures the strength of the singular term of the elastic solution of the stress field. In brittle materials, a crack propagates when the applied stress intensity factor KA reaches a critical value KIC which is a material’s characteristic called toughness. On the other hand, complete relaxation in plastic materials should lead to the disappearance of the singular solution in r − 1/2, i.e. to an effective stress intensity factor KE equal to zero, KE being understood as the decrease of KA to zero induced by plasticity. This is confirmed by all calculations performed on the plastic zone developed at the crack tip in non-hardening materials [8,9]. In materials like fcc metals, characterised by a the high dislocation mobility at any temperature, achievement of a full plastic relaxation at crack tip is practically time independent. In bcc or covalent materials, due to the strong influence of temperature on dislocation generation and/or mobility, there is a restriction on dislocation activity incompatible with a full plastic relaxation at crack tip: KE is different from zero and strongly time dependent under dynamical loading. It has been checked, using equilibrium models [10 – 15], that full relaxation is avoided through the introduction of a dislocation free zone between the crack and the plastic zone. Apart from these equilibrium situations, it is obvious that as soon as a material is unable to provide enough dislocations (low number of sources, high threshold value of their activation stress etc.) or to move them fast enough (temperature limited mobility) to fully relax a dynamically loaded crack, a dislocation free zone will appear. The stress relaxation, i.e. the decrease of the applied stress intensity factor KA to an effective value KE, results from the shielding of the crack by the dislocations. The shielding concept was in fact first introduced on the microscopic level [2] before its applications were extended to continuum mechanics [11]: the isolated dislocation is considered as an internal loading with an associated stress intensity factor kD. The formalism was detailed for mode III shielding [16] and then extended to the general case [17,18]. The physical reality of shielding has been experimentally checked for silicon on plastic zones developed to saturation under creep at high temperature and then frozen by cooling under load. Room temperature fracture implies that there have to be stresses up to three times larger than those required for dislocation free crystals [19]. In materials which exhibit limited plasticity one intuitively see the roles played by dislocations and their discrete sources on fracture, i.e. dislocation modelling should be preferred to continuum mechanics. In these materials, when a crack is loaded under a constant opening or loading rate, there is a competition between the increase in stored elastic energy due to the applied
load on the one hand, and the plastic relaxation induced by emission and motion of dislocations on the other hand. If the plastic zone’s growth rate (i.e. the shielding increase rate) is controlled by dislocation activity (i.e. mainly by temperature) one may expect a transition from a low energy-absorbing cleavage mode (at low temperature) to a high energy-absorbing ductile mode (at high temperature). This BDT temperature TC is controlled both by physical parameters (threshold value Kmin for dislocation emission, initial number of sources, multiplication and mobility of dislocations, material’s toughness) and by geometrical parameters (slip crystallography, crack opening mode). Since the plastic zone’s growth rate is time-dependant, the transition temperature TC is sensitive to the loading rate dKA/dt and thus cannot be a material’s intrinsic characteristic.
3. Influence of dislocation sources on the BDT St John [20] first investigated the BDT in mode I on pre-cleaved silicon single crystals tested under a constant opening rate dd/dt. The transition occurs within a very narrow temperature range and the transition temperature TC becomes opening-rate dependent with an activation energy close to 2 eV over the temperature range 973–1223 K: dd/dt (or dK/dt) a exp(− Q/kTC)
(1)
More recent experiments [21] performed on silicon confirmed this value. The kinetic process of dislocation motion is a thermally activated glide process with an activation energy close to 2 eV in silicon. Thus it has been long ago admitted that the transition was controlled by the dislocation’s mobility. The same conclusion holds true for other materials such as Ge, GaAs, Mo and Al2O3 [22]. Experimentally, large shifts are indeed noticed in the Arrhenius plot between the strain/stress rate and the converse of the BDT temperature TC in silicon (Fig. 1), according to the crystal’s orientation, origin (float-zone or Czochralski), purity (intrinsic, doped, oxygen containing, hydrogen charged etc.) and thermo-mechanical treatments. Simple shifts with a constant slope must result from differences in the source’s activities while shifts with slope variations imply other thermally activated mechanisms (nucleation, interactions of the moving dislocations with obstacles, etc.). Thus the shifts noticed in Fig. 1 mainly result from differences in the dislocation source’s densities. In initially dislocation-free silicon crystals, emission of dislocations is restricted to the crack tip. Direct observations of sources [23] prove that such dislocation nucleation is highly inhomogeneous. The nucleation sites are mainly defects along the crack tip since the
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BDT temperature is raised when the cleavage quality is improved [21].
4. The sharpness of the transition Any attempt to predict the BDT temperature will mean dealing with the evaluation of the dislocation activity. It could be possible to by-pass this difficulty by assuming an equilibrium plastic zone growth but, first of all, this assumption is not realistic (experiments confirm this point) and second of all, in covalent crystals the friction stress must be understood as a dynamic parameter since the flow stress which defines it is very sensitive to the strain rate. The best way is to use dislocation modelling introducing explicitly dislocation mobility and to assume that fracture takes place when the effective stress intensity factor KE(t) =KA(t)+ k (i) D (t) overcomes KIC. If not, a ductile opening will be observed. The individual contribution k (i) D (t) of dislocation loop (i) to crack shielding, depends on its size and position at time t. The total contribution is assumed to be negative. Hirsch et al. [24] developed a non equilibrium two dimensional (2D) simulation on this basis. Even though they could not avoid a few assumptions on the dislocation emission conditions, they pointed out the fundamental concept of a critical distance between sources dC below which all vulnerable points z along the crack profile can be shielded (KE(z) BKIC) under given experimental conditions. The new and fundamental feature of this model is that the transition is now defined by two scales, the dislocation free zone size derived from the simulation, and a microstructural parameter, the spacing between dislocation sources.
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Since usually thermally activated processes exhibit continuous variations over wide ranges of temperatures, it appears difficult to some authors [25] to explain the fast increase in material reinforcement observed in silicon within a few degrees below the transition through a simple dislocation shielding mechanism. Indeed, a continuous and steady increase in shielding should lead to a variation of the effective stress intensity factor KE(t) at the most vulnerable point along the crack front close to that imagined Fig. 2(a). A soft transition is observed since fracture takes place for applied KA values, KC, much larger than the material toughness, KIC, at temperatures just below TC. Conversely, for a sharp transition, a rapid increase in shielding (Fig. 2(b)) is required: how to explain such an increase? The influence of the different parameters (nucleation rate, mobility, loading rate…) on the shielding amplitude have been appreciated through simulation [24]. In order to account for the sharpness of the BDT, the first emission must start at a KA value close to KIC, and be followed by rapid emission of bundles of dislocations under a much lower stress level. However, by expressing the nucleation condition in terms of stress rather than in terms of KE, Maeda [26] obtained a sharper transition with less assumptions, while Brede [27] or Xin and Hsia [28] got the same result by accelerating the shielding process through an increase of the number of activated glide systems. One may think that the number of involved slip planes is important but that alone cannot explain the shielding evolution. Michot et al. raised the point that the 2D simulations do not deliver the proper kinetics since they make abstraction of the evolution of the population of dislocation sources during the test. The
Fig. 1. Arrhenius plot between the loading rate dKA/dt and the inverse of the transition temperature TC, as collected by Hirsch and Roberts [22]. Each curve (A – H) corresponds to specific test samples, procedure of introducing the crack, crystal origin or purity. The activation energy for the brittle to ductile transition (BDT) measured in doped silicon (curves B and C) differs from that measured in intrinsic silicon ( 2 eV, curves A, D– H). The activation energy of the glide process in doped silicon also differs from that measured in intrinsic silicon ( 2 eV). For these reasons it is admitted that the dislocation’s mobility imposes the slope of the plot while differences in the dislocation source’s activities lead to the large shifts observed (from F to H, for instance).
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Fig. 2. Assumed variations with time of the shielding contribution KD induced by the plastic zone (negative) to the effective stress intensity factor KE = KA + KD at the most vulnerable points along the crack front, for a constant loading rate dKA/dt. The assumed test temperature is just above the brittle to ductile transition (BDT), i.e. KE does not overcome the toughness KIC of the material. When the threshold value KMIN for emission of dislocations is low (a), the plastic zone has time enough to develop and shield the crack before to be close to fracture: a soft transition is observed (the critical stress intensity factor for fracture KC can be large). When KMIN is large (b), a very high shielding rate is required in order to get the same instability: a sharp transition is observed (KC is close to KIC).
authors derived from in-situ measurements [7] that, close to TC, the dislocation nucleation rate should be proportional to the product KA dKA/dt, i.e. the number of sources should increase with t 2. Furthermore, recent experiments on stimulated emission [7] show that as soon as a first dislocation is available at a crack tip, dislocations are easy to create, even under a very low stress level. Dislocations multiply through an avalanche mechanism, i.e. the dislocation multiplication rate resulting from the activation of new sources is much higher than that resulting from mobility controlled emission from a single source. This is one of the conditions that leads to a sharp transition according to the Hirsch et al. simulation. Nevertheless, this condition is not sufficient. If this multiplication mechanism always functions, how can one justify a soft shielding evolution like the one proposed in Fig. 2(a)? Obviously, the sharpness of the transition depends on another parameter. According to the stimulation by Hirsch et al. [24] a high threshold value KMIN for emission of the first dislocation is also required to get a sharp BDT. Thus, one has to find a physical microscopic mechanism of creation of new sources whose efficiency will be stress dependent.
5. Multiplication mechanism in silicon Little is known about the nature of nucleation sites and the detailed dislocation mechanisms which operate in order to form the plastic zone that shields the crack
tip, preventing brittle failure above the transition temperature. In the absence of bulk sources, primary nucleation takes place heterogeneously on defects along the crack tip [1,7]. The heterogeneous character of dislocation emission suggests that the BDT is controlled neither by the condition for homogeneous emission [2–5], nor by a co-operative dislocation generation instability linked to thermal fluctuations [25]. The consequence of heterogeneous nucleation is that the BDT is shifted to a higher temperature when the quality of the cleavage surface is improved [21], i.e. when the dislocation source spacing along the tip is increased. Apparently, this effect can be huge since shifts of hundreds of degrees of the transition can be observed (Fig. 1). Thus it would be an exaggeration in itself to say that the BDT is dislocation mobility controlled. A predictive evaluation of the potential sources should combine an experimental inventory of the crack front defects and a theoretical evaluation of their strength. In fact, the experimental observations were insufficient to identify the nucleation sites and the available calculations were not precise enough [5,29,30]. In some cases the emission site coincided with an observable ledge. These sources were called ‘primary’(SPR). In other cases dislocation trails were emitted from spots where no ledges could be detected by optical microscopy or scanning electron microscopy (SEM). However, since dislocations can be emitted at a quasi perfect crack at the intersection point of the front and the slip plane of an attracted dislocation [7], sources are not necessarily associated with pre-existing defects. The new sources activated during plastic zone growth by this ‘stimulated emission process’ are called ‘secondary’(SSEC). Fig. 3 shows the position of the Thompson tetrahedron with respect to the chosen crack orientation. A few a/[2011] dislocations are supposed to be emitted at a defect on the (1( 11( ) slip plane and to travel a given distance from this primary source SPR. If the positive resolved shear stress acting on it is overcome by a negative one acting in the secondary (1( 1( 1) plane, crossslips towards the latter plane (Fig. 4) is mechanically authorised: the P1P2 screw segment of the dislocation bulges and moves to the point SSE where it intersects the crack tip. The secondary source thus triggered emits a new trail of dislocations and the process starts again. Several arguments support this assumption, even if, due to the limited resolution of the X-ray topography, this mechanism has not been directly observed. (i) Firstly, concerning at least one specific crystallographic orientation, many evidences of extensive cross slip during plastic zone growth have been reported in silicon [31]. (ii) Secondly, the mechanical feasibility of the previous cross-slip process has been checked. The attraction or the repulsion of a dislocation by the stress field of
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Fig. 3. One of the crystallographic orientations designed for mode I loading of silicon samples cleaved along the (111) plane [1,21]. The Thomson tetrahedron shows how available slip systems are oriented with respect to the horizontal crack tip. Four dislocations with Burgers vector a/2[011] are supposed to be emitted at a defect along the crack front along a given (1( 11( ) plane. The source associated with a defect is called primary source (SPR). One assumes cross-slipping of the P1P2 screw segment of the fourth emitted dislocation into the (1( 1( 1) plane. This plane cuts the crack front at point SSE.
the crack depends on the sign of the Burgers vector. One assumes that the detected dislocations obey the growing conditions (the high density of dislocations within the plastic zone does not allow the sign’s determination through X-ray topography). Since these requirements cannot be expressed directly through the classical Schmid factor because of the non-uniformity of the stress field, results are expressed in the form of equal-stress contours, as explained in details in Ref. [1]. For instance, for a given a/2[011] Burgers vector and a given orientation of the dislocation line the {a/ 2[011](1( 11( )} slip system fulfils the growing condition
Fig. 4. According to the stress analysis around the crack tip (mode I loading, orientation a, [1]), the P1P2 segment (burgers vector a/2[011]) is attracted by the crack in the (1( 1( 1) plane. Hence cross-slip can take place and the segment P1P2 bends into the (1( 1( 1) plane until it cuts the crack front and triggers a secondary source SSEC, through the stimulated emission process [7]. This source is not necessarily associated with a defect.
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(mode I loading, orientation a [1]) while the {a/ 2[011](1( 1( 1)} slip system fulfils the shrinking conditions. The dislocation is repelled by the crack tip in the first case, attracted in the second case. One assumes that the previously mentioned {a/2[011](1( 11( )} slip system, which extends experimentally far away from the crack tip, is submitted to the highest stress. For what reason should the dislocation move out of this plane? Under such a non-equilibrium situation, the last emitted dislocation is slowed down by the back-stress of the previously emitted dislocations: the higher the number of dislocations in the inverse pile-up, the smaller the stress pushing the internal dislocation in the (1( 11( ) plane. Since the {a/ 2[011](1( 1( 1)} slip system fulfils the shrinking conditions the P1P2 screw segment can be attracted by the crack tip, thus cross-slip takes place. Furthermore, 3D calculations [32] show that the length of crack tip concerned by the shielding effect is smaller than the loop radius: this means that the primary source only is strongly shielded, the dynamical pile-up is no longer feeded with fresh dislocations able to maintain a stress level high enough to keep the dislocation in the primary plane. Depending on the experimental conditions (temperature, load and loading rate) there must be a critical number of emitted dislocations above which cross-slipping takes place and leads to stimulated emission at point SSE. (iii) Lastly, this formal reasoning is partly supported by observation of etch pits on a cracked sample relaxed under creep conditions [33]. On the outer part of the plastic zone, a small number of rows of pits, parallel to the intersection of the (1( 11( ) plane with the sample surface and containing less that a few tens of a/2[011] dislocations are observed: these well defined rows are connected with the activity of individual primary sources. Oppositely, it is difficult to find traces of slip along the primary plane in the inner part of the plastic zone where pits distribution is much more homogeneous. One can conclude that plastic relaxation starts by the triggering of a few primary sources emitting a few tens of dislocations, rapidly followed by stimulation of many new secondary sources emitting less dislocations still. Is this mechanism stress sensitive (second requirement of the conclusion of Section 4)? Experimentally, etch pits counting on silicon samples loaded under creep condition for 2 h [33], indicate a very fast increase of the number N of dislocations within the plastic zone with the applied load: NK 8A! The rate of creation of sources (the converse of the lapse of time needed by the dislocation to travel from the primary source SPR to the secondary one SSE (Figs. 3 and 4)) is proportional to the dislocation’s mobility (stress exponent close to one in silicon) and to the converse of the distance d travelled by the dislocation. When the threshold value for source activation, KMIN, is small, growth conditions are
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close to some ‘equilibrium’ conditions, i.e. all the dislocations feel nearly the same stress field, cross-slip is note likely to occur the rate of activation of new sources is very low. For higher KMIN values, non-equilibrium conditions coupled with a higher mobility enhance cross-slipping and thus increase the rate of activation of new sources. Under given conditions of temperature and loading rate, the lower the threshold value for source activation, KMIN, the lower the multiplication rate: thus the shielding rate increases steadily and a soft transition is expected (Fig. 2(a)). Oppositely, for a high KMIN value, the multiplication rate is instantly very high, so is the shielding rate: a sharp transition is expected (Fig. 2(b)). One can conclude that the sharp transition observed in silicon results from the fact that the few sources available at the crack tip are activated under a high KMIN value and that they multiply through an avalanche mechanism whose efficiency strongly depends on the stress level. Is it possible to decrease KMIN? Usually, in metals, the Frank net is able to generate dislocations under a low stress level, i.e. stimulated emission at the crack tip will take place under a low KMIN value and lead to a soft transition. In silicon an equivalent net can be developed through pre-deformation. Effectively, the shape of the BDT changed from sharp to soft when dislocation-free silicon crystals are replaced by warm pre-stressed samples [34]. A similar increase in fracture toughness upon deformation has also been observed in NiAl single crystals [35].
6. Comments and conclusions When two independent but alternative activated mechanisms are in competition during any microstructural transformation the global evolution will be controlled by the fastest mechanism: that with the highest activation energy at high temperature, that with the highest pre-exponential factor at low temperature. On the contrary, when two independent but simultaneous mechanisms are required to achieve such a transformation the evolution will be controlled by the slowest mechanism. BDT depends on the material’s ability to both generate and move dislocations. These mechanisms are not independent in the sense that there first has to be emission of dislocations before they are made to move, but, the slowest mechanism still controls the global transition. The shielding rate must be, in a first approximation, proportional to the product: dislocation nucleation rate times the dislocation mobility. Under the assumption of heterogeneous nucleation the number of sources depends either on the density of defects along the crack tip for dislocation free crystals or on the
density of bulk sources at the origin of stimulated emission at the crack tip. The nucleation rate depends not only on the stress level but also on temperature throughout the dislocation’s mobility, as is explicitly included in the multiplication model. The assumption of homogeneous nucleation cannot totally be ruled out, but, since experimentally nucleation appears more easily on defects, one could imagine an activated nucleation process only at stresses closed to the toughness of the material. The second term of the product, the dislocation’s mobility, controls the activity of the sources and, as mentioned before, partly the nucleation rate. Therefore, it is not surprising that in most cases the BDT is interpreted as a thermally activated process controlled by the mobility of dislocations [22,35]. In fact, except for dislocation free silicon samples, there are no evidences of a BDT dominated by nucleation. In a recent work on tungsten single crystal [36] dislocation nucleation is identified as a controlling process only in a low temperature semi-brittle fracture regime well below the transition. The present analysis of a dislocation multiplication mechanism at a crack tip has a strong physical basis and experimentation supports some features of this model. Obviously a good knowledge of the sources location and activity is required in order to predict in a quantitative way the behaviour of semi-brittle materials. A complete exploitation of the available experimental data in the light of 3D numerical simulations, now on the way, should help to understand the influence of temperature, KMIN and loading rate on dislocation source’s activity. One important fact to point out is that the rate of generation of new sources along the crack front depends on the dislocation’s mobility. Therefore it is not surprising that all the experiments performed on silicon lead to a common activation value (Fig. 1), i.e. that required for activation of dislocation glide. One can conclude that the BDT is nucleation controlled in dislocation free silicon crystals in the sense that the absolute value of the transition temperature is imposed by the primary source’s densities along the crack front.
Acknowledgements The authors wish to thank the staff of L.U.R.E. for provision of laboratories facilities.
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A model of dislocation multiplication at a crack tip: influence on the brittle to ductile transition Ge´rard Michot a,*, M.A. Loyola de Oliveira b,1, G. Champier a,2 b
a Ecole des Mines de Nancy, Parc de Saurupt, 54042 Nancy, France Uni6ersidade Federal de Espirito Santo, C.T., A6. Fernando Ferrari, Goiabeiras, Vitoria 29000 E.S., Brazil
Abstract It has been shown long ago experimentally that primary nucleation of dislocations in silicon takes place heterogeneously on defects along the crack tip. More recently, it has been observed that a source is easily activated at the intersection point of the crack front and of an attracted dislocation. The authors offer and discuss a source multiplication mechanism based on this stimulated emission process. One of the dislocations emitted at a primary source on the plane of maximum resolved shear stress cross-slips to a plane where it is attracted by the crack: the intersection event gives rise to a secondary source. Because shielding is very low at this point, there occurs an emission of a new bundle of dislocations. The process then starts again giving rise to an ‘avalanche multiplication’ of dislocations which strongly shield the crack. Soft/sharp brittle to ductile transitions (BDT) observed in semi-brittle materials result from such a high shielding rate coupled with a low/high threshold stress intensity factor for the activation of the primary sources. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Brittle to ductile transition; Crack; Dislocation source; Shielding; Silicon
1. Introduction For a cracked body without bulk sources, the emission of dislocations is restricted to the crack tip. Such a generation has been successfully investigated experimentally in silicon through in-situ X-ray topography [1]. It has been shown that the crack front’s ability to generate dislocations is decreased by improving the crack quality. This means that nucleation takes place heterogeneously on defects along the tip, such as the intersections of cleavage ledges with the front. This statement, which has been claimed long ago by one of
This paper is dedicated to Professor Herbert Herman on the occasion of his 65th birthday. * Corresponding author. Tel.: + 33-3-8358-4158; fax: + 33-3-83579794. E-mail addresses: [email protected] (G. Michot), [email protected] (M.A. Loyola de Oliveira) 1 Tel.: +55-27-335-2851; fax: +55-27-335-2650. 2 Tel.: +33-3-8358-4158; fax: +33-3-8357-9794.
the authors, is contrary to the usual assumption of homogeneous nucleation [2–4]. For this reason, introduction of defects in the emission models is relatively recent and still incomplete [5,6]. This is probably due to some reluctance to accept the fact that a material’s property, as important as the brittle to ductile transition (BDT), could be extrinsic and, as a consequence, could not be modelled in a one-to-one way. With regards to sources, only a few experimental results are available and most of them have been obtained on silicon by the authors. They have derived in particular that the number of dislocation sources increases with time during the test [7]. In this paper, the main features of crack/dislocation interactions (Section 2) will be recalled first before presenting outstanding results about the BDT as is reported in the literature (Section 3). Then, emphasis will be put on sharpness of the transition (Section 4) and its link with dislocation source multiplication at the crack tip (Section 5). Finally, conclusions will be given (Section 6).
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2. Crack and plasticity In a loaded cracked body the stress intensity factor K measures the strength of the singular term of the elastic solution of the stress field. In brittle materials, a crack propagates when the applied stress intensity factor KA reaches a critical value KIC which is a material’s characteristic called toughness. On the other hand, complete relaxation in plastic materials should lead to the disappearance of the singular solution in r − 1/2, i.e. to an effective stress intensity factor KE equal to zero, KE being understood as the decrease of KA to zero induced by plasticity. This is confirmed by all calculations performed on the plastic zone developed at the crack tip in non-hardening materials [8,9]. In materials like fcc metals, characterised by a the high dislocation mobility at any temperature, achievement of a full plastic relaxation at crack tip is practically time independent. In bcc or covalent materials, due to the strong influence of temperature on dislocation generation and/or mobility, there is a restriction on dislocation activity incompatible with a full plastic relaxation at crack tip: KE is different from zero and strongly time dependent under dynamical loading. It has been checked, using equilibrium models [10 – 15], that full relaxation is avoided through the introduction of a dislocation free zone between the crack and the plastic zone. Apart from these equilibrium situations, it is obvious that as soon as a material is unable to provide enough dislocations (low number of sources, high threshold value of their activation stress etc.) or to move them fast enough (temperature limited mobility) to fully relax a dynamically loaded crack, a dislocation free zone will appear. The stress relaxation, i.e. the decrease of the applied stress intensity factor KA to an effective value KE, results from the shielding of the crack by the dislocations. The shielding concept was in fact first introduced on the microscopic level [2] before its applications were extended to continuum mechanics [11]: the isolated dislocation is considered as an internal loading with an associated stress intensity factor kD. The formalism was detailed for mode III shielding [16] and then extended to the general case [17,18]. The physical reality of shielding has been experimentally checked for silicon on plastic zones developed to saturation under creep at high temperature and then frozen by cooling under load. Room temperature fracture implies that there have to be stresses up to three times larger than those required for dislocation free crystals [19]. In materials which exhibit limited plasticity one intuitively see the roles played by dislocations and their discrete sources on fracture, i.e. dislocation modelling should be preferred to continuum mechanics. In these materials, when a crack is loaded under a constant opening or loading rate, there is a competition between the increase in stored elastic energy due to the applied
load on the one hand, and the plastic relaxation induced by emission and motion of dislocations on the other hand. If the plastic zone’s growth rate (i.e. the shielding increase rate) is controlled by dislocation activity (i.e. mainly by temperature) one may expect a transition from a low energy-absorbing cleavage mode (at low temperature) to a high energy-absorbing ductile mode (at high temperature). This BDT temperature TC is controlled both by physical parameters (threshold value Kmin for dislocation emission, initial number of sources, multiplication and mobility of dislocations, material’s toughness) and by geometrical parameters (slip crystallography, crack opening mode). Since the plastic zone’s growth rate is time-dependant, the transition temperature TC is sensitive to the loading rate dKA/dt and thus cannot be a material’s intrinsic characteristic.
3. Influence of dislocation sources on the BDT St John [20] first investigated the BDT in mode I on pre-cleaved silicon single crystals tested under a constant opening rate dd/dt. The transition occurs within a very narrow temperature range and the transition temperature TC becomes opening-rate dependent with an activation energy close to 2 eV over the temperature range 973–1223 K: dd/dt (or dK/dt) a exp(− Q/kTC)
(1)
More recent experiments [21] performed on silicon confirmed this value. The kinetic process of dislocation motion is a thermally activated glide process with an activation energy close to 2 eV in silicon. Thus it has been long ago admitted that the transition was controlled by the dislocation’s mobility. The same conclusion holds true for other materials such as Ge, GaAs, Mo and Al2O3 [22]. Experimentally, large shifts are indeed noticed in the Arrhenius plot between the strain/stress rate and the converse of the BDT temperature TC in silicon (Fig. 1), according to the crystal’s orientation, origin (float-zone or Czochralski), purity (intrinsic, doped, oxygen containing, hydrogen charged etc.) and thermo-mechanical treatments. Simple shifts with a constant slope must result from differences in the source’s activities while shifts with slope variations imply other thermally activated mechanisms (nucleation, interactions of the moving dislocations with obstacles, etc.). Thus the shifts noticed in Fig. 1 mainly result from differences in the dislocation source’s densities. In initially dislocation-free silicon crystals, emission of dislocations is restricted to the crack tip. Direct observations of sources [23] prove that such dislocation nucleation is highly inhomogeneous. The nucleation sites are mainly defects along the crack tip since the
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BDT temperature is raised when the cleavage quality is improved [21].
4. The sharpness of the transition Any attempt to predict the BDT temperature will mean dealing with the evaluation of the dislocation activity. It could be possible to by-pass this difficulty by assuming an equilibrium plastic zone growth but, first of all, this assumption is not realistic (experiments confirm this point) and second of all, in covalent crystals the friction stress must be understood as a dynamic parameter since the flow stress which defines it is very sensitive to the strain rate. The best way is to use dislocation modelling introducing explicitly dislocation mobility and to assume that fracture takes place when the effective stress intensity factor KE(t) =KA(t)+ k (i) D (t) overcomes KIC. If not, a ductile opening will be observed. The individual contribution k (i) D (t) of dislocation loop (i) to crack shielding, depends on its size and position at time t. The total contribution is assumed to be negative. Hirsch et al. [24] developed a non equilibrium two dimensional (2D) simulation on this basis. Even though they could not avoid a few assumptions on the dislocation emission conditions, they pointed out the fundamental concept of a critical distance between sources dC below which all vulnerable points z along the crack profile can be shielded (KE(z) BKIC) under given experimental conditions. The new and fundamental feature of this model is that the transition is now defined by two scales, the dislocation free zone size derived from the simulation, and a microstructural parameter, the spacing between dislocation sources.
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Since usually thermally activated processes exhibit continuous variations over wide ranges of temperatures, it appears difficult to some authors [25] to explain the fast increase in material reinforcement observed in silicon within a few degrees below the transition through a simple dislocation shielding mechanism. Indeed, a continuous and steady increase in shielding should lead to a variation of the effective stress intensity factor KE(t) at the most vulnerable point along the crack front close to that imagined Fig. 2(a). A soft transition is observed since fracture takes place for applied KA values, KC, much larger than the material toughness, KIC, at temperatures just below TC. Conversely, for a sharp transition, a rapid increase in shielding (Fig. 2(b)) is required: how to explain such an increase? The influence of the different parameters (nucleation rate, mobility, loading rate…) on the shielding amplitude have been appreciated through simulation [24]. In order to account for the sharpness of the BDT, the first emission must start at a KA value close to KIC, and be followed by rapid emission of bundles of dislocations under a much lower stress level. However, by expressing the nucleation condition in terms of stress rather than in terms of KE, Maeda [26] obtained a sharper transition with less assumptions, while Brede [27] or Xin and Hsia [28] got the same result by accelerating the shielding process through an increase of the number of activated glide systems. One may think that the number of involved slip planes is important but that alone cannot explain the shielding evolution. Michot et al. raised the point that the 2D simulations do not deliver the proper kinetics since they make abstraction of the evolution of the population of dislocation sources during the test. The
Fig. 1. Arrhenius plot between the loading rate dKA/dt and the inverse of the transition temperature TC, as collected by Hirsch and Roberts [22]. Each curve (A – H) corresponds to specific test samples, procedure of introducing the crack, crystal origin or purity. The activation energy for the brittle to ductile transition (BDT) measured in doped silicon (curves B and C) differs from that measured in intrinsic silicon ( 2 eV, curves A, D– H). The activation energy of the glide process in doped silicon also differs from that measured in intrinsic silicon ( 2 eV). For these reasons it is admitted that the dislocation’s mobility imposes the slope of the plot while differences in the dislocation source’s activities lead to the large shifts observed (from F to H, for instance).
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Fig. 2. Assumed variations with time of the shielding contribution KD induced by the plastic zone (negative) to the effective stress intensity factor KE = KA + KD at the most vulnerable points along the crack front, for a constant loading rate dKA/dt. The assumed test temperature is just above the brittle to ductile transition (BDT), i.e. KE does not overcome the toughness KIC of the material. When the threshold value KMIN for emission of dislocations is low (a), the plastic zone has time enough to develop and shield the crack before to be close to fracture: a soft transition is observed (the critical stress intensity factor for fracture KC can be large). When KMIN is large (b), a very high shielding rate is required in order to get the same instability: a sharp transition is observed (KC is close to KIC).
authors derived from in-situ measurements [7] that, close to TC, the dislocation nucleation rate should be proportional to the product KA dKA/dt, i.e. the number of sources should increase with t 2. Furthermore, recent experiments on stimulated emission [7] show that as soon as a first dislocation is available at a crack tip, dislocations are easy to create, even under a very low stress level. Dislocations multiply through an avalanche mechanism, i.e. the dislocation multiplication rate resulting from the activation of new sources is much higher than that resulting from mobility controlled emission from a single source. This is one of the conditions that leads to a sharp transition according to the Hirsch et al. simulation. Nevertheless, this condition is not sufficient. If this multiplication mechanism always functions, how can one justify a soft shielding evolution like the one proposed in Fig. 2(a)? Obviously, the sharpness of the transition depends on another parameter. According to the stimulation by Hirsch et al. [24] a high threshold value KMIN for emission of the first dislocation is also required to get a sharp BDT. Thus, one has to find a physical microscopic mechanism of creation of new sources whose efficiency will be stress dependent.
5. Multiplication mechanism in silicon Little is known about the nature of nucleation sites and the detailed dislocation mechanisms which operate in order to form the plastic zone that shields the crack
tip, preventing brittle failure above the transition temperature. In the absence of bulk sources, primary nucleation takes place heterogeneously on defects along the crack tip [1,7]. The heterogeneous character of dislocation emission suggests that the BDT is controlled neither by the condition for homogeneous emission [2–5], nor by a co-operative dislocation generation instability linked to thermal fluctuations [25]. The consequence of heterogeneous nucleation is that the BDT is shifted to a higher temperature when the quality of the cleavage surface is improved [21], i.e. when the dislocation source spacing along the tip is increased. Apparently, this effect can be huge since shifts of hundreds of degrees of the transition can be observed (Fig. 1). Thus it would be an exaggeration in itself to say that the BDT is dislocation mobility controlled. A predictive evaluation of the potential sources should combine an experimental inventory of the crack front defects and a theoretical evaluation of their strength. In fact, the experimental observations were insufficient to identify the nucleation sites and the available calculations were not precise enough [5,29,30]. In some cases the emission site coincided with an observable ledge. These sources were called ‘primary’(SPR). In other cases dislocation trails were emitted from spots where no ledges could be detected by optical microscopy or scanning electron microscopy (SEM). However, since dislocations can be emitted at a quasi perfect crack at the intersection point of the front and the slip plane of an attracted dislocation [7], sources are not necessarily associated with pre-existing defects. The new sources activated during plastic zone growth by this ‘stimulated emission process’ are called ‘secondary’(SSEC). Fig. 3 shows the position of the Thompson tetrahedron with respect to the chosen crack orientation. A few a/[2011] dislocations are supposed to be emitted at a defect on the (1( 11( ) slip plane and to travel a given distance from this primary source SPR. If the positive resolved shear stress acting on it is overcome by a negative one acting in the secondary (1( 1( 1) plane, crossslips towards the latter plane (Fig. 4) is mechanically authorised: the P1P2 screw segment of the dislocation bulges and moves to the point SSE where it intersects the crack tip. The secondary source thus triggered emits a new trail of dislocations and the process starts again. Several arguments support this assumption, even if, due to the limited resolution of the X-ray topography, this mechanism has not been directly observed. (i) Firstly, concerning at least one specific crystallographic orientation, many evidences of extensive cross slip during plastic zone growth have been reported in silicon [31]. (ii) Secondly, the mechanical feasibility of the previous cross-slip process has been checked. The attraction or the repulsion of a dislocation by the stress field of
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Fig. 3. One of the crystallographic orientations designed for mode I loading of silicon samples cleaved along the (111) plane [1,21]. The Thomson tetrahedron shows how available slip systems are oriented with respect to the horizontal crack tip. Four dislocations with Burgers vector a/2[011] are supposed to be emitted at a defect along the crack front along a given (1( 11( ) plane. The source associated with a defect is called primary source (SPR). One assumes cross-slipping of the P1P2 screw segment of the fourth emitted dislocation into the (1( 1( 1) plane. This plane cuts the crack front at point SSE.
the crack depends on the sign of the Burgers vector. One assumes that the detected dislocations obey the growing conditions (the high density of dislocations within the plastic zone does not allow the sign’s determination through X-ray topography). Since these requirements cannot be expressed directly through the classical Schmid factor because of the non-uniformity of the stress field, results are expressed in the form of equal-stress contours, as explained in details in Ref. [1]. For instance, for a given a/2[011] Burgers vector and a given orientation of the dislocation line the {a/ 2[011](1( 11( )} slip system fulfils the growing condition
Fig. 4. According to the stress analysis around the crack tip (mode I loading, orientation a, [1]), the P1P2 segment (burgers vector a/2[011]) is attracted by the crack in the (1( 1( 1) plane. Hence cross-slip can take place and the segment P1P2 bends into the (1( 1( 1) plane until it cuts the crack front and triggers a secondary source SSEC, through the stimulated emission process [7]. This source is not necessarily associated with a defect.
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(mode I loading, orientation a [1]) while the {a/ 2[011](1( 1( 1)} slip system fulfils the shrinking conditions. The dislocation is repelled by the crack tip in the first case, attracted in the second case. One assumes that the previously mentioned {a/2[011](1( 11( )} slip system, which extends experimentally far away from the crack tip, is submitted to the highest stress. For what reason should the dislocation move out of this plane? Under such a non-equilibrium situation, the last emitted dislocation is slowed down by the back-stress of the previously emitted dislocations: the higher the number of dislocations in the inverse pile-up, the smaller the stress pushing the internal dislocation in the (1( 11( ) plane. Since the {a/ 2[011](1( 1( 1)} slip system fulfils the shrinking conditions the P1P2 screw segment can be attracted by the crack tip, thus cross-slip takes place. Furthermore, 3D calculations [32] show that the length of crack tip concerned by the shielding effect is smaller than the loop radius: this means that the primary source only is strongly shielded, the dynamical pile-up is no longer feeded with fresh dislocations able to maintain a stress level high enough to keep the dislocation in the primary plane. Depending on the experimental conditions (temperature, load and loading rate) there must be a critical number of emitted dislocations above which cross-slipping takes place and leads to stimulated emission at point SSE. (iii) Lastly, this formal reasoning is partly supported by observation of etch pits on a cracked sample relaxed under creep conditions [33]. On the outer part of the plastic zone, a small number of rows of pits, parallel to the intersection of the (1( 11( ) plane with the sample surface and containing less that a few tens of a/2[011] dislocations are observed: these well defined rows are connected with the activity of individual primary sources. Oppositely, it is difficult to find traces of slip along the primary plane in the inner part of the plastic zone where pits distribution is much more homogeneous. One can conclude that plastic relaxation starts by the triggering of a few primary sources emitting a few tens of dislocations, rapidly followed by stimulation of many new secondary sources emitting less dislocations still. Is this mechanism stress sensitive (second requirement of the conclusion of Section 4)? Experimentally, etch pits counting on silicon samples loaded under creep condition for 2 h [33], indicate a very fast increase of the number N of dislocations within the plastic zone with the applied load: NK 8A! The rate of creation of sources (the converse of the lapse of time needed by the dislocation to travel from the primary source SPR to the secondary one SSE (Figs. 3 and 4)) is proportional to the dislocation’s mobility (stress exponent close to one in silicon) and to the converse of the distance d travelled by the dislocation. When the threshold value for source activation, KMIN, is small, growth conditions are
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close to some ‘equilibrium’ conditions, i.e. all the dislocations feel nearly the same stress field, cross-slip is note likely to occur the rate of activation of new sources is very low. For higher KMIN values, non-equilibrium conditions coupled with a higher mobility enhance cross-slipping and thus increase the rate of activation of new sources. Under given conditions of temperature and loading rate, the lower the threshold value for source activation, KMIN, the lower the multiplication rate: thus the shielding rate increases steadily and a soft transition is expected (Fig. 2(a)). Oppositely, for a high KMIN value, the multiplication rate is instantly very high, so is the shielding rate: a sharp transition is expected (Fig. 2(b)). One can conclude that the sharp transition observed in silicon results from the fact that the few sources available at the crack tip are activated under a high KMIN value and that they multiply through an avalanche mechanism whose efficiency strongly depends on the stress level. Is it possible to decrease KMIN? Usually, in metals, the Frank net is able to generate dislocations under a low stress level, i.e. stimulated emission at the crack tip will take place under a low KMIN value and lead to a soft transition. In silicon an equivalent net can be developed through pre-deformation. Effectively, the shape of the BDT changed from sharp to soft when dislocation-free silicon crystals are replaced by warm pre-stressed samples [34]. A similar increase in fracture toughness upon deformation has also been observed in NiAl single crystals [35].
6. Comments and conclusions When two independent but alternative activated mechanisms are in competition during any microstructural transformation the global evolution will be controlled by the fastest mechanism: that with the highest activation energy at high temperature, that with the highest pre-exponential factor at low temperature. On the contrary, when two independent but simultaneous mechanisms are required to achieve such a transformation the evolution will be controlled by the slowest mechanism. BDT depends on the material’s ability to both generate and move dislocations. These mechanisms are not independent in the sense that there first has to be emission of dislocations before they are made to move, but, the slowest mechanism still controls the global transition. The shielding rate must be, in a first approximation, proportional to the product: dislocation nucleation rate times the dislocation mobility. Under the assumption of heterogeneous nucleation the number of sources depends either on the density of defects along the crack tip for dislocation free crystals or on the
density of bulk sources at the origin of stimulated emission at the crack tip. The nucleation rate depends not only on the stress level but also on temperature throughout the dislocation’s mobility, as is explicitly included in the multiplication model. The assumption of homogeneous nucleation cannot totally be ruled out, but, since experimentally nucleation appears more easily on defects, one could imagine an activated nucleation process only at stresses closed to the toughness of the material. The second term of the product, the dislocation’s mobility, controls the activity of the sources and, as mentioned before, partly the nucleation rate. Therefore, it is not surprising that in most cases the BDT is interpreted as a thermally activated process controlled by the mobility of dislocations [22,35]. In fact, except for dislocation free silicon samples, there are no evidences of a BDT dominated by nucleation. In a recent work on tungsten single crystal [36] dislocation nucleation is identified as a controlling process only in a low temperature semi-brittle fracture regime well below the transition. The present analysis of a dislocation multiplication mechanism at a crack tip has a strong physical basis and experimentation supports some features of this model. Obviously a good knowledge of the sources location and activity is required in order to predict in a quantitative way the behaviour of semi-brittle materials. A complete exploitation of the available experimental data in the light of 3D numerical simulations, now on the way, should help to understand the influence of temperature, KMIN and loading rate on dislocation source’s activity. One important fact to point out is that the rate of generation of new sources along the crack front depends on the dislocation’s mobility. Therefore it is not surprising that all the experiments performed on silicon lead to a common activation value (Fig. 1), i.e. that required for activation of dislocation glide. One can conclude that the BDT is nucleation controlled in dislocation free silicon crystals in the sense that the absolute value of the transition temperature is imposed by the primary source’s densities along the crack front.
Acknowledgements The authors wish to thank the staff of L.U.R.E. for provision of laboratories facilities.
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