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Simulation of deformation twins and their interactions with cracks
Computational Materials Science 89 (2014) 224–232
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Simulation of deformation twins and their interactions with cracks Anke Stoll a,b, Angus J. Wilkinson a,⇑ a b
Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom Clinic for General, Visceral and Vascular Surgery, University Clinic Magdeburg, Leipziger Straße 44, 39120 Magdeburg, Germany
a r t i c l e
i n f o
Article history: Received 29 December 2013 Received in revised form 16 March 2014 Accepted 21 March 2014
Keywords: Deformation twins Dislocations Cold work Stress intensity factor Nuclear materials Austenitic stainless steel
a b s t r a c t A discrete dislocation model was used to simulate residual stress fields close to deformation twins in stainless steels. Dislocation pairs were distributed along an initially elliptical twin boundary and an iterative scheme used to allow the dislocations to relax towards positions where the internal shear stresses were below a friction stress. The relaxed twin shape was close to elliptical but the flanks were flatter and the twin tips where the grain boundaries are situated were somewhat blunted compared to an ellipse. A dislocation-based boundary element model was then used to assess the interaction between the stress field from such twins and a crack. The stress field near the tip of the crack was characterized in terms of mode I and mode II stress intensity factors. The effects of twin width, length, orientation and distance from the crack tip on the strength of the interaction were studied. Wider, shorter twins were found to induce the largest stress intensity at the crack tip when close to the crack tip and aligned perpendicular to the crack plane. The influence of a pair of deformation twins does not significantly exceed the influence of a single deformation twin. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Slip and twinning are the major deformation modes which enable a crystalline solid to generate a permanent shape change under the action of an applied stress. An extensive review on deformation twinning has been given by Christian and Mahajan [1]. Considerably more attention has been paid to understanding slip than twinning but there is currently interest in twinning. The classical definition of twinning requires that the twin and matrix lattices are related by either a reflection in some plane or by a rotation of 180° about some axis [1]. In many cases especially for the FCC, HCP and BCC metals both of these conditions are met and such twins are termed compound twins. Deformation twins can form by a homogeneous simple shear of the parent lattice. Deformation twinning has been observed in FCC, BCC and HCP metals and the characteristic shear strains can be quite large. The large lattice rotation between a twin and the parent lattice makes deformation twinning have a significant impact on both the intensity and nature of deformation textures [2,3]. Deformation twins often take up a lenticular shape in order to best accommodate the large shear strain while minimizing strain energy generated within the surrounding matrix [1]. In the FCC metals deformation twinning is perhaps most studied in steels where a class of austenitic alloys known as TWIP ⇑ Corresponding author. Tel.: +44 1865 273792; fax: +44 1865 273764. E-mail address: [email protected] (A.J. Wilkinson). http://dx.doi.org/10.1016/j.commatsci.2014.03.041 0927-0256/Ó 2014 Elsevier B.V. All rights reserved.
(twinning induced plasticity) steels have been developed primarily for the automotive industry e.g. [4–6]. In the austenitic stainless steels there is also evidence of intense twin-like deformation bands that form in increasing number during cold deformation [7–10] and are characterized by the classic 60° rotation about h1 1 1i compared to the matrix grain. There is some evidence that significant plastic deformation is required before these twins are observed and the threshold strain is dependent on the alloy composition [7–11]. These deformation twin bands are of considerable technological importance and are currently of particular interest within the nuclear industry because they have been found to be present in alloys that have shown susceptibility to intergranular stress corrosion cracking (SCC) in the cold worked state. In these alloys the SCC cracks mostly follow grain boundaries which have been oxidized and embrittled by the environmental exposure (hydrogenated water at moderate 300 °C temperatures) prior to cracking. Segregation during sensitization through either thermal exposure in service or accelerated heat treatment before laboratory tests and irradiation damage may worsen the situation but neither are necessary. This is of very significant concern because steels such as 304 had until relatively recently been largely considered immune to SCC in these environments [9]. Detailed TEM work by Lozano-Perez et al. [9] showed that deformation twins intersecting a free surface are preferred SCC nucleation sites while crack tips growing along grain boundaries branch into intersecting twin-like deformation bands which are preferentially oxidised.
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Deformation twinning has been modeled using a variety of approaches including molecular dynamics simulation [12–15], dislocation [16], disclination dipole [17] and finite element [18,19] modeling. The dislocation modeling approach is of interest here as it allows us to combine modeling of the twin with dislocation modeling of the crack. Mitchell and Hirth [16] gave an elegant model using a continuous density distribution of infinitesimal partial dislocations within a double-ended dislocation pile-up to represent the deformation twin. An analytical solution in which dislocations were at equilibrium was found for the density distribution and shape of the deformation twin via Hilbert transformations. The shape of an isolated twin in their 2 dimensional analysis was found to be an ellipse to first approximation though higher order terms show that twin tips are blunter with increasing stress while the twin boundary is populated with edge dislocations, while higher stress tends to sharpen the twin tip for the screw case. Dislocation mechanics has a more extensive history in the modeling of cracks and fracture. The seminal work by Billby Cotterell and Swinden [20,21] found equilibrium solutions for continuous density distributions of infinitesimal dislocation arrays representing the crack and associated plastic zone. A second type of solution to this problem in which dislocation densities are allowed to increase at the far end of the plastic zone which is blocked by a microstructural feature has found considerable use in modeling short fatigue crack propagation [22–24]. However, more complex geometries and loading cases have required a move to discrete dislocation treatments and numerical simulation rather than analytical solutions [25–34]. The question we seek to address in this paper is whether or not localized stresses near the tips of deformation twins seen in cold worked austenitic stainless steels could provide significant enhancement of the mechanical driving force for intergranular cracks and thus, in part, contribute to the recently observed susceptibility to stress corrosion cracking. To this end we develop a discrete dislocation version of the Mitchell and Hirth [16] model of deformation twins using arrays of edge dislocations which are allowed to relax to an equilibrium configuration. Stresses from the deformation twin are then used to load a dislocation-based boundary element model of a crack, which is assumed to be growing along a straight section of grain boundary at which the twin has terminated on one side of the boundary. The effects of the size, proximity and inclination of the deformation twin on the stress intensity factors (modes I and II) induced at the crack tip are investigated. For ease of computation we work within an elastically isotropic medium. We also assume that the ideal twinning shear for twins in FCC crystals is present and does not relax h1 1 2i{111} through dislocation mediated plasticity and that the twin shape is not perturbed by the presence of the crack. These assumptions make our calculations an upperbound assessment of the stress intensity that could be generated by the deformation twin.
2. Methodology 2.1. Simulation of deformation twins To simulate twins, we start from an elliptical shape with partial dislocations positioned along the elliptical twin boundary as proposed in [16] (see Fig. 1). For an ellipse centred at the origin with its major axis along the x0 -direction partial dislocations in the right hand half of the ellipse have positive Burgers vector while those at positions mirrored across the y0 -axis have negative Burgers vector. The Burgers vectors are of 1/6 a[1 1 2] type (i.e. the x0 -axis is p parallel to [1 1 2]) with magnitude b = a/ 6. The twin is on the 0 with the partial (111) plane (i.e. the y -axis is along [111])
225
Fig. 1. Discrete dislocation array representation of a twin.
p dislocations on planes separated along the y0 -axis by a/ 3. The characteristic twinning shear c is then given by b/d (where d is p the spacing of slip planes along the y0 -axis) and is 1/ 2 for the FCC system. In order to find a relaxed twin shape an elliptical starting geometry was chosen which is known to be a reasonably good approximation. The twin was then divided up into a number of slip planes spaced much further apart in the simulation than the planes in the crystal. To keep the required twinning shear (111) p of 1/ 2 for our chosen material the magnitude of the Burgers vector for all dislocation partials at the boundary of the twin are changed in proportion to the plane spacing. The shear stress generated by all the partial dislocations positioned on the twin boundary is then calculated at the dislocation positions. Dislocations are then moved according to their shear stress. If the shear stress is higher than a fixed critical shear stress (or friction stress) + sc, the dislocation is moved outward, away from the centre of the ellipse. The distance of movement was set at one quarter of the distance from the current position of the dislocation to that of the neighboring dislocation (which is further outward). The movement was constrained to be along the x0 -direction i.e. along a horizontal plane (constant y0 ). Dislocations are not permitted to move beyond the original two ends of the twin which are considered pinned by impenetrable grain boundaries. If the shear stress at a dislocation position is below sc it is moved toward the centre. Again the distance between the dislocation location and the neighboring dislocation (closer to centre) is calculated and the dislocation is allowed to move one quarter of that distance toward x0 = 0, again with y0 constant. Once dislocations at x0 = 0 start to pile-up, distances between dislocations of the two twin halves on the same planes are calculated and dislocation pairs are allowed to eliminate once their separation is smaller than 1 nm (at which point the mutual interaction causes stresses in excess of 1 GPa). Using the approach of moving dislocation partials along horizontal planes according to their shear stresses does not result in a twin shape with zero shear stresses along the twin boundary (calculated at the positions of dislocations). Therefore, a friction stress sc has to be introduced as mentioned earlier to define a maximum and minimum shear stress. If the shear stress at a dislocation position is within the range sc to +sc then the dislocation at that position does not move. Once a relaxed twin shape is obtained a coordinate transformation is conducted and the twin is turned to allow the deformation twin to be placed in front of the crack tip. 2.2. Simulation of crack opening and stress field For the simulation of crack propagation a dislocation-based boundary element model BEM approach was developed as had previously been used by Riemelmoser et al., Nowell et al. and Schick et al. [25–28,30]. The most important principles of the modeling
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semi-minor axis bt (which is half the twin width wstart) varies depending on the relaxation process of the twin shape to the relaxed configuration. Generally we find the flanks of the relaxed twin are slightly flatter compared to the ellipse which is in accord with Mitchell and Hirth’s results [16]. The relaxed twin geometry is influenced by various factors including: the slip plane spacing, the size of the friction stress, the starting width of the ellipse and the twin - grain boundary inclination angle. These influences were evaluated in sensitivity studies which are reported in the following sub-sections.
Fig. 2. Discrete dislocation array representation of a crack.
approach will be shown here while for a more detailed review, the reader is referred to [35,36]. Fig. 2 shows the general configuration of the dislocation model for a planar crack along the x-axis. Plane strain conditions are assumed and the sample dimensions are large compared to the length of the crack. Each boundary element in the crack contains a collocation point cei and dislocation dipoles of climb and glide character in order to simulate mode I and mode II displacements respectively. The boundary elements are made gradually smaller closer to the crack tip to guarantee a high accuracy there. The length li of boundary elements were calculated knowing the crack length and the number of boundary elements and follow the geometric progression li = 1.08 li1. A stress matrix G is generated and GIij , GIIij are the stresses at collocation point i generated by climb and glide dipoles of element j when the Burgers vectors of dislocations in the glide or climb dipole are each set to unity. Head [37] gave the full equations for stresses of dislocations in a homogeneous, elastically isotropic half space which are used in the analysis. To solve the fracture problem for the crack opening and shearing and the stress field around the crack a set of linear equations are constructed and solved numerically. At each collocation point we require that the stress rapp (=ryy, rxy) applied to the crack is negated by stresses generated by the dislocation dipoles representing the crack so that the normal stress and shear stress across the crack faces are brought to zero. Then the solution for the opening displacements (bi) of an elastic crack with loading rapp can be calculated solving the system of equations
Gji bi ¼ rapp ! bi ¼ ðGji Þ1 rapp j j :
ð1Þ
In this case rapp is obtained from the deformation twin simulation described in Section 2.1. The solution of the elastic problem is found with the calculation of bi, from which the stress at any point can be calculated using superposition and Head’s solution for the stresses from a dislocation in an semi-infinite half space [37]. The stress intensity factors KI and KII are calculated at the crack p tip by fitting the ryy and rxy stress variations to 1/ r distributions with r the distance from the crack tip along a line directly in front of the crack tip as indicated in Fig. 2 [35,36].
3. Results 3.1. Equilibrium twin geometry The ellipse has the middle point (0, 0). Its semi-major axis at is chosen as 50 lm to mimic a twin of total length lt = 100 lm. The
3.1.1. Influence of slip plane spacing First of all, the influence of the twin slip plane spacing was investigated to assure that the number of partial dislocations chosen to describe the twin leads to accurate results without having to work with larger computation times. The elliptical starting geometry of a twin was described by 100, 200, 400, 600, 800 dislocations with total twin length lt = 100 lm and starting total width wt = 500 nm. Fig. 3a shows that the overall relaxed twin shape results are consistent and do not depend strongly on the number of dislocations used to describe the initial elliptical twin shape. However, at the tips of the relaxed twins slight differences appear (Fig. 3b). Generally, the twin tips are broader when a larger number of dislocations is used to represent the twin geometry. In order to calculate a relaxed twin with 100 dislocations 135 iterations had to be performed while for 800 dislocations 997 iterations are needed. The relaxed shape of a twin initially described by 200 dislocations is found within reasonable calculation time and shows the broad twin tip typical for higher accuracy calculations with many more dislocations. Therefore, in the following, the initial elliptical twins were described by 200 dislocations unless explicitly stated otherwise. Fig. 3c shows that the s0xy shear stress variation along the twin boundary. Only at the position of the outermost dislocation (located at the twin tip which is positioned at the grain boundary) the shear stress is above the friction stress of 100 MPa. The general situation is that dislocations towards the tip of the twin experience shear stresses tending to move them outwards and to extend the twin. This can result in a blunted tip as dislocations are prevented from moving beyond the grain boundary which acts as an impenetrable barrier in these simulations. In contrast the dislocations further back from the tip experience repulsion from those at the tip and so are pushed back against the friction stress. 3.1.2. Influence of friction stress The relaxed twin geometry is influenced by the level of friction stress. The simulations are always initiated from an ellipse that has the required length with dislocations at the twin tips pinned by grain boundaries and a width that is larger than that expected for the relaxed system. Relaxation then leads to a narrowing of the twin as the central dislocations move back and opposing pairs of dislocations collapse back and are annihilated. For a very large friction stress this process has a minimal influence and the overall twin shape appears almost unaltered from the starting geometry. However, close inspection generally reveals some outward movement of dislocations very near the twin tip where stresses are very high leading to a slight blunting of the tip compared to the starting ellipse, while a small amount of narrowing at the centre of the twin also occurs. In these situations of very high friction stress the majority of the dislocation may not have moved during the relaxation which is undesirable as dislocation positions are then those imposed by the initial setup rather than truly relaxed. We thus work with lower friction stress levels which are more representative of flow stresses in the target stainless steels and at which all the dislocations move during the relaxation. Tip blunting is still evident but the major effect of decreasing the friction stress is that
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Fig. 3. Effect of number (given in legend) of dislocations initially in the array on the relaxed twin shape. (a) Overview of whole twin and (b) near tip region. The black lines shows the initial ellipse shape. Different x0 and y0 length scales are used to emphasise difference in shape (true shape is longer and thinner). (c) Shear stress remaining after relaxation process.
more of the central dislocations collapse back and are annihilated causing the twin width to reduce. The effect is demonstrated in Table 1 which gives relaxed twin widths as a function of friction stress for a starting geometry of a 100 lm long, 1000 nm wide twin represented by 200 dislocations (i.e. 100 pairs of dislocations) on planes separated by 10 nm. At the lower friction stresses, which are more realistic for our target austenitic stainless steel materials, the twin narrows to a small fraction of the starting configuration width and consequently is made up of rather few dislocations. A better representation might be obtainable if the initial ellipse width were reduced while maintaining the same number of dislocations. 3.1.3. Influence of starting geometry Getting the initial ellipse width close to, but larger than, the relaxed twin width is advantageous in both reducing the number of iterations in the relaxation calculation and in keeping a large number of dislocations representing the relaxed twin. Providing the initial twin is larger than the relaxed twin width then the overall shape of the relaxed twin is not too strongly dependent on the initial twin width. However, for the wider initial configurations the relaxed twin is represented by fewer dislocation pairs spaced more widely apart and so do not represent the fine detail as correctly. In
particular the shape near the twin tip is not well captured if the initial configuration is too wide so that too few dislocations remain in the relaxed configuration. Having looked at a range of twin lengths and widths has shown that a good twin starting width is twin length/350 for friction stress sc = 100 MPa which gives a value of 286 nm and results in relaxed twin widths within a range of 100 nm- to 500 nm which is consistent with experimental observations in austenitic stainless steel [7–11]. 3.1.4. Influence of grain boundary inclination angle As dislocations at the twin tip are pushed outward to the grain boundary it was necessary to examine the influence of the inclination of the grain boundary plane relative to the twin axis and Burgers vector direction. As might be expected the overall relaxed twin shape is not strongly affected by the angle between the twin axis and the grain boundary. The twin widths stay essentially unchanged as the angle is varied, remaining at 118 nm for all cases other than for the twin at an angle of 22.5° when the twin is one simulated slip plane narrower at 113 nm. However, the geometry at the twin tips is of course affected. The mirror symmetry across the major axis of the twin found when the twin is normal to the grain boundary is removed when the boundary is tilted. 3.2. Influence of a deformation twin on crack tip stress fields
Table 1 Twin width for varying friction stresses for 100 lm long twins. Friction stress sc (MPa)
Twin width (nm)
Start 900 600 400 100 50
1000 980 737 475 131 71
From experimental observations in the literature we know that twins in austenitic stainless steels can span entire grains that can in turn be of order 100 lm in diameter. At such lengths the deformation twin bands range in width from 100 nm to 500 nm thick [9,17]. Our simulations of twin-crack interactions will thus be centred on such twin sizes. The crack is assumed to be 10 mm long and located in a homogeneous medium under an applied far-field
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tensile stress of 100 MPa along the crack normal. It is described by 250 boundary elements of decreasing size towards the crack tip as described in Section 2.2. Simulations with only this far field externally applied stress loading the crack generate a mode I stress p intensity factor KI of 14.4 MPa m, while symmetry of course results in a mode II stress intensity factor KII of zero. This is a useful baseline as it approximates the threshold stress intensity level required to initiate stress corrosion cracking in high temperature water for the austenitic stainless steels of interest [9,38,39]. 3.2.1. Influence of distance between crack tip and twin The first calculations are for a twin inclined at 90° to the crack, with a twin length of 100 lm and width of 118 nm. Fig. 4 shows how KI and KII vary as the distance between the crack tip and the twin changes from 10 nm to 10 mm. As expected the stress intensity factors take opposite sign but maintain the same magnitude when the sense of the shearing direction for the deformation twin is reversed. The increase in KI is highest when the twin is positioned very close to the crack tip and the twin induces a KI of p 2.3 MPa m at the closest distance of 10 nm considered here. This is a significant increase of about 16% above the baseline level for the mode I stress intensity factor. There is a steady monotonic decrease in KI induced by the twin as the distance to the crack tip is increased. The mode II stress intensity factor induced is p rather more modest at 0.2 MPa m in magnitude at most. The variation of KII with distance between twin and crack is more complex with two changes of sign occurring as the crack is brought close to the crack tip from far away. The range at which there is significant interaction is very large compared to the twin width. We define, arbitrarily, the length of interaction as being the furthest distance of the twin tip from the crack tip at which stress intensity p p of 0.2 MPa m is induced for mode I, and 0.1 MPa m for mode II. At these threshold levels both mode I and mode II have the same interaction length of 27 lm for this particular case of a 100 lm long, 118 nm wide twin with a friction stress of 100 MPa.
Fig. 4. Influence of distance between crack tip and twin tip on stress intensity factor on crack for twin running perpendicular to crack plane. (a) Mode I and (b) mode II. In each case the solid and dotted lines show the effect of reversing the sign of the shear direction in the twin.
3.2.2. Influence of twin inclination angle Intergranular cracks will encounter twins running through grains at a range of different inclination angles. Fig. 5 shows the mode I and II stress intensity factors induced by twins inclined at different angles to the crack plane as a function of their distance from the crack tip. Only one sense of the twin shear direction is shown. The twins were 100 lm long and the widths were 118 nm for all except the twin inclined at 22.5° for which a slightly narrower width of 113 nm was found. The normal stresses induced by the twin across the crack plane are largest when the twin is at 90° and so the highest KI values are achieved then. However, there are a considerable range of angles (112.5° to 45°) over which a similarly large KI is induced. The interaction lengths, at which the p p induced KI > 0.2 MPa m and KII > 0.1 MPa m, are given in Table 2 from which it is seen that the high KI induced by twins over the angular range 112.5° to 45° persists to considerable lengths (>15 lm). By contrast the mode II interaction is stronger for deformation twins that are less steeply inclined to the crack plane as greater shear stresses are then generated by the twin on the crack plane. The mode II stress intensity factors are generally smaller than for p mode I though KII values can exceed 1 MPa m when the twins are highly inclined and very close to the crack tip. KII values are generally smaller in magnitude as the twin inclination angle is increased toward 90° and when the twin is inclined beyond 90° the interaction increases in strength again but the sign of KII becomes negative. 3.2.3. Influence of twin width (varying friction stress) Section 3.1.2 showed how the friction stress opposing motion of the partial dislocations bounding the twin has a strong influence on the twin width. Here we keep the twin length fixed at 100 lm and at 90° to the crack plane while varying the friction stress from 50 MPa to 300 MPa. The width of the initial ellipse was scaled in proportion to the friction stress while the length of
Fig. 5. Influence of twin inclination angle [legend in (a)] on the stress intensity factor induced at a crack tip as a function of distance between crack tip and twin tip. (a) Mode I and (b) mode II.
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A. Stoll, A.J. Wilkinson / Computational Materials Science 89 (2014) 224–232 Table 2 Influence of twin inclination angle on range of interaction with crack (100 lm long 118–113 nm wide twins, 100 MPa friction stress). Angle
Range for KI (lm)
Range for KII (lm)
112° 90° 67.5° 45° 22.5°
29 27 22 18 6
8 27 45 55 70
the Burgers vectors and their spacing along the y0 axis was reduced to maintain a constant twinning shear strain and number of dislocations. Table 3 details the starting conditions and resulting twin width after relaxation from which it is seen that the twin width increases linearly with friction stress. Fig. 6 shows that both the mode I and mode II stress intensity factors induced by the twins increase in proportion with the twin width and hence also in proportion to the friction stress. For the widest twin wt = 355 nm considered here, which is within the range seen experimentally in austenitic stainless steels, a very sigp nificant KI increase of 6 MPa m occurs when the twin is very close to the crack tip. 3.2.4. Influence of twin length (varying friction stress) Twins of different length can be obtained by moving the barriers to outward motion of the dislocations at the twin tips further apart. If the friction stress is reduced as the twin length is increased then it is possible to maintain a constant relaxed twin width as indicated by Table 4. The initial ellipse width and spacing of slip planes and Burgers vector length were held constant for this series of simulations. Fig. 7 shows the variation of stress intensity factors induced by such twins as a function of their distance from the crack tip. Despite the reduced twin volume for the shorter twin, the more compact twin shape leads to higher stresses within the surrounding matrix as demonstrated by the need for increased friction stresses in these cases. This results in larger KI and KII values while the shorter twins are close to the crack tip. However the longer twin length does extend the interaction length (Table 5) and results in the shorter twins having a weaker interaction with the crack at longer distances (Fig. 7). 3.2.5. Influence of twin length & thickness (constant friction stress) If the friction stress is held constant then any changes to the twin length will also cause changes in the relaxed twin width. Table 6 shows how the relaxed twin width decreases in proportion to the prescribed twin length for a fixed friction stress of 100 MPa. The width of the initial ellipse and the Burgers vector and slip plane spacing are also adjusted so that the final relaxed configuration is represented by a similar number of dislocations for each of the simulations (Table 6). The variation in stress intensity factors induced by these twins are shown as a function of distance between the twin and the crack tip in Fig. 8. As expected the larger twins generate larger KI and KII Table 3 Simulation parameters for twins with varying width. Twin length lt (lm)
Initial width wstart (lm)
Burgers vector b (nm)
Friction stress sc (MPa)
Relaxed width wt (nm)
100 100 100 100
286/2 286 286 2 286 3
1 2 4 6
50 100 200 300
59 118 237 355
Fig. 6. Influence of twin width [legend in (a)] on the stress intensity factor induced at a crack tip as a function of distance between crack tip and twin tip. Twin perpendicular to crack plane. (a) Mode I and (b) mode II.
Table 4 Simulation parameters for twins with varying length. Twin length lt (lm)
Initial width wstart (lm)
Burgers vector b (nm)
Friction stress sc (MPa)
Relaxed width wt (nm)
50 100 200 300
286 286 286 286
2 2 2 2
200 100 50 33
118 118 118 118
values. Fig. 8 also indicates that the interaction persists to larger distances from the larger twins and is confirmed by the interaction lengths tabulated in Table 7. 3.3. Influence of an array of twins on a mode I crack In cold worked stainless steels the deformation twins bands often occur as a series of parallel features across a given grain [7,9,10]. The next two sub-sections describe some simulations that have been made to briefly investigate how twin-twin interactions affect first the shape of the relaxed twins and second the strength of the interactions with a crack. 3.3.1. Interaction of two twins The procedures for generating an array of dislocations relaxed against a friction stress and described in Section 2.1 were extended to a pair of twins. As before, dislocations were allowed to move outward if the shear stress at their location is positive and exceeds the friction stress or to move inward if shear stress is negative but larger than the friction stress. The shear stress at a particular dislocation includes contributions from all other dislocations within that twin and all dislocations representing the other twin. Fig. 9 depicts the shape of the lower twin of the pair as a function of
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Fig. 7. Influence of twin length [legend in (a)] on the stress intensity factor induced at a crack tip as a function of distance between crack tip and twin tip. Twin perpendicular to crack plane. (a) Mode I and (b) mode II.
Fig. 8. Influence of twin length and width [legend in (a)] on the stress intensity factor induced at a crack tip as a function of distance between crack tip and twin tip. Twin relaxed against a constant friction stress of 100 MPa and aligned perpendicular to crack plane. (a) Mode I and (b) mode (II).
Table 5 Influence of twin with varying lengths.
Table 7 Effect of twin size at constant friction stress (100 MPa) on the interaction range.
Twin length lt (lm)
Range for KI (lm)
Range for KII (lm)
50 100 200 300
17 27 42 55
20 27 50 51
Twin lengthlt (lm)
Relaxed width wt (nm)
Range for KI (lm)
Range for KII (lm)
50 100 200 300
59 118 237 355
10 27 60 100
10 27 84 140
Table 6 Simulation parameters for twins with varying length and width obtained for a constant friction stress of 100 MPa. Twin length lt (lm)
Initial width wstart (nm)
Relaxed width wt (nm)
Burgers vector b (nm)
50 100 200 300
143 286 572 858
59 118 237 355
1 2 4 6
the separation between the centre lines of the two twins. The twin length was fixed at 100 lm and the friction stress 100 MPa. The twin-twin interaction strengthens the inward force on the innermost dislocations and so leads to a narrower relaxed twin width. The reduction in twin width increases as the twins are brought closer together. The earlier analytical work of Mitchell and Hirth [16] also indicated that twin-twin interactions generate more slender sharper twin shapes. 3.3.2. Variation of distance between crack tip and two twins Fig. 10 shows the stress intensity factors induced by a pair of twins at different twin–twin separations as a function of the distance between the crack tip and the closest twin. The infinite
Fig. 9. Influence of separation between twins in an array on the relaxed twin shapes.
twin-twin separation of course corresponds to the single twin case that has already been described. Perhaps surprisingly, the addition of the second twin actually slightly decreases the stress intensity factors at the crack tip. This is because the twin-twin interaction reduces the twin width which has already been shown to significantly reduce the KI and KII values
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Fig. 10. Influence of spacing between a pair of twins [legend in (a)] on the stress intensity factor induced at a crack tip as a function of distance between the closest crack tip and twin tip. Twins relaxed against a constant friction stress of 100 MPa and aligned perpendicular to crack plane. (a) Mode I and (b) mode II.
achieved. This reduction is more significant than the additional contributions to KI and KII made by the second twin which is further from the crack tip. 4. Discussion The initial part of this paper used a discretized version of the analytical model presented by Mitchell and Hirth [16]. The numerical approach was efficient and the calculations were fast and had little computational demand. Several observations from the earlier work by Mitchell and Hirth were confirmed by the numerical simulations including the approximately elliptical shape of the twin but slightly blunted at the twin tip. A dislocation-based boundary element method was then used to investigate the interaction between the simulated deformation twins and long straight cracks. There are numerous examples of models which capture the generation of deformation twins at the tips of cracks under loading (recent examples being [40,41]). Here the twin is a means of lowering the stress intensity at the crack tip through a plastic deformation mechanism. Deformation twinning has also been associated with crack initiation and fracture, particularly in BCC metals at low temperatures. Very early experimental work by Cahn in molybdenum [42] and Hull in Fe–Si [43] showed that large local stresses near the intersections of deformation twins can be relieved by the generation of a crack which can propagate to fracture the sample. Despite the long association of deformation twins with cracks the situation in which a crack interacts with a pre-existing deformation twins has not been studied in detail. To our knowledge there have not previously been any calculations of stress intensity factors induced by pre-existing deformation twins. Neither have there been experimental measurements of such stress fields with which the simulations can be compared. The motivation for this work was largely provided by the observation that cold work has a significant deleterious effect on the
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stress corrosion cracking resistance of austenitic stainless steels used in the nuclear industry [7–11,39]. Deformation twin bands are seen in these materials after moderate reductions in cold rolling and even ‘warm’ rolling at 200 °C [10]. The twins can typically traverse the entire widths of parent grains and have widths that vary in the range from 100 nm to 500 nm, and become more frequently observed as plastic strain levels increase. Kruska et al. [8] have shown in 304-type stainless steel that some grain boundaries and deformation twin bands that intersect the sample surface become preferentially oxidized when exposed to simulated pressurized water reactor (PWR) primary water at 360 °C. In similar environmental conditions but without cold work and therefore with no deformation twins present these steels are generally considered immune to stress corrosion cracking. One role that the cold work and resulting deformation twins may have is in providing a rapid diffusion path allowing the ingress of oxygen. However, grain boundary diffusion paths would still be available without cold work which suggests that this is not the only effect. A possible additional contribution to the increased susceptibility to stress corrosion cracking could be the local residual stresses associated with these twin deformation bands. Our calculations show that the deformation twins can induce significant stress intensity factors at the tips of cracks growing along grain boundaries which have also blocked the twin propagation. The largest KI values were obtained for the widest twin width considered (wt = 355 nm) and p amount to 6 MPa m when the twin was very close to the crack tip. This is 40% of the threshold stress intensity factor of p 15 MPa m required to initiate stress corrosion cracking of these steels in high temperature water environments of interest to the nuclear industry [9,38,39]. This result suggests that local stresses near the deformation bands must be seen as significant contributions to the unexpected SCC response of these systems. The high dislocation densities seen within the twin bands in TEM observations [7–10] suggest that some of the twinning shear strain p (c = 1/ 2) might be relaxed by dislocation slip processes within the twins themselves. Similarly some dislocation mediated plasticity may occur in the neighboring grain ahead of the blocked deformation twin which in our simulations we have treated as impenetrable. Both of these factors would tend to reduce somewhat the stress intensities generated and must be the subject of further work. However the mechanism of deformation twin assisted SCC still seems plausible.
5. Conclusions In this paper we combine a discrete dislocation representation of deformation twins with a dislocation-based boundary element model of a crack. The system was used to evaluate the mode I and II stress intensity factors induced by the simulated deformation twins at the tip of a long straight crack. The main results are summarized in the following points: 1. The relaxed twins were approximately ellipse shaped, however the flanks of the twins tended to be a little flatter and the twin tips blunted at impenetrable planar barriers representing grain boundaries. 2. The mode I stress intensity factor was largest when the twin was perpendicular to the crack plane but remained reasonably high provided the twin was at 45° or more from the crack plane. Wider shorter twins tended to induce larger stress intensities. 3. For the majority of geometries considered here the stress intensity factors induced by the twins tended to be larger for mode I than for mode II. However, mode II was enhanced and mode I reduced when the twin was at a small angle to the crack plane.
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4. The induced mode I stress intensity factor tended to be larger when the twin was close to the crack tip and reduced steadily as the twin was moved away from the crack tip. For the mode II case a more complicated variation was found with the sign of KII depending on both position and inclination of the twin relative to the crack. 5. The mode I stress intensities induced by the twin were sufficiently large to make it plausible that such interactions make a significant contribution to the reduced stress corrosion cracking resistance observed for austenitic stainless steels in high temperature water environments of concern to the nuclear industry.
Acknowledgement We thank Dr. Sergio Lozano-Perez (University of Oxford) for insightful discussion and encouragement to undertake these calculations. References [1] J.W. Christian, S. Mahajan, Prog. Mater Sci. 39 (1995) 1–157. [2] M.R. Barnett, M.D. Nave, C.J. Bettles, Mater. Sci. Eng. A – Struct. Mater. Properties Microstruct. Process. 386 (2004) 205–211. [3] C.N. Tome, R.A. Lebensohn, U.F. Kocks, Acta Metall. Mater. 39 (1991) 2667–2680. [4] D. Barbier, N. Gey, S. Allain, N. Bozzolo, M. Humbert, Mater. Sci. Eng. A – Struct. Mater. Properties Microstruct. Process. 500 (2009) 196–206. [5] I. Gutierrez-Urrutia, S. Zaefferer, D. Raabe, Mater. Sci. Eng. A – Struct. Mater. Properties Microstruct. Process. 527 (2010) 3552–3560. [6] H. Idrissi, K. Renard, L. Ryelandt, D. Schryvers, P.J. Jacques, Acta Mater. 58 (2010) 2464–2476. [7] W. Karlsen, G. Diego, B. Devrient, J. Nucl. Mater. 406 (2010) 138–151. [8] K. Kruska, S. Lozano-Perez, D.W. Saxey, T. Terachi, T. Yamada, G.D.W. Smith, Corros. Sci. 63 (2012) 225–233. [9] S. Lozano-Perez, T. Yamada, T. Terachi, M. Schroder, C.A. English, G.D.W. Smith, C.R.M. Grovenor, B.L. Eyre, Acta Mater. 57 (2009) 5361–5381. [10] S. Roychowdhury, V. Kain, S. Neogy, D. Srivastava, G.K. Dey, R.C. Prasad, Acta Mater. 60 (2012) 610–621.
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Simulation of deformation twins and their interactions with cracks Anke Stoll a,b, Angus J. Wilkinson a,⇑ a b
Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom Clinic for General, Visceral and Vascular Surgery, University Clinic Magdeburg, Leipziger Straße 44, 39120 Magdeburg, Germany
a r t i c l e
i n f o
Article history: Received 29 December 2013 Received in revised form 16 March 2014 Accepted 21 March 2014
Keywords: Deformation twins Dislocations Cold work Stress intensity factor Nuclear materials Austenitic stainless steel
a b s t r a c t A discrete dislocation model was used to simulate residual stress fields close to deformation twins in stainless steels. Dislocation pairs were distributed along an initially elliptical twin boundary and an iterative scheme used to allow the dislocations to relax towards positions where the internal shear stresses were below a friction stress. The relaxed twin shape was close to elliptical but the flanks were flatter and the twin tips where the grain boundaries are situated were somewhat blunted compared to an ellipse. A dislocation-based boundary element model was then used to assess the interaction between the stress field from such twins and a crack. The stress field near the tip of the crack was characterized in terms of mode I and mode II stress intensity factors. The effects of twin width, length, orientation and distance from the crack tip on the strength of the interaction were studied. Wider, shorter twins were found to induce the largest stress intensity at the crack tip when close to the crack tip and aligned perpendicular to the crack plane. The influence of a pair of deformation twins does not significantly exceed the influence of a single deformation twin. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Slip and twinning are the major deformation modes which enable a crystalline solid to generate a permanent shape change under the action of an applied stress. An extensive review on deformation twinning has been given by Christian and Mahajan [1]. Considerably more attention has been paid to understanding slip than twinning but there is currently interest in twinning. The classical definition of twinning requires that the twin and matrix lattices are related by either a reflection in some plane or by a rotation of 180° about some axis [1]. In many cases especially for the FCC, HCP and BCC metals both of these conditions are met and such twins are termed compound twins. Deformation twins can form by a homogeneous simple shear of the parent lattice. Deformation twinning has been observed in FCC, BCC and HCP metals and the characteristic shear strains can be quite large. The large lattice rotation between a twin and the parent lattice makes deformation twinning have a significant impact on both the intensity and nature of deformation textures [2,3]. Deformation twins often take up a lenticular shape in order to best accommodate the large shear strain while minimizing strain energy generated within the surrounding matrix [1]. In the FCC metals deformation twinning is perhaps most studied in steels where a class of austenitic alloys known as TWIP ⇑ Corresponding author. Tel.: +44 1865 273792; fax: +44 1865 273764. E-mail address: [email protected] (A.J. Wilkinson). http://dx.doi.org/10.1016/j.commatsci.2014.03.041 0927-0256/Ó 2014 Elsevier B.V. All rights reserved.
(twinning induced plasticity) steels have been developed primarily for the automotive industry e.g. [4–6]. In the austenitic stainless steels there is also evidence of intense twin-like deformation bands that form in increasing number during cold deformation [7–10] and are characterized by the classic 60° rotation about h1 1 1i compared to the matrix grain. There is some evidence that significant plastic deformation is required before these twins are observed and the threshold strain is dependent on the alloy composition [7–11]. These deformation twin bands are of considerable technological importance and are currently of particular interest within the nuclear industry because they have been found to be present in alloys that have shown susceptibility to intergranular stress corrosion cracking (SCC) in the cold worked state. In these alloys the SCC cracks mostly follow grain boundaries which have been oxidized and embrittled by the environmental exposure (hydrogenated water at moderate 300 °C temperatures) prior to cracking. Segregation during sensitization through either thermal exposure in service or accelerated heat treatment before laboratory tests and irradiation damage may worsen the situation but neither are necessary. This is of very significant concern because steels such as 304 had until relatively recently been largely considered immune to SCC in these environments [9]. Detailed TEM work by Lozano-Perez et al. [9] showed that deformation twins intersecting a free surface are preferred SCC nucleation sites while crack tips growing along grain boundaries branch into intersecting twin-like deformation bands which are preferentially oxidised.
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Deformation twinning has been modeled using a variety of approaches including molecular dynamics simulation [12–15], dislocation [16], disclination dipole [17] and finite element [18,19] modeling. The dislocation modeling approach is of interest here as it allows us to combine modeling of the twin with dislocation modeling of the crack. Mitchell and Hirth [16] gave an elegant model using a continuous density distribution of infinitesimal partial dislocations within a double-ended dislocation pile-up to represent the deformation twin. An analytical solution in which dislocations were at equilibrium was found for the density distribution and shape of the deformation twin via Hilbert transformations. The shape of an isolated twin in their 2 dimensional analysis was found to be an ellipse to first approximation though higher order terms show that twin tips are blunter with increasing stress while the twin boundary is populated with edge dislocations, while higher stress tends to sharpen the twin tip for the screw case. Dislocation mechanics has a more extensive history in the modeling of cracks and fracture. The seminal work by Billby Cotterell and Swinden [20,21] found equilibrium solutions for continuous density distributions of infinitesimal dislocation arrays representing the crack and associated plastic zone. A second type of solution to this problem in which dislocation densities are allowed to increase at the far end of the plastic zone which is blocked by a microstructural feature has found considerable use in modeling short fatigue crack propagation [22–24]. However, more complex geometries and loading cases have required a move to discrete dislocation treatments and numerical simulation rather than analytical solutions [25–34]. The question we seek to address in this paper is whether or not localized stresses near the tips of deformation twins seen in cold worked austenitic stainless steels could provide significant enhancement of the mechanical driving force for intergranular cracks and thus, in part, contribute to the recently observed susceptibility to stress corrosion cracking. To this end we develop a discrete dislocation version of the Mitchell and Hirth [16] model of deformation twins using arrays of edge dislocations which are allowed to relax to an equilibrium configuration. Stresses from the deformation twin are then used to load a dislocation-based boundary element model of a crack, which is assumed to be growing along a straight section of grain boundary at which the twin has terminated on one side of the boundary. The effects of the size, proximity and inclination of the deformation twin on the stress intensity factors (modes I and II) induced at the crack tip are investigated. For ease of computation we work within an elastically isotropic medium. We also assume that the ideal twinning shear for twins in FCC crystals is present and does not relax h1 1 2i{111} through dislocation mediated plasticity and that the twin shape is not perturbed by the presence of the crack. These assumptions make our calculations an upperbound assessment of the stress intensity that could be generated by the deformation twin.
2. Methodology 2.1. Simulation of deformation twins To simulate twins, we start from an elliptical shape with partial dislocations positioned along the elliptical twin boundary as proposed in [16] (see Fig. 1). For an ellipse centred at the origin with its major axis along the x0 -direction partial dislocations in the right hand half of the ellipse have positive Burgers vector while those at positions mirrored across the y0 -axis have negative Burgers vector. The Burgers vectors are of 1/6 a[1 1 2] type (i.e. the x0 -axis is p parallel to [1 1 2]) with magnitude b = a/ 6. The twin is on the 0 with the partial (111) plane (i.e. the y -axis is along [111])
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Fig. 1. Discrete dislocation array representation of a twin.
p dislocations on planes separated along the y0 -axis by a/ 3. The characteristic twinning shear c is then given by b/d (where d is p the spacing of slip planes along the y0 -axis) and is 1/ 2 for the FCC system. In order to find a relaxed twin shape an elliptical starting geometry was chosen which is known to be a reasonably good approximation. The twin was then divided up into a number of slip planes spaced much further apart in the simulation than the planes in the crystal. To keep the required twinning shear (111) p of 1/ 2 for our chosen material the magnitude of the Burgers vector for all dislocation partials at the boundary of the twin are changed in proportion to the plane spacing. The shear stress generated by all the partial dislocations positioned on the twin boundary is then calculated at the dislocation positions. Dislocations are then moved according to their shear stress. If the shear stress is higher than a fixed critical shear stress (or friction stress) + sc, the dislocation is moved outward, away from the centre of the ellipse. The distance of movement was set at one quarter of the distance from the current position of the dislocation to that of the neighboring dislocation (which is further outward). The movement was constrained to be along the x0 -direction i.e. along a horizontal plane (constant y0 ). Dislocations are not permitted to move beyond the original two ends of the twin which are considered pinned by impenetrable grain boundaries. If the shear stress at a dislocation position is below sc it is moved toward the centre. Again the distance between the dislocation location and the neighboring dislocation (closer to centre) is calculated and the dislocation is allowed to move one quarter of that distance toward x0 = 0, again with y0 constant. Once dislocations at x0 = 0 start to pile-up, distances between dislocations of the two twin halves on the same planes are calculated and dislocation pairs are allowed to eliminate once their separation is smaller than 1 nm (at which point the mutual interaction causes stresses in excess of 1 GPa). Using the approach of moving dislocation partials along horizontal planes according to their shear stresses does not result in a twin shape with zero shear stresses along the twin boundary (calculated at the positions of dislocations). Therefore, a friction stress sc has to be introduced as mentioned earlier to define a maximum and minimum shear stress. If the shear stress at a dislocation position is within the range sc to +sc then the dislocation at that position does not move. Once a relaxed twin shape is obtained a coordinate transformation is conducted and the twin is turned to allow the deformation twin to be placed in front of the crack tip. 2.2. Simulation of crack opening and stress field For the simulation of crack propagation a dislocation-based boundary element model BEM approach was developed as had previously been used by Riemelmoser et al., Nowell et al. and Schick et al. [25–28,30]. The most important principles of the modeling
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semi-minor axis bt (which is half the twin width wstart) varies depending on the relaxation process of the twin shape to the relaxed configuration. Generally we find the flanks of the relaxed twin are slightly flatter compared to the ellipse which is in accord with Mitchell and Hirth’s results [16]. The relaxed twin geometry is influenced by various factors including: the slip plane spacing, the size of the friction stress, the starting width of the ellipse and the twin - grain boundary inclination angle. These influences were evaluated in sensitivity studies which are reported in the following sub-sections.
Fig. 2. Discrete dislocation array representation of a crack.
approach will be shown here while for a more detailed review, the reader is referred to [35,36]. Fig. 2 shows the general configuration of the dislocation model for a planar crack along the x-axis. Plane strain conditions are assumed and the sample dimensions are large compared to the length of the crack. Each boundary element in the crack contains a collocation point cei and dislocation dipoles of climb and glide character in order to simulate mode I and mode II displacements respectively. The boundary elements are made gradually smaller closer to the crack tip to guarantee a high accuracy there. The length li of boundary elements were calculated knowing the crack length and the number of boundary elements and follow the geometric progression li = 1.08 li1. A stress matrix G is generated and GIij , GIIij are the stresses at collocation point i generated by climb and glide dipoles of element j when the Burgers vectors of dislocations in the glide or climb dipole are each set to unity. Head [37] gave the full equations for stresses of dislocations in a homogeneous, elastically isotropic half space which are used in the analysis. To solve the fracture problem for the crack opening and shearing and the stress field around the crack a set of linear equations are constructed and solved numerically. At each collocation point we require that the stress rapp (=ryy, rxy) applied to the crack is negated by stresses generated by the dislocation dipoles representing the crack so that the normal stress and shear stress across the crack faces are brought to zero. Then the solution for the opening displacements (bi) of an elastic crack with loading rapp can be calculated solving the system of equations
Gji bi ¼ rapp ! bi ¼ ðGji Þ1 rapp j j :
ð1Þ
In this case rapp is obtained from the deformation twin simulation described in Section 2.1. The solution of the elastic problem is found with the calculation of bi, from which the stress at any point can be calculated using superposition and Head’s solution for the stresses from a dislocation in an semi-infinite half space [37]. The stress intensity factors KI and KII are calculated at the crack p tip by fitting the ryy and rxy stress variations to 1/ r distributions with r the distance from the crack tip along a line directly in front of the crack tip as indicated in Fig. 2 [35,36].
3. Results 3.1. Equilibrium twin geometry The ellipse has the middle point (0, 0). Its semi-major axis at is chosen as 50 lm to mimic a twin of total length lt = 100 lm. The
3.1.1. Influence of slip plane spacing First of all, the influence of the twin slip plane spacing was investigated to assure that the number of partial dislocations chosen to describe the twin leads to accurate results without having to work with larger computation times. The elliptical starting geometry of a twin was described by 100, 200, 400, 600, 800 dislocations with total twin length lt = 100 lm and starting total width wt = 500 nm. Fig. 3a shows that the overall relaxed twin shape results are consistent and do not depend strongly on the number of dislocations used to describe the initial elliptical twin shape. However, at the tips of the relaxed twins slight differences appear (Fig. 3b). Generally, the twin tips are broader when a larger number of dislocations is used to represent the twin geometry. In order to calculate a relaxed twin with 100 dislocations 135 iterations had to be performed while for 800 dislocations 997 iterations are needed. The relaxed shape of a twin initially described by 200 dislocations is found within reasonable calculation time and shows the broad twin tip typical for higher accuracy calculations with many more dislocations. Therefore, in the following, the initial elliptical twins were described by 200 dislocations unless explicitly stated otherwise. Fig. 3c shows that the s0xy shear stress variation along the twin boundary. Only at the position of the outermost dislocation (located at the twin tip which is positioned at the grain boundary) the shear stress is above the friction stress of 100 MPa. The general situation is that dislocations towards the tip of the twin experience shear stresses tending to move them outwards and to extend the twin. This can result in a blunted tip as dislocations are prevented from moving beyond the grain boundary which acts as an impenetrable barrier in these simulations. In contrast the dislocations further back from the tip experience repulsion from those at the tip and so are pushed back against the friction stress. 3.1.2. Influence of friction stress The relaxed twin geometry is influenced by the level of friction stress. The simulations are always initiated from an ellipse that has the required length with dislocations at the twin tips pinned by grain boundaries and a width that is larger than that expected for the relaxed system. Relaxation then leads to a narrowing of the twin as the central dislocations move back and opposing pairs of dislocations collapse back and are annihilated. For a very large friction stress this process has a minimal influence and the overall twin shape appears almost unaltered from the starting geometry. However, close inspection generally reveals some outward movement of dislocations very near the twin tip where stresses are very high leading to a slight blunting of the tip compared to the starting ellipse, while a small amount of narrowing at the centre of the twin also occurs. In these situations of very high friction stress the majority of the dislocation may not have moved during the relaxation which is undesirable as dislocation positions are then those imposed by the initial setup rather than truly relaxed. We thus work with lower friction stress levels which are more representative of flow stresses in the target stainless steels and at which all the dislocations move during the relaxation. Tip blunting is still evident but the major effect of decreasing the friction stress is that
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Fig. 3. Effect of number (given in legend) of dislocations initially in the array on the relaxed twin shape. (a) Overview of whole twin and (b) near tip region. The black lines shows the initial ellipse shape. Different x0 and y0 length scales are used to emphasise difference in shape (true shape is longer and thinner). (c) Shear stress remaining after relaxation process.
more of the central dislocations collapse back and are annihilated causing the twin width to reduce. The effect is demonstrated in Table 1 which gives relaxed twin widths as a function of friction stress for a starting geometry of a 100 lm long, 1000 nm wide twin represented by 200 dislocations (i.e. 100 pairs of dislocations) on planes separated by 10 nm. At the lower friction stresses, which are more realistic for our target austenitic stainless steel materials, the twin narrows to a small fraction of the starting configuration width and consequently is made up of rather few dislocations. A better representation might be obtainable if the initial ellipse width were reduced while maintaining the same number of dislocations. 3.1.3. Influence of starting geometry Getting the initial ellipse width close to, but larger than, the relaxed twin width is advantageous in both reducing the number of iterations in the relaxation calculation and in keeping a large number of dislocations representing the relaxed twin. Providing the initial twin is larger than the relaxed twin width then the overall shape of the relaxed twin is not too strongly dependent on the initial twin width. However, for the wider initial configurations the relaxed twin is represented by fewer dislocation pairs spaced more widely apart and so do not represent the fine detail as correctly. In
particular the shape near the twin tip is not well captured if the initial configuration is too wide so that too few dislocations remain in the relaxed configuration. Having looked at a range of twin lengths and widths has shown that a good twin starting width is twin length/350 for friction stress sc = 100 MPa which gives a value of 286 nm and results in relaxed twin widths within a range of 100 nm- to 500 nm which is consistent with experimental observations in austenitic stainless steel [7–11]. 3.1.4. Influence of grain boundary inclination angle As dislocations at the twin tip are pushed outward to the grain boundary it was necessary to examine the influence of the inclination of the grain boundary plane relative to the twin axis and Burgers vector direction. As might be expected the overall relaxed twin shape is not strongly affected by the angle between the twin axis and the grain boundary. The twin widths stay essentially unchanged as the angle is varied, remaining at 118 nm for all cases other than for the twin at an angle of 22.5° when the twin is one simulated slip plane narrower at 113 nm. However, the geometry at the twin tips is of course affected. The mirror symmetry across the major axis of the twin found when the twin is normal to the grain boundary is removed when the boundary is tilted. 3.2. Influence of a deformation twin on crack tip stress fields
Table 1 Twin width for varying friction stresses for 100 lm long twins. Friction stress sc (MPa)
Twin width (nm)
Start 900 600 400 100 50
1000 980 737 475 131 71
From experimental observations in the literature we know that twins in austenitic stainless steels can span entire grains that can in turn be of order 100 lm in diameter. At such lengths the deformation twin bands range in width from 100 nm to 500 nm thick [9,17]. Our simulations of twin-crack interactions will thus be centred on such twin sizes. The crack is assumed to be 10 mm long and located in a homogeneous medium under an applied far-field
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tensile stress of 100 MPa along the crack normal. It is described by 250 boundary elements of decreasing size towards the crack tip as described in Section 2.2. Simulations with only this far field externally applied stress loading the crack generate a mode I stress p intensity factor KI of 14.4 MPa m, while symmetry of course results in a mode II stress intensity factor KII of zero. This is a useful baseline as it approximates the threshold stress intensity level required to initiate stress corrosion cracking in high temperature water for the austenitic stainless steels of interest [9,38,39]. 3.2.1. Influence of distance between crack tip and twin The first calculations are for a twin inclined at 90° to the crack, with a twin length of 100 lm and width of 118 nm. Fig. 4 shows how KI and KII vary as the distance between the crack tip and the twin changes from 10 nm to 10 mm. As expected the stress intensity factors take opposite sign but maintain the same magnitude when the sense of the shearing direction for the deformation twin is reversed. The increase in KI is highest when the twin is positioned very close to the crack tip and the twin induces a KI of p 2.3 MPa m at the closest distance of 10 nm considered here. This is a significant increase of about 16% above the baseline level for the mode I stress intensity factor. There is a steady monotonic decrease in KI induced by the twin as the distance to the crack tip is increased. The mode II stress intensity factor induced is p rather more modest at 0.2 MPa m in magnitude at most. The variation of KII with distance between twin and crack is more complex with two changes of sign occurring as the crack is brought close to the crack tip from far away. The range at which there is significant interaction is very large compared to the twin width. We define, arbitrarily, the length of interaction as being the furthest distance of the twin tip from the crack tip at which stress intensity p p of 0.2 MPa m is induced for mode I, and 0.1 MPa m for mode II. At these threshold levels both mode I and mode II have the same interaction length of 27 lm for this particular case of a 100 lm long, 118 nm wide twin with a friction stress of 100 MPa.
Fig. 4. Influence of distance between crack tip and twin tip on stress intensity factor on crack for twin running perpendicular to crack plane. (a) Mode I and (b) mode II. In each case the solid and dotted lines show the effect of reversing the sign of the shear direction in the twin.
3.2.2. Influence of twin inclination angle Intergranular cracks will encounter twins running through grains at a range of different inclination angles. Fig. 5 shows the mode I and II stress intensity factors induced by twins inclined at different angles to the crack plane as a function of their distance from the crack tip. Only one sense of the twin shear direction is shown. The twins were 100 lm long and the widths were 118 nm for all except the twin inclined at 22.5° for which a slightly narrower width of 113 nm was found. The normal stresses induced by the twin across the crack plane are largest when the twin is at 90° and so the highest KI values are achieved then. However, there are a considerable range of angles (112.5° to 45°) over which a similarly large KI is induced. The interaction lengths, at which the p p induced KI > 0.2 MPa m and KII > 0.1 MPa m, are given in Table 2 from which it is seen that the high KI induced by twins over the angular range 112.5° to 45° persists to considerable lengths (>15 lm). By contrast the mode II interaction is stronger for deformation twins that are less steeply inclined to the crack plane as greater shear stresses are then generated by the twin on the crack plane. The mode II stress intensity factors are generally smaller than for p mode I though KII values can exceed 1 MPa m when the twins are highly inclined and very close to the crack tip. KII values are generally smaller in magnitude as the twin inclination angle is increased toward 90° and when the twin is inclined beyond 90° the interaction increases in strength again but the sign of KII becomes negative. 3.2.3. Influence of twin width (varying friction stress) Section 3.1.2 showed how the friction stress opposing motion of the partial dislocations bounding the twin has a strong influence on the twin width. Here we keep the twin length fixed at 100 lm and at 90° to the crack plane while varying the friction stress from 50 MPa to 300 MPa. The width of the initial ellipse was scaled in proportion to the friction stress while the length of
Fig. 5. Influence of twin inclination angle [legend in (a)] on the stress intensity factor induced at a crack tip as a function of distance between crack tip and twin tip. (a) Mode I and (b) mode II.
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A. Stoll, A.J. Wilkinson / Computational Materials Science 89 (2014) 224–232 Table 2 Influence of twin inclination angle on range of interaction with crack (100 lm long 118–113 nm wide twins, 100 MPa friction stress). Angle
Range for KI (lm)
Range for KII (lm)
112° 90° 67.5° 45° 22.5°
29 27 22 18 6
8 27 45 55 70
the Burgers vectors and their spacing along the y0 axis was reduced to maintain a constant twinning shear strain and number of dislocations. Table 3 details the starting conditions and resulting twin width after relaxation from which it is seen that the twin width increases linearly with friction stress. Fig. 6 shows that both the mode I and mode II stress intensity factors induced by the twins increase in proportion with the twin width and hence also in proportion to the friction stress. For the widest twin wt = 355 nm considered here, which is within the range seen experimentally in austenitic stainless steels, a very sigp nificant KI increase of 6 MPa m occurs when the twin is very close to the crack tip. 3.2.4. Influence of twin length (varying friction stress) Twins of different length can be obtained by moving the barriers to outward motion of the dislocations at the twin tips further apart. If the friction stress is reduced as the twin length is increased then it is possible to maintain a constant relaxed twin width as indicated by Table 4. The initial ellipse width and spacing of slip planes and Burgers vector length were held constant for this series of simulations. Fig. 7 shows the variation of stress intensity factors induced by such twins as a function of their distance from the crack tip. Despite the reduced twin volume for the shorter twin, the more compact twin shape leads to higher stresses within the surrounding matrix as demonstrated by the need for increased friction stresses in these cases. This results in larger KI and KII values while the shorter twins are close to the crack tip. However the longer twin length does extend the interaction length (Table 5) and results in the shorter twins having a weaker interaction with the crack at longer distances (Fig. 7). 3.2.5. Influence of twin length & thickness (constant friction stress) If the friction stress is held constant then any changes to the twin length will also cause changes in the relaxed twin width. Table 6 shows how the relaxed twin width decreases in proportion to the prescribed twin length for a fixed friction stress of 100 MPa. The width of the initial ellipse and the Burgers vector and slip plane spacing are also adjusted so that the final relaxed configuration is represented by a similar number of dislocations for each of the simulations (Table 6). The variation in stress intensity factors induced by these twins are shown as a function of distance between the twin and the crack tip in Fig. 8. As expected the larger twins generate larger KI and KII Table 3 Simulation parameters for twins with varying width. Twin length lt (lm)
Initial width wstart (lm)
Burgers vector b (nm)
Friction stress sc (MPa)
Relaxed width wt (nm)
100 100 100 100
286/2 286 286 2 286 3
1 2 4 6
50 100 200 300
59 118 237 355
Fig. 6. Influence of twin width [legend in (a)] on the stress intensity factor induced at a crack tip as a function of distance between crack tip and twin tip. Twin perpendicular to crack plane. (a) Mode I and (b) mode II.
Table 4 Simulation parameters for twins with varying length. Twin length lt (lm)
Initial width wstart (lm)
Burgers vector b (nm)
Friction stress sc (MPa)
Relaxed width wt (nm)
50 100 200 300
286 286 286 286
2 2 2 2
200 100 50 33
118 118 118 118
values. Fig. 8 also indicates that the interaction persists to larger distances from the larger twins and is confirmed by the interaction lengths tabulated in Table 7. 3.3. Influence of an array of twins on a mode I crack In cold worked stainless steels the deformation twins bands often occur as a series of parallel features across a given grain [7,9,10]. The next two sub-sections describe some simulations that have been made to briefly investigate how twin-twin interactions affect first the shape of the relaxed twins and second the strength of the interactions with a crack. 3.3.1. Interaction of two twins The procedures for generating an array of dislocations relaxed against a friction stress and described in Section 2.1 were extended to a pair of twins. As before, dislocations were allowed to move outward if the shear stress at their location is positive and exceeds the friction stress or to move inward if shear stress is negative but larger than the friction stress. The shear stress at a particular dislocation includes contributions from all other dislocations within that twin and all dislocations representing the other twin. Fig. 9 depicts the shape of the lower twin of the pair as a function of
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Fig. 7. Influence of twin length [legend in (a)] on the stress intensity factor induced at a crack tip as a function of distance between crack tip and twin tip. Twin perpendicular to crack plane. (a) Mode I and (b) mode II.
Fig. 8. Influence of twin length and width [legend in (a)] on the stress intensity factor induced at a crack tip as a function of distance between crack tip and twin tip. Twin relaxed against a constant friction stress of 100 MPa and aligned perpendicular to crack plane. (a) Mode I and (b) mode (II).
Table 5 Influence of twin with varying lengths.
Table 7 Effect of twin size at constant friction stress (100 MPa) on the interaction range.
Twin length lt (lm)
Range for KI (lm)
Range for KII (lm)
50 100 200 300
17 27 42 55
20 27 50 51
Twin lengthlt (lm)
Relaxed width wt (nm)
Range for KI (lm)
Range for KII (lm)
50 100 200 300
59 118 237 355
10 27 60 100
10 27 84 140
Table 6 Simulation parameters for twins with varying length and width obtained for a constant friction stress of 100 MPa. Twin length lt (lm)
Initial width wstart (nm)
Relaxed width wt (nm)
Burgers vector b (nm)
50 100 200 300
143 286 572 858
59 118 237 355
1 2 4 6
the separation between the centre lines of the two twins. The twin length was fixed at 100 lm and the friction stress 100 MPa. The twin-twin interaction strengthens the inward force on the innermost dislocations and so leads to a narrower relaxed twin width. The reduction in twin width increases as the twins are brought closer together. The earlier analytical work of Mitchell and Hirth [16] also indicated that twin-twin interactions generate more slender sharper twin shapes. 3.3.2. Variation of distance between crack tip and two twins Fig. 10 shows the stress intensity factors induced by a pair of twins at different twin–twin separations as a function of the distance between the crack tip and the closest twin. The infinite
Fig. 9. Influence of separation between twins in an array on the relaxed twin shapes.
twin-twin separation of course corresponds to the single twin case that has already been described. Perhaps surprisingly, the addition of the second twin actually slightly decreases the stress intensity factors at the crack tip. This is because the twin-twin interaction reduces the twin width which has already been shown to significantly reduce the KI and KII values
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Fig. 10. Influence of spacing between a pair of twins [legend in (a)] on the stress intensity factor induced at a crack tip as a function of distance between the closest crack tip and twin tip. Twins relaxed against a constant friction stress of 100 MPa and aligned perpendicular to crack plane. (a) Mode I and (b) mode II.
achieved. This reduction is more significant than the additional contributions to KI and KII made by the second twin which is further from the crack tip. 4. Discussion The initial part of this paper used a discretized version of the analytical model presented by Mitchell and Hirth [16]. The numerical approach was efficient and the calculations were fast and had little computational demand. Several observations from the earlier work by Mitchell and Hirth were confirmed by the numerical simulations including the approximately elliptical shape of the twin but slightly blunted at the twin tip. A dislocation-based boundary element method was then used to investigate the interaction between the simulated deformation twins and long straight cracks. There are numerous examples of models which capture the generation of deformation twins at the tips of cracks under loading (recent examples being [40,41]). Here the twin is a means of lowering the stress intensity at the crack tip through a plastic deformation mechanism. Deformation twinning has also been associated with crack initiation and fracture, particularly in BCC metals at low temperatures. Very early experimental work by Cahn in molybdenum [42] and Hull in Fe–Si [43] showed that large local stresses near the intersections of deformation twins can be relieved by the generation of a crack which can propagate to fracture the sample. Despite the long association of deformation twins with cracks the situation in which a crack interacts with a pre-existing deformation twins has not been studied in detail. To our knowledge there have not previously been any calculations of stress intensity factors induced by pre-existing deformation twins. Neither have there been experimental measurements of such stress fields with which the simulations can be compared. The motivation for this work was largely provided by the observation that cold work has a significant deleterious effect on the
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stress corrosion cracking resistance of austenitic stainless steels used in the nuclear industry [7–11,39]. Deformation twin bands are seen in these materials after moderate reductions in cold rolling and even ‘warm’ rolling at 200 °C [10]. The twins can typically traverse the entire widths of parent grains and have widths that vary in the range from 100 nm to 500 nm, and become more frequently observed as plastic strain levels increase. Kruska et al. [8] have shown in 304-type stainless steel that some grain boundaries and deformation twin bands that intersect the sample surface become preferentially oxidized when exposed to simulated pressurized water reactor (PWR) primary water at 360 °C. In similar environmental conditions but without cold work and therefore with no deformation twins present these steels are generally considered immune to stress corrosion cracking. One role that the cold work and resulting deformation twins may have is in providing a rapid diffusion path allowing the ingress of oxygen. However, grain boundary diffusion paths would still be available without cold work which suggests that this is not the only effect. A possible additional contribution to the increased susceptibility to stress corrosion cracking could be the local residual stresses associated with these twin deformation bands. Our calculations show that the deformation twins can induce significant stress intensity factors at the tips of cracks growing along grain boundaries which have also blocked the twin propagation. The largest KI values were obtained for the widest twin width considered (wt = 355 nm) and p amount to 6 MPa m when the twin was very close to the crack tip. This is 40% of the threshold stress intensity factor of p 15 MPa m required to initiate stress corrosion cracking of these steels in high temperature water environments of interest to the nuclear industry [9,38,39]. This result suggests that local stresses near the deformation bands must be seen as significant contributions to the unexpected SCC response of these systems. The high dislocation densities seen within the twin bands in TEM observations [7–10] suggest that some of the twinning shear strain p (c = 1/ 2) might be relaxed by dislocation slip processes within the twins themselves. Similarly some dislocation mediated plasticity may occur in the neighboring grain ahead of the blocked deformation twin which in our simulations we have treated as impenetrable. Both of these factors would tend to reduce somewhat the stress intensities generated and must be the subject of further work. However the mechanism of deformation twin assisted SCC still seems plausible.
5. Conclusions In this paper we combine a discrete dislocation representation of deformation twins with a dislocation-based boundary element model of a crack. The system was used to evaluate the mode I and II stress intensity factors induced by the simulated deformation twins at the tip of a long straight crack. The main results are summarized in the following points: 1. The relaxed twins were approximately ellipse shaped, however the flanks of the twins tended to be a little flatter and the twin tips blunted at impenetrable planar barriers representing grain boundaries. 2. The mode I stress intensity factor was largest when the twin was perpendicular to the crack plane but remained reasonably high provided the twin was at 45° or more from the crack plane. Wider shorter twins tended to induce larger stress intensities. 3. For the majority of geometries considered here the stress intensity factors induced by the twins tended to be larger for mode I than for mode II. However, mode II was enhanced and mode I reduced when the twin was at a small angle to the crack plane.
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4. The induced mode I stress intensity factor tended to be larger when the twin was close to the crack tip and reduced steadily as the twin was moved away from the crack tip. For the mode II case a more complicated variation was found with the sign of KII depending on both position and inclination of the twin relative to the crack. 5. The mode I stress intensities induced by the twin were sufficiently large to make it plausible that such interactions make a significant contribution to the reduced stress corrosion cracking resistance observed for austenitic stainless steels in high temperature water environments of concern to the nuclear industry.
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