Ductility by shear band delocalization in the nano-layer of gradient structure

ABSTRACT Nanostructured (NS) metals typically fail soon after yielding, starting with the formation of narrow shear bands. Here we report the observation of shear band delocalization in gradient metals. Shear bands were nucleated and delocalized in the NS layers by propagating along the gage length soon after yielding, converting the shear band into a localized strain zone (LSZ). Synergistic work hardening was developed in the LSZ by regaining dislocation hardening capability, and by back-stress hardening from the strain gradients in the axial and depth directions, which helped with enhancing global ductility. GRAPHICAL ABSTRACT

Ductility is generally defined as uniform elongation in the specimen gage length during tensile testing [20]. It can be determined by the Considère criterion [21], (dσ/dε)/σ ≥ 1, where σ is true stress and true strain, or by the Hart's criterion [20], (dσ/dε)/σ + m ≥ 1,where m is the strain rate sensitivity. Neither criterion literally specifies whether homogeneous strain in the gage section is a must for ductility, which raises a question: is it possible to maintain high ductility if strain is not uniform?
We investigated this issue using a gradient structured (GS) interstitial-free (IF) steel, consisting of NS surface layers with a continuous increase in grain sizes along the depth to the central coarse-grained (CG) layer [11,12]. It is found that shear bands were formed in the NS layers at a very early stage during tensile testing, as NS metals typically do. Unexpectedly, the shear bands became stabilized due to strain gradient and propagated slowly along the gage length to become a localized strain zone (LSZ), which produced synergistic strain hardening to help with retaining ductility. In other words, the shear bands helped with retaining ductility, contrary to our conventional understanding of strain localization [22][23][24].
CG IF steel plate of 1 mm thick with a mean grain size of 26 μm was used as the initial material. Gradient structure (GS) was prepared using surface mechanical attrition treatment (SMAT) technique [25]. The NS layer is ∼ 40 μm thick, with a mean grain size of 200 nm.
The tensile samples were dog-bone shaped with the gauge section of 1 mm × 2 mm × 10 mm. Uniaxial tensile tests were conducted at a strain rate (ε app )of 8 × 10 −4 s −1 under room temperature. Digital image correlation (DIC) imaging was performed on the top NS surface layer (see Supplementary Material). Microstructures, texture, and Vickers micro-hardness (H V ) were characterized on samples subjected to various tensile strains. H V was measured with load of 25 g and dwell period of 15 s on NS surface after grinding off surface roughness by ∼ 10 μm deep. Focused ion beam was used to cut the cross-sectional transmission electron microscopy (TEM) samples precisely in the shear band and LSZ in the NS layer at varying tensile strains according to the DIC image. Figure 1(a) shows the microstructure of the central CG layer with an average grain size of 26 μm, and Figure 1(b) shows the nanostructure in the NS surface layer at ∼ 40 μm depth, which reveals entangled dislocations in grain interiors and the average grain size is 200 nm, typical of severely deformed metals [3,4,26]. The corresponding H V gradient is visible from the surface to the center (Figure 1(c)). Figure 2(a) shows the true stress-strain (σ -) curves of GS and CG samples. The GS sample shows uniform elongation (E U ) of 20.6%, which retained about 80% of that (26%) of the CG sample, while doubled yield strength. Figure 2(b) shows a set of typical contour maps showing longitudinal (axial) strain (ε L , %) distribution at varying applied tensile strains (ε app , %). At ε app = 1%, two shear bands crossing each other were formed in the upper part of the sample. The shear bands are measured at ∼ 45°to tensile axis, with the orientation of maximum resolved shear stress. Thus, the plastic response in NS layer begins with the nucleation of shear bands. These two shear bands propagated downward with increasing ε app along the gauge length and continually broadens to form an LSZ, see maps with increasing ε app from 2.7% to 15.8% until necking at ε app = 20.6% (E U ) (Figure 2(a)). A weaker LSZ is also visible in the lower part but fail to propagate much. In every sample tested, there is only one dominant LSZ, which led to the failure of whole sample.
The LSZ accumulated plastic strains continuously. Figure 2(c) shows heterogeneous ε L at varying ε app as a result of the propagating LSZ. ε L was measured along a longitudinal line which goes through the maximum ε L (ε max ) in each contour, e.g. white line at ε app = 1% in Figure 2(b). The LSZ is defined, here, as that with ε L > ε app , e.g. the segment bounded by two × marks in the curve at ε app of 15.8%. Figure 2(d) shows the evolution of ε max and minimum ε L (ε min ) in NS layer. Note that ε max is always in the center of the LSZ. Figure 2(e) shows the evolution of average ε L in LSZ,ε LSZ . Figure 2(f) shows the axial maximum strain rateε L (ε max ) in NS layer calculated byε L = ∂ε/∂t.ε max is found always at the propagating front of the LSZ and can be used as an indicator for the propagating rate of LSZ.
From Figure 2(b-f), several features of the plastic deformation can be drawn. Firstly, the shear band/LSZ sustained more strain in its interior than outside with increasing ε app (Figure 2(c,d)), typical of strain localization. The left and right peaks of ε L , e.g. at ε app ≤ 6.8% (Figure 2(c)), represent the upper and the lower shear bands, while the latter disappeared at ε app = 15.8%. Secondly, ε L is not uniform in the NS layer during the whole testing (Figure 2(b-d)). Moreover, ε max in shear band/LSZ is larger than ε app , even than E U (shadowed area in Figure 2(d)). In contrast, ε L is equal to ε app in CG (see Figure S1 in the Supplementary Material), as represented by the diagonal dotted line, due to uniform deformation in the stable CG layer before necking. Thirdly, the ratio ofε LSZ /ε app in the LSZ (Figure 2(e)) can be seen as an indicator on the severity of strain localization. As seen, strain localization started at low  ε app of ∼ 1%, reached the maximum at ε app = 2.0%, and then decreased until the end of uniform elongation (E U ). Fourthly,ε max started about one order of magnitude larger thanε app = 8 × 10 −4 s −1 (Figure 2(f)) and dropped monotonously. In other words, the shear band/LSZ propagated fast initially, but slowed down later. In contrast, the CG sample shows a nearly constantε L until necking (Figure 2(f) and Figure S1 in SI). Most importantly, the applied strain is mostly sustained in the LSZ. The deformation fraction supplied by LSZ is calculated by (ε LSZ × area ofLSZ)/(ε app × gauge area). As seen in the inset of Figure 2(e), the LSZ accommodated the majority of the applied tensile strains.  Figure 2(b). In other words, the LSZ triggered localized lateral shrinkage at the very early stage. This is unexpected because the sample was still globally stable with strong strain hardening. Figure 2(h) shows the average lateral shrinkage rate (ε T ). It reached the peak at ε app ∼ 5%, decreased subsequently, and increased again until global necking (E U ). In contrast, the CG sample has constantε T , equal to half ofε app , typical of uniform plastic deformation.
The heterogeneous ε L (Figure 2(c)) caused axial strain gradient, dε L /dL, near the LSZ boundaries in the NS layer. The maximum strain gradient lies always at the front of propagating LSZ, and increased with ε app (Figure 3(a)). Strain softening occurred in the center of the LSZ at the early stage of shear band formation (Figure 3(b)), showing dramatic drop of H V . H V increased later with increasing ε app from 6.8% to 15.8%, indicating that it recovered some strain hardening capability (Figure 3(c)).
Strain gradient arises from mechanical incompatibility. The propagating front of the LSZ demarcates its boundary (Figure 3(b)). As shown, there is a steep strength gradient at the LSZ boundary, which will lead to strain gradient during tensile deformation. Geometrically necessary dislocations need to be produced to facilitate the strain gradient [27][28][29], which will produce strong back-stress hardening [14,[30][31][32][33][34][35][36]. The back-stress hardening will impede the axial rapid propagation of the LSZ, which helps with the stabilization and delocalization of the shear band/the LSZ, leading to the drop ofε L with ε app (Figure 2(g)).
The initial drop and later rise of H V in the center of the LSZ (Figure 3(b,c)) indicate the recovery of dislocation strain hardening. Figure 4(a) shows a weak compression shear texture with (110) orientation parallel to compressive axis due to SMAT before tensile testing (a-1), which evolved later into strong tensile textures with (110) orientation parallel to tensile axis at ε app of 15.8%. The tensile texture strength is especially strong at the center of the LSZ, as indicated in Figure 4(a-3), as compared with locations outside of the LSZ (as indicated in Figure 4(a-2). This indicates strong dislocation activities in nanograins in the LSZ during tensile deformation.
The dislocation activity is corroborated by TEM observations. At ε app of 4% (Figure 4(b)), the dislocations are hardly seen inside most grains in the center of the LSZ, in contrast to high density of dislocations before tensile testing (Figure 1(b)). Inset at the top left corner of Figure 4(b) reveals dislocation debris in a few grains. This indicates dismantlement of original dislocation substructure due to the change of strain path and stress state [12,37]. This is the reason of the observed strain softening in the shear band/LSZ (Figure 3(b)). Further straining led to the formation of new dislocation networks near grain boundaries, as shown in Figure 4(c) (ε app = 20.6%). This is caused by the complex stress state [12,37] in the LSZ, where multiple slip systems are activated, which in turn forms new dislocation entanglements and accumulation (see the inset). The change in dislocation density coincides with that of H V (Figure 3(c)), suggesting their close relationship. Furthermore, the mean grain size is maintained close to ∼ 200 nm in the LSZ during the tensile testing, indicating no grain growth in the LSZ.
It should be noted that initial softening and recovered hardening observed in the LSZ has some similarity and difference from the reported hardness fluctuation observed during the severe plastic deformation of a nanocrystalline Ni-Fe alloy [38]. Although both cases were linked to dislocation density change, the initial softening and recovered hardening in the LSZ was caused by the change of strain path, while the latter was observed during severe straining in the strain direction.
The NS layer and central CG layer in the GS sample were subjected to the same amount applied tensile strain. The NS layer deformed by propagating shear bands (Figure 2(b,c)), while the CG layer deformed uniformly. Strain delocalization in the shear bands/LSZ helped with retaining ductility in NS layer. In contrast, shear bands would have failed homogeneous NS metals quickly [2,3,11,12]. The LSZ regained strain hardening capability after the initial softening (Figure 4(c)), a phenomenon that would rarely occur in homogeneous NS metals [38]. Both forest dislocation hardening and  back-stress hardening occurred to stabilize the shear band [14,33], making it possible for its delocalization. The propagation of shear banding into the sample depth is deterred because the underneath CG layer is stable. This induces strain gradient in the depth direction and back-stress hardening to prevent the LSZ from propagating into the depth [12,31,34].
The strategy of utilizing the shear band delocalization to develop synergistic work hardening for improving the ductility is expected also applicable to other heterostructured metals consisting of NS and CG domains. Another way to delocalize strains caused by shear bands is to develop high-density of them all over the NS domains so that no individual shear band will fail the specimen. Similar synergistic work hardening as discussed above should also work in this situation to improve ductility. In fact, such types of LSZs have been observed in layered heterostructures although their effect on work hardening was not discussed [39].
In conclusion, strain localization by shear bands seems unavoidable in nanostructures. However, in GS materials the detrimental shear bands could be harnessed to benefit ductility. Specifically, in a gradient structured specimen shear bands nucleated early in the NS and propagated along the gage length, instead of across the specimen cross-section as normally observed in homogeneous materials. This delocalized the shear bands to form an LSZ. Strain gradient was produced in the propagating front of the LSZ, which produced back-stress hardening [14,34] to stabilize the propagating shear bands. In addition, strain gradient were also produced near the interfaces between the LSZ and CG central layer, which produces more back-stress hardening. Dislocation hardening capability in the LSZ was recovered after initial strain softening, which, along with back-stress hardening, induces synergistic strain hardening to help with ductility in NS layer.

Impact statement
We propose strategies for synergistic strain hardening in gradient structure by stabilizing shear band and strain gradients, turning harm of shear band into benefit of ductility.

Disclosure statement
No potential conflict of interest was reported by the authors.