Synergetic-Deformation-Induced Strengthening in Gradient Nano-Grained Metals: A 3D Discrete Dislocation Dynamics Study

: Gradient nano-grained (GNG) metals have shown high synergetic strength and good duc-tility due to their unique gradient microstructure. In this study, the mechanical behavior of gradient nano-grained metals was investigated by three-dimensional discrete dislocation dynamics. The simulation results show a trend that the successive yielding and gradual “transmission” of dislocations along the gradient direction result in a gradient distribution of stress and plastic strain. The distribution of geometrically necessary dislocations is more inhomogeneous in the gradient nano-grained (GNG) sample compared with those homogenous counterparts. The non-uniform deformation response of component layers induces the synergetic-deformation-induced (SDI) strengthening in the GNG sample. The back stress originates from geometrically necessary dislocations that pile up near the interface of gradient layers and leads to a signiﬁcant hardening while there is a slight softening in different gradient layers in the GNG sample. This study provides a deeper insight into the SDI strengthening in gradient structure from the submicron scale.


Introduction
Making materials showing both high strength and good ductility has been an enduring endeavor for materials scientists.Grain refinement produced by severe plastic deformation (SPD) techniques has been extensively used to improve the strength of materials in the past several decades.However, the improvement of strength is usually accompanied by the loss of tensile ductility.
In recent years, gradient nano-grained (GNG) structures, which have a macroscopic gradual change in grain size from the surface to the core, have been introduced into metals and produce superior strength and ductility.To date, there has been a lot of research on GNG materials, both from the views of experiments [1][2][3][4][5][6][7][8] and numerical simulations [9][10][11][12][13][14][15][16].It was reported in several previous experimental studies [17][18][19][20][21] that the strength of GNG materials is higher than the value predicted from the rule of the mixtures (ROM), i.e., the volume-weighted sum of the strength of homogenous-grained components in GNG materials, indicating that a synergetic-deformation-induced (SDI) strengthening exists in the materials with GNG structures.It is believed that this unique property in GNG materials originates from multiple strengthening mechanisms, including the macroscopic stress gradient and plastic incompatibility between gradient layers, the occurrence of strain gradient and thus the generation of geometrically necessary dislocations (GNDs) and back stress due to the gradient grain size [17][18][19].However, the detailed analysis and further validation of these strengthening mechanisms in GNG materials are still limited.
Discrete dislocation dynamics (DDD) is an efficient method for investigating the plastic deformation of crystalline materials at the submicron scale.Recently, Lu et al. [22] constructed a GNG polycrystalline model within the DDD framework and then used it to study the mechanical response of GNG materials.However, details about the mechanism of SDI strengthening in GNG materials still need to be elucidated.Hence, in this study, the three-dimensional (3D) DDD was used to further investigate the mechanical properties of GNG materials.The strain hardening rate, the effects of softening and hardening, the "transmission" of dislocations along gradient direction, and the distribution of GNDs, were analyzed.

Simulation Method
In the present simulations, the 3D multiscale discrete dislocation dynamics framework, which was initially developed by Zbib et al. [23] and Zbib and de la Rubia [24], and later improved by Huang et al. [25,26], was extended to incorporate a dislocation-penetrable grain boundary (GB) model.In the 3D DDD simulations, the curved dislocation lines are discretized into straight dislocation segments, and these segments are connected by dislocation nodes.The force (per unit length) acting on each dislocation segment can be expressed as follows: where b and ξ are the Burgers vector and the line direction of the dislocation segment, respectively.σ is the total stress acting on the dislocation segment.F sel f is the self-force of the dislocation.The dislocation segment will move under the force in Equation ( 1) and then generate a plastic strain: where n is the normal vector of the slip plane.A is the area swept by the dislocation segment.
V is the volume of the homogenization region.The detail of the multiscale DDD framework can be found elsewhere [23][24][25][26].The GB model used in the present simulations considers the mechanisms of dislocation absorption at GBs and dislocation emission from GBs based on a "coarse graining" approach, as has been presented in detail in our previous work [22,27].The GNG model constructed within the DDD framework is shown in Figure 1a.The model has four component layers with grain sizes of 1600b, 800b, 400b and 200b.To examine the unique deformation behavior of the GNG sample, four polycrystalline counterparts with homogenous nano-grained (HNG) structures are also simulated for comparison.The dimensions of these samples are shown in Figure 1a-e.All samples contain multiple cubic-shaped grains with randomly distributed grain orientations.Note that Huang et al. [28] have validated that the results obtained by a model with cubic-shaped grains are consistent with those by a more realistic Voronoi model.Representative parameters of isotropic aluminum are chosen, including shear modulus G = 26 GPa, Poisson ratio v = 0.33 and magnitude of Burgers vector b = 0.25 nm.The bottom surface of the samples is constrained, and the top surface is uniaxially compressed in a displacement-controlled manner with a strain rate of 5000 s −1 .Four vertical surfaces are treated as free surfaces, i.e., no external stress or displacement is applied on these surfaces.Frank-Read (F-R) sources are initially distributed in the whole volume in a random way.Similar to Ref. [22], the length of F-R sources varies from d/4 to d/3 (d is the grain size) randomly.The dislocation density is fixed to be 10 µm −2 for GNG and HNG samples.

Results and Discussion
The representative stress-strain curves for the GNG sample and HNG counterparts are presented in Figure 2a.It shows that the flow stress increases with decreasing grain size for four HNG samples, exhibiting a typical grain size effect in polycrystals.To present a more statistically reliable result, the average stress-strain curves for all samples are plotted in Figure 2b.Note that all these curves are obtained by averaging five realizations with randomly distributed FR sources and grain orientations.For the GNG sample, if the component's gradient layers are separate or the interaction between adjacent layers is weak, the flow stress can be predicted by the rule of mixture (ROM), i.e., the volume-weighted sum of the flow stress of all independent layers with homogenous grains in the GNG sample [17]: where  and Vi are the flow stress and the volume of the i-th layer, respectively.n is the number of component layers, and V is the whole volume of the GNG sample.In the present simulations, n equals 4 and the flow stress  is taken as the flow stress of HNG-1, HNG-2, HNG-3 or HNG-4.The flow stress calculated by the ROM method is also plotted in Figure 2b.It shows that the flow stress of the GNG sample is up to 18% higher than the value obtained from the ROM, suggesting that there is a strong interaction between gradient layers during deformation.Figure 2c shows that the strain hardening rate of the GNG case is larger than the ROM one, which is another indicator that the gradient structure has an SDI strengthening due to the strong interaction between layers with gradient grain sizes.Figure 2d presents the stresses at  = 0.48% and  = 0.82% for four HNG samples and the corresponding four layers in the GNG sample.It shows that the stress of Layer-1 is significantly higher than that of HNG-1, indicating that the gradient structure can greatly improve the strength of the "soft" domain (large-grain region); while for Layer-3 and Layer-4, their stresses are slightly lower than their corresponding HNG samples, suggesting that the strength of the "hard" domain (small-grain region) reduces in the gradient structure.This extraordinary hardening and softening in gradient structures is consistent with the experimental result [20].Shimokawa et al. [29] investigated harmonic structure materials by atomic and dislocation simulations and found that the stress in the core region (soft region) of the harmonic structure is larger than that in the homogeneous structure with the same grain size, while the shell region (hard region) has lower stress in comparison to that in corresponding homogeneous structure.This hardening and softening in the harmonic structure are similar to our simulation results.

Results and Discussion
The representative stress-strain curves for the GNG sample and HNG counterparts are presented in Figure 2a.It shows that the flow stress increases with decreasing grain size for four HNG samples, exhibiting a typical grain size effect in polycrystals.To present a more statistically reliable result, the average stress-strain curves for all samples are plotted in Figure 2b.Note that all these curves are obtained by averaging five realizations with randomly distributed FR sources and grain orientations.For the GNG sample, if the component's gradient layers are separate or the interaction between adjacent layers is weak, the flow stress can be predicted by the rule of mixture (ROM), i.e., the volume-weighted sum of the flow stress of all independent layers with homogenous grains in the GNG sample [17]: where σ i and V i are the flow stress and the volume of the i-th layer, respectively.n is the number of component layers, and V is the whole volume of the GNG sample.In the present simulations, n equals 4 and the flow stress σ i is taken as the flow stress of HNG-1, HNG-2, HNG-3 or HNG-4.The flow stress calculated by the ROM method is also plotted in Figure 2b.It shows that the flow stress of the GNG sample is up to 18% higher than the value obtained from the ROM, suggesting that there is a strong interaction between gradient layers during deformation.Figure 2c shows that the strain hardening rate of the GNG case is larger than the ROM one, which is another indicator that the gradient structure has an SDI strengthening due to the strong interaction between layers with gradient grain sizes.Figure 2d presents the stresses at ε = 0.48% and ε = 0.82% for four HNG samples and the corresponding four layers in the GNG sample.It shows that the stress of Layer-1 is significantly higher than that of HNG-1, indicating that the gradient structure can greatly improve the strength of the "soft" domain (large-grain region); while for Layer-3 and Layer-4, their stresses are slightly lower than their corresponding HNG samples, suggesting that the strength of the "hard" domain (small-grain region) reduces in the gradient structure.This extraordinary hardening and softening in gradient structures is consistent with the experimental result [20].Shimokawa et al. [29] investigated harmonic structure materials by atomic and dislocation simulations and found that the stress in the core region (soft region) of the harmonic structure is larger than that in the homogeneous structure with the same grain size, while the shell region (hard region) has lower stress in comparison to that in corresponding homogeneous structure.This hardening and softening in the harmonic structure are similar to our simulation results.Note that the degree of hardening or strengthening is greater than the softening in the GNG sample; this can be verified by the fact that the overall flow stress of the GNG sample is larger than the value predicted by the ROM, as mentioned above.To investigate the response of plastic strain and stress with applied strain at different layers, the average equivalent plastic strains and equivalent stresses of different layers are calculated.Figure 3 gives the results of GNG and HNG-2 samples.The figures on the left side label the positions of the layers.For the GNG sample, as shown in Figure 3a, the strain at which the equivalent plastic strain starts to deviate from zero for four layers satisfies: Layer-1 < Layer-2 < Layer-3 < Layer-4, indicating a trend of successive yielding of layers with gradient grain sizes during deformation, which can also be verified by the corresponding evolution curves of equivalent stress in Figure 3b.From Figure 3a, it can also be seen that the changing rate of equivalent plastic strain increases as the grain size of the component layer increases, implying that the layers with a larger grain size can produce a larger plastic strain and thus accommodate more plastic deformations.Unlike the GNG sample, the evolution curves of equivalent plastic strain and equivalent stress with applied strain at different layers are approximately identical in the HNG-2 sample (see Figure 3c,d).The inserts in the plots in Figure 3a-d present the contours of equivalent plastic strain/stress at the strain of  = 0.87%, which further indicates the gradient distribution of plastic strain/stress in the GNG sample along the gradient direction while it is uniform in the whole volume in the case of HNG-2.To investigate the response of plastic strain and stress with applied strain at different layers, the average equivalent plastic strains and equivalent stresses of different layers are calculated.Figure 3 gives the results of GNG and HNG-2 samples.The figures on the left side label the positions of the layers.For the GNG sample, as shown in Figure 3a, the strain at which the equivalent plastic strain starts to deviate from zero for four layers satisfies: Layer-1 < Layer-2 < Layer-3 < Layer-4, indicating a trend of successive yielding of layers with gradient grain sizes during deformation, which can also be verified by the corresponding evolution curves of equivalent stress in Figure 3b.From Figure 3a, it can also be seen that the changing rate of equivalent plastic strain increases as the grain size of the component layer increases, implying that the layers with a larger grain size can produce a larger plastic strain and thus accommodate more plastic deformations.Unlike the GNG sample, the evolution curves of equivalent plastic strain and equivalent stress with applied strain at different layers are approximately identical in the HNG-2 sample (see Figure 3c,d).The inserts in the plots in Figure 3a-d present the contours of equivalent plastic strain/stress at the strain of ε = 0.87%, which further indicates the gradient distribution of plastic strain/stress in the GNG sample along the gradient direction while it is uniform in the whole volume in the case of HNG-2.The incompatibility in different layers in the GNG sample leads to the gradient d tribution of stress and plastic strain, which is believed to be a crucial contributor to superior strength-ductility synergy for gradient structures [17].To further understa the nature of this gradient distribution of plastic strain and stress, the evolution of dislocation structure with applied strain is investigated.Figure 4a-f presents the rep sentative dislocation structures at varied strains for the GNG sample.It shows that d locations start to activate and multiplicate in larger grains and then gradually propag to smaller grains, resulting in a trend of subsequential yielding of layers with gradi sizes; this is in line with the results presented in Figure 3.With increasing applied stra more and more dislocations are accumulated in front of grain boundaries.Consequen dislocation emissions from grain boundaries occur, as indicated by the red lines in Figu 4d-f.The incompatibility in different layers in the GNG sample leads to the gradient distribution of stress and plastic strain, which is believed to be a crucial contributor to the superior strength-ductility synergy for gradient structures [17].To further understand the nature of this gradient distribution of plastic strain and stress, the evolution of the dislocation structure with applied strain is investigated.Figure 4a-f presents the representative dislocation structures at varied strains for the GNG sample.It shows that dislocations start to activate and multiplicate in larger grains and then gradually propagate to smaller grains, resulting in a trend of subsequential yielding of layers with gradient sizes; this is in line with the results presented in Figure 3.With increasing applied strain, more and more dislocations are accumulated in front of grain boundaries.Consequently, dislocation emissions from grain boundaries occur, as indicated by the red lines in Figure 4d-f Figure 4a-f also shows a trend that dislocations emitted from grain boundaries at th layer interface mainly move into the interior of the adjacent layer that has smaller grain sizes.In other words, dislocations are "transmitted" from the layers with larger grains to the ones with smaller grains.Figure 4g displays the quantitative results of emitted dis locations from different layer interfaces into different layer interiors.The positions o three interfaces and four layers are labeled in the insert.The legend labeled "I→2" in th plot means the dislocations are emitted from Interface-I into the interior of Layer-2, th meanings of other labels are the same.Figure 4g shows that, during the elasto-plasti deformation stage, the length of the emitted dislocations is larger when the grain size o the layer that the dislocations emit into is larger, i.e., "I→2" > "II→3" > "III→4" and "I→1" > "II→2" > "III→3".Note that for the cases of "III→4" and "III→3", the emitted dislocation length is nearly equal to zero; this is because few dislocations are activated in Layer-3 and Layer-4, which is also demonstrated by Figure 3a,b.Comparing "I→2" with "I→1" and "II→3" with "II→2", it can be seen that, for Interface-I and Interface-II, the length of th dislocations that emit to the layers with smaller grain sizes is nearly an order larger than that of the ones with larger grain sizes.This is direct evidence showing a trend that th dislocations are gradually "transmitted" from the larger-grain layer to the smaller-grain layer.
This phenomenon can be interpreted as follows.For the layers with larger grain sizes, the dislocation activation and multiplication are more intense and thus lead to more dislocation pile-ups in front of GBs.As a result, more dislocations are absorbed in the GBs, and thus more dislocations will emit into the adjacent layers with smaller grain sizes.Both the successive activation of dislocation sources and gradual "transmission" o dislocations along the gradient direction contribute to the gradient distribution o equivalent plastic strain and stress for the GNG sample.
Geometrically necessary dislocation (GND), which accommodates the non-uniform deformation of samples, is a key factor contributing to the strength of a material.Th Figure 4a-f also shows a trend that dislocations emitted from grain boundaries at the layer interface mainly move into the interior of the adjacent layer that has smaller grain sizes.In other words, dislocations are "transmitted" from the layers with larger grains to the ones with smaller grains.Figure 4g displays the quantitative results of emitted dislocations from different layer interfaces into different layer interiors.The positions of three interfaces and four layers are labeled in the insert.The legend labeled "I→2" in the plot means the dislocations are emitted from Interface-I into the interior of Layer-2, the meanings of other labels are the same.Figure 4g shows that, during the elasto-plastic deformation stage, the length of the emitted dislocations is larger when the grain size of the layer that the dislocations emit into is larger, i.e., "I→2" > "II→3" > "III→4" and "I→1" > "II→2" > "III→3".Note that for the cases of "III→4" and "III→3", the emitted dislocation length is nearly equal to zero; this is because few dislocations are activated in Layer-3 and Layer-4, which is also demonstrated by Figure 3a,b.Comparing "I→2" with "I→1" and "II→3" with "II→2", it can be seen that, for Interface-I and Interface-II, the length of the dislocations that emit to the layers with smaller grain sizes is nearly an order larger than that of the ones with larger grain sizes.This is direct evidence showing a trend that the dislocations are gradually "transmitted" from the larger-grain layer to the smaller-grain layer.
This phenomenon can be interpreted as follows.For the layers with larger grain sizes, the dislocation activation and multiplication are more intense and thus lead to more dislocation pile-ups in front of GBs.As a result, more dislocations are absorbed in the GBs, and thus more dislocations will emit into the adjacent layers with smaller grain sizes.Both the successive activation of dislocation sources and gradual "transmission" of dislocations along the gradient direction contribute to the gradient distribution of equivalent plastic strain and stress for the GNG sample.
Geometrically necessary dislocation (GND), which accommodates the non-uniform deformation of samples, is a key factor contributing to the strength of a material.The GND density can be extracted from the dislocation network by the following equation [30]: where V is the considered volume and n is the number of dislocation segments within this volume.l i is the line vector of a dislocation segment i located within the volume V, and b i is the Burgers vector of the i-th segment.
To examine the strain gradient along the gradient direction, the sample is first divided into several slices with an equal thickness along the Z-axis (the normal of the slice is parallel to the Z-axis).Then, Equation ( 4) is used to calculate the GND density of each slice.It should be mentioned that the thickness of the slice will affect the value of GND density, as indicated in Figure 5a,b.However, the trends of the GND density distribution along the Z-axis are the same.There is a significant peak value at the position of the layer interface, and the magnitudes of these peak values increase as the surrounding grain size increases.According to Zhu and Wu [31], in heterostructured materials, the piled-up GNDs at the domain interface create back stress and forward stress, which, respectively, make the soft domain appear stronger and the hard domain appear weaker.In the present simulation, the strengthening of Layer-1 at ε = 0.48% (Figure 2d) is due to the high back stress from the significant piled-up GNDs at Interface-I, while the slight softening of Layers-2, -3 and -4 results from the corresponding forward stress.At the larger strain of ε = 0.82%, Layer-2 turns to exhibit a hardening effect (Figure 2d); this is because the GND density at Interface-II increases and thus Layer-2 turns to experience back stress.The back stress and the associated forward stress form a hetero-deformation-induced (HDI) stress, which is the main contributor to the extra strengthening in heterostructured materials [31].
Figure 5c further compares the GND density distribution in the GNG sample with those in the HNG counterparts.Each data point is scaled by the sample's width l z and the maximum GND density ρ GND,max to compare the distribution of GND density for these samples.This figure shows that the HNG-2 and HNG-3 samples also have GND density peaks in the vicinity of the layer interfaces.However, one difference compared with the GNG sample is that the peak values in the HNG-2 and HNG-3 samples are very close.On the other hand, for the GNG sample, the value of ρ GND /ρ GND,max in the region between the adjacent two peaks is lower than those in the HNG-2 and HNG-3 samples, which means that the difference in GND density between the layer interface and layer interior is more significant in the GNG sample.Such a heterogeneous distribution of GNDs is due to the gradient grain size in the GNG sample, both at the interface-interface level and interface-layer level, which will generate higher back stress and hence contribute to a substantial SDI effect.It should be mentioned that the quantitative relationship among GND density, back stress and strengthening in gradient nano-grained materials needs further study, which might be the goal of our future work.
Finally, the GND density along the Y-axis for the GNG sample was also examined; the result is presented in Figure 5d.Comparing Figure 5d with Figure 5b, it can be seen that the distribution of GND along the Y-axis is far more uniform than that of the Z-axis.= 0.82%, Layer-2 turns to exhibit a hardening effect (Figure 2d); this is because the GND density at Interface-II increases and thus Layer-2 turns to experience back stress.The back stress and the associated forward stress form a hetero-deformation-induced (HDI) stress, which is the main contributor to the extra strengthening in heterostructured materials [31].

Conclusions
In summary, a 3D discrete dislocation dynamics framework was employed to further probe the deformation mechanisms of GNG metals.The stress-strain responses indicate an SDI strengthening effect resulting from gradient grain sizes.The relatively more inhomogeneous distribution of GNDs in the GNG sample also contributes to an SDI strengthening effect.In addition, the extraordinary hardening in layers with larger grain sizes and softening in layers with smaller grain sizes were observed in gradient-grained samples; this phenomenon is related to the back stress and corresponding forward stress resulting from the piled-up GNDs at the "interface" of gradient layers.

Figure 2 .
Figure 2. (a) Representative stress-strain curves and (b) average stress-strain curves (the results are obtained by averaging five realizations) of the GNG sample and four HNG samples.(c) The corresponding strain-hardening rate versus strain curves.(d) Stress at the strains of 0.48% and 0.82% for four HNG samples and the corresponding four layers in the GNG sample; the insert shows different layers in the GNG sample.The curves calculated from the rule of mixtures (ROM) are also plotted in (b,c).

Figure 2 .
Figure 2. (a) Representative stress-strain curves and (b) average stress-strain curves (the results are obtained by averaging five realizations) of the GNG sample and four HNG samples.(c) The corresponding strain-hardening rate versus strain curves.(d) Stress at the strains of 0.48% and 0.82% for four HNG samples and the corresponding four layers in the GNG sample; the insert shows different layers in the GNG sample.The curves calculated from the rule of mixtures (ROM) are also plotted in (b,c).

Figure 3 .
Figure 3. Averaged equivalent plastic strain and equivalent stress versus strain at different lay for (a,b) the GNG sample and (c,d) the HNG-2 sample.The positions of different layers are in cated in the left figures.Inserts in the plots are corresponding contours of the equivalent pla strain/stress at the applied strain of  = 0.87%.

Figure 3 .
Figure 3. Averaged equivalent plastic strain and equivalent stress versus strain at different layers for (a,b) the GNG sample and (c,d) the HNG-2 sample.The positions of different layers are indicated in the left figures.Inserts in the plots are corresponding contours of the equivalent plastic strain/stress at the applied strain of ε = 0.87%.

Figure 4 .
Figure 4. (a-f) Dislocation structures at different applied strains for the GNG sample.The disloca tion lines activated from initial dislocation sources are colored blue, while the red lines indicate th ones emitted from grain boundaries.(g) The length of emitted dislocations from different layer in terfaces to different layer interiors; the insert shows the positions of four "layers" (1, 2, 3 and 4) and three "interfaces" (I, II and III).The legend labeled "I→2" in the plot means dislocations emit from Interface-I into Layer-2; the meanings of other labels are the same.

Figure 4 .
Figure 4. (a-f) Dislocation structures at different applied strains for the GNG sample.The dislocation lines activated from initial dislocation sources are colored blue, while the red lines indicate the ones emitted from grain boundaries.(g) The length of emitted dislocations from different layer interfaces to different layer interiors; the insert shows the positions of four "layers" (1, 2, 3 and 4) and three "interfaces" (I, II and III).The legend labeled "I→2" in the plot means dislocations emit from Interface-I into Layer-2; the meanings of other labels are the same.

Figure 5 .Figure 5 .
Figure 5. GND density along the Z-axis at four different applied strains for the GNG sample when the thicknesses of the divided slice are (a) 50b and (b) 100b.(c) Normalized GND density  GND  GND,max ⁄ at  = 0.87% versus normalized position Z/lz for GNG, HNG-2 and HNG-3 samples. GND,max is the maximum value of the GND density of all slices in each sample, lz is the width of the sample in the Z-direction.The large symbols at the peaks indicate the values of  GND  GND,max ⁄ at the interfaces.The thickness of the divided slice in (c) is 50b.(d) GND density