A New Model That Considers the Gradual Change in Local Crystalline Geometry to Account for Grain Boundary Hardening Open Access

Ever since the dependency of the yield strength on grain size was recognized, a significant amount of research has been devoted to understand and model grain boundary strengthening. Accordingly, a considerable number of equations, models, and theories were proposed among which the Hall–Petch equation is the most widely known and accepted one. Yet, it seems grain boundary hardening is not understood to a full extent. Consequently, grain boundary hardening is still widely researched in the field of materials science.

Nonetheless, polycrystalline strengthening mainly arises from two main reasons: the first one is due to the necessity of multiple slips that take place during plastic deformation, and the second arises from the presence of grain boundaries within polycrystalline materials [1].

Therefore, the deformation and yielding of polycrystalline materials are quite different than that of single crystals. The only constraint on the deformation of a single crystal comes from the grips of the testing machine, and accordingly, single crystals can deform freely by single slip depending on the orientation of the crystal with respect to the loading axis, whereas the deformation of each grain within a polycrystal should be compatible with the deformation of neighboring grains. Strain compatibility between adjacent grains requires more than one slip system to operate within each polycrystal grain during deformation. Consequently, compared to single crystals, the onset of yielding takes place at higher stresses. According to Taylor, to maintain strain continuity across the grain boundaries, at least five independent slip systems should be active and for FCC materials an average orientation factor of 3.06 is computed when all possible orientations are considered [2,3]. For the most favorably oriented single crystals, the orientation factor takes the lowest value of 2, whereas when all possible orientations are considered, an average orientation factor of 2.238 is computed for a single slip of FCC single crystals [2,3]. The higher orientation factor for polycrystalline materials on average results in higher flow stress when compared to that of single crystals that deform by a single slip. Associated with it, the flow curves of the single crystals that are oriented for multiple slips resemble more than that of polycrystals [3,4]. Nevertheless, this type of polycrystalline strengthening is known to be independent of grain size and arises solely from the fact that the deformation within the grains of polycrystals occurs on more than one slip system and, according to Taylor, at least five slip systems are necessary.

Here, σ is the yield strength and σ_o and k are material constants. The former is apparently the yield strength of a theoretical polycrystalline material that has a grain size that goes to infinity, which represents the yield strength for a polycrystalline material for which the effect of the grain boundaries is negligible. According to the theory that is used to explain the Hall–Petch effect, the latter is related to the stress required to cause slip in the vicinity of the grain boundaries. In the current discussions, the abbreviated terms, HP stress and HP slope, are used interchangeably with the two Hall–Petch constants, σ_o and k, respectively.

Here, τ_o is the lattice friction stress which also corresponds to the critical resolved shear stress for a single crystalline material and m is the average orientation factor of the active slip systems within the grains of a polycrystal. Seemingly, the multiplication of τ_o with m gives the term σ_o of the Hall–Petch relationship. This term takes into account the effect of multiple slips on the yield strength and reflects the mentioned polycrystalline hardening behavior which is independent of grain size, whereas the term k_s is related to the shear stress that is required to activate dislocation sources that are in close proximity to the grain boundaries, which address the hardening due to the presence of grain boundaries. Apparently, the generalized Hall–Petch equation allows one to predict the yield strength of a polycrystalline material from the lattice friction stress when the Hall–Petch constant, k_s, and grain size of the material are known.

From now on, to avoid confusion between the two Taylor models: the one that requires activation of five slip systems to maintain strain compatibility for the deformation of polycrystalline materials is called the Taylor polycrystalline deformation model and the other that correlates strength to the dislocation density is called the Taylor hardening model.

Here, ρ is the dislocation density, b is the burgers vector, and x is the average distance traveled by the dislocations. This equation, which is known as the Orowan equation, can be used to estimate the necessary dislocation density for a given plastic strain amount.

Nevertheless, to the best of the author's knowledge, so far, in all studies where grain boundary hardening was investigated, an average grain diameter was taken into account to predict dislocation density or to predict dislocation pile-ups at the grain boundaries. Thus, in studies where grain boundary hardening was studied, the average grain size was considered the characteristic microstructural length at which dislocations glide and are stored. However, within polycrystalline grains, the local crystalline size and the corresponding local crystalline volume that are constrained by the grain boundaries, where the dislocations glide and are stored, vary gradually. In this context, a novel approach has been devised in which the gradual variation both in the local length, i.e., the local size of the crystal, and in the corresponding local crystalline volume within polycrystalline grains, is taken into account. Computations were made for a representative spherical grain for given average grain sizes. The representative grain is divided into segments in the form of crystalline strips. The local density of dislocations, accordingly the local strength developed within each crystalline segment during deformation, is computed for a given grain size. Thus, rather than considering a polycrystalline grain as a whole, the local dislocation densities and the corresponding local shear strengths developed within different parts of a polycrystalline grain are computed. The local crystalline size and volume, i.e., local geometry depends on the average grain size, the computed local dislocation densities, and corresponding local shear strengths for the crystalline strips, show dependency on the grain size, as well.

In addition, it was assumed that, for a given plastic strain, rather than with strain, the dislocation density that is stored in each crystalline segment within the grains is correlated with stress [2,9–11]. During plastic deformation, this forces different parts within a grain to have about the same dislocation density and correspondingly to have about the same strength for a given macro-strain. This also assures less strain within the crystalline segments with smaller local crystalline sizes and volumes that lie in close proximity to the grain boundaries for a given macro-strain, whereas a larger strain is developed at the strips with larger sizes and larger volumes which lie relatively at the center of the grain. The strain experienced by each crystalline strip is governed by the state of stress equilibrium.

Based on this assumption and the new approach that considers the local crystalline geometry, a new model was developed to predict the 0.2% yield strength for a material in terms of reciprocal of the average grain size. The Orowan equation was implemented in conjunction with Taylor polycrystalline deformation and Taylor hardening models to estimate the necessary dislocation density for 0.2% plastic strain and the corresponding yield strength for a material. The Orowan equation allows one to estimate the necessary dislocation density for a given shear strain, and considering the local geometry of the crystal allows, the utilization of the Orowan equation to estimate the necessary local dislocation density, and accordingly, the local strength developed within the crystal for a given average grain size.

The model is verified by comparing the predicted yield strength values with the experimental ones for several pure FCC materials, namely, pure copper, nickel, and aluminum. A good agreement is achieved for pure copper and nickel, whereas for pure aluminum, a good agreement is achieved to a limited degree.

The model developed requires excessive computations. To overcome this, based on the knowledge provided by the model, an alternative equation was devised to compute the yield strength in terms of the reciprocal of grain size. The equation derived turned out to be almost the same as an equation that was presented and used in some previous studies [12–14]. Just, there exists an uncertainty in terms of the value of a parameter, the parameter C which appears in the equation that was used previously. In this study, the equation, and so the parameter C, is derived as a natural outcome of the current model. Hence, according to the currently developed model, the parameter C is definite and a value of 49.6 is computed for its value for FCC metals. The predicted results by the equation and the model are consistent. Just, there exists a slight gradual deviation between the predicted results by the equation and by the model as the grain size gets finer. This occurs due to the imprecision in the computed local crystalline geometry, which is related to the selected location of the basal plane of the crystalline strip where computations were made. Thus, a second equation was also derived which gives improved results in terms of consistency with the ones predicted by the model. It was also shown that the exact value of parameter C exhibits a weak dependency on the grain size. Hence, by adding an appropriate exponential term to the equation that is derived initially, which considers this weak dependency, a third alternative equation was also derived which gives the best results in terms of consistency with the predicted ones by the model. Of the three equations in terms of simplicity and accuracy, the third one seems to be the optimum option. Nevertheless, by using the equations derived in this study, the yield strength for a given pure FCC material can be predicted with ease.

The model and equations developed here are limited to the computation of yield strength values corresponding to very small plastic strains, such as 0.2% proof stress. The increase in the plastic strain results in an increase in the dislocation density. Consequently, the flow stress values for larger plastic strains will be overpredicted by the model. Thus, it is expected that the prediction capability of the model and the equations will decrease significantly for larger strain values, i.e., when dislocation density increases significantly. Further elaboration is needed to extend the utilization of the model and equations to increase the prediction capability further into the flow stress region. In addition, as mentioned, the model and so the equations are verified just for several pure FCC metals. Further elaboration is most probably necessary to predict yield strength for alloys other than pure FCC metals, as well.

Nevertheless, the proposed model provides new insights into understanding the dependency of the yield strength of a polycrystalline material on the grain size. In the model, the gradual change in the local length of the crystal and the gradual change in the corresponding local crystalline volume which are constrained by the grain boundaries in which dislocations glide and stored are taken into account. This allows utilization of the fundamental equations and models proposed earlier by Orowan and Taylor to predict local necessary dislocation densities and corresponding local strengths, and even local plastic strains, in terms of grain size and plastic strain for different parts within the polycrystalline grains. To mention, the model does not take into account how slip and dislocation structure evolve and how much dynamic recovery takes place in the course of the plastic deformation. It just estimates the necessary dislocations for a given small plastic strain amount under certain assumptions.

Nevertheless, the current study shows that when utilized suitably in conjunction with Taylor polycrystalline deformation and Taylor hardening models, and when the local crystalline geometry is considered, the yield strength of polycrystalline materials can be estimated by the Orowan equation. In essence, the current study emphasizes the importance of the consideration of the local crystalline geometry of the polycrystalline grains to account for grain boundary hardening.

In the proposed model, a spherical representative grain is used to model the deformation of a polycrystalline material which represents the overall material behavior. Owing to the geometrical symmetry of the sphere, the theoretical calculations were made for the quarter spherical volume.

First, the representative grain volume is divided into strips that are separated by planes that are assumed to lie parallel to a hypothetical active slip plane, which has an orientation factor equal to the average orientation factor of the active slip systems (Fig. 2(a)). The basal diameter for each strip varies, and for the largest strip at the center of grain, the basal diameter is equal to the diameter of the representative grain which corresponds to the average grain size. When going away from the grain center, in the slip plane’s normal direction, the diameter for each crystalline strip decreases gradually. Accordingly, for the crystalline strips that lie in close proximity to grain boundaries, the diameters and volumes are extremely small when compared to those for strips that lie near the center of the grain.

Nonetheless, next, the deformation of a grain within a polycrystalline volume is considered. The mechanical behavior of each polycrystalline grain is anisotropic by nature and orientation variations are present among the neighboring grains. Furthermore, the deformation of each grain within the polycrystalline material is constrained by its neighbors which leads to a delayed yielding that occurs at higher stress levels when compared to the yielding of single crystals that can deform by a single slip, in particular. Yielding will take place when the compatibility of deformation among the neighboring grains is satisfied. This requirement urges grains within a given cross section to deform almost simultaneously in the same manner. The deformation experienced by each grain should resemble the macro-deformation. According to the Taylor model, it was not until five slip systems are activated on average for grains in a given cross section of a polycrystalline material that this requirement is fulfilled.

Thus, according to the Taylor polycrystalline deformation model, five active slip systems should operate simultaneously. Hence, in the foregoing computations, an average orientation factor for the five active slip systems is utilized which is 3.06 for FCC materials [2,3].

In the model, first, an initial edge dislocation density of 10⁶/mm² is assigned. As mentioned, according to the Taylor hardening model, the strength of a material is related to its dislocation density. In his model, Taylor considered only the edge dislocations. Thus, the initial shear strength of the material at the annealed condition is computed by implementing the assigned value of edge dislocation density in Eq. (4). Nevertheless, for polycrystalline materials, grain boundaries also contribute to the strength. Thus, the computed shear strength value only corresponds to the lattice friction stress, τ_o, which appears in the generalized Hall–Petch equation (Eq. (2)).

Next, dislocation loops are introduced into each crystalline section, i.e., into each crystalline strip within the representative grain volume, so that the density of edge dislocations within the grain volume prior to deformation would be equal to 10⁶/mm² in total. In Fig. 2(a), dislocations are presented for the quarter of the representative grain where the computations were made. Circular dislocation loops are assumed, and for the sake of simplicity, a stress-based strategy is adopted to introduce dislocation loops within the representative grain volume. Dislocation loops are introduced in each stripe, so that the stress that arises from the presence of edge dislocations within each stripe would be the same. Thus, the total length of dislocations initially present within each crystalline segment was governed by the stress equilibrium among the strips. In fact, in real crystalline materials, during annealing and afterwards, dislocations might assume a configuration that satisfies stress equilibrium within the polycrystalline grains through thermally activated phenomena, as well.

So, according to the model, for a given material, t is proportional to the burgers vector and inversely proportional to the plastic strain and to the average orientation factor of the active slip systems. In fact, Eq. (10) gives a seeming minimum thickness for the crystalline strips, so that any given small plastic strain could be produced by the movement of a dislocation or a number of dislocations which produce a slip step at the grain boundary region which is equal to burgers vector. A larger thickness value can also be assigned. A larger value of t will require a larger total length of dislocation loops for each stripe to produce a given strain. When compared to grain size if strip thickness is sufficiently fine, a similar but just a little higher yield stress value will be predicted.

Nevertheless, according to the model, as mentioned, the spacing between slip steps, i.e., the thickness of the strips, is an inverse function of the shear strain and it is independent of the grain size. Next, the total length of dislocation loops present in each strip could be calculated. To accomplish this, equations that give the diameter and the volume for each crystalline segment and an equation that gives the total dislocation length within each segment are needed.

Here, D_i is the diameter of the ith stripe and i takes the value from 1 to D/2t. The other variables are mentioned in Eq. (11).

Here, ρ_i and V_i are the dislocation density and the volume for the ith strip, respectively. As mentioned, i takes values from 1 to D/2t. λ_i is the length of dislocation loops present in the ith strip in total, and n_i is a number that is required to calculate the total length of dislocation loops or dislocation loops within the ith strip. In fact, when n_oi which is used to compute to total initial dislocation length is subtracted from n_i, the remainder will correspond to $nx¯$ that appears in Eq. (5), which is the number of dislocations multiplied by a fraction of the average distance traveled by dislocations. In addition, here, it should be noted that the dislocation density, ρ_i, and the total length of dislocation loops, λ_i, in Eq. (13) refer to the density and length of circular dislocation loops, which are larger when compared to that of edge dislocations, i.e., edge dislocation component of the circular loops.

As mentioned, dislocations initially present within the grains are assumed to satisfy stress equilibrium among the different strips which enforces the same dislocation density for all crystalline strips. n_i is a variable that is required to calculate the total length of dislocation loops that are present within the ith strip. The initially assigned edge dislocation density was 10⁶/mm²$(ρ⊥,o=106/mm2)$⁠; thus, an initial total dislocation density of ½π × 10⁶/mm² is implemented in Eq. (15). Therefore, the initial dislocation density is approximately estimated by multiplying the initial edge dislocation density by $π/2(ρo=1/2πρo,⊥)$⁠. By implementing ρ_i (or for initial state ρ_o) in Eq. (15), the n_i (or n_oi) for each crystalline segment can be calculated. The calculated values for the initial state could be labeled as n_oi, since they are for the initial dislocation loops that are already present within each crystalline segment before deformation.

Here, λ_i is the total length of dislocations within the ith strip and i takes values from 1 to D/2t. D_i is the diameter of the ith crystalline strip and, as mentioned, n_i is a variable that is required to calculate the total length of dislocations within the ith strip. It is also indicative of the possible locations for the dislocation loops as a fraction of the diameter for the ith crystalline strip. However, as for the computations, rather than the locations of dislocations, the total length of dislocations is of importance. The strip thickness is assigned so that any given plastic strain can be produced by a movement of a single dislocation or by movements of many dislocations, provided that the total distance traveled by the dislocation or dislocations is equal to D_i/2. D_i/2 corresponds to slip length L in Eq. (6). Thus, if n_oi is equal to 0.3, for any given strain, it would increase to 1.3, or if n_oi is equal to 1.4, to produce any given strain, it would increase to 2.4 by definition. To mention, any motion of dislocation loops in this manner is accompanied by an increase in the total dislocation length, i.e., an increase in the dislocation density.

Nonetheless, as mentioned, according to the model, the distance between slip steps, i.e., the thickness of strips, is an inverse function of plastic strain and its value ensures that for a given plastic strain, the value of n_i would be equal to n_oi + 1. As explained in Eq. (5), regardless of the number of dislocations present or generated within the crystalline strip, the total length traveled by the loops should be equal to the radius of the strip. Thus, independent from the number of dislocations loops present within a crystalline strip, the total distance traveled by the dislocations is invariable and is equal to the radius of the crystalline strip which is governed by Eq. (9). Thus, according to the model, any given small strain amount is produced when n_oi increases by 1 for each crystalline strip. Since n_ois are computed for all strips, the yield strength of a polycrystalline material for a given small plastic strain could now be computed.

Here, ε_p,i is the plastic strain that is produced by the movement of dislocation loops for the ith strip. Thus, if computations were made for 0.2% proof strength, i.e., if n_i is calculated for 0.2% yield strength, then the computed value of ε_p,i for each polycrystalline strip should be 0.002, as well. In Eq. (21), the initial n_oi is subtracted from the final n_i, since n_oi represents the initial state of dislocation loops within a crystalline strip, i.e., since n_oi represents the state of dislocation loops for zero plastic strain.

Here, it should be noted that when the model is utilized in the prescribed manner, the computed dislocation densities and, accordingly, the calculated strengths for each crystalline strip will increase gradually, when going away from the grain center, in the normal direction of the active slip plane. For the strips that lie in close proximity to the grain boundaries, significantly higher strength values would be computed when compared to the strengths of the ones that lie at about the center of the grains, whereas the strips that are away from the grain boundaries would have relatively similar and lower strengths.

Alternatively, it is possible that, rather than strain, the plastic deformation which takes place within polycrystalline grains is controlled by the stress [2,9–11]. It is a well-known fact that, regardless of the uniformity of macroscopic deformation, materials’ mechanical response at the microscopic level is inhomogeneous due to the microstructural inhomogeneities. As a result, during deformation, the strain amount experienced by different portions of a strain-hardening material at an infinitesimal time or strain interval is governed by the local strength of the material, which is also a function of local strain and local strain rate. Thus, when a segment of a material strain hardens more and becomes harder than other regions, deformation at that segment is halted, and deformation will continue at the softer regions. With ongoing deformation, strain hardening takes place in other regions, as well, and the difference between the strengths among all segments diminishes, and, eventually deformation at segments, which previously come to a stop, will continue again [11]. The progress of deformation in this manner most probably occurs rather quickly, at infinitesimal time or strain intervals. Within this context, it is assumed that local deformations within the polycrystalline grains are controlled by stress, as well. Hence, it is possible that, at a micro-scale, a similar situation takes place almost simultaneously between adjacent crystalline strips of polycrystalline grains. Such that, during plastic deformation when a larger amount of dislocation density is stored within a segment of crystal, i.e., when a crystalline strip strain hardens more than others, the deformation at that segment is halted. With ongoing deformation, the number of segment where deformation comes to a stop increases, whereas the number of segments where deformation is carried decreases. As the difference in the stored dislocation density, i.e., the difference in instantaneous shear strength among the strips, diminishes, deformation will continue at the segments where it has come to an end previously.

Nevertheless, if the deformation within the grains is controlled by stress as assumed, the plastic strain experienced by the strips with smaller diameters that are in close proximity to grain boundaries will be much less than the ones experienced by the strips with larger diameters that lie closer to the center of the grain. This is because, for crystalline strips with smaller diameters and volumes, a much more rapid increase in the dislocation density occurs when compared to ones with larger diameters and volumes during straining. Thus, according to the model, for a given plastic strain, the strain experienced by each crystalline strip within the grains shows a variation to satisfy stress equilibrium among the crystalline strips while the average strain value for all strips should still be equal to the given macro-strain. Accordingly, in the model, next, an alternative approach is adopted to calculate the dislocation density and yield strength.

Here, $ρεp$ is the dislocation density which is valid for all crystalline strips within the representative grain for a given plastic strain. ε_ph represents a hypothetical plastic strain value, which is larger than the macro-plastic strain value, ε_p, at which computations are made. The value of ε_ph is larger than ε_p since its value reflects the pseudo strain effect that arises from initial dislocations present in the crystal before deformation in terms of computations. Except for β, other variables in Eq. (22) were stated before. The value of n is equal to D/2t (or Dmε_p/2b). β is a coefficient that has a value very close to unity.

The variables in Eq. (23) were already mentioned.

If the terms, ε_ph and ε_p, are omitted in Eq. (23), a higher value β will be employed. This time, the pseudoplastic strain effect of initial dislocation loops present within the crystalline strips would be addressed with a larger value of β. In either way, the exact value of β can be computed by trial and error for each grain size.

Since the density of only edge dislocations is considered in the Taylor hardening model, the edge dislocation density for each crystalline strip can be calculated by using Eq. (17). For this purpose, first, the n_i value for each crystalline strip is computed. To achieve this, the newly computed dislocation density by Eq. (22) is implemented in Eq. (15). Then, these calculated n_i values are implemented in Eq. (17).

Here, $σεp$ is the yield strength of polycrystalline material which is computed for any given small plastic strain amount, such as 0.2% proof strength.

Here, it should be noted that the computations were made for a single slip system which has an orientation factor that is equal to the average orientation factor of the five active slip systems. The increase in the edge dislocation density which is calculated for this hypothetical active single active system is multiplied by five to consider the increase in the edge dislocation density for all active systems. To note that, in these computations, the spatial orientation differences among the five active slip systems are not considered for the sake of simplicity.

The model developed in this study is verified by comparing the predicted results with the experimental ones for three different pure FCC metals, copper, nickel, and aluminum, namely. The necessary material properties of these materials for computations are given in Table 1.

Verification is first achieved for copper which is one of the most studied materials when grain size–strength dependency is the consideration [12,13,15]. In addition, when it comes to strength–dislocation density dependency, copper by far is the most studied material [16]. Thus, validation of the model is first made for pure copper.

As mentioned first, an initial edge dislocation density of 10⁶/mm² was assigned. By using the Taylor hardening model, i.e., Eq. (4), first the lattice friction stress was computed for copper and a friction shear stress of 2.35 MPa was predicted. The reported values for the lattice frictional stress for copper vary in the range of 2.5–9.2 MPa [17,18]. Thus, the prediction seems reasonable. Then, by using Eq. (10), the thickness of strips is computed for 0.2% plastic strain. By doing the other required computations for the model, the 0.2% yield strength values for pure copper were computed for various given grain sizes. The predicted results were then compared to the experimental ones found in the literature. In a study, 0.2% yield strength values and corresponding grain sizes were reported for pure copper [13]. The predicted results and experimental results are compared in Fig. 4(a). A good agreement is found. The experimental yield strength values were predicted within a 7.8% error margin. The difference between the predicted and the experimental results is 0.77%, 6.25%, and 7.8% for the 3.4-, 15-, and 150-µm grain sizes, respectively. The slopes of the Hall–Petch curves fitted in the predicted and experimental results are also in good agreement. Just, a lower Hall–Petch stress, σ_o is predicted when compared to the experimentally determined one. The predicted and experimentally determined values of σ_o are 1.82 and 3.75 MPa, respectively. The difference between the two is 1.92 MPa.

In addition, a figure present in another study shows the yield strength values with respect to the inverse root of grain size for five-grain sizes, including the three that are used in the previous comparison [19]. The reported grain sizes and corresponding yield strength values for the two additional grain sizes, i.e., for 6.9- and 40-µm grain sizes, were read from the figure provided in that study. Just for the 6.9-µm grain size, according to the data points given in the figure, the grain size seemed to have a value of 6.2 µm. Thus, a grain size of 6.2 µm is used in predictions. The predicted and experimental results are compared in Fig. 4(b). Again, a good agreement is found. The experimental yield strength values are predicted again within a 7.8% error margin. The slope of the predicted and experimentally obtained Hall–Petch curves are in good agreement, as well. The predicted and experimentally obtained σ_o values are 1.64 and 2.7 MPa, respectively. Again, a lower Hall–Petch stress, σ_o, is predicted. Just the difference between the predicted and experimentally determined HP-stress values is reduced to 1.17 MPa. Therefore, a better prediction in terms of Hall–Petch stress, σ_o, is accomplished when five experimental data points are taken into consideration. As can be seen, both predicted and experimentally obtained HP constants show dependency on the selected data points, i.e., grain sizes. Thus, to compare the predicted and experimental results, not only for yield strength but also in terms of k and σ_o, the prediction was made for almost the same grain sizes. Furthermore, the predicted and experimental results are also compared in Fig. 4(c). For this comparison, the experimental data for 6.2-µm grain size are excluded. When this data point is excluded, the prediction accuracy in terms of yield strength values is still in the range of less than 7.8% error and again a similar HP slope is obtained. The predicted and experimentally obtained values for σ_o become 1.63 and 2.81 MPa, respectively.

In addition, a higher initial edge dislocation density of 1.63 × 10⁶/mm² was assigned which gives a lattice friction stress of 3 MPa. The predicted results were compared to experimental results reported by Thompson et al. [13] and an improved agreement was obtained (Fig. 5(a)). The difference between the predicted and the experimental results falls in the range of less than 5.14%. The differences are reduced to 0.5%, 5.14%, and almost zero for the 3.4-, 15-, and 150-µm grain sizes, respectively. When compared to the experimental one, about the same HP slope is predicted, and the difference between the predicted and experimental σ_o reduced to 0.75 MPa. When the predicted and experimental ones were compared for the five data points that were presented in the other study, a significant agreement was found in Fig. 5(b). The predicted and experimental σ_o values are almost the same, and the difference between the predicted and experimental HP slopes is reduced to 0.13 MPa, whereas the prediction accuracy in terms of yield strength values is reduced to 6.2%. An alternative comparison was also made for which the experimental data for which 6.2-µm grain size is excluded (Fig. 5(c)). For this case, a good match between the predicted and experimental values for the HP slope and for the σ_o is achieved, and the maximum difference between the predicted and experimental yield strength values is reduced to 5.1%.

Next, the model is verified for pure nickel, as well. Initially, an edge dislocation density of 10⁶/mm² is assigned for the annealed state which will result in lattice friction shear stress of 3.8 MPa. The reported values for the lattice frictional stress for nickel vary in the range of 5.5–19.6 MPa [20]. And 0.2% yield strength values for various grain sizes were predicted by using the currently developed model. Next, the predicted results were compared with the experimental results provided in the literature. In a study, 0.2% yield stress for pure nickel for four different grain sizes was presented with respect to the inverse root of grain size [21]. In the figure at which experimental results are taken, experimental results from four other studies for pure nickel were also shown. However, considerable differences are present among the experimental results obtained by different studies. Especially, one of the experimental results significantly deviates from the other four. Thus, predicted results were compared with experimental results by Thompson [21]. The 0.2% yield strengths and corresponding inverse root of grain sizes were read from the figure provided by Thomson. As mentioned first experimental results are compared to predictions that were made for 10⁶/mm² initial edge dislocation density (Fig. 6(a)). Although there is a good agreement between the predicted and experimental results in terms of HP slope, there exists an appreciable discrepancy between the predicted and experimentally determined σ_o values, which results in a considerable difference in the predicted and experimental yield strength values. The maximum difference between the predicted and experimental results in terms of yield strength is 21%.

For a better prediction, an initial edge dislocation density of 3.4 × 10⁶/mm² is also assigned which gives a lattice friction shear stress of 7 MPa according to the Taylor hardening model. In this case, the computed lattice friction stress falls in a range of the ones reported for pure nickel [20]. Predicted and experimental results were compared in Fig. 6(b). Except for one data point, a good agreement between the predicted and experimental yield strength values is obtained, as well. The 0.2% yield strength values for 10-, 87-, and 130.7-µm grain sizes are predicted by 2%, 0.9%, and 0.1% differences, whereas for 23.6-µm grain size, there is about a 15% difference between the predicted and experimental yield strength values. A reasonably good match between the predicted and experimentally obtained k and σ_o values is achieved. The difference between the predicted and experimentally determined k and σ_o values is about 5.5% and 5.7%, respectively.

As can be seen from Figs. 6(a) and 6(b), one experimental data point does not follow the Hall–Petch relationship, i.e., the linear relationship between the yield strength and the inverse root of grain size, and this is the reason why this experimental data point is predicted poorly by the model. The deviation and scatter of experimental data are generally attributed to orientation differences among the materials being tested [14]. Nevertheless, since there are only four data points present, this data point seems to have an appreciable impact on the slope of the experimentally determined HP curve, which in turn affects the experimentally obtained σ_o value, as well. Thus, when this experimental data point is excluded, a remarkably good match between the predicted and experimentally determined k and σ_o values is achieved (Fig. 6(c)). The difference between the predicted and experimentally determined k and σ_o values is reduced to 2.5% and 2.8%, respectively.

Finally, the model is verified for pure aluminum, as well. As it is the case for copper and nickel, first an initial edge dislocation density of 10⁶/mm² is assigned which will give a lattice friction shear stress of 1.5 MPa. Next, the 0.2% yield strength values for various grain sizes were computed. These results were then compared with some experimental results found in the literature. However, for pure aluminum, there exists a significant variation in the reported results in terms of experimentally determined 0.2% yield strengths, HP-slope, and HP-stress values [14,22–24]. The predicted and experimental results are shown in Fig. 7. The experimental results by Fujita and Tabata and by Al-Haidary and Petch seem to mark an upper and lower bound for pure aluminum, respectively. The predicted HP curve by the model lies between these two extremes and seems to match more with the ones reported by Hansen [14] and by Wyrzykowski and Grabski [22]. Thus, predicted results were compared to the ones provided by these studies.

When the experimental results from these two studies are considered, they seem to be in agreement with each other and seem to follow a similar trend to a certain degree. Thus, to a limited extent, a single HP curve can be used to represent the experimental results for these two studies. Just, when compared to the trend for the other data points, the yield strength value reported for the finest, 2 µm, grain size by Wyrzykowski and Grabski seems to be a little off. Thus, this data point is excluded and the predicted results were compared with the experimental results for grain sizes that range from about 1250 to 4 µm (Fig. 8(a)). For pure aluminum, experimental results for significantly coarse grain sizes are presented, as well. As it can be seen from Fig. 8(a), for these considerably coarse grain sizes, i.e., for grains with an average grain size larger than about 600 µm, the predicted yield strength values begin to level off. Interestingly, a similar trend is observable in the experimental results, as well. Most of the experimental data for these coarse grain sizes lie above the HP curve fitted into experimental results. Nevertheless, there is a significant scatter in the experimental results. Thus, rather than comparing predicted yield strength values directly with the experimental ones, the yield strength values computed with the predicted and experimental HP constants are compared. At the left end of predicted and experimental HP curves, a maximum difference of 28% is present in terms of yield strength values that are computed by using HP constants which were determined with respect to predicted and experimental results, on the right-hand side, a maximum difference of 9.3% is present and at about 32-mm grain size, the difference between the two curves reduced to zero. In addition, a fair agreement between the predicted and experimental k and σ_o values is present. A higher k and a lower σ_o values were predicted. The difference between the predicted and experimental σ_o values is 2 MPa.

In Fig. 8(b), the predicted and experimental results for grain sizes finer than about 155 µm were compared. At this grain size range, maximum differences are about 3.2% and 7.3% at the left and right ends of the HP curves, whereas at about 75-µm grain size, the difference between the two curves reduces to zero. Much better agreement is present between the k values whereas the difference between the σ_o values is reduced to 0.89 MPa. To mention again, the predicted and experimentally determined k and σ_o values show variation with respect to the grain size range of data points. Since experimental data points for pure aluminum span from 1250-µm to 4-µm grain size, and the yield strength values begin to level off significantly for the grain size is larger than about 600 µm, and σ_o values that are determined for 1250- to 4-µm grain sizes and that are determined for 155- to 4-µm grain sizes are considerably different. In fact, as will be discussed later, for the former case, a fourth-order polynomial would be a better fit for the predicted results. This may be true for experimental results, as well.

In addition, an initial edge dislocation density of 1.96 × 10⁶/mm² was assigned. A lattice friction shear stress of 2.1 MPa is computed. The predicted and experimental results and HP curves were compared (Fig. 9(a)). For this case, the difference between the HP curves that were fitted into predicted and experimental results at the left end and the right end of the curves becomes 4.8% and 9.1%. In addition, a much better agreement in terms of k and σ_o values is achieved. The difference between the σ_o values is reduced to 0.52 MPa.

In Fig. 9(b), the predicted and experimental results and HP curves are compared for grain sizes finer than about 155-µm grain size. A maximum of 5% and 7.6% differences is achieved at the left and right ends of the predicted and experimental HP curves. A similar k value is predicted and the difference between σ_o values is reduced to 0.29 MPa. For relatively coarse grain sizes, the predictions by the model match well with the experimental ones to a certain degree. However, for relatively finer grain sizes, higher yield strength values are predicted by the model. Overall, it can be said that a good agreement is achieved for pure aluminum, but to a limited degree.

Nevertheless, the newly developed model is verified for three different pure FCC metals. A good agreement is achieved for pure copper, pure nickel, and pure aluminum; this is achieved to a certain degree.

The predicted results for pure nickel, copper, and aluminum for a wide range of grain sizes that were computed for initial edge dislocation densities of 3.4 × 10⁶, 1.63 × 10⁶, and 1.96 × 10⁶/mm², respectively, are shown in Fig. 10(a). Hall–Petch curves are fitted into the predicted yield strength values for each material. As can be seen, the data points approach to vertical axis with a shallower slope; thus, according to predicted results, a fourth-order polynomial curve might be a better fit when a wide range of grain sizes including significantly coarse grain sizes are considered (Fig. 10(b)).

The last term in the fourth-order polynomial equations given in the figure corresponds to the Hall–Petch constant, σ_o. Thus, according to the model, when a fourth-order polynomial curve is fitted into experimental results for a wide grain size range that includes significantly coarse grain sizes, the last term in the polynomial equation will give the σ_o value, which will approach mτ_o. mτ_o is the first term in the generalized Hall–Petch equation, i.e., in Eq. (2). To obtain a σ_o value that is even closer to mτ_o, a higher-order polynomial curve can also be utilized. Furthermore, the predicted results for grain sizes finer than 150 µm are also given (Fig. 10(c)). For this relatively narrow grain size range, a linear HP curve fits well into the predicted results for all materials presented. As can be seen, the predicted σ_o values are considerably lower than the ones predicted by the fourth-order polynomial curves. In fact, as mentioned, predicted σ_o and k values show dependency on the range of the grain sizes considered. Thus, there is an appreciable difference between the σ_o and k values presented in Fig. 10(c) and the previously presented in Figs. 5(a)–5(c), 6(b), 6(c), 9(a), 9(b), and 10(a). Thus, within this context, it seems that the experimentally determined σ_o and k values show dependency on the range of the grain sizes considered to a certain degree, as well.

Furthermore, as mentioned when a linear curve is fitted into experimental data points, it is likely that a σ_o value which is considerably lower than mτ_o is determined, and it seems this also depends on the range of grain size being considered. When a grain size range that extends to relatively coarse grain sizes is considered, a shallow HP slope, i.e., a smaller k value with a relatively higher σ_o value, is determined. In addition, as mentioned, when a fourth-order or even a higher-order polynomial curve is fitted into the predicted yield strength values that were computed for a wide range of grain sizes, including significantly coarse ones is employed, a σ_o value approaching to mτ_o could be obtained. Nevertheless, it seems the reported experimental σ_o and k values depend on the range of grain size that is considered.

Finally, when the predicted data are closely examined, it can be seen that a slight deviation from linearity is present for the fine grain size region, as well, which is the case for pure nickel, in particular (Figs. 10(a) and 10(c)). Thus, a polynomial curve will be slightly a better fit in this region, as well, which will exhibit a slight gradual increase in the yield strength with respect to the inverse root of grain size for fine grain sizes, as well. An experimental result for commercially pure aluminum 1060 alloy exhibits this trend for very fine grain sizes ranging from 30 µm to 0.7 µm [25]. Just, according to reported yield strength values, the gradual increase in yield strength values with respect to grain size is much more pronounced. To mention, the reported yield strength values for that alloy were significantly higher than high-purity aluminum. As it was shown by Hansen, impurity levels as much as 0.5% result in significantly higher yield strength values [14]. There is about a 10-MPa difference between the experimentally determined yield strength values for 99.5% and 99.999% pure aluminum materials, which increases the yield strength approximately by a factor of 2.

The effect of initial dislocation density on the predicted yield strength values is demonstrated for pure copper in Fig. 11(a). Fourth-order polynomial curves are fitted into predicted results. For most metals, at the annealed state, the dislocation densities are on the order of 10⁴ and 10⁶/mm² [26]. Accordingly, the predictions were made for various initial edge dislocation densities ranging from 7.5 × 10⁶ to 1 × 10⁴/mm². As can be seen from the figure, the predictions for 10⁵ and 10⁴/mm² initial edge dislocation densities give almost the same curve. Thus, although there seems to be a considerable dependency of HP results on the initial dislocation density, it seems below a certain level; a decrease in the dislocation density does not cause an appreciable difference in terms of predicted yield strength values. Accordingly, the curve for 10⁵/mm² initial dislocation density is arbitrarily taken as a lower bound, whereas 7.5 × 10⁶/mm² initial dislocation density is arbitrarily taken as an upper bound to show how the predictions made by the model vary with respect to implemented initial dislocation density.

Nonetheless, according to the model, since the increase in the dislocation density during plastic deformation is significantly less for coarse grain sizes, as can be seen from Fig. 11(a), the implemented initial dislocation density has a significant impact on the yield strength values for relatively coarse grain sizes, whereas for fine grain sizes, the influence is less pronounced. Thus, according to the model, the determined σ_o and k_y values seem to be dependent on the initial dislocation density, as well. The initial dislocation density level for the coarse grain sizes seems to have a more pronounced effect on the Hall–Petch constants. For coarse grain sizes, a higher initial edge dislocation density shifts the predicted yield strength values to higher stress levels, whereas for fine grain sizes, the predicted yield strength values are weakly dependent on the initial dislocation density. Hence, it can be said that the predicted results for very fine grain sizes are nearly insensitive to initial dislocation density since the initial dislocation density level is almost negligible when compared to predicted necessary dislocation densities for 0.2% plastic strains (Figs. 11(a)–11(c)).

In Figs. 11(b) and 11(c), linear curves are fitted into predicted results. When the predicted data points for various initial dislocation densities are examined closely, it can be seen that, for low initial dislocation density values, a better fit by a linear curve is obtained. Predicted results that can be well-fitted by a linear curve are extended to relatively coarse grain sizes (Figs. 11(b) and 11(c)). Thus, for initial dislocation densities of 7.5 × 10⁶, 1.63 × 10⁶, and 1 × 10⁶/mm^1/2, computations are made for grain sizes ranging from 150 µm to 1 µm (Fig. 11(c)). This grain size range is the one that can be well-fitted by a linear curve for nearly all these initial dislocation densities, whereas for a lower initial density of 10⁵/mm^1/2, computations are made for a larger grain size range, where 1000-µm grain size is arbitrarily chosen as the largest one that is implemented in the computations. Thus, for the lowest initial dislocation density of 10⁵/mm^1/2, predictions are made for grain sizes ranging from 1000 µm to 1 µm, and for this wide grain size range, the predicted results are still well-fitted by a linear curve.

In addition, for higher initial dislocation densities, it can be seen that a slight deviation from linearity is present at relatively fine grain size regions. Correspondingly, as initial dislocation density is increased, the deviation from linearity for the finer grain sizes occurs, as well, but to a certain degree, whereas the yield strength values predicted for a low initial dislocation density are almost perfectly fitted by a linear curve for fine grain size region, as well.

Nevertheless, according to the model, the predicted yield strength values and corresponding σ_o and k values are dependent on the initial dislocation density to a limited extent. As can be seen from Fig. 11(a), for reasonable initial dislocation density levels, the predictions made by the model fall into a considerably narrow region that is limited by the curves that are computed for the lowest possible and for a reasonably high initial dislocation density value, which is arbitrarily taken as 7.5 × 10⁶/mm². In addition, as mentioned, for relatively finer grain sizes, the two curves representing the upper and lower bounds approach each other and begin to give similar predictions (Fig. 11(a)). Thus, for relatively fine grain sizes, the effect of initial dislocation density is almost negligible, and similar yield strength values are predicted even when considerably different initial dislocation densities are assigned, whereas a significant increase in the initial dislocation density will result in a considerable increase in the predicted yield strength values for relatively coarse grain sizes and, correspondingly, a significant decrease in the initial dislocation density will result in an appreciable decrease in the predicted yield strength values. However, as mentioned, an initial dislocation density below a certain level produces no significant changes, in terms of predicted yield strength values. Thus, it seems there exists a lower bound for predicted yield strength values.

As noted, for pure aluminum, there is a significant scatter in the experimental results presented in the study by Hansen, which was attributed to orientation differences among the various specimens [14]. To see the prediction capability in terms of possible variations with respect to various initial dislocation densities, computations were made for relatively high and a low initial dislocation density of 6.9 × 10⁶ and 4 × 10⁴/mm², respectively. In Fig. 12, corresponding predicted results were demonstrated together with the previously predicted ones for 1.96 × 10⁶/mm² initial edge dislocation density and with the experimental results. For coarse grain sizes larger than 40 µm, all experimental results fall into the region that is predictable by the model, whereas for finer grain sizes, the experimental results are overpredicted by the model, which is true for grain sizes finer than 20 µm, in particular. To mention, for pure copper and pure nickel, the predicted and experimental results are in a good agreement for nearly all grain sizes, including fine grain sizes.

Nevertheless, the difference in the reported values for σ_o and k_y and the scatter in the experimental results were generally attributed to orientation difference or to the impurity level of the materials being tested. According to the current model, the yield strength values and accordingly the σ_o and k_y values seem to be dependent on the initial dislocation density levels, as well. Thus, it seems, in addition to orientation difference, possible variations in the initial dislocation densities especially for coarse grain sizes result in a scatter in the experimental results in terms of yield strength values, which in turn may result in a variation in experimentally determined σ_o and k_y values, as well. For the completeness of the discussion, other than these, it is also known that, when the share of surface grains becomes considerable, alterations in the deformation behavior and corresponding yield strength values occur. Thus, the specimen size may have an influence on the experimentally determined σ_o and k_y values, as well [3,14,27]. For a given specimen size, this can be the case for coarse grain sizes, in particular.

As for the experimental results reported by Wyrzykowski and Grabski for pure aluminum, the experimental result for the finest, 2 µm, grain size level off and a deviation from the increasing yield strength trend with respect to the increase in the inverse root of grain size occurs. Thus, this data point was excluded and wasn’t taken into account for the comparison of predicted results with experimental ones. In addition, according to the model, for a given plastic strain, predicted dislocation density is inversely proportional to grain size, and for relatively fine grain sizes, the predicted necessary dislocation densities for 0.2% plastic strain increase significantly. For grain sizes finer than 20 µm, the experimental results are overpredicted by the model for pure aluminum, and the difference between the two increases as grain size becomes finer which is manifested by the underpredicted HP-stress and overpredicted HP-slope values. Within this context, when compared to pure copper and nickel, it seems for aluminum a different behavior that is not truly considered by the model occurs during deformation for relatively fine grain sizes, in particular. As mentioned, in the model, Eq. (10) gives a seeming minimum thickness for the crystalline strips, so that any given small plastic strain could be produced by the movement of dislocations which produce a slip step at the grain boundary region which is equal to Burgers vector. As grain size gets finer, the dislocation density necessary for a given plastic strain increases. Therefore, it seems crystalline strips with finer thickness are to be considered. A finer strip than that is computed by Eq. (9) reduces the necessary dislocation density and the shear strength that is developed for a given portion of the grain. Thus, rather than producing a slip step equal to Burgers vector, by movement of dislocations within crystalline strips that have thicknesses that are finer than computed by Eq. (9), any small plastic strain can be produced by displacement of each strip that is less than the Burgers vector. In that case, the total length that is traveled by the dislocations would be less than the radius of the strip D_i/2. That means, any given small strain amount is produced with less necessary dislocation density, which in turn produces less shear strength for a given portion of the crystalline grain.

It seems slip steps at the grain boundaries produce discontinuities at the grain boundary regions, and as grain size gets finer, this probably becomes more significant. As mentioned, strips with finer thickness mean reduced slip step length, which in turn produces less discontinuity at the grain boundary region. This can be the reason why strips with finer thickness are required for fine grain sizes.

Accordingly, this can be the reason why the experimental result for the finest, 2 µm, grain size for pure aluminum levels off and why for grain sizes finer than 20 µm the experimental results are overpredicted by the model for pure aluminum, as well. Within this context, it is possible that as grain size gets finer at some point, the model will begin to predict larger yield strength values for copper and nickel, as well. Thus, for significantly fine grain sizes, consideration of strips with finer thicknesses is required for improved predictions. To mention again, for strips that are finer than the ones predicted by Eq. (9), the total distance traveled by the dislocations, and accordingly, the total length and density of dislocations within a given strip become less. This probably occurs as grain sizes get finer and/or as plastic strain increases.

Again, within this context, it can be said for relatively higher plastic strain levels such as 0.3% or 0.5% plastic strain amounts, and/or for significantly fine grain sizes, and the experimental flow stress values will be overpredicted by the model developed for pure copper and nickel, as well. Thus, the model developed in this study is limited to the prediction of yield strength values corresponding to small plastic strains, such as 0.2%. Otherwise, an overprediction is expected for fine grain sizes, in particular. Thus, to extend the prediction capability of the model further into the flow stress region elaborations are required.

Here, it should be mentioned that the model does not consider how slip and dislocation density have evolved and how much dislocation is annihilated due to dynamic recovery during the course of plastic deformation. It just gives a prediction about the required dislocation density for a given small amount of plastic strain based on the assumptions that rather than strain, deformation within grains is controlled by stress, and the strain is produced by the displacement of crystalline strips, which is equal to the Burgers vector. To do this, the Orowan equation is utilized in conjunction with the Taylor polycrystalline deformation model and the Taylor hardening model.

The model proposed here requires excessive computations. Thus, based on the knowledge provided by the model, a simple equation is derived to compute the increase in the dislocation density in terms of plastic strain and the reciprocal of grain size, as well.

Here, $Δρ⊥i$ is the increase in the edge dislocation density for a given strain, $Δλ⊥i$ is the increase in the total edge dislocation length within the half volume, D_i is the diameter of the ith strip, and t is the thickness of the strip. As can be seen, Eq. (10) is implemented in Eq. (26) to eliminate t from the equation. Thus, according to the equation to compute the increase in the dislocation density in terms of any given macro-plastic strain amount, the diameter of the strip where the plastic strain is equal to the macro-plastic strain should be known.

where L_NR refers to normalized radial position. Thus, for the crystalline segment that is located at a distance of √(1 − π²/16) × D/2 from the grain center, the plastic strain value is equal to the macro-plastic strain which is equal to 0.002 for the current discussion, in particular.

Almost the same equation which was presented and utilized in some previous studies is obtained. In the equation that is used and presented in some previous studies, there exists an uncertainty in terms of the value of parameter C, since the value of parameter C is ambiguous, whereas Eq. (30) is derived and the value of parameter C is computed as a natural outcome of the model. Here, in Eq. (30), the parameter C is a well-defined parameter by the model and a value of about 49.6 is computed for FCC metals. The current model considers the gradual change in the local geometry of crystals. Accordingly, the plastic strain developed varies among the different sections within the grains. For the crystalline section located at the normalized radial distance of √(1 − π²/16), the plastic strain that is developed is equal to the macro-strain which is almost true for all grain sizes. Employing the Orowan equation in conjunction with Taylor polycrystalline deformation and Taylor hardening models for the corresponding local crystalline geometry gives Eq. (30) and the parameter C takes a value of 49.6 for FCC metals since the multiplication 5 × 4 × 8 × m/π² is 49.6. Thus, it seems the current model provides useful insights into understanding the physical basis for the equation that was previously utilized to compute yield strength in terms of the plastic strain and the reciprocal of grain size and also in understanding the physical basis for the parameter C present in the equations.

If Fig. 13 closely examines one might see that all curves converge and seem to intersect at about 0.385 normalized radial distance. Thus, regardless of the grain size, for the strip that is located at about 0.385 normalized distance, almost the same plastic strain with a value of 0.00235 occurs which is higher than 0.002 by a factor of 1.175. The diameter for that strip is approximately equal to 0.9237D. If computations were made for this strip about the same value of 49.6 will be computed for the parameter C, as well.

When the predicted results by Eq. (30) and the ones predicted by the model are compared, it is seen that the yield strength values predicted by Eq. (30) are just slightly lower than the ones predicted by the model (Fig. 14). For relatively very coarse grain sizes, almost the same yield strength values are predicted. As the grain size gets finer, a slight but gradual increase in the discrepancy between the two predicted results occurs. For about 2.5-µm grain size, the difference is increased to 1%.

The discrepancy between the results predicted by Eq. (30) and by the model seems to arise due to an imprecision present at the computed volume and assigned location of the basal plane of the crystalline strip at which the computations are made. Thus, an alternative equation is also developed to predict the increase in the dislocation density for a given strain for the aforementioned crystalline strip. Again, the radial location of the crystalline strip is equal to √(1 − π²/16)D/2. Here, just a slight adjustment is made in terms of geometry and the radial location for the basal plane of the strip. The comparison of the geometry and the exact radial location for the crystalline strip for the two equations is shown in Fig. 15.

The increase in the edge dislocation density calculated by using Eq. (31) should be multiplied by five to account for the increase in five active slip systems. To predict the yield strength, this value is then implemented in Eq. (25). When the results predicted by using Eq. (31) are compared to the ones predicted by the model, it is seen that the predicted ones by Eq. (31) are also slightly lower than the ones by the model (Fig. 16). The difference between the predicted results increases slightly with decreasing grain size in general. A maximum 0.6% difference occurs for 3.4-µm grain size, and then the difference is reduced to 0.25% for 2.5-µm grain size. Hence, by using Eq. (31) an improved consistency is achieved. To note that this is accomplished at the expense of simplicity of the equation. Hence, Eq. (30) seems to offer a handy alternative, for a quick evaluation of yield strength in terms of grain size, whereas in terms of precision and consistency with the model, Eq. (31) seems to give better results. However, as mentioned, this is achieved at the expense of simplicity in terms of the ease of computation.

Computations are made for the crystalline section where the plastic strain is equivalent to the macro-strain (ϒ_p = mε_p) which has a value of 0.002, in particular. The value of parameter C depends on the exact location of the basal plane of the crystal and on the exact geometry of the crystalline section at which computations are made. Thus, it seems the value of parameter C in Eq. (30) is weakly dependent on the grain size which is approximately taken into account by multiplying parameter C with the term e^0.00125/√D.

When Eq. (33) is employed, no observable trend seems to be present in terms of the discrepancy between the results predicted by Eq. (33) and by the model (Fig. 17). For some grain sizes almost, the same predictions were made. For some grain sizes, a slight over and, for others, a slightly less prediction are present. The maximum deviation between the two occurs for the fine grain sizes ranging from 5 µm to 2.5 µm. Maximum differences of 0.16%, −0.14%, and 0.17% are observed between the predictions made by the equation and by the model for 5-µm, 3.4-µm, and 2.5-µm grain sizes, respectively.

Hence, in terms of improved consistency with the model, Eq. (33) seems to be a better option when compared to Eq. (30) and even when compared to Eq. (31), whereas, in terms of simplicity, Eq. (33) seems to be a better option when compared to Eq. (31), as well.

Finally, predicted yield strength values by Eqs. (30), (31), and (33) are compared with the ones predicted by the model for pure nickel and pure aluminum in Figs. 18(a)–18(c) and 19(a)–19(c), respectively. The predicted results by the equations and the model are consistent with each other for these materials, as well. As in the case of pure copper, the predicted results by Eq. (30) are slightly lower than the ones predicted by the model. There is again a slight gradual increase in the discrepancy between the results as grain size gets finer. An improved consistency with the predicted results by the model is achieved by using Eqs. (31) and (33).

In this study, a new model is proposed to account for grain boundary hardening. In this respect, a novel approach is devised to predict the density of dislocations stored and plastic strain developed within the volume of the polycrystalline grains. Correspondingly, the grains within polycrystals are subdivided into segments in the form of crystalline strips. Thus, the gradual change in the characteristic crystalline length and corresponding volume at which dislocations glide and are stored during plastic deformation within the polycrystalline grains is taken into account.

In the model, computations were carried out for each crystalline segment within the representative grain. Since the local size and the local volume of the crystal are dependent on the grain size, the computed dislocation densities and the corresponding yield strengths show a dependency on grain size, as well.

In addition, it was assumed that the deformation within each polycrystalline grain is controlled by the stress, that is, rather than strain the dislocations that are stored within each strip are correlated with stress, so that the strain experienced by each crystalline strip is governed by the state of stress equilibrium.

In the model, the Orowan equation is implemented in conjunction with the Taylor polycrystalline deformation model to predict dislocation density for a given plastic yield strain. The predicted dislocation density is then implemented in the Taylor hardening model to compute the yield strength of a material. The model seems to work only for the prediction of dislocation densities and corresponding yield strength values for small plastic strains, such as 0.2%.

The verification of the model is accomplished by comparing the predicted and experimental results for three different pure FCC metals. For pure copper and nickel, a good agreement between the predicted and experimental results is achieved, whereas, for pure aluminum, a good agreement is achieved to a limited degree. For aluminum, the experimental results are overpredicted for relatively fine grain sizes.

Since the model requires excessive computations, based on the model, an alternative simple equation was also derived to predict dislocation density and corresponding yield strength values in terms of plastic strain and reciprocal of grain size, at ease. The equation derived turned out to be almost the same as an equation presented and utilized in some previous studies. However, there was an uncertainty in the value of the parameter that appears in the equation. Thus, in that equation, the value of the parameter C seemed to be indefinite. However, in the current study, the equation is derived and the value for the mentioned parameter is computed as a natural outcome of the currently developed model. Thus, according to the model, the value of the parameter is definite and a value of 49.6 is computed for FCC metals. Therefore, it seems the model provides useful insights both in understanding the physical basis for the equation that is used in these previous studies and in understanding the physical basis of the parameter C. The predictions made with the equation and the model are consistent. However, as grain size decreases, there exists a slight gradual increase in the discrepancy between the two. An alternative, second equation was also devised at the expense of the simplicity of the equation with improved consistency. Finally, it was shown that the exact value of parameter C which appears in the first equation is weakly dependent on grain size. Thus, a third alternative equation is also devised which considers this weak dependency. In this case, again an improved consistency is achieved between the results predicted by the equation and the model. Nevertheless, devised equations provide means for the prediction of the yield strength values for pure FCC materials with ease.

The verifications of the model and equations are achieved for several pure FCC metals. For other alloys, i.e., other than pure FCC metals, further elaboration of the model is probably necessary.

Nevertheless, the model developed in this study provides new insights in understanding of dependency of yield strength on the grain size. The model takes into account the fact that within the polycrystalline grains, there exists a gradual change in the local crystalline geometry constrained by the grain boundaries where dislocations glide and are stored. Owing to that, the fundamental equations and models proposed earlier by Orowan and Taylor are utilized to develop a model, and three alternative equations are developed, as well, to predict the yield strength of a material in terms of plastic strain and the reciprocal of average grain size.

In addition, it was shown that, according to the model, the yield strength values are dependent on the initial dislocation density of material at the annealed state, which is especially true for relatively coarse grain sizes. In accordance with that, for materials with relatively high initial dislocation densities, the experimentally determined k and σ_o values may exhibit appreciable variation with respect to the grain size range that is considered.

To summarize, the Orowan equation allows one to estimate the necessary dislocation density for a given plastic strain. By implementing the local dislocation densities computed with the Orowan equation into the Taylor hardening model, local yield strengths developed within the polycrystalline grains were computed.

Thus, the current study shows that when local crystalline geometry is considered, in conjunction with Taylor polycrystalline deformation and Taylor hardening models, the Orowan equation can be utilized to compute the yield strength of polycrystalline materials. Thus, in essence, the current study emphasizes the significance of consideration of the local crystalline geometry to account for grain boundary hardening.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.