Intrinsic material length scales and plasticity

Dislocation lines moving within a crystal structure are hindered by microstructural obstacles. These obstacles can be roughly classified by their dimensionality: solute atoms (0D), dislocations (1D), grain boundaries (2D), and precipitates (3D) [Reference Ehrler, Dunstan, Zhu, Hou, P’ng and Bushby8, Reference Gladman9, Reference Morris10, Reference Zhu, Bushby and Dunstan11]. For a dislocation to overcome an obstacle and “escape,” it requires a critical shear stress, τ_c, known as the Orowan stress [Reference Zhu, Bushby and Dunstan11] equivalent to

(5)$${\tau _{\rm{c}}} = \beta {{Gb} \over L}\quad ,$$

where L is the distance between the obstacles and β is a mechanism-dependent strengthening factor [Reference Ehrler, Dunstan, Zhu, Hou, P’ng and Bushby8, Reference Zhu, Bushby and Dunstan11, Reference Ehrler, Hou, Zhu, P’ng, Walker, Bushby and Dunstan12, Reference Greer and Nix13, Reference Lefebvre, Devincre and Hoc14, Reference Dunstan, Ehrler, Bossis, Joly, P’ng and Bushby15]. The intrinsic material length scales of a crystal are divided into two categories. Three of the obstacle types, dislocations, solute atoms and precipitates, are classed as bypassable. This is because dislocations can get past them if the stress is high enough. As grain boundaries completely contain a dislocation, they are considered to be not bypassable. Note that the Hall–Petch relationship can be classically derived in two different ways: one through the assumption of forest hardening of grains, which assumes dislocations remain within a grain and multiply to accommodate the applied shear strain, and one which looks at the critical stress required to be produced by a pile-up of dislocations at a grain boundary to promote slip in an adjacent grain [Reference Morris10]. This paper adopts the former approach, consistent with the assumption of boundaries as non-bypassable objects, although this assumption it is not considered critical to this derivation, as both approaches give the same inverse root grain size dependency. Forest (or Taylor) hardening depends on the average spacing between dislocations, ${L_{\rm{D}}} = {1 \over {\sqrt \rho }}$, where ρ is the dislocation density. This is assumed to be the only material length scale that changes during plastic deformation and leads directly to Eq. (1). Solid solution strengthening depends on the average spacing between solute atoms. This is taken to be ${L_{\rm{S}}} = {1 \over {{c^n}}}$, where c is the solute strengthening element concentration and n can be either ${1 \over 2}$ for dilute concentrations or ${2 \over 3}$ for higher concentrations [Reference Morris10, Reference Jones and Ashby16, Reference Labusch17]. Precipitation hardening depends on the mean spacing between precipitates. If these are assumed to be spherical, then it is reasonable to assume that ${L_{\rm{P}}} = 1.6{r_{\rm{P}}}\left( {{{\sqrt \pi } \over {4f}} - 1} \right)$, where r _p is the mean radius of the precipitate and f is the volume fraction of precipitates [Reference Atkinson, Gill, Shirzadi and Jackson18]. The contributions from these three stress hardening mechanisms are combined by taking the average effective spacing between the whole field of obstacles, L _τ, such that the total shear yield stress is

(6)$${\tau _{\rm{Y}}} = {\tau _0} + {{Gb} \over {{L_\tau }}}\quad ,$$

where the additional intrinsic (friction) strength of the crystal, τ₀, has been introduced. If the width of the obstacles is ignored, the average spacing between them is inversely proportional to their number density. The total number density of obstacles is therefore proportional to $L_\tau ^{ - 1} = L_{\rm{D}}^{ - 1} + L_{\rm{S}}^{ - 1} + L_{\rm{P}}^{ - 1}$ [Reference Ehrler, Hou, Zhu, P’ng, Walker, Bushby and Dunstan12]. Each type of obstacle has a different level of effectiveness in blocking dislocation movement. Therefore, in practice, each mechanism is weighted accordingly such that the effective length scale is actually taken to be

(7)$${1 \over {{L_\tau }}} = {{{\beta _{\rm{D}}}} \over {{L_{\rm{D}}}}} + {{{\beta _{\rm{S}}}} \over {{L_{\rm{S}}}}} + {{{\beta _{\rm{P}}}} \over {{L_{\rm{P}}}}}\quad ,$$

where β_S = 0.01–0.05, β_D = 0.1–0.3, and β_P = 0.5–0.8 are the estimated range of values for the strengthening factors for solid solution strengthening, forest hardening, and precipitate hardening, respectively [Reference Morris10]. The actual value of β_i depends on the properties of the material, the distribution of obstacles, and the size of the obstacle, e.g., β_P will be reduced if the precipitate size is too small to prevent precipitates being “by-passed” by cutting. Other methods than (7) have been proposed for combining strengthening contributions, principally root mean squared approaches when the length scales are similar [Reference Queyreau, Monnet and Devincre19].

The contribution from grain size strengthening is still to be introduced. The grain diameter, d, does not appear directly in (7) because grain boundaries are not bypassable obstacles in the Orowan sense. This hardening contribution to (6) must therefore be associated with the dislocation spacing as this is the only length scale that can change during deformation. This will be considered in “A hardness model with intrinsic and extrinsic length scales” section.

A hardness model with intrinsic and extrinsic length scales

The hardness model is initially developed for an indenter of general shape and then later explicitly derived for the two cases of self-similar and spherical indenters. The surface profile of the plastic indent is assumed to be commensurate with that of the indenter itself. The indent profile is assumed to be axisymmetric and defined by the indentation depth function, δ(r), a function of the horizontal radial distance r, as shown in Fig. 1. For the conceptual purposes of this derivation, it is assumed that GNDs are injected downward from the surface beneath the indenter during deformation [Reference Rester, Motz and Pippan20]. GNDs are necessary to create the atomic steps that form the plastic indent profile. In practice, GNDs may nucleate below the surface and move upward to form the steps, but this does greatly affect the conclusions of the model. Following Nix and Gao [Reference Nix and Gao5], the plastic zone beneath the indenter is assumed to be a hemisphere whose radius is the contact radius of the indenter, a, defined by δ(a) = 0. A number of authors have shown that the radius of the plastic region is in fact larger than the contact radius. Durst et al. [Reference Durst, Backes and Göken21], for instance, amended Nix–Gao using the assumption that the plastic zone radius is related to the contact radius a _pz = f _aa, where the scaling factor can be in the range of f _a = 0.5–3.5. The profile of the indentation defines the distance, s, between surface steps of height b to be

(8)$${1 \over {s\left( r \right)}} = - {1 \over b}{{{\rm{d}}\delta } \over {{\rm{d}}r}}\quad .$$

Figure 1:

The residual indentation profile in a substrate due a self-similar (conical) indenter after indentation and retraction.

A circular surface step of height b at radius r will generate a GND with a line length of l = 2πr. This is assumed to travel to the perimeter of the plastic zone, to a depth below the original reference surface y. For a hemispherical plastic zone of radius a, we have r ² + y ² = a ². The dislocation line density due to a single GND at radius r can be described by the following equation:

(9)$${\rho _{\rm{G}}}\left( r \right) = {l \over V} = {1 \over {s\left( r \right)y\left( r \right)}}\quad ,$$

where each of the dislocation line loops of length l injected from the surface are contained within a thin hollow cylinder of radius r, thickness s, and depth y, such that the volume is V = 2πrsy. The average dislocation density is taken to be

(10)$${\bar{\rho }_{\rm{G}}} = {1 \over {\bar{s}\bar{y}}}\quad ,$$

where $\bar{y}$ is the average depth of the hemispherical plastic zone,

(11)$$\bar{y} = {1 \over {\pi {a^2}}}\int_0^a {2\pi ry{\rm{d}}r} = {{2a} \over 3}\quad ,$$

given rdr = −ydy, and the average step spacing $\bar{s}$ is determined from the indenter profile.

Utilization of Eq. (10) generates the standard Nix–Gao models for single crystalline pure metals as expected. Here, we make two adaptations to account for the nanoindentation response of polycrystalline alloys. First, it is assumed that GNDs cannot freely travel from the surface to a depth y in the presence of obstacles. As a dislocation of length l moves through an array of obstacles, it is assumed that an additional dislocation length αl will be created. If the obstacles are an average (vertical) distance $\bar{L}$ apart, then the volume they occupy is $2\pi rs\bar{L}$. These will be SSDs, such that the average SSD density will become

(12)$${\bar{\rho }_{\rm{S}}} = {\alpha \over {\bar{s}\bar{L}}}\quad .$$

For a combination of different obstacles, we follow the same process as (7) and write the effective intrinsic length scale due to strain hardening as follows:

(13)$${1 \over L} = {{2{\alpha _{\rm{G}}}} \over d} + {{{\alpha _{\rm{D}}}} \over {{L_{\rm{D}}}}} + {{{\alpha _{\rm{s}}}} \over {{L_{\rm{S}}}}} + {{{\alpha _{\rm{P}}}} \over {{L_{\rm{P}}}}}\quad .$$

The first term on the right hand side contributes to grain size strengthening and assumes that the mean free path of a dislocation within a grain is equivalent to its radius, ${1 \over 2}d$. Note that, when combined with (15), the strengthening effect of grain boundaries conforms to the Hall–Petch relationship, ${1 \over {\sqrt d }}$. It is assumed that dislocations cannot cross grain boundaries and that a new dislocation must be generated to further accommodate the deformation. For this reason, we expect that α_g ≈ 1, although we note that the mean length of a random cut through a spherical grain of diameter d is ${{2d} \over 3}$ [as in Eq. (11)], suggesting values of α_G up to 1.5 are not physically unreasonable. The effectiveness of the other obstacles in contributing to strain hardening is expected to follow the order predicted for the stress hardening factors, β_i, such that α_G > α_P > α_D > α_s. A first initial estimate of α_i ≈ β_i would appear to be reasonable, although small obstacles such as solute atoms can be by-passed without significant dislocation multiplication (through the motion of jogs), suggesting that α_s ≪ β_S. Also, dislocation multiplication due to movement past precipitates is likely to be related to the particle size. Overall, we have

(14)$$\rho = {\bar{\rho }_{\rm{G}}} + {\bar{\rho }_{\rm{S}}} = {\rho _{{\rm{s0}}}} + {1 \over {\bar{s}}}\left( {{3 \over {2a}} + {1 \over L}} \right)\quad ,$$

where ρ_s0 is the initial SSD density in the material and a and L are the extrinsic and intrinsic length scales, respectively. The indenter shape defines $\bar{s}$ and will be determined later.

From (2) and (14), we can therefore write

(15)$$H = {H_{{\rm{SP0}}}} + {H_{\rm{D}}}\sqrt {1 + {1 \over {{\rho _{{\rm{s0}}}}\bar{s}}}\left( {{3 \over {2a}} + {1 \over L}} \right)} \quad ,$$

where ${H_{\rm{D}}} = 3\sqrt 3 Gb{\beta _{\rm{D}}}\sqrt {{\rho _{{\rm{s0}}}}}$ is the hardness contribution from dislocation (forest) hardening and ${H_{{\rm{SP0}}}} = 3\sqrt 3 Gb\left( {{{{\beta _{\rm{S}}}} \over {{L_{\rm{S}}}}} + {{{\beta _{\rm{P}}}} \over {{L_{\rm{p}}}}}} \right)$ is the contribution from the intrinsic strength of the crystal and solute and precipitate hardening.

Before proceeding further, it is worth noting the Nix–Gao model (3) and (4), as well as (15), predict that the dislocation density tends toward infinity as the contact area tends to zero. This is not physically reasonable. Here, we avoid this problem by restricting the maximum dislocation density that can occur in a crystal to a fixed value, ρ_max. An upper bound of the estimates for this value must be ${\rho _{{\rm{max}}}} \ll {1 \over {{b^2}}} = 16 \times {10^{18}}\;{{\rm{m}}^{ - 2}}$ for b = 0.25 nm, as this represents the (impossible) scenario of a dislocation at every atomic site. Hence, it is typically assumed here that ρ_max = 10¹⁵–10¹⁶ m⁻² is the expected range. If we have two lengths, x and y, then we can approximate the maximum of these two values using $\max \left( {x,\;y} \right) \approx {\left( {{x^n} + {y^n}} \right)^{{1 \over n}}}$ for n > 1, where the approximation becomes exact as n → ∞. Here, we replace the dislocation spacing ${L_{\rm{D}}} = {1 \over {\sqrt \rho }}$ by

(16)$${L_{\rm{D}}} = \max \left( {{1 \over {\sqrt \rho }},\;{1 \over {\sqrt {{\rho _{\max }}} }}} \right) \approx \sqrt {{1 \over \rho } + {1 \over {{\rho _{\max }}}}} \quad ,$$

where we have taken n = 2 as a weaker approximation of the maximum value to allow for statistical variations. Huang et al. [Reference Huang, Zhang, Hwang, Nix, Pharr and Feng22] proposed a two-stage model, which effectively took a sharp maximum, i.e., n = ∞. The forest hardening term, ρ/ρ_s0, is therefore replaced in (2) by ${{\rho {\rho _{\max }}} \over {{\rho _{{\rm{s0}}}}\left( {\rho + {\rho _{\max }}} \right)}} = {{{\rho \over {{\rho _{{\rm{s0}}}} \cdot 1}}} \over {\left( {1 + {{{\rho \over \rho }}_{\max }}} \right)}}$. With (14) and assuming that ρ_s0 ≪ ρ_max, we can therefore modify (15) to account for very small indentation effects:

(17)$$H = {H_{{\rm{SP0}}}} + {H_{\rm{D}}}\sqrt {1 + {{{1 \over {{\rho _{{\rm{s0}}}}\bar{s}}}\left( {{3 \over {2a}} + {1 \over L}} \right)} \over {1 + {1 \over {{\rho _{\max }}\bar{s}}}\left( {{3 \over {2a}} + {1 \over L}} \right)}}} \quad .$$

Equation (17) is now evaluated for two different types of indenter profile, self-similar and spherical.

• (i) Self-similar indenters.

In this case, the indenter is assumed to be conical in shape, although this method is relevant to the analysis of any pointed indenter with a constant slope. The geometry is shown in Fig. 2 and is defined by the function δ(r) = h − r tan θ, where h is the maximum indentation depth, θ is the indenter angle, and r is the radial distance from the axis of symmetry of the indenter.

Figure 2:

Idealized profile of a self-similar (conical) indent after indentation to a depth h into the surface.

Equation (8) yields

(18)$${1 \over s} = {{\tan \;\theta } \over b} = {1 \over {\bar{s}}}\quad .$$

Noting that ${h \over a} = \tan \;\theta$ means that substituting (18) into (17) gives the full expression for the ISE for a self-similar indenter as

(19)$$H = {H_{{\rm{SP0}}}} + {H_{\rm{D}}}\sqrt {1 + {{{{{h^*}} \over h} + {{{L^*}} \over L}} \over {1 + {{{h_0}} \over h} + {{{L_0}} \over L}}}} \quad ,$$

where ${h^*} = {{3\;{{\tan }^2}\;\theta } \over {2b{\rho _{{\rm{s0}}}}}}$ is the Nix–Gao parameter as usual, ${h_0} = {{{\rho _{{\rm{s0}}}}} \over {{\rho _{\max }}}}{h^*}$ is another fitting parameter, and the two other lengths are simply related to these two parameters by ${L^*} = {2 \over {3\;\tan \;\theta }}{h^*}$ and ${L_0} = {2 \over {3\;\tan \;\theta }}{h_0}$.

It is worth noting at this stage that (19) can be rewritten as follows:

(20)$$H = {H_{{\rm{SP0}}}} + {\hat{H}_{\rm{D}}}\sqrt {1 + {{{{\hat{h}}^*}} \over {h + {{\hat{h}}_0}}}} \quad ,$$

where ${\hat{H}_{\rm{D}}} = {H_{\rm{D}}}\sqrt {{f \over e}}$, ${\hat{h}^*} = e{h^*} + \left( {e - f} \right){h_0}$, and ${\hat{h}_0} = f{h_0}$ with $e = {L \over {L + {L^*}}}$ and $f = {L \over {L + {L_0}}}$. If the intrinsic material length scale L and the indenter angle θ are constant across all tests to be analyzed, then (19) can be written in the same form as (20) with the length scale removed (i.e., L → ∞). This indicates that self-similar indenters cannot be used to uniquely differentiate the intrinsic length scale L from the hardness of a material under these test conditions.

• (ii) Spherical indenters.

Figure 3 shows the geometry for a spherical indenter of radius R. The indenter contact radius is defined by $a = \sqrt {2Rh - {h^2}} \approx \sqrt {2Rh}$, assuming that the indentation depth h ≪ R.

Figure 3:

Geometry for a spherical indent.

Equation (8) yields

(21)$${1 \over s} = {r \over {b\sqrt {{R^2} - {r^2}} }} \approx {r \over {bR}}\quad ,$$

giving an average surface step spacing of

(22)$${1 \over {\bar{s}}} = {1 \over {\pi {a^2}}}\int_0^a {2\pi r \cdot {r \over {bR}}} = {{2a} \over {3bR}}\quad .$$

Using (22) in Eq. (17) then provides the full expression for the hardness measured by spherical indenters:

(23)$$H = {H_{{\rm{SP0}}}} + {H_{\rm{D}}}\sqrt {1 + {{{{{R^*}} \over R}\left( {1 + {{2a} \over {3L}}} \right)} \over {1 + {{{R_0}} \over R}\left( {1 + {{2a} \over {3L}}} \right)}}} \quad ,$$

where ${R^*} = {1 \over {b{\rho _{{\rm{s0}}}}}}$ is the usual Nix–Gao parameter and ${R_0} = {1 \over {b{\rho _{\max }}}}$.

Validation against published results

In this section, the predictions of the proposed models are tested against data from the literature. The full expressions for the hardness of polycrystalline alloy samples are given in Eqs. (20) and (23) for self-similar and spherical indenters, respectively. These equations are simple enough to give practical and physical interpretation of many features of hardness test results. However, (20) and (23) can also generally be simplified for a given situation, where a particular mechanism dominates. A number of such subcases are considered in this section. Unlike the basic Nix–Gao equations, the proposed hardness functions do not typically allow for a selection of plotting axes that allow for a simple linear fit to the data. Hence, nonlinear regression analysis is performed using the Excel Solver function to find the best fit of the model parameters to the data.

Hardness of single crystals with intrinsic strength for a self-similar indenter

The Nix–Gao model for self-similar indenters (3) has been very successful in interpreting experimental hardness data, particularly for single crystalline metals. This model assumes that there is a population of SSDs of density ρ_s0 in the crystal, which the GNDs, generated by the plastic indentation process, interact with. However, the preexisting dislocation density in these samples (typically annealed) is low. Here, the consequences of assuming that the SSD dislocation population is small (ρ_s0 ≈ 0) are examined in this context. In this case, (20) can be simplified to

(24)$$H = {H_{{\rm{SP0}}}} + {A \over {\sqrt h }}\quad ,$$

where $A = 9\;\tan \;\theta {\beta _{\rm{D}}}G\sqrt {{b \over 2}}$ and ${H_{{\rm{SP0}}}} = 3\sqrt 3 {\tau _0}$. For a pure metal, the intrinsic material length scale is very large (L → ∞), and it is assumed that GND saturation is not reached (ρ_max → ∞). Figure 4 compares the best fit of the standard Nix–Gao model (3) and Eq. (24) with single crystal copper data, where the grain size was large compared with the indent. It is interesting to note that (24) provides a slightly better fit to the data than (3) and that the principal difference between the two models is in the macroindentation regime [near the origin in Fig. 4(b)], where (24) extrapolates to a lower macroscale hardness than (3).

Figure 4:

Comparing a forest hardening model (3) against an intrinsic hardening model (24) with hardness ISE data for annealed single crystalline copper [Reference McElhaney, Vlassak and Nix23]. (a) H versus h, and (b) H ² versus ${1 \over h}$.

Best fits are obtained for (3) with h* = 1.78 μm and H ₀ = 0.56 GPa. Given the definitions of these parameters in (3), and taking G = 48 GPa, b = 0.25 nm, and θ = 25°, then yields ρ_s0 = 7 × 10¹⁴ m⁻² and β_D = 0.33. This is a high dislocation density, comparable to that found in a cold-worked sample [Reference Sousa, Sordi and Brandão24]. Typically, the dislocation densities found using (3) are high and often close to the maximum value expected ρ_max. This is investigated further in “Hardness of single crystals with a maximum dislocation density for a self-similar indenter” section.

Best fits for (24) give H _SP0 = 0.365 GPa and A = 18.5 GPa nm^1/2 which predict that τ₀ = 70 MPa and β_D = 0.26 for the value for the intrinsic strength. The intrinsic strength is typically of the order of 1–3 MPa, so this fitting value is unphysically large. In practice, it is expected that both the intrinsic strength and the SSD population contribute to the hardening of materials. The conclusion shows that (3) and (24) are quite similar for typical hardness tests, although the physical constants the fitting parameters relate to are quite different. It is difficult to say if one approach is more valid than another on this basis. For the remainder of this analysis, only one of the (almost indistinguishable) curves will be shown, but the physical parameters relating to each fit will be quoted.

Hardness of single crystals with a maximum dislocation density for a self-similar indenter

Figure 5(a) shows H versus h data for electropolished polycrystalline iridium with a grain size of 30 μm using a Berkovich indenter [Reference Pharr, Herbert and Gao7]. This is compared in Fig. 5(b) with the Nix–Gao Eq. (3), which predicts that a plot of H ² versus ${1 \over h}$ should be linear. It can be seen that this model works for larger scale indents (to the left) but fails at smaller test scales (on the right). Equation (20) is simplified to just include the effect of a maximum dislocation density ρ_max, so that

(25)$$H = {H_{\rm{D}}}\sqrt {1 + {{{h^*}} \over {h + {h_0}}}} \quad ,$$

where we take L → ∞ again for a pure metal and follow Nix–Gao in ignoring strengthening effects not due to forest hardening (H _SP0 = 0). This is identical in form to (20) indicating that, for self-similar indenters, (25) also applies to polycrystals as well as single crystals. The best fit of (25) in Fig. 5 shows that it captures the essential features of the material response much better than Eq. (3).

Figure 5:

Hardness data as a function of indentation depth for iridium [Reference Pharr, Herbert and Gao7] for a Berkovich indenter comparing predictions of Eqs. (3) and (25). (a) H versus h, and (b) H ² versus ${1 \over h}$.

The Nix–Gao Eq. (3) gives H ₀ = 2.2 GPa and h* = 3.48 μm, yielding predictions of ρ_s0 = 3.75 × 10¹⁴ m⁻² and β_D = 0.42 assuming G = 208 GPa for iridium.

Equation (25) is best fitted by H _D = 2.47 GPa, h* = 1.97 μm, and h ₀ = 150 nm. This yields a prediction of ρ_s0 = 6.6 × 10¹⁴ m⁻², which is about 8% of the maximum value of ρ_max = 8.75 × 10¹⁵ m⁻².

To extend the comparison of “Hardness of single crystals with intrinsic strength for a self-similar indenter” section, if we assume that the initial strength is not due to forest hardening (ρ_s0 = 0), then (20) gives

(26)$$H = {H_{{\rm{SP0}}}} + {A \over {\sqrt {h + {{\tilde{h}}_0}} }}\quad ,$$

where ${\tilde{h}_0} = {{3\;{{\tan }^2}\;\theta } \over {2b{\rho _{\max }}}}$. Equation (26) is best fitted when H _SP0 = 1.83 GPa, A = 2.43 GPa nm^1/2, and ${\tilde{h}_0} = 93\;{\rm{nm}}$. It is not shown in Fig. 5 as it is very close to the curve of (25). The theory then predicts that τ₀ = 352 MPa, β_D = 0.25, and ρ_max = 1.40 × 10¹⁶ m⁻².

An additional set of test data for single crystal MgO obtained using a Berkovich indenter, shown in Fig. 6, also supports this conclusion. Nix–Gao (3) gives H ₀ = 9.3 GPa and h* = 90 nm, yielding predictions of ρ_s0 = 1.5 × 10¹⁶ m⁻² and β_D = 0.38 assuming G = 155 GPa for MgO. This indicative SSD density is very close to the maximum possible density expected.

Figure 6:

Hardness data as a function of indentation depth for MgO [Reference Feng and Nix25] for a Berkovich indenter comparing predictions of Eqs. (1) and (25) for H versus h.

Equation (25) is best fitted by H _D = 9.0 GPa, h* = 180 nm, and h ₀ = 160 nm. This yields a prediction of ρ_s0 = 0.75 × 10¹⁶ m⁻² with ρ_max = 0.84 × 10¹⁶ m⁻², suggesting that the initial SSD density is almost at the maximum dislocation density, a situation which is most unlikely.

Equation (26) is best fitted when H _SP0 = 8.2 GPa, A = 1.45 GPa nm^1/2, and ${\tilde{h}_0} = 80\;{\rm{nm}}$. The theory predicts that τ₀ = 1.58 GPa and ρ_max = 1.6 × 10¹⁶ m⁻². While the intrinsic strength is very high, the maximum dislocation density is physically reasonable without requiring an excessive residual SSD density. In practice, the macroscopic hardness will arise from a combination of SSDs and other strengthening mechanisms. This would require using all terms in Eq. (20), but as the two effects are hard to differentiate, a best fit between either of the two cases presented here is valid.

Hardness of polycrystals using a spherical indenter

Hou et al. [Reference Hou, Bushby and Jennett26] have published hardness data for the spherical indentation of copper with different grain sizes. In this case, the intrinsic material length scale (grain size) is known but varies from sample to sample. Here, we model these data taking the minimum number of degrees-of-freedom required to capture the response, such that intrinsic strengthening (H _SP0 = 0) and GND saturation (R ₀ = 0) are ignored. In this case the full expression for spherical indentation hardness (23) can be simplified to

(27)$$H = {H_{\rm{D}}}\sqrt {1 + {{{R^*}} \over R}\left( {1 + {{2a} \over {3L}}} \right)} \quad .$$

Figure 7 shows the results of the best fit. The four curves are defined by (27) for a single set of parameters by minimizing the mean square error with the data across all four different indenter sizes simultaneously. In this case, the hardness is measured at a fixed strain of ${a \over R} = 0.25$ and the intrinsic material length scale is taken to be ${1 \over L} = {{2{\alpha _{\rm{G}}}} \over d}$, where d is the measured grain size. An important feature of these hardness measurements is that the indent is positioned within the center of a grain. If the positioning is random, then the grain size can influence the hardness even if the indentation depth is very much less than d. The best fit parameters are H _D = 309 MPa, R* = 21.3 μm, and α_G = 1.41. It can be seen from Fig. 7 that (27) fits the data well. The value of α_G ≈ 1 in line with expectation, given the variation in grain shape and the approximations with the model and the microstructural measurements. The values predict that ρ_s0 = 1.88 × 10¹⁴ m⁻² and β_D = 0.36. An equally good fit (not shown) can be achieved assuming intrinsic strengthening such that

(28)$$H = {H_{{\rm{SP0}}}} + \tilde{A}\sqrt {{1 \over R}\left( {1 + {{2a} \over {3L}}} \right)\quad ,}$$

where $\tilde{A} = 3\sqrt 3 {\beta _{\rm{F}}}G\sqrt b$. The best fit is obtained when H _SP0 = 0.18 GPa, A = 1.21 GPa µm^1/2, and the intrinsic length scale factor α_G = 1.45, similar to before. This predicts that τ₀ = 35 MPa and β_D = 0.31.

Figure 7:

Hardness data for four differently sized spherical indenters as a function of the grain size in polycrystalline copper [Reference Hou, Bushby and Jennett26] compared with a fit from (27). The strain is fixed such that ${a \over R} = 0.25$.

Hardness of a polycrystalline alloy with a maximum dislocation density using a spherical indenter

Swadener et al. [Reference Swadener, George and Pharr6] published data on indentation of a polycrystalline iridium alloy using a range of small indenter sizes. Unlike “Hardness of polycrystals using a spherical indenter” section, the intrinsic material length scale in this case is the same for all indents. This is therefore now a fitting parameter to be determined and compared with microstructural observations. For the analysis of these data, we initially ignore strengthening mechanisms in (23) to write

(29)$$H = {H_0}\sqrt {1 + {{{{{R^*}} \over R}\left( {1 + {{2a} \over {3L}}} \right)} \over {1 + {{{R_0}} \over R}\left( {1 + {{2a} \over {3L}}} \right)}}} \quad .$$

The best simultaneous fit for the suite of five curves for different indenter radii is shown in Fig. 8. This is achieved for H ₀ = 0.21 GPa, R* = 6.0 mm, R ₀ = 26 μm, and L = 12.5 μm. These values predict that ρ_s0 = 6.7 × 10¹¹ m⁻², ρ_max = 1.5 × 10¹⁴ m⁻², and β_D = 0.95. The latter is a larger value than expected as is the value of R*. This may be due to increased scatter in the hardness measurements for very small spherical indenter radii.

Figure 8:

Hardness versus strain for polycrystalline iridium alloy [Reference Pharr, Herbert and Gao7] for a wide range of spherical indenter radii compared with a fit of Eq. (29).

The grain radius was measured by Swadener et al. [Reference Swadener, George and Pharr6] as ${1 \over 2}d = 15\;{\rm{\mu m}}$. This compares very favorably with the predicted intrinsic material length scale of L = 12.5 μm. The iridium is also alloyed with 0.3% W. This indicates a solid solution strengthening spacing of L _S = 5 nm. The two length scales can be combined through (13), such that ${1 \over L} = {{2{\alpha _{\rm{G}}}} \over d} + {{{\alpha _{\rm{s}}}} \over {{L_{\rm{S}}}}}$. It is not possible to separate the contribution from each mechanism, but this suggests values of α_G < 1.2 and α_s < 4 × 10⁻⁴, both of which are realistic.