Unified Hall-Petch description of nano-grain nickel hardness , flow stress and strain rate sensitivity measurements

It is shown that: (i) nano-grain nickel flow stress and hardness data at ambient temperature follow a Hall-Petch (H-P) relation over a wide range of grain size; and (ii) accompanying flow stress and strain rate sensitivity measurements follow an analogous H-P relationship for the reciprocal “activation volume”, (1/v*) = (1/A*b) where A* is activation area. Higher temperature flow stress measurements show a greater than expected reduction both in the H-P ke and in v*. The results are connected with smaller nano-grain size (< ∼20 nm) measurements exhibiting grain size weakening behavior that extends to larger grain size when tested at very low imposed strain rates.It is shown that: (i) nano-grain nickel flow stress and hardness data at ambient temperature follow a Hall-Petch (H-P) relation over a wide range of grain size; and (ii) accompanying flow stress and strain rate sensitivity measurements follow an analogous H-P relationship for the reciprocal “activation volume”, (1/v*) = (1/A*b) where A* is activation area. Higher temperature flow stress measurements show a greater than expected reduction both in the H-P ke and in v*. The results are connected with smaller nano-grain size (< ∼20 nm) measurements exhibiting grain size weakening behavior that extends to larger grain size when tested at very low imposed strain rates.

The yield stress, flow stress and hardness of conventional mm-scale grain size face-centered cubic and related metal structures are known to be raised by an order of magnitude at nano-scale grain size dimensions, , say, at ≈ 20 nm.At smaller grain size, the strength is reversed to decrease with further grain size reduction.Not so well known is that the strain rate dependence of the flow stress increases also by an order of magnitude during the same range of grain size strengthening.In the regime of grain size weakening, moreover, the strain rate dependence had been predicted on a grain size weakening constitutive equation basis to increase by a further order of magnitude.Our purpose here is to present a compilation of flow stress, hardness and strain rate sensitivity measurements for conventional and nano-grain nickel material, covering both grain size weakening and strengthening behaviors, and to provide an explanation of the measurements on the basis of a combined Hall-Petch inverse square root of dependence and thermal activation dislocation model description.
An early description of a Hall-Petch dependence for diamond pyramid hardness measurements obtained on electrodeposited polycrystalline nickel with microstructural coverage beginning from conventional sub-mm grain sizes and extending down to a smallest grain diameter of 12 nm had been given by Hughes et al. 1 The linear dependence had been drawn recently in Fig. 1 onto a compilation of conventional and ultrafine grain size nickel measurements 2 reported by Matsui et al. 3 in accordance with the relationship: (1) The left-side listing of references is that given by Matsui et al. while the right-side listing applies for measurements added here and representing three rather diverse studies reported for nano-grain size nickel material: first, by Y.J. Li et al. 4 on stress-strain and strain rate "jumps" obtained in compression tests as a function of temperature; secondly, for comparable same type strain rate jumps obtained by Mohanty et al. 5 in miniature size specimen compression tests; and thirdly, by H. Li et al. 6 on strain rate sensitivity measurements made on mostly smaller nm grain size material tested in tension over an exceptionally wide range in strain rates and leading at smallest nm grain sizes to grain size weakening.In each case for the newer measurements, the initial 298 K flow stresses have been multiplied by 3 to calculate the hardness in the manner done by Hughes et al. 1 The pair of filled square points in Fig. 1 came from representative true stress -strain curves reported by Y.J. Li et al. 4 at two nano-grain sizes.The material was produced by pulsed electrodeposition.The respective grain sizes were determined by x-ray diffraction.Strain rate "jumps" of 3 to 6 order-of-magnitude rate changes were conducted during the stress -strain tests.In general, larger stress jumps were observed at higher T; a different result from conventional grain size measurements.Minor or even negative strain hardening behavior was exhibited in the overall deformation curves.The filled-right-side-up triangle point in the figure for Mohanty et al. 5 had been obtained on micropillars of ∼1.5 micron diameter fabricated with a focused ion beam technique.An average 30 nm grain size was determined in this case also by x-ray diffraction.Evidence of grain boundary sliding was reported from scanning electron microscope observations.The filled-downside-triangle point was obtained in tension by H. Li et al. 6 at a 40 nm grain size determined by transmission electron microscopy.This measurement was the only one credited to show "dislocation based deformation" when tested over higher strain rates of 10 -4 to 2 s -1 .At smaller grain sizes extending down to 6 nm and at lower strain rates in the range of 10 -10 to 10 -6 s -1 , the reported stress -strain curves may be observed to show grain size weakening.
An explanation of the coupled grain size/strain rate/temperature dependencies of face-centered cubic (fcc) metals has been determined by combination of the dislocation pile-up model for the H-P relationship with a thermal activation description of dislocation motion.In Eq. 1, the H-P based flow stress, σ ε , at strain, ε, relates to the hardness parameters through σ 0ε = H 0 /3 and k ε = k H /3.And the dislocation pile-up model description of a concentrated stress, τ C , in k ε is given as 2,7 In Eq. ( 2), m T and m S are respectively Taylor and Sachs orientation factors, G is shear modulus, b is dislocation Burgers vector, and α ≈ 0.8 is an average factor for the dislocation character.For fcc copper, nickel and aluminum, the τ C value in k ε had been correlated with the cross-slip shear stress, τ III , for example, as measured in single crystal tests. 2 Very importantly, the value of τ C is assumed to be composed of τ CG + τ CTH terms for athermal and thermal components of the shear stress in the grain boundary region.
For quantitative assessment of k ε , we begin with the Hughes et al. 1 straight line H-P type hardness dependence in Fig. 1 at ambient temperature.A value of k ε = 6.9 MPa.mm 1/2 is obtained when employing the multiplier of (1/3)k H . Hughes et al. had assigned the k ε value to an equivalent flow stress dependence at an effective strain, ε = 0.075.The value compares with k ε = 7.3 MPa.mm 1/2 for the two points at 298 K of Y.J. Li et al. 4  type model evaluation of k ε = 5.3 MPa.mm 1/2 by employing independently reported single crystal cross-slip measurements.Very notably, as will be shown, the k ε values for the Li et al. 4 flow stress measurements are determined to be reduced from the larger 7.3 MPa.mm 1/2 level to 4.84 MPa.mm 1/2 at 373 K.The estimated reduction in k ε is larger than can be accounted for by a weaker temperature dependence of the single crystal cross-slip shear stress in Eq. ( 2) that was used by Hughes et al. 1 and so gives indication of an extent of grain boundary weakening having occurred at the higher temperature.
The strain rate jumps reported by Y.J. Li et al. provide an additional basis for investigation of the H-P behavior at nanoscale dimensions.Such connection had been established for copper and nickel materials on a thermal activation -strain rate (TASRA) basis by Armstrong and Rodriguez. 8In this framework, the strain rate sensitivity is monitored by an activation volume parameter v* = A*b, in which A*, the activation area containing the dislocation Burgers vector, b, is given as v* = kT[∂ln(dγ/dt)/∂τ] T . ( In Eq. ( 3), k is Boltzmann's constant, T is temperature, (dγ/dt) is shear strain rate, and τ is shear stress.The shear strain rate follows the relationship (dγ/dt) = m T (dε/dt) for which m T is the Taylor orientation factor of ∼3.1.On the basis of both σ 0 ε and k ε being temperature and strain rate dependent, an H-P relationship for (1/v*) obtains 8 as In Eq. ( 4), the product and this explains (1/v*) following an analogous H-P type equation.Calculations of (b 3 /v*) derived from the measurements of Y.J. Li et al., 4 Mohanty et al. 5 and H. Li et al. 6 are given in Table I.For H. Li et al., one might note that we used m T in place of their √ 3 factor in determining v*.This measurement is particularly important because at the same grain size and at smaller values all tested at much lower strain rates, grain size weakening was measured.The tabulated v* values were computed for stress and strain rate changes done for the first jumps in the stress -strain curves except in the case of the two larger strain results for the 80 nm grain size material at 298 K in which case the v* values are for the first two jumps at ε = 0. 20 and ε = 0.27.The four ambient temperature measurements are added in Fig. 2 to the compilation of measurements previously reported by Armstrong and Rodriguez. 7In general, v* is seen to be reduced in Table I at increasing ε for all of the measurements, as normally is associated with a reduced spacing of dislocation intersections, thus being contained in (1/v*) 0 , however, such effect is now shown in accordance with Eq. ( 4) to be contained in the H-P grain size term and so the same mechanism is taken to apply in the grain boundary regions.Furthermore, the result shown in Table I of v* decreasing at the lower stresses measured at higher temperatures is not usual and this requires explanation because of the otherwise well-established 9 reciprocal dependence of v* on τ Th .
The v* calculations were obtained on the assumption of ∂τ in Eq. ( 3) being equal to the change in thermal stress component, ∆τ Th , and this may not be true for the current measurements Other measurements reported by Rajaraman et al. 12 on H-P type grain size strengthening for pulse-deposited nickel material of 17 nm provide a useful example for less favorable comparison with those measurements shown in Figs. 1 and 2. Comparable H-P values of σ 0ε = 426 MPa and k ε = 5.5 MPa.mm 1/2 were reported at ambient temperature for the initial yield behavior that was followed by negative strain hardening thereafter over a broad range of strain rates.A flow stress of 2.05 GPa and multiplier of 3 for hardness value at a lowest strain rate of 0.04 s -1 produces reasonable agreement with the H-P line in Fig. 1 despite separate measurement being reported of lower nanoindentation hardness values in the range of ∼4.8-5.1 GPa.A reported logarithmic strain rate sensitivity of 0.0064 at the 2.05 GPa flow stress gives a low value of (b 3 /v*) = 0.017 for comparison with the Fig. 2 results.
In summary, the assembled flow stress, hardness and strain rate sensitivity measurements demonstrate via their related Hall-Petch dependencies a unifying assessment of nano-grain nickel material deformation behavior covering the regimes of grain size strengthening and weakening.Quite reasonable agreement has been obtained, first, for a Hall-Petch description of strength properties obtained from three rather diverse investigations.Secondly, the (1/v*) characterization of strain rate sensitivity measurements has been connected in a reasonably quantitative manner to the dislocation pile-up model description for the H-P term involving k ε and which thermal activation parameter is shown to provide a sensitive marker for eventual transition to the onset of weakening at smaller nano-grain sizes, especially, in tests at lower creep-related strain rates.

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FIG. 1. Hall-Petch description of hardness dependence on grain size for nickel materials.

TABLE I .
Activation "volume", v* = A*b, values determined from strain-rate-sensitivity (SRS) measurements made on nanopolycrystalline nickel materials.