**1. Introduction**

Over recent decades, instrumented indentation tests have been recognized as an effective tool for probing the thermally activated deformation of metallic materials [1–3]. Typical experimental methods for the study of indentation creep contain the constant load and hold (CLH) test [4], constant strain rate (CSR) test [5], constant loading rate (CLR) test [6], strain rate jump (SRJ) test [7], etc. Being different from the conventional creep tests, the strain rate sensitivity (SRS) and activation volume (the two critical rate sensitive parameters) measured by these creep tests have been noticed to exhibit an obvious size effect [2,7–9]. The comprehension of these size-dependent parameters is critically essential for the interpretation of the fundamental creep deformation mechanisms [1,10–12].

So far, there exist two types of size effect as informed from the tests of indentation creep, including the interface-dominant creep size effect [7,9,11,13–16] and indentation depth- or force-related creep size effect [2,8,17–23]. For the former, previous literature has indicated that both the SRS and activation volume are affected by the intrinsic microstructures like grain and twin boundaries at the micro- or nano-scale [24,25]. For face-centered cubic (FCC) materials, enhanced SRS with decreasing grain size has been observed for nanocrystalline gold [14], copper [13] and nickel [11]. As for nano-twinned materials, a similar scaling relation has also been noticed for the dependence of SRS on the twin thickness [26,27]. It is, therefore, realized that there exists an intrinsic length scale for the SRS and activation volume, which is determined by the grain size and twin width of nanostructured materials [7,9,11,13–15] or the

film thickness of nanocrystalline films [28–30]. In order to interpret the thermally activated mechanisms for this length scale, Asaro and Suresh [26] proposed an analytical model by considering the emission of partial dislocations from grain and twin boundaries. Following this idea, a non-homogeneous nucleation model was later developed that can rationalize the size-dependent SRS and activation volume for nanocrystals and nano-twinned materials [31].

Besides the influence of intrinsic microstructures, there exists another form of size effect when addressing the indentation creep of single crystals and polycrystals with large grain size, i.e., the SRS decreases or the activation volume increases with increasing indentation depth or loading force, and this phenomenon has been widely observed in the CLH [19,21,22,32,33], CSR [8,23,34,35] and SRJ [36–38] test. For example, the size effect of indentation creep has been studied for polycrystalline pure aluminum through CLH tests at room temperature, which exhibits an obvious decreasing tendency of the SRS with increasing loading force even after the correction of thermal drift effects [22]. Similarly, the SRS of both annealed and 80% cold-worked 70/30 brass has been noticed to decrease with increasing indentation depth when performed under CSR tests [35]. Moreover, when applying SRJ tests on sintered silver nanoparticles, the SRS decreases from 0.04 to 0.024 with the increase in indentation depth from 1100 nm to 1700 nm [37]. Therefore, it is anticipated that there exist some different mechanisms for the depth- or force-related creep size effect of single or polycrystals, when compared with the interface-dominant creep size effect of nanocrystals or nano-twinned polycrystals.

In recent years, several possible explanations have been proposed for addressing the depthor force-related creep size effect, including the consideration of the free surface effect [21], thermal drift [2,19] and the evolution of geometrically necessary dislocations (GNDs) [2,8,39,40]. Sadeghilaridjani et al. [21] attribute this creep size effect to the high diffusion and mobility of dislocations near the sample surface, which result in a comparatively high SRS at shallow indents. However, even when the indentation depth extends 100 nm so that the influence of the free surface can be ignored, the size effect can still be observed in brass [35] and Al alloys [23]. Another possible explanation is considered to be the influence of thermal drift [2,19]. It is believed that the measurement error could exceed 100% when the indentation displacement rate gets close to the thermal drift rate [2]. However, even if the thermal drift is artificially inhibited or corrected during the indentation creep tests, the creep size effect still exists, especially at shallow indentation depths [23]. Actually, it is interesting to note that the depth- or force-related creep size effect seems to follow a similar evolution tendency as the hardness–force (or depth) relation of polycrystalline aluminum and alpha brass [8]. For the latter, it is with the well-known indentation size effect that the hardness decreases with increasing indentation depth due to the influence of GNDs [41]. Consequently, the fundamental mechanisms addressing the creep size effect are believed to originate from the thermally activated interaction between GNDs, which could become comparatively difficult as the density of GNDs becomes high at shallow indents [8,39,40].

In this work, we intend to propose a mechanistic model scaling the depth- or loading force-dependent SRS and activation volume of FCC materials, as corresponding theoretical analyses addressing this creep size effect have been seldomly reported in the literature. The outline of this paper is given as follows: in Section 2, the theoretical model is proposed in detail for the creep size effect. In Section 3, the experimental data of alpha brass, aluminum and austenitic steel are considered to verify the rationality and accuracy of the model results. Finally, we close with a brief conclusion in Section 4.
