**1. Introduction**

With increasing requirements for the application of advanced structural materials under high temperature environments, it becomes necessary and vital to effectively characterize the thermal-related materials properties at elevated temperatures to ensure structural safety and device function [**?** ]. However, for the materials with limited size and characteristic microstructures, for example, multilayer thin-films and ion-irradiated materials with defect damage, the direct application of traditional mechanical tests seems invalid, and the consideration of small-scale testing techniques becomes inevitable [**?** ]. Among several promising candidates, the technique of nano-indentation has been well developed over the last decades, and taken as a valid method to characterize the localized materials properties at elevated temperatures due to the development of advanced high temperature indentation systems [**???** ].

So far, plenty of experimental works have been performed in the field of high temperature nano-indentation [**??????** ], and some principal features observed in these experiments indicate that the well-known indentation size effect, that is, the increase of materials hardness with decreasing indentation depth, still exists even when the testing temperature *T* increases up to *T*m/3 (*T*<sup>m</sup> is the melting temperature) for most materials [**?????** ]. However, when compared with the test performed at room temperature, both the increasing rate of materials hardness and ultimate bulk hardness are noticed to get weakened at elevated temperatures [**? ?** ]. For instance, Lee et al. [**?** ] investigated the dependence of indentation size effect on *T* for [0 0 1] -oriented single crystalline Nb, W, Al and Au, and demonstrated that for all of them both the hardness at infinite indentation depth and intrinsical materials length scale are strong functions of *T*. Similar experimental phenomena [**?** ] have also been observed in the indentation test of polycrystalline Co, Ni and Pt that the indentation size effect becomes moderated when *T* increases from room temperature to *T*m/3.

In order to interpret the fundamental mechanisms related to the above observed experimental results, several theoretical models have been developed in the past years [**?????** ]. Thereinto, the most widely applied model, was proposed by Nix and Gao [**?** ], which indicates that the intrinsic length scale is ascribed to the formation of geometrically necessary dislocations (GNDs) within the plasticity affected region. Later, Durst et al. [**?** ] modified the Nix-Gao model by redefining the volume of the plasticity affected region, and pointing out that its radius should scale linearly with the contact radius. In addition, the contribution of intrinsic lattice friction resistance was noticed to play a dominant role in determining the material's hardness, especially for body-centered cubic (BCC) materials at low temperatures [**?** ]. When further addressing the temperature effect on the fundamental deformation mechanisms, it is noted that increasing temperature can not only help enhance the dislocation mobility and expansion of the plasticity affected region [**?** ], but also lead to the decrease of lattice friction at elevated temperatures for most crystalline materials [**?** ]. However, most existing hardness models are proposed at room temperature, and the temperature effect on both microstructural evolution and lattice friction has not yet been systematically addressed [**? ?** ].

In this work, we intend to propose a theoretical framework for the hardness-depth relationship with temperature effect for single crystals. In Section **??**, a detailed derivation of the hardness model with temperature effect is presented. The dominant deformation mechanisms cover the dependence of lattice friction and network dislocation interaction on temperature. In Section **??**, the experimental data of four single crystals (Cu, Al, CaF2 and W) with different crystalline structures is applied to verify the accuracy and rationality of the proposed model. Moreover, corresponding deformation mechanisms and microsturcutres evolutions are discussed. Finally, we close with a conclusion in Section **??**.
