*Article* **Hardness-Depth Relationship with Temperature Effect for Single Crystals—A Theoretical Analysis**

**Hao Liu 1, Long Yu <sup>2</sup> and Xiazi Xiao 3,\***


Received: 23 January 2020; Accepted: 11 February 2020; Published: 13 February 2020

**Abstract:** In this paper, a mechanistic model is developed to address the effect of temperature on the hardness-depth relationship of single crystals. Two fundamental hardening mechanisms are considered in the hardness model, including the temperature dependent lattice friction and network dislocation interaction. The rationality and accuracy of the developed model is verified by comparing with four different sets of experimental data, and a reasonable agreement is achieved. In addition, it is concluded that the moderated indentation size effect at elevated temperatures is ascribed to the accelerated expansion of the plasticity affected region that results in the decrease of the density of geometrically necessary dislocations.

**Keywords:** temperature effect; indentation size effect; theoretical model; nano-indentation
