**1. Introduction**

With excellent electronic characteristics such as a large band gap, high critical breakdown strength, high electronic saturation rate, high thermal conductivity, and high irradiation resistance, silicon carbide (SiC) has become an outstanding representative of the third generation of semiconductor materials and has been increasingly widely applied in a variety of fields, including computer, aviation, power, and nuclear energy development [1–4]. In addition, it is an attractive option to use SiC to produce space mirrors and large ground-based reflectors thanks to its remarkable advantages of large sti ffness, small thermal deformation coe fficient, and good thermal stability [5]. However, it is di fficult to obtain SiC parts with high forming accuracy and ideal machined surface quality due to its characteristics of high hardness (with a Mohs hardness between 9.0 and 9.5) and brittleness [6,7].

The traditional cutting force models tend to ignore the e ffect of elastic recovery on the macroprocessing of ductile materials. However, numerous studies [8–10] have shown that the role of elastic recovery is significant in the micro–nano-machining of brittle materials. Wasmer et al. [11] proposed a typical scratch pattern for brittle materials using an increasing load along a scratch path that is divided into five stages: elastic regime, ductile regime, subsurface crack regime, surface crack regime, and micro-abrasive regime. Elastic and ductile deformations were observed before the ductile regime, and brittle fractures appeared and began to dominate the latter deformation with increasing cutting force. Therefore, there is a cutting depth, referred to as the critical depth of cut, where the ductile region transitions to the brittle region. Lawn et al. [12] established a microfracture model under the point indentation of brittle materials consisting of the following processes: (1) the sharp indenter induces a zone of elastic and ductile deformation around the contact point; (2) a median crack suddenly initiates below the contact point; (3) the median crack stably extends with increasing indenter load; (4) the median crack begins to close during the initial unloading process; (5) lateral cracks begin to appear due to residual stress; and (6) lateral cracks continue to extend and cause chipping.

Methods such as grinding and chemical mechanical polishing are used in traditional processing and are characterized by low production e fficiency, high production cost, and, particularly, surface damage caused by the contamination of the polishing slurry [13]. In 1951, researchers found that hard and brittle materials show the characteristics of ductile removal under certain processing conditions [14]. Since the 1990s, researchers have conducted many studies into removal mechanisms in silicon carbide ductile regime processing, such as ductile regime grinding [7,15], ductile regime laser-assisted processing [16], ductile regime diamond cutting [17], ductile regime diamond wiresaw [18], and ductile domain ultrasonic-assisted processing [19]. These studies indicated that in the course of ductile regime processing, the chip is removed by ductile deformation, causing no damage or cracks to the machined surface of the workpiece, and the surface processing quality can be maintained [20–22]. The critical depth of cut at the ductile–brittle transition, the maximum cutting depth where no cracks appeared on the surface or subsurface of the sample, is a fundamental parameter of all methods of ductile regime processing. A formula was obtained by Bifano according to Gri ffith's principle through a quasi-static scratch test of several kinds of common brittle materials [23]:

$$d\_{\mathbb{C}} = a \left( \frac{E}{H} \right) \left( \frac{K\_{\mathbb{C}}}{H} \right)^2 \tag{1}$$

where α is a constant, *E* is the elasticity modulus, *H* is the hardness, and *K*c is the fracture toughness. This formula was amended by later scholars; for example, Gaobo's study on the critical depth of cut of 6H-silicon carbide indicated that the experimental results were not in line with the calculation of Formula (1) and the amended constant α [24].

Based on the above, most research has focused on the critical depth of cut using Gri ffith's principle and experimental method, which is based on cracks in the ductile extension which would have an influence on the performance of devices. With the development of nanotechnology, particularly the scanning electron microscope (SEM) and nanomechanical testing technology, a number of powerful tools has been provided to investigate the properties of silicon carbide at the nanoscale. This paper proposes a method considering the elastic recovery of the workpiece surface in nanoscratching in order to obtain the critical depth of cut for SiC using scratching stress and cleavage strength.
