# An Improved Cutting Force Model for Ultrasonically Assisted Grinding of Hard and Brittle Materials
## 1 Introduction  
With the development of science and technology, many advanced materials such as hard and brittle materials, super alloys and composites are widely used in the aerospace, electronics, medical and automotive industries [1]. Among the advanced materials, the hard and brittle materials, such as the silicon carbide ceramics, sapphire and optical glass, exhibit outstanding properties, including high hardness, excellent corrosion resistance, high temperature resistance and wear resistance. The traditional processing methods do not meet the industrial requirements. These methods have a low material removal rate and suffer from serious tool wear and even breakage, which limits their applications [2]. The K9 optical glass is a typical hard and brittle material that is difficult to process. Extensive research demonstrates that UAG can effectively reduce tool wear and suppress breakage in the machining of hard and brittle materials at a high machining efficiency and reduced material damage [3, 4]. Ultrasonically assisted grinding (UAG) is traditionally a hybrid machining process in which ultrasonic vibration is imposed on a grinding tool or workpiece for alternative material removal mechanisms [1]. UAG has become one of the advanced processing techniques for the advanced machining processes in the machining of hard and brittle materials. The advantages include: reduced grinding force [5], reduced grinding temperature [6], reduced surface and subsurface damage [7, 8]. In the machining of hard and brittle materials, the machining force is an essential factor affecting the surface integrity of a workpiece [9, 10], the wear and lifetime of a grindingwheel [11; 12], and the material removal rate [13], etc. Therefore, it is necessary to establish a model for predicting grinding forces so as to achieve good surface integrity, grinding efficiency and extended tool life. In the existing literature, there are extensive studies on the modelling material removal rate during UAG [14]. [PERSON] [15] proposed a modeling method of material removal rate based on the ultrasonic machining of ceramic materials, carried out experimental verification, and further deduced the relationship between the material removal rate and machining parameters. [PERSON] [16] deduced the model of cutting force and material removal rate of ultrasonic machining ceramic matrix composites and verified that the increase in machining parameters would lead to the increase in material removal rate and the coarsening of the machined surface. [PERSON] [17] proposed an approach to modeling the ductile-mode removal in rotary ultrasonic machining. [PERSON] [18] carried on a varied-depth nano-scratch test of single grain on a nano indentation system and based on the varied-depth nano scratch test and the grain trajectory of ultrasonic vibration-assisted grinding (UVAG), a theoretical model of the normal grinding force is acquired using the material removal in unit time as a bridge. [PERSON] [19] developed the mathematical model of cylindrical grinding and obtained the grinding wheel blunt and abrasive removal rate. However, there are few reports on the predictive models of cutting forces during the UAG of hard and brittle materials. During the axial ultrasonically assisted grinding, the diamond grains in the end face of a tool is forced to generate the intermittent hammering action [12; 15]. Based on the impulse theory and brittle fracture theory, [PERSON] et al. established a cutting force model of a core tool for machining holes and tested it on titanium alloy [20], hard and brittle materials [21] and composite materials [22] with the rotary UAG. [PERSON] [23] proposed a grinding force model considering both ductile the removal force and brittle removal force, which was verified by experiments. [PERSON] [24] developed a grinding force model for micro-grinding of reaction-bonded silicon carbide ceramics. [PERSON] et al.