# Simulation and Experimental Study on the Shrub-Cutting Performance of Quasi-Planetary Cutter
## 2 Materials and Equipment
### Materials
#### 2.1.2 Physical and Mechanical Properties  
The constitutive model describes the relationship between a material's physical and mechanical properties and external loads using mathematical expressions. It reflects the deformation behaviour and mechanical response of materials under different loading conditions, particularly the derivation of mechanical properties from the relationship between stress and strain. The constitutive model can represent a material's elasticity,  
Figure 1: Shrub materials and collection areas. plasticity, viscosity, and other properties, allowing for the prediction of its response under various loading conditions. Orthotropic constitutive models are commonly used to describe the physical and mechanical properties of materials. Transverse isotropy is a specific type of orthotropic behaviour [16]. This means that in a particular direction, usually referred to as the longitudinal or axial direction, the mechanical properties differ from those in the perpendicular direction. However, within the plane perpendicular to this direction, the properties remain uniform. Based on the assumptions of the shrub model, shrub material can be regarded as a transversely isotropic material. In the section perpendicular to the longitudinal direction, the mechanical properties in the radial and tangential directions are approximately the same. This interface is known as a homogeneous plane, whereas a plane with differing mechanical properties is called a heterogeneous plane. The transversely isotropic constitutive model includes various mechanical parameters on the homogeneous and heterogeneous planes, such as density (\(\rho\)), elastic modulus (\(E_{\text{L}}\), \(E_{\text{R}}\), and \(E_{\text{T}}\)), [PERSON]'s ratio (\(\mu_{\text{LR}}\), \(\mu_{\text{LT}}\), and \(\mu_{\text{RT}}\)), and shear modulus (\(G_{\text{LR}}\), \(G_{\text{LT}}\), and \(G_{\text{RT}}\)). Appropriate methods must be employed for the calculation and measurement of the parameters in the constitutive model, as shown in the density test and compression test of shrubs given in Figure 2. An electronic balance and a graduated cylinder were used to determine the density of each shrub material using the displacement method. An LD23-104 microcomputer-controlled electronic universal testing machine was utilised to conduct axial and radial compression tests on the shrubs. The experiment was conducted in accordance with GB/T 1935-2009 Method of Testing in Compression Parallel to Grain for Wood [17]. Bush samples were sequentially placed at the centre of a spherical sliding support. To apply pressure to the samples, a flat compression plate was loaded downward at a constant rate of 5 mm/min until structural deformation and failure occurred, marking the end of the test. From these tests, the slope of the stress-strain curve was obtained, representing the shrub's elastic modulus. The anisotropic (\(E_{\text{L}}\), \(G_{\text{RT}}\), and \(\mu_{\text{RT}}\)) and isotropic (\(E_{\text{R}}\), \(E_{\text{T}}\), \(\mu_{\text{LR}}\), \(\mu_{\text{LT}}\), \(G_{\text{LR}}\), and \(G_{\text{LT}}\)) plane parameters of the bush constitutive model were derived from axial and radial compression testing, respectively. For isotropic materials, as indicated by Equation (1), the relationship between shear strain and shear stress, along with the expression of shear strain in the generalized Hooke's law, leads to the derivation of the relationship between shear modulus, Young's modulus, and Poisson's ratio, as shown in Equation (2). \[\left\{ \begin{array}{c}\gamma=\frac{\tau}{G}\\ \gamma=\frac{2(1+\mu)}{E}\tau\end{array} \right. \tag{1}\]  
\[\left\{ \begin{array}{c}G_{\text{RT}}=\frac{E_{\text{R}}}{2(1+\mu_{\text{RT }})}\\ G_{\text{LR}}=\frac{E_{\text{L}}}{2(1+\mu_{\text{LR}})}\end{array} \right. \tag{2}\]  
In these equations, \(\tau\) represents the shear strain (MPa). \(\gamma\) represents the shear strain. \(G_{\text{LR}}\) represents the shear modulus of the shrub's homogeneous plane (MPa), where \(G_{\text{LR}}\) = \(G_{\text{LT}}\).