# The Mesoscopic Numerical Simulation of GAP/CL20/AP Composite Solid Propellant Based on MPM and FEM
## 3 The Modeling Process
### Interface Model of Binder/Grain  
According to the experimental results of the propellant, there was an obvious interface between binder and solid grains, and the interfacial debonding was one of its main damage modes [11]. Generally, cohesive elements can be used to model the interfacial debonding process [13; 14]. The bilinear model of the cohesive element is adopted in this project because of its simple structure and easy acquisition of the parameters. The traction-displacement laws of the normal direction \(n\) and the tangent directions \(t\) and \(n\) of the interface satisfy the following:  
\[ \begin{array}{c}T_{n}=K(1-D)\delta_{n}\\ T_{t}=K(1-D)\delta_{t}\\ T_{s}=K(1-D)\delta_{s}\end{array}  \tag{15}\]  
where \(K\) is the initial interface stiffness; \(\delta_{n},\delta_{t},\delta_{s}\) denote the displacement component; \(T_{n},T_{t},T_{s}\) represent the stress component; and the damage factor \(D\) is as follows:  
\[D=\left\{ \begin{array}{c}0,\ \delta\leq\delta^{0}\\ \frac{\delta^{\prime}(\delta-\delta^{0})}{\delta(\delta^{\prime}-\delta^{0})},\delta>\delta^{0}\end{array} \right. \tag{16}\]  
where \(T_{c}\) is the maximum fracture stress, \(\delta^{0}\) denotes the critical opening displacement of the interface, \(\delta\) represents the opening displacement, and \(\delta^{\prime}\) is interface failure displacement. In addition, this project used the MAXS fracture criterion to describe the damage evolution of cohesive elements. \[\max\left\{\frac{\langle T_{n}\rangle}{T_{n}^{c}},\frac{T_{s}}{T_{s}^{c}}, \frac{T_{t}}{T_{t}^{c}}\right\}=1 \tag{17}\]  
where \(T_{n}^{c},T_{s}^{c},T_{t}^{c}\) represent the peak values of the nominal stress when the deformation is either purely normal to the interface or purely in the first or the second shear direction, respectively. The bilinear model of cohesive elements in the commercial software Abaqus mainly includes three parameters: \(K,T_{c},\delta^{\prime}\). Since the interface parameters between the micro-particles of the propellant and the binder matrix could not be obtained directly by experiments but the exact value of the parameters needs to be obtained, an inverse analysis method was adopted to solve the parameters by using the experimental results such as the macroscopic stress-strain curve. This method needs to constantly adjust the parameters in the numerical simulation to achieve a good agreement between the numerical simulation and the experimental results. Additionally, the adjusted parameter values are regarded as accurate values of the parameters. In recent years, the inverse analysis method has been widely used in many fields such as structural optimization and material parameter identification. In this paper, the experimental stress-strain curve is expressed as a set of a certain number of points:  
\[F(x)=(x_{i},F(x_{i}))\ i=1,2,3,\cdots,n \tag{18}\]  
Next, the stress-strain curve calculated by Abaqus is also expressed as a set of \(n\) points:  
\[Q(x)=(x_{i},Q(x_{i}))\ i=1,2,3,\cdots,n \tag{19}\]  
The objective interpolation function, which can reflect the closeness between experiment and numerical simulation, is expressed as follows:  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline C\({}_{10}\) & C\({}_{01}\) & C\({}_{20}\) & C\({}_{11}\) & C\({}_{02}\) & \(D_{1}\) & \(D_{2}\) \\ \hline
0.1481 & 0 & 0.011527 & 0 & 0 & 0 & 0 \\ \hline \(E_{1}\) & \(\tau_{1}\) & \(E_{2}\) & \(\tau_{2}\) & & & \\\[\text{min}R=\frac{1}{n}\sum_{i=1}^{n}[F(x_{i})-Q(x_{i})]^{2}\;i=1,2,3,\cdots,n \tag{20}\]  
where \(n\) is the number of evenly distributed discrete points, \(i\) represents the i-th point, \(F(x_{i})\) denotes the experimental result value of the \(i\)-th point, and \(Q(x_{i})\) is the simulation result of the i-th point. The number of discrete points selected in this paper is 34.