## 2 Development of Full-Process Numerical Analyzing Method
### Development of the Progressive Residual Strength Prediction Method  
#### 2.2.1 Stress Analysis of Composite of Residual Strength  
Consider the composite laminates have been loaded incrementally up to the \(n\)th step. By only considering thermal stress and external load, the equilibrium equation at the current load \(p^{n}\) can be expressed, as follows:  
\[\sigma_{ij,j}^{n}=0\ (i,j=1,2,3;\text{ in }V_{n-1}) \tag{10}\]  
\begin{table}
\begin{tabular}{c c} \hline
**Damage Modes** & **Expression** \\ \hline Fiber tension failure (FTF) & \(\left(\frac{\sigma_{1}}{X_{T}}\right)^{2}+\left(\frac{\sigma_{12}}{S_{f}} \right)^{2}+\left(\frac{\sigma_{13}}{S_{f}}\right)^{2}\geq 1\ (\sigma_{1}\geq 0)\) \\ Fiber compression failure (FCF) & \(\left(\frac{\sigma_{1}}{X_{C}}\right)^{2}+\left(\frac{\sigma_{12}}{S_{f}} \right)^{2}+\left(\frac{\sigma_{13}}{S_{f}}\right)^{2}\geq 1\ (\sigma_{1}<0)\) \\ Matrix tension failure (MTF) & \(\left(\frac{\sigma_{1}}{Y_{T}}\right)^{2}+\left(\frac{\sigma_{12}}{S_{12}} \right)^{2}+\left(\frac{\sigma_{13}}{S_{23}}\right)^{2}\geq 1\ (\sigma_{2}>0)\) \\ Matrix compression failure (MCF) & \(\frac{1}{4}\left(\frac{\sigma_{12}}{S_{12}}\right)^{2}+\left(\frac{\sigma_{12} }{S_{12}}\right)^{2}+\frac{Y_{C}\sigma_{12}}{S_{12}S_{12}S_{12}S_{12}S_{12}S_{1 2}S_{  
The force boundary condition can be written, as follows:  
\[(\sigma_{ij,j}^{n})n_{j}^{n}-\overline{T}_{i}^{n}=0(\text{on }S_{\sigma}) \tag{11}\]  
According to the incremental virtual displacement principle, transforming the Equations (10) and (11) as  
\[\int_{V_{n-1}}\sigma_{ij}^{n}\delta\big{(}\Delta\varepsilon_{ij}\big{)}\text{ d}V-\int_{S_{\sigma}}\overline{T}_{i}^{n}\delta(\Delta u_{i})\text{d}S=0 \tag{12}\]  
where \(\sigma_{ij}^{n}\) is mechanical stress in each material principal direction at step \(n\), \(V_{n-1}\) is the configuration of the body at step \(n\), and \(\Delta\varepsilon_{ij}\) and \(\Delta u_{i}\) are the increment of strain and displacement between the \((n-1)\)-th and \(n\)-th load step, respectively. The total stresses \(\sigma_{ij}^{n}\) can be expressed by the sum of previous stresses and the stress tensor increments, as follows:  
\[\sigma_{ij}^{n}=\sigma_{ij}^{n-1}+\Delta\sigma_{ij}^{n} \tag{13}\]  
Substituting Equation (13) into Equation (12), the following expression can be obtained:  
\[\int_{V_{n-1}}\Delta\sigma_{ij}^{n}\delta\big{(}\Delta\varepsilon_{ij}\big{)} \text{ d}V=\int_{S_{\sigma}}\overline{T}_{i}^{n}\delta(\Delta u_{i})\text{d}S -\int_{V_{n-1}}\sigma_{ij}^{n-1}\delta\big{(}\Delta\varepsilon_{ij}\big{)} \text{ d}V \tag{14}\]  
Moreover, the increment of strain \(\Delta\varepsilon_{ij}\) in the loading process can be expressed as:  
\[\Delta\varepsilon_{ij}=\Delta\varepsilon_{ij}+\Delta\eta_{ij}, \tag{15}\]  
where \(\Delta\varepsilon_{ij}\) and \(\Delta\eta_{ij}\) are the linear and quadratic term, respectively. Assuming that the increment of the external load is small enough, the stress-strain relationship between any two adjacent load steps can be considered linear, i.e.,  
\[\Delta\sigma_{ij}^{n}=Q_{ijlk}^{n-1}\Delta\varepsilon_{kl} \tag{16}\]  
where \(Q_{ijlk}^{n-1}\) is the reduced material stiffness matrix element at step \(n-1\). Subsequently, substituting Equations (15) and (16) into Equation (14) and omitting high-order items, one obtains  
\[\int_{V_{n-1}}Q_{ijlk}^{n-1}\Delta\varepsilon_{kl}\delta\big{(}\Delta \varepsilon_{ij}\big{)}\text{ d}V+\int_{V_{n-1}}\sigma_{ij}^{n-1}\delta\big{(} \Delta\eta_{ij}\big{)}\text{ d}V=\int_{S_{\sigma}}\overline{T}_{i}^{n}\delta( \Delta u_{i})\text{d}S-\int_{V_{n-1}}\sigma_{ij}^{n-1}\delta\big{(}\Delta \varepsilon_{ij}\big{)}\text{ d}V \tag{17}\]  
Since the material properties, \(Q_{ijlk}^{n-1}\), depend on the current stresses and strains, Equation (17) has to be solved by a finite element discrete method combined with the Newton-Raphson iteration scheme.