## 3 Uncertainty Quantification of Macro Elastic Parameters Based on Non-Probabilistic Thought
Here, the interval arithmetic operations of two interval numbers \(b_{i}\in[\underline{b_{i}},\overline{b_{i}}]\) and \(b_{j}\in[\underline{b_{j}},\overline{b_{j}}]\) can be expressed as  
\[[\underline{b_{i}},\overline{b_{i}}]+[\underline{b_{j}},\overline{b_{j}}]=[ \underline{b_{i}}+\underline{b_{j}},\overline{b_{i}}+\overline{b_{j}}] \tag{13}\]  
\[[\underline{b_{i}},\overline{b_{i}}]-[\underline{b_{j}},\overline{b_{j}}]=[ \underline{b_{i}}-\overline{b_{j}},\overline{b_{i}}-\underline{b_{j}}] \tag{14}\]  
\[[\underline{b_{i}},\overline{b_{i}}]\cdot[\underline{b_{j}},\overline{b_{j}}]= [\min\Bigl{\{}\underline{b_{i}}\cdot\underline{b_{j}},\underline{b_{i}}\cdot \overline{b_{j}},\overline{b_{i}}\cdot\underline{b_{j}},\overline{b_{i}}\cdot \overline{b_{j}}\Bigr{\}},\max\Bigl{\{}\underline{b_{i}}\cdot\underline{b_{j}},\underline{b_{i}}\cdot\overline{b_{j}},\overline{b_{i}}\cdot\underline{b_{ j}},\overline{b_{i}}\cdot\overline{b_{j}}\Bigr{\}}] \tag{15}\]  
\[[\underline{b_{i}},\overline{b_{i}}]/[\underline{b_{j}},\overline{b_{j}}]=[ \min\Bigl{\{}\underline{b_{i}}/\underline{b_{j}},\underline{b_{j}},\overline{b _{i}}/\overline{b_{j}},\overline{b_{i}}/\underline{b_{j}},\overline{b_{i}}/ \overline{b_{j}}\Bigr{\}},\max\Bigl{\{}\underline{b_{j}}/\underline{b_{j}}, \underline{b_{j}}/\overline{b_{j}},\overline{b_{i}}/\underline{b_{j}},\overline {b_{j}}/\overline{b_{j}}\Bigr{\}}] \tag{16}\]  
with \(i,j=1,2,\ldots,m\). As a result, the lower and upper bounds of macro elastic mechanical parameters can be calculated. However, it should be noted that a natural interval expansion of functions may occur. That is to say, the obtained interval will be overlarge. Therefore, one can take good advantage of the parameter vertex solution theorem to precisely calculate the lower and upper bounds of macro elastic mechanical parameters. In other words, one can obtain that:  
\[ \begin{array}{l}\underline{y}=\min\bigl{\{}f(\hat{\mathbf{b}})\bigr{\}}=\min \Bigl{\{}f(\hat{b}_{1},\hat{b}_{2},\cdots,\hat{b}_{m})\Bigr{\}}\\ \overline{y}=\max\bigl{\{}f(\hat{\mathbf{b}})\bigr{\}}=\max\Bigl{\{}f(\hat{b}_{1}, \hat{b}_{2},\cdots,\hat{b}_{m})\Bigr{\}}\end{array}  \tag{17}\]  
where \(\hat{\mathbf{b}}\) is the uncertain micro input parameter vector including all the vertices. Here, for the m-dimensional interval variables, there exist \(2^{m}\) vertices, in which  
\[\hat{b}_{i}=\underline{b_{i}}\ or\ \hat{b}_{i}=\overline{b_{i}}\ i=1,2, \cdots,m \tag{18}\]  
The parameter vertex solution theorem can provide an exact solution for macro elastic parameters of composites because of the monotonic functions. However, the two aforementioned methods cannot take the correlations between any two uncertain mechanical properties into account. Furthermore, one can make use of the Monte Carlo simulation (MCS) method to carry out uncertainty quantification of macro elastic parameters, where a lot of sample points in feasible domains of interval variables are generated. The MCS method refers to the procedure of assigning a value to an independent interval variable in a chosen quantification model and dependent output variables can be observed and modeled. As a result, the distributed domains of macro elastic mechanical parameters for composites can be obtained and the correlations between elastic parameters can be predicted. The procedure of the MCS method is as follows:  
1. Based on non-probabilistic thoughts, make use of the computer and sampling techniques to generate and select sample points of the independent input variables, such as constituent elastic properties, fiber and void volume ratios and model parameters. 2. Calculate dependent output variables including macro elastic mechanical properties \(E_{1},E_{2},v_{12}\) and \(G_{12}\) by using the chosen quantification model. 3. Replicate the experiment and each time adopt a new set of input parameters. 4. Take good advantage of appropriate analysis methods to calculate characteristic parameters and distributions of the macro elastic properties of composite laminates.