# Design and Mechanical Properties of Negative Poisson's Ratio Structure-Based Topology Optimization
## 3 Structural Geometric Construction and Equivalent Elastic Properties
### Theoretical Calculation  
The corner deformation compatibility condition at free end D satisfies:  
\[\delta_{11}M_{0}+\delta_{1F}=0 \tag{14}\]  
In the given formula, \(\delta_{1F}\) represents the angle of rotation at point D when it is solely subjected to the load F/2 and \(\delta_{11}\) denotes the angle of rotation at point D when it is subjected to a unit moment specifically applied at point D.  
As shown in Figure 9c, the bending moments of segments AB, BC, and CD can be calculated when only subjected to F/2. As shown in Figure 9d, when only subjected to a unit moment, the bending moments of each segment can be obtained. Using [PERSON]'s theorem [45] to perform Mohr integral for each beam, \(\delta_{1F}\) and \(\delta_{11}\) can be respectively determined:  
\[M_{AB}(x_{1})=\tfrac{F}{2}(\text{bcos}\alpha-\text{cos}\beta)+\tfrac{F}{2}x_{1 }\quad x_{1}\in(0,\text{a})\quad\overline{M_{AB}}(x_{1})=-1; \tag{15}\]  
\[M_{BC}(x_{2})=-\tfrac{F}{2}c\cos\beta+\tfrac{F}{2}x_{2}\cos\alpha\quad x_{2} \in(0,\text{b})\quad\overline{M_{BC}}(x_{2})=-1; \tag{16}\]  
\[M_{CD}(x_{3})=-\tfrac{F}{2}x_{3}\cos\beta\qquad x_{3}\in(0,\text{c})\quad \overline{M_{BC}}(x_{3})=-1; \tag{17}\]  
Figure 8: Concave beetle-shaped structure. \[\delta_{1F} =\int_{0}^{a}\frac{M_{AB}(x_{1})\overline{M_{AB}}(x_{1})}{E_{m}l_{m} }dx_{1}+\int_{0}^{b}\frac{M_{BC}(x_{2})\overline{M_{BC}}(x_{2})}{E_{m}l_{m}}dx_{ 2}+\int_{0}^{c}\frac{M_{CD}(x_{3})\overline{M_{CD}}(x_{3})}{E_{m}l_{m}}dx_{3} \tag{18}\] \[\delta_{11} =\int_{0}^{a}\frac{\overline{M_{AB}}(x_{1})\overline{M_{AB}}(x_{1} )}{E_{m}l_{m}}dx_{1}+\int_{0}^{b}\frac{\overline{M_{BC}}(x_{2})\overline{M_{BC} }(x_{2})}{E_{m}l_{m}}dx_{2}+\int_{0}^{c}\frac{\overline{M_{CD}}(x_{3})\overline {M_{CD}}(x_{3})}{E_{m}l_{m}}dx_{3} \tag{19}\]  
In Equations (18) and (19), \(E_{\rm m}\) represents the elastic modulus of the material and \(I_{\rm m}\) represents the moment of inertia of the cross-section with respect to the neutral axis. From Equations (18) and (19), it follows that:  
\[M_{0}=-\frac{\delta_{1F}}{\delta_{11}}=\frac{F}{4}\left(\frac{a^{2}+b^{2}\cos \alpha-c^{2}\cos\beta+2 ab\cos\alpha-2 ac\cos\beta-2 bc\cos\beta}{a+b+c}\right)=F\theta \tag{20}\]  
where \(\theta=\frac{1}{4}\left(\frac{a^{2}+b^{2}\cos\alpha-c^{2}\cos\beta+2 ab\cos \alpha-2 ac\cos\beta-2 bc\cos\beta}{a+b+c}\right)\). According to [PERSON]'s theorem [46], the displacement deformation of each segment in the \(y\)-axis direction can be calculated under the symmetric load of F/2 in the \(y\)-axis direction:  
\[ \begin{array}{l}M_{AB}(x_{1})=\frac{F}{2}(\mathrm{bcos}\alpha-\cos\beta)+ \frac{F}{2}x_{1}-M_{0}\quad x_{1}\in(0,\mathrm{a})\\ \overline{M_{AB}}(x_{1})=(\mathrm{bcos}\alpha-\mathrm{ccos}\beta)+x_{1}-2\theta ;\end{array}  \tag{21}\]  
\[ \begin{array}{l}M_{BC}(x_{2})=-\frac{F}{2}\mathrm{ccos}\beta+\frac{F}{2}x_{ 2}\cos\alpha-M_{0}\quad x_{2}\in(0,\mathrm{b})\\ \overline{M_{BC}}(x_{2})=-\mathrm{ccos}\beta+x_{2}\cos\alpha-2\theta;\end{array}  \tag{22}\]  
\[ \begin{array}{l}M_{CD}(x_{3})=-\frac{F}{2}x_{3}\cos\beta-M_{0}\quad\quad x_ {3}\in(0,\mathrm{c})\\ \overline{M_{CD}}(x_{3})=-x_{3}\cos\beta-2\theta;\end{array}  \tag{23}\]  
Figure 9: (**a**) The \(1/4\) cellular structure. (**b**) The internal forces of a \(1/4\) structure. (**c**) The \(1/4\) structure when only F/2 load is applied. (**d**) The \(1/4\) structure when acting only on unit bending moment. (**e**) Only horizontal unit forces acting on \(1/4\) of the structure. \[\Delta_{YY}=\frac{2}{E_{m}I_{m}}\left[\int_{0}^{a}M_{AB}(x_{1})\overline{M_{AB}}(x_ {1})dx_{1}+\int_{0}^{b}M_{BC}(x_{2})\overline{M_{BC}}(x_{2})dx_{2}+\int_{0}^{c}M _{CD}(x_{3})\overline{M_{CD}}(x_{3})dx_{3}\right] \tag{24}\]  
\[\Delta_{YY}=\frac{F}{3E_{m}I_{m}}\left[