## 3 Pull-Out Performance of Cephalomedullary Nail
### Finite Element Analysis with Damage Model  
In order to precisely evaluate the pull-out behavior of the synthetic bone material, commercial FEM software, ABAQUS/Explicit, has been utilized. The screw and helical blade are pre-installed in the synthetic bone material as demonstrated in Figure 8. Since the porous materials show substantially different material behaviors in tension and compression test as represented in Figure 4, it is highly required to adopt the special material model which can express tension-compression anisotropy characteristics. In this paper, the concrete damaged plasticity model [26,28,29] has been applied to numerical simulation for the pull-out test, which does not only represent tensile and compressive material behaviors, simultaneously, but also describes the degradation of material hardening for failure, successfully. The concrete damaged plasticity model assumes the non-associated potential plastic flow where the flow potential, G, used for this model is the Drucker-Prager hyperbolic function as expressed in Equation (1). \[G=\sqrt{\left(e\sigma_{t0}tany\right)^{2}+\overline{q}^{2}}-\overline{p}tan\psi \tag{1}\]  
Figure 6: Pull-out specimen at the final failure: (**a**) screw (0.14 g/cm\({}^{3}\)); (**b**) helical blade (0.14 g/cm\({}^{3}\)); (**c**) screw (0.35 g/cm\({}^{3}\)); (**d**) helical blade (0.35 g/cm\({}^{3}\)). Figure 7: Comparison of peak force in pull-out test. \(\psi(\theta,f_{i})\) is the dilation angle measured in the p-q plane at high confining pressure, \(\sigma_{t0}(\theta,f_{i})=\sigma_{t}|_{\frac{\theta^{pl}}{\theta_{i}}=0,\tilde{ \varepsilon}_{t}^{pl}=0}\) is the uniaxial tensile stress at failure, \(\epsilon(\theta,f_{i})\) is a parameter, referred to as the eccentricity, which defines the rate at which the function approaches the asymptote, \(\overline{p}=-\frac{1}{3}\sigma\mathbf{I}\) is the effective hydrostatic pressure, \(\sigma\) is the effective stress, \(\overline{q}=\sqrt{\frac{3}{2}(S:S)}\) is the von Mises equivalent stress, where \(S=\sigma+\ \overline{p}\mathbf{I}\) is the deviator of the Cauchy stress tensor, \(\sigma\), [30; 31; 32]. In terms of effective stresses, the yield function, \(F\), takes into consideration the yield strength of tension and compression with the form as follows [31; 33],  
\[F=\frac{1}{1-\alpha}\Big{(}\overline{q}-3\alpha\overline{p}+\beta\Big{(} \tilde{\varepsilon}^{pl}\Big{)}\!<\!\widehat{\sigma}_{max}>-\gamma\!<\! \widehat{\sigma}_{max}>\Big{)}-\overline{\sigma}_{c}\Big{(}\tilde{ \varepsilon}^{pl}\Big{)}=0 \tag{2}\]  
\[\alpha=\frac{\Big{(}\frac{\sigma_{t0}}{\sigma_{t0}}\Big{)}-1}{2\Big{(}\frac{ \sigma_{t0}}{\sigma_{t0}}\Big{)}-1};0\leq\alpha\leq 0.5,\beta=\frac{\overline{ \sigma}_{c}\Big{(}\tilde{\varepsilon}_{t}^{pl}\Big{)}}{\overline{\sigma}_{t} \Big{(}\tilde{\varepsilon}_{t}^{pl}\Big{)}}(1-\alpha)-(1+\alpha),\ \gamma=\frac{3(1-K_{C})}{2K_{c}-1} \tag{3}\]  
where \(\tilde{\varepsilon}^{pl}\) is the equivalent plastic strain tensor, \(\widehat{\sigma}_{max}\) is the maximum principal effective stress, \(\frac{\sigma_{t0}}{\sigma_{t0}}\) is the ratio of initial equibiaxial compressive yield stress to initial uniaxial compressive yield stress, and \(K_{C}\) is the ratio of the second stress invariant on the tensile meridian, \(\overline{\sigma}_{t}\) and \(\overline{\sigma}_{C}\) are the effective tensile and compressive cohesion stress, respectively. In addition, \(\alpha\) and \(\gamma\) both are dimensionless material constants. To represent the degradation of the material, the concrete damaged plasticity model adopts the constitutive curves for uniaxial damage in tension and compression, separately, as shown in Figure 9. The evolution of the compressive damage parameter, \(d_{c}\), is linked to the corresponding plastic strain \(\tilde{\varepsilon}_{c}^{pl}\) which is determined proportionally to the inelastic strain \(\tilde{\varepsilon}_{c}^{in}=\varepsilon_{c}-\sigma_{c}/E_{0}\) using a constant factor \(b_{c}\) with \(0<b_{c}\leq 1\)[34].