工程科学与技术   2017, Vol. 49 Issue (3): 209-216

Computer-aided Modeling Analysis and Manufacturing Approach for Silicone Prosthesis
WANG Su, WANG Pengju, XU Liang, YANG Yangyang, CHENG Donghua, WU Xiaonan
School of Manufacturing Sci. and Eng., Sichuan Univ., Chengdu 610065, China
Abstract: Computer-aided manufacturing approach for silicone prosthesis with two-component silicone rubber was studied and modeled by 3D modeling.The model was based on temperature,material parameters and geometric features for body and normal mice.The temperature difference between mold and prosthesis surface was discussed by numerical simulation on the temperature field of die and prosthesis.The feasibility of the modeling in repairing the soft tissue defect of chest wall was validated by applying phantom load from softness and fatigue life.The verification was based on finite element method.The test feasibility was based on printing material and manufacturing process.The mold were manufacturing by 3D printing.The test were carried out based on the effects of mucosal and crosslinking reaction efficiency in die at 5,25 and 60 ℃.The effects of prosthesis softness,viscosity and stripping time were discussed by choosing different ratio of silicone and curing agent and temperature.The ratio and temperature were determined by optimizing data in the least square method and examing the regression equation prominence.The results showed that the prosthesis is affected by ratio,but not by temperature within certain range.The soft prostheses are generated when the ratio is set in 10∶0.43,and at 55 ℃.
Key words: silicone prosthesis    3D modeling    finite element method    3D printing    ratio    mold temperature    optimization analysis

1 参数化建模

1.1 3维模型构建

 图1 乳房假体植入术的临床效果 Fig. 1 Clinical effect of breast prosthesis implantation

 图2 模具截面 Fig. 2 Section of mould

 图3 假体模型 Fig. 3 Prosthesis model

 $r = \frac{{{a^2} + {b^2}}}{{2a}}$ (1)
 $S = 2{\rm{\pi }}ar$ (2)
 $V = {\rm{\pi }}a{r^2} - \frac{{{\rm{\pi }}{r^3}}}{3} + \frac{{{\rm{\pi }}{{(r - a)}^3}}}{3}$ (3)
 $m = \rho V$ (4)

 图4 设计流程图 Fig. 4 Flow chat of design

 图5 模具模型 Fig. 5 Mold model

1.2 模具温度场模拟

 ${\rm{0 = }}{s_1} < {s_2} < \cdot \cdot \cdot < {s_i} < \cdot \cdot \cdot < {s_M} = H,$

 ${\rm{0 = }}{\delta _1} < {\delta _2} < \cdot \cdot \cdot < {\delta _i} < \cdot \cdot \cdot < {\delta _N} = H,$

 $\,\,\,\,\,\,\,{\rm{0 = }}{\tau _1} < {\tau _2} < \cdot \cdot \cdot < {\tau _j} < \cdot \cdot \cdot < {\tau _L} = H\text{。}$

 $K\frac{{{\partial ^2}T}}{{\partial {s^2}}} = \rho c\frac{{\partial T}}{{\partial \tau }}$ (5)
 $K\frac{{{\partial ^2}T'}}{{\partial {\delta ^{\rm{2}}}}} = \rho c\frac{{\partial T'}}{{\partial \tau }}$ (6)

$T_i^n$ ${T^n}'$ 依次表示tn ${t'_n}$ 时刻siδi处模温，式（5）、（6）分别由中心差商和向前差商近似得：

 $\left\{ \begin{array}{l}\frac{\displaystyle{T_{i + 1}^n - 2T_i^n + T_{i - 1}^n}}{{{\displaystyle{\left( {\Delta s} \right)}^2}}} - \frac{\displaystyle{\rho c}}{\displaystyle{K}}\frac{\displaystyle{T_i^{n + 1} - T_i^n}}{\displaystyle{\Delta \tau }} = 0 ,\\[9pt]\frac{\displaystyle{{{T}_{i + 1}^n}\!\!\!\!\!'- {{2T}_i^n}\!' + {T}_{i - 1}^n}\!\!\!\!\!'}{{{\displaystyle{\left( {\Delta \delta } \right)}^2}}} - \frac{\displaystyle{\rho c}}{\displaystyle{K}}\frac{\displaystyle{{{T}_i^{n + 1}}\!' - {{T}_i^n}\!'}}{\displaystyle{\Delta \tau }} = 0\end{array} \right.$ (7)

1.3 制品温度场模拟

 图6 球坐标系下的导热微分方程 Fig. 6 Heat conduction differential equation in spherical coordinate system

 \begin{aligned}[b]& \frac{\displaystyle{1}}{{\displaystyle{r^2}}}\frac{\displaystyle{\partial }}{\displaystyle{\partial r}}\left( {\lambda {r^2}\frac{\displaystyle{\partial T}}{\displaystyle{\partial r}}} \right) + \frac{\displaystyle{1}}{\displaystyle{{r^2}{\rm{si}}{{\rm{n}}^2}\theta }}\frac{\displaystyle{\partial }}{\displaystyle{\partial \varphi }}\left( {\lambda \frac{\displaystyle{\partial T}}{\displaystyle{\partial \varphi }}} \right) + \\[2pt]& \frac{\displaystyle{1}}{{\displaystyle{r^2}\sin \theta }}\frac{\displaystyle{\partial }}{\displaystyle{\partial \theta }}\left( {\lambda \sin \theta \frac{\displaystyle{\partial T}}{\displaystyle{\partial \theta }}} \right) = \rho c\frac{\displaystyle{\partial T}}{\displaystyle{\partial \tau }} \end{aligned} (8)

 ${T^ * }(A,B) = \frac{{pc}}{{4{\rm{\pi }}Kr\left( {A,B} \right)}}$ (9)

$\frac{\displaystyle{\partial T}}{\displaystyle{\partial \tau }} = \sum\limits_{j = 1}^N {{f_j}\left( X \right)\frac{\displaystyle{\partial {\beta _j}\left( T \right)}}{\displaystyle{\partial \tau }}}$ ，其中，坐标函数fj(X)为函数ψ的二阶导数， $\frac{\displaystyle{\partial {\psi _j}}}{\displaystyle{\partial n}} = {P_j}$ ${Q^ * } = \frac{\displaystyle{\partial {T^ * }}}{\displaystyle{\partial n}}$

 \begin{aligned}[b]& - \int_\varGamma {\left[ {{T^ * }\left( {A,B} \right) \cdot Q(B) - {Q^ * }(A,B)T\left( B \right)} \right]} {\rm{d}}s + \\& \varphi \left( A \right)T\left( A \right) = \frac{\displaystyle{\rho c}}{\displaystyle{K}}{ \sum\limits_{{{j = 1}}}^N } {\frac{\displaystyle{\partial {\beta _j}\left( T \right)}}{\displaystyle{\partial \tau }}} \times \\& \left\{ {\int_\varGamma {\left[ {{T^ * }\left( {A,B} \right) \cdot {P_j}(B) - {Q^ * }(A,B){\psi _j}\left( B \right)} \right]} {\rm{d}}s + } \right.\\& \left. {{\psi _j}(A){\varphi _j}(A)} \right\}\end{aligned} (10)

 $\varphi (A) = \left\{ \begin{array}{l}\!\!\! 1 , \;{A \in \varOmega }\text{；}\\[7pt]\!\!\! 1/2 , \; {A{\text{在}}\varGamma {\text{中且光滑}}}\text{；}\\[7pt]\!\!\! \theta /2{\rm{\pi }} , \; {A{\text{在}}\varGamma {\text{中不光滑且内角为}}\theta }\text{。}\end{array} \right.$

 $\frac{{\rho c}}{K}\left( {{\mathit{\boldsymbol{H\psi }}} + {\mathit{\boldsymbol{G\psi }}}} \right)\left( {\frac{{\partial \beta }}{{\partial \tau }}} \right) = {\mathit{\boldsymbol{HQ}}} + {\mathit{\boldsymbol{GT}}}$ (11)

 $T = {T_0}\left( {r,\varphi ,\theta } \right)$ (12)

 $\left\{ \begin{array}{l}\displaystyle\left( {\frac{{\partial {T_{\rm{i}}}}}{{{\partial _{\rm{n}}}}}} \right) = \frac{h}{k}\left( {{T_{\rm{a}}} - {T_{\rm{i}}}} \right),\\\displaystyle{T_{\rm{i}}} = {T_{\rm{e}}}\end{array} \right.$ (13)

1.4 计算结果

 图7 模具温度场 Fig. 7 Temperature field of mold

 图8 假体温度场 Fig. 8 Temperature field of prosthesis

2 有限元模型构建与分析

2.1 材料参数的设定

 图9 硅橡胶的疲劳寿命曲线 Fig. 9 Fatigue life curve of silicone rubber

2.2 有限元网格划分

 图10 有限元网格划分 Fig. 10 Finite Element Meshing

2.3 施加载荷及约束

 图11 假体表面受力图 Fig. 11 Forces FBD of prosthesis surface

2.4 求解结果及分析

 图12 Fy= –1 N时的应变云图 Fig. 12 Total deformation when Fy= –1 N

 图13 不同Fy作用时的应变值 Fig. 13 Corresponding total deformation when Fy take different value

 图14 Fx=1 N时的应变云图 Fig. 14 Total deformation when Fx=1 N

 图15 y=0.2 mm时的寿命分布图 Fig. 15 Life distribution when y=0.2 mm

3 模具制作 3.1 材料可行性分析

ABS树脂是丁二烯、苯乙烯和丙烯腈3种单体的接枝共聚物，具有优异的热稳定性和耐化学性能，熔点为175 ℃[3]。双组分液体硅橡胶通常由含氢硅油、乙烯基生胶及铂催化剂等组成。白凡士林系从石油中得到的多种烃的半固体混合物，熔点为45～60 ℃，相对密度为0.815～0.880[4]。白凡士林和ABS树脂对假体影响尚待研究。

3.2 模具制备

4 实验案例

 图16 硅胶假体制备流程图 Fig. 16 Flow chart of the preparation of silicone prosthesis

ABS型塑料模具具有热胀冷缩效应，其使用温度为–40～80 ℃，相应收缩率为0.4%～0.8%。硅胶假体制备过程中，需在室温下浇注硅胶与固化剂的混合液于涂抹有白凡士林的模具型腔，并静置一段时间，消除气泡。假体成型后体积有较小缩变，假体与模具间产生较小缝隙，导致粘模。因此假体成型前后模具温差不宜过大，以降低模具变形对实验的影响。温度低于5 ℃时，硅胶与固化剂不发生交联反应。文献[9]中，ABS树脂变形温度为62~95 ℃，玻璃化转变温度为90～100 ℃，医学上对假体表面精度要求较高，为防止模具显著变形，设置电加热套温度不超过60 ℃。模温取5～60 ℃时，假体成型前后模具温差变化在实验允许范围内，模具变形对制备假体影响较小。模温取室温时，假体成型前后模具温差近似不变，模具变形几乎为零，假体与模具间缝隙微小，便于脱模。

5 数据优化分析

 $Y{\rm{ = }}a\ln X + b,\;Z{\rm{ = }}c\ln X + d\text{。}$

 $\begin{array}{l}\varphi \left( {a,b} \right) = \sum\limits_{i = 1}^4 {{{\left( {a\ln {X_i} + b - {Y_i}} \right)}^2}}\text{，} \\[8pt]\varphi \left( {c,d} \right) = \sum\limits_{i = 1}^4 {{{\left( {c\ln {X_i} + d - {Z_i}} \right)}^2}} \text{。}\end{array}$

 $\left\{ \begin{array}{l}\!\!\!\!\frac{\displaystyle{\partial \varphi \left( {a,b} \right)}}{\displaystyle{\partial a}} = 2\sum\limits_{i = 1}^n {\left( {a\ln {X_i} + b - {Y_i}} \right)\ln {X_i}} = 0\text{，}\\[8pt]\!\!\!\! \frac{\displaystyle{\partial \varphi \left( {a,b} \right)}}{\displaystyle{\partial b}} = 2\sum\limits_{i = 1}^n {\left( {a\ln {X_i} + b - {Y_i}} \right)} = 0\end{array} \right.$ (14)
 $\left\{ \begin{array}{l}\!\!\!\! \frac{\displaystyle{\partial \varphi \left( {c,d} \right)}}{\displaystyle{\partial c}} = 2\sum\limits_{i = 1}^n {\left( {c\ln {X_i} + d - {Z_i}} \right)\ln {X_i}} = 0\text{，}\\[8pt]\!\!\!\! \frac{\displaystyle{\partial \varphi \left( {c,d} \right)}}{\displaystyle{\partial d}} = 2\sum\limits_{i = 1}^n {\left( {c\ln {X_i} + d - {Z_i}} \right)} = 0\end{array} \right.$ (15)

 $Y = - 117.6\ln X - 60.6$ (16)

 $Z = - 242.9\ln X - 107.86$ (17)

 $\begin{array}{l}\displaystyle{L_{XX}} = \sum\limits_{i = 1}^n {{{\left( {\ln {X_i}} \right)}^2}} - \frac{1}{n}{\left( {\sum\limits_{i = 1}^n {\ln {X_i}} } \right)^2} = 0.27,\\[10pt]\displaystyle{L_{YY}} = \sum\limits_{i = 1}^n {Y_i^2} - \frac{1}{n}{\left( {\sum\limits_{i = 1}^n {{Y_i}} } \right)^2} = 4\,\,033.50,\\[10pt]\displaystyle{L_{ZZ}} = \sum\limits_{i = 1}^n {Z_i^2} - \frac{1}{n}{\left( {\sum\limits_{i = 1}^n {{Z_i}} } \right)^2} = 18\,\,631.25\text{。}\end{array}$

 $\begin{array}{l}\displaystyle {L_{XY}} = \sum\limits_{i = 1}^n {\left( {\ln {X_i}} \right){Y_i}} - \frac{1}{n}\left( {\sum\limits_{i = 1}^n {\ln {X_i}} } \right)\left( {\sum\limits_{i = 1}^n {{Y_i}} } \right) = - 31.04,\\[10pt]\displaystyle {L_{XZ}} = \sum\limits_{i = 1}^n {\left( {\ln {X_i}} \right){Z_i}} - \frac{1}{n}\left( {\sum\limits_{i = 1}^n {\ln {X_i}} } \right)\left( {\sum\limits_{i = 1}^n {{Z_i}} } \right) = - 64.20\text{。}\end{array}$

 $\begin{array}{l}{R_1} = \frac{\displaystyle{{L_{XY}}}}{\displaystyle{\sqrt {{L_{XX}}} \sqrt {{L_{YY}}} }} = - 0.94,\\[12pt]{R_2} = \frac{\displaystyle{{L_{XZ}}}}{\displaystyle{\sqrt {{L_{XX}}} \sqrt {{L_{ZZ}}} }} = - 0.91,\end{array}$
 $\begin{array}{l}{F_1} = \left( {n - 2} \right)\frac{\displaystyle{R_1^2}}{\displaystyle{1 - R_1^2}} = 15.18,\\[12pt]{F_2} = \left( {n - 2} \right)\frac{\displaystyle{R_2^2}}{\displaystyle{1 - R_2^2}} = 9.63\text{。}\end{array}$

F[10]知：F0.1(1,2)=8.53；

 ${F_\alpha }(1,n - 2) = {F_{0.1}}(1,2) = 8.53 < {F_2} < {F_1}\text{。}$

6 结　论

1）模温为55 ℃时，一定范围内，假体的柔软度和黏性随配比增加降低。

2）配比为10∶0.43时，一定范围内，假体的柔软度和黏性随模温变化甚微。

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