Parametric modeling method and mechanical behavior of 3D woven composites with warp insertion
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摘要:
为了能够更为准确地描述机织复合材料的细观结构和力学行为，建立了一种适用于含衬经的三维机织复合材料的参数化模型构建方法，讨论了几何模型内部各参数之间的等式约束关系，并以此建立了几何约束模型以实现对几何模型的参数化描述。在几何约束模型的基础上，重点讨论了参数化模型的建立方法，以输入参数建立了几何模型与实际结构的映射关系，提高了几何模型对实际结构的还原程度，对材料的分析与设计具有指导意义。采用渐进损伤模型预测了三维机织复合材料的经向强度，并与试验结果进行对比。在拉伸载荷作用下，经纱和接结纱以纵向断裂失效为主，纬纱以横向开裂失效为主，富树脂区的失效模式为拉伸断裂，经向强度预测值为853.6 MPa，相对误差为2.3%，与试验结果基本吻合。
Abstract:To describe the microstructure and mechanical behavior of woven composites more accurately, a parametric model construction method for 3D woven composites with warp insertion was established. The equality constraints among the parameters in the geometric model were investigated, and the geometric constraint model was developed to realize the parametric description of the geometric model. Based on the geometric constraint model, the construction method of the parametric model was discussed, and the mapping relationship between the geometric model and the actual structure was established by input parameters, which not only improves the reduction degree of the geometric model to the actual structure but also provides guidance for material analysis and design. The progressive damage model was applied to predict the strength of 3D woven composites in the warp direction, and the prediction results were compared with the experimental results. Under the tensile load, the failure mode of the warp and binder yarn is mainly longitudinal fracture, the failure mode of the weft yarn is transverse fracture, and the failure mode of the resin-rich region is tensile fracture. The predicted strength of the material in the warp direction is 853.6 MPa, with a relative error of 2.3%, which is basically consistent with the experimental results.
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图 4 三维机织复合材料纱线主节点和纱线路径
Figure 4. The yarn master nodes and paths of 3D woven composites
图 8 不同输入参数的三维机织复合材料RVE模型
Figure 8. 3D woven composites RVE model with different input parameters
图 9 三维机织复合材料经向拉伸预测结果
Figure 9. Prediction results for 3D woven composite under tensile load in the warp direction
纱线 材料 纱线纤度/（g/km） 接结纱 TG800HXC-6K 250 经纱 TG800X-12K×2 1000 纬纱 TG800X-12K×2 1000 
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材料 参数 数值 T800碳纤维 纵向模量Ef11/GPa 294.0 横向模量Ef22/GPa 19.3 剪切模量Gf12/GPa 43.5 泊松比νf12 0.29 纵向拉伸强度Xft/MPa 5500 纵向拉伸失效应变$\varepsilon _{{\text{lt}}}^{{\mathrm{final}}}$ 0.0232 密度ρ/（g/cm3） 1.8 TDE-86树脂 树脂模量Em/GPa 3.55 树脂泊松比νm 0.33 拉伸强度Xmt/MPa 80.0 压缩强度Xmc/MPa 241.0
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参数 数值 纬纱机织密度Mwf/（根/cm） 4.0 经纱机织密度Mwp/（根/cm） 4.0 接结纱机织密度Mbi/（根/cm） 4.0 机织复合材料平均厚度T/mm 4.0 纬纱总层数Nwf/层 5 经纱总层数Nwp/层 4 接结纱总层数Nbi/层 4 表层纬纱横截面平均面积Awfs/mm2 0.551 内层纬纱横截面平均面积Awfi/mm2 0.852 经纱横截面平均面积Awp/mm2 0.760 接结纱横截面平均面积Abi/mm2 0.230 表层纬纱平均厚度Hwfs/mm 1.78 接结纱平均宽度Wbi/mm 0.81
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表 4 三维机织复合材料RVE模型参数
Table 4. Parameters of 3D woven composites RVE model
参数 数值 RVE长度/mm 5.0 RVE宽度/mm 5.0 RVE厚度/mm 4.0 表层纬纱宽度/mm 1.820 表层纬纱厚度/mm 0.303 内层纬纱宽度/mm 1.784 内层纬纱厚度/mm 0.559 经纱宽度/mm 1.767 经纱厚度/mm 0.430 接结纱宽度/mm 0.733 接结纱厚度/mm 0.314 表层接结纱倾斜角度/rad 1.109 内层接结纱倾斜角度/rad 1.056
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表 5 纱线路径曲线及横截面方向
Table 5. The yarn path curve and cross-section direction
纱线 Fi（t） Vi（t） Ni（t） 接结纱 $ {{\boldsymbol{F}}_i} （t） = \left\{ \begin{gathered} x{\boldsymbol{i}} + 0{\boldsymbol{j}} + {\textit{z}}{\boldsymbol{k}} （y = 0） \\ x{\boldsymbol{i}} + （{{{L_{{y}}}} / 2） }{\boldsymbol{j}} + {\textit{z}}{\boldsymbol{k}} （y = {{{L_{{y}}}} /2） } \\ x{\boldsymbol{i}} + {L_{{y}}}{\boldsymbol{j}} + {\textit{z}}{\boldsymbol{k}} （y = {L_{{y}}}） \\ \end{gathered} \right. $ $ {{\boldsymbol{V}}_i} （t） = u{\boldsymbol{i}} + 0{\boldsymbol{j}} + w{\boldsymbol{k}} （y = 0,{{{L_{{y}}}} /2},{L_{{y}}}） $ $ {{\boldsymbol{N}}_i} （t） = - w{\boldsymbol{i}} + 0{\boldsymbol{j}} + u{\boldsymbol{k}} （y = 0,{{{L_{{y}}}} /2},{L_{{y}}}） $ 经纱 $ {{\boldsymbol{F}}_i} （t） = \left\{ \begin{gathered} x{\boldsymbol{i}} + {{{L_{{y}}}} /4}{\boldsymbol{j}} + {\textit{z}}{\boldsymbol{k}} （y = {{{L_{{y}}}}/4}） \\ x{\boldsymbol{i}} + （{{3{L_{{y}}}} / 4}） {\boldsymbol{j}} + {\textit{z}}{\boldsymbol{k}} （y = {{3{L_{{y}}}} / 4） } \\ \end{gathered} \right. $ $ {{\boldsymbol{V}}_i} （t） = 1{\boldsymbol{i}} + 0{\boldsymbol{j}} + 0{\boldsymbol{k}} （y = {{{L_{{y}}}} / 4},{{3{L_{{y}}}} / 4}） $ $ {{\boldsymbol{N}}_i} （t） = 0{\boldsymbol{i}} + 0{\boldsymbol{j}} + 1{\boldsymbol{k}} （y = {{{L_{{y}}}} /4},{{3{L_{{y}}}}/ 4}） $ 纬纱 $ {{\boldsymbol{F}}_i} （t） = \left\{ \begin{gathered} 0{\boldsymbol{i}} + y{\boldsymbol{j}} + {\textit{z}}{\boldsymbol{k}} （x = 0） \\ （{{{L_{{x}}}}/2}） {\boldsymbol{i}} + y{\boldsymbol{j}} + {\textit{z}}{\boldsymbol{k}} （x = {{{L_{{x}}}} /2） } \\ {L_{{x}}}{\boldsymbol{i}} + y{\boldsymbol{j}} + {\textit{z}}{\boldsymbol{k}} （x = {L_{{x}}}） \\ \end{gathered} \right. $ $ {{\boldsymbol{V}}_i} （t） = 0{\boldsymbol{i}} + 1{\boldsymbol{j}} + 0{\boldsymbol{k}} （x = 0,{{{L_{{x}}}} / 2},{L_{{x}}}） $ $ {{\boldsymbol{N}}_i} （t） = 0{\boldsymbol{i}} + 0{\boldsymbol{j}} + 1{\boldsymbol{k}} （x = 0,{{{L_{{x}}}} /2},{L_{{x}}}） $
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表 6 几何模型输入参数
Table 6. Input parameters of the geometric model
模型 细观结构输入参数 宏观结构输入参数 几何模型[26] ①表层纬纱宽度、②表层纬纱厚度、③内层纬纱宽度、④内层纬纱厚度、
⑤经纱厚度、⑥经纱宽度、⑦接结纱宽度、⑧接结纱厚度、
⑨表层接结纱倾斜角度、⑩内层接结纱倾斜角度参数化模型 ①表层纬纱面积、②内层纬纱面积、③经纱横截面面积、
④接结纱横截面面积、⑤接结纱宽度、⑥纬纱机织密度、⑦经纱机织密度、
⑧接结纱（经纱）层数、⑨材料厚度
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损伤模式 判定准则 纱线纵向拉伸损伤 ${\varphi _{{\text{lt}}}} = { （ {{{{\sigma _{11}}} / {{X_{\text{t}}}}}} ） ^2} + {{ （ {\tau _{12}^2 + \tau _{13}^2} ） } /{S_{{\text{lt}}}^2}} \geqslant 1$ 纱线纵向压缩损伤 ${\varphi _{{\text{lc}}}} = { （ {{{{\sigma _{11}}} \mathord{\left/ {\vphantom {{{\sigma _{11}}} {{X_{\text{c}}}}}} \right. } {{X_{\text{c}}}}}} ） ^2} \geqslant 1{\text{ }}$ 纱线横向拉伸损伤 $ {\varphi _{{\text{tt}}}} = {[ {{{ （ {{\sigma _{22}} + {\sigma _{33}}} ） } / {{Y_{\text{t}}}}}} ]^2} + {{ （ {\tau _{23}^2 - {\sigma _{22}}{\sigma _{33}}} ） } /{S_{{\text{tz}}}^2}} + {{ （ {\tau _{12}^2 + \tau _{13}^2} ） } /{S_{{\text{lt}}}^2}} \geqslant 1 $ 纱线横向压缩损伤 $ {\varphi _{{\text{tc}}}} =[{{{Y_{\text{c}}^2}/ {{{ （ {2{S_{{\text{tz}}}}} ） }^2}}} - 1} ]{{ （ {{\sigma _{22}} + {\sigma _{33}}} ） }/ {{Y_{\text{c}}}}} + {{{{ （ {{\sigma _{22}} + {\sigma _{33}}} ） }^2}}/ {{{ （ {2{S_{{\text{tz}}}}} ） }^2}}} + {{ （ {\tau _{23}^2 - {\sigma _{22}}{\sigma _{33}}} ） } / {S_{{\text{tz}}}^2}} + {{ （ {\tau _{12}^2 + \tau _{13}^2} ） } /{S_{{\text{lt}}}^2}} \geqslant 1{\text{ }} $ 树脂基体拉伸损伤 $ {\varphi _{{\text{mt}}}} = {{\left[ {3{J_2} + {I_1} （ {{X_{{\text{mc}}}} - {X_{{\text{mt }}}}} ） } \right]}/ { （ {{X_{{\text{mc}}}}{X_{{\text{mt}}}}} ） }} \geqslant 1 $ 树脂基体压缩损伤 $ {\varphi _{{\text{mc}}}} = {{\left[ {3{J_2} + {I_1} （ {{X_{{\text{mc}}}} - {X_{{\text{mt }}}}} ） } \right]} / { （ {{X_{{\text{mc}}}}{X_{{\text{mt}}}}} ） }} \geqslant 1 $ 注：σii和τij （i, j = 1, 2, 3）为正应力与切应力分量，I1为应力第一不变量，J2为应力偏量第二不变量；Xt和Xc为纱线纵向拉伸和压缩强度，Yt和Yc为纱线横向拉伸和压缩强度，Slt、Stz为纱线剪切强度，Xmt和Xmc为树脂基体的拉伸和压缩强度。
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表 8 不同损伤模式的等价位移和等价应力[34]
Table 8. Equivalent displacement and equivalent stress for different damage modes[34]
损伤模式 等价位移 等价应力 纱线纵向拉伸损伤 $\chi _{{\text{eq}}}^{{\text{lt}}} = \sqrt {\varepsilon _{11}^2 + \varepsilon _{12}^2 + \varepsilon _{13}^2} {l_{{\text{eq}}}}$ $\sigma _{{\text{eq}}}^{{\text{lt}}} = {{\left( {{\sigma _{11}}{\varepsilon _{11}} + {\tau _{12}}{\varepsilon _{12}} + {\tau _{13}}{\varepsilon _{13}}} \right){l_{{\text{eq}}}}} \mathord{\left/ {\vphantom {{\left( {{\sigma _{11}}{\varepsilon _{11}} + {\tau _{12}}{\varepsilon _{12}} + {\tau _{13}}{\varepsilon _{13}}} \right){l_{{\text{eq}}}}} {\chi _{{\text{eq}}}^{{\text{lt}}}}}} \right. } {\chi _{{\text{eq}}}^{{\text{lt}}}}}$ 纱线纵向压缩损伤 $\chi _{{\text{eq}}}^{{\text{lc}}} = \left\langle { - {\varepsilon _{11}}} \right\rangle {l_{{\text{eq}}}}$ $ \sigma _{{\text{eq}}}^{{\text{lc}}} = {{\left\langle { - {\sigma _{11}}} \right\rangle \left\langle { - {\varepsilon _{11}}} \right\rangle {l_{{\text{eq}}}}} \mathord{\left/ {\vphantom {{\left\langle { - {\sigma _{11}}} \right\rangle \left\langle { - {\varepsilon _{11}}} \right\rangle {l_{{\text{eq}}}}} {\chi _{{\text{eq}}}^{{\text{lc}}}}}} \right. } {\chi _{{\text{eq}}}^{{\text{lc}}}}} $ 纱线横向拉伸损伤 $\chi _{{\text{eq}}}^{{\text{tt}}} = \sqrt {\varepsilon _{22}^2 + \varepsilon _{12}^2 + \varepsilon _{23}^2} {l_{{\text{eq}}}}$ $\sigma _{{\text{eq}}}^{{\text{tt}}} = {{\left( {{\sigma _{22}}{\varepsilon _{22}} + {\tau _{12}}{\varepsilon _{12}} + {\tau _{23}}{\varepsilon _{23}}} \right){l_{{\text{eq}}}}} \mathord{\left/ {\vphantom {{\left( {{\sigma _{22}}{\varepsilon _{22}} + {\tau _{12}}{\varepsilon _{12}} + {\tau _{23}}{\varepsilon _{23}}} \right){l_{{\text{eq}}}}} {\chi _{{\text{eq}}}^{{\text{tt}}}}}} \right. } {\chi _{{\text{eq}}}^{{\text{tt}}}}}$ 纱线横向压缩损伤 $\chi _{{\text{eq}}}^{{\text{tc}}} = \left\langle { - {\varepsilon _{22}}} \right\rangle {l_{{\text{eq}}}}$ $ \sigma _{{\text{eq}}}^{{\text{tc}}} = {{\left\langle { - {\sigma _{22}}} \right\rangle \left\langle { - {\varepsilon _{22}}} \right\rangle {l_{{\text{eq}}}}} \mathord{\left/ {\vphantom {{\left\langle { - {\sigma _{22}}} \right\rangle \left\langle { - {\varepsilon _{22}}} \right\rangle {l_{{\text{eq}}}}} {\chi _{{\text{eq}}}^{{\text{tc}}}}}} \right. } {\chi _{{\text{eq}}}^{{\text{tc}}}}} $ 树脂基体拉伸损伤 $\chi _{{\text{eq}}}^{{\text{mt}}} = \left| {{\varepsilon _1}} \right|{l_{{\text{eq}}}}$ $ \sigma _{{\text{eq}}}^{{\text{mt}}} = {{\left| {{\sigma _1}{\varepsilon _1}} \right|{l_{{\text{eq}}}}} \mathord{\left/ {\vphantom {{\left| {{\sigma _1}{\varepsilon _1}} \right|{l_{{\text{eq}}}}} {\chi _{{\text{eq}}}^{{\text{mt}}}}}} \right. } {\chi _{{\text{eq}}}^{{\text{mt}}}}} $ 树脂基体压缩损伤 $\chi _{{\text{eq}}}^{{\text{mc}}} = \left| {{\varepsilon _3}} \right|{l_{{\text{eq}}}}$ $ \sigma _{{\text{eq}}}^{{\text{mc}}} = {{\left| {{\sigma _3}{\varepsilon _3}} \right|{l_{{\text{eq}}}}} \mathord{\left/ {\vphantom {{\left| {{\sigma _3}{\varepsilon _3}} \right|{l_{{\text{eq}}}}} {\chi _{{\text{eq}}}^{{\text{mc}}}}}} \right. } {\chi _{{\text{eq}}}^{{\text{mc}}}}} $ 注：leq为单元特征长度，六面体单元特征长度为单元体积的立方根。
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表 9 三维机织复合材料参数化模型特征参数
Table 9. Characteristic parameters of the parametric model for 3D woven composites
特征参数 参数化模型预测值 测量值[26] 相对误差/% 三维机织复合材料RVE厚度/mm 4.0 4.0 0 表层接结纱倾斜角度/rad 1.109 1.134 −2.2 内层接结纱倾斜角度/rad 1.056 1.065 −0.8 经纱宽度/mm 1.767 1.70 3.9 经纱厚度/mm 0.430 0.45 −4.4 表层纬纱宽度/mm 1.820 1.78 2.2 表层纬纱厚度/mm 0.303 0.28 8.2
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