*3.4. Free-Vibration Analysis*

Tables 10 and 11 show the first three vibration modes and natural frequencies of 3D-TCP under the boundary conditions in Cases 4 and 7. It is clear that the vibration modes predicted by 2D-EPM agree with those predicted by 3D-FEM. For example, the first, second buckling modes, respectively, have one and two half-waves along the *x*<sup>1</sup> axis, and the third buckling mode has two half-waves along the *x*<sup>2</sup> axis under the boundary condition in Case 4. The maximum natural frequency error is 8.67%, indicating 2D-EPM has high accuracy in free-vibration analysis of 3D-TCP.

**Table 10.** Comparison of the first three free vibration characteristics predicted by two models under the boundary condition in Case 4.

**Table 11.** Comparison of the first four free vibration characteristics predicted by two models under the boundary condition in Case 7.

#### **4. Influence of Structural Parameters on Equivalent Stiffness**

The structure of 3D-TCP is complex and has many parameters (see Figure 3). In Section 3.2, we can see that the local stress and strain in binder yarns are relatively greater, indicating the binder yarns plays a very important role in preventing interlayer separation in 3D-TCP. Therefore, it is very important to study the influence of binder yarn width on the equivalent stiffness, which can also provide guidance for the design of 3D-TCP. Secondly, the influence of warp (weft) yarn width on the equivalent stiffness are also investigated. Table 12 lists the structural parameters of 3D-TCP used in the parameter analysis.

**Table 12.** Structural parameters of 3D-TCP used in the parameter analysis.


<sup>1</sup> *n* is the multiplier, and can be chosen as eight different values: 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8 and 0.9, respectively.

Figures 13 and 14 show that the binder width and warp (weft) width have great influence on the equivalent tensile stiffness *A*<sup>11</sup> and bending stiffness *D*11. The values of *A*<sup>11</sup> and *D*<sup>11</sup> increase with the increasing warp (weft) width, but decrease with the increasing binder width. The main reason is that the increase of binder width will lead to the decrease of yarn content, further resulting in the decrease of *A*<sup>11</sup> and *D*11. However, the smaller the binder yarn width is, the smaller the constraint in the thickness direction will be, resulting in the easy delamination. Therefore, the binder yarn width should be adjusted to ensure enough stiffness and integrity in the design of 3D-TCP.

**Figure 13.** Influence of the binder yarn width on equivalent stiffness of 3D-TCP. (**a**) *Aij*; (**b**) *Dij*.

**Figure 14.** Influence of the warp (weft) yarn width on equivalent stiffness of 3D-TCP. (**a**) *Aij*; (**b**) *Dij*.

#### **5. Comparison of Effective Performance between 2D-PWL and 3D-TCP with the Same Thickness**

The 3D-TCP is developed from the 2D plain-woven laminate (2D-PWL), and has better mechanical properties. To compare the effective performance of the two textile composite plate, we establish the 2D-EPM of 2D-PWL and 3D-TCP with the same plate thickness and yarn content.

The structure parameters of unit cell within the 6-layered 2D-PWL as shown in Figure 15 are: *L* = 1.2 mm, *D* = 0.8 mm, *H* = 0.2 mm. The structure parameters of unit cell within the 3D-TCP are: *b*<sup>1</sup> = 1.0 mm, *h*<sup>1</sup> = 0.2 mm, *l*<sup>1</sup> = 1.2 mm, *b*<sup>2</sup> = 0.6 mm, *h*<sup>2</sup> = 0.1 mm, *l*<sup>2</sup> = 1.2 mm, *t* = 0 mm. The thickness of both plates is 1.32 mm and the yarn content is 50.47%. The obtained equivalent stiffness matrix of 2D-PWL and 3D-TCP are shown in Tables 13 and 14, respectively. Based on the equivalent stiffness matrix, the 2D-EPM of 150 mm × 150 mm is established to study the difference of effective performance between 2D-PWL and 3D-TCP.

**Figure 15.** The 6-layered 2D plain-woven laminate.



**Table 14.** Equivalent stiffness matrix of 3D-TCP (unit: SI).


#### *5.1. Comparison of Bending Behaviors*

The boundary conditions used in bending analysis are shown in Figure 16, and the predicted bending displacements are shown in Table 15. It can be observed that the bending displacement of 3D-TCP is greater than that of 2D-PWL under uniform load (Case 8 and Case 9). This may due to the fact that the warp yarns and weft yarns of 3D-TCP are not interlaced, and are only constrained by the binding yarns in the thickness direction. While the displacement of 3D-TCP is smaller than that of 2D-PWL under the concentrated force and bending moment (Case 10 and Case 11), indicating that the torsion resistance of 3D-TCP are better than 2D-PWL. The above results are consistent with the obtained equivalent stiffness in Tables 13 and 14. That is, the bending stiffness *D*<sup>11</sup> and *D*<sup>22</sup> of 2D-PWL are greater than those of 3D-TCP, while the torsional stiffness *D*<sup>66</sup> of 2D-PWL is smaller than that of 3D-TCP.

**Figure 16.** Boundary and load conditions for bending behavior analysis of 2D-PWL and 3D-TCP. (**a**) Case 8; (**b**) Case 9; (**c**) Case 10; (**d**) Case 11.

**Table 15.** Comparison of bending displacement between 3D-TCP and 2D-PWL under different conditions.
