Curing Deformation Analysis for the Composite T-shaped Integrated Structures

Abstract

Curing deformation of the T-shaped integrated structures is discussed in this paper. The mechanism of the deformation is analyzed for the T-shaped integrated structures, and a simple mathematical model for the deformation of the T-shaped integrated structures is established. Compare the mathematical model with the finite element analysis, the results show a good agreement. From the simple mathematical model, it can be seen that both cure shrinkage and thermal expansion are the major factors to produce the deformation of the typical T-shaped integrated structures and the tool-part contraction is the secondary factor. Therefore, it is important for the T-shaped integrated structures to select suitable fabrication process and the appropriate tool. The different geometry and material parameters of the T-shaped integrated structures are studied, and then a regression model is established.

Appendix: Continuous Fiber Micromechanics Model

In the equations presented below, the subscripts 1, 2 and 3 refer to the directions in the principal coordinate system of the lamina. Subscripts m, f and c correspond to the matrix(resin),fiber and composite properties, respectively. Fiber volume fraction of the lamina is denoted as V _f and k is the isotropic plane strain bulk modulus defined by

$$\left\{ \begin{aligned}& k_m = \frac{{E_m }}{{3\left( {1 - 2v_m } \right)}} \\& k_{2f} = \frac{{E_{2f} }}{{2\left( {1 - v_{23f} - 2v_{12f}^2 \frac{{E_{2f} }}{{E_{1f} }}} \right)}} \\ \end{aligned} \right.$$

(A.1)

The following equations define the transversely isotropic engineering constants of the lamina. The longitudinal Young’s modulus:

$$E_{1c} = E_{1f} V_f + E_m \left( {1 - V_f } \right) + \frac{{4\left( {v_m - v_{12f} } \right)k_m k_{2f} G_m \left( {1 - V_f } \right)V_f }}{{\left( {k_{2f} + G_m } \right)k_m + \left( {k_{2f} - k_m } \right)G_m V_f }}$$

(A.2)

The major Poisson’s ration:

$$v_{12c} = v_{13c} = v_{12f} V_f + v_m \left( {1 - V_f } \right) + \frac{{\left( {v_m - v_{12f} } \right)\left( {k_m - k_{2f} } \right)G_m \left( {1 - V_f } \right)V_f }}{{\left( {k_{2f} + G_m } \right)k_m + \left( {k_{2f} - k_m } \right)G_m V_f }}$$

(A.3)

The in-plane shear modulus:

$$G_{12c} = G_{13c} = G_m \frac{{\left( {G_{12f} + G_m } \right) + \left( {G_{12f} - G_m } \right)V_f }}{{\left( {G_{12f} + G_m } \right) - \left( {G_{12f} - G_m } \right)V_f }}$$

(A.4)

The transverse shear modulus:

$$G_{23c} = \frac{{G_m \left[ {k_m \left( {G_m + G_{23f} } \right) + 2G_{23f} G_m + k_m \left( {G_{23f} - G_m } \right)V_f } \right]}}{{k_m \left( {G_m + G_{23f} } \right) + 2G_{23f} G_m - \left( {k_m + 2G_m } \right)\left( {G_{23f} - G_m } \right)V_f }}$$

(A.5)

The transverse Young’s modulus:

$$E_{2c} = E_{3c} = \frac{1}{{\left( {\frac{1}{{4K_{2c} }}} \right) + \left( {\frac{1}{{4G_{23c} }}} \right) + \left( {\frac{{v_{12c}^2 }}{{E_{1c} }}} \right)}}$$

(A.6)

where K _2c is the effective plane strain bulk modulus of the composite given by

$$K_{2c} = \frac{{\left( {k_{2f} + G_m } \right)k_m + \left( {k_{2f} - k_m } \right)G_m V_f }}{{\left( {K_{2f} + G_m } \right) - \left( {k_{2f} - k_m } \right)V_f }}$$

(A.7)

The transverse Poisson’s ration:

$$v_{23c} = \frac{{2E_{1c} K_{2c} - E_{1c} E_{2c} - 4v_{12c}^2 K_{2c} E_{2c} }}{{2E_{1c} K_{2c} }}$$

(A.8)

The following equations define the transversely isotropic thermal expansion coefficient of the lamina. The longitudinal direction thermal expansion coefficient:

$$CTE_{1c} = \frac{{CTE_{1f} E_{1f} V_f + CTE_{1m} E_m \left( {1 - V_f } \right)}}{{E_{1f} V_f + E_m \left( {1 - V_f } \right)}}$$

(A.9)

The transverse direction thermal expansion coefficient:

$$\begin{array}{*{20}l} {{CTE_{{2c}} = CTE_{{3c}} } \hfill} \\ {{ = {\left( {CTE_{{2f}} + v_{{12f}} CTE_{{1f}} } \right)}V_{f} + CTE_{m} {\left( {1 + v_{m} } \right)}{\left( {1 - V_{f} } \right)} - {\left[ {v_{{12f}} V_{f} + v_{m} {\left( {1 - V_{f} } \right)}} \right]}CTE_{{1c}} } \hfill} \\ \end{array} $$

(A.10)