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Process Modelling of Curing Process-Induced Internal Stress and Deformation of Composite Laminate Structure with Elastic and Viscoelastic Models

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Abstract

In this paper, two kinds of transient models, the viscoelastic model and the linear elastic model, are established to analyze the curing deformation of the thermosetting resin composites, and are calculated by COMSOL Multiphysics software. The two models consider the complicated coupling between physical and chemical changes during curing process of the composites and the time-variant characteristic of material performance parameters. Subsequently, the two proposed models are implemented respectively in a three-dimensional composite laminate structure, and a simple and convenient method of local coordinate system is used to calculate the development of residual stresses, curing shrinkage and curing deformation for the composite laminate. Researches show that the temperature, degree of curing (DOC) and residual stresses during curing process are consistent with the study in literature, so the curing shrinkage and curing deformation obtained on these basis have a certain referential value. Compared the differences between the two numerical results, it indicates that the residual stress and deformation calculated by the viscoelastic model are more close to the reference value than the linear elastic model.

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Acknowledgements

The author would like to thank “Jing Hou Xin” research group for many fruitful discussions. This work is supported by the National Natural Science Foundation of China (Grant No. 11664023).

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Authors and Affiliations

  1. School of Materials Science and Engineering, Lanzhou University of Technology, Lanzhou, 730050, China

    Dongna Li & Xudong Li

  2. School of Science, Lanzhou University of Technology, Lanzhou, 730050, China

    Jianfeng Dai

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  1. Dongna Li
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Correspondence to Dongna Li.

Appendix

Appendix

This appendix presents micromechanical homogenization equations which used to determine the overall composite properties of unidirectional lamina. Noted that subscripts f and r represent fiber and resin, respectively. The thermo-physical properties of the composites, including the density, ρ, the specific heat capacity, C p , and the longitudinal thermal conductivity, k L , (x -direction in Eq. (1)), can be respectively calculated by the following rule of mixtures [33]:

$$ \rho ={v}_f{\rho}_f+\left(1-{v}_f\right){\rho}_r $$
(15)
$$ {C}_p=\frac{\rho_f{v}_f{C}_{pf}+{\rho}_r\left(1-{v}_f\right){C}_{pr}}{\rho } $$
(16)
$$ {k}_L={v}_f{k}_f+\left(1-{v}_f\right){k}_r $$
(17)

The transverse thermal conductivity of the composites, k T , (y and z -direction in Eq. (1)) can be calculated based on the E-S model [34]. It is given as:

$$ \frac{k_T}{k_r}=\left(1-2\sqrt{v_f/\pi}\right)+\frac{1}{2B}\left[\pi -\left(4/\sqrt{1-\beta}\right){\mathrm{tan}}^{-1}\left(\sqrt{1-\beta }/1+\beta \right)\right] $$
(18)

in which:

$$ \beta ={B}^2{v}_f/\pi $$
(19)
$$ B=\mu \left({k}_r/{k}_f-1\right) $$
(20)
$$ \mu =a/b $$
(21)

where a and b are the axial lengths of the elliptic section of the fiber along the y -axis and z -axis, respectively. In this paper, the cross section of the fiber is round, so a is equal to b, i.e.μ = 1.

Self-consistent micromechanics homogenization equations [32] used to determine the composite properties are listed in this appendix. Subscripts 1, 2 and 3 represent the principal directions of unidirectional lamina. The Young’s modulus of the composites in the longitudinal direction of the fiber is expressed as:

$$ {E}_1={E}_{1f}{v}_f+{E}_r\left(1-{v}_f\right)+\frac{4{\left({\upsilon}_r-{\upsilon}_{12f}\right)}^2{K}_r{K}_f{G}_r{v}_f\left(1-{v}_f\right)}{\left({K}_f+{G}_r\right){K}_r+\left({K}_f-{K}_r\right){G}_r{v}_f} $$
(22)

in which:

$$ {K}_f=\frac{E_{1f}{E}_{2f}}{2\left(1-{\upsilon}_{23f}\right){E}_{1f}-4{\upsilon}_{12f}^2{E}_{2f}} $$
(23)
$$ {K}_r=\frac{E_r}{2\left(1-{\upsilon}_r\right)-4{\upsilon}_r^2} $$
(24)

where E 1f and E 2f are the Young’s modulus of the fiber in the longitudinal direction and transverse direction; E r is the Young’s modulus of the resin; G r is the shear modulus of the resin; υ 12f and υ 23f are the in-plane Poisson’s ratio and transverse Poisson’s ratio of the fiber; υ r is the Poisson’s ratio of the resin; K f and K r are the bulk modulus of the fiber and resin, respectively.

The in-plane shear modulus of the composites is expressed as:

$$ {G}_{12}={G}_{13}={G}_r\frac{\left({G}_{12f}+{G}_r\right)+\left({G}_{12f}-{G}_r\right){v}_f}{\left({G}_{12f}+{G}_r\right)-\left({G}_{12f}-{G}_r\right){v}_f} $$
(25)

The out-of-plane shear modulus of the composites is expressed as:

$$ {G}_{23}={G}_r\frac{\left({G}_{23f}+{G}_r\right){K}_r+2{G}_{23f}{G}_r+\left({G}_{23f}-{G}_r\right){K}_r{v}_f}{\left({G}_{23f}+{G}_r\right){K}_r+2{G}_{23f}{G}_r-\left({G}_{23f}-{G}_r\right)\left({K}_r+2{G}_r\right){v}_f} $$
(26)

where G 12f and G 23f are the in-plane shear modulus and out-of-plane shear modulus of the fiber, respectively.

The plane strain bulk modulus of the composites is expressed as:

$$ {K}_2=\frac{\left({K}_f+{G}_r\right){K}_r-\left({K}_f-{K}_r\right){G}_r{v}_f}{\left({K}_f+{G}_r\right)-\left({K}_f-{K}_r\right){v}_f} $$
(27)

The Young’s modulus of the composites in the transverse direction of the fiber is expressed as:

$$ {E}_2={E}_3=\frac{1}{1/4{K}_2+1/4{G}_{23}+{\upsilon}_{12}^2/{E}_1} $$
(28)

The in-plane Poisson’s ratio of the composites is expressed as:

$$ {\upsilon}_{12}={\upsilon}_{13}={\upsilon}_{12f}{v}_f+{\upsilon}_r\left(1-{v}_f\right)+\frac{\left({\upsilon}_r-{\upsilon}_{12f}\right)\left({K}_r-{K}_f\right){G}_r{v}_f{v}_r}{\left({K}_f+{G}_r\right){K}_r+\left({K}_f-{K}_r\right){G}_r{v}_f} $$
(29)

The out-of-plane Poisson’s ratio of the composites is expressed as:

$$ {\upsilon}_{23}=1-\frac{E_1{E}_2+4{\upsilon}_{12}^2{E}_2{K}_2}{2{E}_1{K}_2} $$
(30)

The longitudinal CTE of the composites is expressed as:

$$ {\phi}_1=\frac{v_f{\phi}_{1f}{E}_{1f}+\left(1-{v}_f\right){\phi}_m{E}_{1m}}{v_f{E}_{1f}+\left(1-{v}_f\right){E}_{1m}} $$
(31)

The transverse CTE of the composites is expressed as:

$$ {\displaystyle \begin{array}{c}\hfill {\phi}_2={\phi}_3={v}_f\left({\phi}_{2f}+{\upsilon}_{12f}{\phi}_{1f}\right)+{v}_m\left(1+{\upsilon}_m\right){\phi}_m\hfill \\ {}\hfill -\left({\upsilon}_{12f}{v}_f+{\upsilon}_m{v}_m\right)\frac{\phi_{1f}{E}_{1f}{v}_f+{\phi}_{1m}{E}_{1m}{v}_m}{E_{1f}{v}_f+{E}_{1m}{v}_m}\hfill \end{array}} $$
(32)

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Li, D., Li, X. & Dai, J. Process Modelling of Curing Process-Induced Internal Stress and Deformation of Composite Laminate Structure with Elastic and Viscoelastic Models. Appl Compos Mater 25, 527–544 (2018). https://doi.org/10.1007/s10443-017-9633-5

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