| Liquid crystal elastomers are cross-linked polymers with liquid crystalline mesogens, combining the elasticity of polymers with the liquid crystalline properties so their mechanical behaviors have the characteristic of thermo-mechanical-order coupling. Liquid crystal elastomers have the potential of various applications. The study on the mechanical behaviors is a preparation for the future engineering application. The properties of multi-physics coupling enrich the theory in mechanics and the method of analysis. In this paper, we have studied some typical mechanical behaviors of the liquid crystal elastomers under the thermo-mechanical-order coupling by the stress-strain constitutive equation and the mechanical-order coupling equation.First, we studied the elastic moduli of the liquid crystal elastomers under the infinitesimal deformation. We obtain the formulae for the elastic moduli, analyze the temperature dependence of the moduli and the influence of some materials parameters. Exact formulae for the elastic moduli are obtained by the implicit function method based on somewhat general energy functions. The formulae indicate that both the moduli parallel and perpendicular to the director of the liquid crystal elastomers are smaller than the modulus of the classical elastomers because of the mechanical-order coupling. Moreover, the moduli are generally anisotropic due to the biaxiality induced by stretching the liquid crystal elastomers perpendicular to the director. Then we have obtained the explicit analytical expressions of the parallel and perpendicular moduli by making use of the Landau-de Gennes free energy and the neo-classical elastic energy. Very different from the classical elastomers, they are both strongly nonlinear functions of the temperature in the nematic phase. Furthermore, their ratio, the degree of anisotropy, changes with the temperature as well. The results agree qualitatively with some experiments. Better quantitative agreement is obtained by some modifications of the constitutive relations of the elastic energy.Then we studied the nonlinear larger deformation behaviors of uniaxial stretch parallel to the director of the liquid crystal elastomers, including the stress-strain relation with different temperatures, stretch-temperature relation and stress-temperature relation. Due to the mechanical-order coupling, the order parameter is increased by the stretch. This influences the stress-strain relation of the liquid crystal elastomers and leads to the result that the stress with the same stretch is reduced. The influence of the change of the order parameter varies with the temperature; the stress is decreasing with increasing stretch at the temperature slightly higher than the phase transition temperature. Because the order parameter changes with the temperature, the stretch decreases with increasing temperature when the nominal stress is fixed; the stress increases with increasing temperature when the stretch is fixed. The phase transition temperature increases linearly with increasing stresses.Further we studied the mechanical behaviors in the biaxial stretch at the temperature far below the phase transition temperature, including two kinds:the control of displacement and the control of stress. We focus on properties of the stress, stretch, order parameter, biaxial order parameter in the biaxial stretch. While the stretch parallel to the director and the stretch perpendicular to the director are equally increasing, the order parameter first decreases then increases with increasing stretch, and the stress parallel to the director is larger than the stress perpendicular to the director. While the stretch perpendicular to the director is increasing and the stretch parallel to the director is fixed, the director rotates when the stretch is increasing above the critical value. When the director rotation happens, the stress parallel to the director changes its value with the stress perpendicular to the director. Then they both increase with increasing stretch. While the stress parallel to the director and the stress perpendicular to the director are equally increasing, the order parameter decreases with increasing stress, and the stretch perpendicular to the director is larger than the stretch parallel to the director. While the stress perpendicular to the director is increasing and the stress parallel to the director is fixed, the director rotates when the stress is increasing above the critical value. When the director rotation happens, the order parameter 'jumps' to a higher value and then increases with increasing stress. In the same kind of deformation at a high temperature, when the director rotation happens, the order parameter 'jumps' to a lower value and then decreases with increasing stress.We also studied the effective quasiconvex elastic energy of the liquid crystal elastomers; analyze the phase diagram for the effective quasiconvex elastic energy. When the director rotation happens, the liquid crystal elastomers will form internal unsymmetrical micro-structures in order to reduce the overall energy in the macroscopic deformation. Based on this physical background, we obtain the effective quasiconvex elastic energy of the liquid crystal elastomers by the quasiconvex method. The effective elastic energy is a convex function. It has four phases under various deformations, including the liquid phase, the intermediate phase 1, the intermediate phase 3 and the solid phase. In the liquid phase, the effective elastic energy is zero, thus the material behaviors are completely soft, like a liquid; in the solid phase, the effective elastic energy is very similar to the one describing the classical elastomers; in the intermediate phases, the energy depends only on one eigenvalue of the effective left Cauchy-Green tensor, thus the material behaviors are intermediate between the completely soft one and a solid-like one. The biaxiality is the reason for the intermediate phase 3 of the effective quasiconvex elastic energy. |
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