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We propose a theoretical approach and a numerical method for determining the Frank elastic constants based on the experimental dependence of the effective values of the permittivity components on the magnetic field. The theoretical task was to find the minimum of the free energy and then to solve the inverse problem on finding elastic constants by the least squares root minimizing with experimental data. The proposed approach combines strong and weak models with various pretilt conditions at the boundaries. This model also describes the inhomogeneity of the electric field inside the sample. The proposed method allows to achieve higher accuracy using a small amount of experimental data. This statement is confirmed by the error estimation study, which is also presented in this research. As an experimental sample, we used the gadolinium-based liquid crystal complex, since there are no data on the Frank elastic constants for this complex.

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The orienting effect of a magnetic field on an erbium-based liquid crystal complex was studied by a capacitance method. The dependence of the effective dielectric permittivities of the complex on the magnetic field is obtained. A theoretical approach and a numerical method for determining the Frank elastic constants are proposed based on the experimental dependence of the effective values of the permittivity on various magnetic fields.

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