Abstract
Identification of mechanical properties within the framework of a transversely isotropic model is a relevant problem in biomechanics and geophysics. The use of one-dimensional rod models for interpreting laboratory test results often leads to significant systematic errors due to neglected end effects and inhomogeneity of the stress-strain state. The aim of this study is to develop an applied theory of deformation for slender prisms and to solve the inverse problem of reconstructing elastic constants from displacement measurements on the lateral surface of the specimen. To address this problem, Lagrange’s variational principle was combined with the Kantorovich method. This approach made it possible to reduce the three-dimensional equations of elasticity to a system of ordinary differential equations and to obtain analytical solutions for specimens of arbitrary cross-section. Two types of specimen orientation are analyzed in detail: longitudinal and transverse relative to the axis of elastic symmetry. Based on the obtained solutions, an algorithm for identifying the five elastic constants is proposed, consisting of two stages. In the first stage, four elastic constants are determined from displacement measurements in the region away from the ends. In the second stage, the shear modulus \(c_{44}\) is calculated using an approximate Saint-Venant relation. The effectiveness of the proposed approach is confirmed by a series of numerical experiments for cortical bone tissue and shale using the finite element method. The results demonstrate good accuracy in parameter reconstruction. The developed model can significantly improve the reliability of interpreting data from standard compression tests on transversely isotropic materials.