On the influence of friction during compression of elongated prism specimens

The contact problem of axial compression of an elastic, slender prismatic specimen by loading plates is considered, taking into account friction in the contact zone. The relevance of the work stems from the need for accurate evaluation of the elastic properties of materials with complex structures – such as functionally graded materials, composites, and biomaterials, where classical rod models produce systematic errors of up to 20–30 % due to neglected friction and contact effects. To solve the problem a variational approach combined with the Kantorovich method is used; this reduces the three-dimensional linear elasticity problem to a system of second-order ordinary differential equations with variable coefficients for generalized displacements. Analytical solutions are obtained for specimens with arbitrary and square cross-sections under constant elastic properties. An effective integral friction parameter is introduced – the ratio of tangential to normal generalized forces on the end faces. The inverse problem of reconstructing the elastic properties and the friction parameter from displacement data at several points on the lateral surface is solved. An iterative algorithm based on the contraction-mapping principle is developed, providing sufficiently accurate recovery of the unknown parameters in 3–5 iterations for the considered set of initial guesses. In addition, approximate asymptotic formulas based on solutions of the direct problem are proposed for rapid parameter recovery from measurements at a single point, together with recommendations for selecting measurement locations on the lateral surface. Numerical experiments on synthetic data generated by the direct problem confirm the effectiveness of the proposed methods and their robustness to noise (in the case of the iterative algorithm), demonstrating superiority over rod models. The approach enables improved processing of experimental compression test data, which is especially important for samples made of biomaterials and composites where friction significantly influences result interpretation.