Analytical solutions for long cylindrical shells under radial deformations based on the isotropic relaxed micromorphic continuum

Abstract

This study presents a closed-form analytical solution for the elastostatic response of long cylindrical shells composed of microstructured materials within the framework of the isotropic relaxed micromorphic continuum. The formulation accounts for microstructural effects by introducing an independent micro-distortion tensor field in addition to the classical displacement field. Under the assumptions of axisymmetric deformation and plane strain conditions, the governing equilibrium equations reduce to a coupled system of ordinary differential equations in the radial coordinate. By introducing suitable auxiliary variables, the system is reformulated into a non-homogeneous modified Bessel equation, which admits an exact analytical solution. Explicit expressions are derived for the radial displacement field and the non-zero components of the micro-distortion tensor. Numerical examples are presented to illustrate the influence of material parameters and the characteristic length on the displacement. The results demonstrate that the relaxed micromorphic model predicts deviations from classical elasticity where microstructural effects are more pronounced. The obtained solution provides valuable physical insight into the mechanics of cylindrical shells and serves as a benchmark for validating numerical implementations of relaxed micromorphic models.

Figures (8)

• Figure 1: A long cylindrical shell.
• Figure 2: Normalized radial displacement of a thick shell with an inner-to-outer radius ratio $\beta=0.15$, comparing the classical elasticity and relaxed micromorphic models for three values of $G_1$ with $r_o/L_{\rm c}=2$, $G_2=5$, $G_3=2$ and $U_i/U_o=0$.
• Figure 3: Normalized radial displacement of a thin shell with an inner-to-outer radius ratio $\beta=0.85$, comparing the classical elasticity and relaxed micromorphic models for three values of $G_1$ with $r_o/L_{\rm c}=2$, $G_2=5$, $G_3=2$ and $U_i/U_o=0$.
• Figure 4: Influence of $G_2$ on the normalized radial displacement of a thick cylindrical shell ($\beta = 0.15$) with $G_1 = 2$, $G_3 = 1.3$, $r_o/L_{\rm c} = 2$, and $U_i/U_o = 0$.
• Figure 5: Influence of $G_3$ on the normalized radial displacement of a thick cylindrical shell ($\beta = 0.15$) with $G_1 = 2.5$, $G_2 = 3.5$, $r_o/L_{\rm c} = 2$, and $U_i/U_o = 0$.