基于等向性放松微结构连续体的长圆柱壳体在径向变形下的解析解
Analytical solutions for long cylindrical shells under radial deformations based on the isotropic relaxed micromorphic continuum
AI总结：
本文基于等向性放松微结构连续体，推导了长圆柱壳体在径向变形下的解析解，揭示了微结构效应与经典弹性力学的差异。
摘要：
本研究在等向性放松微结构连续体框架下，提出了长圆柱壳体微结构材料弹性静力学响应的闭式解析解。通过引入独立的微结构变形张量场，补充经典位移场，考虑微结构效应。在轴对称变形和平应变条件下，平衡方程简化为径向坐标中的耦合常微分方程组。通过引入合适的辅助变量，将系统改写为非齐次修正贝塞尔方程，该方程具有精确解析解。推导出径向位移场和非零微结构变形张量分量的显式表达式。通过数值示例展示了材料参数和特征长度对位移的影响。结果表明，放松微结构模型在微结构效应显著时预测偏离经典弹性力学。所获解为圆柱壳体力学提供了有价值的物理见解，并可作为验证放松微结构模型数值实现的基准。
英文摘要：
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.
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