Mathematical Solution for Buckling of Tunable Auxetic/FG-CNTRC Sandwich Hyperboloid Structures in Thermal Environment using an Improved Fourier Series Approach

The present study develops a rigorous analytical framework to explore the mechanical stability and buckling characteristics of advanced sandwich hyperboloid shell systems featuring tunable auxetic behavior. The proposed structural configuration integrates functionally graded carbon nanotube (CNT) reinforced composite layers as the outer face sheets, combined with an auxetic core whose negative Poisson's ratio can be systematically adjusted. The effective mechanical response of the face layers is dictated by the prescribed variation of CNT content across the thickness direction, allowing for tailored stiffness and stability properties. The influence of thermal conditions is incorporated as an external environmental field acting on the structure, while the instability phenomenon itself is treated as fundamentally driven by mechanical loading. The governing stability equations are formulated on the basis of the minimum potential energy principle within an advanced continuum mechanics setting and are solved in closed analytical form under simply supported edge constraints using an enhanced Fourier series based solution strategy. This analytical procedure provides rapid convergence and maintains a high level of accuracy, even in the presence of the pronounced geometric complexity associated with hyperboloid shell curvatures. An extensive parametric assessment is conducted to quantify the effects of auxetic adjustability, carbon nanotube volume fraction and grading patterns, sandwich layer arrangement, key geometric ratios, and thermal conditions on the critical buckling load. To verify the reliability of the developed formulation, the obtained results are systematically benchmarked against established solutions reported in the literature, revealing excellent consistency. The outcomes of this investigation clearly demonstrate that deliberate auxetic design and functional grading of CNT reinforcements play a decisive role in improving the mechanical buckling resistance and overall stability of sandwich hyperboloid structures operating under combined mechanical and thermal environments.