Lagrange Multiplier and Penalty Methods for the Thermal Buckling Analysis of Higher-Order Composite Laminates

\hl{Advances in composite structures with improved mechanical properties enable a better balance between strength and weight, benefiting high-demand fields like civil and aerospace engineering. Higher-order theories effectively model such structures but are often complex owing to the multitude of parameters involved. To ease this burden, internal constraints are introduced, but they can complicate numerical methods like the finite element method (FEM) by requiring higher-order shape functions. Driven by the challenges of finite element modeling for constrained composites, this study focuses on the thermal buckling behavior of higher-order GPLRC plates. Internal constraints in higher-order plate theories are introduced by employing the Lagrange Multiplier Method (LMM) and the Penalty Method (PM). These methods disable the interpolation of displacement parameters using Lagrange shape functions with} $C^0$ \hl{continuity, ensuring simple shape functions, a well-posed weak formulation, compatibility with standard FEM software, and avoiding the need for complex formulations in a classical finite element framework}. The theoretical formulation of the GPLRC laminate plate is derived based on the General Third-order Shear deformation plate Theory (GTST). The Halpin–Tsai model and rule of mixtures are used to determine the laminated plate material properties. Four GPL distribution patterns across composite layers are analyzed for their impact on laminate buckling behavior. \hl{The LMM and PM results are systematically compared by varying parameters such as element count, GPL weight fraction, and geometrical dimensions, demonstrating the effectiveness and accuracy of constraint enforcement techniques in FEM-based thermal buckling analysis of composite plates.} \hl{The study provides a robust framework for modeling complex composite structures, enabling efficient solutions in computational environments.}