Free vibration of functionally graded graphene platelets-reinforced porous thin plates in a nonlinear temperature field

Functionally graded graphene platelets-reinforced porous materials (FG-GPL porous materials) comprise a porous matrix and graphene platelets (GPL) as reinforcing nanoparticles. With a continuous spatial gradient in both porosity and material properties, the FG-GPL porous materials offer advantages of delamination resistance, light weight, high strength, and high thermal resistance. They are among the most promising materials for integrated thermal protection and load-bearing structures in hypersonic vehicles. As a critical structural component, the study of thermal vibration in plates provides strong guidance for engineering applications.

Currently, research on FG-GPL porous materials mainly focuses on normal temperature conditions, with little consideration of high-temperature working environments. However, the natural frequency and mechanical properties are significantly influenced by the thermo-mechanical coupling effect in thermal environments.

Moreover, the vibration characteristics of FG-GPL porous plates are mainly studied using numerical methods, which typically do not provide analytical expressions, making it challenging to establish a clear mathematical-physical correlation.

In summary, existing research faces challenges in providing direct references for FG-GPL porous plate structures operating at high temperatures due to the neglect of thermal environments and the temperature-dependent characteristics of material properties. Furthermore, a mathematical-physical analytical expression is difficult to obtain due to the widespread reliance on numerical methods.

The thermal vibration governing equations for an FG-GPL porous rectangular thin plate under a nonlinear temperature field were established for the first time with consideration of the temperature-dependent material properties. The frequency was analytically solved using the state-space method (SSM). A model of FG-GPL porous material was established based on the defined porosity distributions and GPL dispersion patterns. Then, the thermal vibration governing equations of FG-GPL porous thin plates were derived based on refined first-order shear deformation theory under a nonlinear temperature field. The equations were analytically solved using the SSM, and the natural frequencies were calculated.

The correctness and accuracy of the thermal vibration model of the FG-GPL porous thin plates were verified. Then, the effects of the thermal environments and material distribution patterns (including the porosity distribution, the GPL dispersion pattern, the porosity coefficient, and the weight fraction of GPL) on frequency were analyzed. Furthermore, the material distribution pattern most suitable for high-temperature environments was identified. Subsequently, for this pattern, the influence of thermal gradient in combination with porosity coefficients, thermal gradient with GPL weight fractions, and the comprehensive effects of all three factors were further investigated.

The thermal vibration governing equations of an FG-GPL porous thin plate with temperature-dependent material properties under a nonlinear temperature field were established based on a micromechanical model and a refined first-order shear deformation theory. The natural frequencies were analytically solved using the state-space method. The combined influence of the thermal environment and material distribution patterns was mainly analyzed. For an FG-GPL porous plate composed of PMMA and GPL, the main conclusions are as follows:

(1) With high precision, the state-space method is suitable for analytically solving the vibration governing equations of functionally graded graphene platelet-reinforced porous plates.

(2) The porosity distribution and GPL dispersion pattern significantly affect the frequency. When the pores are concentrated near the geometric midplane or on the high-temperature side, the natural frequency increases. A symmetric GPL dispersion pattern results in the highest frequency under both normal and high-temperature conditions.

(3) For the combined effects of porosity and thermal gradient, the plate with symmetric porosity distribution has a higher frequency when the thermal gradient is below 280K and the porosity coefficient exceeds 0.8. The frequency increases with the porosity coefficient. However, when the thermal gradient exceeds 280K, the plate with a gradient porosity distribution (higher porosity on the high-temperature side) exhibits a higher frequency, making it more suitable for high-temperature environments.

(4) For the combined effects of GPL dispersion and thermal gradient, the plate with a symmetric GPL dispersion pattern (reinforced on both upper and lower surfaces) maintains the highest natural frequency under both normal and high-temperature conditions.

(5) For the combined effects of porosity, GPL, and thermal gradient, an appropriate porosity distribution and porosity coefficient can increase the frequency at different temperatures. The plate with gradient porosity and symmetric GPL dispersion pattern exhibits the highest frequency when the thermal gradient is 40K and the porosity coefficient is less than 0.3. As the thermal gradient increases to 130K, the highest frequency is achieved with a porosity coefficient of less than 0.65. Whereas at a thermal gradient of 220K, only a porosity coefficient below 0.85 is needed.