## 1 Introduction  
Hypersonic vehicles suffer increasingly severe operating environments of a combination of aerodynamic, mechanical, acoustic, and thermal loads during cruise and re-entry flights, which present a significant challenge for the structural integrity and durability. Several experimental studies have shown that thermal-acoustic loading conditions could cause thermal protection systems (TPS) to exhibit snap-through response, which can significantly reduce the mechanical properties and bearing capacity of the structure. The random motion of plates under acoustic loadings at elevated temperatures were investigated on metallic structures [1, 2] and on composite structures [3]; acoustic fatigue tests were performed to demonstrate the bearing capability of structures at elevated temperatures [4, 5]. Acoustic or sonic fatigue and the deterioration of material and structural strength from combined thermal-acoustic loadings have been of significant concern in the structural design and safety of TPS of hypersonic vehicles. Continuous-fiber reinforced ceramic matrix composites such as C/C and C/SiC composites have shown themselves to be promising as potential high temperature materials because of their excellent strength-to-weight and stiffness-to-weight ratios, and high toughness and wear resistance at both room and elevated temperatures [4]. However, the limited amount of reliable experimental data related to their performance in the presence of severe operating environments has to date impeded the development of the evaluation of mechanical properties technology. Thus, analytical techniques for predicting the residual strength of TPS subjected to combined thermal-acoustic loadings are urgently needed. In earlier studies of thin-walled structures under thermal and acoustic loadings, much research was focused on developing models and methods of nonlinear dynamic response, but now more attention is paid to modeling the dynamic behaviors of the system accurately by the reduced order method. [PERSON] [6] predicted the dynamic response of plates subjected to thermal-acoustic loading by the reduced order model. The reduced order methods were investigated on metal beams [7], curved metal panels [8], and composite plates [9; 10] by [PERSON]. The influence of boundary condition on the choice of basis for composite panels under thermal-acoustic loadings was studied by [PERSON] [11; 12]. The non-white Gaussian random excitations were studied by [PERSON] [13]. The prediction of the life of composites and structures under variable amplitude fatigue loading has been recognized as an important problem for more than three decades [14; 15]. A number of distinguishing characteristics of this problem make it different from the fatigue modeling and life prediction of their isotropic counterparts: their anisotropic and inhomogeneous nature leads to various damage mechanisms and models during the entire fatigue life stage. In addition to the material, the loading characteristics, such as the amplitude, the ratio of cyclic stress, and the load sequence, have an effect on damage mechanisms and models [15; 16]. This complexity is the cause of no methodology being consistent in robustly and accurately predicting life of various material systems. For methodologies based on phenomenological modeling to predict the fatigue life and the deterioration of structural strength under variable amplitude loads, the most widely used damage metric that does not always lead to accurate results is the simple empirical linear Palmgren-Miner rule, which cannot take into account the effect of load sequence [15]. Alternatively, the advantages of residual strength or static strength degradation as a damage metric make it promising in a scheme of fatigue life and deterioration of structural strength [17; 18]. The reduction in static strength of the material as a parameter during cycling is used to measure the damage, which is related to cyclic stress peak, cyclic stress ratio and number of cycles.