*2.1. FEM-Calculated Elastic Hydraulic Component*

In the traditional HST hybrid model, the ageing component only contains the irreversible component. The reversible ageing displacement follows the same evolution rule as the instantaneous hydraulic displacement due to the common causal factor of reservoir water pressure, these two types of displacements are separated into the hydraulic component in the paper [28]. In contrast to the instantaneous elastic modulus determined by the material properties, the inverted elastic modulus of dam body concrete provides a thorough reflection of the immediate and hysteretic elastic deformation ability through the analysis of the traditional HST hybrid model [3]. According to the theoretical derivation above, the hysteretic water pressure component is subsumed into the ageing component in this paper; therefore, only the elastic water pressure component needs to be calculated by the finite element method [29] as follows:

$$
\delta\_{\text{He}} = \mathcal{X} \delta'\_{\text{He}}.\tag{2}
$$

$$E\_0 = \frac{E\_0'}{X} \,\prime \tag{3}$$

where *δHe* is the actual water pressure component. *δ He* is the water pressure component value obtained from the elastic finite element calculation; *X* is the total water pressure component adjustment factor, and *E*<sup>0</sup> is the initial value of the instantaneous elastic modulus used in the calculation.

In the conventional elastic inversion method, the resulting modulus of elasticity of the dam concrete is a composite reflection of the instantaneous elastic deformation and viscoelastic hysteresis deformation. Therefore, a combined model of mechanical elements is required to separate the instantaneous elastic modulus and viscoelastic modulus when calculating the elastic hydrodynamic component. The Burgers model is a four-parameter fluid model that is commonly used to describe viscoelastic behaviour. It is a hybrid of the Maxwell and Kelvin–Voigt models. The Maxwell model is used to describe stress relaxation, but only in the case of irreversible flow. Although the Kelvin–Voigt model can represent creep, it cannot represent instantaneous deformation. It is not able to account for the stress

relaxation [28]. After the combination of these two models, the Burgers model can express both relaxation and creep effects after combining these two models [30]. Many researchers have used it extensively in recent years to describe the dynamic behaviour of viscoelastic materials [31,32]. Thus, it is used in this paper to calculate the instantaneous and hysteretic hydraulic displacements by the FEM and in the inversion analysis.
