## 3 Methodology  
In order to simulate the thermomechanical stresses during thermal cycling of a packed bed, a methodology was developed aiming for time efficient calculations. This methodology is based on the utilization of a particle discrete model for identifying mechanical properties (the effective Young's modulus) of a specific packed bed (detailed in [12]), as well as thermal profiles of an exemplarily large-scale application (detailed in [4]). This information is implemented into a continuum model, which enables time efficient calculations of the thermo-mechanical induced stresses of the packed bed. Figure 1 displays a flow chart of the involved steps of this methodology and the sequence in which those steps are carried out.  
Based on a selected packed bed of ceramic pebbles, a particle discrete model for simulating uniaxial compression tests was implemented into a Discrete Element Method tool. The mechanical properties of the pebbles, e.g., the Young's modulus and the particle diameter, are used as input for the particle discrete model. During uniaxial compression, the selected packed bed is confined in all directions, while being compressed and sequentially relaxed with a top cover plate. This procedure of compression/decompression is repeated for multiple cycles. The resulting stress/strain relations of the packed bed are recorded, which enable the determination of mechanical packed bed properties, e.g., the effective Young's Modulus. The model was validated with the experimental results of a custom-built test rig for uniaxial compression tests. Further information on these steps can be gathered from the preceding work [12], in which the modelling approach and experimental setup are described in detail.  
Based on the stress/strain results of the uniaxial compression simulations, it is possible to calculate specific continuum parameters [16]. Those parameters are used as central elements for the implementation of a continuum model in a Finite Element Model (FEM) tool. The work at hand is based on a simplified approach, in which only the effective Young's modulus is implemented. The implementation of the continuum model is presented in the next section. In addition to the mechanical aspects, thermally induced forces need to be taken into consideration as well. For this purpose, the thermal profile shown in Figure 2 is applied. The selected profile shows the temperature spread between the end of thermal charging and discharging process of the large-scale TES described in [4]. This specific profile was chosen since it represents the maximum temperature spread during TES operation in [4] and thus can be considered as the \"worst-case\" profile, leading to maximum thermo-mechanical loads.  
With the combined mechanical input from the particle discrete simulations and the thermal profiles derived from the external source [4], it is possible to utilize the continuum model for calculating  
Figure 1: Flow chart illustrating process from particle to stresses in large scale application. Green frame: previously discussed in [12]; red frame: Subject of the present work; black frame: External input [4].  
Figure 2: Temperature profile in axial direction of the packed bed.  
the contact stresses between the selected packed bed and a container wall of the large scale application described in [4]. Due to the lack of large scale packed bed TES in commercial applications, there currently are no literature data available regarding the thermo-mechanical behavior of large scale TES. Therefore, the data available in [4] is the only source for validation of the model at the moment.