## 2 Mechanical Packed Bed Modelling  
This paragraph covers the theoretical background of the present study. It will give a brief overview of the mechanical modelling approaches for packed beds. These modeling techniques are the basis for the implementation of a simplified approach for calculating thermo-mechanical stresses of the large-scale TES application [4]. An early approach to calculate packed bed stresses was derived by [PERSON] [13] in the context of silo technology. By analyzing a theoretical disc element of a silo packed bed, he derived the following analytical correlation of vertical stresses (\(\sigma_{v}\)) acting on the container wall:  
\[\sigma_{v}=\frac{g\cdot p_{b}\cdot A}{\lambda\cdot\tan(\varphi_{x})\cdot U} \left(1-e^{\frac{-\lambda\cdot\tan(\varphi_{x})\cdot U}{A}}\right). \tag{1}\]  
This correlation is dependent on the bulk density (\(\rho_{b}\)), cross-section (\(A\)), horizontal load ratio (\(\lambda\)), wall friction angle (\(\varphi_{x}\)), the containment circumference (\(U\)), and gravity (\(g\)). The required bulk properties can be determined with various soil mechanical tests. More information on this approach can be gathered from multiple sources, e.g., [13]. While this model enables the calculation of stresses in stationary and thermally uncharged packed beds, it is not employable for applications with additional thermal loads. For a more detailed modelling of the packed bed, two fundamentally different approaches exist: the first is a particle discrete model, where each particle is modelled as independent physical object; the second is a continuum model, where the packed bed is perceived as a macroscopic, cohesive material. Particle discrete models can be implemented using the Discrete Element Method (DEM). The DEM enables the computation of forces between interacting particles. Here, each particle is represented as a friction-spring-damper system. Thus, particle-particle and particle-wall contacts will result in spring, friction, and damping forces. The algorithm used in DEM enables the determination of time dependent movement and rotation of particles. More details on the governing equations and algorithms are given in [12]. Previous work has shown that the DEM produces highly accurate results and allows valuable insights into the basic phenomena [14; 15]. The particle discrete approach is well suited for packed beds with a relatively small number of particles or for a short simulation time. However, due to the high computing time required to solve the multitude of underlying numerical operations for a large number of particles, the DEM is unfit to simulate large-scale applications. As discussed in the previous work [12], a continuum based finite element model approach for packed beds offers a significant speed up in calculation time in comparison to discrete models. The major challenge of the application of a continuum model is the adequate parametrization of the packed bed properties. In the case of continuous packed bed modeling, \"effective\" material properties are needed, instead of bulk material properties. Especially in the area of soil mechanics and nuclear fusion technology, different constitutive equations have been developed to describe the continuous characteristics of packed beds. In the context of the investigation of pebble beds for the use in ceramic breeder blankets, [PERSON] [16] developed a power law function for the effective Young's Modulus,dependent on stress and temperature, by conducting uniaxial compression tests. The function can be converted to the following form:  
\[E_{eff}=C_{1}\Big{(}1+C_{2}\cdot T^{3}\Big{)}\cdot\sigma^{C_{3}} \tag{2}\]  
A similar approach was developed by [PERSON] et al [17]. The effective Young's Modulus is a key parameter when describing a packed bed as a continuum and provides insight into the mechanical behavior of the packed bed. Further information about continuum models and relevant parameters can be gathered from multiple sources [11; 18; 19].