## 1 Introduction  
This study investigates the impact of _Volume Element_ (VE) size on the elastic and plastic properties of polycrystalline materials. The mechanical responses of polycrystalline materials are influenced by their microstructural characteristics, including grain size, shape, orientation, and phase distribution. State-of-the-art microstructure characterization techniques enable us to capture statistics of morphological and crystallographic descriptors. The _Crystal Plasticity Finite Element Method_ (CPFEM), a popular choice in which the _Finite Element Method_ (FEM) is used for the solution of _Crystal Plasticity_ (CP) equations, has facilitated the study of size dependency in polycrystalline materials [1]. We can thoroughly understand the influence of these microstructural features by analyzing the material at multiple scales with different volume elements. Continuum thermomechanics is founded on the concept of the _Representative Volume Element_ (RVE) [2], which is well defined for two cases: (1) a unit cell in a periodic microstructure, and (2) a statistical representative volume containing a large number of microscale elements. Under such conditions, the material in an RVE can be represented in continuum form using classical homogenization approaches to obtain the _effective_ material properties. When an RVE is not well-defined, _Statistical Volume Elements_ (SVEs) can instead be formulated to represent the material in continuum form. For deviation from case 1, e.g., [3; 4; 5; 6], SVEs are used to investigate the perturbations to the geometry and material properties of the periodic unit cell. If the VE is not large enough in case 2, the boundary condition type [7], clustering and position of microstructures [8], and variations to the observation window [9; 10] all result in randomness in the SVE. SVE-homogenized propertiesare referred to as _apparent_ properties. Similar to RVEs, the applied _Boundary Conditions_ are often required to satisfy the Hill-Mandel energy equivalency condition [11]. SVEs provide several advantages in the context of CPFEM for case 2. First, analyzing VEs that are equal to or larger than the RVE size for elastic and plastic properties is often computationally prohibitive given the complexity of CP equations. Second, SVE-based homogenization maintains underlying material inherent randomness and spatial heterogeneity, both of which are important in multi-scale failure simulations [2; 12; 13]. Efficiency and accuracy are dependent on SVE size when using the moving window method, because apparent homogenized properties are assigned at the centers of the SVEs that traverse the domain [10; 14]. If the SVEs are too small, the choice of BCs significantly affects the form and values of the apparent properties. If the SVEs are too large, the randomness and heterogeneities of the properties disappear as the SVE size tends to the RVE size. The RVE size is an important parameter that determines the size beyond which a property is homogeneous, size-independent, and/or isotropic (when applicable) depending on the criterion [13; 15]. The RVE size for a given property can be determined by an analysis of apparent properties for varying SVE sizes. One popular approach is to determine the RVE size when the coefficient of variation in the homogenized properties over all VEs falls below a user-defined tolerance [16]. Alternatively, calibration of an empirical power relation eliminates the need to simulate an RVE-size VE. Rather, the extrapolation of the best fit estimates the RVE size. This approach is particularly useful for CP simulations, when the simulation cost of RVE size domains can be prohibitively high. For various ductile materials, the RVE size is needed in the prediction of elastoplastic and viscoplastic properties [17; 18; 19; 20]. The required number of grains for microstructure and property-based statistically equivalent RVEs were determined through the creation of virtual microstructures and the application of finite element simulations [21; 22].