## 3 The Sciences of \"Admissible\" Things
### Modeling and Design
#### 3.4.2 Voxel Assembly  
In this context, bilayer actuators are highlighted as a notable example, as depicted in Figure 9, with researchers considering such devices as a potential foundation for complex modeling involving the assembly of two voxels exhibiting distinct behaviors. A bilayer actuator comprises two cohesive layers of materials with differing mechanical properties, typically characterized by distinct coefficients of expansion. It is the divergent properties of these materials that enable the bimorph to deform and execute mechanical actions. Bilayers can induce bending moments or form annular structures capable of generating displacement perpendicular to the element through shear forces [101]. A 4D printing model incorporating bimorphs was detailed in [102]. Furthermore, [PERSON] et al. [103] proposed evolutionary systems involving multi-layer or multi-bimorph configurations. The deformation of a bilayer in response to applied loads can be readily understood using classical principles.  
Figure 9: Deformation of a bilayer actuator ((1) active material and (2) passive material).  
For instance, consider material 1, which initially has a length \(L_{0}\) at temperature \(T_{0}\). When its temperature changes from \(T_{0}\) to \(T\), it increases by following Equation (1):  
\[\Delta L=L-L_{0}=L_{0}\cdot\alpha(T-T_{0}), \tag{1}\]  
where \(\alpha\) is the linear expansion coefficient of material 1. [PERSON]'s modulus relates the normal stress \(\sigma\) experienced by the material to the resulting elastic deformation \(\epsilon\). In the material's elastic limit, [PERSON]'s law is expressed in Equation (2):  
\[\sigma=E\epsilon, \tag{2}\]  
with \(\epsilon\) representing the relative elongation. When two materials are bonded together, they deform under stress created by the temperature increase, which is proportional to the coefficient of thermal expansion (CTE) and Young's modulus of the material, as expressed in Equation (3):  
\[\sigma=E\cdot\alpha(T-T_{0}). \tag{3}\]  
With \(R_{0}\) the initial radius of the curvature at temperature \(T_{0}\), \(\alpha_{1}\) and \(\alpha_{2}\) representing the CTE of materials 1 and 2, respectively (with \(\alpha_{1}\) typically associated with the less expandable material), and \(E_{1}\) and \(E_{2}\) representing the Young's moduli of the two materials (usually close in value), the thicknesses of the two materials are denoted by \(s_{1}\) and \(s_{2}\), with \(s=s_{1}+s_{2}\) representing the total of the bimorph (typically, \(s_{1}=\frac{1}{2}s\)). The simplified calculation for the curvature radius was expressed in [104] and via Equation (4):  
\[\frac{1}{R}-\frac{1}{R_{0}}\approx\frac{3}{2}(\alpha_{2}-\alpha_{1})\frac{T-T _{0}}{2s} \tag{4}\]  
From an experimental point of view, utilizing, for example, a multi-material 3D printer allows for the creation of such objects, and their deformation can be monitored based on the amplitude of the applied energy. However, the lack of complete parameter knowledge, including the interactions between bimorph components, presents challenges in further analysis, particularly given that a 4D-printed object typically comprises more than two voxels.