# Numerical Modeling of Non-Uniformly Reinforced Carbon Concrete Lightweight Ceiling Elements
## 3 Smeared Model of a Cross Section  
### Characterization of the Applied Anisotropic Damage Model of a Shell Layer  
The classification of modeling approaches introduced in Figure 1 considers the inelastic material behavior of the composite layers in a general way. Even though a damage model is chosen in the last row of Figure 1 as an example, the classification can be used in combination with plasticity models describing the strain-hardening or strain-softening behavior as well. This classification provides the framework for the explanation of the scaling and mixture rules for inelastic material parameters presented in the sequel. To formulate and to validate the scaling and mixture rules in combination with a particular example of an inelastic material model, a microplane damage model introduced in [25] has been chosen to reflect the nonlinear material behavior of the composite layers. Inelastic effects governing the stress-strain response are thus represented by a damage function as exemplified in the last row of Figure 1. The key idea behind the microplane models is to reflect the material state on a set of planes on a unit hemisphere or circle around a material point instead of using the usual tensorial representation. The microplanes are used to establish the mapping between the strain and stress tensors as proposed in the original model by [PERSON] et al. [26] sketched in Figure 6a. The macroscopic strain tensor \(\mathbf{\varepsilon}\) is decomposed onto the microplane directions by a geometric projection, delivering the microplane strain vectors \(\mathbf{e}\). A constitutive law at the microplane level defines the relation between the apparent and effective microplane strain and stress vectors. Finally, the macroscopic stress tensor \(\mathbf{\sigma}\) is obtained using the condition of energetic equivalence between the microplane discretization and the tensorial representation of stress at the material point. As the original version was detected to be thermodynamically inconsistent, several refinements have been proposed as documented e.g., in [27]. Currently, the microplane discretization of a material state provides a well established framework for sound formulation of anisotropic inelastic material models [28]. In the microplane damage model used here [25], each microplane is associated with a projected strain and damage state variables. Its important property is the separation between the apparent and effective stresses and strains. As indicated in the algebraic structure of the microplane damage model depicted in Figure 6b, the apparent stress tensor \(\mathbf{\varepsilon}\) is first projected onto the microplanes rendering the microplane strain vectors \(\mathbf{e}\). Then, the effective microplane strains \(\mathbf{\varepsilon}\) is obtained with the help of the damage/integrity function. By employing the condition of energetic equivalence, i.e., the principle of virtual work, the corresponding effective strain tensor \(\mathbf{\varepsilon}\) is obtained by integrating the strains over the microplane discretization. The relation between the effective strain tensor \(\mathbf{\varepsilon}\) and the effective stress tensor \(\mathbf{\sigma}\) is provided explicitly using the elasticity tensor \(\mathbf{D}^{\mathbf{e}}\). Analogous to the described mapping of strain, the mapping between the effective stress tensor \(\mathbf{\sigma}\) and apparent stress tensor \(\mathbf{\sigma}\) is performed with the help of the microplane state discretization governed by the prescribed integrity function \(\phi=1-\omega\). In the applied implementation of the model, only the normal component of the microplane  
Figure 6: Algebraic structure of the microplane models (**a**) basic principle of microplane model; (**b**) constitutive stress-strain relation in the microplane damage model. strain and stress is considered. The model is thermodynamically consistent and implicitly captures the [PERSON]'s effect of the undamaged, still an effective skeleton of a material structure in a material point.