# Numerical Study of Concrete: A Mesoscopic Scale Simulation Methodology
## 2 Materials and Methods
### Stage 2: Numerical FEM Model
#### 2.2.3 Material Constitutive Laws and Properties  
Drawing from previous research aimed at simulating concrete cracking and crushing [23; 24; 25; 26; 27; 28; 29], the authors employed the following constitutive models:  
* The Concrete Damage Plasticity (CDP) model used combines the theory of isotropic elastic damage with isotropic plasticity in tension and compression to simulate the inelastic behavior of concrete, given its quasi-brittle nature. Moreover, this model is suitable for both mass and reinforced concrete and accommodates static, cyclic, and dynamic loads. The CDP model addresses two primary fracture mechanisms: tensile cracking and compression crushing. The formulation of continuous damage in elasticity [30] can be expressed as follows: \[\sigma=(1-d)D^{\epsilon}:\left(\epsilon-\epsilon^{i}\right),\] (4) where \(\sigma\) is the Cauchy stress tensor, \(d\) is the damage variable, \(D^{\epsilon}\) is the undamaged elastic stiffness tensor, \(\epsilon\) is the total strain tensor, and \(\epsilon^{i}\) is the inelastic strain tensor, which accounts for the irreversible nature of concrete cracking. * For mortar and ITZ, a nonlinear combined mathematical model of concrete was utilized, incorporating the Drucker-Prager material model [31; 32; 33] for compression stresses and the Willam and Warnke model [34; 35; 36] for tension stresses. This model predicts both material failure modes: cracking and crushing. Figure 3: (**a**) Meshing the Representative Volume Element (RVE). (**b**) Distribution and meshing of aggregates within the specimen. The Drucker-Prager yield criterion, used to determine if a material has exceeded its elastic limit, is defined by the equations:  
\[\text{F}\ \equiv\ \alpha\ I_{1}+\sqrt{I_{2}}-\kappa\ =\ 0, \tag{5}\]  
where \(J_{2}\) and \(I_{1}\) are stress invariants:  
\[I_{1}=\sigma_{x}+\sigma_{y}+\sigma_{z}, \tag{6}\]  
\[J_{2}=\frac{1}{6}\left[\left(\sigma_{x}-\sigma_{y}\right)^{2}+\left(\sigma_{y} -\sigma_{z}\right)^{2}+\left(\sigma_{z}-\sigma_{x}\right)^{2}\right]+\tau_{xy} ^{2}+\tau_{zx}^{2}+\tau_{yz}^{2} \tag{7}\]  
\(\alpha\) is a parameter related to the internal friction angle \(\varphi\):  
\[\alpha=\frac{2\sin\varphi}{\sqrt{3}(3+\sin\varphi)}, \tag{8}\]  
and  
\[\kappa=\frac{6\text{ccos}\varphi}{\sqrt{3}(3+\sin\varphi)}, \tag{9}\]  
is related to cohesion angle \(c\) according to the expression:  
\[c=0.231 ln\left(Ed_{max}^{2}\right)-0.60 \tag{10}\]  
This correlation is established based on experience gained from finite element applications to concrete. The Drucker-Prager yield surface, depicted as a cone of revolution with its axis along the hydrostatic line (\(\sigma_{x}=\sigma_{y}=\sigma_{z}\)) (refer to Figure 4), governs plastic deformations, given by:  
\[d\varepsilon_{ij}^{p}=d\lambda\ \frac{\partial F}{\partial\sigma_{ij}}, \tag{11}\]  
Here, \(d\lambda\) is a scalar determining the magnitude of plastic deformation. The vector of plastic deformations \(d\overset{\to p}{\varepsilon}\) is perpendicular to the yield surface. In this context, the volumetric plastic deformation is:  
\[d\varepsilon_{vol}^{p}=d\varepsilon_{x}^{p}+d\varepsilon_{y}^{p}+d\varepsilon _{z}^{p}=3\alpha d\lambda, \tag{12}\]  
This model can be incorporated into the equations governing the behavior of mortar and ITZ as a yield condition in the nonlinear constitutive model [37]. The failure surface described by [PERSON] and [PERSON] indicates that the compressive strength of concrete under multiaxial compressive stresses can surpass its uniaxial compressive strength significantly. When considering \(\sigma_{x}\geq\sigma_{y}\geq\sigma_{z}\), the sign of \(\sigma_{z}\) determines the mode of concrete failure. To tend towards tension-induced fracture, \(\sigma_{z}\) must be positive, leading to cracks perpendicular to this principal stress. Conversely, when no stress is positive, crushing failure occurs.