2.4. Parameters of Concrete Damage Plasticity (CDP) Model

For modelling the cement-based matrix, i.e., a hardened mixture of cement, silica fume, water, and superplasticizer, we adapted the concrete damage plasticity (CDP) model as described in the influential papers [59,60,61] and Abaqus documentation [56]. Let us recall that the CDP model captures the nonlinear behavior of concrete by accounting for simultaneous development of permanent (plastic) deformation and elastic stiffness degradation using two damage variables, one for tensile damage (dt) and the other for compressive damage (dc), to account for different damage responses of concrete in tension and in compression. The plastic-damage model [56] assumes nonassociated potential flow with the Drucker–Prager hyperbolic function that is continuous and smooth and asymptotically approaches the linear Drucker–Prager yield condition [62]. The evolution of the elastic-plastic-damage deformation process is complex from the numerical viewpoint because it is described by inequality relations and Kuhn–Tucker complementarity conditions (loading/unloading conditions), e.g., [61,63].

The parameters of the CDP model and their values used in the calculations are gathered in Table 2, wherein

β—the internal friction angle of concrete. In the CDP model, β is defined as the inclination angle of the Drucker–Prager surface asymptote to hydrostatic axis of the meridional plane;

m—eccentricity of the surface of the plastic potential. This is the distance measured along the hydrostatic axis between the apex of the Drucker–Prager hyperbola and the intersection of the asymptote of this hyperbola, calculated in practice as a ratio of tensile strength to strength for compression;

fb0/fc0—number specifying the compressive strength ratio in a two-axis state for the strength in a single-axis state;

Kc—parameter defining the shape of the surface of the plastic potential on a deviatoric plane;

η—viscoplasticity parameter, used to regularize the concrete constitutive equations.

Mechanical parameters of the cement matrix in the plastic range.

The used stress–strain relationships for the cement matrix in compression and in tension are shown in Figure 3 and Figure 4, respectively, with the compressive strength equal to 200 MPa and the tensile strength equal to 20 MPa. These relations are described with the function proposed by Saenz [64] as

where

and

Compressive behavior of the cement matrix (concrete damage plasticity (CDP) model).

Tensile behavior of the cement matrix (CDP model).

The lower index γ∈{c,t} stands for a type of stress, with c standing for compression and t standing for tension, respectively. Furthermore, εγu is the ultimate strain in the matrix and fγu is its corresponding stress, whereas fγm is the extremal stress sustained by the matrix with its corresponding strain εγ1. In the calculations, we assumed for compression E=55 GPa, fcm=200 MPa, fcu=196 MPa, εcu=0.0028, and εc1=0.0025 and for tension ftm=20 MPa, ftu=19 MPa, εtu=0.00028, and εt1=0.00025. The values of the parameters of Equation (1) calculated by Equation (2) for compression and tension are collected in Table 3.

The values of the coefficients in Equation (1).

The evolution of the compression damage scalar dc for the cement matrix as a function of strain and that of the tension damage scalar dt are illustrated in Figure 5 and Figure 6 and Equation (4), respectively.

Cement matrix compression damage (CDP model).

Cement matrix tension damage (CDP model).