# Experimental and Parametric Analysis of Pull-Out Resistance of Notched T-Perfobond Connectors
## 4 Numerical Simulation
### Material Constitution of Concrete  
Figure 11 illustrates the nonlinear response of concrete under compression and tension, characterized by the uniaxial compressive stress-strain curves and the relationships between tensile stress and crack width. Figure 11: Material constitution of concrete. (**a**) Compression; (**b**) tension. Figure 10: Interaction and boundary conditions. Figure 9: Finite-element type and mesh. The intrinsic compressive behavior of concrete is articulated by Equation (1) [27]. As depicted in Figure 11a, the stress-strain curve commences with an initial linear elastic branch. Subsequently, the curve transitions into a nonlinear parabolic segment, followed by a descending branch, each reflecting different phases of the material's response under compression. \[\left\{ \begin{aligned} \sigma_{c}&=f_{c}\cdot\frac{n \cdot\xi-\xi^{2}}{1+(n-2)\cdot\xi}&\quad\varepsilon_{c}\leq \varepsilon_{cp}\\ \sigma_{c}&=f_{c}\cdot\frac{\xi}{2\cdot(\xi-1)^{2}+ \xi}&\quad\varepsilon_{c}>\varepsilon_{cp}\end{aligned} \right. \tag{1}\]  
The model outlined by [PERSON] (1993) governs the ascending segment of the curve, where \(\sigma_{c}\) denotes the compressive stress in MPa and \(f_{c}\) represents the concrete's axial compressive strength, also in MPa. The strain at the peak stress is defined as \(\varepsilon_{cp}=0.0035\), and \(\xi\) quantifies the strain relative to the peak strain, expressed as \(\xi=\varepsilon_{c}/\varepsilon_{cp}\). Furthermore, \(n\) is the ratio of the initial modulus of elasticity (\(E_{c}\)) to the modulus of the descending branch at the peak point (\(E_{cp}\)), calculated as \(n=E_{c}/E_{cp}\)[27].For uncracked concrete, a linear stress-strain relationship captures the material's elongation characteristics. In contrast, the cracked section, as illustrated in Figure 11b, necessitates a nonlinear description of the stress-crack width relationship. This is articulated through Equation (2) [27]. \[\frac{\sigma_{t}}{f_{t}}=\left[1+\left(c_{1}\cdot\frac{w}{w_{c}}\right)^{3} \right]\cdot\exp\left(-c_{2}\cdot\frac{w}{w_{c}}\right)-\frac{w}{w_{c}}\cdot \left(1+c_{1}^{3}\right)\cdot\exp(-c_{2}) \tag{2}\]  
In the provided model, \(\sigma_{t}\) signifies the tensile stress in MPa, while \(f_{t}\) is the concrete's tensile strength, also expressed in MPa. The crack width is denoted by \(w\) in mm, and \(w_{c}\) represents the crack width at complete stress relief, defined as \(w_{c}=5.14\ G_{F}/f_{t}\) (mm), where \(G_{F}\) is the fracture energy required to generate a unit area of stress-free crack, as expressed by \(G_{F}=0.073\,f_{c}^{0.18}\) (N/mm). The model incorporates constants of \(c_{1}=3\) and \(c_{2}=6.93\)[27]. The concrete damage plasticity model encapsulates the degradation response of concrete under stress. This model employs two independent uniaxial damage variables, namely \(d_{c}\) and \(d_{t}\), to quantify concrete damage resulting from compressive crushing and tensile cracking, respectively [27]. For concrete under compression, the progression of \(d_{c}\) is linked to the plastic strain (\(\varepsilon_{c}\)). It is determined to be proportional to the inelastic strain (\(\varepsilon_{c}\)_in_ = \(\varepsilon_{c}-\sigma_{c}/E_{c}\)) with a constant factor of \(b_{c}\) (0 \(<b_{c}<\) 1), as delineated in Equation (3) [27]. \[d_{c}=1-\frac{\sigma_{c}}{E_{c}\cdot\varepsilon_{c}^{pl}\cdot(1/b_{c}-1)+ \sigma_{c}} \tag{3}\]  
In this context, \(d_{c}\) represents the compressive damage component of concrete. The ratio of plastic to inelastic strain is denoted by \(b_{c}\), as defined by \(b_{c}=\varepsilon_{c}\)_pl_/\(\varepsilon_{c}\)_in_, and \(b_{c}\) is assigned a value of 0.7 according to [27]. For concrete under tension, the damage evolution component (d) is correlated with the \"plastic\" crack width (\(w^{pl}\)). This component is proportional to the actual crack width (\(w\)) with the proportionality governed by the constant factor (\(b_{t}\), where 0 \(<b_{t}<\) 1), as described by Equation (4) [27].