# Development of Analytical Model for Bonding of CFRP Rod in Concrete Subjected to Cyclic Loads
## 7 Development of Interface Element  
The interface element emerged as a fundamental model that explicitly captures the failure mechanism of CFRP bonding. In finite element analysis, interface elements depict a thin or zero-thickness layer bridging continua. The interface element's constitutive relationship hinges upon a traction-displacement law that generally exhibits elastic behaviour up to a stress-based failure criterion (initiation) and then transitions to softening behaviour, characterised by an area under the curve equal to the critical fracture energy at complete failure (propagation). The formulation of an interface element suitable for planar problems is provided, enabling its linkage with six-node isoperimetric elements possessing rectilinear sides (Figure 21). In the context of planar problems, the geometry of an interface element is represented by a line connecting two solid elements. [PERSON] et al. introduced an interface element tailored for analysing delamination fatigue onset or fatigue increase. The authors demonstrated its ability to fully release one or more elements for each time increment while accurately accounting for the number of cycles required to induce fatigue growth over that length [32]. [PERSON] and [PERSON] developed a novel interface element specifically designed to investigate cyclic crack propagation under mixed-mode loading and applied it to analyse an example of cyclic crack propagation under mixed-mode loading [33]. \[\text{N}_{\text{a}}=-\frac{1}{2}\xi(\xi-1)\text{, }\text{N}_{\text{b}}=+ \frac{1}{2}\xi(\xi+1) \tag{12}\]  
\[ \begin{bmatrix}\Delta_{\text{a}}\\ \Delta_{\text{c}}\end{bmatrix} = \begin{bmatrix}\frac{\Delta_{\text{Na}}}{ \Delta_{\text{v}}\Delta_{\text{a}}}\\ \frac{\Delta_{\text{v}}}{\Delta_{\text{v}}\Delta_{\text{c}}}\end{bmatrix} _{ \begin{array}{c}\Delta_{\text{v}}\\ \Delta_{\text{v}}\end{array} }=[\tau]_{(4\times 8)}[\delta]_{(8\times 1)}\text{, } \begin{cases}\Delta_{\text{a}}\\ \Delta_{\text{c}}\end{cases} = \begin{bmatrix}\text{N}_{\text{a}}&0&\text{N}_ {\text{c}}&0\\ 0&\text{N}_{\text{a}}&0&\text{N}_{\text{c}}\end{bmatrix} [\Delta]=[\text{N}][\Delta] \tag{13}\]  
\[\begin{bmatrix}\text{N}][\Delta]=[\text{N}]_{\delta}= \begin{bmatrix}B_{\text{ ij}}\end{bmatrix} = \begin{bmatrix}B_{\text{ij}}\end{bmatrix} [\delta]= \begin{cases}\varepsilon_{\text{t}}\\ \varepsilon_{\text{n}}\end{cases} , \begin{cases}\varepsilon_{\text{t}}\\ \varepsilon_{\text{n}}\end{cases} =\frac{1}{\text{t}} \begin{cases}\Delta_{ \text{u}}\\ \Delta_{\text{v}}\end{cases} =[\text{N}_{\text{s}}]\{\delta\}= \begin{bmatrix}B_{ \text{ij}}\end{bmatrix} \{\delta\} \tag{14}\]  
\[\varepsilon_{\text{ij}}=\frac{\delta_{\text{ij}}}{\text{T}_{\text{o}}},\text {T}_{\text{o}}=\text{thickness of }\text{ }\text{ }\text{inter face element} \tag{15}\]  
\[t= \begin{cases}t_{n}\\ t_{s}\\ t_{t}\end{cases} = \begin{bmatrix}E_{nn}&E_{ns}&E_{nt}\\ E_{ns}&E_{ss}&E_{st}\\ E_{nt}&E_{st}&E_{tt}\end{bmatrix}\begin{cases}\epsilon_{n}\\ \epsilon_{s}\\ \epsilon_{t}\end{cases} =E\epsilon \tag{21}\]  
However, the analysis performed on the 2D element solely analyses the Z-direction, which is the direction of loading. \[ \begin{split}\text{t}=\begin{cases}\text{t}_{n}\\ \text{t}_{s}\end{cases}=\begin{bmatrix}\text{E}_{nn}&0\\ 0&\text{E}_{ss}\end{cases}\begin{cases}\epsilon_{n}\\ \epsilon_{s}\end{cases}=\text{E}\epsilon\\ (\text{Elastic behaviour matrix for 2D element})\end{split}  \tag{22}\]  
\[E_{ij}=\frac{Young^{\prime}s\text{ }modulus\text{ }in\text{ } normal\text{ }direction}{Thickness\text{ }of\text{ }closure\text{ }element}\quad E_{nn}=\frac{1800}{0.2}=9000 \tag{23}\]  
This basic architecture allows for the integration of multiple damage mechanisms that can operate on the same material simultaneously. Each failure mechanism consists of three components: a damage initiation criterion, a damage evolution law, and a decision regarding element deletion when the system reaches a fully damaged state. Damage initiation marks the beginning of degradation in the response of a material point.