PSI - Issue 73

Roman Vodička et al. / Procedia Structural Integrity 73 (2025) 163 –169 Author name / Structural Integrity Procedia 00 (2025) 000 – 000

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4

the system, the pertinent functional reads ( ) = − ∫ ∙ d . Γ N

(3)

The damage evolution is then controlled by the following system of nonlinear variational inclusions with initial conditions (corresponding to an undamaged state) ( ; , , ) + ( ) ∋ 0, | =0 = 0 , ( ; , , ) + ̇ ( ̇, ̇)∋0, | =0 =0. (4) The first relation determines the stress equilibrium, while the second provides the flow rule for damage evolution, which can provide the damage condition locally ⟦ ⟧ 2 c I + ⟦ ⟧ 2 c I I + ⟦ ⟧ 2 c II I ≤− 2 ′ ( ) . (5) Thus, to start degradation, the interface strain expressed in terms of the displacement jump has to reach the equation ⟦ ⟧ 2 c I + ⟦ ⟧ 2 c I I + ⟦ ⟧ 2 c II I =− 2 ′ (0) . It may provide the stress limit. 3. An example The computational analysis of the connector in relation to its parameter is performed using the described computational model in a particular computer code implemented in MATLAB as described in (Vodička, R. et al. (2018)), here, only a basic characteristic of the computational approach is provided. The solution within a given time range is implemented inside a staggered time stepping algorithm which allows to understand the relations in Eq. (4) as minimisation conditions of energy functionals at each time instant. For those minimisations, algorithms of quadratic programming are implemented for discretised forms of energy functionals in Eqs. (1-3). The discretisation of the structural element is advantageously made by the boundary element method due to only nonlinearity in calculations assumed along the interface and thus related to the boundaries of the domains. Fig. 1 shows a scheme of the solved problem. The parameter which is being varied in the calculation is the radius of curvature r pertinent to the shape of the connector. The scheme also shows the prescribed displacement u which linearly increases during loading as u ( t ) = 0.001 t [mm] with pseudo time step t = 1. Simultaneously, the vertical pressure p = 5 kPa is applied constantly simulating the weight of the above material. The material parameters (included in C ) are considered as follows: the concrete part (the upper one) obeys E = 32 GPa,  = 0.2, the lower FRP part is considered as a transversally isotropic material with characterisation E 1 = 39 GPa, E 2 = E 3 = 8.6 GPa, G 12 = G 13 = 3.8 GPa,  12 =  13 = 0.28,  23 = 0.35. The parameters of the interface are set according to experimental data provided in (Kvočák et al. (2024)) which tested adhesion between FRP and concrete. Based on that, fracture energy was adjusted to G c = 0.5mN m -1 , interface stiffness, to be at the range of material stiffness, to k = 1 TPa m -1 , and influence of fibre bridging is considered in assuming  =  These data provide critical interface stress  = 9.5 kPa based on the relation (5) expressed in terms of stress. The graphs in Fig. 2 show the total horizontal force acting at the right side of the top domain. Up to a limit of adhesive forces, the results are almost the same, there are small differences caused by small variations of the structural stiffness due to various geometries of the connectors (various r ). The next part, up to the intersection point of all data curves, pertains to debonding at the right end of the interface which appears in sliding mode. Then, the data start to vary because the subsequent debonding activates the connector’s geometry. The final state in any case reflects the situation where the interface is fully debonded and the two material domains hold together due to contact forces. These states of the process can be distinguished also in a representative plot chosen for the case of r = 8.5 mm.