Issue 46

A. Kostina et alii, Frattura ed Integrità Strutturale, 46 (2018) 332-342; DOI: 10.3221/IGF-ESIS.46.30 336   ' 1 i    С С (12) In this case, inelastic work of viscous forces averaged over the time   2 / can be calculated as     1 : ( : ) 2 h Q real conj ε C ε (13) Therefore, energy dissipation in this work is described by two mechanisms: intrinsic damping in the main material (hysteretic damping) h Q (13) and viscous mechanical dissipation v Q (11). Temperature distribution in the specimen is calculated according to the heat transfer equation with heat sources induced by material damping or viscous behavior of the material:            p T C k T Q t (14) where T is the temperature, p C is the heat capacity under the constant pressure, k is the thermal conductivity, Q is the heat source. Eqns. (3)-(7), (10)-(11), (14) were used for the simulation of a temperature rise induced by the viscous behavior of the coating while the Eqns. (3)-(7), (12)-(14) were applied to the simulation of the material damping. Boundary conditions are specified individually for each considered case and given in the section below. R ESULTS OF NUMERICAL SIMULATION & DISCUSSION Model verification he data reported in [5] were used in order to verify the correctness of the presented model. For this purpose, a finite-element simulation of the heat generation in a damaged aluminum plate was performed. The plate has dimensions of 150x150x3 mm and contains an edge crack with a length of 20 mm. The mechanical and thermophysical properties of the material have been taken from [5]. It is assumed that the load is applied on a circular surface with a radius of 5 mm located at the center of the plate. The loading direction is perpendicular to the plate, has a frequency of 20000 Hz and the amplitude of 0.05 mm. The duration of the loading is 1 second. A crack tip is modelled as an area with an initial strain localization due to the presence of the large number of microdefects. The initial strain amplitude at the crack tip is equal to 0.165, the isotropic loss factor is set to 6 0.45 10   . The COMSOL Multiphysics® software is used to perform the modeling. To calibrate the size of the mesh, the convergence study has been performed. The considered area was discretized by a number of tetrahedral finite elements having various sizes. A more refined mesh at the crack tip was used. The maximum element size was varied from 45 mm to 2mm. A temperature increase at the crack tip has been obtained as a result of each simulation and the relative error has been calculated. The convergence plot is presented in fig. 1 (a). It can be concluded that the element size should be smaller than 5 mm because in this case the relative error does not exceed 15%. A finite- element mesh with the maximum element size of 2 mm has been chosen to carry out numerical simulation. Fig. 1 (b) shows a comparison of the obtained result to the data presented in [5]. The obtained results show that the model can describe adequately the temperature rise at the crack tip. Numerical simulation of a heat generation in a layered material with coating defects In this section an ultrasonic vibration of a rectangular steel bar (the sizes of the bar are 200x10x3 mm) with a copper coating of a 50 μm thickness is considered. The coating of the specimen has an edge crack with a length of 3 mm and a depth of 50 μm, which is simulated as an acute-angled notch. Schematic representation of the specimen is presented in fig. 2. The considered area was discretized by a number of tetrahedral finite elements. The convergence analysis has shown that the length of the element should not exceed 5 mm. Therefore, a finite-element mesh with a maximum element length of 1.8 mm and a minimal size of 0.408 mm was used (fig. 3). This fine resolution provides an accurate simulation of a heat localization at the crack tip. A total number of the finite elements was approximately 250000. T