Crack Development Assessment Using Modal Analysis in Peridynamic Theory
2019
Andris Freimanis

Defending
29.11.2019. 14:30, Rīgā, Ķīpsalas ielā 6B, 422. telpā

Supervisor
Ainārs Paeglītis

Reviewers
Sandris Ručevskis, Jānis Andersons, Dariuss Bačinskas

Structural damage is caused by design faults, construction quality shortcomings, or external effects. If such damage is not discovered and is allowed to grow, structure’s load-bearing capacity deteriorates, which can lead to costly repairs or in extreme cases structure’s collapse. However, accurate damage detection techniques can extend the life of these structures and provide measurable economic benefit. Modal analysis is widely used to detect structural damage, for quality control in manufacturing, to validate numerical models, etc. In principle, engineers should be able to detect introduced damage from shifts in modal frequencies and changes in mode shapes, yet in practice, damage detection by modal analysis is difficult. Modeling systems with evolving discontinuities e.g. cracks, still present a challenge, because classical theory uses partial differential equations, which are undefined when discontinuities are present in the displacement field. Contrary, peridynamic theory, which is a non-local reformulation of the classical theory, represents forces and displacements using integral equations, which are defined even with discontinuous displacement fields. Therefore, discontinuities are a natural part of a peridynamic solution rather than a burden, thus making this theory an attractive option for damage modeling. Peridynamic modal analysis, however, is still undeveloped. This thesis aims to implement a novel massively-parallel open-source modal solver for peridynamic modal problems, verify its results against finite-element modal analysis results and validated them against experimental modal analysis results, and demonstrate how peridynamic modal analysis can be used together with peridynamic damage simulations to obtain modal parameters of damaged structures. Peridynamic modal solver was implemented in an open source peridynamics software Peridigm. The implementation used the shift-invert transform, in which block Krylov-Schur eigensolver solved the eigenvalue problem and flexible block GMRES linear solver with ILU preconditioner solved the substitution. Solver parameters were optimized using four different peridynamic modal problems. The optimization cases were generated using Latin Hypercube method and the optimal solver parameters for the considered cases are presented. Peridynamic modal analysis was used to compute modal properties of a 100 × 50 × 8 mm plate and nine different crack configurations were considered. Linear peridynamic solid material model, which is equivalent to an elastic material in the classical mechanics theory, was used and no boundary conditions were applied thus simulating the free-free boundary condition case. The first 12 modes were computed. The first six were rigid-body-motion modes, so only the second six were used for verification. Lastly, convergence studies were performed using four different mesh densities and four different horizon lengths. Finite-element results were used to verify the accuracy of peridynamic modal analysis. A model with the same dimensions as in peridynamics was created in finite-element software Ansys using 8-node SOLID 185 elements. Linear-elastic material model was used and no BCs were applied. Cracks were created by not connecting solid element nodes that lay on the crack plane. The experimental modal analysis results, measured with a 2D Polytec PSV-400 scanning laser vibrometer, were used for the experimental validation of the peridynamic results. Test specimens were 100 × 50 × 8 mm plates manufactured from polymethyl methacrylate (PMMA) sheets using laser cutting. The same nine crack configurations as used in peridynamic simulations were created and tested. Specimens were excited using a loudspeaker and they were suspended in two cotton thread loops to ensure free-free BC. Results show excellent agreement between the peridynamic and the finite-element modal frequencies. The differences ranged between 0.00 % and -4.00 %. The peridynamic modal frequencies were lower as the horizon increased. This behavior can be explained by the increasing peridynamic “surface effect”. Peridynamic results also agreed well with the experimental results. The modal frequencies were within ±3.2 % of the experimental results. Moreover, the peridynamic frequency shifts are similar to frequency shifts in the finite-element and experimental analyses. The largest difference between the peridynamic and the finite-element frequency shifts was -1.38 percentage points, but between the peridynamic and the experimental frequency shifts +1.95 percentage points. The peridynamic and the finite-element mode shapes agreed well and were in the same order at each crack configuration. Also, peridynamic and experimental mode shapes agreed well, furthermore, the change in the mode shapes from the introduced damage was similar in both analyses. Two convergences – δm and δ – were considered. In the δm convergence study, results asymptotically approached a single value as the mesh density increased. The convergence is faster than linear, but slower than quadratic. Moreover, it was shown that the asymptotic range of convergence is reached with the largest model. The δ-convergence plots were not asymptotic and showed that the difference between the non-local and the local solution does not shrink smoothly as the horizon decreases. Lastly, a practical application of p modal analysis is demonstrated. The developed modal solver was coupled with a peridynamic fatigue simulation and modal parameters were obtained before and after fatigue damage growth. The difference in modal frequencies was between 2.91 % and 0.45 %. The fatigue simulation created a measurable shift in modal frequencies, which can be used for damage assessment, model validation, and quality assurance.


Keywords
Peridynamics, modal analysis, fracture, fatigue, damage detection

Freimanis, Andris. Crack Development Assessment Using Modal Analysis in Peridynamic Theory. PhD Thesis. Rīga: [RTU], 2019. 114 p.

Publication language
English (en)
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