# Simulation von Rissen in höchstfesten Stahlblechen mit einem Trefftz-Element für Mixed-Mode-Beanspruchungen

Translated title of the contribution: Simulation of cracks in high strength steel sheets using a mixed-mode Trefftz-element

Johannes Hartmann

Research output: ThesisDoctoral Thesis

## Abstract

Within the present work, a special purpose hybrid Trefftz-element for mode III cracks in thin plates is developed. The element is used as an extension of well-established mode I/II elements. Therefore, the proposed element enables the simulation of arbitrary mixed mode crack tip loading within the framework of linear elastic fracture mechanics. The element formulation is based on the analytical solution of the bipotential equation of the Kirchhoff plate theory. Using complex analysis the general solution is adapted to fulfill the natural boundary conditions at the crack edges, too. The essential boundary conditions along the remaining part of the element boundary are taken into account for by an extended elastic potential. The remaining degrees of freedom of the solution are calculated from the minimization of the extended potential. Validation of the element formulation shows convergence of the solution to the finite element reference solution by increasing the number of orders included in the calculation. Within linear elastic theory mode I/II is decoupled from mode III, and the mixed mode Trefftz element is formed by assembling both parts into one element. For the simulation of crack growth in a structural component the special purpose element is used within a standard finite element model, where some of the elements are replaced by the Trefftz element. The crack starts growing inside the element at first and after some crack growth the element position is changed in the direction of crack growth, replacing additional standard elements. Along with the growing crack the Trefftz element can move through the entire structure. For a crack growth algorithm the specific resistance of the material against crack growth is needed as well. Since there are no appropriate standard procedures for the evaluation of the mode III fracture resistance available, within this work a direct evaluation of the deformed crack edges is performed. The data aquisition is carried out using three dimensional digital image correlation. With a newly developed evaluation algorithm, crack propagation and crack tip opening angle are calculated from the geometry data. Using these results the critical stress intensity factor K_IIIc is found by performing a detailed simulation of the experiment. For the evaluation of the crack growth criterion the actual stress intensity factor is calculated from the Trefftz mixed mode element and compared to the critical value of the material obtained from experiments. Finally, the proposed algorithm is demonstrated by the simulation of a simple structural component within an explicit finite element analysis. It is shown, that the concept is capable of simulating the crack behaviour accurately. Finally, future enhancement of the crack propagation algorithm with a crack tip plasticity model is outlined.
Translated title of the contribution Simulation of cracks in high strength steel sheets using a mixed-mode Trefftz-element German Dr.mont. Montanuniversität Pippan, Reinhard, Supervisor (internal)Antretter, Thomas, Assessor A (internal) Published - 2018

### Bibliographical note

embargoed until null

## Keywords

• Hybrid Trefftz-element
• T-complete solutions
• Particular solution
• Strip yield model
• Mode III crack
• Conformal mapping
• Complex analysis