How tough is toughness?

The concept of toughness was introduced by Chvátal [34] more than forty years ago. Toughness resembles vertex connectivity, but is different in the sense that it takes into account what the effect of deleting a vertex cut is on the number of resulting components. As we will see, this difference has major consequences in terms of computational complexity and on the implications with respect to cycle structure, in particular the existence of Hamilton cycles and k-factors

Self-Evaluation Applied Mathematics 2003-2008 University of Twente

This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008

Mixed finite elements with independent strain interpolation for isotropic and orthotropic damage

Tesi per compendi de publicacionsThe numerical modelling of fracture has been an active topic of research for over five decades. Most of the approaches employed rely on the use of the Finite Element Method, which has shown to be an effective and cost-efficient tool for solving many physical phenomena. However, the issue of the spurious dependency of the computed solution with the mesh orientation in cracking problems has raised a great concern since its early reports in the 1980s. This matter has proved to be a major challenge in computational solid mechanics; it affects numerous methods employed to solve the problem, in which the computed crack trajectories are spuriously dependent on the arrangement of the finite element (FE) mesh employed. When performing a structural analysis and, in particular, when computing localized failure, it is fundamental to use a reliable and mesh objective method to be able to trust the results produced by the FE code in terms of the fracture paths, bearing capacity, collapse mechanism and nonlinear responses. In this doctoral thesis, the mixed e/u strain/displacement finite element method is used together with multiple isotropic and orthotropic damage constitutive laws for the numerical modelling of quasi-brittle fracture with mesh objectivity. The independent interpolation of the strains increases the accuracy of the computed solution, guaranteeing the local convergence of the stress and strain fields. This feature is a crucial improvement over the standard FE formulation in solid mechanics where the strains are computed as local derivatives of the displacements and the local convergence of the resulting stresses and strains is not ensured. The enhanced precision provided by the mixed formulation in the area near the crack tip is decisive for obtaining unbiased numerical results with regard to the orientation of the FE mesh. The strain-driven format of the mixed formulation enables to readily consider different constitutive laws defined in a stress-strain structure in the numerical simulations. The thesis includes the study of the effect of the material model employed in the resulting crack trajectories as well as the analysis of the relative performance of several isotropic and orthotropic damage behaviors in mode I, mode II, mode III and mixed mode fracture problems. In this work specific isotropic and orthotropic damage laws are proposed for the numerical modelling of fracture under cyclic loading, which include tensile and compressive damage, stiffness recovery due to crack closure and reopening, as well as irreversible strains. Also, the capacity of the proposed model in reproducing the structural size effect is examined, which is an essential requirement for models aiming at computing quasi-brittle behavior. In this thesis, a comprehensive comparison of the mixed FE formulation with other techniques employed for computing fracture, specifically the Extended Finite Element Method (XFEM) and the Phase-field model, is made, revealing the cost-efficiency of the proposed Mixed Finite Element Method for modelling quasi-brittle cracking with mesh objectivity. This allows to perform the analysis of real-scale structures, in 2D and 3D, with enhanced accuracy, demonstrating the applicability of this method in the engineering practice. The validation of the model is performed with an extensive comparison of computed results with existing experimental tests and numerical benchmarks. The capacity of the mixed formulation in reproducing force-displacement curves, crack trajectories and collapse mechanisms with enhanced accuracy is demonstrated in detail.En esta tesis doctoral, el método de los elementos finitos mixtos e/u deformación/desplazamiento es utilizado junto con varias leyes constitutivas de daño isótropo y ortótropo para la modelización numérica de la fractura cuasi-frágil de forma objetiva con respecto a la orientación de la malla. La interpolación independiente de las deformaciones aumenta la precisión de la solución calculada, garantizando la convergencia local de los campos de tensiones y deformaciones. Esta característica representa una mejora crucial con respecto a la formulación estándar de elementos finitos de la mecánica de sólidos, donde las deformaciones se calculan como derivadas locales de los desplazamientos y la convergencia local de las tensiones y deformaciones resultantes no está garantizada. La mayor precisión aportada por la formulación mixta en la zona cercana a la punta de la fisura es decisiva para obtener resultados numéricos que no presenten una dependencia espuria con la orientación de la malla de elementos finitos. El formato expresado en función de la deformación de la formulación mixta permite considerar directamente diferentes leyes constitutivas que siguen una estructura tensión-deformación para su uso en las simulaciones numéricas. La tesis incluye el estudio del efecto que tiene la ley constitutiva utilizada en la trayectoria de las fisuras resultantes, así como el análisis del desempeño relativo de varias leyes de daño isótropas y ortótropas en problemas de fractura en modo I, modo II, modo III y modo mixto. En este trabajo se proponen leyes de daño isótropo y ortótropo específicas para la modelización numérica de la fractura bajo carga cíclica, que incluyen daño a tracción y a compresión, recuperación de la rigidez por el cierre y reapertura de fisuras, así como deformaciones irreversibles. Además, se comprueba la capacidad del modelo propuesto para reproducir el efecto tamaño, que es un requisito esencial para los modelos que tengan como objetivo calcular el comportamiento cuasi-frágil de los materiales. En la tesis se realiza una comparación exhaustiva de la formulación mixta de elementos finitos con otras técnicas que se utilizan para calcular el problema, específicamente el Método de los Elementos Finitos Extendidos (XFEM) y el modelo Phase-field, revelando la eficiencia computacional del Método de los Elementos Finitos Mixtos propuesto para modelizar la rotura cuasi-frágil de forma objetiva con respecto a la malla. Ello permite realizar el análisis de estructuras de tamaño real, en 2D y 3D, con mayor precisión, demostrando la aplicabilidad del método a problemas reales de ingeniería. La validación del modelo se realiza con una comparación de resultados calculados con ensayos de laboratorio existentes y con simulaciones de casos teóricos de referencia. Se demuestra la capacidad de la formulación mixta para reproducir curvas fuerza-desplazamiento, trayectorias de fisuras y mecanismos de colapso con precisión mejorada.Postprint (published version