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It has been known for decades that fatigue crack propagation in elastic-plastic media is very sensitive to load history since the nonlinear behavior of the material can have a great influence on propagation rates. However, the raw computation of millions of fatigue cycles with nonlinear material behavior on tridimensional structures would lead to prohibitive calculation times. In this respect, we propose a global model reduction strategy, mixing both the a posteriori and a priori approaches in order to drastically decrease the computational cost of these types of problems. First, the small scale yielding hypothesis is assumed, and an a posteriori model reduction of the plastic behavior of the cracked structure is performed. This reduced model provides incrementally the plastic state in the vicinity of the crack front, from which the instantaneous crack growth rate is inferred. Then an additional a priori model reduction technique is used to accelerate even more the time to solution of the whole problem. This a priori approach consists in building incrementally and without any previous calculations a reduced basis specific to the considered test-case, by extracting information from the evolving displacement field of the structure. Then the displacement solutions of the updated crack geometries are sought as linear combinations of those few basis vectors. The numerical method chosen for this work is the finite element method. Hence, during the propagation the spatial discretization of the model has to be updated to be consistent with the evolving crack front. For this purpose, a specific mesh morphing technique is used, that enables to discretize the evolving model geometry with meshes of the same topology. This morphing method appears to be a key component of the model reduction strategy. Finally, the whole strategy introduced above is embedded inside an adaptive approach, in order to ensure the quality of the results with respect to a given accuracy. The accuracy and the efficiency of this global strategy have been shown through several examples; either in bidimensional and tridimensional cases for model crack propagation, including the industrial example of a helicopter structure.

There is currently a knowledge gap regarding the plastic material properties of many steel types that Atlas Copco use in their high torque power tools. This makes it difficult to fully utilise the capabilities of the Finite Element Method (FEM) to aid the developmental process. Case hardened steel is of special interest as there is not any established method on how to model it numerically.Test specimens of steel type 9195 and 2541 has been developed with two different geometries; hollow and solid. Specimen were heat treated to create case hardened specimens with different Case Depths (CD) as well as specimens made to mimic the material behaviour of the case and core of case hardened steel.Monotone torsion tests were conducted to generate stress-strain data for material models. Hardness tests of case hardened steel were made to determine hardness profiles. By combining these results, a spatially dependent case hardened steel material in an FE-model was created.Test results of the homogeneous specimen tests were successfully modelled using FEA. The case hardened steel could also be modelled, but with a discrepancy against the test results as the model core material were not representative actual core. However, the spatially dependent material model has high potential for simulating the case hardened steel, given the correct inputs. / Det finns för närvarnade bristfällig kännedom gällandes plastiska materialegenskaper hos många av de stål Atlas Copco använder i sina högmomentsverktyg för åtdragning av förband. Detta gör det svårt att fullt ut använda finita elementmetoden (FEM) som ett verktyg i produkters utvecklingsprocess. Sätthärdat stål är av intresse då det inte finns någon etablerad metod för hur det ska modelleras numeriskt.Provstavar gjorda av ståltyperna 9195 och 2541 har tagits fram med två olika geometrier; ihåliga och solida. Provstavarna var värmebehandlade vilket gav sätthärdade material med olika sätthärdningsdjup samt provstavar i material som skulle efterlikna materialet hos höljet och kärnan av sätthärdat stål.Monotona vridprov utfördes för att generera spännings-töjningsdata för användning i materialmodeller. Hårdhetstester gjordes för att fastställa hårdhetsprofiler hos sätthärdat stål. Genom att kombinera dessa provresultat skapades en numerisk material-model av sätthärdat stål med FEM.Testresultaten från de homogena provstavarna återskapades numeriskt med goda resultat. De sätthärdade provstavarnas beteende kunde också återskapas, men med ett något bristfälligt resultat då kärnmaterialet hos modellen inte var representativt av riktigt kärnmaterial hos sätthärdat stål. Modellen har dock hög potential för att återskapa sätthärdat stål, givet korrekt materialdata.

Il est connu depuis des décennies que la propagation des fissures de fatigue dans les matériaux élastoplastiques est très sensible à l’histoire du chargement car le comportement non-linéaire du matériau peut avoir une grande influence sur les vitesses de propagation. Cependant, le calcul brut de millions de cycles de fatigue avec des comportements matériaux non-linéaires sur des structures tridimensionnelles réalistes conduirait à des temps de calcul prohibitifs. Ainsi, nous proposons de coupler deux approches de réduction de modèle a priori et a posteriori, afin de diminuer considérablement le coût de calcul de ce type de problèmes. Tout d’abord, considérant l’hypothèse de plasticité confinée, une stratégie de réduction de modèle a posteriori du comportement plastique de la structure fissurée est proposée. Le modèle réduit ainsi obtenu fournit incrémentalement l’état plastique autour du front de fissure, duquel est déduite la vitesse instantanée de la fissure. De plus, une seconde approche de réduction de modèle, a priori cette fois, est aussi mise en place afin d’accélérer encore plus les temps de résolution du problème global. Cette approche a priori consiste à construire incrémentalement —et sans calculs préalables— une base réduite spécifique à chaque cas-test, en extrayant de l’information des champs de déplacement de la structure au cours du temps et pendant la propagation éventuelle de la fissure. Ainsi, les champs de déplacement solutions de la géométrie fissurée réactualisée sont vus comme une combinaison linéaire de cette base réduite de vecteurs. La méthode numérique considérée ici est la méthode des éléments finis. De fait, pendant la propagation de la fissure, la discrétisation spatiale du modèle doit être réactualisée afin d’être conforme avec le front de la fissure. Dans ce but, une technique spécifique de déformation de maillage est utilisée, et permet de discrétiser la géométrie variable du modèle avec des maillages de même topologie. Cette technique de déformation de maillage apparaît comme une étape clé de la stratégie de réduction de modèle. Finalement, une approche adaptative est construite autour de cette stratégie. Elle permet de garantir la qualité des résultats obtenus par rapport à un critère de précision donné. La précision et l’efficacité de cette stratégie globale sont démontrées à travers de nombreux exemples bidimensionnels et tridimensionnels dans le cadre de propagation de fissure en model, de même que pour un exemple industriel d’une pièce fissurée d’hélicoptère. / It has been known for decades that fatigue crack propagation in elastic-plastic media is very sensitive to load history since the nonlinear behavior of the material can have a great influence on propagation rates. However, the raw computation of millions of fatigue cycles with nonlinear material behavior on tridimensional structures would lead to prohibitive calculation times. In this respect, we propose a global model reduction strategy, mixing both the a posteriori and a priori approaches in order to drastically decrease the computational cost of these types of problems. First, the small scale yielding hypothesis is assumed, and an a posteriori model reduction of the plastic behavior of the cracked structure is performed. This reduced model provides incrementally the plastic state in the vicinity of the crack front, from which the instantaneous crack growth rate is inferred. Then an additional a priori model reduction technique is used to accelerate even more the time to solution of the whole problem. This a priori approach consists in building incrementally and without any previous calculations a reduced basis specific to the considered test-case, by extracting information from the evolving displacement field of the structure. Then the displacement solutions of the updated crack geometries are sought as linear combinations of those few basis vectors. The numerical method chosen for this work is the finite element method. Hence, during the propagation the spatial discretization of the model has to be updated to be consistent with the evolving crack front. For this purpose, a specific mesh morphing technique is used, that enables to discretize the evolving model geometry with meshes of the same topology. This morphing method appears to be a key component of the model reduction strategy. Finally, the whole strategy introduced above is embedded inside an adaptive approach, in order to ensure the quality of the results with respect to a given accuracy. The accuracy and the efficiency of this global strategy have been shown through several examples; either in bidimensional and tridimensional cases for model crack propagation, including the industrial example of a helicopter structure.