Festigkeitsrechnung bei mehrachsiger,stochastischer Schwingbeanspruchung mittels der Energiedissipationshypothese

Strength calculation for multiaxial stochastic load by the energy-dissipation hypothesis The energy dissipation hypothesis is further developed to be applicable to multiaxial, stochastic load cases. The static and dynamic parts of the stress components were determined using the hydrostatic and the deviatoric stress components with regard to closed hysteresis loops. The influence of the mean stresses is considered and the total energy dissipation rate is calculated as a base for a cumulative fatigue damage evaluation.     

Invarianten‐basierte Mehrachsigkeitshypothese zur Anwendung bei Schwingbeanspruchung

Multiaxial hypothesis based on invariants for the application of fatigue loading Increasing deviations of experimentally determined and calculated fatigue lives can be observed for multiaxially loaded specimens with increasing contribution of shear stresses. An improvement of this situation can be gained by linking the calculation procedure to both constant amplitude life curves for pure push/pull and shear loading. A hypothesis is presented in this paper which is formulated strictly using only the invariants of the stress tensor to interpolate between the border cases. A modification of this hypothesis is able to take nonproportional loading due to a phase shift between the stress components into account.     

Dauerschwingfestigkeit bei nichtsynchroner mehrachsiger Beanspruchung

Fatigue Strength under Nonsynchronous Multiaxial Stresses In [1] various multiaxial criteria for synchronous stresses as well as for combined and alternating stresses have been reviewed. This paper discusses various criteria for nonsynchronous stresses, e. g. out-of-phase stresses, nonsinusoidal stress time functions and different frequencies between stress components. The criteria aim at the predictions of high cycle fatigue only. Two groups of multiaxial criteria are considered:

• — critical plane criteria that means fatigue failure is governed by stresses on certain critical planes and
• — criteria of integral straining (Anstrengung) that means the stresses in all interference planes of a volume element effect the fatigue behaviour.

Only the last groups, i. e. Effective Shear Stress Criteria [8, 32], Shear Stress Intensity Hypothesis (SIH) [10] allow a reliable fatigue evaluation both for constant and changing direction of principal stresses. An extensive comparison between test results (phase difference, different frequencies, complex stress time functions) and prediction, according to the SIH, shows good agreement.     

Statistischer Größeneinfluß bei zufallsartiger Schwingbeanspruchung

Statistical Size Effect at Random Loading Results of random tests are used to show the applicability of the theory of statistical size effect. The specimens of all the tests of this report were made from the aluminium alloy 7075 T 7351. The specimens were machined from plates (2000 × 1000 × 40 mm). The main dimension of the specimens corresponds with the rolling direction. A standard random load sequence of Gaussian type which is recommended for general application in fatigue testing was used as loading program [19]. In a first test series notched round bar specimens (Fig. 2) were used to investigate the sequence effect. 30 specimens were tested with the standard load sequence, 30 specimens with the variable load sequence. All the other test conditions were kept constant. Fig. 4 shows that the scatter of lifetime (cycles to fracture) is not influenced by the load sequence. In a second test series flat specimens with central holes were tested (Fig. 5) in order to show the existence of a pure statistical size effect at random loading. If the specimen has 1, 3 or 7 holes and all the other test conditions are kept constant, only the size of the area where cracks can initiate is enlarged. The results which are plotted in Fig. 6 show a very clear size effect. The results of the 1-hole specimen (A = A₀) were used to determine the constants of Eq. (5). By inserting A = 3 A₀ resp. A = 7 A₀ the other test results can be predicted. In a third test series round bar specimens with a circumferential notch with different notch root radius (ρ = 1 mm resp. ρ = 6 mm) were tested (Fig. 2, Table 1). The specimens were loaded in such a way that the maximum stress in the notch root and its time-dependence was equal. In this very common case, the stress integral must be determined by applying Eq. (3). Additional tests to determine k and numerical calculations were necessary to evaluate Eq. (3). The final results are plotted in Fig. 8. By inserting A = A₀ resp. A = 5 A₀ into Eq. (7) the results can be described. The basis of the applicability of the theory of statistical size effect is the weakest link concept. The examination of the fracture surface and X-ray analyses showed that there was a higher content of iron (Fig. 11) in those regions where the fatigue cracks initiated. It is supposed that inclusions of intermetallic Al-Fe particles are those weakest links in the structure, where fatigue cracks nucleate.     

Inhomogene Verformungserscheinungen bei Schwingbeanspruchung des Vergütungsstahls 42 CrMo 4

Inhomogeneous Deformation Processes During Cyclic Loading of Quenched and Tempered Steel SAE 4140 The cyclic deformation behavior of a normalized steel SAE 4140 in the loading range σ_a < R_eS is characterized by softening and hardening phenomena. Cyclic softening processes are combined with inhomogeneous plastic deformations and the propagation of a fatigue Lüdersband. The following hardening process occurs homogeneously. At stress amplitudes σ_a < R_p0.2 the quenched and tempered SAE 4140 worksoftens continuously until crack initiation. Plastic deformation processes are extremely inhomogeneous and are concentrated on small fatigue zones. The inhomogeneity increases with decreasing tempering temperatures und stress amplitudes.     

Versuchs- und Rechendaten zur Dauerschwingfestigkeit von metallischen Werkstoffen unter mehrachsiger Beanspruchung

Experimental Data and Calculated Results about the Fatigue Endurance Limit of Metals under Multiaxial Alternating Load An extensive catalogue of present available experimental data about the fatigue endurance limit of metallic materials under multiaxial loading conditions and thereupon determined deviation ratios between experimental results (long life fatigue tests) and calculated values by five newer computation methods is demonstrated (a further showed statistical analysis of these devations indicates, that in relation to the other failure criterions the ?Quadratische Versagenshypothese”? QVH is preferably recommended for a reliable application). The tabulated data-catalogue totally includes 530 referenced loading cases (limited to various biaxial states of combined normal and torsional alternating stress with sinusoidal synchronous or out-of-phase amplitudes and superimposed mean stresses) with experimental results (probability of survival of P_S = 50%) on metallic materials (unalloyed and alloy steels, non-ferrous metals and wrought aluminium alloys, cast irons and sintered metals).     

Berechnung der Dauerschwingfestigkeit bei mehrachsiger Beanspruchung — Teil 1

Estimation of Endurance Limit under Multiaxial Loading Multiaxial criteria for endurance limit prediction of metallic materials will be discussed in this paper. The Shear Stress Intensity Hypothesis SIH is improved under consideration of the variable fatigue limit ratio τ_w/σ_w and mean stress effects. Fatigue behaviour under synchronous and nonsynchronous stresses and influences such as mean stresses, out-of-phase stresses, different frequencies, nonsinusoidal stress time function are analysed theoretically and verified with test results. The prediction according to SIH shows good agreement with the test results. Based on an extensive statistical investigation it can be recommended to use the SIH for ?crack free materials”? and the normal stress hypothesis NSH for ?materials with cracks”?. For components the hypothesis SIH and NSH are developed under consideration of the notch effect. In this study a concept for the prediction of endurance limit under multiaxial loads has been developed.     

Einfluß von Druckvorspannungen auf die Dauerfestigkeit metallischer Werkstoffe bei ein‐ und mehrachsiger Beanspruchung

Effect of compression stresses on the fatigue strength of metals at uniaxial and multiaxial state of loading The calculation of the effect of multiaxial mean stresses on the fatigue strength required to consider the relative direction between the constant and cyclic parts of stresses. For this a new concept for the assessment of stresses has been presented for any fracture behaviour of materials. Experimental investigations on the effect of compression stresses on the fatigue strength of hollow cylinder samples from the steel 20MnCr5 has been used to support the parameter determination of the strength model     

Langzeit- und Dauerschwingfestigkeit des Vergütungsstahls 25CrMo4 bei mehrachsiger Beanspruchung durch drei schwingende Lastspannungen

High-Cycle and Long-Life Fatigue of 25CrMo4 under Multiaxial Load Conditions by three Alternating Stresses Statistically verified experimental results from high-cycle and long-life fatigue tests (HCF and LLF) with altogether 537 unnotched solid cylindrical and thin-walled hollow specimen are demonstrating the fatigue behaviour (S-N-characteristics, scatterband) of 25CrMo4 under uniaxial loading with superimposed static stresses (consideration of the mean stress effect) and under biaxial loadings in variation of phase differences between the three combined normal and torsional stresses σ_x, σ_y, τ_xy The fatigue strength is commonly decreasing with life time in the high-cycle regime until reaching the fatigue endurance limit in the transition range to infinite life. The “ductility level” τ_w/σ_w and the “mean stress sensibility” p = p (σ_w, σ_zSch, R_m) are relatively independent of the intensity by stress amplitudes and fatigue life to failure. In comparision with the specific case of biaxial combined loading with synchroneous amplitudes, the fatigue resistance characteristics are detrimentally influenced by out-of-phase normal stresses σ_x, σ_y; a phase difference of 180° between the normal stress amplitudes is the most critical state of combination, especially in the lower cycle regime caused by a greater slope coefficient (probability of survival P_s = 50%). On the contrary is there in the high-cycle regime as well as in the long-life range no significant influence to the fatigue strength by biaxial load conditions of simultaneously normal stresses with out-of-phase torsional stress τ_xy     

Ermüdung von Keramik unter Schwingbeanspruchung

Fatigue of Ceramics under Cyclic Loading Fatigue of ceramics attracts more attention due to the expected availability of high-performance ceramic components e.g. for engines, However, the knowledge in fatigue of brittle materials is still inadequate, the actual standpoint as taken from literature is shortly reviewed. In this study two experimental-analytical ways are presented which manifest the chances and difficulties in this part of the fatigue research. First, the probabilistic approach to identify a true cyclic fatigue effect in brittle ceramics is envisaged. The second way applies direct observation of crack extension in static and cyclic loading mode in order to define the conditions and characteristic differences of both types of crack propagation. Results are presented from both test methods for several Si₃N₄, SiC and ZrO₂ materials and possible mechanisms are discussed. It is concluded, that toughened ceramics are more prone to cyclic fatigue effects than conventional and pure brittle ceramics (e.g. glass).     

Heuristiken zur Dienstplanung bei flexibler Personalkapazität

Die Untersuchung von Dienstplanungsproblemen erfolgt differenziert nach Problemstellungen. Diese Aufteilung hat dazu geführt, da? Gemeinsamkeiten der Teilgebiete nicht mehr hinreichend gewürdigt werden. Unter Verwendung verschiedener neuer Modellierungskonzepte ist ein auf dem Multi-Mode Resource-Constrained Project Scheduling Problem basierendes allgemeines Modell zur Dienstplanung formuliert worden, in das die verschiedenen Teilgebiete durch geeignete Parameterinstanziierungen eingebettet werden k?nnen. Der vorliegende Beitrag stellt ein speziell an die Problemstruktur angepa?tes Priorit?tsregelverfahren vor, dessen Eignung in einer umfangreichen experimentellen Untersuchung nachgewiesen wird.     

Statistischer Größeneinfluß unter mehrachsiger Wechselbelastung

Statistical Size Effect Under Alternating Load The results of 41 series of tests with smooth cylindrical and notched specimens, subjected to uni- and multiaxial alternating loads (tensioncompression, torsion, rotating bending and combined torsion and bending), are compared with theoretically predicted data. The prediction are based on the ?Weakest-Link-Concept”?. Multiaxial stresses are included either by means of suitable strength criteria, or by a fracturemechanical consideration of a crack in a multidimensional stress field. For the evaluation of the experimental data the 2-parametric Weibull's equation was used. The experimental and theoretical results may deviate by less than 20%, when the ratio between the strength at cyclic torsion and cyclic bending ist taken into account in the function of survival probability [eqns. (2), (7)].     

Optimierte Overlay-Modelle zur Beschreibung des Werkstoffverhaltens bei zyklischer Beanspruchung

Zusammenfassung Das Verhalten von metallischen Werkstoffen unter zyklischer Belastung wird mit Hilfe von Overlay-Modellen beschrieben, die durch parallele Anordnung von isotrop oder kinematisch verfestigenden Elementen gebildet werden. Die Modellparameter der Elemente (Elastizit?tsmoduli, Tangentenmoduli und Streckgrenzen) werden durch eine Optimierungsrechnung bestimmt. Mit einem 2-Elemente-Overlay-Modell werden die Versuchsabl?ufe von sechs Proben, die bei unterschiedlichen Belastungszyklen durchgeführt worden sind, mit sehr guten Genauigkeiten simuliert.     

Kurzzeitschwingfestigkeit von duktilen Stählen unter mehrachsiger Beanspruchung

Low-Cycle Fatigue of Ductile Steels under Multiaxial Deformations To investigate the fatigue behaviour of cyclically softening and hardening steels under multiaxial elastic-plastic strains, axial strain and shear strain controlled fatigue tests under constant amplitude loading were carried out. S-N curves under axial strain and torsional pure shear as well as under combined axial strain and shear, in and out of phase, were obtained for the cyclically softening tempered steel 30 CrNiMo 8 (similar to AlSI-Type 4340) and the cyclically hardening quenched stainless steel X 10 CrNiTi 189 (AISI-Type 321) in the region of low-cycle fatigue. For both steels, used in the design of vessels, pipings, shafts, etc. the fatigue life to crack initiation is reduced by an out of phase (δ = 90°) shearing of the strained specimens in comparison to the in phase loading. The decrease of fatigue life under out of phase strains is caused by changing direction of principal strains resulting in an interaction of the deformations in all directions of the surface. This interaction is taken into account by a calculation procedure deriving an equivalent strain and predicting the fatigue life under combined strain on the base of S-N curves for unaxial strain.     

Beitrag zur Bestimmung der Schwingungsrichtung bei ruhenden Dampfturbinenschaufeln mit Deckband

Zusammenfassung Für ruhende Dampfturbinenschaufeln mit einem Deckband wird ein Verfahren zum Berechnen der Richtung der Schaufeleigenschwingungen unter der Annahme entwickelt, da? bei den Schwingungen die gesamte Form?nderungsarbeit einen Kleinstwert annimmt. Ein Vergleich mit Versuchsergebnissen zeigt, da? diese Theorie bessere Werte liefert als bisher verwendete Verfahren.     

Ein Rechenverfahren zur Robotersteuerung bei vorgegebener Greiferbewegung mit sechs Freiheitsgraden

Zusammenfassung Um das Roboter-Teach-In zu vermeiden, werden die Daten für die Bahnsteuerung eines Greifers berechnet. Als Ergebnisse erh?lt man die Drehwinkel bzw. Schubl?ngen, die Winkelgeschwindigkeiten bzw. Schubgeschwindigkeiten und die Winkelbeschleunigungen bzw. Schubbeschleunigungen für die Antriebsmotoren in jeder Greiferstellung.     

Eine Theorie zur Berechnung von Flatterschwingungen bei Unterschallströmungen

Zusammenfassung Ausgehend von der Stromfadentheorie für ein durchströmtes Rohr wird ein Druckgesetz angegeben, welches angewandt auf die umströmte Membranplatte, einem nichtkonservativen System, zu einer kritischen Strömungsgeschwindigkeit für das Flattern der Membran führt. Die Stabilitätsuntersuchung zeigt, daß die Dämpfung destabilisierend wirkt, und im Stabilitätskriterium tritt eine Unstetigkeit bezüglich der Dämpfungsziffer, ein für nichtkonservative Systeme charakteristisches Merkmal, auf. Nach der vorliegend erläuterten Theorie läßt sich das Flattern der Windfahnen und Segel und der dabei auftretenden wandernden Transversalwellen erklären.

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Summary Starting from the theory of filament of flow in a tube containing flowing fluid it is stated a law of pressure which applied to the blowed membrane plate, a nonconservative system, leads to a critical flutter velocity for the membrane. It is shown, that the damping forces have a destabiling effect and in the stability criterion there appears a discontinuity depend on the damping coefficient, which is a caracteristical feature of nonconservative systems. With the help of the above menhoned theory the flutter of a vane or persenning and the herewith connected travelling transversal waves can be explained.

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Mit 6 Textabbildungen

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Die vorliegende Arbeit wurde vom Verfasser auf der GAMM-Tagung in Zürich (1967) in gekürzter Form unter dem Titel Ein Beitrag zur Berechnung von Flatterschwingungen vorgetragen.