Effects Of Nonproportional Loading Research Articles

On how S-N curves for nonproportional loading can be determined from S-N curves for proportional loading – An exemplary evaluation of numerous fatigue tests using scaled normal stresses

The fatigue design of safety–critical components necessitates a delicate balance between sustainability and safety. This study explores the challenge of achieving a balanced design for components subjected to multiaxial loads, focusing on nonproportional loading effects such as an increasing or decreasing fatigue life compared to proportional loading. Existing studies present divergent views on the correlation between nonproportional hardening and fatigue life shift, with some proposing a connection while others treat them independently. It is still unclear which of the research groups may be right. In this article, the effects are considered separately as independent effects, with the focus on the shift in fatigue life.This paper compares various methods of representing fatigue test results under proportional and nonproportional loading. An attempt is made to establish comparability of the tests from different references. One challenge here is, on the one hand, to establish comparability with different stresses on individual material groups and, on the other hand, to consider the second material effect, namely nonproportional hardening.For this purpose, a strength hypothesis is used on the one hand and a plastic correction in combination with a damage parameter is applied on the other. The procedure described in this paper can also be used with other suitable strength hypotheses and damage parameters. In the following, the scaled normal stress hypothesis in combination with the damage parameter according to Smith, Watson and Topper is used as an example.The aim of this paper is to quantitatively describe the shift in fatigue life to be able to include this material effect in a fatigue assessment depending on the material group. A comprehensive database comprising steel, wrought aluminum, cast aluminum, and spheroidal graphite cast iron is utilized to evaluate the shift in fatigue life under nonproportional loading. The database covers various control types, specimen geometries, and amplitude ratios, extending from low cycle to very high cycle fatigue regime.Based on the evaluation, material group-dependent properties can be derived, which can finally be used in a fatigue assessment as an averaged correction of the calculated fatigue life or to predict S-N curves for nonproportional loading based on these for proportional loading.

Machine learning enabled identification of sheet metal localization

The Forming Limit Curve (FLC), which describes the maximum applicable strain before localization, depends on the particular material, but also on the applied load and history of the load. Recent investigations have shown that the non-proportional loading effect on the FLC can be predicted with data-driven or machine-learning-based methods. Here we compare different ML methods to their applicability in predicting localization points under multi-segmented non-proportional loading. Therefore, an FE-based metamodel is developed that allows imposing an arbitrary loading history on sheet metal to predict the point of localization. A series of virtual experiments are conducted with this metamodel to generate a database of bi-linear loading paths that are used for training. Different ML-based methods were used to predict the localization point based on the strain history data. The 1D-Convolutional Neural Network (1D-CNN), with the ability to learn dependency between input features, has the best accuracy in predicting the localization point.

Effects of nonproportional loading and residual stress on creep crack tip fields

Nonproportional loading and residual stresses effects on creep crack tip fields are important topics in fracture mechanics at elevated temperature. In this paper, nonproportional loading and residual stresses effects on creep crack tip fields are systematically investigated based on the novel interpretation of C(t)-integral considering residual stress. With the numerical implementation of the corrected C(t)-integral considering residual stress, the eigenstrain and mechanical induced residual stress effects on the creep crack tip fields are studied. We find that C(t)-integral level is affected remarkably in the presence of residual stress regardless of eigenstrain or mechanical induced residual stresses for creep crack tip under small scale creep. However, level of C*-integral is not significantly influenced under the effects of residual stresses in presence of large-scale creep although the transition creep time is remarkably changed. Otherwise, C(t)-integral is highly affected by loading path under small scale creep. However, C*-integral is independent of the loading path if the creep time is long enough. Nonproportional loading effect on C*-integral is very limited for creep crack tip field according to our study. The investigation given in this paper could deepen the understanding the effects of residual stress and nonproportional loading on the creep crack tip field.

Load path sensitivity and multiaxial fatigue life prediction of metals under non-proportional loadings

Engineering components often operate under complex loadings, in which the variable amplitude multiaxial stresses are raised by geometric discontinuities including holes, grooves, fillets and shoulders, etc. Besides, the non-proportional loading will lead to the rotation of maximum principal stress/strain and additional fatigue damage of structural elements in service. Consequently, the multiaxial and non-proportional loading have attracted increasing attentions. In this study, for distinguishing the effects of different load paths on fatigue life, a simple and applicable method to quantify the non-proportionality is defined, which comprehensively considers the influence of loadings on all material planes. A new equivalent strain damage parameter based on critical plane is proposed in this study. Specifically, to quantify the effect of non-proportional loading on fatigue damage, a non-proportional degree of loading on the generalized plane is developed. Coupling with the existing non-proportional coefficient, a novel fatigue damage parameter is derived by combining a non-proportional factor. Experimental data of 304 stainless steel, sintered porous iron and CuZn37 brass are utilized for model validation and comparison. Fatigue lives with different load paths are evaluated respectively. In comparison with the Fatemi-Socie (FS), Chen and Itoh models, proposed method more effectively evaluate the multiaxial fatigue life of materials under non-proportional loadings.

On multiaxial creep–fatigue considering the non-proportional loading effect: Constitutive modeling, deformation mechanism, and life prediction

In this paper, a series of strain-controlled fatigue and creep–fatigue tests under proportional/non-proportional loadings were performed for type 304 stainless steel at 873 K. Then, post-test metallographic observations were performed through the electron back scattered diffraction (EBSD) and transmission electron microscope (TEM) combinative characterizations. In this aspect, the wavy slip dominated deformation mechanism under non-proportional loadings was considered as the essence for additional hardening, while the introduction of creep resulted in further microstructure evolutions by facilitating recrystallization. Afterward, a unified viscoplasticity constitutive model was proposed to simulate the cyclic stress–strain responses, in which an additional hardening parameter combined with a loading-path parameter was used to describe the cyclic hardening curves. Concurrently, stress triaxiality was introduced to provide accurate descriptions for the stress relaxation behavior. Semi-physical continuum damage models involving multiaxial damage factor and non-proportional strain energy parameter was proposed to predict the multiaxial creep–fatigue damage evaluations. Good agreements between experimental data and simulated results were achieved with the help of the proposed numerical procedures.

Multiaxial low cycle fatigue of notched 10CrNi3MoV steel and its undermatched welds

It is a fundamental step to quantify the multiaxial fatigue damage considering the complex service conditions for notched components or welded structures. In our study, multiaxial Low Cycle Fatigue (LCF) tests of notched specimens were conducted on 10CrNi3MoV high strength steel and related undermatched welds under proportional and non-proportional cyclic loadings. The characteristic states of normal/shear stress, strain and energy around notch roots are analyzed considering the elasto-plastic material behavior under low cyclic loadings by the cyclic plasticity model. The predicted fatigue life curves of the test data were constructed by five strain-based and energy-based parameters combining with the critical plane approach and notch Theory of Critical Distance (TCD). According to Smith-Watson-Topper (SWT) damage parameters definitions, the paper proposed a simplified SWT model incorporating the effect of non-proportional loading, notch energy concentration characteristics and cyclic hardening behaviors for the LCF assessment of notch specimens. The proposed model shows good agreements between predictions and test data by comparing different multiaxial fatigue damage models for notch specimens.

Neural network model predicting forming limits for Bi-linear strain paths

The Forming Limit Curve (FLC) describes the maximum in-plane strains that can be applied to sheet metal before necking initiates. It is well known that the FLC depends on the yield locus, the hardening behavior, strain rate, temperature, superposed out-of-plane stresses and the shape of the loading path. There is an infinite number of possible paths for reaching a particular state of strain which makes the formulation of simple analytical models predicting the effect of non-linear loading paths on the FLC particularly challenging. Here, it is demonstrated that a neural network model is able to provide a computationally-efficient approach to describe the effect of non-proportional loading. A finite-element-based Marciniak-Kuczynski model is developed to characterize the forming limits of a DP780 steel through virtual experiments. Subsequently, a large data base is generated to serve as a basis for machine learning. Using a standard backpropagation algorithm with Bayesian regularization, a compact neural network model is trained and tested based on the results from virtual experiments. Even though the experimental data base includes only a finite number of random loading paths, the results show that the neural network model is able to predict the FLCs for a given proportional pre-straining history.

Low cycle fatigue performance of DP600 steel under various pre-straining paths

The present work concerns the assessment of low cycle fatigue performance of dual phase steel subjected to change in uniaxial pre-strain path. An uniaxial pre-strain of 12.5% in the rolling or transverse direction were imposed on the dual phase steel blanks, thereafter fully reversed strain controlled fatigue tests were conducted parallel or orthogonal to the initial pre-straining direction. As received specimens depict mild cyclic hardening in the initial cycles followed by gradual cyclic softening until failure. However, all pre-strained materials exhibit continuous cyclic softening with higher magnitude and rate throughout the life as compared to the as received specimen. Relaxation of mean stresses were observed for the fatigue tests with pre-straining. Rotation of principal axis due to orthogonal reloading (i.e. low cycle fatigue) after initial pre-straining introduces non-proportionality, thereby leading to significant reduction of fatigue lives. In-grain misorientation and lattice rotation analyses are noticed during electron backscattered diffraction investigation, and which is due to the non-proportional loading effect.

Taking into account the non-proportional loading effect on high cycle fatigue life predictions obtained by invariant-based approaches

Industrial structures are often subjected to multiaxial fatigue loadings. If the multiple stress signals are not synced the loading is said to be non-proportional. Most of the multiaxial fatigue criteria give highly inaccurate lifetime predictions when used in the case of such loadings. The scalar equivalent stress defined by the criteria does not take into account the non-proportional nature of the multiaxial loading and leads to non-conservative predictions. Moreover a multiaxial fatigue criterion can only be applied on a stress cycle which has no clear definition when multiple unsynced signals are to be considered. This study addresses these issues by proposing a correction of an invariant based multiaxial fatigue criterion through the definition of a non-proportional degree indicator. A definition of multiaxial cycle is also given based on the Wang-Brown method. Finally a complete chain of invariant based lifetime prediction for non-proportional multiaxial fatigue is validated.

Forming Limits Under Stretch-Bending Through Distortionless and Distortional Anisotropic Hardening

The necking behavior of sheet metals under stretch-bending process is a challenge for the forming limit prediction. State-of-the-art forming limit curves (FLCs) allow the prediction under the in-plane stretching but fall short in the case under out-of-plane loading condition. To account for the bending and straightening deformation when sheet metal enters a die cavity or slide along a radius, anisotropic hardening model is essential to reflect the nonproportional loading effect on stress evolution. This paper aims to revisit the M-K analysis under the stretch-bending condition and extend it to accommodate both distortionless and distortional anisotropic hardening behavior. Furthermore, hardening models are calibrated based on the same material response. Then the detailed comparison is proposed for providing better insight into the numerical prediction and necking behavior. Finally, the evolution of the yield surface and stress transition states is examined. It is found that the forming limit prediction under stretch-bending condition through the M-K analysis strongly depends on the employed anisotropic hardening model.

Analysis of soil liquefaction using centrifuge tests of a site subjected to biaxial shaking

This study investigated the dynamic response and liquefaction of level saturated-sand deposits subjected to biaxial shaking using a number of centrifuge tests. Dense and loose sandy soil deposits were built in a 2D laminar container and subjected to a series of biaxial base excitations that approximate in somewhat realistic fashion the conditions of a site subjected to earthquake shaking. A dense array of accelerometers and pore pressure sensors was used to monitor the deposit response. The recorded accelerations and pore pressures were employed along with a non-parametric identification procedure to estimate the corresponding dynamic shear stress-strain histories. In turn, these histories were employed to assess the effects of non-proportional loading on soil contraction and dilation mechanisms. Pore water pressure buildup was found to be affected by load non-proportionality and a direct function of the phase angle of the induced shear stresses. The loose and dense soil deposits had contrasting as well as similar response patterns.

Fretting fatigue damage nucleation under out of phase loading using a continuum damage model for non-proportional loading

Abstract Damage nucleation involves creation of micro cracks, which are discontinuities in a material considered as continuous at a larger scale. Continuum Damage Mechanics (CDM) approach provides a tool to study damage nucleation under plane and fretting fatigue conditions. Under fretting fatigue conditions, the loading sequence may produce non-proportional stresses. This paper aims to investigate the effect of non-proportional loading on damage nucleation. For this purpose, a CDM based damage model for non-proportional loading is developed and applied to cylindrical pad and flat specimen configuration. The numerical results are also compared with experimental results from literature. It is found that, taking into account the triaxiality function variation in damage law, improves crack initiation lifetime estimation. In addition, a sensitivity analysis is performed by varying stress range and triaxiality function. It is found that for in phase loading only the stress range affects the initiation life, whereas, for out of phase loading both stress range and triaxiality function affect initiation life, especially at higher ranges.

Multiaxial fatigue study on steel transversal attachments under constant amplitude proportional and non-proportional loadings

In this study, the multiaxial fatigue strength of full-scale transversal attachment is assessed and compared to original experimental results and others found in the literature. Mild strength S235JR steel is used and an exploratory investigation on the use of high strength S690QL steel and the effect of non-proportional loading is presented. The study focuses on non-load carrying fillet welds as commonly used in bridge design and more generally between main girders and struts. The experimental program includes 33 uniaxial and multiaxial fatigue tests and was partially carried out on a new multiaxial setup that allows proportional and non-proportional tests in a typical welded detail. The fatigue life is then compared with estimations obtained from local approaches with the help of 3D finite element models. The multiaxial fatigue life assessment with some of the well-known local approaches is shown to be suited to the analysis under multiaxial stress states. The accuracy of each models and approaches is compared to the experimental values considering all the previously cited parameters.

Multiaxial fatigue strength of common structural steel and the response of some estimation methods

This paper describes the multiaxial fatigue strengths obtained from a series of load-controlled fatigue experiments in the high-cycle fatigue domain. The experiments were carried out on unnotched specimens manufactured from common structural steel CSN 41 1523. Various load combinations were induced, including some rare combinations, e.g. a combination of pressurizing and plane bending. The stress states on the inner surface and also on the outer surface of hollow specimens are therefore evaluated in the subsequent fatigue strength analyses. The goal of the paper is to derive a new solution for estimating multiaxial fatigue strength that will better conform to the described experiments than the Manson-McKnight method, but which will keep its simplicity of calculation. Our final proposal – the MMP criterion – is compared with various versions of the Manson-McKnight method, and also with four multiaxial methods (Dang Van, Liu-Zenner, Crossland and Papuga PCr). The results for the MMP criterion are quite motivating, thanks to the simplicity of the computational algorithm and the quality of the prediction output, which is comparable with the output of the evaluated multiaxial methods. However, the set of experiments described here is not large enough either for more thorough statistical analyses of prediction quality, or for more enhanced improvement of the MMP criterion, which would include e.g. the effect of non-proportional loading.

Fretting fatigue crack propagation lifetime prediction in cylindrical contact using an extended MTS criterion for non-proportional loading

Fretting fatigue is a phenomenon that may lead to premature failure of many common mechanical components. Under fretting conditions, the crack growth phase is considerably different from plain fatigue, because of the effect of the contact stresses distributions on the crack and vice-versa. Being able to accurately model those stresses and the propagation phase, finite element method has become an interesting tool to predict lives of components under fretting conditions. For the modelling of propagation stage, it is known that classical linear elastic fracture mechanics orientation criteria do not predict correct crack paths under fretting conditions. In the present paper, an extension of the maximum tangential stress (MTS) orientation criterion taking into account the effects of non-proportional loading and crack face interactions is considered with the goal of improving the predictions of crack path and propagation lives for a cylindrical pad configuration. It is found that the extended MTS criterion provides better crack path prediction than the classical MTS criterion in case of cylindrical fretting fatigue problem.

An equivalent stress parameter for multi-axial fatigue evaluation of welded components including non-proportional loading effects

This paper presents a comprehensive investigation into non-proportional multi-axial fatigue of welded components by introducing an equivalent structural stress parameter that takes into account of load-path non-proportionality in addition to plate thickness and stress state effects. This is accomplished by formulating a “moment of load path” or “MLP” based fatigue damage parameter that provides a consistent treatment of load-path non-proportionality under arbitrary multi-axial loading conditions for which cycle counting can be consistently performed by means of a previously developed path-dependent maximum range (PDMR) cycle counting procedure. To examine its broad applicability and effectiveness, non-proportional multi-axial test data obtained using different components, joint types, and loading conditions from various sources are analyzed using the newly developed equivalent stress parameter. The results show that the new equivalent stress parameter enables not only an effective consolidation of all multi-axial test data (up to about 300 tests) analyzed in this paper into a narrow band, but also the demonstrated transferability between the master S-N curve (dominated by test data under uniaxial cyclic loading conditions) adopted by the 2007 ASME Div 2 and API 579 RP/ASME FFS-1 Codes and the consolidated S-N curve dominated by severe non-proportional multi-axial cyclic loading conditions. As a result of the present development, a unified fatigue evaluation procedure based on the newly proposed effective stress parameter and a single master S-N curve can be implemented for arbitrary cyclic loading conditions regardless of stress multi-axiality or load path proportionality.

The effects of nonproportional loading on the elastic-plastic crack-tip fields

In this paper, the biaxial loading path effects on the mode I plane strain elastic-plastic crack-tip stress fields are investigated computationally. First, three different loading sequences including one proportional loading and two non-proportional loading paths are applied to the modified boundary layer (MBL) model under small-scale yielding conditions. For the same external displacement field applied at the outer boundary of the MBL model, the mode I K field and T-stress field combined as the different loading paths are applied to investigate the influence of the nonproportional loading. The results show that for either the compressive or tensile T-stress, the loading path which applied K field first followed by T-stress field generates the lower crack-tip constraint comparing to proportional loading. There is only minor difference between the results from proportional loading path and that with the T-stress field applied first following by K field. Next, two finite width specimens under non-proportional biaxial loading conditions that generate the same three loading paths are analyzed. Similar crack tip characteristics are observed in these specimens as these obtained from the MBL model, and it is demonstrated that the near-tip behavior in specimens can be predicted accurately using the results from MBL models. The present results show that it is very important to include the load sequence effects in elastic-plastic fracture analysis when dealing with nonproportional loading conditions.

The Effects of Nonproportional Loading on the Elastic-Plastic Crack-Tip Fields

In this paper, the loading path effects on the plane strain elastic-plastic crack-tip stress field are investigated computationally. Three different loading sequences include one proportional loading and two non-proportional loading paths are applied to the modified boundary layer (MBL) model under small-scale yielding conditions. For the same external displacement field applied at the outer boundary of the MBL model, the mode I K field and T-stress field combined as the different loading paths are applied to investigate the influence of the nonproportional loading. The results show that for either the compressive or tensional T-stress, the loading path which applied K field followed by T field generates the lower crack-tip constraint. There is only slightly difference between the proportional loading path and that with the T-stress field following by K field. The results show that it is very important to include the load sequence effects in fracture analysis when dealing with nonproportional loading conditions.

Ductile fracture of aluminum 2024-T351 under proportional and non-proportional multi-axial loading: Bao–Wierzbicki results revisited

The effect of stress state and loading path on the ductile fracture of aluminum 2024-T351 is characterized through tension–torsion experiments on tubular specimens. The experimental program includes proportional and non-proportional loading paths leading to the onset of fracture at nearly plane stress conditions at stress triaxialities between 0 and 0.6. Stereo digital image correlation is used to measure the displacements and rotations applied to the specimen shoulders. An isotropic non-quadratic Hosford plasticity model with combined Voce–Swift hardening is used to obtain estimates of the local stress and strain fields within the specimen gage section. The hybrid experimental–numerical results indicate a higher strain to fracture for pure shear than for uniaxial tension. The calibration of a Hosford–Coulomb fracture initiation model suggests that the ductility of aluminum 2024-T351 decreases monotonically as a function of the stress triaxiality, whereas it is a non-symmetric convex function of the Lode angle parameter. It is shown that a simple non-linear damage accumulation rule can describe the effect of non-proportional loading on the strain to fracture.

High temperature nonproportional low cycle fatigue using fifteen loading paths

This paper discusses effects of nonproportional loading on low cycle fatigue lives for Type 304 stainless steel hollow cylinder specimens at 923K. Strain controlled axial–torsion low cycle fatigue tests were performed using fifteen proportional and nonproportional strain paths at a strain rate of 0.1%/s. Nonproportional straining significantly reduced fatigue lives of the steel at 923K. Reduction in fatigue life due to nonproportional straining reached about 80% compared with the proportional straining in specific strain paths. The nonproportional strains, ΔɛNP and ΔεNP∗, and energy parameter, ΔσIΔɛI, successfully correlated proportional and nonproportional fatigue lives but ASME equivalent strain gave significantly unconservative estimates.