Non-proportional Loading Research Articles

Numerical Study on a Ductile Fracture Model in Pre-Cracked Tension Tests of SUS304L.

The effectiveness of a ductile fracture model in accurately predicting fracture initiation has been demonstrated. In this study, we concentrate on applying the ductile fracture model to pre-cracked structures constructed from SUS304L stainless steel with experimental and numerical analyses. The Swift hardening law was employed to extend the plastic behavior beyond the onset of necking. Additionally, the Hosford-Coulomb model, integrated with a damaged framework, was utilized to predict ductile fracture behavior, particularly under non-proportional loading conditions. Tension tests were conducted on various specimens designed to illustrate various fracture modes resulting from geometric effects. Numerical analyses were conducted to explore the loading histories, utilizing an optimization process to calibrate fracture model parameters. The proposed fracture model is validated against pre-cracked structures detailed in a reference paper. The results convincingly demonstrate that the fracture model effectively predicts both fracture initiation and propagation in pre-cracked structures.

Phase field fracture in elasto-plastic solids: Considering complex loading history for crushing simulations

Phase field models have gained growing popularity in fracture analysis. However, phase field modelling remains lacking for complex real-life applications such as structural crushing analyses to date. The novelty of this study is to propose an explicit phase field modelling framework considering complex loading history to predict the crushing behaviour of additively manufactured metallic structures. First, the explicit phase field model is formulated by considering stress state dependent fracture initiations with non-proportional loading. Then, five types of material specimens with a wide range of stress states are experimentally tested to calibrate material parameters for additively manufactured 316L steel and Ti-6Al-4V titanium. Last, the square tubes are 3D-printed and tested under three-point bending and axial compression to demonstrate the capacity of the proposed model. The modelling results unveil that the crushing behaviour of both materials could be properly reproduced in terms of force-displacement curves and crack paths. Finally, the fracture mechanism of crushing deformation is analysed. For the three-point bending of Ti-6Al-4V tubes, cracks are mainly induced by the medium stress triaxiality tension and shear loading. For the axial compression of 316L tubes, the stress states of critical elements are relatively diverse yet slightly concentrated at the shear, compression-shear, and high stress triaxiality tension regions. Remarkably, non-proportional loading is significant in crushing deformation as a material point may experience a significant change in stress state with loading. By considering a non-proportional loading dependent threshold, the proposed phase field model can predict the crushing behaviour of structures even when the fracture initiations do not locate on the fracture locus. This study provides an effective explicit phase field framework featured with an insightful fracture mechanism for quasi-static crushing of additively manufactured metallic materials.

Assessing Fatigue in Materials with Small Defects: A New Multiaxial Model Based on Principal Stress Amplitudes

A new multiaxial fatigue model for materials containing small defects proposed by the authors is presented, which one can be considered as a modification of the SWT model. This new model relates the fatigue limit of the material obtained from the √area parameter with values associated with the principal stresses. These values are the amplitude of the principal stresses and the maximum principal stress observed. The amplitude value associated with the principal stresses is defined by using the Maximum Variance Method. This amplitude only can be easily obtained under uniaxial loading conditions, but its calculation for torsion, proportional and non-proportional multiaxial loadings is not trivial. Therefore, the calculation of the principal stress amplitude posed a challenge not yet addressed by other authors. The multiaxial fatigue model was evaluated with experimental data from AISI 4140 steel, with several different loading conditions, including uniaxial loading, combined loads in in-phase and out-of-phase configurations. In addition, two types of specimens were used: smooth cylindrical specimens and specimens with a surface micro hole. Comparing the experimental data with the prediction of the new model it was observed that the predictions are slightly conservative with average error not exceeding 6%.

Determination of fracture mechanics parameters by digital image correlation method of surface crack for 7075 Al alloy under nonproportional cyclic loading (Growth life prediction method considering shear stress on the crack surface)

Determination of fracture mechanics parameters by digital image correlation method of surface crack for 7075 Al alloy under nonproportional cyclic loading (Growth life prediction method considering shear stress on the crack surface)

A technique for estimation of cyclic stress-strain curves using an optical strain measurement and its application to additively manufactured AlSi10Mg

The article describes a new accelerated technique for estimation of cyclic stress-strain curves of cyclically stable materials based on a full-field measurement of strain during low-cycle testing on a large amplitude of loading. Any optical method suitable for capturing the strain contours on the curved part of the specimen during cyclic loading is applicable in the approach developed. In this study, the digital image correlation method is used to measure the deformation on the curved part of the specimen. The technique is presented on chosen results of low-cycle fatigue tests performed on AlSi10Mg printed with selective laser melting technology. Proportional loading cases were performed in tension-compression and pure torsion. The case of non-proportional loading was realized in the form of 90 degree out-of-phase tests. The stress-strain behavior is evaluated using a standard procedure and an accelerated technique with a good correlation for the three loading cases.

Stress-strain Deformation Analysis of Conventional Vehicle Shock Absorber Materials under In-service Multi-translated Non-proportional Loading Conditions

Stress-strain Deformation Analysis of Conventional Vehicle Shock Absorber Materials under In-service Multi-translated Non-proportional Loading Conditions

Develop A new mathematical approach to predict the crack propagation rate in a thin plate subjected to in-plane biaxial loading

The mechanical design of metals must deal with structures collapsing due to cracks occur in the material. This study focused on the edge crack at non-proportional cycle loading because it propagates quickly. This research attempted to determine two mathematical models that could predict the crack propagation rate for thin samples of aluminum alloy types 6061 and 5052 under constant tensile stresses and cyclic shear stress applied within the elastic limits of the material using “Griffith energy of dynamic fracture”. This was performed to evaluate if these models can predict crack growth rates and compare with the numerical results of a computer system in the ANSYS program R1 2021. The direction of the crack path was calculated and compared with the analytical program. The results of the two mathematical models in predicting the dynamic growth rate in the studied alloys gave low error rates to the numerical solution.

Comparison of critical plane models for multiaxial fatigue life prediction

Background. The operation of numerous machines and units takes place under conditions of multi-axial cyclic loading, which, as a rule, is non-proportional. Evaluating the fatigue durability of metal alloys under conditions of multi-axial non-proportional loading is a relevant task in modern engineering. Solving this problem requires fatigue calculation methods that would consider operating conditions and properties of structural materials, including factors such as the type of stress state, loading trajectory, material sensitivity to non-proportional loading, and so on.Objective. To conduct a comparative analysis of a range of fatigue life models based on the concept of the critical plane, including the Fatemi-Socie, Wang-Brown, Smith-Watson-Topper, Liu I, and Liu II approaches, and to identify the limits and peculiarities of their application.Methods. The fatigue lives calculated using the selected models were compared with experimental results obtained for various metal alloys subjected to uniaxial tension-compression, alternating torsion, and proportional and non-proportional multiaxial loading.Results. The applicability limits of fatigue life models based on the critical plane concept were analyzed for different metal alloys under conditions of proportional and non-proportional multiaxial loading.Conclusions. The research results demonstrated that models requiring the use of material constants obtained from tests in both tension-compression and alternating torsion provide reliable fatigue life estimates for various types of metal alloys. Calculations based solely on fatigue curves from alternating torsion better correlate with the results of tests on ductile materials, while calculations based on criteria utilizing fatigue curves from tension-compression align more closely with results from tests on brittle materials.

Fatigue assessment of a FSAE car rear upright by a closed form solution of the critical plane method

Material fatigue is extensively discussed and researched within scientific and industrial communities. Fatigue damage poses a significant challenge for both metallic and non-metallic components, often resulting in unexpected failures of in-service parts. Within multiaxial fatigue assessment, critical plane methods have gained importance due to their ability to characterize a component's critical location and detect early crack propagation. However, the conventional approach to calculate critical plane factors is time-consuming, making it primarily suitable for research purposes or when critical regions are already known. In many real-world scenarios, identifying the critical area of a component is difficult due to complex geometries, varying loads, or time limitations. This challenge becomes particularly crucial after topological optimization of components and in the context of lightweight design. Recently, the authors proposed an efficient method for evaluating critical plane factors in closed form, applicable to all cases that necessitate the maximization of specific parameters based on stress and strain components or their combination. This paper presents and validates the proposed methodology, with reference to a rear upright of a FSAE car, which is characterized by a complex geometry and is subjected to non-proportional loading conditions. The efficient algorithm demonstrated a substantial reduction in computation time compared to the standard plane scanning method, while maintaining solution accuracy.

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.

Mechanical Responses of Ductile Aluminum Alloy under Biaxial Non-Proportional Tensile Reverse Loading Patterns

This paper deals with the study of the mechanical responses of ductile metals under biaxial non-proportional cyclic loading tests. The biaxially loaded HC specimens manufactured from 4 mm thick aluminum alloy sheets (EN AW 6082-T6) are subjected to various loading paths, including monotonic and cyclic loading conditions. The aim is to investigate the plastic, damage, and fracture behavior of the material under these different loading scenarios. In terms of numerical aspects, a modified anisotropic two-surface cyclic plastic–damage continuum model is used to predict the material behavior in the load-displacement field and different strain fields. Numerically predicted stress states are analyzed in detail to gain a better understanding of the damage mechanisms. Moreover, the scanning electronic microscopy (SEM) pictures taken from the fracture surfaces confirm the dependency of the damage mechanisms on the loading histories. The present work indicates the importance of considering different loading conditions for the accurate prediction of material responses.

Numerical analysis of non-proportional biaxial reverse experiments with a two-surface anisotropic cyclic plasticity-damage approach

This paper deals with the numerical analysis of ductile damage and fracture behavior under non-proportional biaxial reverse loading conditions. A two-surface anisotropic cyclic elastic–plastic-damage continuum model is adequately presented, which takes into account the Bauschinger effect, the stress-differential effect, and the change of hardening rate after reverse loading. An efficient Euler explicit numerical integration algorithm, based on the inelastic (plastic or plastic-damage) predictor-elastic corrector approach, is utilized to analyze the stress and finite strain loading histories. Detailed discussions are provided on different numerical integration-related consistent tangent operators that achieve convergence within the global Newton–Raphson scheme. The proposed continuum model is implemented into the commercial software Ansys as a user-defined subroutine (UMAT). Furthermore, the novel non-proportional biaxial tensile reverse experiments are performed to validate the proposed continuum model. The associated numerical simulations investigate the stability and accuracy of the proposed algorithm and material model.

Damage and fracture behavior under non-proportional biaxial reverse loading in ductile metals: Experiments and material modeling

This paper deals with a modified anisotropic stress-state-dependent plastic-damage continuum model incorporating combined hardening and softening rules to predict the plastic, damage, and fracture behavior of ductile metals under monotonic and reverse loading conditions. In the experimental part, a newly designed biaxially loaded cruciform flat HC-specimen of aluminum alloy EN AW 6082-T6 was subjected to biaxial non-proportional loading to generate a wide range of stress triaxialities during experiments. Thus, it is enabled to perform shear monotonic and reverse loading superimposed by various preloads without unloading processes. The experimental data reveal the Bauschinger effect, the strength-differential (SD) effect, and the change in the hardening ratio after shear reverse loading. This is captured by a Drucker–Prager-type yield criterion with an extended Voce isotropic hardening and modified Chaboche’s kinematic hardening rule. In addition, the damage surface is assumed to be translated in the direction of the rate of the damage strain. The numerically predicted global force–displacement curves and local strain fields are verified in terms of the digital image correlation (DIC) technique. Moreover, the scanning electron microscopy (SEM) is used to examine the fracture surfaces and to validate the proposed damage constitutive model.

Multiaxial fatigue of water pipe grey cast iron

Grey Cast Iron (GCI) water pipes are subjected to multiaxial, cyclic stresses caused by combinations of loads such as internal water pressure and road vehicle weight. However, the multiaxial fatigue performance of this material has not previously been characterised. To address this gap more than 45 fatigue tests, including some under non-proportional tension-torsion loading, were completed using a GCI material very similar to water pipe GCI. Of the four multiaxial fatigue criteria tested, the Smith-Watson-Topper (SWT) criterion provided the best predictions by a narrow margin, supporting the idea that a tensile cracking mode dominates the fatigue life of GCI.

A new multiaxial fatigue endurance model for high strength steels taking into account the presence of small defects

In the present work, a new multiaxial fatigue model is proposed for metallic material with small defects. This model correlates the uniaxial fatigue limit obtained from the area parameter of a high strength steel containing small defect with values associated with the amplitude of the principal stresses, whose calculation for non-proportional loadings is not trivial. The performance of the model was evaluated with experimental data from AISI 4140 steel, under various loading conditions, in which smooth cylindrical bars were considered, regarding: (i) the effect of non-metallic inclusions in the material, and (ii) superficial micro hole with a straight bottom machined in the material (550 μm in diameter and depth). The proposed model has provided accurate fatigue strength estimates with average error not exceeding 16 %.

A novel approach to multiaxial fatigue life prediction using the critical plane and phase difference angle

Multiaxial loads are the main cause of fatigue failure in engineering applications, where this failure affects the components and structure of these applications directly. The probability of failures can be decreased by using multiaxial fatigue analysis to assess the service life of components. This study proposes a simple and effective approach for measuring fatigue life in the presence of multiaxial loading. The proposed damage parameter includes terms such as the critical plane and phase difference angles. Calculating fatigue life using this model under proportional and nonproportional loading conditions is possible. The accuracy and validity of the model were tested by comparing the experimental results of four experimental datasets with the life estimation results. The analysis results of four metal materials show that the predicted life correlated well with the experimental observations. In addition to the proposed model, five famous fatigue prediction models were used, and the results of the proposed model were compared with these models. The suggested damage parameter is better for predicting fatigue life under different loads. This was shown through model comparison and prediction error analysis. The findings show good agreement between the suggested model's life estimation results and the experimental findings. Predicting fatigue life becomes more realistic with the suggested approach for any angle between 0 and 90°. The proposed model showed the highest accuracy among the tested models. Afterwards, the Fatemi and Socie FS, Kandil Brown and Miller (KBM), Smith Watson and Topper (SWT), maximum share stress (MSS), and ES models demonstrated lower levels of prediction error, respectively.

Phase transitions of carbon nanotube bundles under non-proportional triaxial compressions

Lateral compressions of (n, n) and (n, 0) single-walled carbon nanotube (SWCNT) bundles are simulated by density functional theory. Feasible transition pathways are verified by the body-centered tetragonal C4 (bct-C4) phase from the (4, 4) bundle and the carbon-centered orthorhombic C8 (Cco-C8) phase from (6, 6) and (8, 8) bundles. Three new phases, the sp3-hybridized phases Cco-C16 from the (4, 0) bundle and Cco-C32 from the (8, 0) bundle, the hexagonal phase Hex-C24 composed of sp2- and sp3-hybridized carbons from the (6, 0) bundle, under non-proportional triaxial loading are predicted. Measured hardness values for Cco-C16/C32 exceed 40 GPa, indicating super-hardness. Electronic band structures of Cco-C16/C32 exhibit a 3.64 eV bandgap, while Hex-C24 exhibits metallic carbon properties. Our results provide the potential phase transition pathways of SWCNT bundles under non-proportional compression.

A large strain anisotropic ductile damage model — Effective driving forces and gradient-enhancement of damage vs. plasticity

A novel anisotropic ductile damage model at finite strains is proposed in this work. The model is formulated based on the fictitious configuration concept and includes deformation induced evolution of anisotropic damage properties. As a key contribution, different effective driving forces for both, plasticity and damage are elaborated in the context of a so-called two-surface formulation, i.e. the consideration of two admissible (but not independent) domains — one related to plastic flow and one related to damage evolution. The investigations include the introduction of effective stresses driving plastic flow on the one hand, and effective driving forces related to the evolution of anisotropic damage on the other. In addition and as an alternative to the introduction of effective driving forces, contributions influencing the respective threshold values related to plastic flow and damage evolution are proposed. Specification of the underlying constitutive equations together with the introduction of a suitable algorithmic implementation allows the model to predict experimentally observed material behaviour. Moreover, fundamental properties of the model are discussed by means of non-proportional loading paths generated by combinations of loading under uniaxial tension and simple shear. In view of the simulation of inhomogeneous boundary value problems, the implementation of the model into a finite element formulation is discussed. Since softening, respectively material degradation and necking phenomena, may results in mesh-dependent simulation results, an appropriate regularisation must be included in the model. In this work, gradient-enhancements in the form of a so-called micromorphic approach are included. In particular, gradients of scalar-valued hardening variables related to plasticity and damage are accounted for. A detailed analysis of the necessity of these gradient-enhancements of both, damage and plasticity is carried out based on the proposed multi-surface anisotropic ductile damage model. It turns out that gradient-enhanced regularisation of the damage contribution alone is not sufficient to regularise the problem (and vice versa, i.e. gradient-enhanced regularisation of the plasticity contribution alone is not sufficient as well) but contributions related to both phenomena, i.e. damage and plasticity, need to be regularised.

Ductile fracture of anisotropic QP980 steel sheet by using the isotropic/anisotropic modified Mohr-Coulomb models

The modified isotropic Mohr-Coulomb (iso-MMC) fracture model, which utilizes the Mises yield function and isotropic hardening law, was calibrated through tensile tests on various specimens. The resulting calibrated model was constructed in (η, L, ε¯f) space to represent the isotropic fracture locus of QP980 steel sheet. Subsequently, this calibrated model was utilized to predict the ductile fracture behavior of QP980 steel sheet loaded along rolling direction under different stress states. To comprehensively consider the impact of anisotropic properties on fracture evolution, an anisotropic MMC (aniso-MMC) ductile fracture model was developed in terms of strain space. Under the assumption of plane stress, the Mohr-Coulomb (MC) criterion was converted from stress space to strain space through an anisotropic constitutive model in order to characterize ductile fracture in QP980 steel sheet. The quadratic yield function of Hill's 48 yield function with the isotropic hardening law was employed to characterize the plastic behavior during loading. The anisotropic fracture loci of QP980 steel sheet were obtained in various loading directions. The scanning electron microscope (SEM) was utilized to characterize the microscopic morphology of the fracture surfaces of QP980 steel sheet, investigating the fracture mechanism of anisotropic ductility under different stress states. The results of both tensile experiments and simulations have confirmed that the plastic anisotropy of materials plays a critical role in determining ductile fracture behavior. The aniso-MMC fracture model was ultimately validated through hole expansion tests and equi-biaxial tensile tests, demonstrating its capability to predict ductile fracture behavior under both proportional and non-proportional loading conditions.

Topology optimization for infinite fatigue life of cyclic symmetric structures subjected to non-proportional loading

This paper presents a density based topology optimization method for infinite fatigue life constraints of non-proportional load cases, with a specific focus on parts with cyclic symmetry. Considering non-proportional loads in topology optimization significantly broadens the types of design problems that can be handled. The method estimates the local variation in Signed von Mises stress using a smooth min/max function and constrains the resulting stress amplitude using established stress based topology optimization methods. Accounting for non-proportionality of loading significantly increases the computation cost with respect to existing proportional methods, as the time-varying stress field needs to be computed. Inertia effects are neglected in the structural analysis. Therefore, a quasi-static analysis is used to obtain the stress history. To reduce the computational cost, advantage is taken of cyclic symmetric properties to reduce the number of necessary time steps to evaluate. This reduces the computational cost roughly proportional to the number of unique load time steps present in the repeated segments as opposed to a standard implementation. The method is tested on numerical examples in 2D and 3D for both proportional and non-proportional loads and was found to be locally accurate up to the accuracy of the constraint aggregation.