Proportional Loading Conditions Research Articles

Topology optimization for fatigue reserve factors

This paper describes a topology optimization approach that applies the common fatigue analysis practices of rainflow cycle counting and critical plane searches to cover both proportional and non-proportional fatigue loading conditions of metals. The existing literature on topology optimization has so far mainly considered fatigue damage under proportional loading conditions and typically uses continuous damage models to avoid the discontinuous nature of fatigue rainflow cycle counting and critical plane searches. Furthermore, previous publications often introduced heuristic schemes to scale the fatigue damage and set the move limits for the design variables rather low to avoid oscillations in the design variables and damage responses during the optimization iterations, because fatigue damage is typically highly localized. Therefore, these approaches cause many optimization iterations. Contrarily, our present approach applies the fatigue reserve factor (FRF) directly in the optimization formulation instead of the fatigue damage where FRF is a fatigue reserve factor for infinite fatigue life. The inverse FRF scales nearly linearly with the stresses. Therefore, the present approach needs no heuristic scaling for the fatigue topology optimization. The numerical implementation applies the semi-analytic adjoint sensitivity method for multiple load cases. Numerically, FRF shows more stable optimization convergence using less optimization iterations. Different FRF topology-optimized designs for a variety of fatigue damage types are validated and compared. Additionally, the optimized FRF designs are compared to both strictly stiffness optimized designs and stress strength optimized designs.

A closed-form solution for evaluating the Findley critical plane factor

The fatigue assessment of structural components is a significant topic investigated both in the academia and industry. Despite the significant progress in comprehension over the past few decades, fatigue damage remains a significant challenge, often leading to unexpected component failures. One commonly used approach for fatigue assessment is the critical plane analysis, which aids in identifying the critical location and early crack propagation direction in a component. However, the conventional method for calculating critical plane factors is computationally demanding and is typically utilized only when the critical regions of the component are already known. In situations where the critical areas are difficult to be identified due to complex geometry, loads, or constraints, a more efficient method is required for evaluating critical plane factors. This research paper introduces an analytical algorithm to efficiently evaluates the widely used Findley critical plane factor. The algorithm operates within the framework of linear-elastic material behavior and proportional loading conditions, relying on tensor invariants and coordinate transformation laws. The algorithm has been tested on different component geometries, including a box-welded joint and a tubular specimen, subjected to proportional loading conditions such as tension, torsion, and a combination of them. The analytical method allowed a significant reduction in computation time while providing the exact solution of critical plane factor and critical plane orientations.

Closed‐form solution for the Fatemi‐Socie extended critical plane parameter in case of linear elasticity and proportional loading

AbstractFatigue damage remains a significant issue for both metallic and non‐metallic components, being the main cause of in‐service failures. Among the different assessment methodologies, critical plane methods have gained significance as they enable identifying the critical location and the early crack propagation orientation. However, the standard plane scanning method used for calculating critical plane factors is computationally intensive, and as a result, it is usually applied only when the component's critical region is known in advance. In the presence of complex geometries, loads, or constraints, a more efficient method would be required. This work presents a closed‐form solution to efficiently evaluate a critical plane factor based on the Fatemi‐Socie criterion, in the case of isotropic linear‐elastic material behavior and proportional loading conditions. The proposed algorithm, based on tensor invariants and coordinate transformation laws, was tested on different case studies under various loading conditions, showing a significant reduction in computation time compared to the standard plane scanning method.

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.

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.

Mesoscopic unit cell analysis of ductile failure under plane stress conditions

Ductile failure under plane stress conditions is analyzed at the meso-scale using periodic unit cell model simulations of void growth in a plastically deforming matrix. Equivalent strains to failure by the onset of plastic instability at the macro-scale are estimated using the loss of ellipticity criterion for the equilibrium equations. Failure loci obtained from the cell model simulations are compared with the predictions of an instability-based ductile failure model and the Hosford–Coulomb damage indicator model, under both proportional and non-proportional loading conditions. The instability-based model is shown to quantitatively predict the shape of the failure locus under proportional loading, including the presence of a cusp at uniaxial tension and a ductility minimum under plane strain tension, in the absence of heuristic adjustable parameters in the failure criterion. It is shown that the characteristic shape of the plane stress failure locus is primarily due to the Lode dependence of the failure criterion, and not the damage growth law as assumed in the damage indicator models. Under non-proportional loading involving a step change in loading direction at an intermediate strain, the instability-based model correctly predicts the non-linear variation of the failure strain as a function of the intermediate strain; unlike a linear variation predicted by the damage indicator models, which is not in agreement with the cell model simulations. Forming limit curves showing the strains to the onset of localized necking in thin sheets are also obtained from the cell model simulations using an appropriate modification of the macroscopic instability criterion.

The fatigue damage squeezing method: A signal compression strategy for accelerated life testing

Fatigue testing using full-length, in-service signals has historically been both time- and cost-prohibitive because the field environments of the test article typically reflect its multiyear service life. Current methods for accelerating fatigue tests do exist, but they often rely on heuristics, do not apply to multiaxial or dynamic loading situations, and/or use truncation methods that remove important load interaction effects, which may ultimately lead to uncharacteristic failure modes during testing. This paper presents the fatigue damage squeezing (FDS) method, a novel approach for systematically compressing fatigue test signals using the continuous wavelet transform. The FDS method uses an integrated fatigue model and predictions from finite element analysis to produce compressed signals that not only exhibit spectral and temporal characteristics that are similar to the original signals but also produce representative failure modes in a fraction of the test time. The method was validated using uniaxial and biaxial experimental tests with proportional loading conditions.

A linearization solution for elastic-plastic torsion problems by Edge-based smoothed finite element method

This paper proposes a new scheme for calculating the elastic-plastic uniform torsion of rods considering the hardening behavior of the material. Using PRANDTL-REUSS equation under Hencky proportional loading condition, we develop an elastic-plastic torsion governing equation that can be applied to materials with arbitrary multilinear hardening forms. The edge-based smoothed finite element method is applied to solve the boundary value problem of the above equation. In addition, the incremental scheme is used to calculate the plastic torque. Using the Lagrange interpolation method, the Mises stress is calculated by calculating the gradient in the cell. The nodal stresses are obtained by averaging the stresses at the centroid of the cells. After each load step, the overall domain is divided into elastic and plastic zones according to the yield stresses of the material, and then different governing equations are applied in the different zones. To summarize, this paper provides linearization solutions for nonlinear torsion.

Planar-Biaxial Thermo-mechanical Fatigue Behavior of Nickel-Base Superalloy Inconel 718 Under Proportional Loading

The planar-biaxial thermo-mechanical fatigue behavior of nickel-base superalloy Inconel 718 was studied for selected proportional loading conditions, in particular biaxial strain ratios of 1.0, 0.6, and − 1.0. The cyclic temperature loading with minimum and maximum temperatures of 400 °C and 630 °C and a duration of 250 seconds was either In-Phase or Out-of-Phase to the mechanical axes. Besides the multiaxial tests, uniaxial thermo-mechanical fatigue tests were conducted In-Phase and Out-of-Phase with the same temperature cycle and cycle duration. The performed thermo-mechanical fatigue tests were analyzed regarding the deformation and lifetime behavior and compared with high-temperature isothermal low-cycle fatigue tests from a previous work of the authors. On the side of the lifetime description, a strain- and a stress-based approach were presented. For the planar-biaxial tests, the crack initiation mechanism and crack paths were shown.

A damage coupled plastic-stress-state model considering proportional and non-proportional loading conditions

A damage coupled plastic-stress-state model considering proportional and non-proportional loading conditions

Dimensionality reduction through convolutional autoencoders for fracture patterns prediction

Dimensionality reduction methods construct a low dimensional or reduced space. Complex structural mechanics problems can be approximated on a low dimensional space with a Reduced-Order Model or surrogate model. For instance, Principal Components Analysis can compute a reduced basis from a database of full-order snapshots. Principal Components Analysis, however, does not reconstruct low dimensional spaces efficiently for highly nonlinear problems, mainly because it is a linear dimensionality reduction method. In this paper, an original approach based on an autoencoder neural network is developed to construct a nonlinear Reduced-Order Model for a highly nonlinear brittle fracture problem. We show on a set of simulations how the autoencoder can be efficient for dimensionality reduction or compression of highly nonlinear data. A complete deep learning framework is proposed to predict crack propagation patterns directly from the loading conditions. Finally, we validate our nonlinear Reduced-Order Model with data sets generated for two problems using proportional and non-proportional loading conditions. In the first problem, the crack always initiates at the same location but propagates in various directions, which may even vary during the simulation if loading conditions are non-proportional. In the second problem, the crack may additionally initiate at various locations. Results for the two problems evaluate the capabilities of the proposed approach.

Multi-axial fatigue life assessment of additively manufactured nickel-based superalloys

In the present work, the microstructures and mechanical properties of additively manufactured IN718 alloy were studied and the dendritic columnar microstructures elongated along the building direction are related to macroscopic mechanical properties. Detailed experiments under both proportional and non-proportional multi-axial loading conditions indicate that the influence of the building orientation from manufacturing on mechanical property and fatigue performance is negligible. Fractography analysis reveals that fatigue crack initiates from carbide and oxygen phases. The shear energy-based models reasonably predict the fatigue life and shear failure is the dominant failure mode for the laser melting material.

A new ductile fracture model for structural metals considering effects of stress state, strain hardening and micro-void shape

In this study, a new micromechanically-motivated phenomenological uncoupled fracture model is proposed for predicting the ductile fracture of structural metals considering the combinative effects of stress state, strain hardening and micro-void shape. The Lee–Mearvoid growth theory, which is used for characterizing the growth of ellipsoidal voids in a hardening matrix, is modified to derive the influence function of stress triaxiality, void shape and strain hardening on material’s ductility. Meanwhile, a J2-J3 based Bai–Wierzbicki’s yield function is used as a failure criterion and incorporated into the power-law hardening function to derive the Lode dependence of fracture strain. Further, a detailed parametric study is performed to demonstrate the influence of the material parameters on the 3D fracture envelope and 2D fracture locus of the new model. It is shown that the unique feature of the new fracture model is that it constructs an explicit and quantitative relationship between the ductility, strength and microstructure of metals. All parameters in the new model have intuitive physical meanings which correspond to specific material property indies controlling the material strength and evolution of micro-void defects. In the meantime, these model parameters can be easily calibrated via conventional tests, which greatly reduces the difficulty of the calibration work and makes the model being more friendly for engineering applications. Finally, test data of three structural metals, i.e. 2024-T351, 6061-T6 aluminum alloys and Q460 high strength structural steel, obtained under two loading conditions, i.e. proportional and non-proportional loading conditions, are used to calibrate and validate the proposed model. The accuracy of the new fracture model is then investigated by comparing its prediction result with that obtained from the modified Mohr–Coulomb and Hosford–Coulomb model. It is shown that the proposed new fracture model can predict the ductility limit of the investigated structural metals accurately under a wide range of stress states and provides the same good prediction accuracy as the other two typical models, which confirms the promising prospect of the new model in practical applications.

Crystal Plasticity Simulation of Yield Loci Evolution of SUS304 Foil.

The deformation process of metal foils is usually under a complex stress status, and the size effect has an obvious influence on the microforming process. To study the effect of grain orientation and grain size distribution on the yield loci evolution of SUS304 stainless steel foils, three representative volume element (RVE) models were built based on the open source tools NEPER and MTEX. In addition, the yield loci with different grain sizes are obtained by simulation with Duisseldorf Advanced Material Simulation Kit (DAMASK) under different proportional loading conditions. The initial yield loci show a remarkable difference in shape and size, mainly caused by the distinct texture characteristics. By comparing the crystal plasticity simulation with the experimental results, the model with normal grain size distribution and initial texture based on Electron Back-scattered Diffraction (EBSD) data can more accurately describe the influence of the size effect on the shape and size of yield loci, which is the result of the interaction of grain size distribution and texture. However, the enhancement of grain deformation coordination will weaken the impact of the size effect on yield loci shape if the grain size distribution is more uniform.

Extended GTN model for predicting ductile fracture under a broad range of stress states

Continuous damage accumulation resulting from void nucleation, growth, and coalescence leads to the final ductile fracture. The Gurson–Tvergaard–Needleman (GTN) model and its extensions, which model these mechanisms, are famous coupled constitutive models for predicting material failure. In this study, the GTN model was extended by incorporating two different damage variables: void volume fraction and shear damage; this was done to describe damage accumulation and ductile fracture under a wide range of stress states from compression-dominated to tension-dominated ones. The shear damage was composed of the shear void nucleation related to secondary voids and shear deformation damage. In addition, the shear void nucleation and void closure effect resulting from compression deformation were introduced to model the change in void volume under different loading conditions. The Lode angle dependence function was updated to express the various stress states and competition mechanism between two damage variables. Furthermore, the effective damage and its incremental form was used to define the extent of damage accumulation and fracture initiation. Subsequently, the proposed model was implemented in finite element software through user subroutines, and three distinct experiments related to high, low, and negative stress triaxiality were performed to calibrate the model parameters by the inverse analysis method for 1045 steel and 2024-T351 aluminum alloy. The experimental results under proportional and path-changing loading conditions were used to analyze the damage evolution and validate the proposed model. The good agreement between the numerical prediction and experimental results for the displacement and location at fracture, indicates good predictive ability of formulated model under arbitrary loading conditions.

Consideration of Fatigue Evaluation of SUS316 and SUS430 Steels under Multiaxial Stresses Combined Internal Pressure and Axial Load using Hollow Cylinder Specimen

This study focuses on discussing fatigue life evaluation method of SUS316 and SUS430 steels under wide ranged proportional multiaxial loading conditions. Fatigue tests under 7 types of loading path which are combined axial stress and inner pressure were carried out using hollow cylinder specimens at room temperature. It has been considered that the proportional loading does not cause additional hardening, however this study shows that the complex loading path causes additional hardening without principal direction change of stress and strain in a cycle. The Itoh-Sakane model which evaluates loading and lives under non-proportional multiaxial fatigue by only considering the maximum and the minimum principal stress or strain and their principal directions. However, this model is also needed to consider middle principal stress because change of the middle principal stress influence fatigue life under current loading paths. Therefore, the Itoh-Sakane model was modified to evaluation fatigue life of SUS316 and SUS430 under combined internal pressure and axial loading conditions.

User-friendly anisotropic hardening function with non-associated flow rule under the proportional loadings for BCC and FCC metals

This paper presents a user-friendly function under non-associate flow rule to capture anisotropic hardening (or evolution of yield surface) and the curvature of yield loci for body-centred cubic (BCC) and face-centred cubic (FCC) under the proportional loading conditions. The calibrated model for BCC and FCC metals is applied to illustrate the hardening behavior of uniaxial tension along different loading directions and of the equibiaxial tension for several steels and aluminum alloys (718AT, QP980, AA5182-O and AA 6xxx-T81) to evaluate its accuracy. The prediction is compared with the experimental results and modeling results of several popular plasticity models. The comparison demonstrates that the proposed model is pressure-insensitive, relatively simple for numerical implementation, capable of illustrating the different yielding behaviors between BCC and FCC metals and, capable of accurate modeling of flow curves under directional loading directions and equibiaxial tension both under the plane stress and triaxiality loading conditions. The proposed model is also flexible on adjusting the curvature of the yield surface between uniaxial tension and equibiaxial tension. Accordingly, the proposed anisotropic hardening model is useful to illustrate anisotropic hardening behaviors under uniaxial and equibiaxial tension.

A data-driven approach to predicting the anisotropic mechanical behaviour of voided single crystals

Machine learning techniques are increasingly used to extract important physical information from a broad range of materials and to identify their processing-structure–property relationships. In this work, a neural network framework is coupled with a crystal plasticity finite element based approach to predict the deformation and ductile failure behaviour of porous FCC single crystals when subjected to multi-axial loading conditions. The work relies on 3D unit cells with a centrally located spherical void to represent the microstructure of the porous single crystal material, and on a crystallographic slip-based crystal plasticity constitutive model to describe its deformation behaviour. Stress–strain data generated by unit cell finite element simulations are relied upon to construct the neural network model. Different strategies for the neural network input and output variables and parameters are first explored so as to optimise performance and accuracy. Both proportional and non-proportional loading conditions resulting from a constant and a varying stress triaxiality during deformation, respectively, are considered. The optimum neural network strategy is shown to successfully predict the behaviour of the porous single crystal under both proportional and non-proportional loading, albeit with the void behaviour at high stress triaxialities better described than that at low triaxialities. The results also reveal that the use of tensorial quantities for both stresses and strains as input and output neural network quantities is more suitable as a form of data representation for multiaxial loading conditions than uniaxial equivalent stress and strain quantities. It was also found that the inclusion of prior knowledge as neural network input quantities in the form of reference stress–strain solutions for the void-free single crystal considerably improves the predictive capabilities of the proposed data-driven approach, even when only a very limited number of training cases was used.

A critical plane-based multiaxial fatigue life prediction method considering the material sensitivity and the shear stress

The material sensitivity coefficient of Fatemi-Socie model (FS model) is proved to be not a constant for different materials and the effect of the shear stress on fatigue life is not considered. In this condition, a new multiaxial fatigue prediction model considering the shear stress and material sensitivity is established based on the critical plane method in this paper. Firstly, the material plane with the maximum shear strain amplitude is defined as the critical plane and a calculating method to determine the location of critical plane is presented. Secondly, a calculating method of material sensitivity coefficient k' is proposed considering the effects of both the stress and strain on the critical plane by investigating the relationship between material sensitivity coefficient and the multiaxial fatigue life. Besides, the effect of the normal strain amplitude on the critical plane is considered because it has a certain impact on multiaxial fatigue life. Finally, four materials, including 45 steel, 316 L stainless steel, GH4169 alloy, and 7075-T651 aluminum alloy, are adopted to verify the feasibility and correctness of the proposed model under the proportional and non-proportional loading condition by comparing the equivalent strain model, the maximum shear strain model, the SWT model, KBM model, and FS model with the experimental result. The comparison results show that the prediction life of the proposed model is only second to the KBM model under multiaxial proportional loadings, but the prediction capability is superior to the other five models and the majority of the prediction results are within the scatter band with factor 3 under non-proportional loadings.

Creep–fatigue life evaluation of type 304 stainless steel under non-proportional loading

The fatigue life of stainless steel under multiaxial loading conditions as a consequence of multiple damage effect occurring at elevated temperature is a topic of great importance. However, the interaction of creep, fatigue, and multiaxial non-proportional loading is poorly understood owing of a lack of experimental data obtained under multiaxial loading conditions. In the present study, data of previous creep–fatigue experiments performed on type 304 stainless steel under multiaxial loading conditions are summarized from available literature. These test results include the effects of strain rate, asymmetric strain waveform, and loading path on the creep–fatigue life of the stainless steel. Then, additional creep–fatigue tests are performed on the stainless steel under non-proportional loading conditions with four strain rates at 873 K in air. The applicability and limitations of several frequency-based life prediction models are discussed on the basis of the experimental results. Finally, an improved version of the frequency modified damage function (FMDF) model considering the effects of various loading path is proposed. The new model shows good correlation with the failure lives under multiaxial proportional and non-proportional loading conditions at elevated temperatures.