Proportional Loading Research Articles

Prediction of multiaxial fatigue life with a data-driven knowledge transfer model

A data-driven model is presented for accurate prediction of multiaxial fatigue life based upon the principle of transfer learning (TL). The Tradaboost framework is explored to adjust the weights of training data from different sources, actuating information transfer from domain knowledge to the data-driven modeling of multiaxial fatigue life. Subsequently, extensive experimental results tested under the proportional and non-proportional circle loadings are collected for model evaluation. The results demonstrate that the proposed model is more accurate than domain knowledge-based, conventional data-driven, and comparable TL-based models, with a low data requirement, showcasing good applicability for multiaxial fatigue life assessment.

A multiscale modeling for progressive failure behavior of unidirectional fiber-reinforced composites based on phase-field method

The effective macroscopic properties of composites are determined by the intricate interactions among the individual components within their microstructure. Preserving these microscopic details during the failure simulation of macrostructures presents significant challenges. This work proposes a multiscale modeling framework to numerically predict the macroscopic fracture properties of unidirectional fiber-reinforced composites based on micromechanical analysis. In this study, 2D representative volume elements (RVEs) combined with the phase-field method are utilized to simulate fiber-reinforced composites under transverse loadings. A series of representative loading conditions are employed to investigate cracking patterns and to construct failure strength envelopes of the composites subjected to different multiaxial proportional loadings. By extracting the softening curve from the uniaxial tensile simulation of the RVE and fitting it with a tenth-order polynomial, the homogenized cohesive law, combined with the phase-field method, is applied to the damage analysis of macroscopic heterogeneous materials. The homogenized model of unidirectional fiber reinforced composites is numerically validated through simulations of a 2D flat plate. The simulation results demonstrate the excellent potential of the proposed multiscale modeling framework to accurately and efficiently predict the progressive failure and fracture behavior of fiber-reinforced composites in engineering applications.

Using physics-informed neural networks to predict the lifetime of laser powder bed fusion processed 316L stainless steel under multiaxial low-cycle fatigue loading

Axial-torsional Low-Cycle Fatigue (LCF) tests were conducted under strain control on Additively Manufactured (AM) 316L stainless steel using laser powder bed fusion. The tests covered various strain amplitudes under tension-compression, proportional, and pure shear loading paths. The AM 316L stainless steel exhibited cyclic softening and transgranular cracking under all the investigated loading conditions. The presence of deposition defects, predominantly the lack of fusion type, was identified as the main factor influencing the crack initiation and propagation, as well as the scatter in the fatigue lifetime. Therefore, to account for the damaging effects of these deposition related defects on fatigue lifetime, a novel physics-informed neural network was proposed. Subsequently, this neural network was combined with the critical plane approach, based on the tensile mode of failure, in order to predict the lifetime of AM 316L stainless steel. The predicted data exhibited a good correlation with the experimental results.

A “poor-man’s” deformation plasticity based approach to topology optimization of elastoplastic structures

This paper presents a topology optimization framework utilizing a deformation plasticity model to approximate the isotropic hardening von-Mises incremental elastoplasticity model under monotone proportional loading. One advantage of the model is that it is based on a yield surface allowing for precise matching to uniaxial elastoplastic isotropic hardening response. The deformation plasticity model and the incremental plasticity model coincides for proportional loading and since the deformation plasticity model is path-independent, the computational cost and implementation complexity reduce significantly compared to the conventional incremental elastoplasticity. To investigate the deformation plasticity model combined with topology optimization, we compare three common elastoplastic optimization objectives: stiffness, strain energy and plastic work. The possibility to limit the peak local plastic work while maximizing the strain energy is also investigated. The consistent analytical sensitivity analysis which only requires the terminal state is derived using adjoint method. Numerical examples demonstrate that the proportionality assumption is reasonable and the deformation plasticity model combined with topology optimization is a competitive alternative to cumbersome incremental elastoplasticity.

Neural network based rYld2004 anisotropic hardening model under non-associated flow rule for BCC and FCC metals

This paper extends the reduced Yld2004 (rYld2004) function to present the anisotropic hardening behavior for body-centered cubic and face-centered cubic metals under the proportional loading conditions based on neural network. The parameters of the rYld2004 anisotropic hardening model (AH_rYld2004) are determined by the uniaxial tensile yield stresses along 0°, 15°, 30°, 45°, 60°, 75° and 90° from the rolling direction as well as equibiaxial tension. The evolution of anisotropic parameters are described by the back propagation neural network optimized by ant colony optimization algorithm. The predicted data by AH_rYld2004 and some common anisotropic models are compared with the experimental results to verify the precision of the AH_rYld2004 in characterizing anisotropic hardening. The comparison proves that the AH_rYld2004 precisely characterize the anisotropic evolution with increasing plastic deformation for AA 3003-O and QP980. Simultaneously, the AH_rYld2004 function based on neural network is used to accurately simulate of circular cup deep drawing for AA 3003-O and uniaxial tension for QP980. The results indicate that the AH_rYld2004 model is capable to accurately represent the plastic anisotropic evolution for uniaxial tension along seven loading directions and equibiaxial tension.

Tensile properties and damage behaviors of the prepreg-RTM co-cured composite bolted T-joint with a novel configuration

The tensile properties and damage behaviors of prepreg-resin transfer molding (RTM) co-cured composite bolted T-joints, consisting of an internal skeleton and an external skin, were investigated through experimental and numerical methods. With the validated finite element model, the effects of adding load-carrying beams, the corner radius and the stacking sequence were investigated. Results indicate that the novel T-joint, weighing only 55.7% of an equivalent sized 2A12 aluminum T-joint, can achieve 82% of the aluminum T-joint’s tensile stiffness and 107% of its proportional limit load. Damage in the joint initiates near the bolt holes and leads to delamination failure within the skeleton. With load-carrying beams including bolt holes, the tensile stiffness and ultimate load increase to 1.11 and 1.51 times that of the original configuration, respectively, with the final failure mode shifting to the fracture of the base panel. With the final failure mode unchanged, the ultimate load remains essentially unchanged with increasing the corner radius. Under tensile loads, the preferred stacking sequence for the skeleton is [0/+45/90/−45]nS. For the skin, the greater the number of ±45° plies, the higher the ultimate load, while the greater the number of 90° plies, the higher the tensile stiffness.

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.

Predicting fatigue life with the IMWCW under complex loading

The Integration Modified Wöhler Curve Method (IMWCW), a novel approach for predicting fatigue life under complex strain-based non-proportional loading paths is introduced in this research. This method integrates concepts from the Modified Wöhler Curve Method (MWCW) with the principle of damage incremental integration to provide a more nuanced prediction of material fatigue. Besides two established E-N curves (strain-number of cycles) for uniaxial tension and torsion, a new fundamental E-N curve that characterizes the strain-life relationship under complete non-proportional loading paths is introduced. A novel strain path decomposition technique is presented, which dissects an infinite number of instantaneous micro-elements along any loading path into components parallel and perpendicular to the proportional loading direction. It is assumed that each instant will cause fatigue damage to the material. A concurrent model for the incremental integration of damage is formulated, allowing for the calculation of cumulative damage. The model is further expanded to provide analytical expressions for proportional loading and non-proportional loading, linking them to three fundamental E-N curve, so as to calibrate the parameters required for the integration model. The empirical validity of the model is confirmed through experimental testing on 7 different metallic materials under 20 loading conditions, with 391 data points analyzed. The results demonstrate that 91% of the model's predictions fall within a scatter band of 2, confirming its robustness and reliability in forecasting fatigue life in complex, real-world scenarios.

Experimental study of the fatigue failure behavior of aluminum alloy 2024-T351 under multiaxial loading

Multiaxial fatigue has been drawing much attention because it is more applicable to engineering practice. In this paper, an experimental study of aluminum alloy (AA) 2024-T351 was carried out under multiaxial loading with an aim to assess the damage evolution process and failure mechanism under in-phase and out-of-phase loadings. Firstly, fatigue life and stress response under multiaxial cyclic loads were obtained, and it found that although there is non-proportional hardening, the fatigue life subjected to proportional loading is significantly shorter than that of under a nonproportional loading, which was tried to be explained by the ductility of the material. Secondly, a detailed analysis of the damage evolution process based on the degradation of the elastic modulus and the fatigue failure process based on the digital image correlation (DIC) method was provided. Next, a micro-analysis of the specimens’ fracture appearance was conducted to obtain the fracture characteristics and found that AA2024-T351 presents a dominant shear fracture mode under proportional loading and a mixed mode of tensile and shear fracture under non-proportional loading. Last but not least, The Fatemi-Socie criterion was modified by considering the material’s ductility and the interaction between normal stress and shear stress acting on the critical plane. The multiaxial life prediction results of the modified FS model for AA2024-T351 in this paper were all within the scatter bands of 3 on life.

Energy hub management for integrated energy systems: A multi-objective optimization control strategy based on distributed output and energy conversion characteristics

To improve the comprehensive utilization efficiency of energy, a multi-objective optimization control strategy applied to the energy hub (EH) within the system is proposed to address the electrical and thermal load distribution of the integrated energy system (IES) and the low-carbon economic operation. First, a model of the electrical and thermal energy outputs is established based on the characteristics of the IES network and the multidimensional “load parameter” evolution law. Moreover, a distributed control strategy is proposed that utilizes the information interaction between neighboring EHs to accurately share the electrical and thermal loads, which reduces the communication burden of the system while avoiding overloads. Furthermore, to coordinate the optimal operation of the devices within the hub, based on the energy conversion characteristics of the EH, a multi-objective optimization model is proposed that considers low-carbon and economic aspects to realize efficient energy use. Simulation results show that the proposed strategy effectively improves the robustness of the system while realizing proportional load distribution and low-carbon economic operation. Under the same load, when focusing on system economics, the operating cost is 2.182/¥ lower than when focusing on low-carbon systems, but carbon emission is 1.6753/kg CO2 higher.

The geomechanical response of the Gulf of Mexico Green Canyon 955 reservoir to gas hydrate dissociation: A model based on sediment properties with and without gas hydrate

Gas hydrate is commonly found concentrated in sand- and silt-rich reservoirs and these are thought to have great potential as an energy resource. To evaluate this potential, it is critical to understand their in-situ stress state and their behavior during production. We explore the geomechanical behavior of sandy silt sediments from a hydrate reservoir at Green Canyon GC 955, deep-water Gulf of Mexico without hydrate and compare these results to previous work on the same sandy silty sediments with hydrate present. Previous results from hydrate-bearing sediments at GC 955 using intact specimens show that the ratio of lateral to axial effective stress under uniaxial strain (K0) is strongly time-dependent and over long timescales approaches 1 (isostatic stress state). New results from sediments without hydrate using reconstituted specimens reveal that K0 is 0.51 and independent of time and stress. The reservoir without hydrate is approximately three times more compressible than with hydrate. In addition, we find that hydrate-free sediments have a normalized creep rate of Cα/Cc of about 0.028 at all stress levels, whereas hydrate bearing sediments exhibit a decrease in Cα/Cc with stress. We use these observations to present a new geomechanical model that describes how stress and porosity evolve during production of the hydrate reservoir. The key insight in our proposed model is that the sediment skeleton and gas hydrate phases share any applied external load in proportion to their relative stiffnesses. The model results predict that hydrate-bearing sediments are stiffer than gas hydrate-free sediments and that K0 reduces dramatically after gas hydrate dissociation, as is observed. At GC 955, a 5 MPa depressurization is needed to cause gas hydrate dissociation. This change in pore pressure increases the vertical and horizontal effective stresses from 3.8 to 8.8 MPa and 3.8–4.4 MPa, respectively. The dramatic vertical effective stress increase will cause significant compaction (7%) and perturb stresses near the wellbore. These experimental observations and the load-sharing model presented have the potential to underpin a new generation of hydrate reservoir simulation models, which may drive more effective production approaches.

A strain components-based Mohr–Coulomb fracture criterion for proportional loading

Ductile fracture criteria are crucial in the application of elastoplastic materials like advanced high-strength steels, particularly due to their tendency to fail without significant localized necking. It is aimed to figure out the unified mechanism for ductile fracture and accurately describe it with phenomenological criteria. In this study, the effect of the stress triaxiality and the Lode parameter is analyzed mathematically. The maximum shear strain and the normal strain to fracture are emphasized in the proposed strain components-based Mohr–Coulomb fracture criterion particularly designed for plane stress states, and it is assumed that fracture occurs when the combination of the maximum shear strain and the normal strain reaches the critical value. It is then extended to describe fracture with the equivalent plastic strain and the normal strain to fracture. The experimental fracture data of AA 2024-T351 and the additively manufactured Ti-6Al-4V titanium alloy are used to verify the proposed models. A fracture forming limit diagram comprising forming limit curves calibrated from stress states under different stress triaxialities is proposed based on the proposed fracture criteria for plane stress states. This study shows that a strong correlation between the maximum shear strain and the normal strain to fracture has been observed, and it can be described piecewise-linearly. Besides, the orientation of the fracture surface under both tension (including plane strain tension) and compression (including plane strain compression) can be qualitatively explained by the Mohr’s strain circles. The results provide a new insight into the intrinsic deformation and fracture mechanism of materials that fail without severe localized necking.

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.

An engineering applicable method for multiaxial fatigue under proportional and non-proportional loads based on the octahedral plane projection

Present multiaxial fatigue models have achieved satisfactory performance in predicting multiaxial fatigue life of different loading paths; however, many of them require critical plane calculations and additional material constants. In this study, a novel multiaxial fatigue life prediction model based on octahedral shear strain energy is proposed. In the proposed model, the maximum octahedral shear strain energy serves as the damage parameter without requiring any additional material parameters. All material constants in this model can be derived from material strength constants and uniaxial fatigue constants, greatly simplifying the life prediction process. Moreover, based on the actual physical phenomenon of non-proportional additional hardening, the multiaxial non-proportional correction factor is incorporated into the plastic stage of the life equation, aligning more closely with fundamental principles in physics. Then, by utilizing the experimental data from the literature for four materials, a comparison is made between the prediction results of the novel model and those of four commonly employed models. The findings demonstrate that the new model exhibits superior simplicity and accuracy. Subsequently, through life prediction outcomes for DZ411 material across various paths, the stability and precision of the octahedral energy model are substantiated, rendering it suitable for practical engineering applications.

Influence of proportional multiaxial fatigue loading on the residual mechanical properties of glass-reinforced plastic pipes

This work is dedicated to the experimental investigation of multiaxial cyclic loading influence on the residual mechanical properties of glass-reinforced plastic pipes (GRP pipes). Static, cyclic and combined (static after the preliminary cyclic loading) tests have been carried out on tubular specimens under tension with torsion. Residual stiffness curves have been made, their approximation has been carried out, using previously developed model, damage accumulation stages’ boundaries have been calculated. A specimen’s individual fatigue life prediction method, based on the initial parts of stiffness degradation curves, has been proposed. Influence of the preliminary multiaxial cyclic loading on residual Young’s modulus and tensile strength has been studied. Patterns of deformation and fracture processes have been studied, using digital image correlation method and acoustic emission method. A significant influence of fatigue damage on the residual properties of GRP pipes has been concluded.

Study on mechanical properties of E-glass mat reinforced basic magnesium sulfate cement thin-slab

To achieve sustainable development and reduce the carbon footprint in building material production, the E-glass fiber mat was impregnated with basic magnesium sulfate cement (BMSC) by lamination impregnation. This process reduces environmental pollution from cement preparation, increases fiber volume content, and enhances the mechanical properties of cementitious composite materials. The study examined axial tensile, four-point bending resistance, drop weight impact resistance and pendulum impact resistance of BMSC thin-slab reinforced with dispersed short glass fiber (SGF), short glass fiber mat (SGFM) and continuous glass fiber mat (CGFM) of varying gram weights. The results show that the mechanical properties of BMSC thin-slab reinforced with fiber mat are much higher than those of SGF reinforced thin-slab. The tensile strength of BMSC thin-slab reinforced with CGFM is higher than that of BMSC thin-slab reinforced with SGFM at the same gram weight, and the tensile strength increases as the weight of CGFM increases. Specifically, the axial tensile strength of BMSC thin-slab reinforced with 450 g of CGFM is nearly 8 times higher than that of SGF reinforced BMSC thin-slab. BMSC thin-slab reinforced with 450 g of CGFM has the maximum proportional limit load and bending strength. The impact strength and toughness indexes of BMSC thin-slab reinforced with CGFM are the highest, followed by SGFM reinforced BMSC thin-slab. The addition of 300 g CGFM significantly enhanced flexural toughness during the early stage of thin-slab cracking, while 450 g CGFM showed significant improvement in flexural toughness during the middle and late stages of cracking. When comparing thin-slabs with the same glass fiber volume ratio, those reinforced with 450 g CGFM exhibited the best impact resistance. SEM results indicated superior adhesion between 450 g CGFM and the cement matrix, as well as between adjacent CGFM layers, with fiber breakage being the main cause of specimen failure.

Biaxial tensile behavior of CoCrFeNi high-entropy alloy under dynamic and proportional loadings

The dynamic failure behavior of CoCrFeNi High-Entropy Alloy (HEA) under plane biaxial stress was investigated in detail. The dynamic biaxial tensile tests were conducted using an Electromagnetic Biaxial Split Hopkinson Tensile Bar (EBSHTB) system. For comparison, the quasi-static uniaxial and biaxial tensile tests, as well as dynamic uniaxial tensile tests, were performed respectively. A cruciform specimen suitable for large plastic deformation was designed and employed in the experiments. The Finite Element Method (FEM) verified that the improved cruciform specimen could satisfy the basic requirements. The feasibility of the proposed specimen was further confirmed through loading tests. Finally, the quasi-static and dynamic yield loci of the HEA in the first quadrant of the principal stress space were plotted. The results indicate that the alloy exhibits obvious strain hardening effect and strain rate strengthening effect, the yield locus and plastic work contours can be accurately described by Hill’48 criterion.

Non-proportionality quantification of noisy stress-time signals using a new rainflow-based noise removal method for fatigue assessment

Non-proportional stresses near welded joints are well known to cause increased fatigue damages as compared to proportional stresses. In this paper, a new method to remove noise from stress-time signals is developed and implemented in a principal component analysis-based approach for non-proportionality quantification. The noise removal method makes it possible to remove noise from stress-time signals causing low stress ranges, while still keeping the overall shape of the signal. When the signal shape is kept it is possible to accurately predict the levels of non-proportionality. The noise removal method is based on the standard rainflow counting method. By choosing a lower stress range limit, the stress ranges assumed to be caused by noise can be excluded and the original signal shape is kept by utilizing simple polynomial fitting. This makes the approach easy to implement and easy to control as it requires only two inputs. The noise removal method and non-proportionality quantification approach are then validated against simulated signals with noise and a simple experiment with proportional loading. The results show that the newly developed method for noise reduction accurately removes noise while keeping the signal shape.

Numerical simulations and testing of notched composite plates under envelope loadings: Validation with the safe life domain method

The certification of aircraft structures specifies various loads, possibly combined, to be considered to demonstrate its strength. In general, structural testing up to static failure is performed using proportional loading. In this study, the effects of loading with combinations of tension-shear-pressure are studied on large notched composite specimens. The VERTEX test rig is used. The Safe Life Domain method is proposed to validate whole domains of loading combination by means of a single envelope test. Envelope validation tests are implemented on tension-shear loadings with and without pressure, based on numerical predictions of the specimen failure and of the bench behaviour. Buckling mode variability and its impact on failure is considered, to show possible path-dependency of failure and thus highlight an ignored complexity of structure failure. Therefore, the numerous proportional tests needed for compliance demonstration can be reduced thanks to a few envelope tests instead.

An anisotropic hypoplastic model for clays considering the non-coaxial behavior

Previous experimental tests on clays have confirmed the non-coincidence of principal strain increments with principal stress axes even though the stress rate is colinear with the current stress. To simulate the non-coaxial behavior of saturated clay subjected to monotonic proportional or non-proportional loading with constant principal stress rate directions, an anisotropic hypoplastic model is presented in this paper. The model is proposed by incorporating an improved anisotropic asymptotic state boundary surface (ASBS) and a non-coaxial asymptotic strain rate direction into the original explicit-ASBS hypoplastic model. The non-coaxial flow results from a non-coaxial stress rate defined by a Gram-Schmidt orthogonalization process based on a reference stress tensor. The capability of the proposed model is demonstrated by simulating the test data performed in hollow cylindrical apparatus (HCA) on Wenzhou clay and Shanghai clay with different drainage conditions and initial consolidation states. In addition, a series of numerical stress probing tests have been conducted to gain further insights into the properties of the proposed model.