Non-proportional Loading Paths Research Articles

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.

Evaluation of crush performance of extruded aluminum alloy tubes based on finite element analysis with ductile fracture modeling

In this study, the crush performance of four aluminum extrusions are investigated through analysis based on finite element (FE) simulation incorporating experiment-numerical hybrid ductile fracture modeling. Tensile tests on standard and non-standard specimens, coupled with digital image correlation and FE analyses, are employed to establish constitutive models for the materials. The evaluation of crush performance is performed based on both axial crush test and corresponding 3-dimensional FE simulation, which could validate the employed plasticity and fracture laws and predict crush performance indicators with reliable accuracy. Additionally, FE modeling enables thorough analysis of highly non-proportional loading paths and potential crack initiation sites, with the crack formation mechanism elucidated through the evolution of a proposed damage variable. Furthermore, the modified crush performance indicators are newly suggested, which could explain the dependence on both the absorbed energy of tubular extrusion and properties of ductile fracture.

Biaxial Testing of Thin Metal Sheets under Non-Proportional Loading Conditions

During the manufacturing and service of thin metal sheets, different stress states occur, often caused by non-proportional loading conditions. They can lead to localization of inelastic deformations as well as to damage and failure processes. In the present paper, a series of experiments with newly designed biaxially loaded specimens is presented to analyze the damage and failure behavior of thin ductile metal sheets under non-proportional load paths. Bands of holes with different orientation have been milled in critical regions of the specimens to localize stresses and strains. In compression tests, a special downholder is used to avoid buckling. During the loading processes, strain fields in critical regions of the specimens are monitored by digital image correlation technique. After the experiments, fracture surfaces are investigated by scanning electron microscopy showing different damage and fracture modes depending on the loading history. The experiments clearly demonstrate the efficiency of the thin specimens and the experimental program.

Determination of the onset of yielding and the Young’s modulus after a change in the loading direction

The onset of plastic deformation is an important parameter for an accurate description of the flow curve and the Young’s modulus. Determining the actual physical start of flow is already experimentally challenging for classic sheet metal materials. In addition to the experimental challenge, the onset of flow depends on numerous parameters such as strain rate, temperature and forming history. Non-proportional load paths in particular can significantly influence the onset of flow. Three different materials, a micro-alloyed steel HC340LA, a dual-phase steel CR330Y590-DP and an aluminium alloy AA6016-T4 are investigated in this publication. The physical onset of flow of the materials is determined at three different pre-strain levels as well as without and with a change in the load direction. Temperature-based approaches are used for this purpose. In-situ synchrotron diffraction is used to validate the results obtained. Those results can help to improve existing material models and springback prediction. Such models rely on material parameters that are as accurate as possible.

Multiscale Thermodynamics-Informed Neural Networks (MuTINN) towards fast and frugal inelastic computation of woven composite structures

The complex behavior of inelastic woven composites stems primarily from their inherent heterogeneity. Achieving accurate predictions of their linear and nonlinear responses, while considering their microstructures, appears feasible through the application of multi-scale modeling approaches. However, effectively incorporating these methodologies into real-scale applications, particularly within FE2 analyses, remains challenging due to the significant computational requirements they entail. To overcome this issue, while considering the scale effects, this study introduces an alternative approach based on Artificial Neural Networks (ANNs) to perform a macroscopic surrogate model of composites. This model, referred to as Multiscale Thermodynamics Informed Neural Networks (MuTINN), is founded on thermodynamic principles and introduces specific quantities of interest that serve as internal state variables at the macroscopic level. This captures efficiently the state and evolution laws governing the history-dependent behavior of these composites while retaining the thermodynamic admissibility and the physical interpretability of their overall responses. Moreover, to facilitate its numerical implementation within a FE code, a Meta-UMat has been developed, streamlining the application of multiscale FE×MuTINN approach for composite structure computations. The prediction capabilities of the proposed approach is demonstrated across the material scales, exemplified through diverse instances of woven composite structures. These applications account for anisotropic yarn damage and an elastoplastic polymer matrix behavior. The numerical results and the related comparison with experimental findings and FE computations demonstrate remarkable consistency across a wide range of non-proportional loading paths. This promises a potential solution to alleviate the computational challenges associated with multiscale simulations of large-scale composite structures.

Uncertainty quantification in multiaxial fatigue life prediction using Bayesian neural networks

In practical engineering scenarios, unforeseen failures may result in significant costs and risks. Assessing the uncertainty related to multiaxial fatigue life prediction is crucial for engineering applications. However, establishing a reliable multiaxial fatigue life prediction model remains challenging due to uncertainties in physical models, material properties, and measurement data. This paper proposes an effective Bayesian Neural Network (BNN) model for quantifying the uncertainty in multiaxial fatigue life prediction. Fatigue parameters in the multiaxial fatigue life prediction model are used as inputs for the BNN. The BNN method utilizes No-U-turn Sampler (NUTS) and Automatic Differentiation Variational Inference (ADVI) to address the inference problem in the model and perform life predictions for unknown non-proportional loading paths and different materials. Validation using experimental data from six different materials confirms the effectiveness of the BNN model. Experiments show that the performance of BNN models is beneficial in most cases compared to classical ML models, assessed through different error standards, statistical tests, Taylor diagrams, uncertainty analysis, and scatter index analysis. The proposed method provides a clear understanding of the uncertainty in multiaxial fatigue life prediction, offering a promising approach for simulating fatigue life under multiaxial loading in engineering applications.

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.

Microstructural evolutions and life evaluation of non-proportional creep-fatigue considering loading path and holding position effects

In this paper, strain-controlled non-proportional creep-fatigue tests under three non-proportional loading paths were performed for type 304 stainless steel. The strain holding period at the peak axial or peak shear strains was introduced to investigate the effect of the holding position on the creep damage accumulation. Compared with the other non-proportional loading paths, the circle loading path generated the most detrimental effect on the material life. In addition, the introduction of the holding period reduced the material life, in which axial holding resulted in a greater creep damage than that of the shear holding. A post-examination method through the electron backscattered diffraction (EBSD) observations was conducted to reveal the damage mechanisms under various loading paths. The variations of six EBSD-based misorientation parameters showed the significant effect of the loading path and holding position on the damage mechanisms. Subsequently, a multiaxial damage factor (MDF)-involved strain energy density exhaustion (SEDE) model and a non-proportional energy parameter (NPEP)-involved Ostergren model were collaborated to provide a precise prediction within a factor of 1.5 for non-proportional creep-fatigue life distributions.

Path-dependent multiaxial fatigue behavior of A319 aluminum alloy under non-proportional loading conditions

Abstract In the multiaxial low fatigue test study, four multiaxial non-proportional fatigue loading paths were used including rectangular, square, rhombic and circular. The effect of loading condition changes on the fatigue behavior and cyclic damage characteristics of A319 aluminum alloy was investigated by means of stress response curves, stress–strain hysteresis lines, fatigue fracture morphology, and dislocation substructure morphology in combination with material microstructure. The non-proportional hardening behavior exhibits strong loading path dependence, and the fatigue plastic deformation in the torsional direction is greater in the circular and rhombic paths, and the fatigue damage is more severe, resulting in a shorter fatigue life of A319 aluminum alloy. The axial directions under non-proportional multiaxial loading all have tensile and compressive asymmetric behavior. The differences in loading paths result in different fatigue crack expansion modes. The non-proportional additional hardening behavior under different loading paths is closely related to the dislocation substructure. In the rhombic and circular paths, the alloys have higher dislocation density and strong interaction between dislocations and precipitation phases, which is macroscopically expressed as stronger non-proportional hardening ability in the rhombic and circular loading paths.

A new simplified distortional hardening model for nonlinear strain paths

The material behaviors under nonlinear strain paths have been successfully explained through distortional hardening model in which the yield surface is mainly rotating and contracting as plastic deformation occurs. In order to apply to industrial applications, it is beneficial for any constitutive model including distortional hardening model to have simplicity of equation, which is related to numerical efficiency, as well as accuracy in describing material behaviors. This study was mainly motivated to construct a simple form of equation for distortional hardening model with reasonable accuracy as the conventional distortional hardening models. In this paper, a new simplified distortional hardening model is proposed by considering one of the pioneering works for distortional hardening (Tozawa, 1978) and definition of homogeneous function. Any yield function could be used for initial anisotropy and asymmetry yielding. In addition, with the combination of isotropic hardening model for consistency condition, the new model could explain both isotropic and anisotropic hardening under proportional and non-proportional loading paths. The proposed model reasonably described the material behaviors including Basuchinger effect, permanent softening, nonlinear transient behavior, and yield surface contraction. Finally, the availability in finite element simulation has been identified from the springback prediction in U-draw bending.

Plasticity and ductility of an anisotropic recrystallized AA2198 Al-Cu-Li alloy in T3 and T8 conditions during proportional and non-proportional loading paths: simulations and experiments

The anisotropic material behaviour of a recrystallized AA2198 Al-Cu-Li alloy in T3 and T8 conditions was assessed by proportional loading of uniaxial-tension specimens in rolling (L), transverse (T) and diagonal (D) orientations. The width and longitudinal strains were measured to quantify plastic anisotropy. Notched-tension samples were tested in L and T directions. The material showed weak anisotropy in terms of stress strain curves and a moderate plastic anisotropy, consistent with its texture obtained by EBSD. An anisotropic Bron-Besson type material model was identified using this data base and a micro shear-only (SO) test. The model succeeded in predicting the behaviour of micro specimens for proportional tension-only (TO) loading and non-proportional load path changes including 'shear to tension' (ST) as well as 'tension to shear' (TS) tests. The non-proportional loading was achieved using a newly designed cross shaped sample. It was loaded in one direction, unloaded and subsequently loaded in the orthogonal direction till fracture. The average stretch to fracture of both alloys measured by a four point frame optical extensometer decreased by 29 % and 16 % for T3 and T8 respectively for the 'shear to tension' experiment compared to the proportional TO experiment. The average stretch to fracture of 'tension to shear' tests was reduced by 10 % for 2198T3 and hardly reduced for 2198T8 compared to the stretch to fracture of the SO tests, but subject to strong scatter. FE simulations showed local accumulated strain to fracture values that were similar for all loading histories for the T8 condition (0.73 − 0.84). Lower strain to fracture values were found in T3 condition (0.45 − 0.73), despite the enhanced macroscopic ductility in tension. This was attributed to larger less localized plastic zones, especially for the ST test. The ductility scatter was attributed to necking and damage development in tension that can affect strain localization, associated fracture path and ductility, as observed by DIC and fractography.

Impact of dynamic strain aging on multiaxial low cycle fatigue behavior of nickel-based alloy 690 at 350 °C

The multiaxial low cycle fatigue behavior of nickel-based alloy 690 at 350 °C was investigated. The multi-slip planar dislocation configuration under non-proportional loading paths is attributed to short-range order (SRO) and dynamic strain aging (DSA). The effect of multiaxial loading paths on DSA is quantitatively evaluated based on average stress drop, which is dominated by axial strain amplitude and weakened by the cross-slip strength. Finally, the preferential oxidation along the slip bands accelerates the nucleation and propagation of microcracks, and the multi-slip cracking for non-proportional loading paths enhances the oxidation damage and further reduces fatigue life.

Multiaxial fatigue prediction and uncertainty quantification based on back propagation neural network and Gaussian process regression

Engineering structures are often suffering multiaxial stress which leads to multiaxial fatigue failures. Accurate and reliable multiaxial life prediction is a challenging issue in fatigue analysis due to the intricate deterioration mechanisms of the fatigue failure and the large uncertainty in the material parameters at the microscopic scale. Moreover, conventional multiaxial fatigue life prediction models are empirical or semi-empirical. To tackle this problem, a data-driven method is presented, which combines the back propagation neural network (BPNN) and the Gaussian process regression (GPR). The BPNN-GPR method can predict the multiaxial fatigue life and quantify the uncertainty simultaneously. This method is validated using six materials including the uniaxial, multiaxial proportional and multiaxial nonproportional loading cases. All the predicted lives fall within a life factor of ± 3, which indicates that the BPNN-GPR method has the satisfactory capability for multiaxial fatigue life prediction. In addition, results also show the prediction capability of the BPNN-GPR method for unknown multiaxial nonproportional loading paths.

Thermodynamically consistent Recurrent Neural Networks to predict non linear behaviors of dissipative materials subjected to non-proportional loading paths

The present work aims at proposing a hybrid physics-AI based model to predict non-linear mechanical behaviors of dissipative materials. By introducing a specific Neural Network architecture called Thermodynamically Consistent Recurrent Neural Networks (ThC-RNN), this study proposes a new paradigm for the simulation of dissipative materials under complex loading conditions. The design of such architecture allows to take into account the material loading history subjected to multi-axial and non-proportional loading paths, similarly to internal variables for homogeneous materials. A special focus has been given to the respect of thermodynamics principles in the ThC-RNN model by introducing specific thermodynamical constraints during the training phase. Finally, the model’s reliability has been tested on different plasticity models once the training is completed. It is shown that thermodynamic consistency improves significantly the prediction capabilities of the ThC-RNN model, considering several outputs such as stress tensor and tangent modulus components, state variables and mechanical work rate partition (recoverable part, irrecoverable part and dissipative part).

Verification of the Tanaka non-proportional isotropic cyclic hardening model under asynchronous loading

Non-proportional cyclic loading path induces intrinsic strain hardening to the low stacking fault energy materials. The hardening is manifested in increased stress response which underestimation leads to non-conservative life prediction. This study aims to verify the Tanaka non-proportional isotropic cyclic hardening model under a wide spectrum of multiaxial asynchronous loadings. The Tanaka model is coupled with the Chaboche kinematic hardening model for stress path estimation. Modeling results are compared with fatigue experiments for two non-alloyed steel grades, E235 and E355. The Tanaka model improves the performance of the Chaboche model and exhibits its applicability to asynchronous loading.

An Anisotropic Continuum Damage Mechanics Model for the Simulation of Yield Surface Evolution and Ratcheting

A new anisotropic continuum damage model was presented on the basis of irreversible thermodynamics. Second rank symmetric damage tensor, kinematic hardening tensor, and damage potential surface were introduced. The yield surface in this study has been modeled to distort in the stress space by the evolution of the damage tensor. The new model was applied in yield surface characterization and led to a well description of the subsequent yield surfaces. This model is then applied to predict experimental results of the cyclic nonproportional loading paths obtained by Kang et al. [2004], Hassan et al. [2002] and Taleb and Cailletaud [2011]. It is demonstrated that the new model can predict the yield surface distortion and the ratcheting strains of multiaxial nonproportional cyclic loading with acceptable accuracy and by using less material constants with respect to the previous models.

Multiaxial fatigue life prediction for metallic materials considering loading path and additional hardening effect

PurposeA large number of data have proved that under the same von Mises equivalent strain condition, the fatigue life under multiaxial non-proportional loading is often much lower than the life under multiaxial proportional loading. This is mainly due to the influence of the non-proportional loading path and the additional hardening effect, which lead to a sharp decrease in life.Design/methodology/approachThe modulus attenuation effect is used to modify the static hardening coefficient, and the predicted value obtained is closer to the additional hardening coefficient obtained from the experiment. A fatigue life model can consider non-proportional paths, and additional hardening effects are proposed. And the model uses multiaxial fatigue test data to verify the validity and adaptability of the new model. The life prediction accuracy and material application range are satisfactory.FindingsBecause loading path and additional hardening of the material affect fatigue life, a new multiaxis fatigue life model based on the critical plane approach is proposed. And introducing a non-proportional additional damage coefficient, the joint influence of the load path and the additional hardening can be considered. The model's life prediction accuracy and material applicability were verified with multiaxial fatigue test data of eight materials and nine loads compared with the prediction accuracy of the Kandil–Brown–Miller (KBM) model and Fatemi–Socie (FS) model.Originality/valueThe physical meaning of the new model is clear, convenient for practical engineering applications.

An improved continuous damage model to estimate multiaxial fatigue life under strain control problems

In this contribution, a modification and application of the Lemaitre damage evolution law is proposed for the problem of fatigue life estimate under low number of cycles. For this, the denominator of damage, considered constant for Lemaitre, was replaced by a function dependent on the stress triaxiality. The new function has a nonlinear and thermodynamically consistent behavior. In addition, the evolution law of the backstress tensor is adopted as proposed by Desmorat, where there is no saturation of the limit value. The formulation is implemented in an academic tool for analysis at one Gauss point, through an implicit integration strategy of the flow equations. The Newton-Raphson method is used to solve a system of nonlinear equations, with two tensor and two scalar equations. Thus, in order to demonstrate the robustness of the model, experimental data from the literature for SAE 1045 and S460N steels are adopted. Proportional and nonproportional axial, torsional and multiaxial loading paths are simulated. Finally, the results have shown that the modification formulation is a good strategy to control the damage evolution in predicting fatigue life. In this sense, the calculated lives by the new proposal are within a scatter band of factor three, both under normal or torsional uniaxial loading conditions, as well as proportional or nonproportional multiaxial conditions.

Buckling of thick elasto-visco-plastic egg shells under external pressure: Experiments and bifurcation analysis

The buckling of rate dependent structures is not a topic often discussed in the literature. Methods to predict the buckling of such structures exist, but only few experimental works have been published, and even less when considering thick shell structures. In this work we are interested in the buckling of thick elasto-visco-plastic hemi-egg shells. We follow a modelling/experimental approach to analyse such problem. Buckling experiments at room temperature on hemi-egg shells subjected to external pressure are performed. 3D digital image correlation is used to measure the displacement fields on the inner surface of the hemi-egg shell during the buckling experiments. Due to the specimen shape and their loading, they experience non-proportional load paths. The effect of a non-proportional load path on the buckling behaviour has already been discussed many times in the literature for elasto-plastic materials. This work also intends to assess this point for a rate dependent material. A buckling prediction model is used to predict the buckling of hemi-egg shells. This model couples Bodner’s approach with the deformation theory. It is used to estimate the buckling critical times and the buckling modes based on a preliminary finite element analysis. The effects of geometrical and loading imperfections on the buckling behaviour of the hemi-egg shells are also discussed. Finally, the experimental results are compared to the buckling predictions. A good correlation is observed between numerical results and experiments.

An evolution of subsequent yield loci under proportional and non-proportional loading path of ‘as-received’ extruded AZ31 magnesium alloy: Experiments and CPFEM modeling

This work presents an experimental investigation of as-received extruded AZ31 magnesium alloy subjected to proportional and non-proportional loading paths at various finite levels of pre-strain. The measured subsequent yield loci evolutions in the σ11−3σ12 stress space, using a 10µε offset linearity definition of yield, with increasing plastic deformation are reported in this paper. Initial yield locus deviates substantially from a von-Mises yield loci. Subsequent yield loci under proportional loading show an usual nose in the loading direction and almost a flat shape in reverse loading direction. In addition, yield loci in all proportional loading paths (tension, free end torsion and combined tension-torsion) have significant kinematic and some isotropic hardening. Subsequent yield loci under non-proportional loading (tension after torsion and torsion after tension) show distortion in addition to significant kinematic and isotropic hardening, as well as, demonstrate a strong path-dependency. During these experiments with the increasing plastic strain, elastic moduli within each subsequent yield loci are also measured. These experimentally determined yield loci of as received AZ31 alloy are significantly different from the observed ones for aluminum alloys by Khan et al. (2009) and Iftikhar et al. (2021). However, these results are comparable to the ones reported recently by Iftikhar and Khan (2021) for annealed extruded AZ31 alloy. Numerical modeling using crystal plasticity models was carried out to study the origin of the distortion of the yield locus. The CP model is calibrated using the tensile and shear flow curves. Using the CP framework, initial and subsequent yield loci are predicted. CP predictions show good predictions. The CP simulations can capture the observed distortion and asymmetry in the yield loci. In addition, the CP simulations can give insight on the origin of the kinematic hardening, distortion and asymmetry. They show that these are correlated with activity of various slip systems and the amount of twinning occurring during the loading along different loading paths.