Proportional Loading Research Articles

A damage-based method to predict crack initiation lifetime of high-strength steel under proportional bending-torsional loadings

The fatigue life of specimen consists of crack initiation and crack propagation life. As the fracture toughness of high strength steel is low, the specimen will fail soon once crack appears. Therefore, the crack initiation life of high strength steel is considered to be its whole life. Based on the evolution of material fatigue damage and the critical plane method commonly used in multiaxial fatigue strength theory, a crack initiation life prediction method for multiaxial specimens is proposed in this paper. Firstly, a uniaxial nonlinear fatigue damage evolution equation is proposed based on the principles of thermodynamics and continuous damage mechanics. Then, a theoretical calculation method for determining the critical plane under multiaxial load is proposed, and the specific calculation process is given. After the critical plane is determined, the multiaxial fatigue damage parameter is constructed from the normal strain amplitude and shear strain amplitude on the critical plane, and a multiaxial nonlinear fatigue damage evolution equation is proposed by replacing the uniaxial damage parameter using the multiaxial damage parameter. Finally, the crack initiation life of fatigue specimens is predicted by using the proposed multiaxial nonlinear fatigue damage evolution equation, and the multiaxial fatigue tests of Q690D steel under different bending-torsion ratios and different amplitudes are validated. The comparison results show that the prediction error of the proposed method is within the two-fold dispersion band and better than that of Manson-Coffin method.

Load Sharing and Voltage Restoration Improvement in DC Microgrids with Adaptive Droop Control Strategy

DC microgrid is a connecting network between distributed DC generators. The converter control on DCMG is used to obtain proportional load sharing and voltage restoration of DC bus. The traditional droop control is a common control technique in DC microgrid networks with multiple unit converters. However, this technique has limited performance when the difference of the line resistance is considered. In this paper, the intelligent adaptive droop control is proposed to improve the accuracy of load sharing, while the distributed secondary loop is utilized to minimize the DC bus voltage deviation. The droop resistance is tuned using fuzzy logic controller so that the load power sharing error compensation is obtained, while the DC bus voltage deviation is reduced by the PI controller. The exchange of output currents between converter units through communication lines is considered to obtain the actual load current conditions. The proposed droop control strategy can guarantee the load power sharing balance between three sources with a ratio close to 1:1:1. The results of the comparison analysis between the proposed technique and the traditional droop control are simulated completely using Matlab/Simulink, even under fluctuating loads and the line resistance changes.

General principles and criteria of failure of solids on different scale-structural levels under static and alternating loading

Experimental and theoretical studies of the fracture regularities under monotonic and alternating loadings, including stress-corrosion fracture, revealed the main scale-structural levels of the brittle fracture (formation of a strip structure, nucleation of microcracks, microstructurally short non-propagating cracks and short propagating cracks, merging of short cracks with a macrocrack nucleation) and viscous fracture (an evolution of dislocation in slip bands, cellular structure formation, nucleation of a pit relief with micropores, merging of pores with a formation of meso-shear bands and initiation of a viscous crack, growing of a viscous crack under inelastic deformation) of crystal bodies. A transition from one level to another is determined by changes in the fracture mechanisms and occurs with varying degrees of probability. The necessity of considering the destruction as a hierarchical random step-by-step process on all scale-structural levels is substantiated. The fracture of plastic materials is considered as the sum of independent or dependent events, namely the initiation and growing of brittle cracks and evolution of pores through various mechanisms. Attaining of the limit states is determined by the statistics of the distribution of inhomogeneities throughout the entire body volume. The results of physical studies (using electron microscopy, X-ray structural analysis, etc.) provide determination of the geometry and density of defects, dislocations, pores, cracks at different levels. Macrocharacteristics of static, long-term and fatigue strength are determined using methods of mechanical tests. The regularities of the fracture evolution of engineering structures are determined using methods of non-destructive testing (ultrasonic analysis, acoustic emission analysis, magnetic flaw detection, etc.). A brief analytical review of the main known physical approaches — dislocation and energy structural theories, dislocation continuous theories, dilaton-frustron models, stochastic physical theories, and phenomenological approaches, including continuum damage theory and fracture mechanics — is presented. The analysis of gained data leads to the necessity of describing fracture in the framework of stochastic multilevel models. We consider the approach according to which the probability of reaching limit sates at each level under arbitrary multiaxial loading is determined by linear functionals of the loading process, kernels of the functionals being random functions of time. For simple proportional loading, the theory of brittle fatigue scale-structural failure is presented.

Analysis of damage and fracture behavior in ductile metal sheets undergoing compression and shear preloading

The present paper comprises the experimental and numerical analysis of damage and fracture behavior of ductile metals under non-proportional loading with compression and shear preloading. For this purpose, biaxial experiments with the H-specimen using a pneumatic downholder for compression loading followed by failure under a tension stress state and corresponding numerical simulations are performed and analyzed. A thermodynamically consistent anisotropic continuum model is presented. It takes the effects of micromechanical damage mechanisms on the macro scale behavior as a motivation for a phenomenological description of damage behavior as a function of stress state. The experimental findings are compared with the results of tests with proportional load paths and corresponding numerical simulations. Thereto, strain fields of the critical areas of the specimens are analyzed by means of digital image correlation (DIC) technique. Compression and shear preloading can lead to significantly reduced ductility of the material with initiation of failure processes in the small damage strain range. Evolution of numerically predicted plastic and damage equivalent strains illustrates their stress state dependence. The numerical results are also confirmed by images of fracture surfaces taken by scanning electron microscopy (SEM). This experimental-numerical methodology is therefore an efficient tool to develop and validate general modeling approaches, and it is characterized by almost arbitrarily definable load paths with changes in stress states occurring in forming processes.

Experimental and Numerical Analysis of Prestrain on the Formability of Zn-Cu-Ti Alloy Zinc Sheet

The forming limit diagrams (FLDs) characterizing the formability of sheet metals are usually obtained by applying proportional loadings. Nevertheless, the industrial processes involve strain path changes that can modify the limit-strain values. In addition, for strongly anisotropic sheet metals such as the Zn-Cu-Ti zinc alloy, large differences in forming limit curves (FLCs) with respect to the sheet rolling direction are observed. In the present work, the analysis of the effect of bilinear strain paths on the FLC is addressed by both experimental measurements and numerical simulations. For this purpose, a miniature testing device was used that allows evaluation of the influence of strain path changes on the limit strain on samples at 0°, 45° and 90° with respect to the sheet rolling direction cut from non-standard large samples previously subjected to a prestrain along the RD up to an early deformation of ~0.12. Numerical simulations were carried out using the well-known Marciniak and Kuczynski (MK) theory in conjunction with the viscoplastic self-consistent (VPSC) crystal plasticity model. In order to account for the grain fragmentation process due to the continuous dynamic recrystallization (CDRX) mechanism, an ad hoc short-range interaction effect (SRE) model was included in the simulations. Additionally, the measured and simulated texture evolution of Zn-Cu-Ti alloy sheets at the different stages of the deformations were shown. The capacity of the MK-VPSC-SRE model was validated, and the limitations to simulating the texture development, flow stress and forming limit curves, including a non-proportional strain path, were discussed.

Acceleration of continuous-time, high-cycle fatigue constrained problems in topology optimization

An efficiency-enhanced procedure to treat continuous-time, high-cycle fatigue (HCF) constraints in topology optimization is presented. The HCF model predicts the evolution of fatigue damage at each point in the design domain using a system of ordinary differential equations. We employ gradient-based optimization and the fatigue sensitivities are determined using adjoint sensitivity analysis. As the predicted damage has history dependence, adjoint variables are solved via a stepwise backward procedure. Therefore, the computational cost increases in proportion to the number of time steps. To reduce this cost, we propose an extrapolation technique which is valid for all forms of periodic, proportional loads and most non-proportional loads and allows treatment of essentially an unlimited number of load cycles. Using this technique, several problems in both 2D and 3D are solved numerically where the objective is to minimize structural mass subjected to a fatigue constraint.

Plasticity damage self-consistent model incorporating stress triaxiality and shear controlled fracture mechanisms – Model verification and validation

The plasticity damage self-consistent (PDSC) model is a damage model developed in the context of continuum damage mechanics in which the damage dissipation potential is formulated to account for damage contributions due to intervoid necking, shearing, and sheeting for an arbitrary stress state. The synergic action of different micromechanisms contributing to ductile fracture is accounted for by their dependence on the plastic deformation, stress triaxiality, third invariant of the deviatoric of the stress tensor, and unilateral conditions. In this work, the PDSC has been implemented in the commercial finite element code MSC MARC via user subroutines and used to predict ductile failure response under different stress state conditions. Firstly, the model implementation has been verified by comparing calculated damage evolution as a function of plastic strain, for prescribed combinations of stress triaxiality and Lode parameter under proportional loading, with an analytical solution. Successively, the model has been validated in predicting ductile fracture for Al6061-T6 in different specimen geometries and comparing with experiment results. Model capability to anticipate fracture initiation conditions, together with specimen global response, and ductile crack propagation also for complex load paths, is demonstrated.

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 multi-featured shape memory alloy constitutive model incorporating tension–compression asymmetric interpolation

The mechanical response of shape memory alloys is heavily governed by phase transformation and plasticity. These inelastic mechanisms present added complexities in the form of asymmetries with respect to the stress state (tension–compression asymmetry) and the direction (forward or reverse) of transformation, leading to the formation of internal loops. With this in mind, a constitutive model is developed incorporating phase transformation and yield plasticity. The model represents the tension–compression asymmetry through a novel approach by interpolating the maximum transformation strain, the Young’s modulus of martensite and a plasticity factor in the deviatoric plane of the transformation strain direction. Internal loops are also considered by scaling the transformation hardening and the hysteresis width based on memory points. The interaction between the processes of reorientation and phase transformation is also taken into account. The model is validated with several proportional and non-proportional loading tests with considerable success. The capability of the model to simulate complex loading scenarios is demonstrated by simulating the manufacturing and loading of a stent structure.

Experimental characterization and modeling of complex anisotropic hardening in quenching and partitioning (Q&P) steel subject to biaxial non-proportional loadings

Quenching and partitioning (Q&P) steels, as one of the third-generation advanced high strength steels, exhibit complex anisotropic hardening behavior under biaxial non-proportional loadings. In this study, a wide range of strain path changes (SPCs) defined as an angle (χ) between stress deviators before and after the SPC is experimentally applied to characterize the anisotropic hardening of a Q&P steel with a strength grade of 980 MPa (QP980). The investigated SPCs include uniaxial tension followed by compression (UT-UC; χ=180°), uniaxial compression followed by tension (UC-UT; χ=180°), uniaxial tension followed by plane strain (UT-PS; χ=30°), equi-biaxial tension followed by plane strain (EBT-PS; χ=30°), followed by uniaxial tension (EBT-UT; χ=60°) and followed by simple shear (EBT-SH; χ=90°). These SPCs are achieved by the program-controlled biaxial tensile testing equipment using novel arm-reinforced cruciforms. The experimental findings in the QP980 steel sheet are summarized as follows: (1) Strong strain-dependent Bauschinger effect, transient hardening and permanent softening are observed in both UT-UC and UC-UT loadings; (2) Bauschinger effect increases as strain increases, and it exhibits obvious tension-compression asymmetry; (3) The transient hardening behaviors are path-dependent under SPCs with various χ; (4) The Bauschinger coefficient is approximately a parabola function of cosχ; (5) The most significant strength softening occurs under EBT-SH loading. In order to describe the complex hardening behavior of the QP980 steel, a distortional hardening model is proposed on the basis of non-associated flow rule. A coupled quadratic and stress-invariant-based yield function is used to describe stress anisotropy, strength differential effect, and anisotropic hardening for proportional loadings. Then, the Bauschinger effect related hardening and path-dependent permanent softening are captured by a modified homogeneous anisotropic hardening model. The calibrated model parameters and experimental validations demonstrate that the strain- and path-dependent Bauschinger effect can be well predicted by the proposed constitutive model. Especially, the model enables to accurately describe both the asymmetric anisotropic hardening and yield surface distortions of the QP980 steel at various biaxial SPCs.

Strain localization analysis in materials containing randomly distributed voids: Competition between extension and shear failure modes

Strain localization is often a precursor to the ductile failure of materials. This paper investigates plastic strain localization phenomena in the case of random microstructures, namely cubic cells made of an elastic-perfectly plastic matrix embedding distribution of identical non-overlapping spherical voids. The consideration of random microstructures allows for a better representation of the interaction between voids and greater diversity of failure modes than single-void (or unit) cells. The cells are simulated by means of the finite element (FE) method for proportional stress loading paths with controlled stress triaxiality and Lode parameters. Strain localization is detected by Rice’s criterion computed at the cell level. This criterion is shown to accurately capture the onset of localization and the type of failure mode, either extension or shear banding. Moreover, the influence of the loading orientation, i.e. the orientation of the principal axes of the applied stress tensor with respect to the microstructure cube, is systematically studied. Significant anisotropy of failure behavior is observed, especially in the case of single void unit cells, which can be attributed to the intrinsic anisotropy of the simulation cells. Finally minimal failure strain values at localization with respect to all loading orientations are found. A zone of reduced ductility is observed under generalized shear loading conditions.

Multiaxial fatigue life prediction model considering stress gradient and size effect

Stress gradient effect (SGE) and notch size effect (NSE) have important influences on the fatigue performance of engineering components. In this paper, a new multiaxial fatigue life prediction model is developed by considering the effects of both. Firstly, with the help of the coordinate transformation principle (CTP), the location of the dangerous critical plane is found according to the energy-critical plane (ECP) method, and the equivalent stress gradient factor (SGF) within the fatigue damage zone is calculated by analyzing the equivalent stress distribution on the critical plane. Secondly, considering the size of the notch, the effect of stress concentration at the notch is analyzed. Meanwhile, a size influence factor is introduced to characterize the NSE and construct a new fatigue strength reduction factor. Further, combining with the Manson-Coffin equation (MC equation), a new fatigue life prediction model for multiaxial notched components is proposed by considering SGE and NSE on fatigue damage. Finally, the experimental data of fatigue specimens with different notch sizes processed by En8 and GH4169 materials are used for model validation and comparison with other three classical models, and the results show that the prediction life of the new model for proportional and non-proportional loads is within two-fold dispersion band and the prediction ability is better than the other three classical models.

The mechanical response of commercially pure copper under multiaxial loading at low and high strain rates

In this paper, we present the dynamic response of commercially pure copper subjected to combined tension-torsion loads representative of real case impact scenarios. Experiments were conducted both quasi statically, at a strain rate equal to 10−3 s−1, and dynamically at strain rates in the region between 500 s−1 and 1000 s−1. All high rate experiments were conducted using a novel Split Hopkinson Tension-Torsion Bar instrumented with high-speed photographic equipment. The dynamic combined loading experiments demonstrate the capability of the apparatus to generate longitudinal and torsional stress waves which are synchronised upon loading of the specimen. The presented data show that dynamic equilibrium conditions and nearly steady strain rates were achieved during the experiments. Additionally, the analyses of the loading paths show that nearly proportional strain loading was attained during testing.The measured experimental results illustrate, for the first time, the failure stress locus of the material over a wide range of stress states including pure torsion, shear-dominated combined tension-shear, tension-dominated combined tension-shear and plain tension. The quasi-static and dynamic failure envelopes are herein presented in the normal stress vs shear stress space to motivate the development of accurate and effective constitutive models. To conclude, the Drucker-Prager criterion was employed to approximate the failure loci and to assess the rate sensitivity of the material. A moderate asymmetry of the uniaxial ultimate stresses in tension and compression is predicted both at quasi-static and dynamic strain rates.

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.

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.

Physics‐constrained hierarchical data‐driven modelling framework for complex path‐dependent behaviour of soils

AbstractThere is considerable potential for data‐driven modelling to describe path‐dependent soil response. However, the complexity of soil behaviour imposes significant challenges on the training efficiency and the ability to generalise. This study proposes a novel physics‐constrained hierarchical (PCH) training strategy to deal with existing challenges in using data‐driven models to capture soil behaviour. Different from previous strategies, the proposed hierarchical training involves ‘low‐level’ and ‘high‐level’ neural networks, and linear regression, in which the loss function of the neural network is constructed using physical laws to constrain the solution domain. Feedforward and long short‐term memory (LSTM) neural networks are adopted as baseline algorithms to further enhance the present method. The data‐driven model is then trained on random strain loading paths and subsequently integrated within a custom finite element (FE) analysis to solve boundary value problems by way of validation. The results indicate that the proposed PCH‐LSTM approach improves prediction accuracy, requires much less training data and has a lower performance sensitivity to the exact network architecture compared to traditional LSTM. When coupled with FE analysis, the PCH‐LSTM model is also shown to be a reliable means of modelling soil behaviour under loading‐unloading‐reloading and proportional strain loading paths. In addition, strain localisation and instability failure mechanisms can be accurately identified. PCH‐LSTM modelling overcomes the need for ad‐hoc network architectures thereby facilitating fast and robust model development to capture complex soil behaviours in engineering practice with less experimental and computational cost.

Axisymmetric V-Bending of Sheet Metal: Determining the Fracture Strain and the Weakest Material Direction for Plane Strain Tension in One Test

Plane strain tension is one of the most critical loading conditions leading to ductile failure in forming and crash applications. Hence knowing the fracture strain and weakest orientation for this stress state is crucial for safe design. A novel axisymmetric V-bending experiment is proposed to identify the strain to fracture and the weakest sheet material orientation for plane strain tension under proportional loading. After clamping its inner and outer boundaries, a disc-shaped specimen with an annular gage section is bent over a tubular knife of diameter 54mm. A single camera takes images of the top surface of the specimen throughout the experiment. It allows for timely crack detection and digital image correlation-based strain measurements. Experiments are performed on a range of modern engineering materials comprising two aluminium alloys (AA7075 and AA6014) as well as two steels (DP1180 and CR4). The experiments are complemented by Finite Element simulations to assess the robustness of the novel experimental approach.

Deformation and mechanical responses of AZ31B magnesium alloy under combined shear-compression loading conditions

This work presents a test method and the analytical procedure for materials under complex stress/strain state without the expensive multi-axis experimental machine. The deformation and mechanical responses of Mg alloy under combined shear-compression loading are investigated through a specific loading device. The deformation process and strain path of specimens with various loading angles are discussed based on digital image correlation (DIC) technology. Results show that the combined strain/stress state can be obtained through introducing the additional shear component. However, the loading angle almost has no effect on strain/stress path. In addition, the traditional testing machine equipped with combined loading device provides a new method to carry out the proportional loading test. A modified yield criterion considers Tresca-type material and tension-compression asymmetry is proposed to capture the mechanical characteristics of AZ31b Mg alloy. Based on the investigation, the combined test procedure can be regards as an ideal method in creation of materials database which focus on accurate description about the mechanical and fracture behaviors of materials.

QoS-Driven Load Balancing in Hybrid LiFi and WiFi Networks

This work studies the quality of service (QoS) performance of wireless networks that integrate light fidelity (LiFi) and wireless fidelity (WiFi). While the hybrid network is potential for improving network capacity, load balancing becomes essential and challenging due to the nature of heterogeneous access points (APs). A number of studies have been conducted to address this issue, focusing on maximising the network capacity with user fairness constraints. However, in practice, QoS metrics including packet loss ratio and latency are important to network services. In this paper, QoS-driven load balancing is studied for hybrid LiFi and WiFi networks (HLWNets) in two scenarios: single-AP association (SA) and multi-AP association (MA). In each case, an optimisation problem is formulated to minimise the packet loss ratio and latency, and a low-complexity iterative algorithm is proposed to solve the problem. Results show that the novel methods, especially MA, can effectively balance the traffic loads among the APs and improve the QoS performance. In addition, the more subflows the better performance MA provides. Targeting the same level of QoS, MA can achieve a system throughput up to 160% higher than the signal strength strategy and 130% higher than the proportional fairness load balancing.

Ductile failure under non-proportional loading

Ductile failure by void growth in an elasto-plastic material subjected to non-proportional loading, involving step changes in the stress triaxiality or the Lode parameter, is investigated using periodic unit cell model simulations. The equivalent strains to failure by the onset of void coalescence, defined as the localization of plasticity along a band of voids at the micro-scale, are determined as a function of the loading path parameters. The observed trends for the ductility under non-proportional loading are found to be inconsistent with the predictions of a continuum damage mechanics model based on the attainment of a constant critical damage variable, even if the model has been calibrated to predict accurate results for the ductility under proportional loading. It is shown that a recently developed failure criterion based on the onset of plastic instability in a porous material, coupled with a micromechanics-based void growth law, predicts the loading path dependence of failure under non-proportional loading, in better agreement with the cell model simulation results than the continuum damage model. In particular, the triaxiality dependence of the ductility is found to be primarily due to the hydrostatic stress dependence of void growth, while the Lode dependence is primarily due to the instability-based failure criterion with a relatively minor effect of the Lode parameter on void growth. Implications of these findings on the current modeling approaches to ductile failure under shear dominated loading are discussed.