Multiaxial Loading Conditions Research Articles

A new approach on constitutive modeling for quasi-brittle materials under multiaxial loading

Abstract This work introduces a new method for expanding unidimensional models to predict material behavior under multiaxial loading. The concept of six-dimensional mechanical space is adopted, and in each basis direction of the space, there is one fiber-bundle model (FBM), so these six FBMs are independent amongst one another but follow the same damage evolution rule. Under triaxial loading, these FBMs have different stress and strain states, so six corresponding damage variables have different values. The anisotropic damage is thus represented by the six damage variables. When it is under uniaxial loading, the six-dimensional space reduces to one-dimensional space, so the proposed model can be calibrated using uniaxial loading test data, and the calibrated model can predict the material behavior under multiaxial loading conditions. Biaxial loading and triaxial loading test data are used to verify the proposed model, and parametric analyses are performed to analyze model predictions under different loading conditions. It shows that model predictions agree well with the test data and are consistent with experimental observations.

Enhanced Cyclically Stable Plasticity Model for Multiaxial Behaviour of Magnesium Alloy AZ31 under Low-Cycle Fatigue Conditions.

Magnesium alloys, particularly AZ31, are promising materials for the modern automotive industry, offering significant weight savings and environmental benefits. This research focuses on the challenges associated with accurate modelling of multiaxial cyclic plasticity at small strains of AZ31 under low-cycle fatigue conditions. Current modelling approaches, including crystal plasticity and phenomenological plasticity, have been extensively explored. However, the existing models reach their limits when it comes to capturing the complexity of cyclic plasticity in magnesium alloys, especially under multiaxial loading conditions. To address this gap, a cyclically stable elastoplastic model is proposed that integrates elements from existing models with an enhanced algorithm for updating stresses and hardening parameters, using the hyperbolic tangent function to describe hardening and ensure a stabilised response with closed hysteresis loops for both uniaxial and multiaxial loading. The model is based on a von Mises yield surface and includes a kinematic hardening rule that promises a stable simulation of the response of AZ31 sheets under cyclic loading. Using experimental data from previous studies on AZ31 sheets, the proposed model is optimised and validated. The model shows promising capabilities in simulating the response of AZ31 sheet metal under different loading conditions. It has significant potential to improve the accuracy of fatigue simulations, especially in the context of automotive applications.

A frequency domain enhanced multi-view neural network approach to multiaxial fatigue life prediction for various metal materials

Fatigue damages and failure widely exist in engineering structures, particularly in industries such as aerospace, automotive, and construction, where components are often subjected to complex multiaxial loading conditions. Accurate prediction of fatigue life is critical for ensuring the safety and longevity of these structures. In this study, a novel multi-view deep learning model incorporating frequency domain analysis for fatigue life prediction is proposed. The proposed model integrates a Convolutional Neural Network (CNN), a Long Short-Term Memory Network (LSTM), and FNet that combines frequency domain analysis, in a parallel structure to extract features from the loading paths of materials. These extracted features are then connected to a fully connected neural network to predict fatigue life. The model was validated using fatigue data collected from 6 different materials, encompassing 17 loading paths and 336 samples. Additionally, ablation experiments were conducted, and the extrapolation capabilities were evaluated using specifically designed test sets. The results demonstrate that the proposed model exhibits excellent predictive performance and extrapolation capabilities. We anticipate that the multi-view approach, along with its accuracy and applicability, can offer potential applications in engineering fields that require reliable, data-driven models to assess material durability under complex loading scenarios.

Experimental investigation and computational modelling of 304LN stainless steel under constant and variable strain path multiaxial loading conditions

This study investigates the fatigue behavior of 304LN stainless steel under constant and variable strain path multiaxial loading conditions. Experimental methods and computational modelling were used to analyze the response of the material to different loading scenarios. Experimental investigations were conducted to gather empirical data, which were subsequently validated using a modified Ohno-Wang and Tanaka cyclic plasticity framework. Two models were used for the analysis: the MOT model and a modified model that incorporated a weighted function and fractional functions for hardening and softening. In contrast, the modified model offers significant improvements, accurately capturing primary hardening and secondary softening across all loading scenarios. Furthermore, the modified model effectively simulates transient behavior, including load alteration and recovery effects, due to the integration of load history memory regulating functions. Overall, this study emphasizes the importance of considering the complexities of cyclic loading and the sensitivity of the material behavior to the loading history.

Ultimate resistance and fatigue performance predictions of woven-based fiber reinforced polymers using a computational homogenization method

PurposeIn order to achieve the desired macroscopic mechanical properties of woven fiber reinforced polymer (FRP) materials, it is necessary to conduct a detailed analysis of their microscopic load-bearing capacity.Design/methodology/approachUtilizing the representative volume element (RVE) model, this study delves into how the material composition influences mechanical parameters and failure processes.FindingsTo study the ultimate strength of the materials, this study considers the damage situation in various parts and analyzes the stress-strain curves under uniaxial and multiaxial loading conditions. Furthermore, the study investigates the degradation of macroscopic mechanical properties of fiber and resin layers due to fatigue induced performance degradation. Additionally, the research explores the impact of fatigue damage on key material properties such as the elastic modulus, shear modulus and Poisson's ratio.Originality/valueBy studying the load-bearing mechanisms at different scales, a direct correlation is established between the macroscopic mechanical behavior of the material and the microstructure of woven FRP materials. This comprehensive analysis ultimately elucidates the material's mechanical response under conditions of fatigue damage.

Fatigue behaviors and life evaluation of AISI304 stainless steel under non-proportional multiaxial random loading

Fatigue tests on AISI304 stainless steel under non-proportional multiaxial random loading conditions were conducted with uniform hollow specimens at room temperature. The results revealed initial hardening and subsequent cyclic softening under uniaxial loadings and additional hardening under non-proportional multiaxial loadings. This shift from softening to additional hardening significantly reduced the failure life. Fractographic analysis and Electron Backscatter Diffraction (EBSD) observations identified planar slip as the dominant deformation mechanism under uniaxial loading. Multiple slip system activations, strong slip interactions, and martensitic transformation were key factors influencing changes in cyclic deformation behavior under non-proportional loadings. A novel life evaluation method based on the Itoh–Sakane (IS) method was established, demonstrating accurate evaluations for both uniaxial and non-proportional multiaxial fatigue life under random loading conditions.

Experimental and theoretical study on high-temperature creep of VT6 titanium alloy under multi-axial loading conditions

In the framework of damage mechanics, we discuss a new mathematical model that describes the kinetics of the stress–strain state and damage accumulation during material degradation by the mechanism of long-term strength under complex multiaxial stress state. An experimental and theoretical technique is proposed for determination of material parameters and scalar constitutive functions for damaged media based on specially set experiments on laboratory specimens. The results of experimental studies and numerical simulations of short-term high-temperature creep of VT6 titanium alloy under uniaxial and multiaxial loading are presented. Numerical results are compared with the data obtained through experiments. Particular attention is paid to simulating the process of unsteady creep for complex deformation modes, accompanied by rotation of main areas of stress, strain and creep strain tensors. It is shown that the developed version of the constitutive relations of the damaged media enables us to describe the processes of unsteady creep and long-term strength of structural alloys under multiaxial stress with the accuracy sufficient for engineering calculations.

Multiaxial low cycle fatigue behavior and life prediction of wire arc additive manufactured 308L stainless steel

Wire arc additive manufacturing (WAAM) is a technique that has gained popularity in various industrial sectors due to its high productivity and flexibility. However, the comprehension of cyclic deformation and fatigue behaviors in WAAM austenitic stainless steel (SS), particularly under complex multiaxial loading conditions, is much less advanced compared to conventional manufacturing methods. This study aims to fill some of these gaps by performing fatigue tests of WAAM 308L SS under axial, torsional and axial–torsional multiaxial loadings. Similar tests were carried out on the hot-rolled 308L SS as a basis for comparison. WAAM 308L SS exhibits notable distinctions in cyclic deformation, fatigue life, cracking mode and fatigue failure mechanisms compared to its hot-rolled counterparts as a result of its distinct manufacturing technologies and resulting microstructures. To assess all fatigue life data, three critical plane criteria, namely FS, SWT and CXH criteria, were employed. The CXH model, which considers the effects of both tensile and shear damage on fatigue properties, has the ability to predict the fatigue life of both hot-rolled and WAAM 308L SS.

Testing the properties of cemented tailings backfill under multiaxial compressive loading

The engineering design of cemented paste backfill (CPB) typically relies on technical data obtained from tests conducted on CPB samples cured under conventional laboratory conditions. However, these methods often fall short in capturing the complex multiaxial stress curing conditions that CPB masses experience in many underground mines, particularly in deep underground mines. In these mining environments, CPB is subjected to multi-axial stress conditions during the curing process that are challenging to replicate in a controlled laboratory setting. No lab-scale apparatus can currently accurately reproduce these actual field multiaxial stress conditions, especially the intricate interplay of horizontal rockwall closure stresses, vertical self-weight stress, and overburden stresses from upper layers. Moreover, the impact of these conditions on the key engineering properties of CPB is unknown. To address this significant challenge and technology gap, this study has developed a new apparatus for curing and monitoring CPB under realistic multiaxial compressive loading conditions, simulating the complex stress environment encountered in underground mines. The developed apparatus was then utilized to assess the impact of multiaxial stress curing conditions, including vertical stress and horizontal stresses induced by rockwall closures, on key engineering properties (strength, deformation behaviour, self-desiccation) of CPB at early ages. The research results clearly indicate that the application of multiaxial stresses leads to a significant enhancement in the mechanical and physical properties of CPB. This enhancement is supported by observable densification of the pore structure in CPB subjected to multiaxial compressive stresses. In addition, a comprehensive analysis of deformation and strain patterns establishes a direct correlation between multiaxial stresses and the progressive increase in stiffness and hardness of CPB over time. Furthermore, the evolution of matric suction and degree of saturation provides compelling evidence of the significant impact of multiaxial stresses on pore water consumption or self-desiccation. The development of this new apparatus fills a substantial void in CPB research capabilities. Furthermore, the results obtained provide a better understanding of the complex behavior of CPB in multi-stress loading environments, paving the way for the design and optimization of backfill solutions that effectively address the unique challenges posed by deep underground mining operations.

Biaxial compressive capacity of textile-reinforced concrete

The structural behaviour of textile-reinforced concrete (TRC) components under uniaxial compressive loading has already been studied. However, for practical applications, multi-axial loading conditions are often of great importance. It is well known from steel-reinforced concrete that multiaxial compressive stresses lead to increased compressive strength. The extent to which this assumption can be transferred to TRC components should be investigated. This paper investigates the influence of textile carbon grid reinforcement on the compressive capacity of disc-shaped concrete specimens. The specimens were subjected to a biaxial compressive load in the plane of the reinforcement. Emphasis was placed on the influence parameters of production type of the disc specimens (laminated or poured), impregnation material of the carbon textile (polystyrene or epoxy resin) and number of reinforcement layers. It was found that the general presence of a textile reinforcement as a two-dimensional interference plane had the greatest influence on the compressive capacity of the specimens, while the other parameters investigated played a minor role. Depending on the load ratio in the two compression axes, there were increases or decreases in capacity compared to the uniaxial tests. This has to be taken into account in the design of the corresponding components. Finally, an outlook is given on future investigations planned to support and extend the results presented in this paper.

Numerical investigation of the influence of defects on the multiaxial fatigue strength of additively manufactured alloys

Laser Powder Bed Fusion (L-PBF) is one of the Additive Manufacturing (AM) techniques that have revealed salient advantages in enabling the fabrication of 3D parts with intricate shapes and added functionalities. Despite many advances, scientific challenges still exist. One of them, especially where the additively manufactured industrial components undergo fatigue loading, is the defects (gas pores and Lack of Fusions (LoFs)) that are induced during the fabrication process. The present work aims to investigate the influence of AM defects on fatigue strength under multiaxial loading conditions. From numerical simulations, the key question addressed is the impact of defects’ morphology on their criticality under multiaxial loadings. Finite Element (FE) elastic simulations at load ratio R = -1 under multiaxial loading conditions (tension, torsion and tension-torsion) have been performed on numerically generated spherical defect and LoFs obtained from micro-CT scans of additively manufactured TA64 alloy. The Crossland criterion (stress-based) was used to evaluate the fatigue strength and the obtained numerical results were compared with the available experimental results.

A macroscopic viscoelastic viscoplastic constitutive model for porous polymers under multiaxial loading conditions

A macroscopic constitutive model, the Porous Eindhoven Glass Polymer (Porous EGP) model, is presented to describe the deformation behavior of cavitated rubber toughened polymers under multiaxial loading conditions. It is shown that the proposed macroscopic constitutive model is able to describe the non-linear pre-yield regime, strain rate dependence, post-yield behavior (strain softening and hardening) and void evolution for loading conditions ranging from shear to equi-triaxial (pure triaxial) tension and compression. The Porous EGP model is a combination of a well established non-linear viscoelastic viscoplastic model, the Eindhoven Glassy Polymer (EGP) model, and the modified Gurson model. The Gurson model is adopted to determine the equivalent stress and plastic rate of deformation tensor making it depending on the void volume fraction, deviatoric and hydrostatic stress. The macroscopic constitutive model is developed based on the response of realistic 3D representative volume elements (RVEs) containing randomly positioned mono-disperse inclusions. The constitutive behavior of the matrix phase in this full-field model is described by the EGP model, and the cavitated inclusions are idealized as voids. Their response is studied for a range of void volume fractions, multiaxial loading conditions, strain rates and thermodynamic states. The yield behavior of the heterogeneous material depends non-linearly on the macroscopic hydrostatic stress. This response is well captured with the proposed macroscopic constitutive model.

Experimental characterization and modeling of cylindrical CFRP structures under quasi-static multiaxial loading conditions

An experimental and numerical investigation of the cylindrical carbon fiber reinforced polymer (CFRP) structures under various loads including tension/torsion loading conditions has been conducted. Various boundary conditions and parameters were taken into account to check the impact of the shear component to obtain the result. The nonlinear shear model proposed by Chang has been implemented to take into account the softening effect of the stress-strain curve caused by damage accumulation. The computational model of the thin-walled tubes contains the geometrical architecture of the material, such as interweaving, which are characteristic of the parts made by filament winding technology. The studies were preceded by preliminary tests of the individual components to predict elastic properties based on the Abolin'sh micromechanical approach. The strength parameters were empirically delivered on the basis of the experimental results and used to determine the failure of the structure. The accuracy of the calibrated nonlinear shear model was validated using strain gauges and digital image correlation techniques. The strain distribution obtained from FEA was compared with that of the optical method. The damage distribution provided by FEA is exhibited in a similar manner to the real one captured by DIC. The proposed model provides a precise prediction of the CFRP tubes under quasi-static loading conditions proven by the experiments.

Multi-scale analysis method for composite-metal hybrid truss nodes using the MPC-submodel approach

Large-scale composite material-metal composite structures subjected to multi-axial loading conditions experience stress concentration and damage due to the dissimilar mechanical properties of the materials. A multi-scale refined modeling analysis is essential for accurately investigating these hybrid truss nodes. However, the existing multi-scale analysis methods face certain challenges, including modeling difficulties, reduced computational efficiency, convergence issues, and inaccurate boundary conditions. In this study, we first proposed an MPC-submodel approach based on the submodeling method and the multi-point constraint (MPC) technique to address these challenges in multi-scale computations of large-scale structures. Subsequently, we conducted experimental tests on composite-metal hybrid trusses and performed finite element simulations on key nodes using the proposed method, validating its accuracy. Finally, we applied the proposed method to investigate the key nodes within a 200-m long-span composite-metal composite truss bridge. By comparing the results with those from finite element simulations using the multi-point constraint method, we effectively demonstrated the advantages of the proposed method. The results indicated that our approach accurately modeled boundary conditions, significantly reduced computational time, and improved efficiency while maintaining computational accuracy.

Deep homogenization networks for elastic heterogeneous materials with two- and three-dimensional periodicity

We present a deep learning framework that leverages computational homogenization expertise to predict the local stress field and homogenized moduli of heterogeneous materials with two- and three-dimensional periodicity, which is named physics-informed Deep Homogenization Networks (DHN). To this end, the displacement field of a repeating unit cell is expressed as two-scale expansion in terms of averaging and fluctuating contributions dependent on the global and local coordinates, respectively, under arbitrary multi-axial loading conditions. The latter is regarded as a mesh-free periodic domain estimated using fully connected neural network layers by minimizing residuals of Navier's displacement equations of anisotropic microstructured materials for specified macroscopic strains with the help of automatic differentiation. Enabled by the novel use of a periodic layer, the boundary conditions are encoded directly in the DHN architecture which ensures exact satisfaction of the periodicity conditions of displacements and tractions without introducing additional penalty terms. To verify the proposed model, the local field variables and homogenized moduli were examined for various composites against the finite-element technique. We also demonstrate the feasibility of the proposed framework for simulating unit cells with locally irregular fibers via transfer learning and find a significant enhancement in the accuracy of stress field recovery during neural network retraining.

Multiaxial fatigue life predictions of additively manufactured metals using a hybrid of linear elastic fracture mechanics and a critical plane approach

This paper introduces a hybrid approach which leverages Fatemi-Socie critical plane analysis to determine an equivalent-damage uniaxial stress amplitude for various multiaxial loading conditions including uniaxial, torsion, and axial-torsion. Then, mode I crack growth constants along with initial defect information are used to make life predictions for multiaxial loadings as a function of their equivalent-damage uniaxial stress amplitudes. Two different Laser-Powder Bed Fusion metals, Ti-6Al-4V and 17-4 PH stainless steel, in a total of nine different surface and heat treatment conditions are analyzed and life predictions based on initial defect size and mode I crack growth properties from the literature are made using the Hartman-Schijve crack growth equation.

Quantitative representation of directional microstructures of single-crystal superalloys in cyclic crystal plasticity based on neural networks

Nickel-based single-crystal alloys undergo microstructural degradation induced by thermal exposure. The directional rafting of microstructures significantly affects the mechanical properties and makes the material anisotropic. For structural design, establishing a quantitative description of microstructural effects in a constitutive model becomes essential and is still a tough research topic in multi-scale materials modeling. In the present work, the fabric tensor was correlated with the anisotropic cyclic crystal plasticity of nickel-based single-crystal alloys with the help of neural networks. The microstructural representative volume elements with various single-crystal morphologies were generated by the phase-field method and the deformation behaviors were studied under different crystal orientations and loading configurations. The neural network analysis confirmed that the fabric tensor can present anisotropic single-crystallographic microstructural features and describe mechanical behavior under both monotonic and cyclic multi-axial loading conditions. The history-dependent anisotropic cyclic hardening or softening behavior of the material can be captured by the introduced microstructural state variable. A principal component analysis (PCA) aided gradient-based attribution method was proposed to evaluate the importance of input variables. The characterization of different material components and their contribution to the stress–strain relationships are investigated and validated. The fabric tensor was verified to be an effective microstructural indicator for the continuum plasticity of single-crystal alloys.

An attempt to design a universal specimen for any multiaxial loading conditions

The design of the multiaxial testing specimen is a challenging for the arbitrary loading conditions. We find that the stress of the inner thin region of biaxial/multiaxial testing specimen is not in proportion to one of the outer thick region. Therefore, a large magnitude of anisotropy in strength of the biaxial/multiaxial testing specimen may result in its failure before realizing the required stress state. Moreover, it makes the design of the multiaxial testing specimen unreliable for the arbitrary loading conditions. We have first attempted to obtain a universal design of a standard biaxial/multiaxial testing specimen for all materials which can also be employed to realize any stress state at its center, and found it impossible. Alternatively, we have adopted a typical composite material and optimized its geometry to realize widely representative stress modes at the center of a testing specimen. We have focused on obtaining the less number of biaxial/multiaxial testing specimen that can be employed for all stress modes. Overall, this work can be used as a guideline of specimen design for a reliable failure prediction of the composite structures under complex loading conditions.

A Micro-Macro mixed GTN-MMC model to study plasticity and fracture of AISI 4340 steel

Experimental and numerical studies were conducted to study the plastic and fracture behaviors of AISI 4340 steel (with three different heat treatments) under multiaxial stress loading conditions, including axial symmetric and plane strain loadings. The classical Gurson-Tvergaard-Needleman (GTN) plasticity model was extended and calibrated to consider Lode angle dependence on the material's matrix plastic strength especially needed for plane strain loadings. The 3D fracture loci of AISI 4340 steels were previously calibrated by the Modified Mohr-Coulomb (MMC) model. Considering the microvoid growth and nucleation mechanisms, GTN model is combined with MMC model to derive a new analytical solution of microvoid volume fraction at fracture (ff) under general loading conditions. The ff should NOT be a constant, and it depends on different stress states. The modified GTN-MMC model was implemented using VMAT in Abaqus, and it gives a very good correlation between experimental data and finite element simulations.

Very High Cycle Fatigue and Ultrasonic Testing in Multiaxial Loading

Abstract Fatigue damage has special relevance on the lifespan of mechanical components and structures, as it takes responsibility for the majority of registered structural failures. Most fatigue testing today still uses uniaxial loads, nonetheless it is generally recognized that multiaxial stresses occur in many full-scale structures. Nowadays, the growing need for greater lifespans forced the study of material behavior under what is nowadays very high cycle fatigue (VHCF) regime, fatigue research beyond 10E07 up to 10E10 cycles. However, most classic fatigue machines are often very costly, both in terms of time and energy, for such an elevated number of cycles. The high costs of equipment and time to conduct such experiments have seen a massive improvement with the introduction of high frequency ultrasonic machines. Presently, it is already possible to characterize materials VHCF response under axial and multiaxial loading conditions in a fraction of the time. Ultrasonic testing machines were first introduced for simpler uniaxial loading and more recently adapted for multiaxial loading conditions. The present work reviews the current background of ultrasonic fatigue testing machines working at 20-kHz frequency, with emphasis on multiaxial fatigue and VHCF. A review of a series of fatigue tests are presented, respectively, axial, shear, axial/shear, and in-plane bi-axial fatigue tests with different combining dimension specimens allowing different shear/axial stress ratios. Special attention will be put into the performance of multiaxial classical cylindrical specimens under tension/torsion and flat cruciform specimens under in-plane bi-axial testing using low-cost piezoelectric transducers. By comparing all so far developed and successfully reached multiaxial specimens and respective methodologies, two base approaches were possible to be established. When designing new specimens or ultrasonic machines, or both, one of the two methods must be followed. To have a well-comprehended methodology and successful design approach will undoubtedly guide future multiaxial ultrasonic method ideas.