Multiaxial Loading Research Articles

A multiaxial thermo-mechanical damage model for concrete including Load Induced Thermal Strain (LITS)

In the current study, a multiaxial LITS model of concrete was innovatively developed. This model can predict the behavior of concrete subjected to wide applied load level ranges from 20 % to 80 % and temperatures up to 800 °C. A 3D LITS model is developed through a triaxial compression stress confinement implementation and Poisson's ratio. Considering the water evaporation of concrete under high temperature, a third order polynomial LITS derivative function and an exponential derivative function were proposed to characterize the effects of the stress confinement and thermal effect. In order to comprehensively predict the performance of concrete subjected to multiaxial compressive loads in fire, an orthotropic triaxial compression thermo-mechanical damage model for concrete is developed. The effect of LITS on the multiaxial stress-strain response was considered in the multiaxial model. The coupling thermo-mechanical contributions containing the development of LITS among thermal effect, stress confinement, damage and plastic hardening evolution were creatively considered in the potential function within thermodynamics framework. The numerical calculations of this proposed model were conducted for validation. The effect of LITS on the multiaxial stress-strain response was studied. The nonlinear stress-strain responses of concrete were accurately captured at temperatures ranges from 20 °C to 1000 °C through the comparative study with test results in literatures, which indicates the accuracy and applicability of the proposed model.

Concurrent predictions of tension, compression, and shear characteristics of epoxy using three-network viscoplastic model

Epoxy resin controls the mechanical behavior of fiber-reinforced epoxy composites, which are extensively deployed in the aerospace and automotive industries. For numerical simulations to accurately predict the mechanical deformations of these components under multi-axial loading, matrix characteristics must account for tension, compression, and shear loads at varying strain rates. The existing literature lacks a comprehensive approach to predicting the experimental outcomes across all three loading conditions in a polymer matrix simultaneously using a unified set of viscoelastic or viscoplastic model parameters. In this study, the Three-Network (TN) viscoplastic model was successfully applied to concurrently predict the tension, compression, and shear experimental data of an epoxy resin, published by Littel et al. ( Journal of Aerospace Engg., 2008). The elastic modulus, strain softening-hardening response after yield for tension-compression deformation, and post-yield stress saturation under shear deformation are all accurately predicted by the model at different rates of loading conditions. The predicted and simulated results matched well when the TN viscoplastic model predicted parameters were employed as material property in Abaqus (a commercial finite element software) to simulate the deformation modes. This integrated approach highlights the potential of the TN model in enabling precise predictions for epoxy-based composites, which is crucial for optimizing their performance and reliability in aerospace engineering.

Dealing with multiaxial non-proportional fatigue with varying mean stress: Invariant and critical plane approaches for component-like structural adhesive joints

A comparative investigation between an invariant-based approach using the signed Beltrami-stress and critical-plane approach using the Findley-stress is carried out for the fatigue lifetime prediction of an epoxy structural adhesive under multiaxial non-proportional loading with varying mean stress. Fatigue experiments with uniaxial and multiaxial loading (proportional and non-proportional) are done at stress ratios of −1 and 0.1. Two adhesive joints are evaluated: a hollow-cylinder butt-joint (HCBJ, sample with homogeneous stresses distribution), and a flange-rod-joint (FRJ, component-like specimen with complex stress state). For both geometries, multiaxial loading and an increasing mean stress lead to fatigue strength reduction, whereas the phase-shift effect is less pronounced. A systematic procedure for parameter determination is implemented for both approaches. For predictions of the HCBJ-sample, stresses are analytically obtained. For validation with the FRJ-sample, FEA-based stress calculation is used. For both joints, the critical plane approach results in more accurate predictions than the invariant-based approach. In comparative terms, the invariant-based approach is experimentally more complex, by requiring a mean stress correction, but its numerically simpler with lower calculation time. The critical-plane approach requires fewer experiments for parameter determination and provides more accurate results for non-proportional loading. However, it is more numerically demanding with larger computing time.

Fatigue Crack Characteristics in Gradient Predeformed Pearlitic Steel under Multiaxial Loading

Rolling contact fatigue of railway rails not only severely deforms the surface material near the rail head, but also induces an anisotropy in the mechanical behavior due to work hardening and alignment of the microstructure along the shear direction. Cracks typically initiate in this region and propagate along the aligned microstructure. The fatigue behavior of rails is evaluated under uniaxial loading in the undeformed material state. However, this is not representative of the contact loading condition and material performance after years of service. Herein, the nonproportional multiaxial fatigue of as‐received and biaxially predeformed pearlitic rail steel R260 is investigated. Four material states are investigated, corresponding to the microstructure found at different depths from the severely deformed surface material at the rail head. A starting notch is machined by electrical discharge machining to control crack initiation and allow for comparable surface crack propagation measurements. The crack path is found to be strongly influenced by the degree of predeformation while the early surface crack propagation rate is found to be similar for all material states.

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

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

Planar biaxial testing of CXL strengthening effects

The stiffening effect of corneal crosslinking (CXL) treatment, a therapeutic approach for managing the progression of keratoconus, has been primarily investigated using uniaxial tensile experiments. However, this testing technique has several drawbacks and is unable to measure the mechanical response of cornea under a multiaxial loading state. In this work, we used biaxial mechanical testing method to characterize biomechanical properties of porcine cornea before and after CXL treatment. We also investigated the influence of preconditioning on measured properties and used TEM images to determine microstructural characteristics of the extracellular matrix. The conventional method of CXL treatment was used for crosslinking the porcine cornea. The biaxial experiments were done by an ElectroForce TestBench system at a stretch ratio of 1:1 and a displacement rate of 2 mm/min with and without preconditioning. The experimental measurements showed no significant difference in the mechanical properties of porcine cornea along the nasal temporal (NT) and superior inferior (SI) direction. Furthermore, the CXL therapy significantly enhanced the mechanical properties along both directions without creating anisotropic response. The samples tested with preconditioning showed significantly stiffer response than those tested without preconditioning. The TEM images showed that the CXL therapy did not increase the diameter of collagen fibers but significantly decreased their interfibrillar spacing, consistent with the mechanical property improvement of CXL treated samples.

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.

Numerical study on the influence of cell gas on the compression behavior of expanded polypropylene

AbstractExpanded polypropylene (EPP) bead foam mainly consists of entrapped gas within closed polymer cells. This numerical study presents a method to account for the influence of this entrapped gas on the compression behavior of EPP foam. The method developed combines a finite element (FE) model of the foam structure with a smoothed particle hydrodynamics model to simulate the effect of the cell gas. The foam structure is modeled using the open‐source software neper, the FE simulations are conducted using the explicit FE solver of LS‐Dyna. Numerically obtained stress–strain curves for the investigated foam materials, both with and without considering the cell gas, are compared with experimental data from tests using a specially designed vacuum test chamber. The comparison shows a good agreement between numerical and experimental results, indicating that entrapped cell gas increases the structural stiffness under compression. However, in load‐hold‐unload tests, the numerical model fails to accurately capture the stress relaxation behavior observed during the hold phase of the experiment. This study highlights the significant impact of cell gas on the compression behavior of EPP foam and the need for further refinement in simulation strategy to capture effects like the stress relaxation and multiaxial loading.

Fatigue life prediction considering conversion of mean stress for titanium alloy under multiaxial thermo-mechanical random loading

This paper presents a thermo-mechanical fatigue (TMF) life prediction method for titanium alloy TA15 under multiaxial random amplitude loading. Through strain-controlled TMF tests, it was found that under full-reversed cyclic loading, there was a significant presence of mean stress in the stress response. The induced mean stress led to the anomalous phenomenon where the fatigue life of thermal phase angle out-of-phase (TOP) was shorter than that of thermal phase angle in-phase (TIP). Therefore, a new damage calculation model was proposed, which can take into account the pure fatigue damage considering mean stress conversion and the cumulative environmental damage. Uniaxial TMF tests, constant amplitude multiaxial TMF tests, and random amplitude multiaxial TMF tests are conducted to verify the proposed method. The predicted results for all types of tests are in a good agreement with the experimental data.

A 3D thermo-mechanical damage model for concrete including Short-Term Thermal Creep Strain (STTCS)

In the current study, a Short-Term Thermal Creep Strain (STTCS) model of concrete was innovatively developed. This model can predict the time-dependent deformation of concrete subjected to wide applied load level ranging from 20 % to 60 % and temperatures up to 900 °C. A creep compliance function was developed to study the nonlinear creep characteristics of concrete with creep parameters. The definition and sensitivity of the creep parameters were investigated. In order to comprehensively predict the performance of concrete subjected to multiaxial compressive loads in fire, an orthotropic triaxial compression thermo-mechanical damage model for concrete is developed. The influence of STTCS on the multiaxial stress-strain behaviors was taken into account in the multiaxial model. The coupling thermo-mechanical contributions containing the development of STTCS with thermal effect, stress confinement, plastic damage and hardening evolution were creatively considered in the potential function within thermodynamics framework. The sensitivity of yield surface was discussed. The numerical operations of this proposed model were conducted for validation. The nonlinear stress-strain responses of concrete were accurately captured at temperatures ranging from 20 °C to 800 °C through the comparative study with test results in literatures, which indicates the accuracy and applicability of the proposed model.

Probabilistic fatigue life prediction of notched specimens based on modified stress field intensity method under multiaxial loading

In practical engineering components, due to the existence of non-uniform stress and strain field near the notch, it brings severe challenges to fatigue life prediction when evaluating the integrity of notched components. In this study, a probabilistic fatigue life prediction model for notched specimens was established by coupling the stress field intensity (SFI) method and Weibull distribution. Firstly, the position of the dangerous point is determined by finite element calculation, and the maximum strain energy density plane through the dangerous point is defined as the critical plane. Secondly, from the perspective of 2D features, the traditional SFI method is modified based on the stress distribution on the critical plane, and a new concept of effective stress is proposed to predict the fatigue life of notched specimens by the experimental data of smooth specimens. Finally, a new non-proportional additional hardening factor is established to characterize the influence of material properties and loading path on fatigue life. The experimental data of Q345 low alloy steel and GH4169 nickel base alloy are used to compare and analyze the proposed model. The results show that the predicted life of the proposed model is in good agreement with the experimental life.

On how S-N curves for nonproportional loading can be determined from S-N curves for proportional loading – An exemplary evaluation of numerous fatigue tests using scaled normal stresses

The fatigue design of safety–critical components necessitates a delicate balance between sustainability and safety. This study explores the challenge of achieving a balanced design for components subjected to multiaxial loads, focusing on nonproportional loading effects such as an increasing or decreasing fatigue life compared to proportional loading. Existing studies present divergent views on the correlation between nonproportional hardening and fatigue life shift, with some proposing a connection while others treat them independently. It is still unclear which of the research groups may be right. In this article, the effects are considered separately as independent effects, with the focus on the shift in fatigue life.This paper compares various methods of representing fatigue test results under proportional and nonproportional loading. An attempt is made to establish comparability of the tests from different references. One challenge here is, on the one hand, to establish comparability with different stresses on individual material groups and, on the other hand, to consider the second material effect, namely nonproportional hardening.For this purpose, a strength hypothesis is used on the one hand and a plastic correction in combination with a damage parameter is applied on the other. The procedure described in this paper can also be used with other suitable strength hypotheses and damage parameters. In the following, the scaled normal stress hypothesis in combination with the damage parameter according to Smith, Watson and Topper is used as an example.The aim of this paper is to quantitatively describe the shift in fatigue life to be able to include this material effect in a fatigue assessment depending on the material group. A comprehensive database comprising steel, wrought aluminum, cast aluminum, and spheroidal graphite cast iron is utilized to evaluate the shift in fatigue life under nonproportional loading. The database covers various control types, specimen geometries, and amplitude ratios, extending from low cycle to very high cycle fatigue regime.Based on the evaluation, material group-dependent properties can be derived, which can finally be used in a fatigue assessment as an averaged correction of the calculated fatigue life or to predict S-N curves for nonproportional loading based on these for proportional loading.

Neural networks for critical load factors of unstiffened plates according to the reduced stresses method of EN 1993-1-5

The critical load factor resulting from the complete stress field is the main input parameter for the verification against plate buckling according to the reduced stresses method (RSM) from Eurocode EN 1993-1-5. Currently, the code provides just a simple and often conservative interaction formula with a limited application range. Realistic values can only be obtained by numerical methods, like the FEM, which is tedious for many load cases. To enhance the ease of use of the RSM, simple-to-use but accurate methods are needed, which can be automated and implemented into EN 1993-1-5. The current research deals with this problem by applying artificial neural networks (ANN) to predict the critical load factors of unstiffened plates under longitudinal, shear, and patch loading stresses, including their interaction behaviour. ANNs were used because approximated solutions with polynomials (as actual in the standard) are not suitable due to the complexity of the problem. The developed ANNs are highly accurate (as shown by an independent verification), can be implemented into the Eurocode, and can easily be used, even without in-depth programming skills. Moreover, a simple Python code is provided to demonstrate the ease of use. Still, the code can also be implemented into other software packages (like Excel), as only basic matrix operations are needed. The interaction surfaces are visualized, gaining a deeper understanding of plates under multiaxial loading situations. Finally, recommendations are given for generating ANNs for plate buckling problems.

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

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

Dynamic biaxial loading of vascular smooth muscle cell seeded tissue equivalents

An intricate reciprocal relationship exists between adherent synthetic cells and their extracellular matrix (ECM). These cells deposit, organize, and degrade the ECM, which in turn influences cell phenotype via responses that include sensitivity to changes in the mechanical state that arises from changes in external loading. Collagen-based tissue equivalents are commonly used as simple but revealing model systems to study cell–matrix interactions. Nevertheless, few quantitative studies report changes in the forces that the cells establish and maintain in such gels under dynamic loading. Moreover, most prior studies have been limited to uniaxial experiments despite many soft tissues, including arteries, experiencing multiaxial loading in vivo. To begin to close this gap, we use a custom biaxial bioreactor to subject collagen gels seeded with primary aortic smooth muscle cells to different biaxial loading conditions. These conditions include cyclic loading with different amplitudes as well as different mechanical constraints at the boundaries of a cruciform sample. Irrespective of loading amplitude and boundary condition, similar mean steady-state biaxial forces emerged across all tests. Additionally, stiffness-force relationships assessed via intermittent equibiaxial force–extension tests showed remarkable similarity for ranges of forces to which the cells adapted during periods of cyclic loading. Taken together, these findings are consistent with a load-mediated homeostatic response by vascular smooth muscle cells.

Near real-time calculation of the critical plane in all surface nodes of large metallic structures under multiaxial fatigue loading by visualization of the solution space

Applications of critical-plane-based fatigue criteria on large FE meshes remain uncommon, primarily due to the absence of a search scheme capable of identifying the critical plane for multiaxial loading scenarios in a way that is both reliable and fast enough. To address this gap, we introduce a three-module optimization scheme based on the Maximum Variance Method (MVM), guided by insights learned from visualizing the solution space. These visualizations not only validate the MVM strategy but also reveal a symmetry axis, which in turn assists in selecting the most critical among reciprocal solutions. Through iterative refinement of optimization parameters on automatically generated loading signals, our algorithm achieves a significant speedup over existing methods, while improving accuracy. The potential that comes with a processing speed of 10k nodes per CPU minute is demonstrated on a design challenge involving an additively manufactured Ti–6Al–4V bearing house subjected to non-proportional loading. A full critical-plane-based fatigue analysis on the 100k surface nodes takes 75 s with C++ PPL across 16 threads, which is fast enough to enable integration into iterative procedures such as topology optimization, promising substantial advancements in both academic research and engineering practices.

Yield locus and texture evolution of AA7475-T761 aluminum alloy under planar biaxial loading: An experimental and analytical study

AA7475 aluminum alloy is used as a structural material in aerospace and automobile applications where complex loading conditions are applicable. The material properties under multi-axial loading are recommended for efficient component design. Hence, the mechanical behaviour of AA7475-T761 aluminium alloy was investigated through planar biaxial tensile tests. The yield locus evolution and plastic strain direction were measured at higher plastic strain values (up to 7 %) at various load ratios. The yield loci were constructed and compared with existing yield criteria. Yld2000–2d yield criterion demonstrated accurate predictions with a deviation of approximately 1 %, highlighting its effectiveness in capturing the yield behavior of AA7475. Microstructure analysis using Electron Backscatter Diffraction (EBSD) was conducted to understand the misorientation evolution under the multi-axial loading. Bulk-texture analysis was performed using X-ray diffraction method. Texture analysis revealed that uniaxial deformation favoured the development of Cube and Inverse-Copper textures, while biaxial deformation yielded split Inverse-Goss texture formation. Fractography study shows very different fracture surfaces of biaxial samples compared to uniaxial samples. Overall, this study contributes to a comprehensive understanding of the yield behavior, microstructure, and texture evolution of AA7475-T761 aluminium alloy under different loading conditions at very large strains.

Multiaxial creep–fatigue failure mechanism and life prediction of a turbine blade based on a unified numerical solution approach

AbstractExposure of turbine blades to cyclic torsional loading at high temperature, stemming from pre‐torque installation and the aerodynamic forces during operation, has the potential to induce substantial creep–fatigue damage, thereby contributing to the likelihood of premature failure. Investigating the deformation mechanisms and proposing a reliable life prediction method aiming at torsional loading is critical to ensure the structural integrity of turbine blades. This study conducted strain‐controlled fatigue and creep–fatigue tests on Inconel 718 superalloy, employing a multiaxial servo‐hydraulic testing machine. Electron backscattering diffraction elucidated deformation and damage mechanisms, forming a basis for subsequent constitutive modeling and life prediction. The lack of creep–fatigue mechanical behavior and microscopic failure mechanism when stress triaxiality equal to 0 is filled, which provides the theoretical basis and data support for the life design and damage assessment of this material under extreme service conditions. The unified viscoplasticity constitutive model effectively characterized macroscopic deformation under torsional loading. Prediction of creep–fatigue life under torsional loading, utilizing the multiaxial ductility factor‐modified strain energy density exhaustion model, demonstrated excellent alignment with experimental findings. Finally, parametric analyses of stress distribution and damage assessment under different conditions were carried out for the example of a turbine blade with relatively rarely considered aerodynamic loading as a variable. It is expected to be popularized and applied in life design and damage assessment of high‐temperature structures under multiaxial loading in engineering.

The anisotropic and region-dependent mechanical response of wrap-around tendons under tensile, compressive and combined multiaxial loads

The present work reports on the multiaxial region and orientation-dependent mechanical properties of two porcine wrap-around tendons under tensile, compressive and combined loads based on an extensive study with n=175 samples. The results provide a detailed dataset of the anisotropic tensile and compressive longitudinal properties and document a pronounced tension-compression asymmetry. Motivated by the physiological loading conditions of these tendons, which include transversal compression at bony abutments in addition to longitudinal tension, we systematically investigated the change in axial tension when the tendon is compressed transversally along one or both perpendicular directions. The results reveal that the transversal compression can increase axial tension (proximal-distal direction) in both cases to orders of 30%, yet by a larger amount in the first case (transversal compression in anterior-posterior direction), which seems to be more relevant for wrap-around tendons in-vivo. These quantitative measurements are in line with earlier findings on auxetic properties of tendon tissue, but show for the first time the influence of this property on the stress response of the tendon, and may thus reveal an important functional principle within these essential elements of force transmission in the body. Statement of significanceThe work reports for the first time on multiaxial region and orientation-dependent mechanical properties of wrap-around tendons under various loads. The results indicate that differences in the mechanical properties exist between zones that are predominantly in a uniaxial tensile state and those that experience complex load states. The observed counterintuitive increase of the axial tension upon lateral compression points at auxetic properties of the tendon tissue which may be pivotal for the function of the tendon as an element of the musculoskeletal system. It suggests that the tendon’s performance in transmitting forces is not diminished but enhanced when the action line is deflected by a bony pulley around which the tendon wraps, representing an important functional principle of tendon tissue.

Structural Analysis of Deck Reinforcement on Composite Yacht for Crane Installation

The crane installation on the deck of a yacht redistributes the stress field and affects the local structural integrity and performance. The safe operation of the yacht is associated with the optimal placement of the crane on the deck and the proper local structural reinforcement. Here, the structural analysis of the bow part of a yacht made of composite materials is studied, considering the retrofit installation of a crane, in three different cases of reinforcing the deck: (a) without any reinforcement, (b) with a T-type reinforcement, and finally, (c) with a longitudinal beam. The T-type connects the longitudinal bulkhead and the deck, reinforced locally with overlamination skin and adhesive-filler. The longitudinal beam works as a local longitudinal stiffener attached to the deck and connects the second, third, and fourth transverse frames. The structural analysis is performed using the finite element method following the classification societies’ rules. The local reinforcements are made from the same composite materials as the unreinforced deck. The maximum deformations, the principal stresses, and the safety factors following Tsai-Wu and Hashin criteria are calculated and compared for the three different cases. The T-type and longitudinal reinforcements reduce deck stresses by 33%, with longitudinal reinforcement reducing deck deformation by 17%. Composite failure analysis shows the structure was near failure, and the reinforcements enhance safety; T-type is better for multiaxial loads (Tsai-Wu), and longitudinal is superior for micromechanical failure (Hashin). By considering the structural performance and safety aspects, designers and engineers can make optimal decisions regarding yacht crane installation and proper reinforcement, leading to safer and more efficient structures.