Multiaxial Loading Research Articles

A Multiaxial Fatigue Life Criterion Including the Mean Stress Effect Based on π-Plane Projection and Walker Equation

Abstract Critical parts of engineering structures are usually subjected to asymmetric multiaxial fatigue loads. In this study, a new multiaxial equivalent stress ratio is defined, combining multiaxial fatigue damage parameters (based on π-plane projection) with the Walker equation to enhance the methodology for considering the effects of mean stress on multiaxial fatigue life. The new mean stress criterion is applicable to biaxial and more complex asymmetric loading conditions. Subsequently, the predictive results of the new criterion are compared with those of three established criteria using experimental data of four commonly used materials reported in the literature. The results indicate that the new mean stress criterion effectively accounts for the mean stress effects in multiaxial fatigue, offering greater simplicity and accuracy than the existing criteria. Furthermore, a good linear correlation is observed between the multiaxial correction exponent and material constants. To further validate the material applicability of the proposed model, multiaxial fatigue tests were conducted on 304 stainless steel under varying loading paths. The results demonstrate that the new criterion possesses good predictive accuracy and applicability, showing significant potential for predicting multiaxial fatigue life in engineering structures.

A finite fracture mechanics approach to estimate the fatigue strength of V-notched bars under multiaxial loading

The current study aims to assess the high cycle fatigue strength of sharply V-notched bars under mixed mode I/III loading by applying the coupled Finite Fracture Mechanics (FFM) approach. FFM provides strength predictions by simultaneously satisfying a stress condition and an energy balance. A novel semi-analytical implementation of the FFM criterion is presented for the first time to account for a multiaxial fatigue loading by assuming that fracture early propagates along the notch bisector plane through a circumferential-shaped crack. To validate the model, fatigue life predictions are then compared with a large variety of experimental data related to several metals, V-notch configurations and different multiaxial loading conditions. Although the adopted hypotheses are simplistic and do not fully encompass all the physical phenomena that occur in the multiaxial fatigue process, the approach reveals to be a reliable tool for obtaining semi-analytical fatigue limit predictions useful for engineering design practice.

Probabilistic fatigue life prediction of small sample properties notched specimens under multiaxial loading

Probabilistic fatigue life prediction of small sample properties notched specimens under multiaxial loading

Deformation response and microstructure evolution in 304LN stainless steel subjected to multiaxial fatigue loading under different strain paths

Deformation response and microstructure evolution in 304LN stainless steel subjected to multiaxial fatigue loading under different strain paths

A simplified finite-strain constitutive model for SMP to simulate foldable and auxetic structures: experiments and numerical implementation in ABAQUS

Abstract Thermo-mechanical one-way shape memory polymers (SMPs) are innovative materials capable of memorizing and recovering predefined shapes upon heating, with applications in aerospace and biomedical devices. Accurate modeling of their response to thermal and physical loads is essential for these applications. We present a simple and efficient modeling framework for polyurethane-based SMP. We conducted tensile tests at both high and low temperatures to measure the stress–strain behavior of the polymer in its rubbery and glassy phases, respectively. A thermomechanical SMP cycle leading to strain recovery was performed on a rectangular strip and an auxetic structure. To capture the shape memory effect, we extended the uniaxial frozen-phase model to a three-dimensional isotropic constitutive model based on Hencky’s law of elasticity, enabling predictions of finite-strain multi-axial loading responses. Young’s moduli for frozen and active phases were calibrated using tensile test data. The temperature-dependent phase function was established based on the shape recovery curve from thermomechanical experiments. A consistent tangent stiffness matrix was developed, and a numerical algorithm was implemented in displacement-based finite element (FE) software. The algorithm’s capability was validated by simulating SMP thermomechanical cycles in ABAQUS and comparing results with experimental data and MATLAB code under uniaxial loading. Additionally, we performed numerical simulations of SMP thermomechanical loading on an auxetic lattice structure and compared these with experimental results. Finally, the model was used for a parametric study on a single fold structure, a building block for self-deployable or origami-inspired structures in aerospace applications. This framework streamlines the design and implementation of complex SMP structures in FE analysis.

Comparative analysis and evaluation of the commonly-used fatigue life prediction models under various multiaxial loadings

This paper uses five materials to compare and analyze the life estimation results of six commonly-used critical plane models for multiaxial fatigue life. It is found that the estimation results of different models differ significantly for the same material. For the different materials, the estimation results of the same model are sometimes lower than the test results and sometimes higher than the test results. Among the six commonly-used critical plane models, the Fatemi-Socie model can provide satisfactory fatigue life prediction results for shear cracking materials. The reliability of life estimation results by different life prediction models depends on whether the fatigue damage parameters in the prediction model are reliable, and also on whether the expression of the prediction model is reasonable. In addition, the prediction results of the model are also related to the cracking form of the material. Based on the verification results, the applicability of each model was evaluated, and it was pointed out that further research is needed on the applicable conditions of commonly-used critical plane models.

A prior knowledge-guided predictive framework for LCF life and its implementation in shaft-like components under multiaxial loading

A prior knowledge-guided predictive framework for LCF life and its implementation in shaft-like components under multiaxial loading

Experimental testing and micromechanical modelling of unidirectional CFRP composite laminae under multiaxial loading conditions

Experimental testing and micromechanical modelling of unidirectional CFRP composite laminae under multiaxial loading conditions

Discrete ply modelling of aeronautical intermediate-scale notched carbon fibre reinforced thermoplastic specimens subjected to multiaxial loading

Discrete ply modelling of aeronautical intermediate-scale notched carbon fibre reinforced thermoplastic specimens subjected to multiaxial loading

Novel method for accelerated multi-axial load spectrum editing with phase preservation based on the fusion of damage and energy domain features

Novel method for accelerated multi-axial load spectrum editing with phase preservation based on the fusion of damage and energy domain features

An experimental methodology to characterize the multiaxial failure of short‐fiber reinforced thermoplastics

AbstractIn this study, an experimental approach is presented to determine strain‐based failure envelopes for short‐fiber reinforced thermoplastics under multiaxial loading conditions. The multiaxial failure behavior is experimentally investigated by Nakajima puncture tests that, via geometrical variations, allow the change of the stress state from uniaxial to biaxial. Local failure initiation can be mostly not identified by the macroscopic force–displacement curve. Instead, failure is assessed using 3D digital image correlation (3D DIC), which is used to measure the local strain and determine local failure. The methodology is demonstrated for systems with a ductile (polypropylene), pseudo‐ductile (polycarbonate), and brittle (polyphenylene sulfide) matrix. The shape of the failure envelopes is found to be very similar for most of the systems and shown to be governed by a critical strain parallel and perpendicular to the main fiber direction. The magnitude of the critical strains varies widely between the various systems and depends on the reinforcing constituent (e.g., fiber volume fraction), matrix characteristics (e.g., yield stress), and the addition of impact modifiers, as illustrated for the reinforced thermoplastics studied. The determined critical strain can be straightforwardly used to predict failure for complex products under multiaxial loading conditions.Highlights Characterization of multiaxial failure for short‐fiber reinforced thermoplastics The local failure envelope is found to be very similar for most systems studied. Failure is governed by a critical strain in two material directions.

Investigation of Fatigue of Glass Fiber–Reinforced Plastic Tubes Under Multiaxial In‐Phase and Out‐Of‐Phase Loading

ABSTRACTIn order to investigate fatigue in fiber‐reinforced plastics under in‐phase and out‐of‐phase multiaxial loading conditions, tube specimens were designed and tested. Initially, tension–compression and torsional moments were individually applied, followed by their superposition of in‐phase and with a 90° phase shift of the amplitudes. With a special clamping device, an inside illumination was possible and backlight images were taken to investigate the specific damage mechanisms for each scenario. For a better understanding of the layer‐wise stress situation in both scenarios, a classic laminate theory approach was conducted. From a 3D digital image correlation system, the strain fields were analyzed and a significant interlaminar effect in the out‐of‐phase testing was identified. The analysis of the measurements and the dissipated energies revealed a significantly lower fatigue life of the out‐of‐phase tested specimens compared with the in‐phase case, associated with a more severe interlaminar damage development.

Fatigue Behavior of the Auxetic Porous Bone Screw Under the Multiaxial Cyclic Loads in Tibiotalocalcaneal Arthrodesis.

The auxetic porous bone screw (AS) has favorable anti-pullout and osseointegration performance, demonstrating application potential in orthopedic surgeries. The uniaxial fatigue behavior of AS has been well understood. Considering that AS will withstand complex physiological loads in practical application, this study aims to investigate the fatigue behavior of AS under the multiaxial loads in tibiotalocalcaneal arthrodesis. AS and nonauxetic bone screw (NS) with the same porosity were designed based on re-entrant and hexagonal units, respectively. Finite element models of tibiotalocalcaneal arthrodesis implanted with AS and NS were established. Based on the curves of ground reaction forces borne by foot during normal gait cycle, the multiaxial loading spectrums were created and applied to the models. The multiaxial fatigue simulations were conducted to calculate the fatigue life and principal stress distributions of bone screws. Under the multiaxial loads in tibiotalocalcaneal arthrodesis, fatigue fracture was prone to occur in the AS and NS implanted in medial calcaneus. The minimum fatigue life and maximum principal stress of AS and NS were all located near the screw caps connected with the fixation plate. The tensile stress concentration of AS was significantly higher. The estimated fatigue life of AS and NS was approximately 46400 and 1820000 cycles, respectively. The fatigue life of AS was significantly lower than that of NS, which could not meet the fatigue resistance requirement during the recovery period of tibiotalocalcaneal arthrodesis. Local optimization should be conducted near the screw cap of AS to improve its multiaxial fatigue resistance.

Fatigue Assessment of Rib–Deck Welded Joints in Orthotropic Steel Bridge Decks Under Traffic Loading

Rib–deck (RD) welded joints in orthotropic steel bridge decks are prone to different fatigue crack mechanisms. Standard fatigue design methods are inadequate for some of these mechanisms under multiaxial non-proportional loading conditions. This study presents a framework to assess fatigue damage at RD welded joints, considering the different crack mechanisms based on the equivalent structural stress method and its extension to multiaxial non-proportional fatigue, which is the path-dependent maximum stress range (PDMR) cycle counting algorithm. The method is validated for uniaxial loading by using experimental data from the literature. Additionally, non-proportional fatigue damage at RD welded joints of a suspension bridge girder is investigated under simulated random traffic loading. The analyses reveal the limitations of the nominal stress approach to account for complex stress field variations. The PDMR method, more suited to capture the stress path dependency of non-proportional fatigue damage than the hot spot and critical plane-based methods, predicts higher fatigue damage. A comprehensive fatigue test campaign of full-scale RD welded joints is necessary to better understand their fatigue behaviour under multiaxial loading. Until more experimental data are available, the PDMR method is recommended for fatigue verifications of welded RD joints as it yields safer predictions.

Quantitative Review of Critical Plane Criteria and Stress Analysis Approaches for Multiaxial Fatigue of Welded Joints

ABSTRACTThis quantitative review evaluates the effectiveness of stress‐based critical plane criteria, specifically Findley's criterion, the approach due to Carpinteri–Spagnoli (CS), and the Modified Wöhler Curve Method (MWCM), in assessing fatigue strength in aluminum and steel welded joints subjected to constant amplitude (CA) and variable amplitude (VA) multiaxial loading. These criteria were analyzed alongside stress analysis approaches, including nominal stress (NS), hot‐spot stress (HSS), effective notch stress (ENS), and the Theory of Critical Distances–Point Method (TCD PM). Results confirm that all criteria effectively estimate fatigue life for steel welded joints under CA loading, with MWCM combined with HSS proving most accurate. For aluminum joints, estimations showed greater conservatism and scatter, highlighting the need for further experimental data to improve accuracy. Experimentally calibrated constants significantly enhanced prediction reliability. Future research should refine these criteria for diverse aluminum grades and thicknesses, ensuring accurate estimations and robust alternatives to established codes.

Notch and Fracture Mechanics‐Based Assessment of Multiaxial Fatigue Thresholds of Defects and Sharp Notches in Metallic Materials

ABSTRACTThis investigation focuses on the constant amplitude (CA) multiaxial fatigue limit of components made of metallic materials weakened by defects, cracks, and sharp U‐ and/or V‐notches. To estimate the multiaxial fatigue thresholds of plain, sharply notched, and cracked materials and defects, a novel theoretical framework based on the well‐known averaged strain energy density (SED) criterion is proposed, which extends the Atzori–Lazzarin–Meneghetti (ALM) diagram to multiaxial loading. The proposed design equations are validated against 128 experimental multiaxial fatigue limits, taken from the literature.

Characterising the mechanical behaviour of dry-formed cellulose fibre materials

Abstract In the dry-forming process, paper pulp is formed without adding water, making it more resource-effective than traditional papermaking. It is a relatively new technology, patented only in recent years, and very few material investigations exist in the literature; hence, little is known of the constitutive behaviour. The stress state during forming is highly complex, including multiaxial loading, extreme densification, friction, large strains, and fibre-joint formation. This paper studies dry-formed materials at different compression levels, from the sparse mat to the highly densified network. Three primary loading modes are investigated: in-plane tension, out-of-plane shear and out-of-plane compression. The results indicate that the tensile modulus and strength scale quadratically and cubically to the density, respectively, while the shear properties start developing after the density passes a threshold value. The compressive properties proved difficult to quantify, mainly because of the discrepancy between the density before and after the compressive test. The dry-formed material was compared to wet-formed paper materials in the literature. This showed that the in-plane (tensile) properties and the out-of-plane shear strength are visibly lower while the shear stiffness is similar, compared to wet-formed materials. Nonetheless, the findings set a starting point for numerical simulations of the dry-forming process.

Emergence of bimodulus (tension–compression asymmetric) behavior modeled in granular micromechanics framework

Many materials exhibit a bimodulus behavior which manifests as tension–compression asymmetry under uniaxial loading conditions. From the viewpoint of engineering analyses and design, attempts have been made to properly account for this asymmetry by defining bimodulus models under various multiaxial loading conditions. The models available in the literature propose a complex set of cases valid for different combination of tensile/compressive loading characterized in either strain or stress space based on classical continuum mechanics. The key drawback of these propositions is the attempt to reduce an inherently nonlinear problem to a simpler linear one in opposition to the clear evidence of loading dependency. Moreover, the proposed approaches are generally conceived in an orthotropic framework even though the material is initially isotropic. It is only upon the deformation that an apparent anisotropy is induced due to the tension–compression asymmetry, which cannot be considered in the initial, reference, configuration. Indeed, from a fundamental viewpoint, the emergence of the macroscale asymmetry and induced anisotropy has its roots in the microscale mechanics of an otherwise isotropic (or other) microstructure. To further expatiate the fundamental view, we exploit the granular micromechanics paradigm to present a simple continuum model within the isotropic framework in reference configuration. Using this approach, the macroscale tension–compression asymmetry and induced anisotropy is shown to emerge naturally without the inconsistencies observed in the methods proposed in the literature.

Assessing the effectiveness of the linear Drucker–Prager material model in predicting the mechanical behavior of acrylonitrile–butadiene–styrene (ABS) under multiaxial loading conditions

AbstractThis study aims to evaluate the effectiveness of the linear Drucker–Prager material model in accurately predicting the impact behavior of amorphous polymers, with a specific focus on acrylonitrile–butadiene–styrene (ABS), a material known for its versatility and wide industrial use. The relevant material model parameters were determined by incorporating test data from uniaxial tension and compression tests conducted at the same strain rate, along with insights from the Mohr–Coulomb model. The approach employed to determine the material model parameters was validated by comparing the results of three‐point bending simulations at varying speeds, incorporating the Cowper–Symonds model to capture the strain rate‐dependent post‐yield behavior of ABS against experimental data, before being applied in the impact simulations. The comparative analysis of the three‐point bending and impact simulation results revealed that the linear Drucker–Prager model provides a more accurate estimation of the three‐point bending response of ABS than its impact behavior, suggesting that the model is better suited for predicting responses under more uniform stress states that align closely with its assumptions. The results further indicated that the material model effectively predicts the impact behavior of ABS under various energy levels, despite its limitations in fully accounting for the complexities of dynamic effects and stress states in impact scenarios. The methodology developed in this study for determining the associated material model parameters of ABS can also be applied to other amorphous polymers and can be employed in impact simulations to effectively predict their impact behavior.

Parameter Identification of the Mooney–Rivlin Model for Rubber Mounts Subject to Multiaxial Load

Parameter Identification of the Mooney–Rivlin Model for Rubber Mounts Subject to Multiaxial Load