Biaxial Loading Conditions Research Articles

Effects of transverse loading on the ultimate strength prediction of stiffened panels under biaxial loads: Potential improvements to the IACS rule formulation

The current International Association of Classification Societies (IACS) rule for designing stiffened panels was found in Wang et al. (2024) to conservatively predict the ultimate strength of panels when transverse loads predominate. This conservatism arises from the overestimation of bending moments acting on stiffeners by directly applying the beam analogy in both longitudinal and transverse directions in the rule formulation when evaluating stiffener yielding.To address this issue, this paper applies the orthotropic plate theory to calculate bending moments experienced by the stiffeners of panels under biaxial loads. Orthotropic properties are derived from equivalent sectional areas and moments of inertia. The predicted bending moments are validated through comparisons with numerical simulations. The effects of boundary conditions and deflection patterns with varying numbers of half-waves are discussed. Based on the analytical results, a modification of the IACS rule formulation is proposed by introducing a correction factor to account for transverse loading effects in the bending moment calculations. The factor is a function of the ratio between longitudinal and transverse loading.The modified IACS rule formulation is verified through numerical simulations of aluminium panels using ABAQUS and comparisons with results from existing literature on steel stiffened panels under various biaxial loading conditions. The modified rule formulation shows improved accuracy in predicting the ultimate strength of stiffened panels, particularly in cases with dominating transverse loads. The modified IACS rule formulation can be useful for more reliable and cost-effective design of ships and offshore structures.

Smeared fixed crack model for quasi-static and dynamic biaxial flexural response analysis of aluminosilicate glass plates

Glass materials are extensively used in load-bearing structures and impact-resistant components because of their distinctive physical and chemical properties. Accurate predictions and assessment of the mechanical responses of glass structures are crucial for structural design and reliability analysis. In this study, quasi-static and dynamic ball-on-ring (BOR) biaxial flexural tests are conducted on aluminosilicate glass. The smeared fixed crack model is calibrated for deformation and failure analyses. First, the model parameters are calibrated carefully, particularly for different failure criteria. Both the deformation field and fracture modes agree well with the experimental observations during the quasi-static tests. For the low-velocity impact biaxial flexural loading condition, three different numerical techniques, namely the initial scaling tensile strength criterion, non-local approach with energy criterion, and rate-dependent failure stress criterion, are implemented in the numerical models for dynamic failure analysis. Finally, the proposed smeared fixed crack model is compared with the widely used Johnson Holmquist Ⅱ (JH-2) model and demonstrates advantages for low-velocity impact response analysis of glass structures.

Crack Growth Patterns of Aluminum Tubular Specimens Subjected to Cyclic Tensile Loads

This study presents a detailed analysis of a fatigue test campaign in order to identify different crack patterns. It was conducted on 6061-T6 aluminum tubular specimens featuring an internal diameter of 10 mm and different thicknesses (2, 3 and 4 mm). These specimens were subjected to cyclic tensile loads with a load ratio of R = 0.1, utilizing a sinusoidal load function at a frequency of 3 Hz. The investigation examines the crack growth rates, the stress intensity factor, and the final and intermediate fracture zones by applying overloads in some cases. The differences with two-dimensional specimens and the importance of this study for the interpretation of results with biaxial loading states are highlighted. The different states of crack growth detected are analyzed using artificial vision techniques. The differences between the exterior and interior faces of the specimen are revealed, and a series of states prior to the formation of the radial crack front expected in these specimens are identified.

Interaction of multiple micro-defects on the strength and failure mechanism of UD composites by computational micromechanics

The mechanical properties of unidirectional fiber-reinforced plastic (UD-FRP) are affected by a variety of micro-defects, such as random fiber arrangement, fiber misalignment and micro-voids. This study aims to investigate how these multiple micro-defects interact with each other and how they affect the strength and failure mechanism of UD-FRP by means of computational micromechanics. The composite behavior was simulated by the finite element analysis of a representative volume element of the composite microstructure in which the random distribution of fibers, the micro-voids, and the fiber misalignment are explicitly included. Both matrix and interface failure were considered for the loadings of transverse tension/compression, longitudinal compression, transverse/longitudinal shear, and their combination. It was found that these three micro-defects significantly weakened the compressive strength of UD-FRP along the longitudinal direction. Especially, the fiber misalignment magnified the effect of fiber arrangement, while the micro-voids reduce the effect. Besides, the fiber arrangement and micro-voids significantly weakened the tensile and compressive strength of UD-FRP along the transverse direction, but their interaction effect was not obvious. Moreover, transverse and longitudinal shear strength are significantly affected by micro-voids, but only longitudinal shear is affected by geometric fiber arrangement, and this effect is also weakened by micro-voids. Finally, the damage envelope under the combined longitudinal compression and transverse loads was obtained and compared with the Tsai-Wu failure criterion. The results showed that the Tsai-Wu criteria can provide an effective estimation for the failure locus under this biaxial loading condition.

Mesogen Organizations in Nematic Liquid Crystal Elastomers Under Different Deformation Conditions.

Liquid crystal elastomers (LCEs) exhibit unique mechanical properties of soft elasticity and reversible shape-changing behaviors, and so serve as potentially transformative materials for various protective and actuation applications. This study contributes to filling a critical knowledge gap in the field by investigating the microscale mesogen organization of nematic LCEs with diverse macroscopic deformation. A polarized Fourier transform infrared light spectroscopy (FTIR) tester is utilized to examine the mesogen organizations, including both the nematic director and mesogen order parameter. Three types of material deformation are analyzed: uniaxial tension, simple shear, and bi-axial tension, which are all commonly encountered in practical designs of LCEs. By integrating customized loading fixtures into the FTIR tester, mesogen organizations are examined across varying magnitudes of strain levels for each deformation mode. Their relationships with macroscopic stress responses are revealed and compared with predictions from existing theories. Furthermore, this study reveals unique features of mesogen organizations that have not been previously reported, such as simultaneous evolutions of the mesogen order parameter and nematic director in simple shear and bi-axial loading conditions. Overall, the findings presented in this study offer significant new insights for future rational designs, modeling, and applications of LCE materials.

A general solution for electromechanical analysis of electroelastic composite plates with arbitrary holes

Stroh formalism provides an elegant method of solution for two-dimensional anisotropic elasticity problems through the generalized eigen relation associated with the dual coordinate system that gives the material eigen values and eigen vectors, and also the fundamental elasticity matrix and its associated Barnett-Lothe tensors. The extended Stroh-like formalism of Hwu and Hsieh has expanded the scope of basic Stroh formalism by addressing the electro-mechanical coupling effects in electro elastic plates. This paper presents an inclusive solution that addresses the two-dimensional problems in infinite electro-elastic, anisotropic, and isotropic plates with an arbitrary hole under remote arbitrary coupled electro-mechanical loading. This is achieved by adopting the Stroh formalism and its subsequent versions to consider the anisotropic and electro-elastic plates, and further, by incorporating the arbitrary biaxial loading condition into the boundary conditions, and the generalized mapping function into the basic formulation to accommodate all types of in plane loading and a variety of hole shapes respectively. Various equations of the solution are derived explicitly for easy understanding and computer implementation. The stress function is derived explicitly to satisfy the condition of traction free hole with remote electro-mechanical loading under generalized plane stress-open circuit condition. The stresses and electric displacements around the hole boundary are obtained by taking the derivative of the stress function with respect to unit normal along the hole boundary. The solution presented can be degenerated to the anisotropic or isotropic case by choosing the corresponding material properties, or appropriate values of complex parameters respectively. This general solution has exactly reproduced the results of other solutions in the literature for piezoelectric plates given by different methods. New results of stresses and electric displacements are obtained for various arbitrary holes in [PZT5H/45/-45/PZT5H]s graphite/epoxy plate and PZT 4 plates. Results for holes in piezoelectric laminates and PZT 4 piezo layer are also presented by FEM.

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.

Toughening of intrinsically brittle materials by inserting arrays of voids

The aim of the current paper is to find optimum arrangements of voids that are able to catch and trap all possible cracks that might originate from a free surface. The idea is to investigate whether it is possible to apply this crack trapping mechanism to enhance the fracture toughness of intrinsically brittle materials. In doing so, the void volume shall be as small as possible. The investigation applies the crack trajectory interpolation (CTI) method, which is a new, computationally efficient approach to approximate crack trajectories near a single void or stiff particle. Under the assumption of linear superposition, the CTI-method is extended to predict the crack trajectories near arrangements of voids. This is done for uniaxial and biaxial loading conditions. The investigation shows that a symmetrical arrangement of voids should be avoided, since cracks near the symmetry line can bypass the array. Staggered void arrangements provide better conditions for crack trapping. By using elliptical voids with an aspect ratio of 2 instead of circular voids, it is possible to achieve a higher crack trapping efficiency and a lower void volume. A damage-based approach is used to determine the improvements in fracture toughness due to crack trapping for various void arrangements. The Python code of the CTI method is presented in the Appendix of this work.

Experiments on Low–Cycle Ductile Damage and Failure Under Biaxial Loading Conditions

BackgroundThe damage and failure behavior of ductile metals depends on the stress state as well as on the loading history. Biaxial experiments with suitable specimens can be used to targeted generate different loading conditions, thus allowing the investigation of a wide variety of load cycles with different stress states.ObjectiveIn the biaxial experiments with the newly presented HC-specimen cyclic shear loads are superimposed by various constant compressive and tensile loads. Buckling during compressive loading in both axes is avoided by an additional newly introduced downholder.MethodsThe strain fields at the surfaces of the biaxial specimens are evaluated by digital image correlation (DIC), and after failure the corresponding fracture surfaces are analyzed by scanning electron microscopy (SEM). Associated numerical simulations employing the presented material model provide information on the current stress states.ResultsThe introduced downholder successfully prevents buckling during compressive loading. The strain fields detect a clear influence of the shear direction and a tensile superposition of the cyclic shear load leads to more brittle and a compressive superposition to more ductile behavior. The accompanying numerical calculations reveal the associated, different stress states.ConclusionsThe new experimental program with biaxially loaded specimens for the investigation of damage and failure behavior under cyclic loading enables the targeted examination of a wide variety of load cycles and is thus suitable for the comprehensive analysis of these phenomena.

The experimental and numerical investigation of fracture behaviour in PMMA notched specimens under biaxial loading conditions – Tension with torsion

This paper presents the results of experimental fracture test of flat PMMA specimens under biaxial loading condition tension with torsion (proportional). The specimens were made in two thicknesses: 5 and 15 mm and were weakened with V-type edge notches with different root radii: 0.5; 2 and 10 mm. Thanks to the ARAMIS 3D 4 M non-contact vision system, measurement of the elongation and twist angle were recorded. During experimental part all of the deformation processes were observed using PHANTOM cameras. Obtained records made it possible to precisely indicate the moment of crack initiation (tensile force and torsional moment values). Using the microscopic observations the location of crack initiations were determined. Results obtained for biaxial loading were compared with those obtained for uniaxial tension and torsion. Based on experimental data the numerical calculation with FEM were carried out. The principal stress and plastic strain distribution under critical load, were obtained. The points of occurrence of stress maxima and plastic strain were indicated, which were taken as potential crack initiation sites. On the basis of the stress and plastic strain values measured at the critical points, a stress–strain fracture criterion was formulated, which was then positively verified. Additionally, new form of stress–strain fracture criterion was proposed. The bilinear form of the fracture criterion can be successfully used to predict fracture in PMMA flat specimens, regardless of the notch root radius, type of load or element thickness.

Biaxial Tensile Testing of Cruciform Samples to Determine the Impact of Pre-Strain on the Forming Limit Diagram of Aluminum Alloy 5052 through Finite Element Simulation

The present work focuses on the effect of pre-strain on the forming limit curves (FLCs) of aluminium alloy 5052 sheet of 2.5 mm thickness using cruciform samples. To identify the effect of pre-strain on the FLC, the finite element simulations are performed on cruciform samples. The cruciform samples are deformed to a small level of pre-strain in different strain paths. Thereafter, the further deformation under the various strain paths to measure the effect of pre-strain on the forming limits are carried out. Pre-strain in uniaxial, plane strain and biaxial loading conditions are considered in this work. For each case, i.e., uniaxial, plane strain and biaxial conditions two pre-strain conditions are considered. The cruciform samples are carefully thinned in the central region to promote development of large plastic strains there and eventually to neck and fail. The output obtained through simulations are presented in terms of forming limit curves for various strain paths.

Mechanical Behavior of Lithium-Ion Battery Separators under Uniaxial and Biaxial Loading Conditions.

The mechanical integrity of two commercially available lithium-ion battery separators was investigated under uniaxial and biaxial loading conditions. Two dry-processed microporous films with polypropylene (PP)/polyethylene (PE)/polypropylene (PP) compositions were studied: Celgard H2010 Trilayer and Celgard Q20S1HX Ceramic-Coated Trilayer. The uniaxial tests were carried out along the machine direction (MD), transverse direction (TD), and diagonal direction (DD). In order to generate a state of in-plane biaxial tension, a pneumatic bulge test setup was prioritized over the commonly performed punch test in an attempt to eliminate the effects of contact friction. The biaxial flow stress-strain behavior of the membranes was deduced via the Panknin-Kruglov method coupled with a 3D Digital Image Correlation (DIC) technique. The findings demonstrate a high degree of in-plane anisotropy in both membranes. The ceramic coating was found to negatively affect the mechanical performance of the trilayer microporous separator, compromising its strength and stretchability, while preserving its failure mode. Derived from experimentally calibrated constitutive models, a finite element model was developed using the explicit solver OpenRadioss. The numerical model was capable of predicting the biaxial deformation of the semicrystalline membranes up until failure, showing a fairly good correlation with the experimental observations.

Using miniature experiments to reveal strength gradients in battery casings

Complex metallic structures are often manufactured through multi-step forming or casting processes that induce severe mechanical property variations within the final part. Instead of using the virgin material properties to predict the final part performance, the local material properties after manufacturing need to be known. A novel comprehensive testing methodology is presented allowing the resolution of spatial gradients in the plasticity and fracture properties in metallic structures. It involves using a recently-developed miniature linear-motion system that probes the material within 2 mm wide gage sections (uniaxial, notched and central hole tension, in-plane shear). Additionally, a novel mini-bulge testing system is presented along with a mini-punch system to provide further insight into the local material response for biaxial loading conditions. A full testing campaign comprising 60 miniature samples is performed on all sides of a prismatic aluminum casing for Li-ion battery cells. The results demonstrate significant property variations with more than 30 % higher yield stresses for the side walls as compared to the bottom section. The miniature specimens allow identifying a location-dependent anisotropy, which would otherwise be missed by standard-sized samples. Furthermore, it is shown that the anisotropic Yld2000 plasticity model and the stress-state dependent Hosford-Coulomb fracture initiation model can be fully identified using results from experiments on miniature specimens only.

Experimental investigation on mechanical behavior of silane‐modified polyurethane sealant under biaxial cyclic loading

AbstractThe biaxial tension–tension cyclic loading experiments were conducted on the cruciform specimens prepared using a silane‐modified polyurethane sealant to investigate their mechanical behavior. The finite element method was employed to validate the mechanical behavior of cruciform specimens under the biaxial cyclic loading conditions. It is shown that the highest stress concentration is in the central region of the cruciform specimens, indicating that the shape design of the specimen is suitable for the biaxial tension–tension cyclic tests. Additionally, the effects of loading path, mean stress, and stress amplitude on ratcheting strain and dissipated energy were analyzed through the experimental data. The experimental results demonstrated that the loading path had an influence on the mechanical behavior of the cruciform specimen. The additional ratcheting strain caused by the non‐proportional loading path was found to be loading path dependent of the adhesive. Moreover, the mean stress and stress amplitude greatly affected the strain response curves of the specimen under biaxial cyclic loading. The ratcheting strain and dissipated energy increased with increasing mean stress and stress amplitude; nevertheless, stress amplitude had more influence on the mechanical behavior than the mean stress.

Effects of loading rate on void growth in amorphous glassy polymers

Initiation, growth, and coalescence of microscopic voids are precursors to failure in amorphous glassy polymers. Hence, void growth analysis is important for understanding the failure in these polymers. Under quasi-static conditions, void growth in polymers is well understood; however, a similar analysis under dynamic conditions is not available in the literature. In the present work, we analyse the effect of loading rate on the void growth behaviour using large deformation finite element simulations. An axisymmetric unit cell model of a spherical void is considered for the analysis. The void is subjected to a range of strain rates (10−3s−1 to 105s−1). Remote triaxialities equivalent to uniaxial loading, biaxial loading, and higher triaxiality conditions are considered. Remote mean stresses, which are important from the perspective of void growth, have been characterized as a function of void size, strain rate, and triaxiality. Results show that the critical mean stresses required for unstable void growth increase with increasing triaxiality. Higher loading rates further amplify the critical mean stresses, and the amplification depends upon the material rate sensitivity of polymers. The void growth mechanism does not change because of material rate-sensitivity. During void growth, inertia effects are found to be significant after a strain rate of 103s−1. It is found that inertia increases with increasing triaxiality and increasing void size. It stabilizes the void growth and makes them expand in a more spherical shape, even at low triaxialities.

Enhancing buckling analysis of functionally graded plates using unified and generalized higher-order shear deformation theories

This paper introduces a comprehensive framework for stability analysis of functionally graded plates under diverse in-plane loading conditions while employing higher-order shear deformation theories. For this purpose, a unified model, including all shear deformation kinematics, has been proposed. This unified model incorporates any number or type of shear transverse functions, encompassing all existing plate shear deformation theories. This inclusivity is achieved through the introduction of a unique parameter, denoted by ”n” representing the number of shear deformation functions in the displacement model. Such a unified model allows great flexibility in choosing among higher order theories when modeling the behavior of plates subjected to shear deformation, making it a valuable tool for applications in engineering and materials science. Besides, this allowed us to easily introduce a fifth-order theory used for the first time in the buckling analysis of FGM plates. The critical buckling loads expressions corresponding to the employed kinematics were derived and the calculated values are compared to those computed using the numerical solver. The comparison demonstrates the effectiveness and accuracy of the formulated expressions in predicting critical buckling loads under uniaxial and biaxial loading conditions.

LCF behavior of 2024AA under uni- and biaxial loading taking into account creep pre-deformation

This study presents the results of experimental low-cycle fatigue (LCF) tests of aluminum 2024 alloy T3511 temper in uni- and biaxial loading states. Tests were carried out on both the as-received material (hardened extruded rods) and material with different pre-deformation histories. These deformations were carried out in the creep process at 200 °C and 300 °C for two different levels of at each temperature. The pre-deformed material’s basic fatigue characteristics were determined and compared with the appropriate characteristics of the as-received material. In-depth macro- and microscopic analysis (SEM) of fracture surfaces was done. The effect of preliminary creep on LCF behavior of investigated alloy was characterized for both uni- and biaxial loading. An increase of fatigue life occurs for large plastic strains in the case of cyclic tension/compression. For in-phase biaxial loading, improvement of life is observed only for material with pre-deformation at 300 °C. Crack initiates in the plane of maximum shear strains for both biaxial loading and pure torsion. For tension/compression – in the plane of maximum principal stress (strain).

Experimental investigations of uniaxial and biaxial cold stretching within PC‐films and bars using optical measurements

AbstractPolycarbonate (PC) is an amorphous polymer that is an extremely robust material with a high tenacity, and thus suitable for a lightweight construction with glass‐like transparency. Due to these advantageous properties, PC is often used in industry for example in medical devices, automotive headlamps, sporting equipment, electronics, and a variety of other products. PC is often subjected to uniaxial and biaxial loading conditions. Therefore, reliable material models have to take into account the various resulting experimental effects. For those reasons, we investigate PC specimens under uniaxial and biaxial loading by using different stretch rates and loading scenarios. In addition to that, we propose methods for optical measurement of local stretches to obtain the approximated local true stress. In future work, the displacement fields and the resulting reaction forces will be used for parameter identification of constitutive equations.

Biaxial bending moments calculation model based on stress distribution for H-section steel members under different loading paths

To explore the biaxial bending development mechanism of H-section steel members under different cyclic loading paths, we conducted a parametric analysis involving various axial force ratios, plate width-to-thickness ratios, and loading angles. Three loading paths were considered, including linear, rectangular, and triangular loading paths. Finite element models were established in ABAQUS and validated for accuracy using existing experimental data. Based on the finite element analysis results, a methodology for determining the form of normal stress distribution was derived by analyzing the mechanism of biaxial interactions in H-section steel members. By establishing a relationship between the normal stress distribution and the biaxial bending moments, a semi-empirical semi-theoretical calculational model was developed to simulate the entire process of biaxial bending and compression loading at reverse point in H-section steel members under different loading paths. The validity of this model was confirmed, as it demonstrated effective prediction of the complete development trends of biaxial bending moments for H-section steel members. This model establishes the basis for seismic design of H-section steel members subjected to biaxial loading conditions.

Упругопластическое моделирование усталостных трещин

The presented work provides a detailed analysis of modern approaches to creating elastoplastic models of surface crack growth that take into account the influence of the type of biaxial loading on the development of stresses and strains at the crack tip and, accordingly, on the crack growth rate. The use of the plastic stress intensity factor as a characteristic of resistance to cyclic deformation and fracture for biaxial loading conditions is substantiated. Continuum plasticity models are proposed to simulate the elastoplastic behavior of metal using numerical methods and, in particular, the finite element method.