Biaxial Loading Research Articles

Stress inversion in waveguides with arbitrary cross sections with acoustoelastic guided waves

Acoustoelasticity or the change in elastic wave speeds with stress is promising for prestress measurements in waveguides. The theory of guided wave propagation in initially isotropic materials with arbitrary cross sections and under homogeneous biaxial stresses is developed using Semi-Analytical Finite Element (SAFE) modeling in this article. Based on the anisotropic effect induced by the applied biaxial load, an inversion method for biaxial force was developed. The acoustoelastic response for a particular mode and frequency is described by only two constants, which can be determined from known uniaxial loading experiments. The magnitude and direction of the biaxial force can be obtained by further coefficient fitting. Stress inversion can be obtained without considering the shape of the cross section and applies to multiple guided wave modes. The inversion has been verified by the results of SAFE and 3D Sweeping Frequency Finite Element Modeling (SFFEM) method, and the Mean Absolute Errors of stresses obtained by different methods are all within 1%. The 3D SFFEM was combined with the Matrix Pencil Method using the time domain information to extract the dispersion curve. Unlike previous finite element modeling, here the inheritance of the solution between the two solvers was set instead of approximating static load conditions by shortening the guided wave travel time. It guarantees the steady state of the force in the time-variant study, ensuring the high precision required for the study of the acoustoelastic effect.

Surface Wrinkling versus Global Buckling Instabilities in Thin Film‐Substrate Systems under Biaxial Loading: Direct 3D Numerical Simulations

AbstractWhen a thin film bonded to a compliant substrate is subject to in‐plane compressive loading, wrinkles on the thin film may form as a consequence of deformation instability. The specimen geometry may also favor global buckling. In this study, a recently developed computational approach is employed to directly simulate 3D wrinkling and buckling from pre‐instability to post‐instability in a straightforward manner, covering a wide biaxial loading spectrum. The finite element models contain an embedded imperfection with perturbed material properties at the film‐substrate interface. This approach is capable of activating the first and subsequent bifurcation modes, along with the deformation patterns in between. It is demonstrated that, in addition to the temporal evolution of wrinkle patterns, surface wrinkling and global buckling can form simultaneously as the substrate becomes thinner. Wrinkles and buckle can either co‐develop, or the buckling‐induced curvature may help facilitate wrinkle formation depending on the specimen thickness and loading biaxiality. Different wrinkle patterns, such as parallel waveform and checkerboard, can even coexist on the same buckled structure. The state of biaxiality can influence the deformation instability significantly, and each bifurcation point can be traced back to certain abrupt changes in the overall load–displacement response.

Transverse biaxial tests on long fibre reinforced composites

The study and prediction of the failure mechanisms affecting long fibre reinforced composites is a fundamental field that provides an essential knowledge for the structural design with composites. Transverse failure, generated at ply level by the mechanism of damage known as inter-fibre/matrix failure, usually determines the damage initiation in any laminate which justifies its relevant role and the need to understand its micromechanical stages. Previous numerical results established that in the case of transverse biaxial loads (T-nT and T-nC cases) the stages of the damage mechanism detected in the uniaxial case could be affected. The most outstanding conclusion anyhow was that the effect of a secondary tension did not significantly alter the ultimate value of the tension nominally responsible for the failure, whereas a secondary compression could advance the failure appearance. In order to experimentally check the validity of these conclusions the performance of transverse biaxial tests is undertaken in this paper. The present investigation begins with the presentation of the cruciform geometry selected for the specimen and is followed by the description of the specific manufacturing procedure employed. The results of the testing campaign carried out agree with the previous numerical prediction and provide relevant information for the prediction of failure initiation.

Thermodynamically-consistent derivation and computation of twinning and fracture in brittle materials by means of phase-field approaches in the finite element method

A theoretical-computational framework is proposed for predicting the failure behavior of two anisotropic brittle materials, namely, single crystal magnesium and boron carbide. Constitutive equations are derived, in both small and large deformations, by using thermodynamics in order to establish a fully coupled and transient twin and crack system. To study the common deformation mechanisms (e.g., twinning and fracture), which can be caused by extreme mechanical loading, a monolithically-solved Ginzburg–Landau-based phase-field theory coupled with the mechanical equilibrium equation is implemented in a finite element simulation framework for the following problems: (i) twin evolution in two-dimensional single crystal magnesium and boron carbide under simple shear deformation; (ii) crack-induced twinning for magnesium under pure mode I and mode II loading; and (iii) study of fracture in homogeneous single crystal boron carbide under biaxial compressive loading. The results are verified by a steady-state phase-field approach and validated by available experimental data in the literature. The success of this computational method relies on using two distinct phase-field (order) parameters related to fracture and twinning. A finite element method-based code is developed within the Python-based open-source platform FEniCS. We make the code publicly available and the developed algorithm may be extended for the study of phase transformations under dynamic loading or thermally-activated mechanisms, where the competition between various deformation mechanisms is accounted for within the current comprehensive modeling approach.

Effective stiffness, wave propagation, and yield surface attributes of Menger sponge-like pre-fractal topologies

In the current work, the effective stiffness, long-wavelength attributes, and yield limits of Menger sponge-like pre-fractal topologies are investigated for the first time. Three types of Menger sponge-like structures with circular (MC), rhombic (MR), and square (MS) cavities are considered up to their third-level fractal topology. For each level, the elastic, shear, and bulk moduli, as well as the Poisson's ratio values are computed. Moreover, third-level Menger sponge-like fractal structures are additively manufactured and experimentally tested. A clear difference between the elastic stiffness and yield strength of the different Menger sponge-like structures is numerically and experimentally identified. It is observed that the cavity type considerably affects the scaling of elastic stiffness attributes upon increasing fractal level. The MS and MC topologies relate to lower relative density structures compared to the MR topology for all fractal levels investigated. The percentage relative density reductions induced by the fractal topology are a multiple of the corresponding decrease in the magnitude of the propagating shear and longitudinal phase velocities. Furthermore, all Menger sponge levels and inner cavity types yield analogous wave propagation directional dependencies, though with different phase velocity magnitudes. It is also found that increasing fractal topologies diminish the yield strength limits of the Menger fractal structures, with the factor associating the normal and shear loading yield strength of two successive Menger topology levels to be lower, but comparable in magnitude with the fractal dimension number. The fractal level evolution of the yield surface is well-captured by the extended Hill's criterion both for bi-axial and axisymmetric loads. The corresponding coefficients are identified for first, second, and third-level MS structures.

Simulation of Mechanical Behaviour of Woven Fabrics at the Scale of Yarns under Multi-Loading

The mechanical behaviour of textile structures is one of their most important characteristics as far as their end use is concerned. Textile structures, fabrics, or yarns are often considered as continuous mediums apart from the fact that they are composed of some discrete elements, individual fibres composing yarns and yarns composing fabrics. This is known as the transition scale, a very important lock to be considered, to evaluate the real structure behaviour. In this context, this work presents some simulations of the mechanical behaviour of a fabric where the yarn is a continuum material. Particular attention was paid to simultaneous loading in uniaxial or biaxial extension and shear loadings. The results of numerical simulations, which show the deformed fabric unit cell under multi-load conditions, are coherent with experimental observations and encourage the authors to continue the present work with parametrical and inverse case studies.

Elastomer-based skins for morphing aircraft applications: Effect of biaxial strain rates and prestretch

There is an emerging trend in the morphing aircraft research where two or more morphing degrees of freedom are used on a wing which leads to the concept of polymorphing. The skin of the morphing wing must be flexible in the morphing direction but stiff in other directions to withstand the aerodynamic loads and maintain the airfoil shape. Polymorphing changes the loadings profile (from uniaxial to biaxial) and increases the complexity of designing suitable morphing skins. Furthermore, elastomeric materials used on morphing wings are usually prestretched to prevent wrinkling and to increase their out-of-plane stiffness. This paper focuses on elastomeric morphing skins and it studies the effect of biaxial strain rates and prestretch ratios on important mechanical properties such as stiffness, hysteresis losses (%), and stress relaxations (%) from an experimental perspective. Three polymeric materials are considered: Latex, Oppo, and Ecoflex. This study provides a mechanical comparative understanding of the three polymers used in the morphing wing under biaxial loading (two morphing degrees of freedom).

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

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

Investigation into multiaxial mechanical behaviors of Kelvin and Octet-B polymeric closed-cell foams

As a class of effective lightweight energy absorption materials, periodic closed-cell foams have been widely applied in engineering, in which the Kelvin and Octet-B foams have demonstrated great value in the research of multiaxial mechanical characteristics. For this reason, this study aims to develop a series of realistic finite element analysis (FEA) models for investigating their uniaxial, compression-shear, and arbitrary triaxial compression performance. Under uniaxial loading conditions, the mechanical responses and deformation modes of the two foams are compared and analyzed with different densities. The influence of different loading angles is also considered under compressive-shear loading. The deformation pattern of foams subject to equal biaxial and hydrostatic loading are compared with uniaxial compression. Based on sufficient simulation data, the initial yield surfaces of the two foams are plotted in the von Mises and mean stress plane, and fitted by three theoretical yield criteria characterized in terms of quadratic functions. It is found that the Miller criterion can better describe the initial yield surface shape of Kelvin foams than the yield models of Deshpande–Fleck and Zhang et al.; while the above yielding models are all of high fitting accuracy for the Octet-B foam. Further, the ability to resist initial yield of the Kelvin foam has proven superior to Octet-B foams by calculating the curve integration. The study is anticipated to provide new insights into novel design and extensive applications of periodic closed-cell foam materials in practice.

A new understanding of radiographic landmarks of the greater trochanter that indicate correct femoral rotation for measurement of femoral offset

ObjectivesTo establish and validate a novel method for aligning femoral rotation to accurately measure femoral offset for preoperative templating and component sizing, and to identify the physical location of two radiographic lines utilized in the described method.Materials and methodsCadaveric proximal femurs were skeletonized and mounted to a biaxial load frame. Two radiographic lines along the greater trochanter were identified fluoroscopically. The femurs were rotated, and images were taken when the lines appeared superimposed, then in 2-degree increments to 10° of internal and external rotation, and at 30°. Radiographic femoral offset was calculated at each angle, and the maximum and aligned offsets were compared. Bone was removed until the radiographic lines disappeared, then a metal wire was inserted in place of the bone to confirm that the lines reappeared.ResultsThe physical locations of the radiographic landmarks were on the anterior and posterior aspects of the greater trochanter. The mean true femoral offset was 38.2 mm (range, 30.5–46.3 mm). The mean aligned femoral offset was 37.3 mm (range, 29.3–46.3 mm), a 2.4% underestimation. The mean angle between aligned and true offset was 3.6° of external rotation (range, 10°ER-8°IR). Intra-rater intraclass correlation coefficient was 0.991.ConclusionAlignment of the radiographic lines created by the anterior and posterior aspects of the greater trochanter is a reliable and accurate rotational positioning method for measuring true femoral offset when using plain films or fluoroscopy, which can aid surgeons with preoperative templating and intraoperative component placement for total hip arthroplasty.

A novel technique to measure the biaxial properties of materials at high strain rates by electromagnetic Hopkinson bar system

A new electromagnetic biaxial Hopkinson bar system is developed for the first time to investigate the true dynamic mechanical properties of materials under biaxial loadings. Four stress waves along two axes are generated by two independent sets of electromagnetic loading system. Each set of the loading system consists of two loading devices which are drove by LC circuits. The variation of the loading ratios between two axes can be achieved by adjusting the charging voltages of capacitors corresponding to each axis. The synchronism of the incident stress waves is controlled by a digital delay generator. A cruciform specimen with slits and thickness reduction is proposed for biaxial Hopkinson bar test and the stress distribution in gauge area is optimized with finite element method. Data process methods using stress wave signals on the four elastic bars are studied. Results show that a reasonable estimation of stress is obtained based on the bar signals. The first true biaxial tensile experiment is performed on oxygen-free copper at high rates of strain. Real-time images captured by a high-speed camera reveal reasonably homogeneous strain distribution in the gauge area. The stress and strain curves of the test piece under equi-biaxial stress condition are obtained and compared with the results of uniaxial test.

Mechanical characterization and material modeling of ascending aortic aneurysm with different bicuspid aortic cusp fusion morphologies

Bicuspid aortic valve (BAV) is frequently associated with ascending thoracic aortic aneurysm (ATAA). Impact of cusp fusion patterns on biomechanics and microstructure of the ATAA remains unknown. This study aims to investigate biaxial mechanical properties of the ATAAs with right-left (RL) and right-noncoronary (RN) cusp fusion patterns. Fresh ATAA samples (n = 26) were obtained from patients who underwent surgical aneurysm repair. Biaxial extension tests were performed to characterize mechanical behaviors of the RL and RN BAV-ATAAs. A material model was fitted to biaxial experimental data to obtain model parameters. Histological and mass fraction analyses were employed to investigate the underlying microstructure and dry weight percentages of elastin and collagen in the ATAA tissue. The RL and RN BAV-ATAAs exhibited nonlinear and anisotropic mechanical responses to biaxial loading. Tissue stiffness of the RN BAV-ATAAs was significantly higher than that of the RL BAV-ATAAs in the circumferential (2679 ± 755 vs 1942 ± 578 kPa, mean ± SD, p = 0.04) and longitudinal (2535 ± 630 vs 1709 ± 512 kPa, mean ± SD, p = 0.02) directions under the equibiaxial stresses. Laminar structure of elastic fibers was disrupted in both RL and RN BAV-ATAAs. Notably, interstitial fibrosis and thinner elastic fibers were identified in the RN BAV-ATAAs. Mass fraction of collagen was significantly higher for the RN BAV-ATAAs than that of the RL BAV-ATAAs. The tissue stiffness in the circumferential direction was significantly increased and strongly correlated with the mass fractions of collagen for both RL and RN BAV-ATAAs. Our results suggest that elastic properties of the RN BAV-ATAAs are more deteriorated than those of the RL BAV-ATAAs. Changes in biomechanical properties may have great impact on ascending aortic dilation.

Improvement of scarf repair patch shape for composite aircraft structures

ABSTRACT Scarf patching is the preferred technique for repairing physically damaged aircraft composite structures. The commonly used patch design is a circular conical frustum with a 3°optimized scarf angle. However, for this repair patch design the volume of material removed from the repaired composite panel can be quite large. This paper examines the effects of the different parameters on the stress state in the patch adhesive, such as ply orientation, stacking sequence, ply thickness and scarf angle. It explores an adaptive design of the repair patch for orthotropic materials and composite laminates, using 3D Finite Element Analysis. The design optimization approach is based on the evaluation of a Failure Criterion Index (FCI) and the minimization of its Relative Standard Deviation (RSD). An optimized scarf shape, which takes into account biaxial tensile loading conditions, fiber direction, and stacking sequence, is an elliptical conical frustum that induces a variable scarf angle along the sweeping angle. This design showed that less parent material is removed than that of the circular shape (decreased by up to 41%) and that the shear stress dispersion in the adhesive was significantly reduced (RSD decreased by up to 18%). The optimization method and the improved shapes are presented and discussed with respect to the influencing parameters.

Analysis of biaxial fatigue limit models for cases with circular notches

This work shows an analysis of several models of multiaxial fatigue for notches: Navarro-Rios’ model, which analyses the interaction between the crack and its associated plastic zone with the material microstructural barriers, and three models that combine a critical volume method for notches with a critical plane model for multiaxial fatigue in unnotched solids. Specifically, the application of these models for the prediction of the fatigue limit for a plate with a circular hole subjected to axial, shear and in-phase biaxial cyclic loading is studied. The effects of two parameters are analysed: the radius of the hole and the relationship between the torsional and axial fatigue limits. For all the analysed models, cases are observed in which an increase in the hole radius produces an increase in the predicted fatigue limit, that is, the evolution of the fatigue limit with an increasing hole radius is not always monotonically decreasing, as would be expected. These effects, which we have called “humps” because of their appearance on the prediction graphs, mainly occur in shear loading. No humps were observed in the studied experimental results, but the number of available experimental results is too small to assure this tendency. The results shown in the work indicate that a greater knowledge of the physics of multiaxial fatigue in notches is necessary to achieve models that are capable of providing increasingly accurate predictions.

Analysis of Buckling Deformation for the Side Plate of Rectangular CSFT Column Based on Plate Theory with Bi-Axial Loads

In this paper, under the condition of bidirectional stress, the buckling deformation of the side plate in a rectangular concrete-filled steel tube (CFST) column has been studied in detail. We have conducted a theocratical analysis, an experimental validation and a finite element simulation to investigate the influences of the height-width ratios and Nominal Poisson’s ratios on the buckling form of the side plate, and we also try to explain the change of buckling form between unidirectional and bidirectional stress, both of them can provide a good reference and basis for design and application of the CFST column. The specific work can be summarized as follows: Firstly, a theoretical analysis has been conducted to study the buckling coefficient solution method of the thin plate under the conditions of axial compress and transverse tension. Then, under the conditions of the unidirectional and the bidirectional stress, a comparative study is carried out to investigate the changing relationship of the buckling coefficient (k) of the side plate; the results indicate that the buckling characteristic is changed due to the bidirectional stress, meanwhile, the buckling coefficient and the number of buckling half-wave will increase. Furthermore, the existing outcomes and the numerical simulations are adopted to study the relevance between the number of the elastic buckling half-wave in the side plate and the corresponding height-width ratio of the component; the results indicate that the former is larger than the latter. Finally, based on the obtained, the buckling relationship curve, the conclusion can be drawn as follows: when the bidirectional stress has been applied to the side plate, there is an equal interval between the different buckling half-waves; meanwhile, the interval shows a quadratic function reduce trend with the increase of nominal Poisson’s ratio.

Buckling of cylindrical concrete tanks and silos due to prestressing — nonlinear approach

The buckling of concrete cylindrical tanks and silos is associated mainly with prestressing, which influences the meridional critical force. Higher compression produces a lower meridional critical force. For three-dimensional structures subjected to biaxial or triaxial loading, the simplified methods of stability analysis proposed in Eurocode standard for concrete structures are inappropriate. The authors try to answer which procedure may be helpful for the assessment of buckling possibility, which results from prestressing of cylindrical structures. The numerical analyses are performed with the use of “Simulia Abaqus system”, which allows the nonlinear stability analysis adopting the plasticity-based damage material model and assuming initial imperfections. Two variants of load distribution (constant and hydrostatic external prestress) and two variants of wall-foundation connection are considered. The results are presented with the use of two parameters. The first one constitutes the ratio of lateral pressure to vertical pressure on the upper edge. The second one describes the ratio of current vertical force – corresponding to real load – to the initial test value of the iterative process. The obtained relationships of critical vertical force and the level of external prestress for cylindrical tank of height twice the diameter can be the guideline for engineers in designing process.

An assessment of anisotropic phase-field models of brittle fracture

In several classes of ductile and brittle materials consisting of different cleavage planes, an orientation dependency of the fracture process is observed. It leads for instance to complex failure behaviours and crack paths in polycrystalline or architected materials. This paper focuses on modelling anisotropy of brittle fracture by means of a variational phase-field approach. More precisely, we study different models including several phase (or damage) variables corresponding to different damage mechanisms. First, we recall a multi-mechanism gradient damage model based on an anisotropic non-local fracture energy. We then consider a model accounting for an anisotropic degradation of the elasticity stiffness tensor. Both types of anisotropies are compared in terms of their influence on analytical homogeneous solutions under uniaxial and biaxial tensile loadings. Weak and strong anisotropies are captured via the chosen multi-mechanism damage framework. The models are implemented numerically by using a finite element discretization. In order to improve numerical performance, we implement an algorithm based on a hybrid direct–iterative resolution of the displacement sub-problem. Accuracy of model prediction is assessed by comparing numerical results to theoretical solutions under uniaxial loading. Benchmark numerical tests on notched and perforated plates highlight the role of material parameters on the fracture anisotropy. Furthermore, both models are able to retrieve zig-zag crack patterns observed in prior numerical and experimental studies. Finally, we discuss the predictions of a model combining both types of anisotropies.

Strain-rate correlation of biaxial tension and compression mechanical properties of HTPB and NEPE propellants

An effective biaxial tension and compression test method is proposed based on the shortcomings of current research for the mechanical properties of solid propellants under complex stress states. The equal proportion biaxial tension and compression test of HTPB (Hydroxyl-terminated polybutadiene) and NEPE (NitrateEster Plasticized Polyether) solid propellants is performed at different rates while at room temperature, and the damage morphology of the tension–compression zone is analyzed using micro-CT. The results show that the failure mode of the solid propellant under biaxial tension and compression loading is similar to that under uniaxial tension. Meanwhile, the compressive strength is much greater than the tensile strength, which will eventually cause tensile failure. With an increased loading rate, the growth trend of the initial modulus, ultimate strength, and maximum elongation of the propellant is gradually flattened, and the damage degree is gradually reduced. Additionally, damage that forms in the HTPB propellant is from dewetting and particle fracture while that for the NEPE propellant is from matrix tearing. The porosity can be used as the meso-damage parameter of the propellant.

Damage ratio strength criterion for cast iron

According to the assumption of small strains and total strain decomposition of cast iron, the expressions of the inelastic principal strain rate and relative energy dissipation rate of cast iron are derived, and the parameters of the tensile and compressive damage ratio of cast iron in the inelastic phase are proposed. Based on the least energy consumption principle, combined with the above expressions and parameters, the damage ratio strength criterion for cast iron under biaxial stress states is established. The damage ratio variable expression of the cast iron damage ratio strength criterion considering the effect of Lode angle is proposed. Furthermore, the values of the damage ratio variable are verified by the experimental stress–strain curves of cast iron under uniaxial stress states. By comparing the predicted values to 104 experimental data points of cast iron specimens, the failure envelopes of the proposed criterion can represent the failure behavior of cast iron under biaxial loading conditions. To facilitate the practical application in engineering, the further simplified expression of the four-parameter damage ratio strength criterion for cast iron under biaxial stress states is suggested and those empirical parameters are recommended. Compared with some main strength criteria, the simplified criterion is in better agreement with the experimental results under every stress state.

Buckling analysis of FGM plate exposed to different loads conditions

In this work, the buckling analysis of functionally graded materials (FGMs) plates exposed to different load conditions is investigated using an efficient and simple refined plate theory. Uniaxial and biaxial compression loads along with simply supported boundary conditions on rectangular FGM plates are investigated. The effective material properties are computed using the simple power law equation of the volume fraction of the plate constituents. The governing equations are obtained from the principle of virtual works. The accuracy of present analytical solution is confirmed by comparing the present results with those available in existing literature. It can be concluded that the present theory is not only accurate, but also simply in predicting the buckling analysis of functionally graded plates.