Equibiaxial Loading Research Articles

Investigation of isothermal and thermo-mechanical fatigue behavior of the nickel-base superalloy IN738LC using standardized and advanced test methods

The uniaxial and biaxial-planar isothermal fatigue behavior as well as the uniaxial thermo-mechanical fatigue behavior of the nickel-base superalloy IN738LC was investigated at high temperatures using standardized and advanced test technologies. Small scale specimens’ tests were carried out in order to study the specimen size effect and biaxial-planar tests under equibiaxial and shear loading to investigate the influence of the multiaxial stress state on the isothermal fatigue behavior both at 750°C. Standardized specimens were used to determine the uniaxial isothermal and thermo-mechanical fatigue behavior under in-phase and out-of-phase loading in the temperature range between 750°C and 950°C. The standard specimens’ results particularly served as reference data as well as for the verification of a recently developed lifetime prediction model for thermo-mechanical fatigue loading. The small scale specimens’ results were in good agreement with standardized specimens. The comparison of fatigue lifetimes of uniaxial and biaxial-planar tests showed that the equivalent strain hypothesis according to von Mises is applicable to correlate the fatigue life of the equibiaxial loading case within a scatter band of factor two, whereas the shear loading case was predicted conservatively. Furthermore, the recently developed lifetime prediction model correlated the fatigue lives of the isothermal and thermo-mechanical tests under uniaxial and biaxial-planar loading as well as literature data. Finally, the deformation and crack growth mechanism were studied.

Equi-biaxial fatigue testing of EPM utilising bubble inflation

This paper describes an equi-biaxial tension fatigue test system which utilises the bubble inflation method to subject elastomers to equi-biaxial fatigue loading between user-defined limits of pressure, volume, stretch ratio or stress. The test system integrates a hydraulic inflation system, a high speed vision system and a control system. The high-speed vision system allows the stretch ratio and stress acting on the test specimen to be evaluated in real-time during testing. This in turn allows either stretch ratio or stress to be used as a direct control limit. In this research, constant maximum engineering stress control tests have been carried out to evaluate the suitability of the developed dynamic bubble inflation system for equi-biaxial fatigue testing of elastomers. The resulting test data was used to produce Wöhler (S/N) curves and stress/stretch ratio plots for ethylene-propylene rubber (EPM).

Reprint of “Multiaxial fatigue property of Ti–6Al–4V using hollow cylinder specimen under push-pull and cyclic inner pressure loading”

This paper studies a multiaxial fatigue crack mode and a fatigue life of Ti–6Al–4V. Load controlled fatigue tests at room temperature were carried out using a hollow cylinder specimen under multiaxial loading with principal stress ratio λ equal to 0, 0.4, 0.5 and 1.0 and loading ratio R kept constant and equal to 0. λ is defined as λ=σ2/σ1, where σ1 and σ2 are maximum and intermediate/minimum principal stresses, respectively. Here, the test at λ=0 is a uniaxial loading test and that at λ=1.0 an equi-biaxial loading test. A testing machine employed was a newly developed multiaxial fatigue testing machine which can apply push-pull and reversed torsion loading with inner pressure into the hollow cylinder specimen. Based on the obtained results in this study, multiaxial fatigue properties are examined, where the fatigue life evaluation and the crack mode are discussed. The fatigue life is reduced with an increase of λ, due to cyclic ratcheting and crack mode in multiaxial loading. The crack mode is also affected by the surface condition resulting from cut-machining.

Lattice Approach in Continuum and Fracture Mechanics

This study reveals some aspects of lattice formulations to analyze the strain concentration of a famous classic problem in solid mechanics at two different mechanics perspectives: continuum and fracture. A 2D plane stress square panel with a single circular hole is discretized based on Voronoi tessellation. A superelliptic formulation is used to distribute lattice computational points over the panel. From the perspective of linear elasticity under uniaxial and biaxial loading, translational and rotational degrees-of-freedom are considered at each computational node of the lattice to obtain strain concentration factors (SCFs) around the circular perforation. It is observed that the lattice approach is able to approximate the elastic SCFs of three, four, and two in uniaxial, biaxial shear, and equibiaxial tension loadings, respectively. To study the linear elasticity and fracture mechanic (LEFM) and Griffith energy balance in uniaxial loading, a brittle lattice erosion technique is used to compute the energy release rate determined by change in the global stiffness matrix of the mesh with respect to crack extension. This fracture energy is then used to determine the mode I stress intensity factor of the crack emanating from the hole which is validated by the analytical formulation for the same problem. The comparison shows that both methods give very close results for the mode I stress intensity factor. Being simple in terms of constitutive formulation and failure criterion for erosion of brittle material, and also using a propagating crack extension approach, the lattice formulation is used to determine the fracture properties of cohesive/frictional material.

Characterization of the nonlinear behavior of nodular graphite cast iron via inverse identification: Analysis of biaxial tests

In this paper the nonlinear behavior of nodular graphite cast iron is investigated under in-plane biaxial loading regimes. Multiaxial experiments enable a wide variety of loading paths to be investigated. In the presented study two loading histories are proposed, namely, equibiaxial and “snail” loading paths corresponding to proportional and nonproportional regimes. The center gauge zone of the testing cruciform specimen is observed. Mean strain fluctuations are measured and presented in order to determine the material response for the prescribed biaxial loading paths. For the equibiaxial and snail loading histories finite element model updating will couple 2D-DIC measurements with 3D-FE analyses to calibrate material parameters describing elasticity, isotropic or kinematic hardening.

Post-bifurcation and stability of a finitely strained hexagonal honeycomb subjected to equi-biaxial in-plane loading

The buckling and crushing mechanics of cellular honeycomb materials is an important engineering problem. Motivated by the pioneering experimental and numerical studies of Papka and Kyriakides (1994, 1999a,b), we review the literature on finitely strained honeycombs subjected to in-plane loading and identify two open questions: (i) How does the mechanical response of the honeycomb depend on the applied loading device? and (ii) What can the Bloch wave representation of all bounded perturbations contribute to our understanding of the stability of post-bifurcated equilibrium configurations? To address these issues we model the honeycomb as a two-dimensional infinite perfect periodic medium. We use analytical group theory methods (as opposed to the more common, but less robust, imperfection method) to study the honeycomb’s bifurcation behavior under three different far-field loadings that produce (initially) the same equi-biaxial contractive dilatation. Using an FEM discretization of the honeycomb walls (struts), we solve the equilibrium equations to find the principal and bifurcated equilibrium paths for each of the three loading cases. We evaluate the structure’s stability using two criteria: rank-one convexity of the homogenized continuum (long wavelength perturbations) and Bloch wave stability (bounded perturbations of arbitrary wavelength). We find that the post-bifurcation behavior is extremely sensitive to the applied loading device, in spite of a common principal solution. We confirm that the flower mode is always unstable, as previously reported. However, our (first ever) Bloch wave stability analysis of the post-bifurcated equilibrium paths shows that the flower mode is stable for all sufficiently short wavelength perturbations. This new result provides a realistic explanation for why this mode has been observed in the finite size specimen experiments of Papka and Kyriakides (1999a).

Micromechanical Analyses of Debonding and Matrix Cracking in Dual-Phase Materials

Failure in elastic dual-phase materials under transverse tension is studied numerically. Cohesive zones represent failure along the interface and the augmented finite element method (A-FEM) is used for matrix cracking. Matrix cracks are formed at an angle of 55 deg−60 deg relative to the loading direction, which is in good agreement with experiments. Matrix cracks initiate at the tip of the debond, and for equi-biaxial loading cracks are formed at both tips. For elliptical reinforcement the matrix cracks initiate at the narrow end of the ellipse. The load carrying capacity is highest for ligaments in the loading direction greater than that of the transverse direction.

Method for determining the true stress of cross-shaped specimens subjected to biaxial tensile loads

For typical cross-shaped specimens subjected to simultaneous biaxial tensile loads, the specimens’ central section would elongate along different directions under plane stress condition. Consequently, the width of the specimens’ gauge length section would gradually decrease along the tensile direction, and true stress, which is based on the instantaneous width, is different from the values obtained from uniaxial tensile tests. To calculate the true stress of cross-shaped specimens, a horizontal biaxial tensile device was developed to apply the equi-biaxial loads. An optical observation method was adopted to measure the specimen’s true deformation. Then, an exponential fitting method was proposed on basis of the true deformation to describe the nonlinear deformation trajectory of the specimen’s gauge length section. Based on a theoretical model, the calculation method of the true stress was established, and the relationships between the true and engineering stress-strain curves were also discussed.

Investigation of hyperelastic models for nonlinear elastic behavior of demineralized and deproteinized bovine cortical femur bone

In this study, the hyperelastic models of demineralized and deproteinized bovine cortical femur bone were investigated and appropriate models were developed. Using uniaxial compression test data, the strain energy versus stretch was calculated and the appropriate hyperelastic strain energy functions were fitted on data in order to calculate the material parameters. To obtain the mechanical behavior in other loading conditions, the hyperelastic strain energy equations were investigated for pure shear and equi-biaxial tension loadings. The results showed the Mooney–Rivlin and Ogden models cannot predict the mechanical response of demineralized and deproteinized bovine cortical femur bone accurately, while the general exponential–exponential and general exponential-power law models have a good agreement with the experimental results. To investigate the sensitivity of the hyperelastic models, a variation of 10% in material parameters was performed and the results indicated an acceptable stability for the general exponential–exponential and general exponential–power law models. Finally, the uniaxial tension and compression of cortical femur bone were studied using the finite element method in VUMAT user subroutine of ABAQUS software and the computed stress–stretch curves were shown a good agreement with the experimental data.

Multiaxial fatigue property of Ti–6Al–4V using hollow cylinder specimen under push-pull and cyclic inner pressure loading

This paper studies a multiaxial fatigue crack mode and a fatigue life of Ti–6Al–4V. Load controlled fatigue tests at room temperature were carried out using a hollow cylinder specimen under multiaxial loading with principal stress ratio λ equal to 0, 0.4, 0.5 and 1.0 and loading ratio R kept constant and equal to 0. λ is defined as λ=σ2/σ1, where σ1 and σ2 are maximum and intermediate/minimum principal stresses, respectively. Here, the test at λ=0 is a uniaxial loading test and that at λ=1.0 an equi-biaxial loading test. A testing machine employed was a newly developed multiaxial fatigue testing machine which can apply push-pull and reversed torsion loading with inner pressure into the hollow cylinder specimen. Based on the obtained results in this study, multiaxial fatigue properties are examined, where the fatigue life evaluation and the crack mode are discussed. The fatigue life is reduced with an increase of λ, due to cyclic ratcheting and crack mode in multiaxial loading. The crack mode is also affected by the surface condition resulting from cut-machining.

Characterizing liver capsule microstructure via in situ bulge test coupled with multiphoton imaging

The characterization of biological tissue at the microscopic scale is the starting point of many applications in tissue engineering and especially in the development of structurally based constitutive models. In the present study, focus is made on the liver capsule, the membrane encompassing hepatic parenchyma, which takes a huge part in liver mechanical properties. An in situ bulge test experiment under a multiphoton microscope has been developed to assess the microstructure changes that arise with biaxial loading. Multiphoton microscopy allows to observe the elastin and collagen fiber networks simultaneously. Thus a description of the microstructure organization of the capsule is given, characterizing the shapes, geometry and arrangement of fibers. The orientation of fibers is calculated and orientation distribution evolution with loading is given, in the case of an equibiaxial and two non equibiaxial loadings, thanks to a circular and elliptic set up of the bulge test. The local strain fields have also been computed, by the mean of a photobleaching grid, to get an idea of what the liver capsule might experience when subjected to internal pressure. Results show that strain fields present some heterogeneity due to anisotropy. Reorientation occurs in non equibiaxial loadings and involves fibers layers from the inner to the outer surface as expected. Although there is a fiber network rearrangement to accommodate with loading in the case of equibiaxial loading, there is no significant reorientation of the main fibers direction of the different layers.

Biaxial fatigue behavior of a powder metallurgical TRIP steel

Multiaxial fatigue behavior is an important topic in critical structural components. In the present study the biaxial-planar fatigue behavior of a powder metallurgical TRIP steel (Transformation Induced Plasticity) was studied by taking into account martensitic phase transformation and crack growth behavior. Biaxial cyclic deformation tests were carried out on a servo hydraulic biaxial tension-compression test rig using cruciform specimens. Different states of strain were studied by varying the strain ratio between the axial strain amplitudes in the range of -1 (shear loading) to 1 (equibiaxial loading). The investigated loading conditions were proportional due to fixed directions of principal strains. The studied TRIP steel exhibits martensitic phase transformation from ?-austenite via ?-martensite into ?‘- martensite which causes pronounced cyclic hardening. The ?‘-martensite formation increased with increasing plastic strain amplitude. Shear loading promoted martensite formation and caused the highest ?‘-martensite volume fractions at fatigue failure in comparison to uniaxial and other biaxial states of strain. Moreover, the fatigue lives of shear tests were higher than those of uniaxial and other biaxial tests. The von Mises equivalent strain hypothesis was found to be appropriate for uniaxial and biaxial fatigue, but too conservative for shear fatigue, according to literature for torsional fatigue. The COD strain amplitude which is based on crack opening displacement gave a better correlation of the investigated fatigue lives, especially those for shear loading. Different types of major cracks were observed on the sample surfaces after biaxial cyclic deformation by using electron monitoring in an electron beam universal system and scanning electron microscopy (SEM). Specimens with strain ratios of 1, 0.5, -0.1 and -0.5 showed mode I major cracks (perpendicular to the axis of maximum principal strain). Major cracks after shear fatigue had partially mode II orientation (tilted 45° to the loading axes) and afterwards bifurcated into two pairs of mode I cracks. Another shear test revealed a major crack of mode I orientation (parallel to the loading axes). These results are in good agreement to the literature. Micro cracks after shear fatigue were longer than those after biaxial fatigue with strain ratios of 1 and 0.5. Major and minor cracks after equibiaxial and shear loading showed crack branching and crack coalescence. The results on fatigue crack behavior support the assumption that the period of stage I (mode II) crack propagation is much longer under shear loading than under other biaxial conditions due to absence of tensile stress normal to the planes of maximum shear strain under shear loading.

Crack mode and life of Ti-6Al-4V under multiaxial low cycle fatigue

This paper studies multiaxial low cycle fatigue crack mode and failure life of Ti-6Al-4V. Stress controlled fatigue tests were carried out using a hollow cylinder specimen under multiaxial loadings of ?=0, 0.4, 0.5 and 1 of which stress ratio R=0 at room temperature. ? is a principal stress ratio and is defined as ?=sigmaII/sigmaI, where sigmaI and sigmaII are principal stresses of which absolute values take the largest and middle ones, respectively. Here, the test at ?=0 is a uniaxial loading test and that at ?=1 an equi-biaxial loading test. A testing machine employed is a newly developed multiaxial fatigue testing machine which can apply push-pull and reversed torsion loadings with inner pressure onto the hollow cylinder specimen. Based on the obtained results, this study discusses evaluation of the biaxial low cycle fatigue life and crack mode. Failure life is reduced with increasing ? induced by cyclic ratcheting. The crack mode is affected by the surface condition of cut-machining and the failure life depends on the crack mode in the multiaxial loading largely.

Development of disc bending fatigue test technique for equi-biaxial loading

The disc bending fatigue test technique was developed to investigate the fatigue life under an equi-biaxial loading condition. In this test, a uniform thickness disc specimen was subjected to a bending load by applying air pressure on the specimen surface. Eleven specimens made of Type 316 stainless steel were tested in a room temperature ambient environment. The crack initiation and growth behaviors during the test were observed through a transparent window. The fatigue life was defined when the peak pressure measured near the specimen surface was reduced to 95% of the supplied air pressure. The fatigue life obtained by the disc bending fatigue test was shorter than that obtained by the uniaxial and plate bending fatigue tests for the same principal strain range. It was confirmed that the equi-biaxial loading condition reduced the fatigue life. The finite element analysis together with test results revealed that the crack was initiated at the edge of the specimen when the specimen thickness was less than 1.0mm. The specimen thickness should be 1.2mm in order to maximize the strain range at the specimen center. It was concluded that the disc bending fatigue test can derive the fatigue life under an equi-biaxial loading condition, for which strain range is measured at the specimen center.

Ductile fracture experiments with locally proportional loading histories

Basic ductile fracture experiments for sheet metal (or flat coupons extracted from bulk material) are presented to characterize the onset of fracture at different stress states. Special emphasis is placed on designing the experiments such that the stress triaxiality and the Lode angle parameter remain constant while the specimen is loaded all the way to fracture. A new in-plane specimen with two parallel gage sections is proposed to determine the strain to fracture for approximately zero stress triaxiality. A FEA based methodology is shown to identify the optimal specimen geometry as a function of the material's ductility and strain hardening. A tension specimen with a central hole is investigated in detail with regard to determining the strain to fracture for uniaxial tension. It is found that the required hole-to-ligament width ratio decreases as a function of the material ductility and increases as a function of the strain hardening exponent. The bending of a wide strip is pursued to prevent the necking prior to fracture under plane strain tension conditions, while an Erichsen-type of punch test is used to characterize the material response for equi-biaxial tension. It is worth noting that the strain to fracture can be directly determined from surface strain measurements in the cases of shear, plane strain tension and equi-biaxial tension loading, thereby removing the need to perform finite element simulations for extracting the loading path to fracture.

On the effect of the backup plate stiffness on the brittle failure of a ceramic armor

The present investigation enquires the role of the backup plate mechanical properties in the brittle failure of a ceramic tile. It provides a full-field solution for the elastostatic problem of an infinite Kirchhoff plate containing a semi-infinite rectilinear crack (the tile) resting on a two-parameter elastic foundation (the backup plate) and subjected to general transverse loading condition. The backup plate is modeled as a weakly non-local (Pasternak type) foundation, which reduces to the familiar local (Winkler) model once the Pasternak modulus is set to zero. The same governing equations are obtained for a curved plate (shell) subjected to in-plane equi-biaxial loading. Fourier transforms and the Wiener–Hopf technique are employed. The solution is obtained for the case when the Pasternak modulus is greater than the Winkler modulus. Superposition and a two-step procedure are employed: First, an infinite uncracked plate subjected to general loading is considered; then, the bending moment and shearing force distribution acting along the crack line are adopted as the (continuous) loading condition to be fed in the solution for the cracked plate. Results are obtained as a function of the ratio of the Pasternak over the Winkler foundation stiffness times the tile flexural rigidity. It is established that the elastic foundation significantly affects the mechanical behavior of the elastic plate. In particular, the Winkler model substantially underestimates the stress state near the crack tip. Stress-intensity factors are determined, and they are employed as a guideline for increasing the composite toughness. The analytical solution presented in this paper may serve as a benchmark for a more refined numerical analysis.

Optimal energy harvesting cycles for load-driven dielectric elastomer generators under equibiaxial deformation

Dielectric Elastomer Generators (DEGs) are promising devices able to convert mechanical to electrical energy by exploiting the variations in the capacitance of a deformable dielectric membrane. In this work the optimal performance of DEGs in-plane equibiaxial loading configuration is investigated. We analyze a four-stroke cycle which is chosen due to its simplicity and availability. Four failure modes of the DEG are accounted for: electric breakdown, electromechanical instability, ultimate stretch, and loss of the in-plane tensile state. Explicit expressions for the electromechanical fields in the membrane and the harvested energy are derived, and a constraint optimization procedure is employed to determine the optimal cycles. We find that if the electric breakdown strength is larger than a certain universal threshold value, the optimal cycle is independent of this parameter. This threshold value depends on the membrane ultimate stretch, and this dependence is different for low, moderate or high ultimate stretches. Analyses of DEGs with membranes belonging to the three regimes demonstrate that the DEG failure depends on the electric breakdown strength and the ultimate stretch. Accordingly, a diagram with six subregions corresponding to the type of failure is introduced as a design tool. DEGs with multi-layered membranes of two commercially available materials, an acrylic elastomer and a natural rubber, are considered. The number of layers is set according to a predefined load capacity parameter. We find that membranes with a moderate ultimate stretch ensure a cycle more efficient than those with high ultimate stretch, since a larger portion of the invested energy is converted.

다양한 하중경로에서의 DP980 강판의 파단변형률 예측에 관한 연구

Abstract The current study is concerned with the prediction of fracture strains for DP980 steel sheets over a wide range of loading paths. The use of DP980 steel is increasing significantly in automotive industries for enhanced safety and higher fuel efficiency. The material behavior of advanced high-strength steels (AHSSs) sheets sometimes show unpredictable and sudden fracture during sheet metal forming. A modified Lou–Huh ductile fracture criterion is utilized to predict the formability of AHSSs because the conventional forming limit diagram (FLD) constructed based on necking is unable to evaluate the formability of AHSSs sheets. Fracture loci were extracted from three dimensional fracture envelopes by assuming the plane-stress condition to evaluate equivalent plastic strains at the onset of fracture for a wide range of loading paths. Three different types of specimens -- pure shear, dog-bone and plane strain grooved -- were utilized for tensile testing to calibrate the fracture model of DP980 steel sheets. Fracture strains of each loading path were evaluated such that there shows little deviation between fracture strains predicted from the fracture model and the experimental measurements. From the comparison, it is clearly shown that the three dimensional fracture envelopes can accurately predict the onset of the fracture of DP980 steel sheets for complicated loading conditions from compressive loading to shear loading and to equi-biaxial tensile loading.

A unified mathematical modelling and simulation for cathodic blistering mechanism incorporating diffusion and fracture mechanics concepts

A novel mathematical model has been developed to understand the mechanism of blister initiation and propagation. The model employs a two-part theoretical approach encompassing the debondment of a coating film from the substrate, coupled with the design components incorporating diffusion and fracture mechanics, where the latter is derived from equi-biaxial tensile loading. Integrating the two components, a comprehensive mathematical design for the propagation of blister boundaries based on specific toughness functions and mode adjustment parameters has been developed. This approach provided a reliable and efficient prediction method for blister growth rate and mechanisms. The model provided a foundation for holistic design based on diffusion and mechanic components to enable better understanding of the debondment of thin elastic films bonded to a metallic substrate.

Deformation mode dependency of stable orientations: from plane strain compression to equibiaxial tension

Prediction of stable orientations is a key subject in the crystal plasticity literature. This work deals with the effects of deformation mode on the resulted stable orientations and texture evolution of FCC materials. The simulations are based on the numerical procedure developed by the authors in previous works in which, a rate-sensitive crystal plasticity model with Secant hardening law was employed. The resulting non-linear system of equations is solved by the modified Newton–Raphson method. In order to obtain the stable orientations for a deformation mode, initial orientations evenly spaced in the Euler space are selected and their evolution into the stable orientations is tracked. The deformation is applied incrementally until all of the lattice spin components become permanently zero. In this way, all preferred orientations for different ratios of strain-rate components = 0, 0.1, 0.25, 0.5 and 1 (from plane strain compression to equibiaxial tension) are calculated. It is observed that, by increasing this ratio, rolling texture components are gradually replaced by stable orientations of equibiaxial tension loading. In other words, the stable orientations of intermediate strain modes can be extracted from summation of stable orientations of plane strain compression and equibiaxial tension. This suggests that, it is not needed to search other intermediate strain modes for new stable orientations. The presented stable orientations can be evaluated by crystal plasticity-based simulation tools to determine which ones have superior characteristics to be used for texture improvement of material.