In-plane Biaxial Tension Research Articles

Effect of nonlinear multi-axial elasticity and anisotropic plasticity on quasi-static dent properties of automotive steel sheets

This study investigates the influence of the elasto-plastic properties of automotive steel sheets on the denting behavior and suggests a constitutive modeling approach for reliable dent analysis. The stress–strain behaviors of three kinds of steel sheets were measured in uniaxial tension, in-plane biaxial tension and forward-reverse simple shear tests. Advanced constitutive models were employed to capture the plastic anisotropy, reverse loading characteristics such as the Bauschinger effect, and elastic modulus degradation. In particular, the biaxial elastic modulus and its degradation behavior were measured and implemented in the constitutive model. The suggested model significantly improved the prediction of dents compared to the conventional model in terms of the load-displacement curve. Sensitivity studies on the constitutive model demonstrated that mainly plastic anisotropy and elastic behavior of a material influence the panel stiffness, whereas the reverse loading behavior strongly affects the permanent dent depth.

In-plane biaxial compression and tension testing of thin sheet materials

Detailed experimental data on the behavior of textured sheet metals under compressive loading is important to describe their tension–compression asymmetry. This is particularly needed for materials that exhibit a strength-differential effect, or in cases where the Bauschinger effect occurs. So far, there is no systematic work describing the third quadrant in the 2D stress space under biaxial compressive loading. This paper presents a new device for biaxial, compressive in-plane testing of thin sheets. Biaxial and uniaxial compression experiments are carried out in the strain controlled device, analyzing the behavior of deep drawing steel sheets with and without skin-pass treatment. Moreover, in order to allow for the experimental description of the yield surfaces, biaxial tensile tests are performed. Detailed numerical validations and experimental strain analysis both for the new specimen for biaxial compressive testing and for the cruciform specimen for biaxial tensile testing show that reasonably homogeneous strain distributions can be achieved. The combined experimental and numerical method presented here allows to evaluate the tension–compression asymmetry of thin sheet materials. The results for the skin-passed condition clearly exhibit a tension–compression asymmetry, which highlights the necessity of biaxial compression tests already in the as-received material condition. The biaxial compression test opens a pathway to a more detailed analysis of the flow behavior of thin sheets under biaxial compression loading.

Crack growth behavior under biaxial fatigue with phase difference

Crack growth behavior of aluminum alloy 7075-T6 was characterized under in-plane biaxial tension–tension fatigue with phase differences of 90° or 180° between the two applied orthogonal cyclic loads. The initial single crack, created under the biaxial fatigue without any phase difference, splits into two symmetric cracks under the biaxial fatigue with the phase difference. The split cracks grow without any further branching. Directions of split cracks deviate sharply from the direction of the initial single crack. Under both phase differences of 90° and 180°, lengths of both split cracks are almost the same at a certain number of cycle. Strain energy release rate versus crack growth rate relationships of the split cracks are almost equal to each other. Further, sum of strain energy release rates at a given crack growth rate of both split cracks is equal to that of a single crack under the biaxial fatigue without phase difference. Analytical and finite element analyses are presented to explain the splitting of a crack due to the phase difference between the applied biaxial cyclic loads.

Effects of Anisotropic Yield Functions on Prediction of Forming Limit Diagram for AHS Steel

In this study, experimental and numerical analyses of Forming Limit Diagram (FLD) for Advanced High Strength (AHS) steel grade 980 were performed. Forming limit curve was first determined by means of the Nakazima stretch-forming test. Then, analytical calculations of the FLD based on the Marciniak-Kuczynski (M-K) model were carried out. Different yield criteria, namely, Hill’48 (r-value and stress-based), Yld89 (r-value and stress-based) and Barlat2000 (Yld2000-2d) were investigated. The strain hardening law according to Swift was applied. To identify parameters of each model, uniaxial tension, balanced bi-axial bulge test and in-plane biaxial tension test were performed. As a result, predicted plastic flow stresses and plastic anisotropies of the AHS steel by various directions were evaluated. In addition, effects of the anisotropic yield functions, strain rate sensitivities, imperfection values and work hardening coefficient on the predicted FLD were studied and discussed. It was found that the FLD based on the Yld2000-2d yield criterion was in better agreement with the experimental curve. Accuracy of the FLD predictions based on the M-K theory, especially in the biaxial state of stress, significantly depended on the applied yield criteria, for which yield stresses and r-values of different loading directions were required.

20am2-F5 ナノ～マイクロ構造体の機械物性を測る

In this presentation, specially developed mechanical testing techniques for micro-and nano-sized specimens are introduced. For thin films, I have designed and developed uniaxial tension test equipment, in-plane biaxial tension test equipment, and Poisson's ratio measurement equipment to directly measure the mechanical properties used for the design of micro devices. All types of thin films, such as metals, ceramics and polymers, have been successfully evaluated so far. For small structures, tension-torsion-bending-combined loading test equipment and three/four point bending test equipment have been developed. The influences of deformation-mode and specimen size on fracture strength have been evaluated. For nanowires, such as silicon nanowires and carbon nanotubes, on-chip tension test MEMS devices have been developed to investigate both size effect and process effect on the mechanical properties. Representative testing techniques and results will be presented.

Constitutive Modeling of Advanced High Strength Steels Characterized by Uniaxial and Biaxial Experiments

Anisotropic plastic behavior of advanced high strength steel sheets of grade DP780 and DP980 were investigated using three different yield functions, namely, the von Mises, Hills 48 and Barlat2000 (Yld2000-2d) criteria. Uniaxial tensile and balanced biaxial (hydraulic bulge) tests were conducted for the examined steels in order to characterize flow behavior and plastic anisotropy for different stress states. Additionally, disk compression and In-plane biaxial tension tests were performed for obtaining balanced r-value of DP780 and DP980, respectively. All these data were used to determine the anisotropic coefficients. According to the different yield criteria, yield stresses and r-values for different directions were calculated corresponding to these yield criteria. The results were compared with experimental data. It was found that the Yld2000-2d model precisely predict well with experimental data than the other models.

Crack growth behavior of 7075-T6 under biaxial tension–tension fatigue

Crack growth behavior of aluminum alloy 7075-T6 was investigated under in-plane biaxial tension–tension fatigue with stress ratio of 0.5. Two biaxiality ratios, λ (=1 and 1.5) were used. Cruciform specimens with a center hole, having a notch at 45° to the specimen’s arms, were tested in a biaxial fatigue test machine. Crack initiated and propagated coplanar with the notch for λ=1 in L–T orientation, while it was non-coplanar for λ=1.5 between L–T and T–L orientations. Uniaxial fatigue crack growth tests in L–T and T–L orientations were also conducted. Crack growth rate in region II was practically the same for biaxial fatigue with λ=1 in L–T orientation and for the uniaxial fatigue in L–T or T–L orientations, while it was faster for biaxial fatigue with λ=1.5 at a given crack driving force. However, fatigue damage mechanisms were quite different in each case. In region I, crack driving force at a given crack growth rate was smallest for biaxial fatigue with λ=1.5 and for uniaxial fatigue in T–L orientation, followed by biaxial fatigue with λ=1 and uniaxial fatigue in L–T orientation in ascending order at a given crack growth rate.

Fatigue behavior of laminated composites with a circular hole under in-plane multiaxial loading

A modified fiber failure fatigue model is presented for characterizing the behavior of laminated composites with a central circular hole under in-plane multiaxial fatigue loading. The analytical model presented is based on minimum strength model and fiber failure criterion under static loading available in the literature. The analysis starts with the determination of location of a characteristic curve around the hole and the stress state along the characteristic curve under in-plane multiaxial fatigue loading. Number of cycles to failure and location of failure are determined under given fatigue loading condition. Based on ply-by-ply analysis, ultimate fatigue failure and the corresponding number of cycles are determined. Analytical predictions are compared with the experimental results for uniaxial and multiaxial fatigue loading cases. A good match is observed. Further, studies are carried out for different in-plane biaxial tension–tension and biaxial compression–compression loading cases.

Modelling anisotropic hardening of an ultra low carbon high strength steel using crystal plasticity

While modelling single crystal deformation, often the hardening behaviour is approximated using a power law type hardening equation, the coefficients of which are determined by fitting predicted stress–strain curve to experimental stress–strain curve in the uniaxial tension. Also, in the literature, for BCC material either 24 or 48 slip systems have been considered with mixed conclusions. In this work, the suitability of such phenomenologically determined coefficients and the effect of using 24 and 48 slip systems on prediction capability for anisotropic hardening of an ultra low carbon interstitial free high strength steel observed during in-plane biaxial tension tests were assessed. It was found that, though there was some difference between the results for 24 and 48 slip systems, neither 24 nor 48 slip systems could predict the deformation behaviour accurately for all the cases considered.

A first-principles study on the elastic properties of single-walled carbon nanotubesg

A first-principles study is carried out to investigate the effects of structure and size on the elastic properties of single-walled carbon nanotubes (SWNTs). Ten zigzag-type and seven armchair-type SWNTs with diameters between 0.551 and 1.358nm are systematically studied. The linear elastic behaviour of the SWNTs when subjected to small deformations is studied by four virtual mechanical experiments: uniaxial strain, uniaxial stress, in-plane pure shear, and in-plane biaxial tension tests. Assuming that a SWNT should be transversely isotropic, a strain energy approach is used to calculate the Young's moduli in axial and transverse directions, major Posson's ratio, plain strain bulk, and in-plane shear moduli of the carbon nanotubes. It is found that the elastic constants are insensitive to the tube size, but show a slight dependence on the helicity. The differences in all the elastic moduli between zigzag and armchair nanotubes are within 10 per cent.

Textural Development of AA 5754 Sheet Deformed under In-Plane Biaxial Tension

Crystallographic texture evolution was quantified as a function of biaxial strain level and strain path for AA5754-O sheet metal in an effort to provide a physical description of grain rotation resulting from in-plane stretching. Samples were incrementally deformed to near-failure in three strain states (equibiaxial, plane strain, and uniaxial modes) with the rolling direction (RD) and transverse direction (TD) of the sheet parallel to the major stress axis for the latter two cases. The macrotextures were measured using X-ray diffraction techniques. Results showed that for a given strain path similar deformation textures developed in the two material sets (RD- and TD-oriented samples), though the evolution of the deformed texture was not homogeneous over the entire strain range. These variations in orientation intensities were related to the feeder components found in the initial texture and the availability of these components to rotate toward the more stable orientations under the particular mode of deformation. The nonuniform texture development of the sheet along different directions appears to contribute to the anisotropic mechanical response of the sheet during stretching as measured in the evolving multiaxial flow surfaces.

Transversely isotropic elastic properties of multiwalled carbon nanotubes

Five independent effective elastic moduli of a transversely isotropic multiwalled carbon nanotube (MWNT) are studied by analyzing its deformations under four loading conditions, i.e., axial tension, torsional moment, in-plane biaxial tension, and in-plane tension-compression stress. Two distributions of the tension loading on the outermost tube and on all tubes are considered, which correspond to the tensile and compressive Young's moduli. The general relations between the interwall stresses and strains are linearized due to the small strain condition, where the interwall stresses correspond to the variation of the interwall van der Waals forces. Three interwall elastic constants are used to characterize the linear relations associated with three basic interwall deformation modes, i.e., normal deformation in radial direction and two shear deformations in axial and circumferential directions. By taking each tube as a single-walled carbon nanotube, the analytical expressions for the interwall shear stress under the tensile loading on the outermost tube and five elastic moduli of a double-walled carbon nanotube are first obtained. Then, a replacement method is proposed to derive the corresponding expressions for the cases of more walls than two. These analytical expressions are plotted for the case of MWNT's composed of armchair tubes, where the interwall elastic constants are approximated as the corresponding ones of the graphite. The effect of the wall number, diameter, chirality, and length of the MWNT on the shear stress and five elastic moduli are displayed and discussed.

A 90-nm logic technology featuring strained-silicon

A leading-edge 90-nm technology with 1.2-nm physical gate oxide, 45-nm gate length, strained silicon, NiSi, seven layers of Cu interconnects, and low-/spl kappa/ CDO for high-performance dense logic is presented. Strained silicon is used to increase saturated n-type and p-type metal-oxide-semiconductor field-effect transistors (MOSFETs) drive currents by 10% and 25%, respectively. Using selective epitaxial Si/sub 1-x/Ge/sub x/ in the source and drain regions, longitudinal uniaxial compressive stress is introduced into the p-type MOSEFT to increase hole mobility by >50%. A tensile silicon nitride-capping layer is used to introduce tensile strain into the n-type MOSFET and enhance electron mobility by 20%. Unlike all past strained-Si work, the hole mobility enhancement in this paper is present at large vertical electric fields in nanoscale transistors making this strain technique useful for advanced logic technologies. Furthermore, using piezoresistance coefficients it is shown that significantly less strain (/spl sim/5 /spl times/) is needed for a given PMOS mobility enhancement when applied via longitudinal uniaxial compression versus in-plane biaxial tension using the conventional Si/sub 1-x/Ge/sub x/ substrate approach.

Transversely isotropic elastic properties of single-walled carbon nanotubes

While it is known that the elastic properties of a single-walled carbon nanotube (SWNT) are transversely isotropic, the closed-form solutions for all five independent elastic moduli have not been solved completely. In this paper, an energy approach in the framework of molecular mechanics is used to evaluate the local and global deformations of a SWNT in a unified manner. This is carried out under four loading conditions: axial tension, torsional moment, in-plane biaxial tension, and in-plane pure shear, respectively, from which the closed-form expressions for the longitudinal Young's modulus, major Poisson's ratio, longitudinal shear, plane strain bulk, and in-plane shear moduli are obtained. It is shown that as the tube diameter increases, the major Poisson's ratio approaches a constant, the longitudinal Young's and shear moduli and the plane strain bulk modulus are inversely proportional to the tube diameter, and the in-plane shear modulus is inversely proportional to the third power of the tube diameter. The dependence of the elastic moduli of a SWNT on the tube diameter and helicity is displayed and discussed.

Meso- and micro-stress analyses in an 8HS graphite/polyimide woven composite subjected to biaxial in-plane loads at room temperature

The stress distributions in an eight harness satin (8HS) graphite/PMR-15 (polyimide) composite subjected to a variety of in-plane simple and biaxial loading conditions at room temperature were determined in this study. The stresses were calculated numerically on the micro- and meso-scales using the finite element method. By considering visco-elastic properties of the polyimide resin the residual thermal stresses from manufacturing were also determined in the stress analysis. It has been shown that there is a critical external in-plane loading condition for which the micro- and meso-stresses in the composite are the highest. Under in-plane biaxial loads consisting of tension along the warp and fill tows in addition to in-plane shear there exists the highest combination of transverse and shear meso-stresses in the tows. The micro-stresses in the matrix between the fibers are also the highest for this biaxial combination of in-plane loads. Therefore, the combination of in-plane biaxial tension and shear is the critical loading condition for the initiation of transverse cracks in the tows. The numerical methodology developed in this study for the evaluation of visco-elastic meso- and micro-stresses in high temperature woven graphite/polymer composites should help aerospace designers optimize in-service loading conditions to prevent fracture in the tows and to increase the strength of aerospace structures utilizing such advanced composite systems.

Pole instability theory for ellipsoidal bulging of rolled sheet

Abstract The circular bulge test has long been used for assessing sheet metal formability under in-plane biaxial tension. The limiting formability is identified with pole failure under a pressure maximum. Theoretical solutions, which apply to bulging isotropic sheet, identify the sub-tangent to an equivalent stress-strain curve with an instability point corresponding to the maximum pressure. When this pressure is coupled with suitable description of the equivalent flow curve, it is possible to determine the in-plane limiting strains. A theory of instability is outlined for ellipsoidal bulging of rolled orthotropic sheet metals, e.g. the CR steels and 6000 series aluminium alloys used in the forming industry. The theory provides the limiting strains, height and peak pressure for a pole failure when a diffuse instability condition applies to these materials. Predictions to the pressure versus bulge height curves are compared with experimental results for an automotive, zinc clad, steel sheet. A plane strain rim failure, where it occurs, is also identified with a rising pressure along this curve. A general approach is adopted in terms of the die axis length ratio, the sheet's r -values, the Hollomon hardening constants an any inclination of the die's minor axis to the material's rolling direction. The theory of instability requires equivalence in the flow behaviour and this has been found to apply more consistently than was previously reported by employing larger dies within a modified experimental procedure.

Strain-free, ultra-high purity ZnSe layers grown by molecular beam epitaxy

ZnSe layers have been grown by molecular beam epitaxy on high-purity, high-quality ZnSe wafers [(100) oriented] cut from ingots grown by the iodine vapor transport method. Photoluminescence (PL) analysis indicates the homoepitaxial ZnSe material to be of ultra-high purity as well as being strain-free relative to ZnSe/(100) GaAs layers which exhibit in-plane biaxial tension. The 10 K PL spectra recorded from homoepitaxial layers exhibit unsplit free- and donor-bound exciton transitions of comparable intensity together with a strong peak at 2.7768 eV believed to be the so-called Iv transition in relaxed ZnSc. The ultra-high purity nature of the homoepitaxial layers is attributed to the high purity of the substrate material in addition to the use of high purity Zn and Se source materials in this work.

Shear Modulus of Precracked R/C Panels

Modeling of the interface shear transfer (aggregate interlock) and dowel action mechanisms as means of transferring shear stresses along preformed cracks is of great importance in determining the stiffness characteristics of reinforced concrete. An expression of an “effective” tangent shear modulus of precracked reinforced concrete panels subjected to combined in-plane shear and biaxial tension is presented based on equilibrium, compatibility, and constitutive relationships for the average stresses and strains in the concrete and steel. The proposed engineering model for shear transfer is compared with experimental results of concrete panels orthogonally reinforced with #6 bars.

Punching shear strength of concrete slabs subjected to in-plane biaxial tension

Synopsis It has been found in three tests that in-plane biaxial tensile strains up to 0·0017 cause no reduction in the strength in punching shear of a reinforced concrete slab 90 mm thick. The test results are compared with the predictions of the recent Bridge Code (BS 5400) and of plastic theory.