Equal Biaxial Tension Research Articles

Deformation theory and nonlinear dynamic behavior of PVC gel actuators

Polyvinyl chloride (PVC) gel generates complex nonlinear vibration behavior under an alternating current voltage excitation, which has potential application as a dynamic electromechanical actuator. However, there are few studies on the deformation theory of PVC gel actuators, especially the dynamic nonlinear response theory. In this paper, a complex dynamic model is established according to the electrodeformation mechanism of PVC film, and the nonlinear dynamic behavior of the actuator is numerically studied by a differential equation. The effects of applied voltage amplitude, voltage frequency, dibutyl adipate content, mechanical tension, and bias voltage on the dynamic properties of PVC film were analyzed under the condition of equal biaxial tension. The variation of amplitude and the generation and disappearance of the beat frequency during vibration are analyzed by using time-domain characteristics. The degree of PVC actuator nonlinearity as well as vibration stability and periodicity is also reflected based on the phase path and Poincare map. Finally, the law of influence of external condition parameters on the dynamic response of the PVC actuator is obtained.

Designing a Cruciform Specimen via Topology and Shape Optimisations under Equal Biaxial Tension Using Elastic Simulations.

Stress uniformity within the gauge zone of a cruciform specimen significantly affects materials’ in-plane biaxial mechanical properties in material testing. The stress uniformity depends on the load transmission of the cruciform specimen from the fixtures to the gauge zone. Previous studies failed to alter the nature of the load transmission of the geometric features using parametric optimisations. To improve stress uniformity in the gauge zone, we optimised the cross-arms to design a centre-reduced cruciform specimen with topology and shape optimisations. The simulations show that the optimised specimen obtains significantly less stress variation and range in the gauge zone than the optimised specimen under different observed areas, directions, and load ratios of von Mises, S11, S22, and S12. In the quantified gauge zone, a more uniform stress distribution could be generated by optimizing specimen geometry, whose value should be estimated indirectly each time through simulations. We found that topology and shape optimisations could markedly improve stress uniformity in the gauge zone, and stress concentration at the cross-arms intersection. We first optimised the cruciform specimen structure by combining topology and shape optimisations, which provided a cost-effective way to improve stress uniformity in the gauge zone and reduce stress concentration at the cross-arms intersection, helping obtain reliable data to perform large strains in the in-plane biaxial tensile test.

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.

New strain-ratio-independent material constant of free surface roughening for polycrystal sheets in metal forming

A series of conventional studies on free surface roughening have shown empirically that the surface roughness increases linearly with the equivalent strain. However, the effect of the deformation mode on the rate of surface roughening has not been clarified. We continuously observed the evolution of surface roughening under the deformation modes of uniaxial tension, plane strain tension, and equal biaxial tension in the metal forming. The deformation modes affect surface roughening, especially as the equivalent strain increases. It is found that this is caused by the change in surface area with plastic deformation. By removing the effect, a new deformation-mode-independent material constant of free surface roughening is derived.

Finite element method with account of singularity for mixed mode cracks

The paper deals with obtaining an analytical solution for stiffness matrix coefficients at a crack tip area for mixed mode cracks in plane strain conditions. The numerical study is focused on an infinite plate with a straight-through central crack under mixed loading. Analytical solutions are obtained as kinematic boundary conditions for plane strain. We analyzed distribution features of the stress-strain state fields and stress intensity coefficients at the top of the crack area, determined using the finite element method taking into account the singularity. The analytical formulas are obtained which set the kinematic conditions for a general and special case of loading a plate with a defect in the elastic setting for the case of plane deformation. The comparative analysis of the numerical results is presented for two cases of forming the design diagram of the top of the crack: the traditional method of creating a mathematical cut and the finite element method taking into account the singularity. The advantage of using the finite element method considering the singularity is found. We used an example of a plate with a through straight rectilinear central crack with the equal biaxial tension to show that setting the boundary conditions at the top of the crack taking into account the singularity allows one to significantly reduce dimensions of a calculation scheme of the finite element method and keep the calculation accuracy. It is concluded that such a formulation can be applied in an elastic-plastic formulation. The comparison between the classical finite element solution and finite element with singularity is presented. The convenience of the finite element method with singular boundary conditions is demonstrated.

Temperature effect on electromechanical properties of polyacrylic dielectric elastomer: an experimental study

The electromechanical properties of dielectric elastomers (DEs) are strongly affected by the environmental factors, such as ambient temperature and humidity. In this paper, by employing the VHB 4910 elastomer, we carry out an experimental investigation on the temperature effect on the electromechanical properties of DEs, including the electromechanical breakdown and voltage-induced deformation. Two kinds of tension modes, i.e. equal-biaxial tension and unidirectional fiber-constrained state, are adopted in the experimental measurements. It is found that both the breakdown voltage and the breakdown electric field of VHB 4910 under equal-biaxial tension reduce gradually with the increasing temperature. For the VHB elastomer under unidirectional fiber-constrained state, the temperature does not show a monotonous effect on the electromechanical properties. Finally, effect of temperature on voltage-induced deformation of VHB 4910 is tested, and the results demonstrate that the electromechanical deformation initially increases and subsequently decreases as the temperature rises.

The development of a novel forming limit diagram under nonlinear loading paths in tube hydroforming

The loading paths of tubular parts containing the characteristic of variable diameter during hydroforming are nonlinear and similar to the loading paths during tube hydro-bulging tests. Therefore, the forming limit determined by hydro-bulging tests is very suitable for guiding the forming of this kind of parts. However, traditional forming limit diagram (FLD) cannot present the nonlinear loading paths. In this paper, a theoretical analysis model for tube hydro-bulging tests is developed first. And then the stress paths and strain paths during bulging with fixed-ends and free-ends are analyzed through the theoretical model and finite element simulations. The results show that as the length-diameter ratio of the bulging zone ranges from 1.0 to 3.0, stress paths and strain paths are obviously nonlinear. For hydro-bulging with fixed-ends, the axial-hoop stress ratio changes gradually from 0.5 towards 1.0, and the strain state changes from plane strain state towards equal-biaxial tension state. For hydro-bulging with free-ends, the stress ratio changes gradually from 0 towards 0.5, and the strain state changes from simple tension state towards plane strain state. In order to give a better characterization of the forming limit of tubes under nonlinear loading paths, a novel FLD combining the geometric parameter of the tubular part is proposed. Two FLDs of an AA6061 aluminum alloy seamless tube under different end conditions are determined by experiments. The results show that both in hydro-bulging with fixed-ends and with free-ends, a higher forming limit is expected when using a smaller length-diameter ratio. As the new FLD can give direct presentation of limit strains together with the end condition and the length-diameter ratio in tube hydroforming, it can be used directly for the analysis and optimization of actual tube hydroforming processes, instead of the simple evaluation of the formability of raw materials.

Anisotropic plasticity characterization of 6000- and 7000-series aluminum sheet alloys at various strain rates

This study aims to characterize the high strain rate constitutive behavior and the anisotropic plasticity of three high strength aluminum sheet alloys, AA6013-T6, AA7075-T6 and a developmental AA7xxx-T76 alloy. A comprehensive series of mechanical characterization tests were performed under uniaxial tension, simple shear and through-thickness compression (representing equal-biaxial tension) at room temperature and various strain rates ranging from 10−3 to 103 s−1. It was observed that all three alloys exhibited appreciable plastic anisotropy in terms of the measured r-values along various sheet orientations, although they exhibited minimal anisotropy in the stress response. For all three alloys, the flow stress slightly increases with increasing strain rate, which indicates mild positive strain rate sensitivity. However, the flow stress ratios and r-values do not show any explicit relation with respect to strain rate suggesting a negligible effect of strain rate on in-plane plastic anisotropy. Measurement of the constitutive response to large strains (beyond onset of diffuse necking) was achieved using shear specimens that are not subject to onset of necking. To model the flow stress dependence on equivalent plastic strain and strain rate, an extended Hockett–Sherby constitutive model is proposed and shown to fit accurately the experimental data over the studied range of strain and strain rates. The yield behavior of the sheet alloys was predicted using the Barlat YLD2000-2D plane stress yield function (Barlat et al., 2003) assuming both associative and non-associative (NA) flow rules, as well as for general stress states employing two 3D anisotropic models, Barlat YLD2004 (Aretz and Barlat, 2004) and modified–Yoshida (Lou and Yoon, 2018). An associative flow rule was assumed for both of the higher order 3D yield functions considered herein. The measured flow stress ratios and r-values from the uniaxial tension and equal-biaxial tension experiments, along with the stress ratios from the shear tests, were used to determine the coefficients of the yield functions. The calibrated yield functions exhibited good agreement with the experimental data for most loading conditions considered. The higher order Barlat YLD2004 and modified–Yoshida yield functions provided the better predictive accuracy. The non-associative Barlat YLD2000 yield function also performed reasonably well, while the associative Barlat YLD2000 yield function had the lowest accuracy. The constitutive models were implemented into the FE simulation, in order to evaluate the accuracy of the model parameters calibrated in this work.

On the Certification of Positive and Convex Gotoh’s Fourth-Order Yield Function

Gotoh proposed in 1977 a fourth-order homogeneous polynomial of three plane stress components as a yield function to model anisotropic yielding and plastic flow of sheet metals. The yield function admits up to eight experimental inputs from uniaxial tension tests and one measurement from an equal biaxial tension test for calibrating its nine material constants and can model the formation of up to eight ears in deep cup drawing. However, the superior Gotoh’s yield function has not been widely adopted in sheet metal forming analyses in both academy and industry, especially outside Japan. One major concern is uncertain about the positivity and convexity of a calibrated Gotoh’s yield function. Here the problem of certifying a calibrated Gotoh’s yield function to be strictly positive and convex is first described and its resolutions are summarized based on some very recent research results. Algebraic necessary and sufficient conditions for a positive and convex Gotoh’s yield function with only two non-zero stress components are first presented. Sufficient conditions in terms of semi-algebraic and algebraic inequalities for establishing the positivity and convexity of tri-component plane stress Gotoh’s yield function are then summarized. Finally, the complete necessary and sufficient conditions for a positive and convex Gotoh’s yield function in plane stress are realized as a fully numerical minimization problem. Examples of successfully applying those conditions in certifying positive and convex Gotoh’s yield functions of various sheet metals are also given.

Prediction of Forming Limit of Dual-Phase 500 Steel Sheets Using the GTN Ductile Damage Model in an Innovative Hydraulic Bulging Test

Advanced high-strength steel has great potential for application in automotive and high-speed rail industries owing to its excellent mechanical properties. To predict the forming limit of dual-phase 500 steel sheets (DP500), Banabic’s innovative hydraulic bulging test was conducted on them, and the Gurson–Tvergaard–Needleman (GTN) ductile damage model was used to determine their failure under different strain paths. GTN damage parameters were identified using a hybrid experimental–numerical method based on the equal biaxial hydraulic bulging test. The results show that fracture of all of the DP500 specimens occurred in their polar region in experiments under different strain paths from uniaxial to equal biaxial tension. Moreover, the GTN ductile damage model could not only provide an accurate prediction of the fracture position, but also give an appropriate evaluation of the sheet forming limit under different loading paths.

Comparative evaluation of non-associated quadratic and associated quartic plasticity models for orthotropic sheet metals

A non-associated quadratic model and an associated quartic model are comparatively evaluated in this study for their description of anisotropic yielding and plastic flow of 21 representative sheet metals. The quartic yield stress function used in the associated model is a particularly reduced version of Gotoh’s fourth-order polynomial that has only seven independent material constants. The same experimental inputs of four yield stresses and three plastic strain ratios of a sheet metal are used via simple algebraic relations to calibrate the seven material constants in each model. The modeling results of these selected sheet metals show that both models give nearly identical performance for sheet metal plasticity under uniaxial and equal biaxial tension and similar predictions under other plane stress states. The quadratic yield and flow stress functions are shown however to have a larger domain of admissible yield stress and plastic strain ratios for the case of planar isotropy and it is much easier to verify their strict convexity conditions in general. Once it has been calibrated for each sheet metal considered in this study, Gotoh’s yield function is verified to be convex via a numerical minimization approach. A procedure is suggested to obtain an approximate but convex Gotoh’s yield function in case an as-calibrated Gotoh’s yield function is found to be non-convex. A reduced but convex 3D fourth-order polynomial yield function based on a convex Gotoh’s 2D yield function is also given that matches the total number of independent material constants of a 3D non-quadratic model as well. The 3D quartic rate-independent associated plasticity model may thus be readily implemented in a finite element analysis code for simulating 3D sheet metal forming processes.

Evaluation and calibration of anisotropic yield criteria in shear Loading: Constraints to eliminate numerical artefacts

Many anisotropic phenomenological yield functions have been proposed in the literature in which their predictive capabilities strongly depend on the experimental calibration data as well as the calibration procedure to identify the anisotropy parameters. In this paper, emphasis is placed upon the experimental and numerical calibration procedure of anisotropic yield functions in the region of shear loading (zero hydrostatic stress with equal and opposite in-plane principal strains and stresses). Conventional anisotropic calibration procedures are shown to introduce non-physical artefacts into constitutive models which manifest as a non-zero hydrostatic stress or through-thickness strains generated under in-plane shear stress that violate the definition of the shear loading condition. To overcome this issue, a new physically necessary constraint is applied on the plastic potential to enforce equal and opposite principal strains in the shear state and correct the shear region of anisotropic yield functions. Using this necessary constraint, the widely used Yld2000-2d anisotropic yield function was calibrated using an associated flow rule for aluminum alloy sheet using published data for AA2090-T3 to demonstrate how enforcing this constraint can be readily implemented to correct the shear region of the anisotropic yield surface. Furthermore, to investigate the influence of the shear constraint, an AA7075-T6 alloy was experimentally characterized in uniaxial tension, equal-biaxial tension and shear. It was revealed that with the additional shear constraints, non-physical artefacts of plane-stress anisotropic yield functions such as Yld2000-2d can be removed during the calibration procedure. However, due to the additional shear constraints, available anisotropic models may become over-constrained and alternate yield functions with more flexibility or non-associated flow rules may be required.

Failure Investigation for QP Steel Sheets under uniaxial and Equal-Biaxial Tension Conditions

The Quenching and Partitioning (QP) steel sheet is new generation material to induce phase transformation for plasticity in forming vehicle parts. The phase transformation is strongly stress state dependent behavior in experiments, which should affect the failure timing and limit strain in forming processes. In this paper, Nakajima test with QP980 and DP1000 steel sheets under equal-biaxial loading condition is performed for failure behavior. X-ray diffraction (XRD) is adopted to obtain the volume fraction of retained austenite (fA). Digital Image Correlation (DIC) is used to record the surface strain field and its evolution during equal-biaxial tension deformation. The same level Dual Phase (DP) steel is also employed for the purpose of comparison. The results show that phase transformation in QP steel gives small impact on failure strain under equal biaxial tension condition which is contradicted with our understanding. It suggests that failure behavior under uniaxial tension of QP980 is strongly phase transformation dependent. But it shows almost independent under equal biaxial tension condition.

Surface and through thickness crack growth in cruciform specimens subjected to biaxial loading

Fatigue surface crack growth are studied through experiments and computations for aluminum alloy D16T. Subjects for studies are cruciform specimens under different biaxial loading with external semi-elliptical surface crack. The variation of fatigue crack growth rate and surface crack paths behaviour was studied under cyclic uniaxial tension, equal biaxial tension and biaxial tension-compression loading. For the experimental surface crack paths in tested specimens the T-stress, the out-of-plane TZ factor, the local triaxiality parameter h and the governing parameter for the 3D-fields of the stresses and strains at the crack tip in the form of In-integral were calculated as a function of aspect ratio by finite element analysis to characterization of the constraint effects along semi-elliptical crack front. Both elastic and plastic stress intensity factor approach was applied to the fatigue crack growth on the free surface of the cruciform specimens as well as in the deepest point of the semi-elliptical surface crack front. As result principal particularities of the fatigue surface crack growth rate as a function of loading conditions were established. The experimental and numerical results of the present study provide an opportunity to explore the suggestion that crack growth rate may be represented by the plastic stress intensity factor, rather than the magnitude of the elastic SIF alone.

Numerical Modelling of Cylindrical Hole Under Equal Biaxial and Uniaxial Tension Loadings at the Infinity

The determination of stresses and displacements is the vital preoccupation in the world of computing structural elements subjected to any loading. In this context, this work focuses on dealing with classical problem of elasticity which has a vested interest. The stress and displacements of a cylindrical cavity in an infinite elastic medium will be analyzed first on a theoretical level by highlighting the existing analytical solution. And then through the enormous development of computers and programming languages, this problem will be solved numerically where the results will be compared with the analytical solutions. Using the software FLAC2D, this work has a numerical analysis of two problems: the first one is to calculate the stresses and displacements around a cylindrical cavity subjected to a biaxial compression loading at the limits, this case is a verification problem in FLAC and a tool for validating the procedure for numeric simulation. The second one corresponds to the numerical analysis of the stress distribution and displacements around the cylindrical hole under uniaxial tension stress at the infinity. The results obtained by this model are in very good agreement with analytical results.

Evolution of mechanical properties for a dual-phase steel subjected to different loading paths

The evolution of the mechanical properties of a dual-phase (DP590) steel sheet after being prestrained by uniaxial tension, plane strain and equal biaxial stretching was investigated. Specimens were first loaded using the three prestraining modes. Then, from the prestrained specimens, a few sub-sized samples were machined along the rolling direction and the transverse direction for further uniaxial tension testing. Six loading paths were provided. Equal biaxial stretching was performed using a cruciform specimen. The evolution of work hardening performance, elastic modulus, yield stress and tensile stress under the six loading paths were discussed in detail. The results indicate that loading paths can affect the latent work hardening performances, strain hardenability, yield stress and tensile stress evolution as well as the elastic modulus decrease during plastic deformation. The uniaxial tension–uniaxial tension path results in a cross-softening phenomenon, the largest yield stress enhancement and a mild maximum tensile stress increase. The equal biaxial stretching-uniaxial tension path leads to a cross-hardening phenomenon, the least yield stress enhancement and the largest tensile strength increase maximum tensile strength. The elastic modulus of DP590 steel not only changes with the accumulated plastic strain but also varies with the loading paths. The largest decrease of the elastic modulus equal biaxial stretching–uniaxial tension can reach 12.7% beyond 8% equivalent strain, which is 5.2% greater than that in the monotonic uniaxial tension path.

Investigation of the Interface Parameters of Fiber-Reinforced Composite Materials

Design of the interfacial properties is a significant fundamental issue in the field of composite materials. Much attention has been paid to the improvement of the interface strength, while there has been a little work concerning the mechanical design of the interface for fiber-reinforced polymers (FRPs) in which the fiber is transversely isotropic thanks to its orientation feature. Based on the conception of neutral inclusion and imperfect interface conditions, the interface parameters are obtained for the FRPs by using Kolosov constant. We demonstrate the effect of the thermal residual stresses on the neutrality for FRPs in this paper. If the interface has the properties of design parameters without consideration of thermal stress, the presence of thermal residual stress/strain would break the neutrality in the cases of equal-biaxial tension and pure shear. In particular, the neutral fiber does not exist unless the thermal stress is eliminated in the equal-biaxial tension. In the case of uniaxial tension, the neutral fiber does not exist in absence of thermal stress, while the neutrality would become possible by properly controlling the thermal residual stress. Because the large thermal residual stress will result in the formation of cracks in matrix, a trade-off design is necessary between the neutrality and the initiation of matrix cracking.

Molecular dynamics study on the equal biaxial tension of Cu/Ag bilayer films

The mechanical behavior, dislocation nucleation and development, and the evolution of interface in Cu/Ag bilayer films during an equal biaxial tension were studied using molecular dynamics simulations. The results show that dislocations are prone to nucleate at the interface of Cu/Ag bilayer film, and then propagate toward the free surface. The deformation defects consist of partial dislocations and intrinsic stacking faults, accompanying with some extrinsic stacking faults and stacking fault tetrahedra which just occur at the interface and propagate in the Cu layer. Compared with Cu and Ag films, Cu/Ag bilayer films have the maximal yield strain. The surface roughness reaches maximum at the yield strain, and then rapidly decreases with the increase of strain. Atoms in stacking faults increase with the strain at the high strain rate, while the rule is reversed at the low strain rate.

Study of Interfacial Stiffness by Using the Structural Interface Model Based on Elastic Mechanics

The recently proposed structural interface model is a realistic model of thick interfaces, but the bars of the structure are assumed iso-stiffness. In the present work, the structural interface consisting of pairs of bars with different stiffness is developed. Based on the neutral inclusion concept and elastic mechanics, the stiffness of bars of the structural interface is studied. It is found that all the bars have the same stiffness in a special loading condition of equal-biaxial tension. The finite element simulations for glass fiber-reinforced epoxy composite reveal that the fiber does not alter the prescribed stress due to the presence of the structural interface. Compared with the iso-stiffness-bar interface, the developed structure interface is more suitable for characterizing a multi-component interface and more flexible to be designed.

Interface Parameters of Composite Materials with an Elliptical Cross-Section Fiber Bundle

How to design the interfacial properties is a significant fundamental issue in the field of the composite materials, while little work was concerned with the mechanical design of the interface for the fiber reinforced polymer. In the present work, a fiber bundle embedded in the matrix was described as a transversely isotropic material. Based on the imperfect interface conditions, the interface parameters were derived to satisfy the neutral conditions for the composite materials reinforced by the elliptical cross-section fiber bundle. It is found that the interface parameter is not always associated with the applied loading in the case of the anti-plane shear. In the state of equal-biaxial tension, the normal interface parameter is merely related to the mechanical properties of components except for the shape of the fiber bundle, but independent of the loading magnitude. In the other cases of pure shear and uniaxial tension, the neutral interface does not exist except that the fiber bundle has a circular cross-section. It is also found that the interface parameters can be expressed in the forms similar to that for an isotropic inclusion by using Kolosov constant in the in-plane deformations.