Directions Of Plastic Strain Rates Research Articles

Parameter identification for the non-associated flow rules representing corner effects through the equivalent tangential shear modulus reduction after abrupt strain-path change

Some specific plastic flow rules have been proposed to describe the experimental observations of “corner effects”—the apparent violation of the normality of plastic flow after abrupt strain-path change, and rotation of the plastic strain rate direction toward the strain rate direction. In this paper, a simple parameter identification scheme for these non-associated flow rules representing corner effects is proposed from the perspective of the equivalent tangential shear modulus reduction. The nonlinear tension–torsion loading experiments with abrupt strain-path changes were performed on tube specimens made of the aluminum alloy A6063-T5, and working conditions of different prestrains and different loading path changes were included. The finite element analysis was used to verify the rationality of the determined parameters and compare the performances of different plastic flow rules in predicting the non-normality of the plastic flow. The experimental and simulation results suggest that the non-associated flow rules can well predict the plastic flow when the determined parameters are used, with the rules proposed by Yoshida and Tsuchimoto (2018) and Zhang et al. (2022) yielding the most accurate predictions. At the same time, the equivalent tangential shear modulus is apparently reduced during the transient process of strain-path change without elastic unloading; however, it does not seem to decrease to near zero as reported before.

Evolution of Strain Gradient Formulation in Capturing Localization Phenomena in Granular Materials

Abstract This paper highlights the use of incorporating strain gradient into flow stress to study localization behavior in materials. Pioneered by Zbib and Aifantis in the late 1980s, the formulation enabled incorporation of length scales into continuum formulations naturally. The formulation has also evolved into being able to study the effects of microstructure and heterogeneity on localization in granular materials. A multi-slip Mohr–Coulomb type plasticity model with the flow stress in the constitutive equation modified with a higher order gradient term of the effective plastic strain is used for this purpose. The possibility of abrupt changes of mobilized friction caused by intense shearing rate often leads to particle breakage. Its effects on localization are accounted for by modifying the material properties such as mobilized friction using a scaling parameter averaged over a representative elementary area. The change of shearing rate in the integration points was monitored through quasi-statistically measure parameter called inertia number. The inertia number was set to be less than l.0 × 10−3 all the time to consider quasi-static. The formulation was implemented into a finite element code and used to simulate plane strain compression tests on dry sand. The model highlights the effects of confining pressure, anisotropic microstructure, and the non-coaxial angle between the direction of principal stress and principal plastic strain rate directions on shear band characteristics.

On the implementation and validation of a three‐dimensional pressure‐dependent bounding surface plasticity model for soil nonlinear wave propagation and soil‐structure interaction analyses

AbstractNumerous experiments and prior analyses have confirmed that soil inelasticity, which is known to come into effect even at very low strain levels, can significantly affect site response and dynamic soil‐structure interaction (SSI) behavior. To date, only a few studies were able to consider multi‐axial wave propagation problems with appropriate models of soil nonlinearity. Most existing works are limited to either homogeneous soil configurations or equivalent linear soil models. The instances wherein soil nonlinearity is accurately considered have been confined to single element tests and one‐dimensional problems. In this study, an improved pressure‐dependent bounding surface plasticity soil model—with appropriate plastic strain rate direction definition and overshooting correction scheme—is implemented in Abaqus, and validated using recordings from both the Lotung borehole array and centrifuge test data on embedded flexible structures. The implemented model is capable of comprehensively reproducing complex soil behaviors, such as stiffness degradation, damping, dilatancy, and compaction while under a wide strain range, and under general loading conditions using only a few material parameters to be calibrated. Consequently, numerically predicted results are observed to be in better agreement with experimentally measured data, in comparison with linear and another plasticity model, which include accelerations, and bending and hoop strains along the walls of the specimen structures, for low‐ as well as high‐amplitude input motions.

Description of anisotropic properties of sheet metal based on spline curves and hole expansion test simulation of high-strength steel

A plane stress yield function with a yield surface described by Bézier curves comprising arbitrary numbers of segments is presented. This yield function can consider an arbitrary number of multiaxial stress states and their plastic strain rate directions as Bézier curve reference data. The planar anisotropy of the yield surface is described by the basis function of the cubic spline. By using the proposed model, the anisotropic properties of sheet metal are captured flexibly and precisely. To demonstrate the applicability of the proposed yield function, the accuracy of the captured anisotropic properties of high-strength steel was investigated. The proposed model was implemented in a commercial finite element analysis code, and hole expansion simulations of the high-strength steel were conducted to show the accuracy of the prediction.

A compact plane stress yield function formulation

The construction of anisotropic yield function by introducing linear transformations is a common practice with many phenomenological yield criteria. Based on the K1 − K2 representation method of principle stresses (represented by Barlat89 yield function), a compact 13-parameter plane stress function is proposed. The new yield function consists of five different linear transformations, and is designed to have a certain symmetry in both the overall structure and the distribution of linear transformation matrix components. The new 13-parameter yield function is applied to predict the yield surfaces (loci), plastic strain rate directions, flow stresses and r-values for both fcc and bcc materials. Compared with the yield functions of Yld2004-18p and CPB06ex2 (both reduce to 14 parameters in symmetric plane stress condition), the new yield function can achieve similar prediction accuracy with fewer parameters.

Material Modeling in High Strain Range and Forming Limit Analysis for 6000 Series Aluminum Alloy Sheet

This study investigates the effects of material modeling in high strain range on the forming limit analysis of a 6000-series alloy sheet 6016-T4. A servo-controlled tension-internal pressure testing machine was used to measure the multiaxial plastic deformation behavior in a strain range from initial yield to fracture. Tubular specimens were fabricated with a weld line thicker than the sheet thickness to prevent fracture from occurring in the weld zone. Proportional loading in the first quadrant of the principal stress space was applied to the specimens to measure the forming limit curve (FLC) and forming limit stress curve (FLSC), in addition to the contours of plastic work and the directions of plastic strain rates. Moreover, forming limit strains were measured using a flat-headed punch stretching test (PST). From the multiaxial tube expansion test (MTET) data, although a differential hardening effect was observed, appropriate coefficients including the exponent of the Yld2000-2d yield function (Barlat et al, 2003) were determined and employed in the Marcińiak-Kuczyski (1967) forming limit analysis (M-K analysis). The calculated forming limit strains in the vicinity of equibiaxial tension were smaller than the measured ones. The M-K analysis predicted that the forming limit strain in the vicinity of equibiaxial tension becomes minimum when the inclination angle of the localized neck from the rolling direction (RD) is 45° (diagonal direction; DD) while the experimental fracture direction was the RD. With the additional MTET, the plastic deformation behavior in the DD was measured and compared with that calculated from the material model. The reason of the discrepancy in the fracture direction between the experiment and the M-K approach is discussed.

Plastic flow behavior of fcc polycrystal subjected to nonlinear loadings over large strain range

Plastic flow rule is a fundamental equation that describes the plastic behavior of polycrystalline aggregate. To clarify plastic response at the point where the loading mode is suddenly changed, uniaxial tension followed by combined tension–torsion was applied to an A3003-O tube. The loading mode was altered at different plastic strains. It was observed that the direction of the plastic strain rate rotated immediately when the loading mode was changed. The greater the deviation in the subsequent loading mode from the uniaxial tension, the higher is the extent of rotation of the direction of plastic strain rate. The extent of instantaneous rotation of the plastic flow direction was the same regardless of the magnitude of the plastic strain at the alteration of the loading mode. Thus, the experimental results showed that the direction of the plastic flow depended on the direction of the stress/strain rate tensor; however, it was not affected by the magnitude of the plastic strain. Then, the effects of the hardening behavior of a slip system on the direction of the plastic flow were examined through a crystal plasticity simulation that considers the latent and kinematic hardenings. The crystal plasticity simulations reproduced the experimentally observed rotation of the plastic flow direction. Moreover, the simulations showed that the latent and kinematic hardenings did not affect the direction of the macroscopic plastic flow, even though they altered the activities of the slip systems. Therefore, it became clear that the plastic flow behavior of the polycrystal depends on the stress state and the direction of the stress/strain rate, and it is not affected by either the magnitude of the plastic strain or the hardening behavior of a slip system.

SANISAND-F: Sand constitutive model with evolving fabric anisotropy

In order to incorporate the very important role of evolving fabric anisotropy on the mechanical response of sand, a constitutive model is developed within the frameworks of Bounding Surface plasticity and Anisotropic Critical State Theory in multiaxial stress space. The main new constitutive ingredient is a fabric anisotropy variable A, a scalar measure of the relative orientation between an evolving fabric tensor F and the deviatoric plastic strain rate direction. The variable A affects scalar ingredients of the model quantifying the plastic strain rate, i.e. the plastic modulus and the dilatancy. A comprehensive calibration procedure is fully described and an extensive validation is performed against a very large dataset from 55 monotonic element tests on Toyoura sand provided by various laboratories, loaded under drained and undrained conditions. The introduction of A into the model is the main reason why successful simulation of data is achieved for loading at various orientations of the stress tensor under otherwise same initial conditions of void ratio and confining pressure, and this even if the data often exhibit huge difference of response because of the difference in loading orientation.

Necessary and sufficient conditions for reaching and maintaining critical state

SummaryAccording to classical critical state theory (CST) of granular mechanics, two analytical conditions on the ratio of stress invariants and the void ratio are postulated to be necessary and sufficient for reaching and maintaining critical state (CS). The present work investigates the sufficiency of these two conditions based on the results of a virtual three‐dimensional discrete element method experiment, which imposes continuous rotation of the principal axes of stress with fixed stress principal values at CS. Even though the fixity of the stress principal values satisfies the two analytical CST conditions at the initiation of rotation, contraction and abandonment of CS occur, which proves that these conditions may be necessary but are not sufficient to maintain CS. But if fixity of stress and strain rate directions in regard to the sample is considered at CS, the two analytical conditions of CST remain both necessary and sufficient. The recently proposed anisotropic critical state theory (ACST) turned this qualitative requirement of fixity into an analytical condition related to the CS value of a fabric anisotropy variable A defined in terms of an evolving fabric tensor and the plastic strain rate direction, thus, enhancing the two CST conditions by a third. In this way, the three analytical conditions of ACST become both necessary and sufficient for reaching and maintaining CS. In addition, the use of A explains the observed results by relating the stress‐strain response, in particular the dilatancy, to the evolution of fabric by means of the relevant equations of ACST.

A fabric-based sand plasticity model with reversal surfaces within anisotropic critical state theory

The paper describes the formulation and capabilities of a new constitutive model that accounts for the effects of fabric anisotropy on the response of granular materials under monotonic loading. It is developed within the framework of bounding surface plasticity in conjunction with the concept of (stress) reversal surfaces, i.e., the use of the last stress reversal point as projection center for defining the image stress on the bounding surface. A key constitutive ingredient is the fabric anisotropy variable A, relating the fabric tensor to the plastic strain rate direction, that acquires the value A = 1 as the third requirement for critical state according to the anisotropic critical state theory. This A is used in the definition of dilatancy, the plastic modulus and the evolution equation of the fabric tensor, thus simulating experimental results that show more dilative and stiff response when the loading is applied along the direction of the fabric. Model performance is verified against a large database of monotonic shearing tests on samples of Toyoura sand prepared with three different methods, as well as similar tests on samples whose (initially horizontal) deposition plane was rotated by up to 90 degrees. All simulations are performed with a single set of constants, thus validating the efficiency of the model to account for density, stress level and, most importantly, fabric anisotropy effects on the monotonic shearing response. The paper shows that considering dependence of strength and dilatancy on Lode angle θ and state parameter ψ does not suffice for simulating fabric anisotropy effects on sand response. It ends with a discussion of how fabric effects on sand response are more pronounced under undrained, versus drained conditions.

Measurement of Differential Hardening under Biaxial Stress of Pure Titanium Sheet

Biaxial stress tests of a commercially pure titanium sheet (JIS #1) using the cruciform specimen and the servo-controlled biaxial testing machine have been carried out in order to elucidate its anisotropic plastic deformation behavior. The geometry of the cruciform specimen is identical to that regulated by the ISO 16842. Nine linear stress paths, σx (rolling direction): σy (transverse direction) = 1:0, 4:1, 2:1, 4:3, 1:1, 3:4, 1:2, 1:4, and 0:1 in the first quadrant of the principal stress space are applied to the cruciform specimens. Contours of plastic work in the principal stress space and the directions of plastic strain rates at selected levels of plastic work have been precisely measured. The range of the equivalent plastic strain applied to the specimens is 0.002 ≤ ≤ 0.01. The shapes of the work contours significantly change with increasing the test material exhibits differential hardening (DH). Using the data of the work contours and the directions of plastic strain rates, the applicability of selected anisotropic yield functions to the accurate prediction of the plastic deformation behavior of the test material is examined.

Influence of hardening functions on earing prediction in cup drawing of AA3104 aluminum alloy sheet

The accuracy of cup drawing simulation of aluminum alloy sheets strongly depends on the hardening and yield functions used in the simulation. In this study, stress-strain curves for an AA3104 aluminum alloy sheet were measured by uniaxial tensile and rolling tensile tests to identify parameters for various hardening functions. In addition, in-plane uniaxial tensile, biaxial tensile, and combined tension-compression tests were carried out to measure the successive contours of plastic work in the stress space and the directions of plastic strain rates under linear stress paths. The anisotropic coefficients, α1-α8, and exponent M of the Yld2000-2d yield function were determined so that the mean square error of the analytical yield locus from the measured plastic work contour was minimized. Finite element simulations of cup drawing of the aluminum alloy sheet were carried out using the determined parameters for hardening functions and the Yld2000-2d yield function implemented in LS-DYNA. By comparing the simulation results with the results of cup drawing experiments, the effects of the hardening functions on the accuracy of earing profile prediction were investigated.

A plastic flow rule representing corner effects predicted by rate-independent crystal plasticity

Elastoplastic responses of rate-independent polycrystals are simulated under linear and various nonlinear strain paths, and the influence of the loading condition on the plastic flow behavior is investigated. It is observed that the shear components of elastoplastic tangent moduli start decreasing when the plastic deformation occurs, and the direction of the plastic flow depends on the direction of the stress/strain rates. A phenomenological plastic flow rule capable of reproducing these behaviors is developed. The flow rule relates the direction of plastic strain rate to the direction of strain rate, and yields the reduction of shear modulus after plastic yielding. The proposed constitutive model is then integrated into a finite element program, and the plane strain tension and pure bending are analyzed. The advantage of the proposed flow rule for simulating the occurrence of a shear band is demonstrated.

SANISAND-Z: zero elastic range sand plasticity model

The theory of zero purely elastic range in stress space within the framework of bounding surface plasticity is applied to sand constitutive modelling. The yield surface shrinks to zero and becomes identical to the stress point itself, and plastic loading occurs for any direction of the stress ratio rate on which the loading and plastic strain rate directions now depend, rendering the model incrementally non-linear. The simplicity of the conceptual structure of the model is particularly attractive as it consists of only one surface, the bounding/failure surface, and the stress point itself in the stress ratio π-plane. The image stress point on the bounding surface is defined analytically in terms of the direction of the rate of the stress ratio, with the latter being inside, on, or outside the surface, so that the model can address consistently hardening, softening and critical state response. An updating scheme of the initial value of stress ratio at unloading–reloading events is proposed in order to avoid the overshooting phenomenon. The model follows the basic premises of the SANISAND family of models that unify the description for any pressure and density within critical state theory. The simulating capabilities of the model are shown to be comparable with those of classical models with very small yield surfaces, and additional simulations of unorthodox loading paths such as rotational shear are successfully compared with experimental data.

Influence of material modeling on earing prediction in cup drawing of AA3104 aluminum alloy sheet

In-plane biaxial tension and combined tension-compression tests are carried out for AA3104 aluminum alloy sheets. Linear stress paths are applied to cruciform specimens to measure the contours of plastic work in the stress space and the directions of plastic strain rates at each stress path. Coefficients α1- α8 and exponent M of the Yld2000-2d yield function are determined to minimize the mean square error of the analytical yield locus from a measured work contour. The values of the weighting coefficients in the evaluation of the error are varied to check the effect of a specific stress state on the earing behavior. The effects of the combinations of the weighting coefficients on the accuracy of earing prediction in the cup drawing process are discussed.

Forming Limit Analyses of 590 MPa High Strength Steel Sheet Using Differential Work Hardening Model

A servo-controlled tension-internal pressure testing machine with an optical 3D deformation analysis system (ARAMIS®, GOM) was used to measure the multiaxial plastic deformation behavior of a 590MPa high strength steel sheet for a range of strain from initial yield to fracture. Tubular specimens were fabricated from the sheet sample by roller bending and laser welding. Many linear stress paths in the first quadrant of stress space were applied to the tubular specimens to measure the forming limit curve (FLC) and forming limit stress curve (FLSC), in addition to the contours of plastic work and the directions of plastic strain rates. It was found that the shapes of the measured work contours changed with the increase of work hardening (plastic work). The observed differential work hardening (DWH) behavior was approximated by changing the material parameters and the exponent of the Yld2000-2d yield function (Barlat et al, 2003) as a function of the equivalent plastic strain. The FLC and FLSC calculated using the Marciniak-Kuczyński-type (M-K) approach with the DWH model were in good agreement with the measurement.

Forming Simulation Considering the Differential Work Hardening Behavior of a Cold Rolled Interstitial-Free Steel Sheet

The multiaxial plastic deformation behavior of a cold rolled interstitial-free steel sheet with a thickness of 0.65 mm was measured using a servo-controlled multiaxial tube expansion testing machine for the range of strain from initial yield to fracture. Tubular specimens were fabricated from the sheet sample by roller bending and laser welding. Many linear stress paths in the first quadrant of stress space were applied to the tubular specimens to measure the contours of plastic work in stress space up to an equivalent plastic strain of 0.289 along with the directions of plastic strain rates. The test material exhibited differential hardening (DWH). A material modeling method for reproducing the DWH in a finite element simulation has been developed. Hydraulic bulge forming simulation results based on the DWH model had a closer agreement with the experimental results than those calculated using the isotropic hardening models with selected yield functions.

Effects of Anisotropic Yield Functions on the Accuracy of Forming Simulations of Hole Expansion

The deformation behavior of a high strength steel sheet in hole expansion tests using a flat-bottomed cylindrical punch has been investigated both experimentally and analytically to clarify the effects of anisotropic yield functions on the accuracy of finite element simulations of hole expansion. The material used in the hole expansion tests is a high strength steel with a tensile strength of 590MPa. The elastic-plastic deformation behavior of the test material has been precisely measured by biaxial tensile tests using cruciform specimens to determine the appropriate anisotropic yield function for the test material. The measured contours of plastic work and the directions of plastic strain rates are in good agreement with those predicted using the Yld2000-2d yield function with an exponent of 6. Moreover, forming simulations and experiments on the hole expansion of the test material have been carried out. The anisotropic yield functions used in the simulation are von Mises, Hill's quadratic and Yld2000-2d non-quadratic yield functions. The result of forming simulations by the Yld2000-2d yield function with an exponent of 6 has given the closest agreement with the experimental ones. It was found that the yield functions significantly affect the predictive accuracy of the deformation behavior of the steel sheet subjected to hole expansion.

二軸バルジ試験法による高強度鋼板の加工硬化特性の測定と材料モデリング

The deformation behavior of high strength steel sheets with a tensile strength of 590 MPa under biaxial tension was investigated for a strain range from initial yielding to fracture. The test material was bent and laser welded to fabricate a tubular specimen with an inner diameter of 44.6mm, a wall thickness of 1.2mm, and an axial length of 200mm. Multiaxial tube expansion tests were performed using a servo-controlled tension-internal pressure testing machine. Many linear stress paths in the first quadrant of stress space were applied to the tubular specimens to measure the forming limit curve (FLC) and forming limit stress curve (FLSC) of the as-received sheet material, in addition to the contours of plastic work and the directions of plastic strain rates from initial yielding to fracture. Results calculated using the Yld2000-2d yield function with exponents of 6 to 8 provided the closest agreement with the measured work contours and directions of plastic strain rates for a reference plastic strain range of 0.002 ＜ε0p＜ 0.20. It was concluded that the multiaxial tube expansion test is effective for measuring the multiaxial deformation behavior of sheet metals for a wide range of plastic strains.

Multiaxial tube expansion test method for measurement of sheet metal deformation behavior under biaxial tension for a large strain range

A servo-controlled multiaxial tube expansion testing machine was developed to measure the multiaxial plastic deformation behavior of sheet metals for the range of strain from initial yield to fracture. The testing machine is capable of applying arbitrary principal stress or strain paths to a tubular specimen using an electrical, closed-loop servo-control system for axial force and internal pressure, in addition to having a novel strain measurement apparatus for tubular specimens. Tubular specimens with an inner diameter of 44.6mm were fabricated from cold rolled interstitial-free steel sheet with a thickness of 0.7mm by roller bending and laser welding. Many linear stress paths in the first quadrant of stress space were applied to the tubular specimens to measure the forming limit curve (FLC) and forming limit stress curve (FLSC) of the as-received sheet material, in addition to contours of plastic work and directions of plastic strain rates. Results calculated using the Yld2000-2d yield function with an exponent of 6 provided the closest agreement with the measured work contours and directions of plastic strain rates for an equivalent plastic strain range of 0.005–0.36. Moreover, a Marciniak–Kuczyński-type forming limit analysis was performed using the most appropriate yield function with the assumption of isotropic hardening; the calculated and measured FLC and FLSC were in fair agreement. Therefore, the multiaxial tube expansion test is effective to measure the multiaxial deformation behavior of sheet metals for a large range of plastic strain.