Plastic deformation analysis of strain-rate sensitive materials under plane strain conditions

The two-dimensional plane strain equation of plastic flow in accordance with the Levy-Mises constitutive relation is expressed in terms of stream functions of complex variables. Expressions for the stress, strain-rate and velocity are derived, assuming the stream function in the forms of both the summation and product of conjugate flow functions, for plastic flow in a nonlinear viscous (strain-rate sensitive) medium. The plastic states are also derived using a mixed mode solution expressed in terms of non-separable, independent conjugate complex variables. Application of the summation form solution is illustrated through the block indentation problem. Calculations are made on the effect of variation of the strain-rate sensitivity exponent on the contact stress. The predicted behavior of the contact stress suggests the possibility of the development of a specially instrumented plane strain block indentation test for the rapid determination of the strain-rate sensitivity of real materials. By reducing the results of the indentation of a perfectly plastic material it is found that the contact stress is uniform and the external load is constant. The stress on the contact surface obtained using the present analysis is identical to that available from a slip line solution to the problem.     

Earing predictions for strongly textured aluminum sheets

Metallic alloy sheets develop crystallographic texture and plastic anisotropy during rolling. Deep drawing of a cylindrical cup from a rolled sheet is one of the typical forming operations where the effect of this anisotropy is most evident. Generally, in the finite element analyses of this process, the evolution of anisotropy during forming is neglected. In this paper, results of an experimental program carried out to quantify the anisotropy of aluminum alloy AA5042-H2 are reported. In addition to tensile tests along seven directions in the plane of the sheet, cup-drawing tests were conducted. It was observed that the material displays eight ears. The effects of the evolution in anisotropy and the directionality in hardening on the predictions of the earing profile for this material are investigated using a new methodology that incorporates multiple hardening curves corresponding to uniaxial tension along several orientations with respect to the rolling direction, and to biaxial tension. Yielding is described using the anisotropic yield function Yld2000-2D [1] and a form of CPB06ex2 yield function [2], which is tailored for metals with no tension–compression asymmetry. It is shown that even if distortional hardening is neglected, the latter yield function predicts a cup with eight ears as was observed experimentally. Consideration of distortional only leads to improved accuracy in prediction of the non-uniformity of the cup height profile.     

Effect of strain path change on limits to ductility of anisotropic metal sheets

Localized necking in thin metal sheets is analyzed by using the M–K-model approach, and the effect of a number of different non-proportional strain paths prior to the occurrence flow localization are considered. The analyses account for plastic anisotropy, using four different anisotropic plasticity models to fit a set of experimental data for cold-rolled steel sheet. The predicted forming limit diagrams show strong dependence on whether or not the load on the sheet is removed between two load steps on a non-proportional strain path. This dependence is investigated in detail for one of the anisotropic plasticity models, and it is shown that elastic straining plays a large role, as the stresses quickly move from one point of the yield surface to another. When the load is removed between steps, the stress point moves in a different manner, which results in quite different flow localization response.     

Yield loci of anisotropic aluminium sheets

AA8014 aluminium sheets were tested in uniaxial, equibiaxial (bulge test) and plane-strain tension at an almost constant strain rate of 2 × 10⁻³s⁻¹. The results were compared with predictions at different levels of strain, based on macroscopic and crystallographic theories of yielding. While most of the previous investigations compare the results at a strain level of 0.10, the present comparison was made in early stages of the deformation, which might be considered as the real yielding strain of the material. It was concluded that, at the high strain levels Hosford's method, which is a modification of Hill's “old” criterion, gives a proper fit to the experimental data. However, at the lower strain levels the experimental results were very close to the crystallographic loci.     

Analytical and numerical investigation of wrinkling for deep-drawn anisotropic metal sheets

An analysis of the onset of wrinkling is first developed for a doubly curved, elastic–plastic shell element submitted to a biaxial plane stress loading. Plastic yielding is described using a criterion recently proposed for anisotropic sheet metals. The wrinkling limit curves obtained with this analysis are compared with previous results based on different yield criteria. Finite element (FE) simulations of a deep-drawing experiment are also performed using the Abaqus/Explicit code with the aim of comparing the FE results relating to the initiation of wrinkling with the predictions of the analytical model and with experiments from the literature.     

Prediction of forming limits for anisotropic sheets containing prolate ellipsoidal voids

In sheet metal forming operations, the formability of sheet metals is limited by the occurrence of internal damage evolution that eventually yields a localized neck. Thus, designing and optimizing a sheet metal forming process, requires the precise prediction of the forming limits of the sheet materials. Accordingly, the current work attempts to theoretically predict the forming limit diagrams (FLDs) of voided anisotropic sheets using a new version of the Marciniak and Kuczynski (M–K) model. The analysis employs Gologanu–Leblond–Devaux's yield function for materials containing axisymmetric prolate ellipsoidal cavities with random orientations in conjunction with Barlat and Lian's 1989 anisotropic yield criterion. The effect of a void shape parameter on a ductile material under biaxial tensile loading is introduced and examined within the framework of the M–K model, along with the effect of including a first-order strain gradient term in the flow stress. To confirm the validity of the proposed M–K model, the predicted FLDs were compared with experimental results for steel sheets. The predicted forming limits for the voided sheets were found to agree well with the experimental data.     

Finite element method for sheet forming based on an anisotropic strain-rate potential and the convected coordinate system

A variational formulation and the associated finite element (FE) equations have been derived for general three-dimensional deformation of a planar anisotropic rigid-plastic sheet metal which obeys the strain-rate potential proposed by [11.]. By using the natural convected coordinate system, the effect of geometric change and the rotation of planar anisotropic axes were efficiently considered. In order to check the validity of the present formulation, a cylindrical cup deep drawing test was modeled for a 2008-T4 aluminum alloy sheet sample. Eating simulations were performed and planar anisotropic material properties were experimentally determined. Even though quantitative agreement was not fully achieved, reasonably good agreement was found between the FE simulation and the experiment in thickness strain distribution and caring. No numerical difficulty due to planar anisotropy was encountered, and the computational procedure was found to be very stable, requiring only moderate computational time. The results have shown that the present formulation for planar anisotropic deformation can provide a good basis for the analysis of sheet metal forming processes for planar anisotropic materials, especially for aluminum alloy sheets.     

Dynamic shear stress of tough-pitch copper at high strain and high strain-rate

Dynamic shear tests for the tough-pitch copper at high strain and high strain rate was performed. The Split Hopkinson Pressure Bar (SHPB) compression test system was modified to yield a shear deformation in the specimen. Hat-shaped specimens for the tough-pitch copper were adopted to generate high strain of γ=3–4 and high strain-rate of γ=10⁴/s. The dynamic analysis by ABAQUS 5.5/EXPLICIT code verified that shear zone can be localized in hat-shaped specimens. A proper impact velocity and the axial length of the shear localization region were determined through the elastic wave analysis. The displacement in a hat-shaped specimen is limited by a spacer ring which was installed between the specimen and the incident bar. The shear bands were obtained by measuring the direction of shear deformation and the width of deformed grain in the shear zone. The decrease of specimen length has been measured on the optical displacement transducer. Dynamic shear stress-strain relations in the tough-pitch copper were obtained at two strain-rates.     

Finite-element simulation of hole-flanging process of circular sheets of anisotropic materials

A hole-flanging operation on a flat circular sheet with a hole in the center is simulated by an incremental elasto-plastic finite-element method, which incorporates strain-hardening and anisotropy in the direction normal to the sheet, with care taken to describe the boundary conditions of penetration, separation and the alternation of the sliding—sticking state of friction. The simulation clearly demonstrates the processes of generation of deformation shape until unloading. The calculated sheet geometries and the relationship of punch load to punch stroke are in good agreement with the experimental data.The stress at the hole periphery in the flange is assumed to a state of circumferential uniaxial tension, in order to simplify the fracture mode as a simple tension test. By making use of the instability of uniaxial tension, an approximate relationship to determine the onset of necking of the hole periphery in the hole-flanging process is derived and it is found to be influenced by the process geometry and the plastic properties of the material, such as the stress-concentration factor K, strain-hardening n and normal anisotropy R, and the estimated value, being obtained by the derived equation, agrees well with the experimental data.It is noted that the derived relationship for estimating the instability of the hole-flanging process can be combined into the developed finite-element model to simulate the critical condition of the limiting deformation of the hole-flanging process. This combined method could possibly be applied towards improving both the manufacturing process and the design of tools for the hole-flanging operation.     

Two-dimensional strain distribution in elastically anisotropic heterostructures

We consider the two-dimensional distribution of elastic strain in semiconductor heterostructures — quantum wires characterized by anisotropy of the elastic properties. The deformation is caused by the mismatch in lattice parameters between the material of the quantum wire and its environment (matrix).Such deformations affect the position of the energy bands, so that they should be taken into account in the calculation of the electronic states. It is shown that the strain distribution in an anisotropic medium is a linear combination of two distributions relating to transversely stretched modifications of the original quantum wire.     

An approximate analysis for studying the plane strain deformation of strain rate sensitive materials

The paper presents a numerical method for analyzing the plane strain deformation of rate sensitive materials. A rate of energy functional is introduced which is thought to take adequate account of the strain rate sensitivity of the material. In the numerical technique the functional is minimized with respect to a kinematically admissible velocity field and used in a discretized form in a finite element analysis.To serve as an illustration the frictionless, plane-strain, side extrusion process was considered. To simulate actual side extrusion processes friction was incorporated into the analysis by assuming a constant fraction, α, of the current shear stress of the material.Data were available from some preliminary experiments on the side extrusion of a superplastic tin-lead alloy. The theoretically predicted forming pressure, taking α = 0·3, showed reasonably good agreement with the experimental values.     

A new cyclic anisotropic model for plane strain sheet metal forming

A finite difference model was developed for sheet metal subjected to plane strain cyclic bending under tension. The model was used to describe the effects on the mechanics of the deformation of the sheet due to mixed isotropic-kinematic cyclic hardening curves. The analytical results were compared with experimental results obtained under testing conditions closely representative of the analytical model. Two tests were used: a pure bending moment device and a bending/unbending under tension device consisting of three cylindrical pins. The model was used to determine a constitutive curve that best characterize the cyclic behavior of the material tested, as compared with the experimental results. The significance of these results were discussed in relation to the prediction of the restraining forces in the sheet as it is drawn through the blank holder drawbeads.     

Vibration analysis of nano-structure multilayered graphene sheets using modified strain gradient theory

In this paper, for the first time, the modified strain gradient theory is used as a new size-dependent Kirchhoff micro-plate model to study the effect of interlayer van der Waals (vdW) force for the vibration analysis of multilayered graphene sheets (MLGSs). The model contains three material length scale parameters, which may effectively capture the size effect. The model can also degenerate into the modified couple stress plate model or the classical plate model, if two or all of the material length scale parameters are taken to be zero. After obtaining the governing equations based on modified strain gradient theory via principle of minimum potential energy, as only infinitesimal vibration is considered, the net pressure due to the vdW interaction is assumed to be linearly proportional to the deflection between two layers. To solve the governing equation subjected to the boundary conditions, the Fourier series is assumed for w=w(x, y). To show the accuracy of the formulations, present results in specific cases are compared with available results in literature and a good agreement can be seen. The results indicate that the present model can predict prominent natural frequency with the reduction of structural size, especially when the plate thickness is on the same order of the material length scale parameter.     

In-plane plane strain formability of adhesive-bonded steel sheets: influence of adhesive properties

In the present work, the influence of adhesive properties on the formability of adhesive-bonded sheets (deep drawing quality (DDQ) steel and SS 316L) has been studied through in-plane plane strain (IPPS) formability tests. The adhesive properties were modified by different hardener to resin ratios and by having epoxy and acrylic adhesives. The formability is quantified by load–extension behaviour, limit strains and strain hardening exponent of adhesive-bonded sheets. It is observed from the work done that with increase in hardener to resin ratio, the elongation of adhesives is found to improve during tensile tests. This forms the basis for the actual formability change of adhesive-bonded sheets at different hardener/resin (H/R) ratios. During IPPS formability tests of adhesive-bonded steel sheets, the total elongation is found to improve with increase in hardener/resin ratio. Likewise, limit strains also improve with increase in hardener/resin ratios. The improvement in elongation and limit strains is due to the conversion of a resin-rich formulation to a hardener-rich formulation, making the sample more ductile. Both the acrylic and epoxy adhesive-bonded sheets show an equal amount of improvement in limit strain. The strain hardening exponent (n) of adhesive-bonded blanks has improved with increase in hardener/resin ratio in all the three regions of deformation. The adhesive-bonded blanks have larger strain hardening exponent values as compared to double sheet in the corresponding regions of deformation, indicating formability improvement as compared to double sheet.     

Some slip line field results for the plane strain extrusion of anisotropic materials through frictionless wedge-shaped dies

Slip line field results for the pressure to secure the plane strain extrusion of anisotropic material through frictionless wedge-shaped dies of low reduction are presented.     

Limit loads for the plastic bending in plane strain of cantilevers containing rectangular holes under end shear

Slip-line field solutions and upper and lower bounds are given for the plane strain plastic yielding of a rectangular section cantilever containing a rectangular hole subjected to bending with shear, the hole taking varying positions along the longitudinal axis of the cantilever. Using mild steel specimens and an etching technique, both the loads and the modes of deformation at plastic collapse are shown to well authenticate the theoretical results.     

An exact solution for the elastic/plastic bending of anisotropic sheet metal under conditions of plane strain

A theoretical analysis for the elastic/plastic bending of sheet metal exhibiting a state of normal anisotropy is considered in this paper, assuming a plane strain condition to exist in the deformation process. The material is supposed to yield according to Hill's quadratic yield criterion and its associated normality rule of plastic flow. The relationship between the bending couple and the curvature of the bent sheet is presented in a graphical form that reveals the influence of anisotropy and strain-hardening on the bending characteristic of the sheet metal. The results indicate that the elementary bending theory significantly overestimates the magnitude of the bending couple to produce a given elastic/plastic curvature of the bent sheet.     

Heterogeneous tensile test on elastoplastic metallic sheets: Comparison between FEM simulations and full-field strain measurements

The aim of this work is to propose a non-standard tensile test suitable for the identification of material parameters using full-field strain measurements and finite element analysis. The shape of the sample to be used in this new test must verify three criteria: (i) large heterogeneity of the strain in the gauge area, (ii) large strain-paths diversity and (iii) a good sensitivity of the strain field to the material parameters. After identifying the mechanical parameters of a dual-phase steel sheet using σ–ε and r–α curves, samples of different shapes were studied in order to choose the one that presents the best compromise between the three criteria. The comparison between simulated and measured fields shows a qualitative accordance. Taking into account the difference between these fields in the expression of the cost-function to minimize is expected to improve the quality of the identified material parameters.     

Analysis of crease-wrinkle interaction for thin sheets

In this paper, geometrically non-linear post-buckling analyses were performed to study the effect of sheet thickness, deployment angle, and load ratio on the crease-wrinkle interaction. A square sheet configuration with a single transverse crease was modeled using thin shell elements. The analysis proceeded by initially providing a realistic deployed state of a creased membrane sheet. Then an uneven corner loading was applied to introduce wrinkling. The effects of the induced anisotropy from the crease on the fine-scale detail of the wrinkle evolution, as a function of sheet thickness, loading, and crease deployment angle were systematically investigated. Significant differences were found in sheet compliance and crease-wrinkle interaction as these parameters were varied.