A hybrid method for detecting the onset of local necking by monitoring the bulge forming load

Abstract

Strain-based Forming Limit Diagrams (FLD), which are typically obtained under linear or quasi-linear loading conditions, describe the limiting strains in terms of the major and minor in-plane strains before the onset of necking or the final failure (FFD). These strains can be detected by analysing the strain field in the vicinity of necking or cracking defects. It has generally been agreed that the loading versus time signal is not suitable for detecting necking processes. A novel hybrid method of detecting the onset of necking based on the experimental and simulated bulging load is presented in this paper. This method consists mainly in comparing the experimental forming load, i.e., a load showing plastic instability, with the numerical predictions obtained by performing finite element simulation. The simulation of the bulging process does not include any damage or failure criteria. A homogeneous forming load can therefore be simulated without requiring any information about the localization. This method was applied to detecting the onset of local necking in circular and elliptic quasistatic bulge tests on sheet material, with a diameter of 200 mm. Two materials were tested, a 0.8 mm thick DP450 Dual Phase steel sheet and a 1 mm thick AA6016-T4 aluminium sheet. The onset of necking observed with our method was compared with the results obtained by performing Hogström’s analysis based on the measured strain field over time and similar necking strains were obtained. Beside, the Bressian Williams Hill (BWH) shear criterion was identified for each test from experimental results. A slight scattering of the shear stress values was observed.

Appendices

Annex 1

• The swelling of the bulges was simulated numerically using the LS-DYNA software program [35] in implicit calculations. The die and barrel were modeled in the form of volumic elements with the MAT_RIGID behavioural law, and were therefore assumed to be non-deformable. Water was modeled in the form of volumic elements (Size: 2*2*2 mm) with the MAT_ELASTIC_FLUID behavioural law. The bulk modulus was set at 2.5GPa. The sheet material was modeled using shell elements of the Tsai-Belytschko type with 7 integration points through the thickness. Element size at bulge center was 1.5 × 1.5 mm² (Fig. 15a). The sample behavioural laws used were the elasto-plastic POWER_LAW_ISOTROPIC_PLASTICITY in the case of Hollomon’s hardening and the 3_PARAMETER_BARLAT in that of Voce’s hardening. Since all the tests were performed at a strain rate of about 10⁻³ s⁻¹, no strain rate sensitivity was taken into account. The boundary conditions round the edges of the sample were clamped conditions. An experimental piston velocity of 3 mm/min was imposed at the nodes of the water-piston interface using BOUNDARY_PRESCRIBED_MOTION_SET. An example of the mesh used for this purpose is shown in (Fig. 15a). The changes with time in the major and minor strains and the displacement of the apex of the bulge predicted in the simulations are compared in (Figs. 6 and 20) with the experimental data obtained on bulge N°2. Good agreement can be observed to have existed between the simulated and experimental values.

  Fig. 20

  

  Calculated and measured pressures versus the displacement of the apex of the dome in the case of the DP450 circular bulge and DP450 elliptic bulge N°2

Annex 2

• γ_F(h) was calculated in two stages. In the first stage, the ratio R(h) was calculated with various values of h. The values of F_expe and F_simu pertaining in the case of identical displacements were not available. Interpolations were therefore performed on the cloud of dots corresponding to the experimental loads, which were simulated using fourth-order polynomials in the windows (width L) centred on the value h* of the displacement at which it was intended to calculate R(h*). F_expe(h*) and F_simu(h*) could then be calculated. R(h* + ∆h) was determined by sliding the overlapping interpolation window. We took ∆h = 0.02 L. The value of the function R(h) was therefore known only at some values of h*. Secondly, we again interpolated the dots in R(h) into the window L* using an order 3 polynomial, where the coefficients a_i: R(h)=a₀+a₁h+a₂h²+a₃h³ were identified. It was then possible to explicitly determine the second derivative of R(h) : γ_F(h) = 2a₂ + 6a₃h. It sufficed to slide the overlapping interpolation window in order to calculate γ_F(h+∆h) in the same way, taking ∆h = 0.01 L*.

The sensitivity of the width of the window was then studied. Decreasing L* was found to result in changes in the value of R(h*), and hence, in that of γ_F(h*). These values stabilised in a range of L* values before becoming sensitive to changes in L*. The influence of L* observed here was probably attributable to the small number of dots available in the window for defining the polynomial satisfactorily. It was therefore decided to select the size of L* in the range in which R(h*) was stabilised.