Numerical study of the square cup stamping process: a stochastic analysis

Abstract

The industrial demand for products with better quality and lower production costs have encouraged the widespread application of the finite element analysis (FEA) in the development and optimization of sheet metal forming processes. To ensure that the FEA solutions are reliable and robust it is important to take into account the uncertainties that inevitably arise in a real industrial environment. In this context, a numerical study on the influence of the material and process uncertainty in the stamping results of a square cup is presented. In this analysis, it is assumed uncertainty in the elasticity properties, hardening law parameters, anisotropy coefficients, blank thickness, friction coefficient and in the blank holder force. The effect of the uncertainty in these input parameters is evaluated in the punch force, equivalent plastic strain, thickness and cup geometry. Firstly, quasi-Monte Carlo method was used to evaluate the variability in the simulation outputs, considering the uncertainty of the input parameters. This analysis shows that the geometry is the output most sensitive to the uncertainty of the input parameters. Afterwards, a variance-based sensitivity analysis was carried out to identify the input parameters that most influence the output variability. It was concluded that the hardening law parameters and the anisotropy coefficients have the most influence in the stamping results variability of a square cup.

Variance-Based Sensitivity Analysis of the Biaxial Test on a Cruciform Specimen

“…In the sensitivity analysis, is evaluated the sensitivity of the biaxial test results to the following material parameters (inputs): 𝑛𝑛, 𝐶𝐶 and 𝑌𝑌 0 of the Swift hardening law; and 𝐹𝐹, 𝐺𝐺 and 𝑁𝑁 of the Hill'48 anisotropic yield criterion. These input parameters have an average value similar to a DC06 steel, whose values are indicated in Table 1 [12]. In the sensitivity analysis it is assumed that the input parameters vary between an upper and lower limit, equal to ±40% of the average value (see Table 1 The influence of the input parameters is evaluated on the results of the biaxial test (outputs), namely, the forces along the 0x (𝐹𝐹 𝑥𝑥 ) and 0y (𝐹𝐹 𝑦𝑦 ) arms, the principal strains (𝜀𝜀 1 and 𝜀𝜀 2 ), the strain path (𝛽𝛽 = 𝜀𝜀 2 /𝜀𝜀 1 ) and the thickness reduction (TR).…”

Sensitivity analysis of the of the square cup stamping process using a polynomial chaos expansion

“…These sources of uncertainty have a significant effect on the quality of the final product [4,5], leading to an inefficient production and, eventually, to the expensive redesign of the forming process. For all these reasons and due to the increasing availability of big data coupled with the growth of computer performance, the uncertainty analysis of these processes is a current scientific and industrial interest [1,[6][7][8].…”

“…Sensitivity analyses are used to quantify the influence of each uncertainty source in the variability of the forming results [3,[11][12][13]. Variance-based sensitivity analysis (Sobol's indices [14]) is one of the most common methods to quantify this influence [6]. However, this analysis usually requires a large computational cost in order to obtain accurate sensitivity results [6].…”

“…Variance-based sensitivity analysis (Sobol's indices [14]) is one of the most common methods to quantify this influence [6]. However, this analysis usually requires a large computational cost in order to obtain accurate sensitivity results [6]. This drawback contributes to the computational inefficiency of the uncertainty analysis, delaying its full and suitable employment in industry.…”

“…This work presents a numerical study on the influence of the material and process uncertainty in the results of a square cup forming process. The square cup test was chosen for two main reasons: (i) it is a commonly used benchmark test to represent sheet metal forming processes [1,6,[15][16][17][18]; (ii) it is a relatively fast process to simulate, which is suitable for performing a large number of numerical simulations. A polynomial chaos expansion (PCE) metamodel is used for reducing the computational cost of the Sobol's indices assessment [19].…”