Response surface method for probabilistic assessment of metal forming failures

Methodology developed for reliability calculations of structures is applied to estimate reliability of sheet metal forming operations. Sheet forming operations are one of the most common technological processes but the tool and process design is still a difficult engineering problem. Product defects are encountered in industrial practice, with forming limit diagrams (FLD) typically used as a criterion of material breakage in the manufacturing process. An FLD is a graphic illustrating the limits of the principal strains which can occur without failure in a particular forming process and particular sheet material. Principal strains obtained for a particular process are compared with the forming limit curve (FLC). An FLC is a boundary between the strain combinations which produce localized necking and/or fracture (points above FLC) and those that are permissible for the forming operation (points below FLC) which limit the safe zone. Sheet forming operations are characterized by a significant scatter of the results. This can be caused by differences that can occur in the forming of each part. Because of uncertainties inherent in the forming process, a marginal zone is usually introduced in the FLD (5–10% below the FLC) where failure can occur with some probability. A zone of a FLD where good results are guaranteed with sufficient probability is considered as a safe zone. In this paper we adopt a more rational (quantitative) approach to assessment of sheet metal failure during the forming operation than the intuitive approach currently in use. The methodology of reliability theory proves useful for studying the influence of parameters characterizing a forming process like friction parameters, material properties, thickness and blankholding force. The effective response surface method (RSM), developed for reliability calculations of structures, is used to estimate the reliability of sheet forming operations. Results estimated by RSM are compared with the probability of failure assessment using the advance Monte Carlo (AMC) simulation technique. Although AMC usually provides good results, the method requires many simulations and can only be applied to relatively small examples. A gradient method combined with the response surface technique is adequate for bigger problems. An example of practical significance is presented as an illustration of the approach discussed