Alternating tangent approach for the optimal vulcanization of 2D-3D EPM/EPDM thick elements

The main problem in industrial practice when dealing with the curing process of thick EPM/EPDM elements is constituted by the different temperatures, which undergo internal (cooler) and external regions. Indeed, while internal layers remain essentially under-vulcanized, external coating is always over-vulcanized, resulting in an overall average tensile strength insufficient to permit the utilization of the items in several applications where it is required a certain level of performance. A possibility to improve rubber output mechanical properties is the utilization of mixtures of at least two peroxides, the first highly active at high temperatures (i.e., for external layers), the second at low temperatures (internal regions). In this framework, in this article, a simple numerical procedure for the optimization of final mechanical properties of vulcanized 2D and 3D thick rubber items is presented. In particular, a so called alternating tangent approach (AT) for the determination of the optimal input parameters to use during the production of complex 2D/3D thick items is presented. Vulcanization external temperature Tc and rubber exposition time t are assumed as input production parameters, whereas output mechanical property to optimize is represented by the average tensile strength of the item. In the algorithm, a sufficiently large interval of exposition times at fixed Tc (or curing temperatures at fixed exposition times) is chosen at the initial iteration, namely evaluating rubber tensile strength at a very under-vulcanized and at a very over-vulcanized exposition time. For each extreme of the interval, first derivatives of final tensile strength with respect to exposition time (or curing temperature) are evaluated numerically. At the successive iteration, search exposition interval is reduced to one-half through bisection, selecting the right or left semi-interval basing the choice on first derivative sign evaluated at the middle point. The approach proposed converges very quickly to the optimal solution, competing favorably both with a very expensive method based on the subdivision of the domain in a refined grid of points and with recently presented GA approaches. Two meaningful examples of engineering interest, consisting of an high voltage electric cable and a 3D thick rubber docks bumper are illustrated. When dealing with the 3D item, due to its thickness, different mixtures of two peroxides (50%-50%, 25%-75% and 75%-25% molar ratios) are also used to improve drastically final mechanical properties. Optimal production Tc and t parameters are obtained for all the cases analyzed.