An arbitrary lagrangian eulerian formulation for reliable and stable 3D tire production simulation

Abstract

During the tire production process, the so-called green tire consisting of reinforcements and uncured rubber is inserted into a heating press. Under large pressure and temperature, the tire is brought in its final shape and the rubber changes its mechanical properties from soft visco-elasto-plastic to relatively stiff visco-elastic. Due to the large pressure, the soft uncured rubber deforms heavily, which leads to a distorted computational mesh in a pure LAGRANGian formulation. These large distortions in the mesh occur especially during the simulation of a 3D tire with a non-axisymmetric tread pattern and lead to convergence issues. Therefore, an arbitrary Lagrangian EULERian (ALE) formulation is presented with an improved projection algorithm to remedy these issues. In a pure LAGRANGian framework, the material points are connected directly to the computational mesh and a distortion of this mesh leads to convergence issues. Therefore, an Arbitrary LAGRANGian EULERian formulation is presented in which the material particles are partially decoupled from the computational mesh to overcome distortions. A staggered algorithm is presented in which three steps are carried out. A re-meshing step evaluates the distortions of the computational mesh and creates a smooth mesh. Subsequently, the history variables of the non-linear material model are projected from the old mesh to the new one. A special treatment is applied, where the values are directly interpolated from the integration points of the old mesh to the integration points of the new mesh. Lastly, a LAGRANGian simulation is performed, where an actual time step of the simulation is carried out. By a numerical example, the advantages of the ALE formulation in comparison to a pure LAGRANGian formulation is shown. A 3D tire sector model with a non-axisymmetric tread pattern is simulated, using the ALE formulation and a rigid surface formulation based on non-uniform rational B-spline (NURBS) functions. The approach is embedded into the production simulation of tires including curing of the rubber material.