A mixed lubrication model of the cold strip rolling process

Lubrication in cold rolling processes was first modelled using the plasto-hydrodynamic theory (3, 6). Short after came the first mixed lubrication models (11). Later on more sophisticated description of the physics of mixed lubrication was included : in (16 17) the so-called "high speed" mixed models, then in (19) the "low speed" models. The present paper describes an effort towards more general models covering the whole range of rolling speeds, see also (21, 24). Basically, the proposed model describes rolling under mixed lubrication with a set of ODE's representing : the elasto - plastic deformation of the rolled strip (slab method) [7]; the formation of a lubricant film and its evolution with Reynolds equation involving flow factors to include the coupling with (evolving) roughness [14a]; the deformation of roughness peaks through microplastic equations [11a] et [12]. The model in fact includes a transition between a "low speed' type model at entry and a "high speed' model starting at some place in the plastic deformation zone where lubricant pressure p(b) reaches the average pressure p. These equations are solved simultaneously by a 4-th order Runge - Kutta method; at each step in x, the real area of contact A(x) and the local average film thickness h(t)(x) are used to compute the local friction stress injected in [7] for the next step. Three embedded iterations loops are necessary (fig. 2) : the inmost one to determine the lubricant throuput Q corresponding to boundary conditions on Reynolds's equation, the middle one to find the entry velocity (or forward slip) in line with the applied strip tensions, and the outmost one to couple the roll elastic deformation (using the FEM). Examples of application are presented to show which kind of information on the process can come out of the model. One fitting parameter only remains, the (supposedly dry) friction coefficient on "plateaux", since lubricant viscosity and strip and roll roughness data (grouped in a composite roughness) are explicitly taken into account as part of the entry data. Application to an experimental campaign then shows that this local plateaux friction coefficient is not intrinsic but depends on rolling speeds, which points to some micro-HD phenomena taking place. This supports a discussion on possible further refinement of the model, first from the physical point of view (several kinds of plateaux, either dry, or under micro-mixed or micro-HD conditions; thermal coupling) then from the numerical point of view (moving to Finite Differences e.g.) as the model remains quite computationnally intensive.