A mixed lubrication model of the cold strip rolling process

Lubrication in cold rolling processes was first modelled using the plasto-h ydrodynamic theory (3, 6). Short after came the first mixed lubrication mod els (11). Later on more sophisticated description of the physics of mixed l ubrication was included : in (16 17) the so-called "high speed" mixed model s, then in (19) the "low speed" models. The present paper describes an effo rt towards more general models covering the whole range of rolling speeds, see also (21, 24). Basically, the proposed model describes rolling under mi xed 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 [1 4a]; 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 defor mation zone where lubricant pressure p(b) reaches the average pressure p. These equations are solved simultaneously by a 4-th order Runge - Kutta met hod; 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 injec ted in [7] for the next step. Three embedded iterations loops are necessary (fig. 2) : the inmost one to determine the lubricant throuput Q correspond ing to boundary conditions on Reynolds's equation, the middle one to find t he 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 vi scosity 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 co efficient is not intrinsic but depends on rolling speeds, which points to s ome micro-HD phenomena taking place. This supports a discussion on possible further refinement of the model, first from the physical point of view (se veral kinds of plateaux, either dry, or under micro-mixed or micro-HD condi tions; thermal coupling) then from the numerical point of view (moving to F inite Differences e.g.) as the model remains quite computationnally intensi ve.