Strain-rate intensity factor in compression of a layer of a plastic material between cylindrical surfaces

Abstract

A model problem for a rigid perfectly/plastic material is obtained. Based on this solution, it is possible to estimate the influence of the friction surface curvature and one of the types of additional rotational motion of the friction surface on the strain-rate intensity factor.The strain-rate intensity factor introduced in [1] as a coefficient at the principal singular term in the expansion of the equivalent strain rate into a series in the vicinity of the maximum friction surface is used to predict the evolution of material properties in a thin layer near surfaces with large friction stresses in metal-forming processes [2,3]. Quantitative dependences between the strain-rate intensity factor and parameters characterizing material properties, however, have not been established. It seems of interest, therefore, to find the dependence of the strainrate intensity factor on parameters of metal-forming processes, which can be used, in particular, to determine the above-mentioned quantitative dependences. One parameter that can be readily changed in an experiment is the curvature of the friction surface. In the present paper, we study the compression of a layer of a plastic material between two concentric cylinders with the maximum friction law being satisfied on the cylinder surfaces. This simple model problem allows the effect of the friction surface curvature on the strain-rate intensity factor to be estimated.It was found that the strain-rate intensity factor is affected by additional rotation of the instrument, which is used in industrial processes for changing the energy and force parameters of the process [4-6]. The effect of one type of additional rotation of the friction surface on the strain-rate intensity factor is considered in the present paper within the framework of the model problem being solved.For some processes, the strain-rate intensity factor was determined in [7][8][9][10], where the solutions were obtained on the basis of the double shear model [11], which is the generalization of the model of a rigid perfectly/plastic material. Note that the qualitative behavior of the solutions obtained on the basis of the double shear model coincides with the behavior of the solutions obtained on the basis of the model of a rigid perfectly/plastic material [12,13].Let us consider a plane flow of a layer of a plastic material compressed between two surfaces, which have the form of concentric circular cylinders and which obey the maximum friction law. We introduce a polar coordinate system (r, θ) whose origin coincides with the cylinder centers. The radius of the outer cylinder determined by the equation r = R 2 is assumed to be constant. The radius of the inner cylinder increases with a velocity U 0 , and its current radius is described by the equation r = R 1 (Fig. 1). As the flow is symmetric about the axis θ = 0, it is sufficient to construct the solution for θ 0. Let σ rr , σ θθ , and σ rθ be the components of the stress tensor in the polar coordinate system, and let u r and u θ be the components of the ve...

Asymptotic Analysis of Plastic Flow along the Generatrix in a Thin Cylindrical Layer

“…These formulas and the results obtained in [2] mark the beginning of the analytical history of the Prandtl problem. This problem and its various generalizations (see, e.g., [3][4][5][6][7][8][9][10][11][12]) are based on natural, experimentally supported, dynamic and kinematic hypotheses that the shear stresses and velocities perpendicular to the layer are linear along the thickness. Does this problem have other asymptotic solutions different from (1) in which the formulated hypothesis (in particular, the Prandtl hypothesis) are not satisfied?…”