An Exact Solution for Instantaneous Axisymmetric Flow of Polymers

: The main objective of the present paper is to study, by means of a problem permitting a semi-analytic solution, the qualitative behaviour of solution for a model that is used for describing deformation of incompressible polymers near frictional interfaces. It is assumed that the regime of sticking occurs at the friction interfaces. The constitutive equations of the model are a pressure - dependent yield criterion and a flow rule. Such features of the solutions as non-existence and singularity are emphasized.


INTRODUCTION
Experimental studies demonstrate that skin layers are often generated in the vicinity of frictional interfaces in injecting molding of polymers. A review of such experimental results is provided in [1]. It is known from the theory of plasticity for metals that the equivalent strain rate approaches infinity near maximum friction surfaces [2][3][4]. This feature of solution behaviour is used to describe the generation of fine grain layers in metal forming processes [5]. It is of interest to extend this approach to processing of polymers. To this end, it is necessary to understand whether or not the behaviour of solutions for models that are used for polymers is similar to that found in [2][3][4]. In the present paper, the yield criterion for polyethylene proposed in [6] is adopted. This yield criterion is pressure-dependent. However, the volume change of polyethylene is similar to that observed in metals [6]. Therefore, it is reasonable to assume that the equation of incompressibility is satisfied.
In order to study the behaviour of solutions near frictional interfaces, it is not necessary to solve a boundary value problem of practical significance. It is more important to study an exact solution (i.e. the solution that exactly satisfies all the equations and boundary conditions). An ideal plane strain boundary value problem consists of a deformation comprising the simultaneous shearing and expansion (or contraction) of a hollow cylinder [7][8][9]. An axisymmetric analogue to this problem is a deformation comprising the simultaneous shearing and expansion (or contraction) of a material between two conical surfaces. This boundary value problem is solved in the present paper. The solution is for instantaneous flow. Most of the analytic and semi-analytic rigid plastic solutions other than solutions for stationary problems belong to this type of flow [10][11][12]. A rare case of the rigid plastic solutions for both instantaneous and continued flows is presented in [13].

CONSTITUTIVE EQUATIONS
The linear and quadratic stress invariants, 1 I and 2 I , may be defined as ( ) where c and c1 are material constants. This material is practically incompressible [6]. Therefore, the plastic potential function of Mises is adopted in the present paper.

STATEMENT OF THE PROBLEM
Consider a conical layer of plastic material between two conical surfaces of total angles 1 2θ and 2 2θ . It is convenient to introduce a spherical coordinate system ( ) , , r θ φ with the axis 0 θ = coinciding with the axis of symmetry of the conical surfaces (Fig. 1). The plastic material occupies the domain 0 r ≤ < ∞ , 1 2 θ θ θ ≤ ≤ and 0 2π φ ≤ ≤ . Let r u , u θ and u φ be the velocity components referred to this coordinate system. The rate of increasing of the angle 1 θ will be denoted by 1 ω . Then, The conical surfaces are friction surfaces. It is assumed that the regime of sticking occurs at these surfaces. The internal conical surface is fixed against rotation. Therefore, The external conical surface rotates with an angular velocity 2 ω .

Figure 1 Deformation of plastic material between conical surfaces
Therefore, The strain rate components are expressed in terms of the velocity components as

GENERAL SOLUTION
It is reasonable to seek the general velocity of the form Therefore, one of the principal strain rate (and stress) directions coincides with the r -direction. It is possible to assume with no loss of generality that It is seen from Eqs. (6) and (11)  It is understood here that ϕ is a function β found from Eq. (22). It follows from Eq. (7) and (11)

ANALYSIS OF THE SOLUTION
The denominator on the right hand side of Eq.