TPT March 2009

the set of continuum mechanics equations, the rheological relations also need to be linearized. The accuracy of this operation, in turn, determines the correctness of the task solution. Currently linearization is usually performed by AA Iliushin's method of variable elasticity. This method takes into consideration the physical equations of the connection between stress and strain conditions that are formulated based on the hypothesis that deviator components of strain and deformation speed tensors are directly proportional. However, in this case the straight lines that linearize rheological relations at each step always go through a grade level which results in a sawtooth nature of linearized rheological relation in the deformation centre volume (figure 3a). For a more correct record of metal rheology, it is suggested that the formulation of the connection between stress and strain equations has to be based on the hypothesis about the linear dependence between the deviator components of strain and the deformation speed tensors. This situation is depicted in the following: s ij = s ij 0 + ke ij ' In this instance, s ij and e ij are deviator components of strain and deformation speed tensors; s ij is a free term. Application of this hypothesis allows the achievement of a considerably smoother rheological curve (figure 3b) and an increase in the accuracy of the boundary-value problem solution. At the same time, it is easy to notice that when setting s ij 0 = 0 we arrive at the existing hypothesis that formed the basis of the formulating equations that show the connection between stress and strain conditions.

and toughness of the non-standard design reducing mill stand containing two pairs of working rolls separated by a minimal distance (figure 2).

The design of theoretical aspects of mathematical simulation using numerical methods

OJSC RosNITI not only uses turnkey software tools but also designs its own software products for the analysis of deformation processes in tube and pipe manufacturing. The simulation process of the stress-strain state is based on a set of continuum mechanics equations. Despite the fact that the basic continuum mechanics equations have been developed more than a century ago, modern numerical methods require their corresponding adaptation at the stage of task description. Thus, to receive the approximate solution, the set of differential equations of continuum mechanics boundary-value problem has to be linearized. The analysis has shown that the problem of linearization can be solved when the approximate solution is made by the finite-element method. In this case the basic set of equations is as follows:

Where T is shear stress intensity, determined with the help of rheological relations; H is shear deformation speed intensity; v i is speed vector components ; s is average normal strain. As the final element ( e ) takes a small volume, the relation T / H within its limits can be considered constant regardless of the complexity of metal rheological characteristic and it ensures linearization of continuum mechanics boundary-value problem. It is possible to undertake theoretical analysis of the projection method algorithm of boundary-value problems. The approximate solution has shown that the generalised form of recording boundary conditions for the case under review is as follows: In this case, G(M) is the known function given at point at the boundary of deformation centre S . If the function G(M) is selected properly, the generalized boundary condition is reduced to the known kinds of boundary conditions of plastic metal working theory [2] . The calculation accuracy of the stress-strain state in many respects depends on the accuracy of rheological description of the continuum in the numerical solution of a boundary-value problem. To linearize

 Figure 3 :

The ways of rheological relations linearisation

109

www.read-tpt.com

M arch 2009

›