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109

M

arch

2009

www.read-tpt.com

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

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.

Figure 3

:

The ways of

rheological

relations

linearisation