A N umerical Solution to the basic Von Kantian Equation 1 Introduction.

A mathematical model has been produced by which predictions of both roll force and torque,developed over a wide range of flat hot rolling conditions, can be made. The basic Von Karman equation,descri­ bing the distribution of roll pressure.along the arc of contact,takes into account the effect of a variable flow stress of the material under­ going deformation and of both homogeneous and inhomogeneous deformation, together with a mixed frictional condition 'Tik and a deformed roll radius. Since the resulting differential equation possesses no analytical solution, the numerical technique involving the fourth-

( 28^ order Runge Kutta method and originally developed by Alexander' 'for the cold rolling process,was extended to the hot rolling case. The following sections will therefore describe the mathematical basis of the model by considering the analysis relevant to the plastic and elastic deformations of flat sections at high temperatures and varying strain rates,

3.2 Basic differential equation under homogeneous deformation.

The analysis presented in this section relies upon the following assumptions:~

(i) Vertical sections of the material in the direction of rolling remain-plane during deformation.

(ii) There is no lateral spread of the material if the thick­ ness of the strip is small compared with its width.

(iii) Plastic deformation occurs throughout the material between the rolls, though in the \ricinity of both the entry and exit planes elastic deformation and elastic recovery take place respectively.

(iv) The yield stress of the material varies throughout the zone of deformation under isothermal conditions.

(v) The frictional stresses, "between the material and roll surfaces, are proportional to the radial pressure but cannot exceed the shear yield stress of the material. (vi) The deforming arc of contact between the strip and the

rolls remains circular.

(vii) The peripheral velocity of the rolls is uniform.

Fig.3.2.1 represents a longitudinal section of a strip in transit through the roll gap. The state of stress of the material between the rolls is triaxial: a vertical stress, q, a horizontal stress acting in the direction of rolling,p ,and a stress parallel to the roll axis preventing free lateral spread 6,2 . Since the assumption of homogeneous deformation implies that no shear stresses arise between the surfaces of vertical sections, the horizontal stress, p , together with the transversal stress <S>2 and the vertical stress, q , are considered principal stresses. Then if plastic strain in the transverse dj.rection

is ignored, . .

The Von Misses maximum shear strain energy criterion states that plastic yielding occurs, when

Hence the use of the Von Misses criterion in a triaxial stress system and under plane strain conditions leads to,

so

The critical value of the shear' strain energy can be determined considering the case of yielding in a pure torsion test when ,

The vertical stress, q , is constant within a vertical section

due to the absence of shear stresses along its surfaces and assuming that the yield stress through a section is uniform, the horizontal

stress, p , must be therefore a constant. Hence the equilibrium of horiz­ ontal forces per unit width acting on the vertical slice AA of the

strip leads to ,

d j/J j = =2R.'ssnfi ±Tcc£0 13.-21)

where the upper sign refers to the exit side and the lower to the entry side of the arc of contact.

A balance of vertical forces within the slice AA and the use of the yield criterion permits the above equation to be expressed in terms of the radial pressure,

Two frictional, conditions were assumed between the rolls and the material surfaces:

i. A dry fricticnal stress proportional to the radial pressure "I-US » in which case 5q,3«2.ia applicable to slipping conditions was written as,

where and

ii A limiting frictional stress ’L - \L which produced sticking conditions, so Sq.3»2.1a was expressed as

The values of the radial pressure at both the entry and exit

.planes can be found by combining the equilibrium equation of vertical forces and the yield criterion; thus at the entry plane, where

K-k.1 and '0-\i.\ (the effective back tension, see Sect.3*7)# the radial pressure becomes,

(3.2A)

and at the exit plane, where § - 0 , and (the effective front tension,see Sect.3*7)»

( 3 • 2. 4a) 3*3 Basic differential equation under innomogeneous deformation.

A radial pressure distribution during inhomogeneous deformation

(4)

was obtained following Orowan's analysis' , in which it was assumed that the stress distribution inside the deforming material could be approximated to that produced by the compression of the slab between two inclined platens, so that the horizontal. compressive stress acting across the strip thickness was given by the following equation (see Sect.2.4),

(3.3.1) ./ *■

It was assumed after Orowan that under slipping friction and small contact angles the coefficient !ju _ 1 \ could be eliminated*

1 ^9 wfr /

Thus, the variation of the horizontal stress along the deformation zone could be expressed as,(replacing the value of s in terras of from Eq.3»3«l into 3«2da),

n !l)\ th . s\ ln\

-jL_ vi w.ybn a s 6 ; c. .

where Cd.s)/h

and 2i6W(Q)fer^,iUC0sQVh

Under sticking conditions the frictional stress contribution to the horizontal stress becomes l'-K so and the inhomogeneity factor^^reaches a constant vaJLue w(a) = tl//i (see Fig.2.4.3) and Eq.3,3.1

reduces to, ^ _ _ / ^ \ l\ \ ' /h U +

2

0

^ L ^ ( o ) (3.3.3)

Finally, the associated radial pressure can be found by means of the yield criterion expressed by Orowan as, S-2^/(0)

(3.3.6)