ELSEVIER Journal of Materiale Procc,sing Technology 70 (1997) 77-82

A slip-line approach to visioplasticity in plane-strain extrusion by the finite flow-line regions technique

J.P. Wang ’

Abstract

A numerical approach is proposed to establish a new procedure for the visioplasticity method. In this procedure, the finite flow-line regions are first produced by the least square’s method and then the velocity field and strain-rate field are obtained for each region. From the calculated strain-rate field, the constructed model of the slip-line can then be developed to draw the slip-line field. Along the:+ slip lines, the stress fields are found easily from Hencky’s equations. Plane-strain extrusion through a cosine-curved die with 2:l extrusion ratio is use to illustrated this new procedure. The results of this approach show more reasonable SOVJWW compared with the traditional method by the integral equation, in this method the error of the integral of the finite-dlfference flow being avoided. Q 1997 Elsevier Science .%A.

Keywords: Finite Row-line regions; Plane-strain extrusion; Visioplasticity

1. Introduction

The traditional visioplasticity method, which was introduced by Thomsen (11, is used to find the complete stress distribution in the deformation zone, according to the deformation of grid lines marked on the surface of the workpiece. From the experimental data (the values cf the flow function for plane-strain extrusion), the velocity, &sin rate and stress fields can be calcu- lated by the finite-difference method from the stream function, equilibrium and plasticity equations. There exist some difficulties for the tr.!ditional method: firstly, the Row lines obtained from experimental data are not smooth, secondly, the velocity field calculated from these flow lines may not satisfy the condition of conti- nuity (incompressibility); thirdly, the accuracy of the result depends on the spacing of marked lines which increases the difficulty of experimental measurement; and finally, owing to the finite-difference integral, great errors of the calculated stress fields are inevitable.

Many numerical methods of smoothing the flow lines of experimental data have been developed. The smooth-

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ing procedure mentioned by Shabaik and Kobayashi [2] is based on a simple averaging of the points but causes difficulties when treating data that are ill-defined. Ishikawa et al. [3] used the hyperbolic tangent curve to fit the Row lines, but there was great discrepancy in the case of non-hyperbolic tangent curve dies. Fanner et al. [4,5] and Dwivedi [6] have used high-order polynomials to smooth the flow lines and their results have been good, but numerical errors are unavoidable. Pehle and Kopp [7] used various marked lines at a spacing of 0. 1 - 1 mm to process the error analysis for visioplastic- ity datd, but it was shown that the actual values of the shear stress on the tool boundary cannot be calculated with sufficiently high accuracy to provide reasonable solutions.

To solve these difficulties, the flow-function elemental technique (FFET) method was proposed by Lin and Wang [8]. Unfortunately, the parameters used in this method to minimize the error-norm are not sufficient to obtain a more accurate solution. Wang et al. [9] pre- sented a new approach called the finite flow-line regions technique, in which approach the flow lines can be increased to an infinite number and the accuracy can bc improved, but numerical errors from the finite-differ- ence integral cannot be avoided.

78 J.P. Wang/Journal of Muwials Processing Technology 70 (1997) 77-8.2

In this paper, a slip-line method combined with the finite-line regions technique is used. After the finite flow-line regions have been produced, the velocity field and strain-rate field can be obtained for each region. From the calculated strain-rate field, a model of the slip-lines is constructed and used to draw the slip-line field. Along these slip lines, the stress field can be found easily from Hcncky’s equatious. In this procedure, nu- merical errors from the finite-difference integral can be avoided.

Plane-strain extrusion through a cosine-curved die at an extrusion ratio of 2:l is used to illustrate the procedure, computed results for the velocity field, the effective-strain field and the stress field being presented.

2. Velocity tiekl from the stream function

For an incompressible material, the velocity field for plane strain can be expressed by the stream function 4(x. y) as follows:

u,a46 89

8Y’ “C --

ax (1)

where II and u are the velocity components in the x- and y-directions. The strain-rates are:

. au a+ h'-~=~

. au 84 E.'.=&=-G

P.~=;~~+~)=~(~-~) (2)

The effective-strain rate is calculated from its definition:

2 &_($, ;’ )I,?

Js ” “’ (3)

The effective-strain I: can be evaluated from the inte- gration of the effective-strain rate B along the flow lines with respect to time:

C= s

Edr (4) 0

The volumetric strain-rate & is:

;&=~,+&!!!+!!.L-_L!%=O a%$

as ay ax ay a* a, (5)

It is to be noted from Eq. (5j that the condition of incompressibility is satisfied automatically.

3. Procedure of this method

3.1. Experimenlul dererminurion of rhe flow ,/irncrion

The test specimens are rectangular blocks of 30 x 20 x I5 mm of pure commercial lead; 2 mm square grids are marked on their surfaces. In order to confine the material flow to plane strain, two plates are placed on the two cud surfaces of the specimen. In order to decrease the frictional drag on these surfaces, the plates are sprayed with Teflon film prior to the test. The cosine proLIe with a 2:1 extrusion ratio is given by:

y=~r,+t,)+$+f)cos y 0

(6)

where I,,, lr are the initial and final width of the strip, respectively, and L is the axial length of the cosine- curve dies, taken as 24 mm.

The test was carried out at a punch speed of 0.1 mm s - I and engine oil was used as lubricant. The extrusion process was stopped when a sufficient length of speci- men was extruded to assure the establishment of steady-state motion. A photograph of the flow pattern of the test is shown in Fig. l(a). Using a digital scanning machine, the coordinate of the different points along the flow lines is tracked. The space between two points is chosen as 2 mm, the experimental points being shown in Fig. l(b) by circles.

3.2. Finite flow-line regions technique

The flow lines of the experimental data are smoothed firstly by high-order polynomials (Fig. l(b)) and then fitted to cubic spline lines. The intermediate flow lines (in what is called the finite flow-line regions) amongst these flow lines carI be produced by interpolation using the least-squares method (Fig. l(c)). It is interesting to note that the finite flow-line regions can be divided as thinly as possible.

The stream funztion 4(x-, y) for each finite flow-line region can be defined as:

rb0.Y) = (7)

where 4,. 4, are the stream function values of the i-th andj-th flow lines; and y,(x), y,(x) are the equations of the i-th and thej-th flow lines fitted by the cubic spline lines.

From Eq. (6), the velocity field and the strain-rate field can be calculated as:

a4 (rf,- 6,) l’=ay= o;-Y,)

79

2y:b’; - Y :) *(Jj - Y, )(J'; - J’:) (Y, - YY Q,-Y,l’

+(Jyy-$J’:‘) ((j,_ti,)

/ 1 From Eq. (7) it IS seen that the velocity of u along the

y-axis for each finite flow-line region is constant and

(8)

that the strain rate of E, will not continuously cross each flow line. This situation can be eliminated when the finite flow-line regions are set to be infinite in

, .,,,,,,,,,,,,,,,,,,,.,,,,.,,,,,I,~l,,,,,,,I

, . ,

P -“A.,. I# t, I.

Fig. 1. Flow patterns: (a) grid diswkm in the strip: (b) the smoolh- ing of Row lines; and (c) the finite Ilow-line regions.

The method of construction of the slip-line field is to find the curvilinear coordinates of the sz and /I lines. The a line, being the first maximum shear direction, is taken 45” clockwise from the first principal direction, as shown in Fig. 2, whilst the /I line, the second maximum shear direction, is 90° counter-clockwise from the first maximum shear direction. The angle 0 is the angle that the first shear direction makes with the x-axis (mea- sured counter-clockwise).

Assuming that the u and p lines are slip-stream lines, the maximum shear stress (k = a/,/3) is the slip value. The components of the slip value can be expressed as:

k cod’=? ' at k sin U = E ax k sin (I = - 3 av' kcosO= -al’ ax-

The numerical derived from:

construction of slip-line field can be

da=$d.v+gdy= -ksinOdx+kcos()dy

d/I=gdx+$dy= -kcosOdx-ksin0d.t (10)

From Eq. (9), the slip lines can be constructed when

(9)

the first shear directions (0) are known at every point of the deforming body. The origin of the curvilinear coordinates to be drawn can be set to any point of the defomting region (the case of this paper is set to the origin (x = 0. y = 0)).

The z and p lines are constructed from the cdkUkitCd strain-rate field and Eq. (9). The stress field is obtained from Hencky’s equations as follows:

o,,, i- 2kll = C, along the first slip line

un, - ZkO = C, along the second slip line (11) where rrm is the mean stress, and C, and C, are con- stants.

The components of stresses in terms of cr, and I can be expressed as:

a, = o, - k sin 20 (i,=u,,,+ksin20

r = k cos 20

Using Eqs. (IO) and (I I), the stress components at any point in the deformation regions can be calculated. A computer program for the calculation of the state of stress at any point is developed. Constants C, and C, are determined by satisfying the condition that the force on the exit slip line l-2 in the extrusion direction (the s-component) should be zero. That is:

J. P. Wang/Journal of Materials Pnmwing Tcclmdogy 70 (1997) 77-8.2

Ci - line

IX

Fig. 2. Slipstream lines and Mohr’s circle appropriale to point P.

F,= bxdy+r,.vdx s

= (or,,-ksinti)dy+(kcos20)dx=O f

= (C,-2k0-ksinB)dy+(kcos2@dx=O s

(13)

As shown in Fig. 3, region A and region B are first calculated using the known constants C, and C, of slip line l-2, and then the constants C, and C, on slip lines 2-3 and 3-4 can be calculated. From the constants C, and C, of slip lines 2-3 and 3-4, region C and region D are obtained.

4 I

4. Results and diiussion

Fig. 4 shows the velocity field and the effective-strain field obtained from by finite flow-line regions technique: these are in good agreement with experimental observa- tions.

Fig. 5 is the construction of the slip lines obtained from Eq. (10). It can be seen that the slip lines are tangential to the canter line of symmetry 0, =0)

Fig. 3. The procedure of calculating the stress from the slip lines. Fig. 4. Showing: (a) the velocity field: and (b) the etktive-strain lieid.

Fig. 5. The construction of the slip-line WI.

boundary dt 45”, thus being in good coincidence with the frictionless-boundary condition.

Stresses oX obtained by the slip-line method and the integral equations in visloplasticity are shown in Fig. 6, from which figure it is seen that there is the same trend for the two methods, although the solutions obtained by slip-lines are more reasonable. For the slip-line meihod, compressive stress can be found at the en- trance which reduces in magnitude along the tool profi!c and becomes a tensile stress over the concave part of the profile. It is interesting that residual stress can be observed at the exit.

Fig. 7. The ~omours slip-line mechcd.

and (b)

Stresses av obtained by the slip-line method and the integral equations in visioplasticity are shown in Fig. 7. It can be seen that there is compressive stress found in all of the forming regions for both methods, but that the solutions obtained by the slip-line method are more reasonable. In this method, compressive stress is found

at the entrance which decreases in magnitude aiong thr. tool profile but then increases in magnitude over the concave part of the profile.

5. Conclusions

This paper builds up a new procedure for rhe visio- plasticity method, in which procedure the concept of finite flow-line regions is proposed to find the velocity field and strain-rate field for each region. From the calculated strain-rate field, a slip-line model is con- structed and used to draw the slip-line field. Along these siip lines, the stress field can be found easily using Hencky’s equations. Plane-strain extrusion through a cosine-curved die at 2:1 extrusion ratio has been used to illustrate the procedure, the computed results for the stress fields being compared with those for the methcd employing the integral equation in visioplasticity.

From the above discussion, the following conclusions can be drawn.

Fig. 6. The EOII~OUI~ of s~~sses or: (a) integral equation; and (b) slip-line method.

(I) This procedure presents a new way to evaluate the stresses for the visioplasticity method.

(2) Using the slip-line model developed in this proce- dure, the stress field can be found without using the stress integration equations. Hence, the error of the finite-difference integral can be avoided.

(3) The constructed slip-lines are seen to be in good agreement with the frictionless-boundary condition.

82

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