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Parametric assessment of torsional springback in members of work-hardening materials
PARAMETRIC ASSESSMENT OF TORSIONAL SPRINGBACK IN MEMBERS OF WORK-HARDENING MATERIALS J. P. Dwtvnnr, P. C. UPADHYAYand N. K. DAS TALUKDER Department of Mechanical Engnxeering, Institute of Technology, Banaras Hindu University, varanasi-2ztooS,
India
Abstract-The paper deals with the torsional springback problem of bars of strain-hardening materials with arbitrary cross-sections. Using the deformation theory of plasticity, a numerical scheme based on the finite differenae approximation has been proposed. The growth of the elastic-plastic boundary and the resuhing stresses while loading, and the torsional springback and the residual stresses after unloading, are calculated. The results am verified experimentahy with mild steef bars having a square cross-section. The experimental rest&s am found to agree welf with the theoretical predictions obtained numerically.
NOTATION area of crm-wtion modulus of elasticity
slope beyond the yield point, for a piecewise linear stress-strain curve modulus of rigidity a characteristic iength of the cross-section twisting moment of torque twisting txtoment&st suffitieut to start yiekbng plastic torque residual tarque deviatoric stress components deviatoric stress components normalized w.r.t. rkail displacement in X-, Y-, Z-din&ions, respectiveiy stress function stress function corresponding to plastic stress and residual stress, respectively shear-strain in the j-direction acting on a face whose normal is in the i-direction normal strain on the face whose normal is in the idirection yield strain in simpIe tension strain normahxed w.r.t. tu angle of twist per unit length just sticient to start yielding total angle of twist per unit length non-dimensionahzed total angle of twist residual angle of twist per unit length nond~m~~on~~~ residual twist recovered angle of twist (spring~ck~ per unit length non-dimensionalized springback strain-hardening parameter for elastic-linearly plastic stress-strain relation Poisson’s ratio yield stress in simple tension effeotive shear stress norm&sod w.r.t. the yield stress shear stress in the j-direction acting on a face whose normal is in the i-direction stress normalized w.r.t. cry tint partial derivatives w.r.t. x and y, respectively
second partial derivatives w.r.t. x end Y, respectively Lapiacian operator %/ax a$iaY mesh size
In sheet-metal working, sheets are deformed to cylindrical and helical shapes by plastic bending and torsion with the help of a punch and die set. During a forming operation, when the load is applied, the sheet is deformed and the contour of the sheet section matches with that of the die. The deformation is partly elastic and partly plastic. On release of the applied load, elastic deformation disappears due to the release of the elastic strain energy and the final contour of the deformed sheet takes a shape which is different from the shape of the die. This elastic recovery ofdefu~~tio~ is ~ornrno~l~ known as springback. Torsional springback is the measure of elastic recovery of the angle of twist on removal of applied torque after twisting of the section beyond elastic limit. While designing the die-set, the springback factor should be taken into account to avoid mismatch while a~mbli~g different formed sections. Initially springback studies were limited to sheetbending operations only. Sachs [l], Schroeder [2], C&diner 131, Shaffer and Ungar [4], Duncan 151, Singh and Johnson [6f and others studied the springback considering bending of she&s of different shapes, and depicted springback as a function of material thickness, length and width of the sheets taken. Their studies were limited to Vand U-shaped dies for applying bending loads and they predicted the springback as a measure of change
422
J. P. DWIVEDIet al.
Fig. l(b).
in the curvature distribution. So far as torsional springback is concerned, almost all the previous studies on elasto-plastic torsion were limited to monotonically increasing loads only. The only work available on torsional springback is by Dwivedi et al. [7,8] in which residual angle of twist and torque relation, etc. have been predicted theoretically for bars with narrow rectangular sections. It has been shown that the theoretical predictions match very well with the experimental results. The work has the limitation that it is valid for thin rectangular strips only. For the calculation of torsional springback of bars with any arbitrary cross-section, a general scheme is necessary for solving the loading and unloading problems. Because of the computational difficulties, earlier solutions of elasto-plastic torsion were limited to elastic perfectly-plastic behaviour [9] only. Later, incorporating the effect of work hardening, Hodge [lo] and Greenberg et al. [l l] proposed different numerical schemes for the solution of the problem. A genera1 scheme which can be used for any type of cross-section and material behaviour was proposed by Upadhyay [12], in which the material behaviour was assumed to be elastic linearly strainhardening and the deformation theory of plasticity was used in the formulation. The solution was based on the finite difference approximation. Using the same numerical scheme [12], Dwivedi et al. [13] dealt with the torsional springback of square-section bars. But their work is limited to square-section bars of non-linear work-hardening materials. There is no literature available in which springback analysis in torsion of square-section bars of elastic-linearly strain-hardening (linear work-hardening materials) materials has been discussed. In the following the same numerical scheme has been used to determine the springback and elastic-plastic boundary in the bars with square crosssection of elastic-linearly strain hardening materials.
The results have been verified experimentally by taking mild steel bars of square cross-section. The experimental results have been found to agree well with the results obtained numerically. 2. BASIC THEORY
Elastic torsion
Consider a prismatic bar under elastic torsion [14] (Fig. la). Let u, v and w be the small displacements of a point (x, y, z), relative to its initial position, in the X-, Y-, Z-directions, respectively. At a section z = constant, the cross-section rotates about the Z axis, and so u =
-yze,
v =xzB
and
w =6f(x,y)
(1)
where 0 is the angle of twist per unit length. For elastic deformation, 0 is small and is constant along the length of the bar; fIf(x, y) is called the warping function and is assumed to be independent of z. If a stress function tj is taken such that
w
and
rxz = ay
r YZ- -2
ax'
(2)
then the elastic torsion equation is given as VZ$ =($+3)=
-2GB
(3)
with JI = a constant (taken to be zero) along the boundary of the cross-section. The torque T is given by T=2
ti ‘LXdx
where A is the cross-section of the bar.
(4)
Torsional springback ia work-~d~ag
materials
423
different for such an idealization and is given by
(as stated above),
TP
C#=((’ fv)rY~-v8&31,+A[I f
To
T
If the section is not hollow one, then inte~ation by parts gives
where (x,,Y~), (xz,Y~), (+n), (xrry4) are the points on the boundary as shown in Fig l(b). Thus if a function @(x, y) can be found which is zero on the boundary and satisfies eqn (3) over the interior of the cross-section of the bar, then the shear stress distribution, throughout the section, and torque T can be determined. The magnitude of the resultant shear stress at any point can be determined from eqn (2), and is given by
z=a,+a,+a,=o
(8)
and hence, the deviatoric stresses in this case become s,=cr,-$r&=611.
(9
eqns (7) give $=(I
A somewhat less realistic representation of the tensile stress-strain behaviour of compressible metals, other than the Ramberg-Osgood relation, is the piecewise linear relation (elastic-linearly strainhardening materials). However, this approximation does model certain features of plastic flow. In particular, the behaviour of altinium alloys is very closely a~proxima~ by this type of an idealization, The assumption of piecewise linearity states that in each plastic regime (i.e. side or comer), there exists a relationship, between the stress rates and strain rates, whose coefficients are material constants which are independent of the strain history of the material. In addition, the effect of Stan-hying upon the behavionr of the yield condition itself is also linear. It has been assumed that basic assumptions in the theory of elastic torsion are still valid for the elasticplastic case expect the stress&rain relation becomes
if 17,G 1,
L being a strain-hardening parameter for elasticlinearly strain-hardening materials and te,( = G) is the effective stress normalized with respect to yield stress (cry). It has also been assumed that the material follows the von Mises criterion of yielding, i.e. J2 = K2, where K is a parameter dependent on the amount of prestrain, J2 =f,S,, S, is the second invariant of deviatoric stresses. For simplicity, the plastic material has been considered to be incompressible and all viscosity effects have been neglected. For moderate strain-rates and tem~rature both the adoptions are well justified. Since only two of the stresses, namely r, and z,,~ have been assumed to be non-zero; the first invariant of stress
~o~quently
Plastic torsion
(7)
where
=o Fig. 2. Torque T vs angle of twist 0 (loading and unloading).
-@,)-‘IS,
-W)$++[l
+(c?,)-‘I&
(10)
Also, the value of effective stress is given by (3,= $
I*:= + Q”.
(11)
Assuming a stress function $ exists such that
the expression for tagreduces to (i.=& bv P: + The generalized stress-strain (10) gives
,y2 ’
(13)
relation given by eqn
J. P. DWIVEDE et al.
424
Since the slopes of the elastic loading line (AB) and that of the unloading line (XY) are the same, hence
(19) Corresponding to this recovered angle of twist, the unloading problem is solved by seeking the solution of corresponding elastic torsion equation
-5
Fig. 3.
which, on substituting the values of the stress components from eqn (I 2), reduces to
where tis is the stress-function corresponding linear and elastic untwisting. If the stress function corresponding to the plastic twisting ep be known solving eqn (171, then after unloading, the resulting surface on the cross-section is given by *lt=tip--3/s
F,,=[(l-tv)+l(l-(d,)_‘)]I’ OY
$_v)+l(l
q.,=[(l
-(a,)-1)]2. ?
to tip by $
(21)
and residual shear stresses, (T{,)~, are given by (15) (Tr;)R= -
(T,Z)R= %,
By expressing the strain component in terms of the displacements and their derivatives, eqn (15) yields
In a non-dimensionalized (TX
)n =
2.
(22)
form, relations (22) become
fr,,>, a,, ’
(fJ,,,
=
iT!jch. Y
(23)
The residual torque T, is determined with the help of
2 &+8x
.E=[(l+v+l)-(5J’]11/,.
(
>
(16)
~~erentiating the first and second parts of eqn (16) w.r.t. y and X, respectively, and subtracting
-BE=
(1 +r+A)-; 1
TR = 2
v*tj
3. SCHEME OF NUMERICAL SOLUTION
(17)
Before coming onto a numerical solution, co-ordinates are non-dimensionalized as 5 = XlL
is obtained. This equation governs the twisting of the bar in the plastic range, and is analogous to eqn (3) for the elastic torsion. Unloading of the bar from a plastic stage corresponding to torque Tp is equivalent to applying the torque Tp in the opposite direction to that of twisting. The untwisting or unloading being elastic, the stress-strain curve during unloading is a straight line parallel to the elastic loading line. From Fig, 2, the amount of springback twist is es=e,-BR.
(24)
This should be zero in the case of loading within elastic limit.
P1 + -$ W&X)X + (d&),f e
(GR) dx dy. ss A
(18)
V =ylL,
(25)
where L is a characteristic length of prismatic bar. Stress function tj is replaced by new stress function 4 defined by (a(LV)=$W.Y). Equations V24
(26)
(13) and (17) take the form
=$+!$=
-33)
(for elastic deformation)
(27)
Torsional springback in work-h~dening materials
-
5
10
20
30
40
50
425
AFTER LOADING -
-
AFTER UNLOADWG
60
f degree) Fig. 4. Linear work-hardening approximation of the experimental stress-strain curve. Angle
of twist
and
012
6.0
OlL
05
o:e
1.b
XlOA
Fig. 6. Equivalent stress c?~afongihe centre line OA of a
square-section for 0, = 3.0.
(for plastic d~fo~ation). The stress components
and therefore, becomes
&
Equation comes
are given by
the equation
5, = -q-
(28)
for equivalent
stress
For the solution of this elasto-plastic problem [eqns (27) and (28)] the finite difference method has been used, which is explained below. Let, 0 be one of the typical nodes (Fig. 3) at which the differential equation has to be satisfied and let 1, 2,3 and 4 be the neighbouring nodes around 0. Then by using the finite difference approximation, the differential equations (27) and (28), for the elastic and plastic torsion, respectively, can be written at point 0 as algebraic equations
KW2+ (d,)211'2.
(4), which gives the value of torque, be-
+& 2
T=2L4
4 d< d?.
4 [
$+$ 2
4
(31)
1
+error
term O(P)
= -2G0,
1.5
-
AFTER LOADING ---
AFTER UNLOADING
(32)
SQUARE SECTION
we-
0.5 -0.2-
60 0.2
0.4
0.6
0.6
1.0
XlOA
Fig. 5. Equivalent stress Cr,along the cents line OA of a square-section for A = 9.
-0.6
-
0.0
AFTER -
-
0.2 I
LOADING
AFTER “~LOA~,~G
’
0.1 I
’
‘\
0.6 /
1
,
0.6
1
. 1.0
XlOA
Fig. 7. Non-dimensionalized shear stress b-along the centre line OA of a square-section for 1 = 6, @,.= 3.75.
-
AFTER -
-
-
AFTER LoAelsR
---
AFrER ~~~
LOALWG
AFTER “NLOAOOW
----_-
-’d,‘bi,‘,1,0
Xl04
Fig. 8. VaIues of q5along the centreline 0.4 tionl-or a = 6.
ofa square-see-
x IO4
Fig. 9. Values af d, along the q&-e line OA of a squiare-seeti0n for e,==
3.5.
and and
+~+,~q$,]= S 0.0
x3 +h,(h,fh,$-n4@4+h)
x4
4bo 1= -8E
I
-&Q2.
03.0
0.76
(35) 1.0
a
(33)
in which
~7~has been denoted by x, and Crl_ h2. h3 and h4 are the dis&xzes of the ~~~~~u~~~ nodes 1, 2, 3 and 4, respectively. The value of ci, = x at the nodal point 0 is written as SRUAaE SECTIRN
If the mesh is assumed to be square, such that h, = hz = hj = I%(= h (say} then, eqns (32) and (33) reduce to 4rp,--,-~82-~~-Ta)4=2GBh2
(34)
Fig. 10. EliwbqdWic boundaries of a square-section for a =9.
Since eqn (35) involves the values on ri; (or x) at the nodal points which have first to be aalcuhited by finding the derivatives of the stress frmetion Q1,it is obvious thixt the s&&on can be obtained by an iterative process only. To start the soiution a current set of values for 4 will have to be generated. 4. F‘xPFmmmNTATrON
The mstetial seWed for the expe&uentaI veriti~+ tion was mild steef. Test pieces of a square crosssection (18x 18mm) of 15Omm length were prepared for the test. The rn~~us of elasticity of the material (2.1 x lob k~/~‘) was found by the beam bending m&hod. The Iinear w~rk-~~~~g parameter SF the material Awas found by ~o~ug the torsion test on an Avery torsion testing ma&me, in which the angle of the twist is measured with a Vernier scale giving an accuracy of up to 0.1”. The torque-twist curve was approximated by two straight lines as shown in Fig. 4, was found using eqn (7) with E/&Yas the ratio of slopes of the line OA amdOB, The work-h~e~ug parameter (;I) for the test piece material was found to be nearly 15.84, The angle of
twist, torque and residual angle of twist were noted for d%erent levels of e~ast~p~as~~loading.
Since the numerical technique is based on finite difference, it becomes necessary to first decide upon an appropriate mesh size (h) which wilf give a coriverging solution. For the square-se&on bar, solution was obtained using difPerentmesh sizes. A mesh size Ir = tJ16 was found to be an optimum choice fern the point of view of a~~acy and the computational time. The proposed numerical scheme for cakulation of torsional springback has been tested on bars of square cross-se&or2. Different vabes of aq$e of twist ~~~~*~and tie linear work-h~dening parameter A were taken, The results have been presented with the help of non-dimensional plots in Figs 5-15.
Figure 5 shows the equivalent stress t?#distribution along the centre line (0~4) of the cross-section for a fixed vahre of A f&Q taking (8,) as the parameter, In Fig- 6, Cs,variation baa been shown for a Exed
value of & taking 2 as the parameter. in both the figures the patter-u of results for the loading problem is similar to that which has been discussed in the literature by several other authors [l 1, 121. It may be seen that, irrespective of the value of angle of twist (8,) and the amount of work-hardening (i.e. value of A), after removal of the torque, the position of zero residual stress aver the cross-section is almost fixed around X/ttA = 0.75. Si~larly~ it is also observed that the position of the maximum residual stress at the cross-section always lies at the point of elastoplastic boundary (i.e. at the point where (7, = 1) reached along CVAat the end of the loading process. Further, this location of the point of maximum residual stress, as is dear from Fig. 6, does not vary with the amount of work-harde~ng and is always fixed for a given 8p. For BP= 3.0 it occurs at X/OA = 0.4 for all the values of 2. Of course for different amount of loading (i.e. 8,) this position will be different, because it lies on the elasto-plastic boundary which wiil keep shifting with the loading. Also it can be seen from Fig. 6 that, for small changes in the work-hardening behaviour, the residual stress distribution is hardly affected. The difference between the residual stress curve for i = 6 and 9 or 2 = 9 and 12 is practically negligible, and with higher vafues of I this difference wilt be lower. Therefore, it may be concluded that the residual stress and residual angle of twist will not be greatly affected even if one commits some error in approximating the work-hardening parameter 1 of the material. In fact, the same conclusion is drawn from Fig. 13 also. Figure 7 shows the distribution of eXZaIong UA for fixed values of ,I = 6 and 8p = 3.75. Since Z,,*is zero all along the centre line OA, only 2’,,has been plotted to depict the region in which it becomes negative after unloading.
Figures 8 and 9 show the variation of the stress function (4) along the centre line OA. Figure 8 has been plotted with gJ as a parameter for a fixed i., whereas in Fig. 9, 8, has been fixed and A is taken as parameter. Location ofelasto-plastic boundaries, taking 8, and A as parameters are shown in Figs 10 and 11, respectively. The area of plastic region keeps increasing with the angle of twist and the elasto-plastic boundaries for different angles of twist are almost parallel. Figure IO shows that the spacing between the e~asto~p~asti~ boundaries, for the same amount of difference in twist, deereases with the increasing of Oj&. This shows that initially plastic zone spreads faster but, as the twist is continued the rate of growth become slower. With the increase in degree of work-hardening, area of plastic zone decreases (see Fig. IO). Therefore, even for the same twist, spread of plastic zone will be different for different materials depending upon the workhardening parameter 2,. Figure 12 shows springback angle 0, as function of Up (total twist) with i as the parameter. As expected, elastic recovery is found to be more with lower values of il. This difference in the springback values for different amount of workhardening becomes more and more as the twist increases. For materiab with higher values of 1, springback (8,) curve almost stabilizes beyond 0,/U, = 3.0 or so. That is, with increasing values of L, the material will approach to an elasto-ideally plastic behaviour. Residual twist
plots are shown as function of i in Fig. 13 with e, as the parameter. ft can be seen that the residual twist is hardly affected with the changes in .I, and it remains almost constant for a given angle of twist 8, (as also explained in Fig. 6). Comparison of theoretical predictions, as obtained from the proposed scheme of springback analysis, and the experimentaf results are shown in Figs I4 and 15. Theoretical predictions match we11with the experimental results. The maximum difference between the experimental and the theoretical values of gR (= tYR/OO) is nearly 5.5%. This error may be attributed mainly to two factors: t. The lowest inurement on the torsion testing machine is 0.1”. This may cause a significant error for lower values of angle of twist. Also the machine becomes less sensitive to torque changes when the twisting moment approaches zero during unloading. This may affect the recording of the true residual angle of twist which in turn a&&s the springback value. 2. The error in linear work-hardening approximation of the experimental stress-strain (torquetwist) curve may affect the work-hardening parameter and hence the theoretical springback value.
Torsional springback in work-hardening
It is evident from Fig. 4 that the maximum possibilities of error in work-hardening approximation is around the yield point. 6. CONCLUSIONS
The proposed numerical scheme is found to predict the torsional springback quite successfully. This is supported by the excellent agreement between the theoretical and the experimental results. The accuracy of the theoretical results, of course, depends on the mesh size. The method can be used for any other cross-sections such as rectangular, elliptical, triangular, etc. REFERENCES
1. G. Sachs, Principles and Methods of Sheet Metal Fabricating. Reinhold, New York (1951). 2. W. Schroeder, Mechanics of sheet metal bending. Tram ASME Nov. (1943). 3. F. J. Gardiner, The springback of metals. Tram ASME Jan., 2 (1948). 4. B. W. Shaffer and E. E. Ungar, Mechanics of sheet bending process. Tram ASME March, 34 (1960). 5. J. L. Duncan, Aspects of draw die forming of sheet metal. Australian Conference on Manufacturing Engineers, August (1977).
materials
429
6. A. N. Singh and W. Johnson, Springback after cylindrically bending metal strips. Dr Karunesh Memorial International Conference, New Delhi, December (1979). 7. j. P.. Dwivedi, A. N. Singb, S. Ram and N. K. Das Talukder, Springback analysis in torsion of rectangular strips. Int. J. Mech. Sci. 28, 505-515 (1986): 8. J. P. Dwivedi, P. K. Sarkar, A. N. Singh, S. Ram and N. K. Das Talukder, Experimental aspects of torsional springback in rectangular strips. .I. Inst. Engrs (India) 67. 70-73 (1987). 9. A.. Nadai, ‘Thedry of Flow and Fracture of Solids, 2nd Edn, Vol. 1, pp. 494497. McGraw-Hill (1950). 10. P. G. Hodge, Elasto-plastic torsion as a problem in non-linear programming. Ini. J. Mech. Sci. 12,985-995 (1970). 11. H. J. Greenberg, W. S. Dom and E. H. Wetherell, A comparison of flow and deformation theories in plastic torsion of a square cylinder. Plasticity, Proceedings of the Second Symposium on Nova1 Structural Mechanics, pp. 279-296. Brown University (1960). 12. P. C. Upadhyay, Elasto-plastic torsion with work-hardening. M.Tech. thesis, Mechanical Engineering Department, I.I.T., Kanpur (1970). 13. J. P. Dwivedi, P. C. Upadhyay and N. K. Das Talukder, Torsional springback in square section bars of nonlinear work-hardening materials. Int. J. Mech. Sci. 32, 863-876 (1990). 14. S. Timoshenko and J. N. Goodier, Theory of Elnsticify, pp. 258-262. McGraw-Hill, Tokyo (1951).
India
Abstract-The paper deals with the torsional springback problem of bars of strain-hardening materials with arbitrary cross-sections. Using the deformation theory of plasticity, a numerical scheme based on the finite differenae approximation has been proposed. The growth of the elastic-plastic boundary and the resuhing stresses while loading, and the torsional springback and the residual stresses after unloading, are calculated. The results am verified experimentahy with mild steef bars having a square cross-section. The experimental rest&s am found to agree welf with the theoretical predictions obtained numerically.
NOTATION area of crm-wtion modulus of elasticity
slope beyond the yield point, for a piecewise linear stress-strain curve modulus of rigidity a characteristic iength of the cross-section twisting moment of torque twisting txtoment&st suffitieut to start yiekbng plastic torque residual tarque deviatoric stress components deviatoric stress components normalized w.r.t. rkail displacement in X-, Y-, Z-din&ions, respectiveiy stress function stress function corresponding to plastic stress and residual stress, respectively shear-strain in the j-direction acting on a face whose normal is in the i-direction normal strain on the face whose normal is in the idirection yield strain in simpIe tension strain normahxed w.r.t. tu angle of twist per unit length just sticient to start yielding total angle of twist per unit length non-dimensionahzed total angle of twist residual angle of twist per unit length nond~m~~on~~~ residual twist recovered angle of twist (spring~ck~ per unit length non-dimensionalized springback strain-hardening parameter for elastic-linearly plastic stress-strain relation Poisson’s ratio yield stress in simple tension effeotive shear stress norm&sod w.r.t. the yield stress shear stress in the j-direction acting on a face whose normal is in the i-direction stress normalized w.r.t. cry tint partial derivatives w.r.t. x and y, respectively
second partial derivatives w.r.t. x end Y, respectively Lapiacian operator %/ax a$iaY mesh size
In sheet-metal working, sheets are deformed to cylindrical and helical shapes by plastic bending and torsion with the help of a punch and die set. During a forming operation, when the load is applied, the sheet is deformed and the contour of the sheet section matches with that of the die. The deformation is partly elastic and partly plastic. On release of the applied load, elastic deformation disappears due to the release of the elastic strain energy and the final contour of the deformed sheet takes a shape which is different from the shape of the die. This elastic recovery ofdefu~~tio~ is ~ornrno~l~ known as springback. Torsional springback is the measure of elastic recovery of the angle of twist on removal of applied torque after twisting of the section beyond elastic limit. While designing the die-set, the springback factor should be taken into account to avoid mismatch while a~mbli~g different formed sections. Initially springback studies were limited to sheetbending operations only. Sachs [l], Schroeder [2], C&diner 131, Shaffer and Ungar [4], Duncan 151, Singh and Johnson [6f and others studied the springback considering bending of she&s of different shapes, and depicted springback as a function of material thickness, length and width of the sheets taken. Their studies were limited to Vand U-shaped dies for applying bending loads and they predicted the springback as a measure of change
422
J. P. DWIVEDIet al.
Fig. l(b).
in the curvature distribution. So far as torsional springback is concerned, almost all the previous studies on elasto-plastic torsion were limited to monotonically increasing loads only. The only work available on torsional springback is by Dwivedi et al. [7,8] in which residual angle of twist and torque relation, etc. have been predicted theoretically for bars with narrow rectangular sections. It has been shown that the theoretical predictions match very well with the experimental results. The work has the limitation that it is valid for thin rectangular strips only. For the calculation of torsional springback of bars with any arbitrary cross-section, a general scheme is necessary for solving the loading and unloading problems. Because of the computational difficulties, earlier solutions of elasto-plastic torsion were limited to elastic perfectly-plastic behaviour [9] only. Later, incorporating the effect of work hardening, Hodge [lo] and Greenberg et al. [l l] proposed different numerical schemes for the solution of the problem. A genera1 scheme which can be used for any type of cross-section and material behaviour was proposed by Upadhyay [12], in which the material behaviour was assumed to be elastic linearly strainhardening and the deformation theory of plasticity was used in the formulation. The solution was based on the finite difference approximation. Using the same numerical scheme [12], Dwivedi et al. [13] dealt with the torsional springback of square-section bars. But their work is limited to square-section bars of non-linear work-hardening materials. There is no literature available in which springback analysis in torsion of square-section bars of elastic-linearly strain-hardening (linear work-hardening materials) materials has been discussed. In the following the same numerical scheme has been used to determine the springback and elastic-plastic boundary in the bars with square crosssection of elastic-linearly strain hardening materials.
The results have been verified experimentally by taking mild steel bars of square cross-section. The experimental results have been found to agree well with the results obtained numerically. 2. BASIC THEORY
Elastic torsion
Consider a prismatic bar under elastic torsion [14] (Fig. la). Let u, v and w be the small displacements of a point (x, y, z), relative to its initial position, in the X-, Y-, Z-directions, respectively. At a section z = constant, the cross-section rotates about the Z axis, and so u =
-yze,
v =xzB
and
w =6f(x,y)
(1)
where 0 is the angle of twist per unit length. For elastic deformation, 0 is small and is constant along the length of the bar; fIf(x, y) is called the warping function and is assumed to be independent of z. If a stress function tj is taken such that
w
and
rxz = ay
r YZ- -2
ax'
(2)
then the elastic torsion equation is given as VZ$ =($+3)=
-2GB
(3)
with JI = a constant (taken to be zero) along the boundary of the cross-section. The torque T is given by T=2
ti ‘LXdx
where A is the cross-section of the bar.
(4)
Torsional springback ia work-~d~ag
materials
423
different for such an idealization and is given by
(as stated above),
TP
C#=((’ fv)rY~-v8&31,+A[I f
To
T
If the section is not hollow one, then inte~ation by parts gives
where (x,,Y~), (xz,Y~), (+n), (xrry4) are the points on the boundary as shown in Fig l(b). Thus if a function @(x, y) can be found which is zero on the boundary and satisfies eqn (3) over the interior of the cross-section of the bar, then the shear stress distribution, throughout the section, and torque T can be determined. The magnitude of the resultant shear stress at any point can be determined from eqn (2), and is given by
z=a,+a,+a,=o
(8)
and hence, the deviatoric stresses in this case become s,=cr,-$r&=611.
(9
eqns (7) give $=(I
A somewhat less realistic representation of the tensile stress-strain behaviour of compressible metals, other than the Ramberg-Osgood relation, is the piecewise linear relation (elastic-linearly strainhardening materials). However, this approximation does model certain features of plastic flow. In particular, the behaviour of altinium alloys is very closely a~proxima~ by this type of an idealization, The assumption of piecewise linearity states that in each plastic regime (i.e. side or comer), there exists a relationship, between the stress rates and strain rates, whose coefficients are material constants which are independent of the strain history of the material. In addition, the effect of Stan-hying upon the behavionr of the yield condition itself is also linear. It has been assumed that basic assumptions in the theory of elastic torsion are still valid for the elasticplastic case expect the stress&rain relation becomes
if 17,G 1,
L being a strain-hardening parameter for elasticlinearly strain-hardening materials and te,( = G) is the effective stress normalized with respect to yield stress (cry). It has also been assumed that the material follows the von Mises criterion of yielding, i.e. J2 = K2, where K is a parameter dependent on the amount of prestrain, J2 =f,S,, S, is the second invariant of deviatoric stresses. For simplicity, the plastic material has been considered to be incompressible and all viscosity effects have been neglected. For moderate strain-rates and tem~rature both the adoptions are well justified. Since only two of the stresses, namely r, and z,,~ have been assumed to be non-zero; the first invariant of stress
~o~quently
Plastic torsion
(7)
where
=o Fig. 2. Torque T vs angle of twist 0 (loading and unloading).
-@,)-‘IS,
-W)$++[l
+(c?,)-‘I&
(10)
Also, the value of effective stress is given by (3,= $
I*:= + Q”.
(11)
Assuming a stress function $ exists such that
the expression for tagreduces to (i.=& bv P: + The generalized stress-strain (10) gives
,y2 ’
(13)
relation given by eqn
J. P. DWIVEDE et al.
424
Since the slopes of the elastic loading line (AB) and that of the unloading line (XY) are the same, hence
(19) Corresponding to this recovered angle of twist, the unloading problem is solved by seeking the solution of corresponding elastic torsion equation
-5
Fig. 3.
which, on substituting the values of the stress components from eqn (I 2), reduces to
where tis is the stress-function corresponding linear and elastic untwisting. If the stress function corresponding to the plastic twisting ep be known solving eqn (171, then after unloading, the resulting surface on the cross-section is given by *lt=tip--3/s
F,,=[(l-tv)+l(l-(d,)_‘)]I’ OY
$_v)+l(l
q.,=[(l
-(a,)-1)]2. ?
to tip by $
(21)
and residual shear stresses, (T{,)~, are given by (15) (Tr;)R= -
(T,Z)R= %,
By expressing the strain component in terms of the displacements and their derivatives, eqn (15) yields
In a non-dimensionalized (TX
)n =
2.
(22)
form, relations (22) become
fr,,>, a,, ’
(fJ,,,
=
iT!jch. Y
(23)
The residual torque T, is determined with the help of
2 &+8x
.E=[(l+v+l)-(5J’]11/,.
(
>
(16)
~~erentiating the first and second parts of eqn (16) w.r.t. y and X, respectively, and subtracting
-BE=
(1 +r+A)-; 1
TR = 2
v*tj
3. SCHEME OF NUMERICAL SOLUTION
(17)
Before coming onto a numerical solution, co-ordinates are non-dimensionalized as 5 = XlL
is obtained. This equation governs the twisting of the bar in the plastic range, and is analogous to eqn (3) for the elastic torsion. Unloading of the bar from a plastic stage corresponding to torque Tp is equivalent to applying the torque Tp in the opposite direction to that of twisting. The untwisting or unloading being elastic, the stress-strain curve during unloading is a straight line parallel to the elastic loading line. From Fig, 2, the amount of springback twist is es=e,-BR.
(24)
This should be zero in the case of loading within elastic limit.
P1 + -$ W&X)X + (d&),f e
(GR) dx dy. ss A
(18)
V =ylL,
(25)
where L is a characteristic length of prismatic bar. Stress function tj is replaced by new stress function 4 defined by (a(LV)=$W.Y). Equations V24
(26)
(13) and (17) take the form
=$+!$=
-33)
(for elastic deformation)
(27)
Torsional springback in work-h~dening materials
-
5
10
20
30
40
50
425
AFTER LOADING -
-
AFTER UNLOADWG
60
f degree) Fig. 4. Linear work-hardening approximation of the experimental stress-strain curve. Angle
of twist
and
012
6.0
OlL
05
o:e
1.b
XlOA
Fig. 6. Equivalent stress c?~afongihe centre line OA of a
square-section for 0, = 3.0.
(for plastic d~fo~ation). The stress components
and therefore, becomes
&
Equation comes
are given by
the equation
5, = -q-
(28)
for equivalent
stress
For the solution of this elasto-plastic problem [eqns (27) and (28)] the finite difference method has been used, which is explained below. Let, 0 be one of the typical nodes (Fig. 3) at which the differential equation has to be satisfied and let 1, 2,3 and 4 be the neighbouring nodes around 0. Then by using the finite difference approximation, the differential equations (27) and (28), for the elastic and plastic torsion, respectively, can be written at point 0 as algebraic equations
KW2+ (d,)211'2.
(4), which gives the value of torque, be-
+& 2
T=2L4
4 d< d?.
4 [
$+$ 2
4
(31)
1
+error
term O(P)
= -2G0,
1.5
-
AFTER LOADING ---
AFTER UNLOADING
(32)
SQUARE SECTION
we-
0.5 -0.2-
60 0.2
0.4
0.6
0.6
1.0
XlOA
Fig. 5. Equivalent stress Cr,along the cents line OA of a square-section for A = 9.
-0.6
-
0.0
AFTER -
-
0.2 I
LOADING
AFTER “~LOA~,~G
’
0.1 I
’
‘\
0.6 /
1
,
0.6
1
. 1.0
XlOA
Fig. 7. Non-dimensionalized shear stress b-along the centre line OA of a square-section for 1 = 6, @,.= 3.75.
-
AFTER -
-
-
AFTER LoAelsR
---
AFrER ~~~
LOALWG
AFTER “NLOAOOW
----_-
-’d,‘bi,‘,1,0
Xl04
Fig. 8. VaIues of q5along the centreline 0.4 tionl-or a = 6.
ofa square-see-
x IO4
Fig. 9. Values af d, along the q&-e line OA of a squiare-seeti0n for e,==
3.5.
and and
+~+,~q$,]= S 0.0
x3 +h,(h,fh,$-n4@4+h)
x4
4bo 1= -8E
I
-&Q2.
03.0
0.76
(35) 1.0
a
(33)
in which
~7~has been denoted by x, and Crl_ h2. h3 and h4 are the dis&xzes of the ~~~~~u~~~ nodes 1, 2, 3 and 4, respectively. The value of ci, = x at the nodal point 0 is written as SRUAaE SECTIRN
If the mesh is assumed to be square, such that h, = hz = hj = I%(= h (say} then, eqns (32) and (33) reduce to 4rp,--,-~82-~~-Ta)4=2GBh2
(34)
Fig. 10. EliwbqdWic boundaries of a square-section for a =9.
Since eqn (35) involves the values on ri; (or x) at the nodal points which have first to be aalcuhited by finding the derivatives of the stress frmetion Q1,it is obvious thixt the s&&on can be obtained by an iterative process only. To start the soiution a current set of values for 4 will have to be generated. 4. F‘xPFmmmNTATrON
The mstetial seWed for the expe&uentaI veriti~+ tion was mild steef. Test pieces of a square crosssection (18x 18mm) of 15Omm length were prepared for the test. The rn~~us of elasticity of the material (2.1 x lob k~/~‘) was found by the beam bending m&hod. The Iinear w~rk-~~~~g parameter SF the material Awas found by ~o~ug the torsion test on an Avery torsion testing ma&me, in which the angle of the twist is measured with a Vernier scale giving an accuracy of up to 0.1”. The torque-twist curve was approximated by two straight lines as shown in Fig. 4, was found using eqn (7) with E/&Yas the ratio of slopes of the line OA amdOB, The work-h~e~ug parameter (;I) for the test piece material was found to be nearly 15.84, The angle of
twist, torque and residual angle of twist were noted for d%erent levels of e~ast~p~as~~loading.
Since the numerical technique is based on finite difference, it becomes necessary to first decide upon an appropriate mesh size (h) which wilf give a coriverging solution. For the square-se&on bar, solution was obtained using difPerentmesh sizes. A mesh size Ir = tJ16 was found to be an optimum choice fern the point of view of a~~acy and the computational time. The proposed numerical scheme for cakulation of torsional springback has been tested on bars of square cross-se&or2. Different vabes of aq$e of twist ~~~~*~and tie linear work-h~dening parameter A were taken, The results have been presented with the help of non-dimensional plots in Figs 5-15.
Figure 5 shows the equivalent stress t?#distribution along the centre line (0~4) of the cross-section for a fixed vahre of A f&Q taking (8,) as the parameter, In Fig- 6, Cs,variation baa been shown for a Exed
value of & taking 2 as the parameter. in both the figures the patter-u of results for the loading problem is similar to that which has been discussed in the literature by several other authors [l 1, 121. It may be seen that, irrespective of the value of angle of twist (8,) and the amount of work-hardening (i.e. value of A), after removal of the torque, the position of zero residual stress aver the cross-section is almost fixed around X/ttA = 0.75. Si~larly~ it is also observed that the position of the maximum residual stress at the cross-section always lies at the point of elastoplastic boundary (i.e. at the point where (7, = 1) reached along CVAat the end of the loading process. Further, this location of the point of maximum residual stress, as is dear from Fig. 6, does not vary with the amount of work-harde~ng and is always fixed for a given 8p. For BP= 3.0 it occurs at X/OA = 0.4 for all the values of 2. Of course for different amount of loading (i.e. 8,) this position will be different, because it lies on the elasto-plastic boundary which wiil keep shifting with the loading. Also it can be seen from Fig. 6 that, for small changes in the work-hardening behaviour, the residual stress distribution is hardly affected. The difference between the residual stress curve for i = 6 and 9 or 2 = 9 and 12 is practically negligible, and with higher vafues of I this difference wilt be lower. Therefore, it may be concluded that the residual stress and residual angle of twist will not be greatly affected even if one commits some error in approximating the work-hardening parameter 1 of the material. In fact, the same conclusion is drawn from Fig. 13 also. Figure 7 shows the distribution of eXZaIong UA for fixed values of ,I = 6 and 8p = 3.75. Since Z,,*is zero all along the centre line OA, only 2’,,has been plotted to depict the region in which it becomes negative after unloading.
Figures 8 and 9 show the variation of the stress function (4) along the centre line OA. Figure 8 has been plotted with gJ as a parameter for a fixed i., whereas in Fig. 9, 8, has been fixed and A is taken as parameter. Location ofelasto-plastic boundaries, taking 8, and A as parameters are shown in Figs 10 and 11, respectively. The area of plastic region keeps increasing with the angle of twist and the elasto-plastic boundaries for different angles of twist are almost parallel. Figure IO shows that the spacing between the e~asto~p~asti~ boundaries, for the same amount of difference in twist, deereases with the increasing of Oj&. This shows that initially plastic zone spreads faster but, as the twist is continued the rate of growth become slower. With the increase in degree of work-hardening, area of plastic zone decreases (see Fig. IO). Therefore, even for the same twist, spread of plastic zone will be different for different materials depending upon the workhardening parameter 2,. Figure 12 shows springback angle 0, as function of Up (total twist) with i as the parameter. As expected, elastic recovery is found to be more with lower values of il. This difference in the springback values for different amount of workhardening becomes more and more as the twist increases. For materiab with higher values of 1, springback (8,) curve almost stabilizes beyond 0,/U, = 3.0 or so. That is, with increasing values of L, the material will approach to an elasto-ideally plastic behaviour. Residual twist
plots are shown as function of i in Fig. 13 with e, as the parameter. ft can be seen that the residual twist is hardly affected with the changes in .I, and it remains almost constant for a given angle of twist 8, (as also explained in Fig. 6). Comparison of theoretical predictions, as obtained from the proposed scheme of springback analysis, and the experimentaf results are shown in Figs I4 and 15. Theoretical predictions match we11with the experimental results. The maximum difference between the experimental and the theoretical values of gR (= tYR/OO) is nearly 5.5%. This error may be attributed mainly to two factors: t. The lowest inurement on the torsion testing machine is 0.1”. This may cause a significant error for lower values of angle of twist. Also the machine becomes less sensitive to torque changes when the twisting moment approaches zero during unloading. This may affect the recording of the true residual angle of twist which in turn a&&s the springback value. 2. The error in linear work-hardening approximation of the experimental stress-strain (torquetwist) curve may affect the work-hardening parameter and hence the theoretical springback value.
Torsional springback in work-hardening
It is evident from Fig. 4 that the maximum possibilities of error in work-hardening approximation is around the yield point. 6. CONCLUSIONS
The proposed numerical scheme is found to predict the torsional springback quite successfully. This is supported by the excellent agreement between the theoretical and the experimental results. The accuracy of the theoretical results, of course, depends on the mesh size. The method can be used for any other cross-sections such as rectangular, elliptical, triangular, etc. REFERENCES
1. G. Sachs, Principles and Methods of Sheet Metal Fabricating. Reinhold, New York (1951). 2. W. Schroeder, Mechanics of sheet metal bending. Tram ASME Nov. (1943). 3. F. J. Gardiner, The springback of metals. Tram ASME Jan., 2 (1948). 4. B. W. Shaffer and E. E. Ungar, Mechanics of sheet bending process. Tram ASME March, 34 (1960). 5. J. L. Duncan, Aspects of draw die forming of sheet metal. Australian Conference on Manufacturing Engineers, August (1977).
materials
429
6. A. N. Singh and W. Johnson, Springback after cylindrically bending metal strips. Dr Karunesh Memorial International Conference, New Delhi, December (1979). 7. j. P.. Dwivedi, A. N. Singb, S. Ram and N. K. Das Talukder, Springback analysis in torsion of rectangular strips. Int. J. Mech. Sci. 28, 505-515 (1986): 8. J. P. Dwivedi, P. K. Sarkar, A. N. Singh, S. Ram and N. K. Das Talukder, Experimental aspects of torsional springback in rectangular strips. .I. Inst. Engrs (India) 67. 70-73 (1987). 9. A.. Nadai, ‘Thedry of Flow and Fracture of Solids, 2nd Edn, Vol. 1, pp. 494497. McGraw-Hill (1950). 10. P. G. Hodge, Elasto-plastic torsion as a problem in non-linear programming. Ini. J. Mech. Sci. 12,985-995 (1970). 11. H. J. Greenberg, W. S. Dom and E. H. Wetherell, A comparison of flow and deformation theories in plastic torsion of a square cylinder. Plasticity, Proceedings of the Second Symposium on Nova1 Structural Mechanics, pp. 279-296. Brown University (1960). 12. P. C. Upadhyay, Elasto-plastic torsion with work-hardening. M.Tech. thesis, Mechanical Engineering Department, I.I.T., Kanpur (1970). 13. J. P. Dwivedi, P. C. Upadhyay and N. K. Das Talukder, Torsional springback in square section bars of nonlinear work-hardening materials. Int. J. Mech. Sci. 32, 863-876 (1990). 14. S. Timoshenko and J. N. Goodier, Theory of Elnsticify, pp. 258-262. McGraw-Hill, Tokyo (1951).