V o l . 19, pp. 2 3 5 - 2 4 0 , 1 9 8 5 P r i n t e d in t h e U . S . A .
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THE ELASTIC UNLOADING OF TORSION BARS SUBJECTED TO PRIOR PLASTIC DEFORMATION J.J. Jonas, F. Montheil]et and S. Shrivastava Faculty of Engineering, McGill University, 3450 University St., Montreal, H3A 2A7 Canada. (Received
When solid cylindrical bars are loaded in torsion, plastic flow is initiated at the outer radius of the specimens, and yielding is propagated progressively inwards with increasing twist (i). When such bars are subsequently unloaded, reverse twisting or 'springback' takes place; in work hardening materials, the amount of elastic 'untwisting' increases with the previously applied plastic twist, although the relationship is not of course linear. In conventional torsion testing, the twist per unit length applied to the sample is carefully measured, as is the developed torque. However, the elastic 'untwist' can also be determined experimentally; attempts have indeed been made to use this quantity to establish the basic shear stress (r) vs. shear strain (y) relation for various materials under both simple torsion testing conditions, as well as when combined stresses are applied (2-7). In the analysis of such data, it has been assumed that, after 'untwisting', the shear stress is zero at the outer radius of the sample (see, for example, Ref. 4), and this method has been employed for the determination of the yield surface of materials subjected to combined loading (7). In the analysis described below, the stress and strain rate states in a metal bar undergoing plastic deformation are considered in detail for the case of homogeneous materials. (These may be plastically anisotropic, as will be discussed below in more detail~ but the interactions between individual grains of different orientations are not taken into account). The relation between the angle of elastic untwisting and the torque at the moment of unloading is then derived. It is shown that, for rate insensitive materials, full unloading involves the presence of forward and backward residual stresses in the outer and inner shells of the bar, respectively. Such residual stresses not only call for the modification of the relations derived and employed in Refs. (2) to (7), but may also be of importance during simulation of the rolling or forging behaviour of metals by repeated twisting and unloading (8, 9). Strain Rate Tensor in Long Cylindrical Bars Let R, @ and Z and r, ~ and z be the cylindrical polar coordinates of a material point in the undeformed and deformed states of the bar, respectively, Fig. i. Then a general axisymmetric deformation of the bar leads to the following relations between the undeformed and deformed coordinates r
where %., kA and ~ are deformation parameters which can depend on both R and Z in the most 1 Z general case. R, 0, Z are convected coordinates; one combination of values signifies a material point and remains unaltered during a given experiment. The corresponding velocity field is given by: Vr
= = =
r i2 zi%2"
From the above, the components of the strain rate tensor can be computed to be
235 0036-9748/85 $ 3 . 0 0 + .00 Copyright (c) 1 9 8 5 P e r g a m o n P r e s s
il a il Ii + r ~r (~1)
Vr r £
OF T O R S I O N
x 2 + z -f-fz (772)
r a~p 2 ar
l[ Vr =
The equations of equilibrium which must be satisfied in the interior of the body, again under the assumption of axisymmetry, are (in the absence of body forces) aa aarr + rz arr a~ ~r --~z + r a°r~ ~°~z 2°r~ ar +--~--z + r ao
gqs. (4) may be written as
a (r °rr) + ~z ~ (r Orz) - o ~ a"7 a (r2Or~) + gag (r2O~z) 3-7
ar (r arz) + T z
Equations (3) to (5) are very general; they continue to apply at large strains and are valid for wide classes of material properties (but not, for example, for initially textured materials which do not deform axisymmetrically). They hold in particular for elastic/plastic loading and elastic unloading, the conditions which are of interest here. Uniform Torsion We now assume that a state of torsion, called uniform torsion, is possible in which the stress and strain rate components are independent of the z coordinate. In practice, this condition generally prevails in the central portion of the specimen length, i.e. well away from the ends, which will be discussed separately below. If no tractions are applied to the cylindrical surface of the bar (R = a), then it follows from relations (5) above that ~r¢
throughout the bar and the only equilibrium equation which remains to be satisfied is d--~d(r arr) - a ~
Thus any axisymmetric distribution of a~z and azz consistent with the strain distribution is admissible, since the latter quantities do not appear in relation (7).
Such z independence involves the absence of flow localization, a condition which applies as long as the torque/twist curve has a positive slope (i0).
OF T O R S I O N
When the strain components are also independent of z, ~ is given by Z
where ~I is the twist per unit of undeformed length along the z axis, and is independent of r, ¢ and z. Under such conditions, X 2 is independent of position, and the strain rate components are ~I
2X 2 R ~I"
As a result, solely on the basis that the stress and strain rate components are independent of z, it is evident that Or~ = Orz R 0 on the one hand, and ~r~ = Crz ~ 0 on the other, regardless of the constitutive relations (e.g. elastic, plastic, etc.) linking these two types of components. The practical significance of the nul values of the two shear strain rate components during uniform torsion is that radial lines and transverse planes must remain radial and planar, respectively, during both loading and unloading. Note also that the diagonal strain rate components differ from zero only when a specimen length change or cavitation is taking place. If the material can be considered incompressible during.the deformation , then it follows from Eqs. (9) and the condition of incompressibility, Crr + [email protected]
+ ~zz = 0, that not on i y X^ but X 1 also must be independent of position, and that they are related by the expression h I X 2 ! I. Distribution of Elastic and Plastic Shear Strains Along the Radii As is evident from Eqs° (9) above, for incompressible materials, the macroscopic at a given material point Yt = 2 f~¢zdt increases linearly with R as given by X1 Yt = 41 ~ 2 R
where ~i is the total twist per unit undeformed length of the specimen. In what follows, for simplicity, we will neglect length changes and take XI, X 2 = I. If yp (R) is the plastic shear, and T = f(yp) the strain hardening law of the material, the elastic shear can be expressed as: Ye(R)
Here p is the shear modulus and isotropic linear elastic behaviour is being considered. that Yt equations
Yp + 7e
(ii) Assuming (12)
(I0) to (12) can be combined to give yp(R) + f(yp(R))/U = ~i R.
In metals, Ye is generally very small with respect to yp, so that the following first order approximations can be made: yp(R) = ~i R - f(~iR)/p
Ye(R) = f(~iR)/p.
Note that, as a consequence of the non-linear nature of most work hardening laws, Eq. (14b) does not correspond to a linear dependence of Ye on R. The Angle of Elastic Unloadin$ For materials obeying the above-mentioned laws, the process of unloading can be assumed to be perfectly elastic. Then the elastic shear strain associated with a (reverse) angle of unloading per unit undeformed length ~*, again assuming uniform torsion, can be expressed as:
OF T O R S I O N
Ye - $
It ~should be pointed out that, in agreement with . Eq. (9). above, the ~total shear strain. * * * * . . Yt = YYe = Y-P + Ye - ~ R = (~I - ~ ) R remalns llnear wzth • respect to R durlng unloadlng " . * Nevertheless, from Eq. (15), and because of the form of (14b), It is clear that Ye cannot vary . . . . . . . * . . llnearly wlth R (see Fig. 2). It is the non-llnearlty of Ye whlch was not taken into account zn the publications referred to above (2-7), and which is the source of the contradictory results obtained by these earlier workers. According to Eqs. (14b) and (15), the shear stress acting during unloading is
PYe = f(~l R) - V~
so t h a t the torque during unloading can be expressed as:
= i la T* • 2~r2dr O
-'= 2"~ 0
if length changes are neglected.
For total unloading, T
= 0, which yields
/o f (~IR) R2dR"
I t must now be r e c a l l e d t h a t t h e t o r q u e p r i o r t o u n l o a d i n g i s g i v e n by a
T(O1) ~ 2~ ~ f(~l R) R2dR
so that Eq. (18) can also be formulated as =
2 T(* I) 4
~a The above relation indicates that the current torque T(~I)_ can be deduced from the unloading a n g l e $*, and v i c e v e r s a , as was i n d e e d c l a i m e d by T~th and Kov~cs and c o - w o r k e r s ( 2 - 7 ) . Note, however, that residual shear stresses remain in the bar after unloading (Fig. 2), which can be deduced from Eqs. (16) and (18): a
f($1 R) - ~
f(~l R) R2dR.
It is also p o s s i b l e t o deduce t h e s t r a i n h a r d e n i n g law T = f ( y p ) from Eq. ( 1 8 ) . of the latter with respect to ~I is
(21) The d e r i v a t i v e
d~ d~ I
4 ~ ' ~a 4 f (~iR) R3dR.
Now Eq. (18) can be i n t e g r a t e d by p a r t s l e a d i n g t o a
44 [f(,la) ~ 3 ~a
@i 3 ~ f ' (~I R) R 3 d ~ .
combining Eqs. (22) and (23),
3"~a f(~l a) - 3 d~ 1
OF T O R S I O N
so that, finally, f(~la)
* [ d(~n~*)] ~a~4 3 + ~ j .
(If the fixed end testing condition is relaxed, then the work hardening function f(~1%1 a/%2) is given by the RHS of Eq. (25) multiplied by %1/%2). By combining Eqs. (20) and (25), the well known Nadai relationship f(~la)
d(~nT) ] T [3 + ~ j 2~a 3
is obtained, indicating the consistency of the present analysis. End Effects At the ends of a torsion specimen, where the cross-section may no longer be constant (if shoulders are present), and where arbitrary surface tractions are necessary for the application of the torque, ~O~z/~Z # 0. Thus, according to Eq. (5), the condition ~Or~/~r # 0 also applies. In this region, given the constitutive relations normally observed, the radii no longer remain straight (~/~r # 0), although the nature of the non-linearity is different during loading (curvature of one sign only) and after unloading (curvature of both signs). Somewhat similar considerations apply to Orz when axial stresses are applied or induced, e.g. when a texture is developed during the twisting of metals (Ii). Thus, when ~Ozz/~Z # O, it follows from Eq. (5) that ~Orz/~r # O. In consequence, the transverse planes near the specimen ends take on locally concave or convex configurations, depending on the characteristics of the axial stress gradient. Acknowledgements The authors are indebted to the Natural Sciences and Engineering Research Council of Canada and the Ministry of Education of Quebec (FCAC program) for financial support. One of the authors (FM) acknowledges with gratitude the research leave granted by the Centre National de la Recherche Scientifique and the Centre de Mise en Forme des Mat~riaux, Sophia Antipolis, France and the VisitingProfessorship provided by McGill University. References i. 2. 3. 4. 5. 6. 7. 8. 9. i0. ii.
L.M. Kachanov, 'Foundations of the Theory of Plasticity', North-Holland Publishing Co., Amsterdam, 1971, p. 123. I. Kovacs and E. Nagy, Phys. Stat. Sol., 3, 726 (1963). I. Kovacs and E. Nagy, Phys. Star. Sol., 8, 795 (1965). I. KovEcs, Acta Metall., 15, 1731 (1967).-I. Kovacs, Rev. Def. Beh. Mat., 2, 211 (1977). L.S. Toth and I. Kovacs, J. Mats~ Sc., 17, 43 (1982). L.S. Toth, I. Kovacs, J. Lendvai and B.-~ibert, J. Mats. So., 19, 683 (1984). A.B. LeBon and L.N. de Saint-Martin, 'Microalloying 75', Proc.--~nt. Symp. on HSLA Steels, Union Carbide Corp., New York, 1977, p. 90. J.R. Everett, A. Gittins, G. Glover and M. Toyama, 'Hot Working and Forming Processes', C.M. Sellars and G.J. Davies, eds., Metals Society, London, 1980, p. 16. E. Rauch, G.R. Canova, J.J. Jonas and S.L. Semiatin, An Analysis of Flow Localization during Torsion Testing, submitted to Acta Metall. F. Montheillet, M. Cohen and J.J. Jonas, Axial Stresses and Texture Development during the Torsion Testing of AI, Cu and a-Fe, Acta Metall., in press.
Z,z Fig. I
Undeformed and deformed coordinates
r 0.2 On
Shear stress distribution in an aluminum bar twisted to a shear strain y = 1 at the outer radius and unloaded elastically. The material is assumed to obey the parabolic work hardening law ~(yp) shown in the figure, the elastic unloading stress associated with the angle of untwist ~*a = 4.5 x 10-s radians is labelled T*, and the residual stress is T = T - ~*. Note that T # 0 at res res