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Influence of loading path on the plastic instability strain in anisotropic plane sheets
Journal of Materials Processing Technology 166 (2005) 218–223
Influence of loading path on the plastic instability strain in anisotropic plane sheets J. Chakrabarty, F.K. Chen∗ Department of Mechanical Engineering, National Taiwan University, Taipei, 10764 Taiwan, ROC Received 14 August 2002; received in revised form 28 April 2004; accepted 19 August 2004
Abstract Although the plastic instability condition for sheet metals under biaxial stress states has been considered by several investigators in the past, the effect of variation of the stress ratio that generally occurs under actual loading conditions does not seem to have been examined before. The present investigation is therefore concerned with the onset of instability in which the variation of the stress ratio is included in the theoretical framework, under the assumption that the ratio of the two applied loads is held constant during the straining. The sheet metal is assumed to be isotropic in its plane, obeying the customary quadratic yield criterion and its associated flow rule. The results are presented in graphical forms, which bring out the essentials features of the solution to the instability problem. The estimation of the limiting drawing ratio in the deep drawing of cylindrical cups is finally considered as an application of the derived results on plastic instability in plane sheets. © 2004 Elsevier B.V. All rights reserved. Keywords: Plastic instability; Loading path; Anisotropic plane sheet; Limit drawing ratio
1. Introduction The mechanical failure that occurs in the biaxial stretching of plane sheets, made of ductile material, is generally caused by the onset of plastic instability, which is characterized by the attainment of stationary values of the applied loads. The critical rate of hardening of the material that corresponds to the instability of the sheet is expressed in terms of the final stress ratio that occurs in the material at the onset of instability. For a given material, the final stress ratio evidently depends on the initial stress ratio as well as on the manner in which the loads are varied during the stretching. In the existing solutions to the plastic instability problem [1–5], based on both the quadratic and non-quadratic yields functions, it has been supposed that the stress ratio is maintained constant throughout the loading. Since a constant stress ratio is difficult to achieve in an experimental investigation of the process, the effect of appropriate variations of the stress ratio needs to be considered in a complete solution to the instability problem. ∗
Corresponding author. Tel.: +886 2 3621 522; fax: +886 2 3631 755. E-mail address: [email protected] (F.K. Chen).
0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2004.08.016
In the present investigation, a complete solution is obtained for the plastic instability of a plane sheet, which is biaxially stretched in such a way that the ratio of the loads applied in two orthogonal directions in the plane is held constant during the uniform straining that precedes instability. For simplicity, the sheet metal is assumed to have a state of normal anisotropy, characterized by a uniform strain ratio R, and the material is assumed to yield according to the customary quadratic yield criterion that involves the planar yield stress as well as the R-value. Since the choice of the hypothesis of hardening of the material does not seem to have a significant effect on the magnitudes of the principal strains at the onset of instability, the relatively simple hypothesis of work equivalence is adopted in the present analysis for instability. Analytical expressions are derived for the effective strain and the principal surface strains at instability as functions of parameters representing the initial and final stress ratios, permitting a numerical evaluation of the physical quantities in a straightforward manner. The limiting drawing ratio in the deep drawing process involving cylindrical cups is also examined to illustrate the influence of variable stress ratios.
J. Chakrabarty, F.K. Chen / Journal of Materials Processing Technology 166 (2005) 218–223
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of the subtangent to the effective stress–strain curve of the material. In order to simplify the analytical expressions for the solution, it is convenient to introduce a parameter α, which is defined in terms of the stresses components as α=
1 − (σ2 /σ1 ) σ1 − σ2 = σ1 + σ 2 1 + (σ2 /σ1 )
(7)
Thus a is a measure of the stress ratio σ 2 /σ 1 . The ratios of the principal stresses and strain increments are expressed in terms of α as Fig. 1. A plane sheet loaded biaxially.
σ2 1 − σ dε2 1 − (1 + 2R)α , = = σ1 1 + σ dε1 1 + (1 + 2R)α
2. Plastic instability condition Consider a plane sheet of anisotropic material that is isotropic in its plane and is loaded by tensile forces of intensities P1 and P2 in a pair of mutually perpendicular directions across straight edges of lengths b2 and b1 , respectively, as indicated in Fig. 1. The normal stresses σ 1 and σ 2 in the plane of the sheet are uniformly distributed across these edges, so that the applied loads are given by P1 = hb2 σ1 ,
P2 = hb1 σ2
(1)
where h denotes the uniform plate thickness at any stage prior to the onset of instability. Denoting the current planar yield ¯ and the uniform strain ratio by R, the quadratic stress by σ, yield criterion and its associated flow rule may be written as 2R σ2 − (2) σ1 σ2 + σ2 2 = σ¯ 2 1+R dε1 dε2 d¯ε = = (1 + R)σ1 − Rσ2 (1 + R)σ2 − Rσ1 (1 + R)σ¯
(3)
where d¯ε is the effective strain increment corresponding to the effective strain σ¯ according to the hypothesis of work equivalence. During an incremental deformation, the stress and strain increments must satisfy the relation dσ¯ d¯ε = dσ1 dε1 + dσ2 dε2
where the second result follows from Eq. (3) after the substitution for σ 2 /σ 1 . Eqs. (2) and (3) also furnish √ σ2 1−σ 1 + R(1 + α) = , , (9) σ1 1+σ 2[1 + (1 + 2R)α2 ] and 1 + (1 + 2R)α dε1 = . d¯ε 2(1 + R)[1 + (1 + 2R)α2 ]
(10)
Using Eqs. (8)–(10), the plastic instability condition (6) is easily reduced to 1 dσ¯ 1 + (1 + 2R)(3 + 2R)α2 . =√ 3/2 σ¯ d¯ε 2(1 + R)[1 + (1 + 2R)α2 ]
(11)
This is seen to be a much simpler expression than that obtained in terms of the stress ratio σ 2 /σ 1 . The instability condition Eq. (11) involves the value of α that is finally attained at the onset of instability, depending on the initial value α0 and the prescribed loading path. The plastic instability condition corresponding a non-quadratic yield function can be similarly derived [6].
3. A complete solution
(4)
which is easily verified by substituting dε1 and dε2 from the flow rule (3), and using the differentiated form of the yield criterion (2). Since the loads P1 and P2 attain their maximum values at the onset of instability, we have dσ1 db1 dσ2 db2 = = dε1 , = = dε2 σ1 b1 σ2 b2
(8)
(5)
in view of Eq. (1), and the condition of incompressibility of the plastic material. The substitution from above into Eq. (4) gives 1 dσ¯ σ1 dε1 2 σ2 dε2 2 = + (6) σ¯ d¯ε σ¯ d¯ε σ¯ d¯ε irrespective of the prescribed loading path. It may be noted in passing that the left-hand side of Eq. (6) is the reciprocal
Consider the situation where the stresses are allowed to vary in such a way that the ratio of the two applied loads is maintained constant during the biaxial stretching right up to the point of instability. The ratio σ 2 b1 /σ 1 b2 then has a constant value, and the logarithmic differentiation of this relation gives dε1 − dε2 =
dσ1 dσ2 − . σ1 σ2
(12)
It may be noted that dσ 1 /σ 1 > d1 and dσ 2 /σ 2 > d2 prior to the point of instability. Combining the above relation with Eq. (8) furnishes 1 + (1 + 2R)α 1 − (1 + 2R)α dε1 dε2 = = , . dα (1 + 2R)α(1 − α2 ) dα (1 + 2R)α(1 − α2 ) (13)
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J. Chakrabarty, F.K. Chen / Journal of Materials Processing Technology 166 (2005) 218–223
Using the initial condition α = α0 when ε1 = ε2 =0, the preceding equations are readily integrated to give 1 α 1+α ε1 = ln + R ln 1 + 2R α0 1 + α0 1 − α0 + (1 + R) ln , (14) 1−α and
1 α 1+α ε1 = ln − (1 + R) ln 1 + 2R α0 1 + α0 1 − α0 −R ln , 1−α
(15)
as the total principal strains in the plane of the sheet are functions of α and α0 . In order to obtain the strains at the onset of instability, the final value of α must be determined for a given initial value α0 . Let σ 1 , and σ 2 represent the major and minor principal stresses, respectively, in the plane of the sheet. Then 0 ≤ ␣ ≤ 1, and Eq. (13) indicates that α steadily increases (dα > 0) from its initial value α0 , except for the extreme cases α = 0 and 1, which correspond to dα = 0. From Eqs. (10) and (13), the variation of the effective strain with α is given by √ d¯ε 2(1 + R) 1 + (1 + 2R)α2 = . (16) dα 1 + 2R α(1 − α2 ) Using the substitution z = 1 + (1 + 2R)α2 and c = √ 2(1 + R), the above equation is transformed into the more convenient form d¯ε cz2 = 2 dz (z − 1)(c2 − z2 )
(17)
and the integration of this equation under the initial condition z = z0 , which corresponds to α = α0 , results in c (c + z)(c − z0 ) (z − 1)(z0 + 1) ε¯ = c ln + ln . 2(1 + 2R) (c − z)(c + z0 ) (z + 1)(z0 − 1) (18)
tube under an axial load that varies in strict proportion to the applied internal pressure [7].
4. Application to deep drawing The results of the preceding section are relevant for the prediction of the limiting drawing ratio in the deep drawing of cylindrical cups. When the drawing ratio, which is defined as the ratio of the initial blank radius to the punch radius, exceeds the limiting value, failure occurs by plastic instability in the most critical element, which is located over the punch profile radius at the edge of its contact with the deforming sheet metal. For a work-hardening material, the corresponding punch load attains a maximum, after some drawing has taken place, and the stress ratio existing in the critically stressed element at the instant of failure may be reasonably considered as that corresponding to zero rate of circumferential extension. However, the initial stress ratio in the same element, which has always been stretched around the punch head, must have a value significantly different from that in plane strain. This is an important point that has been generally overlooked in the published work dealing with the estimation of the limiting drawing ratio. It seems quite reasonable to suppose that the stress ratio which finally attains the plane strain value, continuously varies from an initial value that corresponds to a state somewhere between balanced biaxial tension and plane strain. The nature of variation of the stress ratio in the critical element is therefore similar to that considered earlier in this paper for biaxial loading. Consequently, it assumed that the ratio of the circumferential and meridional loads transmitted across the critical element is maintained constant during the straining. This allows the initial stress ratio to be determined, following the results of the preceding analysis, and the final thickness of the element then follows from the associated principal strains. Indeed, if the thickness changes from an initial value h0 to a final value h, then Eqs. (14) and (15) give ln
If the effective stress–strain curve is represented by the Ludwik power law σ¯ = σ0 ε¯ n , where σ 0 and n are empirical constants, then Eq. (11) furnishes 3/2
ε=
nc[1 + (1 + 2R)α2 ] ncz3 = . 1 + (1 + 2R)(3 + 2R)α2 (3 + 2R)z2 − c2
(19)
Eqs. (18) and (19) must be solved simultaneously to obtain α and ε¯ at the onset of instability for any given α0 , R, and n. The corresponding values of the principal surface strains ε1 and ε2 then follow from Eqs. (14) and (15). √ The effective instability strain has a maximum value of n 2(1 + R) at α = 0, and a minimum value of n(l + R)[3/(3 + 2R)]2 corresponding to α2 = (3 + 4R)/(1 + 2R)(3 + 2R). The numerical procedure is similar to that for the instability of a pressurized isotropic
h = −(ε1 + ε2 ) h0 1 α 1+α = −2 ln + ln 1 + 2R α 1 + α0 1−α + ln 1 + α0
(20)
and the expression for the thickness ratio at the onset of instability becomes h = h0
α0 2 (1 − α2 ) α2 (1 − α0 2 )
1/(1+2R) =
2Rc2 α0 2 1 − α0 2
1/(1+2R) (21)
since α = (1 + 2R)−1 at the point of instability which occurs under plane strain. The initial value for this parameter is given
J. Chakrabarty, F.K. Chen / Journal of Materials Processing Technology 166 (2005) 218–223
by the equation z0 − 1 c + z0 c z0 + 1 c − z0 √ 2c √ (1 + R + 1 + 2R) = exp(−nc 1 + 2R) √ 2 (R)2 (c + 1 + 2R)
221
(22)
which is obtained from equating Eqs. (18) and (19), after setting α = (1 + 2R)−1 and noting the fact that √ 2 √ c+z (1 + 1 + 2R) z+1 2 = , = (c + 1 + 2R) . c−z 2R z−1 The maximum drawing load, attained at the onset of instability is P = 2πahσ 1 , where σ 1 , is the meridional stress in the critical element, h its final thickness, and a the radius of the punch stem. Using Eqs. (11), (19) and (21), and the assumed strain-hardening law, we get P σ1 h = 2πah0 σ0 σ 0 h0 1/(1+2R) 1 + R 1+n n 2Rc2 α0 2 n . (23) = √ 1 − α0 2 1 + 2R To obtain the limiting drawing ratio, the maximum load P causing plastic instability over the punch head must be equated to the maximum radial drawing load attained at the die throat, the numerical procedure being similar to that adopted by the previous investigators [8–10]. For sufficiently small values of n, however, a smaller value of the limiting drawing ratio would be predicted on the basis of a tensile neck occurring at the die throat when the maximum load is attained. A numerical solution of Eq. (22) furnishes z0 , and hence α0 , for any given R and n. The limiting drawing ratio then follows from an analysis of the flange drawing process, that furnishes the left-hand side of Eq. (23).
Fig. 2. Variation of strain ratio due to constancy of the associated load ratio.
biaxial stretching process. The principal instability strains, which depend on the initial and final values of the stress ratio according to Eqs. (14) and (15), are plotted in Fig. 3 as function of α0 . The results corresponding to the special case of constant stress ratios, with R = l, are indicated by the broken lines for the sake of visual comparison. The upper three curves including the broken line, shown in Fig. 3, represent the values of ε1 . It can be seen that the strains corresponding to constant load ratios vary with α0 in an oscillatory manner, which is not exhibited by those for constant stress ratios. Further, the values of both the major and minor principal strains are overestimated in varying degrees by the assumption of constancy of stress ratios, except over a limited range defined by a central peak in the solid curves for ε1 . It is also to be noted that the effect of increasing the R-value is to decrease the major principal strain, while increasing the magnitude of the minor principal strain over its negative range of values. The strains are evidently increased in magnitude by increasing the strain-hardening exponent n. The results discussed here are relevant for an experimental investigation of
5. Discussion of results Numerical results based on the present analysis are displayed in Figs. 2 and 3, assuming n = 0.2, and R = 1 and 2. The variation of the strain ratio due to the constancy of the associated load ratio is indicated in Fig. 2, where the transverse lines represent the variation of the effective strain ε¯ with the dimensionless parameter α, according to Eq. (18), for selected values of the initial stress ratio represented by α0 . The longitudinal curves in Fig. 2 are the graphical plots of ε¯ against α at the onset of instability, given by Eq. (19), corresponding to n = 0.2. The points of intersection of these curves with the transverse lines define the instability strains and the final stress ratios, corresponding to the specified values of α0 , R and n. Since the effective strain is not a physical strain parameter, its magnitude at the onset of instability cannot be considered as a measure of the uniform surface strains obtainable in the
Fig. 3. Principal instability strains as function of α0 (the upper three curves represent the values of ε1 ).
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J. Chakrabarty, F.K. Chen / Journal of Materials Processing Technology 166 (2005) 218–223 Table 1 Experimental and theoretical values of LDR and measured R-values Material
R-value
Experimental LDR
Theoretical LDR
Soft aluminum 70/30 Brass Killed steel
0.51 0.86 1.39
2.25 2.40 2.57
2.28 2.42 2.60
fairly close. The minor differences in the values of LDR can be attributed to the presence of small amounts of friction that is unavoidable in the deep drawing process.
6. Conclusions Fig. 4. Limiting drawing ratios vs. the strain hardening component n for different values of R.
the plastic instability condition where a constant stress ratio cannot generally be achieved. The computed results for the limiting drawing ratio are displayed in Fig. 4, as a function of the strain hardening exponent n for different values of R. The curves exhibit the observed independence of the limited drawing ratio on the strain-hardening property of the material, except for relatively low n-values. The inclusion of the variation of the stress ratio in the critical element is seen to have the effect of keeping the magnitude of the limiting drawing ratio at a realistic level, considering the fact that the effects of friction and bending are neglected in the analysis. Since these effects increase the maximum drawing load for a given drawing ratio, Eq. (23) will be satisfied for an appreciably lower value of the drawing ratio. The variation of the stress ratio in the critical element of the drawn cup, in the manner considered in this paper, is therefore consistent with experimental values of the limiting drawing ratio. In the experimental investigation of the limiting drawing ratio, circular blanks of sheet metal made of soft aluminum, 70/30 brass, and killed steel were deep drawn over a flatheaded cylindrical punch with a small profile radius. The stress–strain curve for each material, determined by the simple tension test, indicated that the rate of hardening was large enough for the limiting drawing ratio to be independent of the n-value. In order to achieve efficient lubrication, thin sheets of ptfe were used on both sides of the blank before placing it between the die and blank holder. By varying the blank size for each material, the greatest drawing ratio for which the cup could be drawn without failure was established. The location of the sheet metal failure in each case was found to be very close to the point where the cup wall was in contact with the punch profile radius. This observation lends support to the assumption that the stress ratio in the element that fails by instability lies between those corresponding to plane strain and equal biaxial tension. The experimental and theoretical values of the limiting drawing ratio for the three materials, together with their measured R-values, are listed in Table 1. For the materials tested in this investigation, the agreement between the theoretical and experimental values is seen to be
The condition of plastic instability in sheet metals under variable stress ratios has been investigated in this paper on the assumption that the ratio of the applied biaxial loads is maintained constant during the deformation. The results have been shown to be directly applicable to cup drawing using a flat-headed punch. Assuming that the failure occurs due to plastic instability in an element with a stress ratio that varies continuously from an approximate initial value of unity to a final plane strain value corresponding to the given R-value, the limiting drawing ratio has been determined for a range of values n and R. The results appear to be more realistic than those based on the usual assumption of instability occurring under a constant stress ratio corresponding to plane strain. The constancy of stress ratio introduces a constraint in the instability condition, and leads to a significant overestimation of the limiting drawing ratio for moderate to high R-value materials. The recognition of the fact that the stress ratio in the critical element must vary during the drawing process makes the analysis more realistic, and this is borne out by the estimated values of the limiting drawing ratio. Although friction has been neglected in the present analysis, its effect can easily be taken into account by introducing a suitable friction factor in an empirical manner.
Acknowledgement The authors wish to thank the National Science Council of the Republic of China for the financial support under the project # NSC 89-2212-E-002-005.
References [1] H. Swift, Plastic instability under plane stress, J. Mech. Phys. Solids 1 (1952) 1. [2] G.G. Moore, J.F. Wallace, The effect of anisotropy on instability in sheet metal forming, J. Inst. Metall. 93 (1964) 33. [3] Y. Yamada, I. Aoki, On the tensile plastic instability in the axisymmetric deformation of sheet metal, Int. J. Mech. Sci. 8 (1966) 665. [4] M.G. Stout, S.S. Hecker, Role of geometry in plastic instability and fracture of tubes and sheets, Mech. Mater. 2 (1983) 23.
J. Chakrabarty, F.K. Chen / Journal of Materials Processing Technology 166 (2005) 218–223 [5] J. Chakrabarty, Plastic instability and limit strains in anisotropic sheet metal, in: Proceedings of the 32 International MATADOR Conference, Macmillan Press, 1997, p. 373. [6] J. Chakrabarty, Applied Plasticity, Springer-Verlag, New York, 2000. [7] J. Chakrabarty, Theory of Plasticity, second ed., McGraw-Hill Book Company, Singapore, 1998. [8] D.C. Chiang, S. Kobayashi, The effect of anisotropy and work-hardening characteristics on the stress and strain distribu-
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tion in deep drawing, J. Eng. Ind., Trans. ASME 88 (1966) 445. [9] B. Budiansky, N.M. Wang, On the swift cup test, J. Mech. Phys. Solids 14 (1966) 357. [10] E.L. Sebaie, P.B. Mellor, Plastic instability conditions in deep drawing of a blank of sheet metal, Int. J. Mech. Sci. 14 (1972) 535.
Influence of loading path on the plastic instability strain in anisotropic plane sheets J. Chakrabarty, F.K. Chen∗ Department of Mechanical Engineering, National Taiwan University, Taipei, 10764 Taiwan, ROC Received 14 August 2002; received in revised form 28 April 2004; accepted 19 August 2004
Abstract Although the plastic instability condition for sheet metals under biaxial stress states has been considered by several investigators in the past, the effect of variation of the stress ratio that generally occurs under actual loading conditions does not seem to have been examined before. The present investigation is therefore concerned with the onset of instability in which the variation of the stress ratio is included in the theoretical framework, under the assumption that the ratio of the two applied loads is held constant during the straining. The sheet metal is assumed to be isotropic in its plane, obeying the customary quadratic yield criterion and its associated flow rule. The results are presented in graphical forms, which bring out the essentials features of the solution to the instability problem. The estimation of the limiting drawing ratio in the deep drawing of cylindrical cups is finally considered as an application of the derived results on plastic instability in plane sheets. © 2004 Elsevier B.V. All rights reserved. Keywords: Plastic instability; Loading path; Anisotropic plane sheet; Limit drawing ratio
1. Introduction The mechanical failure that occurs in the biaxial stretching of plane sheets, made of ductile material, is generally caused by the onset of plastic instability, which is characterized by the attainment of stationary values of the applied loads. The critical rate of hardening of the material that corresponds to the instability of the sheet is expressed in terms of the final stress ratio that occurs in the material at the onset of instability. For a given material, the final stress ratio evidently depends on the initial stress ratio as well as on the manner in which the loads are varied during the stretching. In the existing solutions to the plastic instability problem [1–5], based on both the quadratic and non-quadratic yields functions, it has been supposed that the stress ratio is maintained constant throughout the loading. Since a constant stress ratio is difficult to achieve in an experimental investigation of the process, the effect of appropriate variations of the stress ratio needs to be considered in a complete solution to the instability problem. ∗
Corresponding author. Tel.: +886 2 3621 522; fax: +886 2 3631 755. E-mail address: [email protected] (F.K. Chen).
0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2004.08.016
In the present investigation, a complete solution is obtained for the plastic instability of a plane sheet, which is biaxially stretched in such a way that the ratio of the loads applied in two orthogonal directions in the plane is held constant during the uniform straining that precedes instability. For simplicity, the sheet metal is assumed to have a state of normal anisotropy, characterized by a uniform strain ratio R, and the material is assumed to yield according to the customary quadratic yield criterion that involves the planar yield stress as well as the R-value. Since the choice of the hypothesis of hardening of the material does not seem to have a significant effect on the magnitudes of the principal strains at the onset of instability, the relatively simple hypothesis of work equivalence is adopted in the present analysis for instability. Analytical expressions are derived for the effective strain and the principal surface strains at instability as functions of parameters representing the initial and final stress ratios, permitting a numerical evaluation of the physical quantities in a straightforward manner. The limiting drawing ratio in the deep drawing process involving cylindrical cups is also examined to illustrate the influence of variable stress ratios.
J. Chakrabarty, F.K. Chen / Journal of Materials Processing Technology 166 (2005) 218–223
219
of the subtangent to the effective stress–strain curve of the material. In order to simplify the analytical expressions for the solution, it is convenient to introduce a parameter α, which is defined in terms of the stresses components as α=
1 − (σ2 /σ1 ) σ1 − σ2 = σ1 + σ 2 1 + (σ2 /σ1 )
(7)
Thus a is a measure of the stress ratio σ 2 /σ 1 . The ratios of the principal stresses and strain increments are expressed in terms of α as Fig. 1. A plane sheet loaded biaxially.
σ2 1 − σ dε2 1 − (1 + 2R)α , = = σ1 1 + σ dε1 1 + (1 + 2R)α
2. Plastic instability condition Consider a plane sheet of anisotropic material that is isotropic in its plane and is loaded by tensile forces of intensities P1 and P2 in a pair of mutually perpendicular directions across straight edges of lengths b2 and b1 , respectively, as indicated in Fig. 1. The normal stresses σ 1 and σ 2 in the plane of the sheet are uniformly distributed across these edges, so that the applied loads are given by P1 = hb2 σ1 ,
P2 = hb1 σ2
(1)
where h denotes the uniform plate thickness at any stage prior to the onset of instability. Denoting the current planar yield ¯ and the uniform strain ratio by R, the quadratic stress by σ, yield criterion and its associated flow rule may be written as 2R σ2 − (2) σ1 σ2 + σ2 2 = σ¯ 2 1+R dε1 dε2 d¯ε = = (1 + R)σ1 − Rσ2 (1 + R)σ2 − Rσ1 (1 + R)σ¯
(3)
where d¯ε is the effective strain increment corresponding to the effective strain σ¯ according to the hypothesis of work equivalence. During an incremental deformation, the stress and strain increments must satisfy the relation dσ¯ d¯ε = dσ1 dε1 + dσ2 dε2
where the second result follows from Eq. (3) after the substitution for σ 2 /σ 1 . Eqs. (2) and (3) also furnish √ σ2 1−σ 1 + R(1 + α) = , , (9) σ1 1+σ 2[1 + (1 + 2R)α2 ] and 1 + (1 + 2R)α dε1 = . d¯ε 2(1 + R)[1 + (1 + 2R)α2 ]
(10)
Using Eqs. (8)–(10), the plastic instability condition (6) is easily reduced to 1 dσ¯ 1 + (1 + 2R)(3 + 2R)α2 . =√ 3/2 σ¯ d¯ε 2(1 + R)[1 + (1 + 2R)α2 ]
(11)
This is seen to be a much simpler expression than that obtained in terms of the stress ratio σ 2 /σ 1 . The instability condition Eq. (11) involves the value of α that is finally attained at the onset of instability, depending on the initial value α0 and the prescribed loading path. The plastic instability condition corresponding a non-quadratic yield function can be similarly derived [6].
3. A complete solution
(4)
which is easily verified by substituting dε1 and dε2 from the flow rule (3), and using the differentiated form of the yield criterion (2). Since the loads P1 and P2 attain their maximum values at the onset of instability, we have dσ1 db1 dσ2 db2 = = dε1 , = = dε2 σ1 b1 σ2 b2
(8)
(5)
in view of Eq. (1), and the condition of incompressibility of the plastic material. The substitution from above into Eq. (4) gives 1 dσ¯ σ1 dε1 2 σ2 dε2 2 = + (6) σ¯ d¯ε σ¯ d¯ε σ¯ d¯ε irrespective of the prescribed loading path. It may be noted in passing that the left-hand side of Eq. (6) is the reciprocal
Consider the situation where the stresses are allowed to vary in such a way that the ratio of the two applied loads is maintained constant during the biaxial stretching right up to the point of instability. The ratio σ 2 b1 /σ 1 b2 then has a constant value, and the logarithmic differentiation of this relation gives dε1 − dε2 =
dσ1 dσ2 − . σ1 σ2
(12)
It may be noted that dσ 1 /σ 1 > d1 and dσ 2 /σ 2 > d2 prior to the point of instability. Combining the above relation with Eq. (8) furnishes 1 + (1 + 2R)α 1 − (1 + 2R)α dε1 dε2 = = , . dα (1 + 2R)α(1 − α2 ) dα (1 + 2R)α(1 − α2 ) (13)
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Using the initial condition α = α0 when ε1 = ε2 =0, the preceding equations are readily integrated to give 1 α 1+α ε1 = ln + R ln 1 + 2R α0 1 + α0 1 − α0 + (1 + R) ln , (14) 1−α and
1 α 1+α ε1 = ln − (1 + R) ln 1 + 2R α0 1 + α0 1 − α0 −R ln , 1−α
(15)
as the total principal strains in the plane of the sheet are functions of α and α0 . In order to obtain the strains at the onset of instability, the final value of α must be determined for a given initial value α0 . Let σ 1 , and σ 2 represent the major and minor principal stresses, respectively, in the plane of the sheet. Then 0 ≤ ␣ ≤ 1, and Eq. (13) indicates that α steadily increases (dα > 0) from its initial value α0 , except for the extreme cases α = 0 and 1, which correspond to dα = 0. From Eqs. (10) and (13), the variation of the effective strain with α is given by √ d¯ε 2(1 + R) 1 + (1 + 2R)α2 = . (16) dα 1 + 2R α(1 − α2 ) Using the substitution z = 1 + (1 + 2R)α2 and c = √ 2(1 + R), the above equation is transformed into the more convenient form d¯ε cz2 = 2 dz (z − 1)(c2 − z2 )
(17)
and the integration of this equation under the initial condition z = z0 , which corresponds to α = α0 , results in c (c + z)(c − z0 ) (z − 1)(z0 + 1) ε¯ = c ln + ln . 2(1 + 2R) (c − z)(c + z0 ) (z + 1)(z0 − 1) (18)
tube under an axial load that varies in strict proportion to the applied internal pressure [7].
4. Application to deep drawing The results of the preceding section are relevant for the prediction of the limiting drawing ratio in the deep drawing of cylindrical cups. When the drawing ratio, which is defined as the ratio of the initial blank radius to the punch radius, exceeds the limiting value, failure occurs by plastic instability in the most critical element, which is located over the punch profile radius at the edge of its contact with the deforming sheet metal. For a work-hardening material, the corresponding punch load attains a maximum, after some drawing has taken place, and the stress ratio existing in the critically stressed element at the instant of failure may be reasonably considered as that corresponding to zero rate of circumferential extension. However, the initial stress ratio in the same element, which has always been stretched around the punch head, must have a value significantly different from that in plane strain. This is an important point that has been generally overlooked in the published work dealing with the estimation of the limiting drawing ratio. It seems quite reasonable to suppose that the stress ratio which finally attains the plane strain value, continuously varies from an initial value that corresponds to a state somewhere between balanced biaxial tension and plane strain. The nature of variation of the stress ratio in the critical element is therefore similar to that considered earlier in this paper for biaxial loading. Consequently, it assumed that the ratio of the circumferential and meridional loads transmitted across the critical element is maintained constant during the straining. This allows the initial stress ratio to be determined, following the results of the preceding analysis, and the final thickness of the element then follows from the associated principal strains. Indeed, if the thickness changes from an initial value h0 to a final value h, then Eqs. (14) and (15) give ln
If the effective stress–strain curve is represented by the Ludwik power law σ¯ = σ0 ε¯ n , where σ 0 and n are empirical constants, then Eq. (11) furnishes 3/2
ε=
nc[1 + (1 + 2R)α2 ] ncz3 = . 1 + (1 + 2R)(3 + 2R)α2 (3 + 2R)z2 − c2
(19)
Eqs. (18) and (19) must be solved simultaneously to obtain α and ε¯ at the onset of instability for any given α0 , R, and n. The corresponding values of the principal surface strains ε1 and ε2 then follow from Eqs. (14) and (15). √ The effective instability strain has a maximum value of n 2(1 + R) at α = 0, and a minimum value of n(l + R)[3/(3 + 2R)]2 corresponding to α2 = (3 + 4R)/(1 + 2R)(3 + 2R). The numerical procedure is similar to that for the instability of a pressurized isotropic
h = −(ε1 + ε2 ) h0 1 α 1+α = −2 ln + ln 1 + 2R α 1 + α0 1−α + ln 1 + α0
(20)
and the expression for the thickness ratio at the onset of instability becomes h = h0
α0 2 (1 − α2 ) α2 (1 − α0 2 )
1/(1+2R) =
2Rc2 α0 2 1 − α0 2
1/(1+2R) (21)
since α = (1 + 2R)−1 at the point of instability which occurs under plane strain. The initial value for this parameter is given
J. Chakrabarty, F.K. Chen / Journal of Materials Processing Technology 166 (2005) 218–223
by the equation z0 − 1 c + z0 c z0 + 1 c − z0 √ 2c √ (1 + R + 1 + 2R) = exp(−nc 1 + 2R) √ 2 (R)2 (c + 1 + 2R)
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(22)
which is obtained from equating Eqs. (18) and (19), after setting α = (1 + 2R)−1 and noting the fact that √ 2 √ c+z (1 + 1 + 2R) z+1 2 = , = (c + 1 + 2R) . c−z 2R z−1 The maximum drawing load, attained at the onset of instability is P = 2πahσ 1 , where σ 1 , is the meridional stress in the critical element, h its final thickness, and a the radius of the punch stem. Using Eqs. (11), (19) and (21), and the assumed strain-hardening law, we get P σ1 h = 2πah0 σ0 σ 0 h0 1/(1+2R) 1 + R 1+n n 2Rc2 α0 2 n . (23) = √ 1 − α0 2 1 + 2R To obtain the limiting drawing ratio, the maximum load P causing plastic instability over the punch head must be equated to the maximum radial drawing load attained at the die throat, the numerical procedure being similar to that adopted by the previous investigators [8–10]. For sufficiently small values of n, however, a smaller value of the limiting drawing ratio would be predicted on the basis of a tensile neck occurring at the die throat when the maximum load is attained. A numerical solution of Eq. (22) furnishes z0 , and hence α0 , for any given R and n. The limiting drawing ratio then follows from an analysis of the flange drawing process, that furnishes the left-hand side of Eq. (23).
Fig. 2. Variation of strain ratio due to constancy of the associated load ratio.
biaxial stretching process. The principal instability strains, which depend on the initial and final values of the stress ratio according to Eqs. (14) and (15), are plotted in Fig. 3 as function of α0 . The results corresponding to the special case of constant stress ratios, with R = l, are indicated by the broken lines for the sake of visual comparison. The upper three curves including the broken line, shown in Fig. 3, represent the values of ε1 . It can be seen that the strains corresponding to constant load ratios vary with α0 in an oscillatory manner, which is not exhibited by those for constant stress ratios. Further, the values of both the major and minor principal strains are overestimated in varying degrees by the assumption of constancy of stress ratios, except over a limited range defined by a central peak in the solid curves for ε1 . It is also to be noted that the effect of increasing the R-value is to decrease the major principal strain, while increasing the magnitude of the minor principal strain over its negative range of values. The strains are evidently increased in magnitude by increasing the strain-hardening exponent n. The results discussed here are relevant for an experimental investigation of
5. Discussion of results Numerical results based on the present analysis are displayed in Figs. 2 and 3, assuming n = 0.2, and R = 1 and 2. The variation of the strain ratio due to the constancy of the associated load ratio is indicated in Fig. 2, where the transverse lines represent the variation of the effective strain ε¯ with the dimensionless parameter α, according to Eq. (18), for selected values of the initial stress ratio represented by α0 . The longitudinal curves in Fig. 2 are the graphical plots of ε¯ against α at the onset of instability, given by Eq. (19), corresponding to n = 0.2. The points of intersection of these curves with the transverse lines define the instability strains and the final stress ratios, corresponding to the specified values of α0 , R and n. Since the effective strain is not a physical strain parameter, its magnitude at the onset of instability cannot be considered as a measure of the uniform surface strains obtainable in the
Fig. 3. Principal instability strains as function of α0 (the upper three curves represent the values of ε1 ).
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J. Chakrabarty, F.K. Chen / Journal of Materials Processing Technology 166 (2005) 218–223 Table 1 Experimental and theoretical values of LDR and measured R-values Material
R-value
Experimental LDR
Theoretical LDR
Soft aluminum 70/30 Brass Killed steel
0.51 0.86 1.39
2.25 2.40 2.57
2.28 2.42 2.60
fairly close. The minor differences in the values of LDR can be attributed to the presence of small amounts of friction that is unavoidable in the deep drawing process.
6. Conclusions Fig. 4. Limiting drawing ratios vs. the strain hardening component n for different values of R.
the plastic instability condition where a constant stress ratio cannot generally be achieved. The computed results for the limiting drawing ratio are displayed in Fig. 4, as a function of the strain hardening exponent n for different values of R. The curves exhibit the observed independence of the limited drawing ratio on the strain-hardening property of the material, except for relatively low n-values. The inclusion of the variation of the stress ratio in the critical element is seen to have the effect of keeping the magnitude of the limiting drawing ratio at a realistic level, considering the fact that the effects of friction and bending are neglected in the analysis. Since these effects increase the maximum drawing load for a given drawing ratio, Eq. (23) will be satisfied for an appreciably lower value of the drawing ratio. The variation of the stress ratio in the critical element of the drawn cup, in the manner considered in this paper, is therefore consistent with experimental values of the limiting drawing ratio. In the experimental investigation of the limiting drawing ratio, circular blanks of sheet metal made of soft aluminum, 70/30 brass, and killed steel were deep drawn over a flatheaded cylindrical punch with a small profile radius. The stress–strain curve for each material, determined by the simple tension test, indicated that the rate of hardening was large enough for the limiting drawing ratio to be independent of the n-value. In order to achieve efficient lubrication, thin sheets of ptfe were used on both sides of the blank before placing it between the die and blank holder. By varying the blank size for each material, the greatest drawing ratio for which the cup could be drawn without failure was established. The location of the sheet metal failure in each case was found to be very close to the point where the cup wall was in contact with the punch profile radius. This observation lends support to the assumption that the stress ratio in the element that fails by instability lies between those corresponding to plane strain and equal biaxial tension. The experimental and theoretical values of the limiting drawing ratio for the three materials, together with their measured R-values, are listed in Table 1. For the materials tested in this investigation, the agreement between the theoretical and experimental values is seen to be
The condition of plastic instability in sheet metals under variable stress ratios has been investigated in this paper on the assumption that the ratio of the applied biaxial loads is maintained constant during the deformation. The results have been shown to be directly applicable to cup drawing using a flat-headed punch. Assuming that the failure occurs due to plastic instability in an element with a stress ratio that varies continuously from an approximate initial value of unity to a final plane strain value corresponding to the given R-value, the limiting drawing ratio has been determined for a range of values n and R. The results appear to be more realistic than those based on the usual assumption of instability occurring under a constant stress ratio corresponding to plane strain. The constancy of stress ratio introduces a constraint in the instability condition, and leads to a significant overestimation of the limiting drawing ratio for moderate to high R-value materials. The recognition of the fact that the stress ratio in the critical element must vary during the drawing process makes the analysis more realistic, and this is borne out by the estimated values of the limiting drawing ratio. Although friction has been neglected in the present analysis, its effect can easily be taken into account by introducing a suitable friction factor in an empirical manner.
Acknowledgement The authors wish to thank the National Science Council of the Republic of China for the financial support under the project # NSC 89-2212-E-002-005.
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J. Chakrabarty, F.K. Chen / Journal of Materials Processing Technology 166 (2005) 218–223 [5] J. Chakrabarty, Plastic instability and limit strains in anisotropic sheet metal, in: Proceedings of the 32 International MATADOR Conference, Macmillan Press, 1997, p. 373. [6] J. Chakrabarty, Applied Plasticity, Springer-Verlag, New York, 2000. [7] J. Chakrabarty, Theory of Plasticity, second ed., McGraw-Hill Book Company, Singapore, 1998. [8] D.C. Chiang, S. Kobayashi, The effect of anisotropy and work-hardening characteristics on the stress and strain distribu-
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