Distorted Yield Loci

CHAPTER 3 YIELD CRITERIA

3.2 Yielding of Ductile Isotropic Materials

3.4.3 Distorted Yield Loci

It is apparent that, within this elliptical function, the five yield points may be fitted exactly with an equal number of coefficients in L and Q. Comparisons of this kind are best made at the limit of proportionality since an equivalent, anisotropic, offset strain cannot be defined. Moreover, if yield were to be so defined, anisotropy, initially present in the yield locus, would diminish with increasing offset strain.

To recover an initial Mises condition from eq(3.3O), we identify H

⁹

and H

^m

with the corresponding isotropie tensors:

Hu=6tJ (3.32)

and, for the general fourth-order isotropie tensor [57]:

H

^m

= l

^m

= HA + M44* +

^v

4 A (3.33a) Substituting eqs(3.32) and (3.33a) into eq(3.30) gives

= 05/ + ¥1 (Aaa' % ' + (M^' Oy + v ffs' Ojl)

= tt ( p <^'<V + v <%'«%')

where o~ = u

^tt

' = 0. Putting p.= v =¥1 leads to the von Mises form

/(<%') = Vk oj/ «%'

It follows that, by setting A — 0 and /* = v = Vi in eq(3.33a), a reduced form of fourth order, isotropie tensor is suited to stress deviators:

7^ = ^(44,+44*) (333b)

since this will lead directly to the von Mises condition. Note that eq(3.30) describes an

ellipsoidal yield surface and will not account for any distortion that may initially be present.

Qs 0-n tfa +

YIELD CRITERIA 89

T

6

er

n

a

a2

+ T

7

a

n2

a

a

= I (3.35a)

Figures 3.11e-h apply to eq(3.35a). These contrast with the quadratic predictions (Kg- 3.1 la-d) for four similar cases of anisotropy described previously. Clearly distortion is now a consisitent feature within each case. Equation (3.34) can be written in a dimensionless form when principal, biaxial stress directions are aligned with orthotropic axes 1 and 2:

(3.35b) The coefficients Q and Tin eq(3.35a,b) are derived from H^ and Hi]Vmn in [48]. Equations (3.30) and (3.35b) are applied to an initial yield locus for extruded magnesium in Fig. 3.13.

Figure 3.13 Yield locus for an orthotropie magnesium extrusion

The 5 constants, L and Q in eq(3.30), are determined directly from the five experimental yield points shown. The determination of seven constants in eq(3.35b) requires an additional yield point (interpolated) and a strain vector direction as indicated. It is seen that both predictions represent the pronounced Bauschinger effect along each axis of orthotropy.

Both quadratic and cubic yield functions appear to describe the measured initial yield points equally well. However, eq(3.35b) provides the better account of distortion appearing in a locus connecting these yield points. Such distortion is linked to the direction of the plastic strain increment vector through the normality rule, i.e. this vector is aligned with the direction of the exterior normal to the yield locus. It appears from Fig. 3.13 that additional yield points would be required to test an initial distortion prediction more precisely. In general, the literature reveals far stronger evidence for a strain-induced form of distortion within a subsequent yield locus than there is for distortion within the initial yield surface.

When it is required to recover a condition of isotropy from eq(3.34), HiM is defined from eq(3.33b) and tf9ifcw appears as a general sixth order isotropic tensor [23]:

4 4 4

⁺

4 A 4 ,

⁺

4»4*4

+ 4*4*4 + 44,4* + 4 4 4 * + 4 , 4 4 + 4™44, + 44*4, + 4 4 4 . ) C

3

-

36

)

Substituting from eq(3.36), reduces the second term in eq(3.34) to the third deviatorie invariant: J3' = %Oj' o£ OjJ, when a = b = 0 and c = %. It fallows that the first two terms in eq(3.36) are unnecessary for the recovery of this isotropic stress deviator.

3.5 Choice of Yield Function

It has been seen that the classical, von Mises and Tresca themes of yielding for metals may each be formulated as a function of the stress deviator invariants. Such macroscopic predictions to the initial yield surface provide for all possible stress combinations. However, these do not offer information about the microstructural mechanisms of yielding, discussed in Chapter 8, In consideration of subsequent yielding, where plastic strain paths, calculated from eF = eT - (ME (see Fig. 1.1), remain linear, this indicates that a simple rale of isotropic hardening may be applied for loading within the plastic range. This employs the concept of an expanding yield surface which retains its initial shape and orientation, to which we shall return in Chapter 9 and 10.

It has been seen that the departure from the isotropic function/^1) may either be due to the influence of J3' (for an isotropic material), or to the presence of initial anisotropy.

Clearly, when selecting an appropriate yield function, checks are necessary to establish precisely the initial condition of the material. Anisotropic yielding and flow behaviour may be identified with distinct differences in the stress-sixain curves obtained from testpieces machinal from different directions in the material. For rolled sheets, the off-axis tensile test will reveal directional differences between yield stresses when anisotropy is present, but the more usual measure of anisotropy for sheet metals is the r value. This is the ratio between the plastic components of an incremental width and thickness strain in a tension test.

Anisotropy is revealed when r*\. The formability of sheet metal is enhanced when r > 1.

Bramley and Mellor [59] showed, from eq(3.27), how it was possible to describe the effects of initial anisotropy in transversely isotropic sheets using the constant gradients of linear plastic strain paths (see Chapter 11).

Alternatively, when isotropy is assumed in a tension test the material may be taken to conform to the general isotropic function/(/j', J3% Substituting this for/into eq(3.22) gives the plastic strain increment tensor;

/=<u [ (df/d^wwd^ + of/ajj-xa/j'/a**)] (3.37)

If Ci is the non-zero tensile stress, it follows from eqs(3.10b and c) that BJ^/Sai = 2c,/3, dj^doi = - oj/3, 3/3'/3oi = 2o|2/9 and a/3'/5c% = - a?B. Then, from eq(3.37), the axial and lateral plastic strain increments become

d£t p = d i [C3//aij'X2ffi/3) + (9//aj3')(2oi2/ 9)] (3.38a) dsf = d i [(df/8Jt')(- OJ/3) + {9//5/j'){- Oi2/ 9)] (3.38b) Dividing eqs(3.38a and b) gives the constant ratio d ^ / d s / —~Vt, irrespective of the yield function/(/2', J3'). That is, the gradient of the lateral versus axial plastic strain plot remains linear, with a gradient of - J4. Figures 3.14a-e compares this isotropic prediction with experimental plastic strain paths for aluminium, steel, copper and brass.

YIELD CRITERIA 91

0

- 4

»)

(a$ Alum

4

i i

„»

inium El 8

8

1 1

• s .

?•'(%)

- 4

i i r

WAl-Alloy HE30TF

- 4

- 8

12

(d) Capper C l O !

«/(*)

- 4

•»« Necking

Steel EN3B

- 8

8 12

i l i i r i i

""""•^ ^ y W Brass C 2121 \ *

F ^ u r e 3.14 Lateral versus axial plastic strain pattis under tension

The agreement found between the gradient of each plot and the theoretical gradients value of - % confirms that testpieces machined longitudinally from extruded bars become almost isotropic following heat treatment. In contrast, the interstage anneals employed for rolling sheet metal may not leave the material in an isotropic condition. It will be seen later that it is desirable to retain anisotropy arising from rolling when it enhances formability.

References

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2. von Mises R. Nachr. Ges. Wiss. Gottingen, 1913,582. (also ZAMM 1928,8,161) 3. Tresca H. "Memoire sur reeoulement des corps solides", Memoirs Par Divers Savants,

Paris, 1968, 18,733 and 1972,20,75.

4. Ikegami K. J. Sac. Mat. Sci. 1975, 24(261), 491 and 24(263), 709 (see also BISI translation 14420, The Metals Soc, Sept 1976)

5. Michno M. J. and Findley W. N. Int. J. Non-Linear Mech., 1976,11,59.

6. Banabic D. et al. Proceedings: 4th ESAFORM Cmf on Material Forming, (Ed.

Habraken A. M. ) Vol. 1, p. 297, University of Liege, Belgium, April 2001.

7. Shahabi S. N. and Shelton A. J. Mech. Eng. Sci., 1975,17,82.

8. Johnson K. R. and Sidebottom O. M. Expl Mech., 1972,12,264.

9. Moreton D. N., Moffat, D. G. and Parkinson, D. B. /. Strain Anal, 1981,16,127.

10. MiastkowsM J and Szezepinski W. Int. J. Solids and Structures, 1965,1,189.

11. Marin J., Hu L. W. and Hamburg I. F. Proc. A.S.M., 1953,45,686.

12. Lessels J. M. and MacGregor, C. W. M Franklin Inst, 1940,230,163.

13. Rogan J. and Shelton A. /. Strain Anal, 1969,4,127.

14. Ivey H. J. / . Mech. Eng. ScL, 1961,3,15.

15. Phillips A. and Tang J-L. Int. J. Solids and Struct,, 1972,8,463.

16. KowalewsM Z. L., Dietrich L. and Socha G. Proeeedmp: Third Int Congress on Thermal Stress, 1999,181-184, University of Kracow.

17. Shiratori R , Deegami K. and Kaneko K. Trans Japan Soc. Mech. Engrs, 1973,39,458.

18. Ellyin F. and Grass J-P. Trans Can. Soc for Mech, Engrs, 1975,3,156.

19. Taylor G. I. and Quinney, H. Phil Trans Roy Soc A, 1931,230,323.

20. Shrivastava H. P., Mroz, Z. and Dubey, R. N. ZAMM, 1973,53,625.

21. Rees D. W. A. Proc. R. Soc. Land., 1982, A383,333.

22. Dracker D. C. J. Appl. Mech., 1949,16,349.

23. Betten J. Acta Mechanica, 1976,25,79.

24. Freudenthall A. M. and Gou P. F. Acta Mechanica, 1969,8,34.

25. Rees D. W. A. J. Applied Mech., 1982,49,663.

26. Grassland B. Proc. I Mech. E., 1954,168,935.

27. Hu L. W. Proc: Naval Structures, (eds Lee E. H. and Symonds P. S.) 1960, 924, Pergamon, Oxford, U.K.

28. Pugh H. LI. D. and Green D. Proc, I. Meek E, 1964,179,1.

29. Ratner S. I. Zeitschriftfur Technische Physik, 1949,19,408.

30. Fung P. K , Burns D. J. and Lind N. C. Proc: Foundations of Plasticity, (ed AJawczuk) 1973,287, Noordhoff, Warsaw.

31. Ros M. and Eichinger A. Eidgenoess, Materialpruef. Versuchanstalt Ind. Bauw.

Gewerbe, Zurich, 1929,34,1.

32. Brandes M. in: The Mechanical Behaviour of Materials Under Pressure, (ed Pugh H.

LI.) 1971, Ch. 6,236, Applied Science, London.

33. Stassi-D'Alia F. Meccanica, 1967,2,178.

34. Hill R. Proc. Roy. Soc, 1948, A193,281.

35. Edelman F. and Dracker D. C. J. Franklin Inst., 1951,251,581.

36. Hearmon R. F. S. Applied Anisotropic Elastieity, Clarendon Press, Oxford 1961.

37. Olsak W. and Urbanowski W. Arch of Mech., 1956,8,671.

38. Sobotka, Z. ZAMM, 1969,49,25.

39. Troost A, and Betten J. Mech. Res. Comms, 1974,1,73.

40. Fava F. J. Annals of the C.LR.P, 1967,15,411.

41. Hu L. W. J. Appl. Mech., 1956,23,444.

42. Hazlett T. H., Robinson A. T. and Dom J. E. Trans ASME, 1950,42,1326.

43. Jones S. E. and Gillis P. P. Met Trans A, 1984, ISA, 129.

44. Hu L. W. and Marin J. J. Appl. Mech., 1955,22,77.

45. Takeda T. and Nasu Y. /. Strain Analysis, 1991,26,47.

46. Hill R. Math. Proc. Camb. Phil Soc, 1979, 85,179.

47. Bourne L. and Hill R. Phil. Mag., 1950,41,671.

48. Gotoh M. Int. J. Mech. ScL, 1977,19,505.

49. Stassi-D'Alia F. Meccanica, 1969,4,349.

50. Harvey S. J., Adkin P. and Jeans P. J. Fatigue of Eng Mats and Struct., 1983,6, 89.

51. DoddB. Int J. Mech. Sci., 1984,12,587.

52. Parmar A. and Mellor P. B. Int. J. Mech. ScL, 1978,20, 385.

53. Bailey R. W. Proc. I. Mech. E., 1935,131,131.

YIELD CRHBRIA 93

54. Davis B. A. Trans ASME, 1961,28E, 310.

55. Rees D. W. A. Acta Mechanica, 1982,43,223.

56. Lee D. and Backofen W. A. Trans Met Sac. AIME, 1966,236,1077.

57. Spencer A. J, M. Continuum Mechanics, 1980, Longman, London.

58. Rees D. W. A. Acta Meckanica, 1984,52,15.

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Exercises

3.1 Show that the constants c» b and d in eqs(3.15a,b and c) are defined from the relationship between F and* as follows: (i) F = i ( l / 2 7 - 4C/95)1*, (ii) F=9*/(27 - 4b)m and (iii) F = i k / ( l / 33 a- 2<#27)w. 3.2 Normalise the Stassi fracture criterion (3.2Sb) with the compressive fracture stress ac and plot the family of loci in OJ, a^ space for p = oja, = 2,3 and » .

3 3 Construct the family of yield loci in a,v space from the unsymmetrical yield function:

within the range 1.5 <. p s - 3.

3 4 Construct a family of yield loci for the unsymmetrical yield function (3.16) for n = 2. Determine the range of j» values that will ensure a closed yield surface.

3.5 Establish the right-hand sides of the expressions (iii) and (iv) given in Table 3.3 in terms of the known yield stress o^ along the principal, 1- axis, of orthotropy.

3.6 Compare the yield criteria (i) - (viii), listed in Table 3.3, when they are reduced to a principal biaxial stress space, i.e. for «, = 0 and r9 = rM = TM = 0, as appropriate,

3.7 Derive from eq(3,22) the principal, plastic constitutive relations corresponding to each of the isotropic yield functions given in eqs(3.15a -c).

3.8 The coefficients Lh L^, Qt, Q2 and ft in eq(3.31c) can account for four cases in which the tensile and compie&sive yield stresses in the 1 and 2 directions may be the same or different under a principal biaxial stress state a{, 05. Show each case with a plot similar to Fig. 3.11a-d.

3.9 Show how the coefficients Tlt T2, T,, r4, Qu fi2 and Q5 in eq(3.35b) can account for several conditions of anisotropy under a principal biaxial stress stale OJ, c|. Show each ease with a plot similar to Fig, 3.11e-h.

3.10 Reduce eq(3.27) to criteria of yielding under a principal, biaxial stress state (OJ, cQ for material:

(i) orfhotropy and (ii) transverse isotropy, where yield stresses are the same in the 2- and 3-directions.

3.11 Show tot Hill's yield criterion (3.27) reduces to a plane stress form:

when plane stresses q,, c^ and TV are applied within the plane of a thin sheet and aligned with the orthotropic axes x and y (z is through the thickness). Use this equation to show tot the variation in yield stress F with orientation #to x within the plane of the sheet, is given by:

Y{0) =

. \ F m i20 + Gco$*d + H + -(2N - F - 6 - 4H)sin220

^ 4

3.12 If, in exercise 3.11, the yield stresses in the three orthogonal directions x, y and z are denoted by X, Y and Z respectively and P is the shear yield stress in the x-y plane of the sheet, show that the coefficients F, G, H and N are given by:

X2

JV =

2{X

2

3.13 Using the orthotropic yield function (3.27), examine how you would predict the onset of yielding in an orthotropie sheet metal when a direct stress a is combined with a shear stress r i n each of the foEowing cases: (i) when both a and r are aligned with the in-plane orthotropic axes x and y and (ii) when they are inclined at 0 to x and y, as shown in Fig. 3.15. Note, the presence of complementary shear in each case.

I Answer (i): 2N H + G

v = X 2 where X is the yield stress in the x-direction]

Figure 3.15

3.14 Using Hill's orthotropic function (3.27), examine how you would predict yielding in an orthotropic sheet under in-plane biaxial stresses oj and t% in each of the following cases: (i) when oj and Oj, are aligned with the in-plane orthotropic directions x and y respectively and (ii) when oj and t% are inclined at 6 to * and y, as shown in Fig. 3.16. Note, that axis z is through the thickness.

[

Answer (i> C2 - a a + <r2 = Xi where X is the yield stress in the jc-direction I

K H + G * 7 H + G f

Figure 3.16

CHAPTER 4

NON-HARDENING PLASTICITY