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A general theory of plastic deformation and instability in thin-walled tubes under combined loading
J. Mech. Phys. Solids, Vol. 44, No. 12, pp. 2041-2057, 1996 Copyright 0 1996 ElsevierScienceLtd
Pergamon
Printed in Great Britain. All rights reserved 002225096/96 $15.00+0.00
PII: SOO22-5096(96)00062-2
A GENERAL THEORY OF PLASTIC DEFORMATION AND INSTABILITY IN THIN-WALLED TUBES UNDER COMBINED LOADING RODNEY HILL Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, U.K. (Received
19 April 1996)
ABSTRACT In all previous investigations of the title problem the constitutive model has been very specific. Most often it has involved the Mises yield function, the Levy-Mises flow rule, a geometrically similar family of subsequent yield loci, and the standard postulate of equivalence between different deformation histories. Also specific has been the chosen pair of control variables and the type of loading path in stress space. By contrast the present analysis admits arbitrary control variables as well as arbitrary loading paths. The constitutive framework also is not specific, though still broadly classical. The framework differs in one important respect : it involves no reference whatsoever to yield functions nor to their path dependence. This greatly simplifies the resulting mathematics and enables the analysis to range much more widely than hitherto. It is demonstrated, inter alia, that the customary axiomatic basis of previous theories is mathematically invalid. This basis was to the effect that the onset of diffuse instability is always associated with the simultaneous stationarity of both control variables, or contrariwise with the maximum value of one alone. In place of either of these axioms (which were unsupported by experimental data), the analysis identifies a continuous spectrum of possible behaviours at the instant when the homogeneous regime ceases to be stable. Copyright 0 1996 Elsevier Science Ltd
1.
INTRODUCTION
Theories of plastic instability in thin-walled tubes subjected to biaxial loading have a long history. Most have to do with internal fluid pressure combined with an independent axial load, and this is the case here as well. The initial regime of homogeneous deformation is primarily of interest, after which the conditions leading to instability will be investigated in detail. In that regard it has been customary hitherto to adopt one or other of two arbitrary and mutually-exclusive axioms : namely that the homogeneous regime becomes unstable at a stage when (i) both the pressure and the axial load are stationary, or when (ii) either the pressure or the axial load first attains a maximum. The present analysis starts from a quite different standpoint and is conducted throughout on classical lines. In particular it takes into consideration possible branching paths of deformation and the respective expenditures of energy. It also envisages arbitrary pairs of control variables (pressure and axial load being but one possible choice) together with arbitrary paths of loading. 2041
R. HILL
2042
The constitutive framework itself differs radically from what has been generally accepted until recently. No reference is made to subsequent yield loci, nor hence to their possible path dependences and changing geometries. In consequence the mathematics is not burdened by yield functions of any kind, nor by global postulates of equivalence between different deformation histories. Relieved of that complexity, the analysis is completely transparent and also far more general. This enables detailed conclusions to be drawn for the first time as to the onset of diffuse instability. The material parameters are few in number and remain free throughout. In principle they can vary along a loading path in ways that are peculiar to the particular test. It is ultimately left to experiment to supply whatever data is needful for quantitative realizations of the formulae. The predictive competence of the theory can then be tested across a range of materials and loading programmes.
2.
A CLASS OF CONTROL VARIABLES
A thin-walled circular-cylindrical tube is subjected to an internal fluid pressure P together with an external axial load L and is strained far into the plastic range. The regime of homogeneous deformation and its eventual termination is examined here from a new standpoint. The test apparatus is presumed capable of applying P and L in arbitrary proportions, and moreover varying these at will in the course of a single experiment. The grain texture of the as-received material is required to be such that all macroelements are isotropic or all have orthotropic symmetry relative to their local generator, surface normal, and transverse tangent. In these circumstances, and until the onset of diffuse instability, the tube necessarily remains circular cylindrical within its gauge length. Consequently the developing texture in each macro-element remains at least orthotropic, irrespective of how the associated moduli vary. The current wall radius, thickness, and gauge length at any stage will be denoted by Y,t, and h respectively ; their product is conserved, given that changes in volume are negligible in the overall context. The distribution of stress in the wall is automatically axisymmetric during the stable regime; its principal values are both tensile and will be denoted by (ol, a,), respectively axial and circumferential. Their variations through the wall thickness can be disregarded provided that t/ris less than l/l0 or so. As long as the deformation remains uniform, the stress ratio is given by 0,/O> = ; (1 + L/dP),
(1)
the components separately being such that 2nrtcr, = (L+7w2P),
c2 = Prjt.
(2)
Note that L must be compressive if values of al/a2 less than i are to be obtained. In practice L and P are commonly applied in fixed ratio, even though a,/az is then free to vary with the tube radius: e.g. Davis (1945), Davis and Parker (1948), Stout and Hecker (1983), Stout et al. (1983), and Hill et al. (1994). An arbitrary relation between L and P alone can easily be enforced by servo-control, however, and is
A general theory of plastic deformation and instability
2043
accordingly admitted in the analysis here. A still more versatile type of control over P and L is also envisaged which responds to feedback from a diametral extensometer that continuously monitors the varying tube radius. Any pair of control variables in this wider class will be denoted generically by p and q. Examples of such pairs are (i) PandL;
(ii) PandL+zr*P;
(iii) PandL/nr2;
(iv) rrr*PandL
(3)
and these will be given special prominence. A loading path in the space of any chosen pair (p, q) is definable by a pre-determined relation of type f(P, 4) = 0.
(4)
The control variables may both vary monotonically, or p may have a turning point where dlfii?q= 0, or q may have one where iJJ/lap = 0. In all cases a projected control path effectively terminates when the deformation ceases to be uniform. When p and q involve r as well as P and L, (4) amounts to a pre-determined relation of type F(P, L, r) = 0
(5)
between concurrent values of the arguments. For example, with p and q as in (iii) of (3) a radial path q = cp in control space amounts to L/&P = c and by (1) would generate a radial path o,/az = ;(l +c)
(6)
in stress space also. The same applies when p and q are as in (iv) of (3). On the other hand, with p and q chosen simply as P and L themselves, a radial path q = cp would ordinarily generate a nonlinear path in stress space. Exceptionally the latter path would be linear if r happened not to vary for a particular value of L/P; such cases have been reported in tests by Hill et al. (1994, p. 3013). In terms of the pairs in (3), and of others that similarly involve P and L linearly (with or without a dependence on r also), the resulting stress in the tube wall is expressible generically as 6 = pa+qg
(7)
in symbolic notation (Hill, 1994). Boldface type, here and later, indicates a twodimensional vector ; in particular the components of c itself are ((r,, CJ*).Vectors a and /I depend only on the tube geometry ; their respective forms and dimensions are decided by the prior choices of p and q. For the controls (i) to (iv) in (3) it follows readily from (2) that (i) a =(1,2)r/2t,
/I =(1,0)/2nrt
withp = P,q = L;
(ii) a = (0, l)r/t,
fi =(1,0)/2rrrt
withp = P,q = L+nr*P;
(iii) a = (1,2)r/2t,
/I= (l,O)r/22
with p = P,q = L/m2 ;
(iv) a =(1,2)/2rrrt,
j =(1,0)/2zrt
withp = m*P,q = L.
(8)
In these cases it is apparent that a and fi maintain fixed directions whereas their magnitudes vary, not necessarily proportionately.
2044
R. HILL
At this point it is convenient to record their differentials : (i) da = @(da, +2d&
d/3 = Bds,
withp = P,q = L;
(ii) da = a(d.a, +2d&
d/Yf= /3dE,
withp = P,q = L+nr2P;
(iii) da = a(ds, +2dsz),
d/I = B(da, +2dsZ)
withp = P,q = L/m2 ;
(iv) da = ad&,,
d/I = bde,
with p = nr2P,q = L.
(9)
Here E, and &2are the principal components of logarithmic strain in the axial and circumferential directions respectively. Having regard to the incompressibility constraint rth = rotoh,,,
(10)
they are expressible as e2 = log(r/r,)
81 = log(r&) +log(t,/t),
(11)
in terms of r and t as the favoured pair of geometric variables (subscript zero indicates an initial value). The scalar factors in (9) are thus de, = -(dr/r+dt/t),
d&, +2ds2 = drjr-dtlt
(12)
when evaluated in terms of the differentials of r and t. The relations inverse to (7) can be expressed (Hill, 1994) as p=aa,
q=bu
(13)
where the juxtaposition of two boldface symbols signifies their scalar product. Vectors a and b here stand in reciprocal relation to a and /I?,in that aa = 1 = b/3, ab = 0 = ba.
(14)
The pairs (a, /I) and (a, b) can be viewed, in fact, as covariant and contravariant in tr space. Corresponding to (8) we now have (i) a = (0, I)t/r,
b =(2, -1)nrt
withp = P,q = L;
(ii) a = (0, I)t/r,
b = (2,O)nrt
withp = P,q = L+rcr’P,
(iii) a = (0, l)t/r,
b = (2, - I)t/r
withp = P,q = L/w2 ;
(iv) a = (0, l)mt,
b = (2, - 1)nrt
withp = nr2P,q = L.
bases
;
(15)
Vectors a and b maintain fixed directions also, and moreover their magnitudes vary so that da/a = -da/a,
these ratios being recorded in (9).
dbjb = -dp/b,
(16)
A general theory
3.
of plastic deformation
CONSTITUTIVE
and instability
2045
FRAMEWORK
A typical tube experiment is specified by some pre-determined relation between the chosen control variables. The outcome is a recorded path in (ai, c2) space along with its companion in (E,,E~) space. For descriptive convenience in what follows, the coordinate axes in these spaces will be regarded as having the same orientations. Attention is now focused on a typical pair of corresponding points, 0 and a, on the respective paths. Let m be a unit vector which is co-directional with the tangent at E; it can be regarded indifferently as tied to either of the points 0 and e. A differential increment of strain is then expressible as ds=mdE
(ds>O)
(17)
where E is the arc-length of the strain path.? The direction m varies with E, of course, unless the path happens to be a ray from the origin. Another unit vector n forms with m an orthogonal pair whose components on the axes of coordinates are m =(cos&sin4),
n z(-sin&cos$).
(18)
Here 4 is the anticlockwise angle from the E, axis to m, and is also the anticlockwise angle from the .s2axis to n. The strain increment at E is assumed to depend linearly on the stress increment at 0 according to the classical model of rigid/plastic behaviour : hd& =
(m da)m
when m da > 0
0
whenmda
where h is a positive modulus dependent on the distinctive constituents of this model are (i) that of ds (which is always m) but only its magnitude, component of da along m. Plastic deformation which means that ads > 0 and therefore pma+qmS
d 0
(19)
strain history up to this stage. The da does not influence the direction and (ii) that ds depends only on the involves the expenditure of work,
> 0
GO)
in terms of the representation (7). Now let $ be the acute angle from m to da reckoned positive if anticlockwise. The scalar product of m and da is equal to cos I+$ do where dg is the length of the arc da. A formula for the ratio of the corresponding differential arcs in the respective spaces now follows from (19) : do/de = hsectj
(-in
< $
(21)
The modulus h is a constant of the material in the particular state, so do/de varies directly as set II/and ranges from h when II/ = 0 to infinity when $ = + $c. In particular, if the increments of stress and strain are co-directional (tj = 0), (19) reduces to da = hdE
withda
= hds.
t Note that ds = ldel but not dlel, except on a ray from the origin.
(22)
2046
R. HILL
This provides an immediate interpretation of h as a modulus and indicates how it can be measured experimentally. The remaining constituent of the classical model is the identification of m with the outward normal at 6 to a particular closed domain in stress space. This domain consists of all points which are accessible from u without further plastic strain; the boundary is usually known as a subsequent yield locus. Its geometry necessarily reflects the material properties in the unique state which is common to every point of the domain. It is recalled that the material state in question was generated by a specific sequence of deformation terminating at a. In principle, therefore, the shape of a subsequent locus is path-dependent, even strongly so, and even among different paths terminating at the same 6. Fortunately the present objectives do not require any modelling of yield loci and their varying geometries. Conjectural yield functions, in fact, have no role whatsoever in the analysis. In their place a new constitutive relation is proposed which is analytically compatible with the classical framework as outlined, though never part of it. The proposal concerns the local curvature of a strain path. This is defined by K = d+/dt:
(23)
and is a signed quantity as 4 in (18) is always reckoned anticlockwise relative to the orthotropic axes. The curvature appears naturally in evolution formulae such as dm/de = kn,
dn/ds = -K-m.
(24)
The proposal (Hill, 1994) is that d$ at a given 0 depends linearly on the components of any ray-like increment da causing further flow. An equivalent statement is that K at a given G depends linearly on the components of any da/d8 complying with mda/de > 0. This dependence is formulated as K = (k da)/h da = (k da)/(m da)
(25)
where (in principle) k is a vector functional of the preceding strain path. The modulus h has been introduced in the formula purely as a normalizing factor, and in consequence the state parameter k is dimensionless. Now let da be decomposed as da =(mcos$+nsin$)do
=(m+ntan$)hds,
(26)
having regard to (21). In conjunction with (25) there follows K = d4/da = (km) + (kn) tan $.
(27)
It is seen that km is the local curvature of the particular branching path in Espace along which do = h de as in (22). Taking account of (21) once again, we can complement (27) by h d4/da = (km) cos t,b+ (kn) sin $.
(28)
Consider the particular gradients of 4 here in the respective directions of m and n (namely $ = 0 and $c), and introduce corresponding quantities pm and pn defined by l/p,,, = (km)/h
when ICI= 0,
I/P,, = (kn)/h
when $ = ire.
(29)
2047
A general theory of plastic deformation and instability
It is evident that pm is the local radius of curvature of the special path from u along which + - 0 and da = h ds. Similarly pn is the local radius of curvature at d of a particular yield locus: it is the one that could be generated by a process of neutral loading starting from 6. (Remember that II/ = kirc at every point of a yield locus, according to the normality flow rule and the sense of description.) These two radii of curvature necessarily have the dimensions of stress, as signalled by the presence of h in the definitions. Moreover, because of the sign convention regarding 4, the radius p,, is positive when + = $c but negative when IJ = -$c. Consequently kn > 0
is a necessary restriction on the range of k. The components brought together in a single formula (Hill, 1994) namely
(30)
of k in (29) can be
k/h = m/p, + nlPn,
(31)
which expresses the state parameter k in terms of the local geometries of the subsequent yield locus and the special branching path. Finally, on eliminating tan $ between (26) and (27), there results (kn)(da-hds)
= n(rc--km)hdc.
(32)
This can be regarded as a natural (if unorthodox) inverse of the constitutive relation (19) ; no inverse of any kind was ever part of the classical framework. It establishes da as explicitly dependent on K as well as de, and as uniquely determined by both of them jointly. Conversely, a resolution of (32) on the direction k returns straight away to the original proposition (25).
4.
EIGENSTATES
In terms of the control variables and their prescribed increments, the resulting increment of stress is da =(Pda+qdS)+(adp+Bdq)
(33)
in the format of (7). The accompanying increment of strain is m d&where hds = mda =(pmda+qmdfl)+(madp+mfidq).
(34)
Conversely, in order to generate a desired increment of stress, the control variables must be changed by dp = a(da-pda-qd/?),
dq = b(da-pda-qd/3).
(35)
These come from the formal resolutes of da on a and b respectively, taking account of the orthonormal relations (14). Motivated by the examples in (8) and (9), the strain-induced increments in a and fi will be assumed to conform with the general evolution formulae da = a(Lds),
dfi = /?(pds).
(36)
R. HILL
2048
Here 3, and p are cons&nt vectors whose values depend only on the specific type of control (Hill, 1994). For instance rZ= (1,2) and p = (1,O) in examples (i) and (ii), while L = p = (1,2) in (iii) and Iz = cc = (1,O) in (iv). Re-arrangement of (33) after introducing (36) gives da/d& = [dp/ds + (Am)p]a + [dq/de + (pm)q]j
(37)
which is a decomposition on the basis (a, 8) in the spirit of (7). Equivalent resolutions on the reciprocal basis (a, b) are a da/de = dp/de + (Am)p,
b da/de = dq/ds + (pm)q
(38)
in replacement of (35). Finally (34) can be recast as h = (ma)[dp/ds +
(WA + (m/OWq/d~+ (rmhl
.
(39)
The consequences of these universal formulae will now be pursued in relation to socalled eigenstates of the system taken as a whole (Hill, 1991, 1994). In a typical tube test an eigenstate is defined as the stage when the monotonically decreasing value of the hardening modulus falls to h = Ama) (Jm) + q(mS) (pm).
(40)
The righthand side is a moving target (so to say) which is set by evolving values of the material parameter m, the control variables (p, q), and the geometric variables (a, /3). The critical value of h is affected by the choice of control variables even when different pairs are programmed to generate the same stress and strain paths. This is because the values of a, 8, a and b at a given strain depend on that choice, as in (8) and (15), and moreover so do the actual values of p and q at a given stress, as in (13). Specific examples are listed in (79) below. Taking (39) and (40) together, it is seen that an eigenstate is characterized also by (ma) dp/dE + (m/I) dq/ds = 0.
(41)
It follows that a dplde + jI dqlds =
dqlde = tnb,
(42)
where n is orthogonal to m and 5 can take any value. Use has been made here of the elementary identities n = (na)a+ (nb)&
(ma)(na) + (mjl)(nb) = 0.
In particular na and m/I vanish together (if at all) and so do nb and ma, But na and nb vanish separately (if at all) and so do ma and m/K
The stress rate is correspondingly da/d& = p(Am)a + q(pm)j + 5n
(43)
from (37). Evidently m da/da is compatible with the critical value of h in (40), whatever
A general theory of plastic deformation and instability 5
2049
may be, since mn = 0. The associated curvature of the strain path at an eigenstate
is ~c= {p(k&)(W + q(kS) (am) + ((kn) 1lh
(44)
by (25). Now let the direction of a stress path at any stage be specified by a unit vector 1; its components are thus cos(4 + II/) and sin(4 + I,G)in terms of the angles defined in Section 3. Then da is replaced by Ida and (43) is split into (lm) do/de = h,
(Im)5 = p(lm)la A 11+q(pm)lfl A II.
(45)
The vertical lines enclosing a vector product signify the actual (not numerical) value of its component in the direction of m A n. The first equation of this pair simply repeats (21), while the second links 5 directly with the path tangent in stress space, and hence indirectly with the path tangent in control space. In this group of formulae p and q can be replaced by an and ba if preferred. Two mutually exclusive possibilities now arise : (i) r # 0.
At least one of dp/dE and dq/de is nonzero.
(ii) 4 = 0.
Both dp/de and dq/ds are zero.
(46)
Consider the tangent at a generic point on the path in control space. Treating the coordinate axes there as righthanded and rectangular, let 8 be the anticlockwise orientation of the tangent relative to the p axis. The unit tangent is then (dp/ds, dq/ds) = (cos 8, sin 19)
(47)
where s is the arc length measured along the path. In case (i) the control path tangent at an eigenstate is orthogonal to the direction (ma,mb) according to (41) and is parallel to the direction (na,nb) according to (42). In case (ii), on the other hand, ds/de = 0 and therefore (d2p/de2, d2q/de2) = d/de{(cos 8, sin 13)ds/de} = (cos 9, sin 0) d2s/ds2.
(48)
The direction of the path tangent is now given by the ratio of the second derivatives ofp and q (or the third derivatives if d2s/de2 happens to vanish also, and so on). From (43) or (45) the path direction in stress space must be such that Mm)
= p(Wa
+ q(pm)%,
(49)
while from (44) the associated curvature of the strain path is ~c= {p(ka)(Im)+q(kS)(pm)}lh.
(50)
No restriction is placed now on the angle between the control path tangent and either of the directions (ma, rnfi) and (na, nb). An eigenstate is said to be latent in case (i) but active in case (ii). In (i) ds/dr: is nonzero at the eigenstate itself, so the values of (ds)/h and d.s have comparable magnitudes in its vicinity. In (ii), on the other hand, ds/ds is zero at the eigenstate, so in its vicinity (ds)/h is smaller than ds by one or more orders of magnitude. Furthermore the simultaneous stationarity of p and q indicates that the control path
2050
R. HILL
could return on itself, if permitted to do so by the servo-mechanism accompanying deformation were still homogeneous).
5.
(and if the
WORK AND STABILITY
The internal work rate per unit volume of tube material during the regime of homogeneous deformation is dw/de = mu = pma+qmj?, a positive quantity segment in strain length de between is expressible as a
(51)
as noted already in (20). It is proposed to integrate rnb over a path space. It suffices to define the segment simply as a smooth arc of points E,,and so+ As. The total expenditure of work per unit volume Taylor series in powers of As, namely mu de = moa,A&+ k
(A&)’ +.
'. ,
(52)
0
if the integrand is continuously differentiable. The segment is presumed to be sufficiently short for powers of AE beyond the second to be disregarded. The leading terms can be re-grouped as
or as
using (24) and replacing m da/da by h, as warranted by (19). The terms in square brackets displayed in (53) can be regarded as heading a Taylor series for the chord AE since m = de/da, dm/de = d2s/ds2. A compact expression for the work is hence e,,+Ae ads = ~J~AE+;~~(AE)~+-, s 80
(54)
in which it should be kept in mind that the chord has to be evaluated correctly to to a trapezoid rule of quadrature, namely
second order (in one form or another). The formula is tantamount
sotAs
ads =(~,+;A~)AE+-. s En The equivalence can be recognized at once because the expansions AaAs =(m,Aa)Ae+...
parallel the exact relations
= ho(As)2+...
(55)
A general theory of plastic deformation and instability
2051
da de = (m da) ds = h(ds)2 between differentials. Now imagine the selfsame path between ~0and e,,+ As to be described under loads generated by purely passive controls p = p,, and q = q. at every stage, which continue of course to satisfy (4). The consequent notional stress in the tube wall would vary according to fl* =poa+qoB
(56)
by (7), because the same basis vectors as before are involved at every stage. The value of the stress rate at a0 is then (da*/da), = [P da/de + q 4Wlo
= MWa + drmMlo
(57)
by (36). In general u* is not related constitutively to E along the segment, so the work that would be done under constant p and q is merely virtual (in the customary sense of the word in mechanics). Nevertheless it is still given to second order by (52), but with da*/dc in place of da/de. Therefore a,,+Ae CT*
ds = co As +t(m da*/de)o (AE)~+. . . .
(58)
s 10 The difference between (54) and (58) is e,+Ae
(a-a*)d~
= ~mo(da/ds-da*/ds)o(Aa)2+...
(59)
= $(ma)dp/ds + (m/3) dq/de], (AE)~+. . .
(6’4
s 80
by (37) and (57). In retrospect it is seen that this calculation makes no reference to the geometry of the actual path segment, other than to its initial tangent and curvature. By the same token, the path imagined to be described under the notional loading need only agree with the actual path in these two respects. The previous word “selfsame” should therefore be understood in that sense. The content of (60) at any stage can be summarized as follows, after dropping the subscript zero which is no longer needed. In any increment of deformation consistent with the current stress, the actual expenditure of energy exceeds what could notionally be supplied by the work of passive controls when (ma)dp/de + (mj?) dq/ds > 0
(61)
in that increment. Such considerations tend to the general conclusion that a serzrocontrolled path of deformation is stable against vanishingly-small disturbances (static or dynamic) so long as (61) is satisfied at every stage by the particular control variables in question. Granted the validity of this conclusion, it follows from (39) that the value of h defined by (40) is indeed critical for stability of the operation. At an eigenstate itself the respective stress rates involved above are such that da/de = da*/de + &r, m da/de = m da*/d&
(62)
2052
R. HILL
on comparing (57) with (43). The respective expenditures of energy are hence equal to second order. and the stability is apparently neutral, regardless of whether an eigcnstate is latent or active. In order to assess the situation more exactly, terms of higher order in the Taylor expansions would need to be examined. That calculation goes beyond what is strictly relevant to the central objective here.
6.
SHEET
STRETCHING
UNDER
BIAXIAL
LOADS
At this point it may be helpful to show in detail how the formulae are particularized in a related context which is more straightforward. This is the biaxial deformation of a Bat rectangular sheet by combined loads along the in-plane axes of orthotropy. All previous treatments have been grounded on the supposed existence of a singly-infinite family of subsequent yield loci, 1/ie same whatecer the loading path. These have ranged from self-similar Mises ellipses (Swift, 1952) to any convex contours whatever, selfsimilar or not (Hill. 1991). For an extended bibliography of early work, see Stout and Hecker (1983). As explained in Section 3, the present approach makes no reference at all to complete yield loci. and is hence free from arbitrary postulates as to their number, shapes and path dependence (and consequently free also from simplistic notions of equivalence). The control variables arc taken to bc the components of nominal stress (load per unit undeformed area). namely (p-q) =(a,irl,,r~,ir?~).
(rl,.r?,)
=(expEI,expcz),
the latter pair being the principal stretches (ratio of final to initial length). parison with the representation (7) identities the bases (2, /I) and (a, b) as z =(l,O)q,,/I
=(O, 1)~:
and
a =(l,O)jn,,b
=(O, I)iqZ.
(63)
A com-
(64)
The vectors in each pair are orthogonal in this case, and this simplifies the manipulations. In the evolution formulae (36) we need the differentials dq, = q,dc,, dq: = qZdcz, and can then identify the constant vectors 1. = (I, 0) and I( = (0, I). The directions mentioned after (42) are (ma,mB)
(wnb) =(-mz~~l.m,lr12)
=(m,~,.m,~~).
which are plainly orthogonal as proved more generally stress components in (63) are (da,,da,) since p dq, = 0, dc, and qd+ h =(mfa,
=(m,o,
dE+q,
dp,mzaz
before. The differentials
dc+qz
dq)
(65) of the
(66)
= (T?de?. Then m da = h ds becomes +mjaz)+(m,ql
which corresponds to (39). An eigenstate is hence characterized
by
dp/dc:+nrzqzdq/de)
(67)
A general theory of plastic deformation and instability
h = m:o, +m:oz
and
m,q, dp/ds+mzqz
dq/ds = 0
2053
(68)
as in (40) and (41). In place of (42) there is now dplds = -&%/r?,,
dqlde = @%I?,.
(69)
-5m2,mZaZ+&r,)
(70)
When substituted in (66) these give (da,/de,da,/ds)
=(+a,
which is (43) in components. As in (45) this can be split into (l,m, +fzmz) da/de = h,
(l,m, +12m2)5 = m,a,I, -m2a21,
(71)
where h has the value defined in (68). The first equation of this pair replicates (21), while the second relates 5 to the direction (II, ZJ of the stress path at an eigenstate. Corresponding to the stress rate in (70), the curvature of the strain path at an eigenstate is ~c= {k,m,a, +kzm2a2 +&ml
-klmZ)}/h
(72)
directly from (25), and this replaces (44). When an eigenstate is latent, as defined by (i) in (46), the local tangent to the control path is co-directional with (dp/ds, dq/ds) and hence with ( -m,/ql, m&) as in (69) with 5 # 0. Since stretches are positive quantities, it can be concluded that a latent eigenstate is attained after a turning point on the control path when m, and m2 have the same sign (dplds and dqlde then have opposite signs), but before a turning point when they have opposite signs (dp/ds and dq/de then have similar signs). At an active eigenstate defined by (ii) in (46), on the other hand, the control variables are simultaneously stationary and MI = m2a2/mlal
(73)
by (71) with r = 0. This corresponds to (49) expressed as a ratio of components, and it shows that the stress path must have a specific direction at an eigenstate, if this is to be activated (Hill, 1991, p. 304). As an example, suppose the servo-control is programmed to produce a purely radial path in stress space, so that (a,, aJ = a(/,, &) where I is now a fixed unit vector and only the path length a varies. This requires that the ratio p/q should be altered continuously, namely in fixed proportion 1,/12to the varying stretch ratio q&,r,. The material parameters h and m will generally depend on 1 and may also vary with a. The second equation in (71) reduces to VImI +km&%r
= f,&(m, -mk
(74)
which shows that 5 is zero only when 1, or 1, vanishes, or when m, = m2. Therefore an active eigenstate is encountered along a radial path only when the stress is uniaxial (which is trivial) or when the incremental strain is equibiaxial momentarily (the stress will only be equibiaxial as well if there is in-plane isotropy). The same observation was made by Hill (1991, p. 305) in the context of a singly-infinite family of complete yield loci. Earlier writers mostly followed Swift (1952) in the unsupported proposition that the onset of instability entails both applied loads being stationary together, and further that such a conjunction must inevitably occur on every radial path in stress
2054
R. HILL
space. Other writers took it as axiomatic that instability is initiated at a turning point of either load alone. Neither proposition has general validity, as just shown. Diffuse instability under the biaxial controls (63) is associated “almost always” with nonstationary loads whose rates of change comply with (68) or equivalently with (69).
7.
TUBE STRETCHING
UNDER BIAXIAL LOADS
The particular pairs of control variables listed in (3) will now be combined with the relevant values of a and /I in (8), Iz and p in (9), and (where needed) a and b in (15). The four cases will be treated in parallel in order to illustrate directly the effects of different controls on eigenstates and on tube stability, even with identical paths in stress space. We begin with the formula (37) and number the four cases as in (3). The results in differential format are (i) do, =(m,a,
+m2a,)d~+(w2dp+dq)/2nrt,
do2 =(m, +2m&,
dc+rdp/t;
(75)
(ii) do, = m, CT,de+dq/2nrt, da2 =(m, +2m2)o, (iii) do, =(m, +2mz)o, da, =(m, +2m2)o,
dc+rdp/t;
(76)
dE+r(dp+dq)/2t, dE+rdp/t;
(77)
(iv) do, = m,o, dE+(dp+dq)/2wt, da, = m,u2 defdp/xrt.
(78)
Equations exactly comparable in structure can be seen in Hill (1991, p. 306). The respective contexts and variables differ significantly, however, because basic quantities in the reference are correlated with a notional family of yield loci which are generated by the contours of a single function @(a,, g2). No such function is invoked in the present analysis, as explained before. The free unit vector m here corresponds formally with grad@/1 grad01 there, while ds and mds correspond with pJ grad@1 ds and p( grad 0) ds (p is a shorthand for @/a grad a). The critical values (40) of the hardening modulus are respectively (i) h = mia, +2m,m,a, (ii) h = m:o, +m,m,a, (iii) h =(m, +2m,)(m,a, (iv) h = m,(m,o,
+m,o,).
+2m:a, f2m:az
; ;
+m2(r2); (79)
Comparable formulae also appear in the reference. The symbol h there, however, is equivalent to (p grad @)2 times the symbol h here. The values of h(de)’ in the two notations are therefore identical.
A general theory of plastic deformation and instability
2055
Eigenstates are characterized also by (41) which imposes constraints on the concurrent rates of change of the two control variables. In the format of the second line of (42) these are (i) dp/ds =
dq/dt: = - @ml + 2m,)nrt ;
(ii) dp/ds = trn, t/r,
dqlde = -
(iii) dp/de = trn, t/r,
dq/dt: = -<(ml +2m2)t/r;
(iv) dp/ds = Sm,xrt,
dq/ds = -[(ml
+ 2m2)wt.
(80)
The associated stress rates at any eigenstate are obtainable from (43) or directly from (74-78). By the latter route the terms in dp and dq which contribute to dg, and do, respectively are simply replaced as they stand by - 5rn2de and [m, ds. There follows (0 d(qi,aJds
= [(mlol +m2c2),(ml
+2m&d+5(-m2,mI);
00 d(cr,,adlde = [m,a,,(m, +2mda21+t(-m2,m,); (iii) d(a,,adlde =(mI+2m,)(a,,~2)+5(-m*,mI); (iv) d(o,,ad/de = ml(aI,~d+5(-mz,m,).
(81)
A stage in the analysis has now been reached which parallels (45). The unit vector 1 is introduced here also and the entries in (81) are split into
(i,m, +12mz) da/da = h
(82)
which is (21) with h given by (79), together with [
(I1
1 (ii)
Vim1 +&m,)5 =
(83)
(iii)
respectively. When 5 # 0 in (80) the local direction of the control path is given by the ratio dpldq. In general the derivatives dp/de and dqlde there are nonzero and have opposite signs. Exceptionally one of the derivatives in (80) may vanish at an eigenstate, namely dp/ds if ml = 0, and dq/ds if m2 = 0 in (ii) or m, + 2mz = 0 in the other three. According to (12) the latter means that the circumferential and through-thickness increments of strain are equal, which in turn would require the stress to be uniaxial (g2 = 0) if the material were fully isotropic. When 5 = 0 in (80) both dp/ds and dq and de vanish and the local direction of the control path depends on the second (or higher) derivatives of p and q. From (83) the local direction of the stress path must now accord with (i) Ml, = (ml +2Q~J(m1~i (4
/2/l,
=h
+m2a2),
+2m2)~2/ml~l,
(iii) and (iv) 12/1, = az/g,,
(84)
2056
R. HILL
analogously to (49). In (iii) and (iv) the conclusion is simply that the local direction must be radial in order to activate any eigenstate. Suppose, more particularly, that the servo-control in each case is operated so that L/&P stays constant with the help of feedback as regards the varying radius. Then by (1) the paths in stress space are always radial, which is to say that 1, and 1, are fixed quantities along each path, while o,l, = 01~1~= a,l, at every stage. Consequently (83) reduces to (85) In cases (iii) and (iv) there, consistently with (84), every radial path is automatically terminated by an active eigenstate (Hill, 1994, pp. 13141315). In cases (i) and (ii), by contrast, “almost all” radial paths are terminated by a latent eigenstate. There are indeed a few situations where < vanishes and an eigenstate is active, but these are quite trivial. In both (i) and (ii) this happens under uniaxial tension G, (P = 0, L a maximum) and when de2 = 0 (tube radius stationary, simultaneous maxima in L, P, and L + xr2P, given that L/lrr2P is constant). In case (i) alone it happens also on the ray r~,/a2 = i (L 3 0, P a maximum), and in case (ii) alone under uniaxial tension g2 (L E - 77r2P,P a maximum). These findings contradict the basic premise in Swift’s (1952) analysis of case (i), namely that the onset of diffuse instability always entails the simultaneous stationarity of L and P. As was customary at that time, Swift modelled strain hardening of the tube material via the Levy-Mises flow rule and a simple equivalence relation based on a singly-infinite family of subsequent Mises ellipses. Restricting the analysis to radial paths in stress space, he then calculated a value of the associated hardening modulus by putting dL = 0 = dP in the rate equations of equilibrium. Finally he asserted that the value was critical for tube stability, and in fact this modulus does agree with (40) here (when the formula is particularized for his constitutive model and control variables). That it must do so is evident because the replacement of (41) at the outset by dp/dE = 0 = dq/de leads directly to active eigenstates in the present sense. On the other hand, Swift failed to notice that in case (i) these are not actually encountered along radial paths except in two special cases (dL = 0 = dP being generally incompatible with a constant ratio of L to xr2P, as said before). All other eigenstates admitted by (41) are latent and such that neither control variable is stationary in case (i). Mellor (1962) assumed the same constitutive model and likewise restricted the analysis to radial paths in stress space. His standpoint can be categorized as case (ii) ; see Mellor (1962), p. 254, first column. The basic premise was that stability would be lost when either P or L + nr2P first attained a maximum. Now d(L + nr’P) = 0 trivially entails dL = 0 and d(nr2P) = 0 simultaneously on any path along which L/xr’P is constant. In effect, then, Mellor calculated separately the strain when each of P and nr2P attained a maximum. It is obvious that the maximum in P occurs first if the tube is expanding, while the maximum in nr2P occurs first if the tube is contracting, and that is what Mellor duly found. From the general standpoint of (41) and (60) here, however, the turning point in one control variable alone has no implication in regard to loss of stability.
A general theory of plastic deformation and instability
2057
An interesting comparison of the respective theories of Swift and Mellor with experimental data has been made by Stout and Hecker (1983). When assessing this, it is important to appreciate that the data points were scaled rather arbitrarily [Stout and Hecker (1983), p. 27, near top of second column]. It is probable that a more satisfactory reconciliation would be achievable by means of the present model, wherein the material parameters are completely free to vary from one path to another. For information about the path-dependent properties of the same material, see Hill et al. (1993).
REFERENCES Davis, E. A. (1945) Yielding and fracture of medium-carbon steel under combined stress. J. Appl. Mech. 12, Al 3-A24. Davis, E. A. and Parker, E. R. (1948) Behaviour of steel under biaxial stress as determined by tests on tubes. J. Appl. Mech. 15,201-209. Hill, R. (1991) A theoretical perspective on in-plane forming of sheet metal. J. Mech. Phys. Solids 39, 295-301. Hill, R. (1994) Classical plasticity: a retrospective view and a new proposal. J. Mech. Phys. Solids 42, 1803-1816. Hill, R., Hecker, S. S. and Stout, M. G. (1993) An investigation of plastic flow and differential work-hardening in orthotropic brass tubes under fluid pressure and axial load. ht. J. Solids and Structures 31,2999-3021. Mellor, P. B. (1962) Tensile instability in thin-walled tubes. J. Mech. Engng Sci. 4, 251-256. Stout, M. G. and Hecker, S. S. (1983) Role of geometry in plastic instability and fracture of tubes and sheet. Mech. Mater. 2, 23-3 1. Stout, M. G., Hecker, S. S. and Bourcier, R. (1983) An evaluation of anisotropic effective stress-strain criteria for the biaxial yield and flow of 2024 aluminum tubes. J. Engng Mater. and Technol. 105,242-249.
Swift, H. W. (1952) Plastic instability under plane stress. J. Mech. Phys. Solids 1, l-18.
Pergamon
Printed in Great Britain. All rights reserved 002225096/96 $15.00+0.00
PII: SOO22-5096(96)00062-2
A GENERAL THEORY OF PLASTIC DEFORMATION AND INSTABILITY IN THIN-WALLED TUBES UNDER COMBINED LOADING RODNEY HILL Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, U.K. (Received
19 April 1996)
ABSTRACT In all previous investigations of the title problem the constitutive model has been very specific. Most often it has involved the Mises yield function, the Levy-Mises flow rule, a geometrically similar family of subsequent yield loci, and the standard postulate of equivalence between different deformation histories. Also specific has been the chosen pair of control variables and the type of loading path in stress space. By contrast the present analysis admits arbitrary control variables as well as arbitrary loading paths. The constitutive framework also is not specific, though still broadly classical. The framework differs in one important respect : it involves no reference whatsoever to yield functions nor to their path dependence. This greatly simplifies the resulting mathematics and enables the analysis to range much more widely than hitherto. It is demonstrated, inter alia, that the customary axiomatic basis of previous theories is mathematically invalid. This basis was to the effect that the onset of diffuse instability is always associated with the simultaneous stationarity of both control variables, or contrariwise with the maximum value of one alone. In place of either of these axioms (which were unsupported by experimental data), the analysis identifies a continuous spectrum of possible behaviours at the instant when the homogeneous regime ceases to be stable. Copyright 0 1996 Elsevier Science Ltd
1.
INTRODUCTION
Theories of plastic instability in thin-walled tubes subjected to biaxial loading have a long history. Most have to do with internal fluid pressure combined with an independent axial load, and this is the case here as well. The initial regime of homogeneous deformation is primarily of interest, after which the conditions leading to instability will be investigated in detail. In that regard it has been customary hitherto to adopt one or other of two arbitrary and mutually-exclusive axioms : namely that the homogeneous regime becomes unstable at a stage when (i) both the pressure and the axial load are stationary, or when (ii) either the pressure or the axial load first attains a maximum. The present analysis starts from a quite different standpoint and is conducted throughout on classical lines. In particular it takes into consideration possible branching paths of deformation and the respective expenditures of energy. It also envisages arbitrary pairs of control variables (pressure and axial load being but one possible choice) together with arbitrary paths of loading. 2041
R. HILL
2042
The constitutive framework itself differs radically from what has been generally accepted until recently. No reference is made to subsequent yield loci, nor hence to their possible path dependences and changing geometries. In consequence the mathematics is not burdened by yield functions of any kind, nor by global postulates of equivalence between different deformation histories. Relieved of that complexity, the analysis is completely transparent and also far more general. This enables detailed conclusions to be drawn for the first time as to the onset of diffuse instability. The material parameters are few in number and remain free throughout. In principle they can vary along a loading path in ways that are peculiar to the particular test. It is ultimately left to experiment to supply whatever data is needful for quantitative realizations of the formulae. The predictive competence of the theory can then be tested across a range of materials and loading programmes.
2.
A CLASS OF CONTROL VARIABLES
A thin-walled circular-cylindrical tube is subjected to an internal fluid pressure P together with an external axial load L and is strained far into the plastic range. The regime of homogeneous deformation and its eventual termination is examined here from a new standpoint. The test apparatus is presumed capable of applying P and L in arbitrary proportions, and moreover varying these at will in the course of a single experiment. The grain texture of the as-received material is required to be such that all macroelements are isotropic or all have orthotropic symmetry relative to their local generator, surface normal, and transverse tangent. In these circumstances, and until the onset of diffuse instability, the tube necessarily remains circular cylindrical within its gauge length. Consequently the developing texture in each macro-element remains at least orthotropic, irrespective of how the associated moduli vary. The current wall radius, thickness, and gauge length at any stage will be denoted by Y,t, and h respectively ; their product is conserved, given that changes in volume are negligible in the overall context. The distribution of stress in the wall is automatically axisymmetric during the stable regime; its principal values are both tensile and will be denoted by (ol, a,), respectively axial and circumferential. Their variations through the wall thickness can be disregarded provided that t/ris less than l/l0 or so. As long as the deformation remains uniform, the stress ratio is given by 0,/O> = ; (1 + L/dP),
(1)
the components separately being such that 2nrtcr, = (L+7w2P),
c2 = Prjt.
(2)
Note that L must be compressive if values of al/a2 less than i are to be obtained. In practice L and P are commonly applied in fixed ratio, even though a,/az is then free to vary with the tube radius: e.g. Davis (1945), Davis and Parker (1948), Stout and Hecker (1983), Stout et al. (1983), and Hill et al. (1994). An arbitrary relation between L and P alone can easily be enforced by servo-control, however, and is
A general theory of plastic deformation and instability
2043
accordingly admitted in the analysis here. A still more versatile type of control over P and L is also envisaged which responds to feedback from a diametral extensometer that continuously monitors the varying tube radius. Any pair of control variables in this wider class will be denoted generically by p and q. Examples of such pairs are (i) PandL;
(ii) PandL+zr*P;
(iii) PandL/nr2;
(iv) rrr*PandL
(3)
and these will be given special prominence. A loading path in the space of any chosen pair (p, q) is definable by a pre-determined relation of type f(P, 4) = 0.
(4)
The control variables may both vary monotonically, or p may have a turning point where dlfii?q= 0, or q may have one where iJJ/lap = 0. In all cases a projected control path effectively terminates when the deformation ceases to be uniform. When p and q involve r as well as P and L, (4) amounts to a pre-determined relation of type F(P, L, r) = 0
(5)
between concurrent values of the arguments. For example, with p and q as in (iii) of (3) a radial path q = cp in control space amounts to L/&P = c and by (1) would generate a radial path o,/az = ;(l +c)
(6)
in stress space also. The same applies when p and q are as in (iv) of (3). On the other hand, with p and q chosen simply as P and L themselves, a radial path q = cp would ordinarily generate a nonlinear path in stress space. Exceptionally the latter path would be linear if r happened not to vary for a particular value of L/P; such cases have been reported in tests by Hill et al. (1994, p. 3013). In terms of the pairs in (3), and of others that similarly involve P and L linearly (with or without a dependence on r also), the resulting stress in the tube wall is expressible generically as 6 = pa+qg
(7)
in symbolic notation (Hill, 1994). Boldface type, here and later, indicates a twodimensional vector ; in particular the components of c itself are ((r,, CJ*).Vectors a and /I depend only on the tube geometry ; their respective forms and dimensions are decided by the prior choices of p and q. For the controls (i) to (iv) in (3) it follows readily from (2) that (i) a =(1,2)r/2t,
/I =(1,0)/2nrt
withp = P,q = L;
(ii) a = (0, l)r/t,
fi =(1,0)/2rrrt
withp = P,q = L+nr*P;
(iii) a = (1,2)r/2t,
/I= (l,O)r/22
with p = P,q = L/m2 ;
(iv) a =(1,2)/2rrrt,
j =(1,0)/2zrt
withp = m*P,q = L.
(8)
In these cases it is apparent that a and fi maintain fixed directions whereas their magnitudes vary, not necessarily proportionately.
2044
R. HILL
At this point it is convenient to record their differentials : (i) da = @(da, +2d&
d/3 = Bds,
withp = P,q = L;
(ii) da = a(d.a, +2d&
d/Yf= /3dE,
withp = P,q = L+nr2P;
(iii) da = a(ds, +2dsz),
d/I = B(da, +2dsZ)
withp = P,q = L/m2 ;
(iv) da = ad&,,
d/I = bde,
with p = nr2P,q = L.
(9)
Here E, and &2are the principal components of logarithmic strain in the axial and circumferential directions respectively. Having regard to the incompressibility constraint rth = rotoh,,,
(10)
they are expressible as e2 = log(r/r,)
81 = log(r&) +log(t,/t),
(11)
in terms of r and t as the favoured pair of geometric variables (subscript zero indicates an initial value). The scalar factors in (9) are thus de, = -(dr/r+dt/t),
d&, +2ds2 = drjr-dtlt
(12)
when evaluated in terms of the differentials of r and t. The relations inverse to (7) can be expressed (Hill, 1994) as p=aa,
q=bu
(13)
where the juxtaposition of two boldface symbols signifies their scalar product. Vectors a and b here stand in reciprocal relation to a and /I?,in that aa = 1 = b/3, ab = 0 = ba.
(14)
The pairs (a, /I) and (a, b) can be viewed, in fact, as covariant and contravariant in tr space. Corresponding to (8) we now have (i) a = (0, I)t/r,
b =(2, -1)nrt
withp = P,q = L;
(ii) a = (0, I)t/r,
b = (2,O)nrt
withp = P,q = L+rcr’P,
(iii) a = (0, l)t/r,
b = (2, - I)t/r
withp = P,q = L/w2 ;
(iv) a = (0, l)mt,
b = (2, - 1)nrt
withp = nr2P,q = L.
bases
;
(15)
Vectors a and b maintain fixed directions also, and moreover their magnitudes vary so that da/a = -da/a,
these ratios being recorded in (9).
dbjb = -dp/b,
(16)
A general theory
3.
of plastic deformation
CONSTITUTIVE
and instability
2045
FRAMEWORK
A typical tube experiment is specified by some pre-determined relation between the chosen control variables. The outcome is a recorded path in (ai, c2) space along with its companion in (E,,E~) space. For descriptive convenience in what follows, the coordinate axes in these spaces will be regarded as having the same orientations. Attention is now focused on a typical pair of corresponding points, 0 and a, on the respective paths. Let m be a unit vector which is co-directional with the tangent at E; it can be regarded indifferently as tied to either of the points 0 and e. A differential increment of strain is then expressible as ds=mdE
(ds>O)
(17)
where E is the arc-length of the strain path.? The direction m varies with E, of course, unless the path happens to be a ray from the origin. Another unit vector n forms with m an orthogonal pair whose components on the axes of coordinates are m =(cos&sin4),
n z(-sin&cos$).
(18)
Here 4 is the anticlockwise angle from the E, axis to m, and is also the anticlockwise angle from the .s2axis to n. The strain increment at E is assumed to depend linearly on the stress increment at 0 according to the classical model of rigid/plastic behaviour : hd& =
(m da)m
when m da > 0
0
whenmda
where h is a positive modulus dependent on the distinctive constituents of this model are (i) that of ds (which is always m) but only its magnitude, component of da along m. Plastic deformation which means that ads > 0 and therefore pma+qmS
d 0
(19)
strain history up to this stage. The da does not influence the direction and (ii) that ds depends only on the involves the expenditure of work,
> 0
GO)
in terms of the representation (7). Now let $ be the acute angle from m to da reckoned positive if anticlockwise. The scalar product of m and da is equal to cos I+$ do where dg is the length of the arc da. A formula for the ratio of the corresponding differential arcs in the respective spaces now follows from (19) : do/de = hsectj
(-in
< $
(21)
The modulus h is a constant of the material in the particular state, so do/de varies directly as set II/and ranges from h when II/ = 0 to infinity when $ = + $c. In particular, if the increments of stress and strain are co-directional (tj = 0), (19) reduces to da = hdE
withda
= hds.
t Note that ds = ldel but not dlel, except on a ray from the origin.
(22)
2046
R. HILL
This provides an immediate interpretation of h as a modulus and indicates how it can be measured experimentally. The remaining constituent of the classical model is the identification of m with the outward normal at 6 to a particular closed domain in stress space. This domain consists of all points which are accessible from u without further plastic strain; the boundary is usually known as a subsequent yield locus. Its geometry necessarily reflects the material properties in the unique state which is common to every point of the domain. It is recalled that the material state in question was generated by a specific sequence of deformation terminating at a. In principle, therefore, the shape of a subsequent locus is path-dependent, even strongly so, and even among different paths terminating at the same 6. Fortunately the present objectives do not require any modelling of yield loci and their varying geometries. Conjectural yield functions, in fact, have no role whatsoever in the analysis. In their place a new constitutive relation is proposed which is analytically compatible with the classical framework as outlined, though never part of it. The proposal concerns the local curvature of a strain path. This is defined by K = d+/dt:
(23)
and is a signed quantity as 4 in (18) is always reckoned anticlockwise relative to the orthotropic axes. The curvature appears naturally in evolution formulae such as dm/de = kn,
dn/ds = -K-m.
(24)
The proposal (Hill, 1994) is that d$ at a given 0 depends linearly on the components of any ray-like increment da causing further flow. An equivalent statement is that K at a given G depends linearly on the components of any da/d8 complying with mda/de > 0. This dependence is formulated as K = (k da)/h da = (k da)/(m da)
(25)
where (in principle) k is a vector functional of the preceding strain path. The modulus h has been introduced in the formula purely as a normalizing factor, and in consequence the state parameter k is dimensionless. Now let da be decomposed as da =(mcos$+nsin$)do
=(m+ntan$)hds,
(26)
having regard to (21). In conjunction with (25) there follows K = d4/da = (km) + (kn) tan $.
(27)
It is seen that km is the local curvature of the particular branching path in Espace along which do = h de as in (22). Taking account of (21) once again, we can complement (27) by h d4/da = (km) cos t,b+ (kn) sin $.
(28)
Consider the particular gradients of 4 here in the respective directions of m and n (namely $ = 0 and $c), and introduce corresponding quantities pm and pn defined by l/p,,, = (km)/h
when ICI= 0,
I/P,, = (kn)/h
when $ = ire.
(29)
2047
A general theory of plastic deformation and instability
It is evident that pm is the local radius of curvature of the special path from u along which + - 0 and da = h ds. Similarly pn is the local radius of curvature at d of a particular yield locus: it is the one that could be generated by a process of neutral loading starting from 6. (Remember that II/ = kirc at every point of a yield locus, according to the normality flow rule and the sense of description.) These two radii of curvature necessarily have the dimensions of stress, as signalled by the presence of h in the definitions. Moreover, because of the sign convention regarding 4, the radius p,, is positive when + = $c but negative when IJ = -$c. Consequently kn > 0
is a necessary restriction on the range of k. The components brought together in a single formula (Hill, 1994) namely
(30)
of k in (29) can be
k/h = m/p, + nlPn,
(31)
which expresses the state parameter k in terms of the local geometries of the subsequent yield locus and the special branching path. Finally, on eliminating tan $ between (26) and (27), there results (kn)(da-hds)
= n(rc--km)hdc.
(32)
This can be regarded as a natural (if unorthodox) inverse of the constitutive relation (19) ; no inverse of any kind was ever part of the classical framework. It establishes da as explicitly dependent on K as well as de, and as uniquely determined by both of them jointly. Conversely, a resolution of (32) on the direction k returns straight away to the original proposition (25).
4.
EIGENSTATES
In terms of the control variables and their prescribed increments, the resulting increment of stress is da =(Pda+qdS)+(adp+Bdq)
(33)
in the format of (7). The accompanying increment of strain is m d&where hds = mda =(pmda+qmdfl)+(madp+mfidq).
(34)
Conversely, in order to generate a desired increment of stress, the control variables must be changed by dp = a(da-pda-qd/?),
dq = b(da-pda-qd/3).
(35)
These come from the formal resolutes of da on a and b respectively, taking account of the orthonormal relations (14). Motivated by the examples in (8) and (9), the strain-induced increments in a and fi will be assumed to conform with the general evolution formulae da = a(Lds),
dfi = /?(pds).
(36)
R. HILL
2048
Here 3, and p are cons&nt vectors whose values depend only on the specific type of control (Hill, 1994). For instance rZ= (1,2) and p = (1,O) in examples (i) and (ii), while L = p = (1,2) in (iii) and Iz = cc = (1,O) in (iv). Re-arrangement of (33) after introducing (36) gives da/d& = [dp/ds + (Am)p]a + [dq/de + (pm)q]j
(37)
which is a decomposition on the basis (a, 8) in the spirit of (7). Equivalent resolutions on the reciprocal basis (a, b) are a da/de = dp/de + (Am)p,
b da/de = dq/ds + (pm)q
(38)
in replacement of (35). Finally (34) can be recast as h = (ma)[dp/ds +
(WA + (m/OWq/d~+ (rmhl
.
(39)
The consequences of these universal formulae will now be pursued in relation to socalled eigenstates of the system taken as a whole (Hill, 1991, 1994). In a typical tube test an eigenstate is defined as the stage when the monotonically decreasing value of the hardening modulus falls to h = Ama) (Jm) + q(mS) (pm).
(40)
The righthand side is a moving target (so to say) which is set by evolving values of the material parameter m, the control variables (p, q), and the geometric variables (a, /3). The critical value of h is affected by the choice of control variables even when different pairs are programmed to generate the same stress and strain paths. This is because the values of a, 8, a and b at a given strain depend on that choice, as in (8) and (15), and moreover so do the actual values of p and q at a given stress, as in (13). Specific examples are listed in (79) below. Taking (39) and (40) together, it is seen that an eigenstate is characterized also by (ma) dp/dE + (m/I) dq/ds = 0.
(41)
It follows that a dplde + jI dqlds =
dqlde = tnb,
(42)
where n is orthogonal to m and 5 can take any value. Use has been made here of the elementary identities n = (na)a+ (nb)&
(ma)(na) + (mjl)(nb) = 0.
In particular na and m/I vanish together (if at all) and so do nb and ma, But na and nb vanish separately (if at all) and so do ma and m/K
The stress rate is correspondingly da/d& = p(Am)a + q(pm)j + 5n
(43)
from (37). Evidently m da/da is compatible with the critical value of h in (40), whatever
A general theory of plastic deformation and instability 5
2049
may be, since mn = 0. The associated curvature of the strain path at an eigenstate
is ~c= {p(k&)(W + q(kS) (am) + ((kn) 1lh
(44)
by (25). Now let the direction of a stress path at any stage be specified by a unit vector 1; its components are thus cos(4 + II/) and sin(4 + I,G)in terms of the angles defined in Section 3. Then da is replaced by Ida and (43) is split into (lm) do/de = h,
(Im)5 = p(lm)la A 11+q(pm)lfl A II.
(45)
The vertical lines enclosing a vector product signify the actual (not numerical) value of its component in the direction of m A n. The first equation of this pair simply repeats (21), while the second links 5 directly with the path tangent in stress space, and hence indirectly with the path tangent in control space. In this group of formulae p and q can be replaced by an and ba if preferred. Two mutually exclusive possibilities now arise : (i) r # 0.
At least one of dp/dE and dq/de is nonzero.
(ii) 4 = 0.
Both dp/de and dq/ds are zero.
(46)
Consider the tangent at a generic point on the path in control space. Treating the coordinate axes there as righthanded and rectangular, let 8 be the anticlockwise orientation of the tangent relative to the p axis. The unit tangent is then (dp/ds, dq/ds) = (cos 8, sin 19)
(47)
where s is the arc length measured along the path. In case (i) the control path tangent at an eigenstate is orthogonal to the direction (ma,mb) according to (41) and is parallel to the direction (na,nb) according to (42). In case (ii), on the other hand, ds/de = 0 and therefore (d2p/de2, d2q/de2) = d/de{(cos 8, sin 13)ds/de} = (cos 9, sin 0) d2s/ds2.
(48)
The direction of the path tangent is now given by the ratio of the second derivatives ofp and q (or the third derivatives if d2s/de2 happens to vanish also, and so on). From (43) or (45) the path direction in stress space must be such that Mm)
= p(Wa
+ q(pm)%,
(49)
while from (44) the associated curvature of the strain path is ~c= {p(ka)(Im)+q(kS)(pm)}lh.
(50)
No restriction is placed now on the angle between the control path tangent and either of the directions (ma, rnfi) and (na, nb). An eigenstate is said to be latent in case (i) but active in case (ii). In (i) ds/dr: is nonzero at the eigenstate itself, so the values of (ds)/h and d.s have comparable magnitudes in its vicinity. In (ii), on the other hand, ds/ds is zero at the eigenstate, so in its vicinity (ds)/h is smaller than ds by one or more orders of magnitude. Furthermore the simultaneous stationarity of p and q indicates that the control path
2050
R. HILL
could return on itself, if permitted to do so by the servo-mechanism accompanying deformation were still homogeneous).
5.
(and if the
WORK AND STABILITY
The internal work rate per unit volume of tube material during the regime of homogeneous deformation is dw/de = mu = pma+qmj?, a positive quantity segment in strain length de between is expressible as a
(51)
as noted already in (20). It is proposed to integrate rnb over a path space. It suffices to define the segment simply as a smooth arc of points E,,and so+ As. The total expenditure of work per unit volume Taylor series in powers of As, namely mu de = moa,A&+ k
(A&)’ +.
'. ,
(52)
0
if the integrand is continuously differentiable. The segment is presumed to be sufficiently short for powers of AE beyond the second to be disregarded. The leading terms can be re-grouped as
or as
using (24) and replacing m da/da by h, as warranted by (19). The terms in square brackets displayed in (53) can be regarded as heading a Taylor series for the chord AE since m = de/da, dm/de = d2s/ds2. A compact expression for the work is hence e,,+Ae ads = ~J~AE+;~~(AE)~+-, s 80
(54)
in which it should be kept in mind that the chord has to be evaluated correctly to to a trapezoid rule of quadrature, namely
second order (in one form or another). The formula is tantamount
sotAs
ads =(~,+;A~)AE+-. s En The equivalence can be recognized at once because the expansions AaAs =(m,Aa)Ae+...
parallel the exact relations
= ho(As)2+...
(55)
A general theory of plastic deformation and instability
2051
da de = (m da) ds = h(ds)2 between differentials. Now imagine the selfsame path between ~0and e,,+ As to be described under loads generated by purely passive controls p = p,, and q = q. at every stage, which continue of course to satisfy (4). The consequent notional stress in the tube wall would vary according to fl* =poa+qoB
(56)
by (7), because the same basis vectors as before are involved at every stage. The value of the stress rate at a0 is then (da*/da), = [P da/de + q 4Wlo
= MWa + drmMlo
(57)
by (36). In general u* is not related constitutively to E along the segment, so the work that would be done under constant p and q is merely virtual (in the customary sense of the word in mechanics). Nevertheless it is still given to second order by (52), but with da*/dc in place of da/de. Therefore a,,+Ae CT*
ds = co As +t(m da*/de)o (AE)~+. . . .
(58)
s 10 The difference between (54) and (58) is e,+Ae
(a-a*)d~
= ~mo(da/ds-da*/ds)o(Aa)2+...
(59)
= $(ma)dp/ds + (m/3) dq/de], (AE)~+. . .
(6’4
s 80
by (37) and (57). In retrospect it is seen that this calculation makes no reference to the geometry of the actual path segment, other than to its initial tangent and curvature. By the same token, the path imagined to be described under the notional loading need only agree with the actual path in these two respects. The previous word “selfsame” should therefore be understood in that sense. The content of (60) at any stage can be summarized as follows, after dropping the subscript zero which is no longer needed. In any increment of deformation consistent with the current stress, the actual expenditure of energy exceeds what could notionally be supplied by the work of passive controls when (ma)dp/de + (mj?) dq/ds > 0
(61)
in that increment. Such considerations tend to the general conclusion that a serzrocontrolled path of deformation is stable against vanishingly-small disturbances (static or dynamic) so long as (61) is satisfied at every stage by the particular control variables in question. Granted the validity of this conclusion, it follows from (39) that the value of h defined by (40) is indeed critical for stability of the operation. At an eigenstate itself the respective stress rates involved above are such that da/de = da*/de + &r, m da/de = m da*/d&
(62)
2052
R. HILL
on comparing (57) with (43). The respective expenditures of energy are hence equal to second order. and the stability is apparently neutral, regardless of whether an eigcnstate is latent or active. In order to assess the situation more exactly, terms of higher order in the Taylor expansions would need to be examined. That calculation goes beyond what is strictly relevant to the central objective here.
6.
SHEET
STRETCHING
UNDER
BIAXIAL
LOADS
At this point it may be helpful to show in detail how the formulae are particularized in a related context which is more straightforward. This is the biaxial deformation of a Bat rectangular sheet by combined loads along the in-plane axes of orthotropy. All previous treatments have been grounded on the supposed existence of a singly-infinite family of subsequent yield loci, 1/ie same whatecer the loading path. These have ranged from self-similar Mises ellipses (Swift, 1952) to any convex contours whatever, selfsimilar or not (Hill. 1991). For an extended bibliography of early work, see Stout and Hecker (1983). As explained in Section 3, the present approach makes no reference at all to complete yield loci. and is hence free from arbitrary postulates as to their number, shapes and path dependence (and consequently free also from simplistic notions of equivalence). The control variables arc taken to bc the components of nominal stress (load per unit undeformed area). namely (p-q) =(a,irl,,r~,ir?~).
(rl,.r?,)
=(expEI,expcz),
the latter pair being the principal stretches (ratio of final to initial length). parison with the representation (7) identities the bases (2, /I) and (a, b) as z =(l,O)q,,/I
=(O, 1)~:
and
a =(l,O)jn,,b
=(O, I)iqZ.
(63)
A com-
(64)
The vectors in each pair are orthogonal in this case, and this simplifies the manipulations. In the evolution formulae (36) we need the differentials dq, = q,dc,, dq: = qZdcz, and can then identify the constant vectors 1. = (I, 0) and I( = (0, I). The directions mentioned after (42) are (ma,mB)
(wnb) =(-mz~~l.m,lr12)
=(m,~,.m,~~).
which are plainly orthogonal as proved more generally stress components in (63) are (da,,da,) since p dq, = 0, dc, and qd+ h =(mfa,
=(m,o,
dE+q,
dp,mzaz
before. The differentials
dc+qz
dq)
(65) of the
(66)
= (T?de?. Then m da = h ds becomes +mjaz)+(m,ql
which corresponds to (39). An eigenstate is hence characterized
by
dp/dc:+nrzqzdq/de)
(67)
A general theory of plastic deformation and instability
h = m:o, +m:oz
and
m,q, dp/ds+mzqz
dq/ds = 0
2053
(68)
as in (40) and (41). In place of (42) there is now dplds = -&%/r?,,
dqlde = @%I?,.
(69)
-5m2,mZaZ+&r,)
(70)
When substituted in (66) these give (da,/de,da,/ds)
=(+a,
which is (43) in components. As in (45) this can be split into (l,m, +fzmz) da/de = h,
(l,m, +12m2)5 = m,a,I, -m2a21,
(71)
where h has the value defined in (68). The first equation of this pair replicates (21), while the second relates 5 to the direction (II, ZJ of the stress path at an eigenstate. Corresponding to the stress rate in (70), the curvature of the strain path at an eigenstate is ~c= {k,m,a, +kzm2a2 +&ml
-klmZ)}/h
(72)
directly from (25), and this replaces (44). When an eigenstate is latent, as defined by (i) in (46), the local tangent to the control path is co-directional with (dp/ds, dq/ds) and hence with ( -m,/ql, m&) as in (69) with 5 # 0. Since stretches are positive quantities, it can be concluded that a latent eigenstate is attained after a turning point on the control path when m, and m2 have the same sign (dplds and dqlde then have opposite signs), but before a turning point when they have opposite signs (dp/ds and dq/de then have similar signs). At an active eigenstate defined by (ii) in (46), on the other hand, the control variables are simultaneously stationary and MI = m2a2/mlal
(73)
by (71) with r = 0. This corresponds to (49) expressed as a ratio of components, and it shows that the stress path must have a specific direction at an eigenstate, if this is to be activated (Hill, 1991, p. 304). As an example, suppose the servo-control is programmed to produce a purely radial path in stress space, so that (a,, aJ = a(/,, &) where I is now a fixed unit vector and only the path length a varies. This requires that the ratio p/q should be altered continuously, namely in fixed proportion 1,/12to the varying stretch ratio q&,r,. The material parameters h and m will generally depend on 1 and may also vary with a. The second equation in (71) reduces to VImI +km&%r
= f,&(m, -mk
(74)
which shows that 5 is zero only when 1, or 1, vanishes, or when m, = m2. Therefore an active eigenstate is encountered along a radial path only when the stress is uniaxial (which is trivial) or when the incremental strain is equibiaxial momentarily (the stress will only be equibiaxial as well if there is in-plane isotropy). The same observation was made by Hill (1991, p. 305) in the context of a singly-infinite family of complete yield loci. Earlier writers mostly followed Swift (1952) in the unsupported proposition that the onset of instability entails both applied loads being stationary together, and further that such a conjunction must inevitably occur on every radial path in stress
2054
R. HILL
space. Other writers took it as axiomatic that instability is initiated at a turning point of either load alone. Neither proposition has general validity, as just shown. Diffuse instability under the biaxial controls (63) is associated “almost always” with nonstationary loads whose rates of change comply with (68) or equivalently with (69).
7.
TUBE STRETCHING
UNDER BIAXIAL LOADS
The particular pairs of control variables listed in (3) will now be combined with the relevant values of a and /I in (8), Iz and p in (9), and (where needed) a and b in (15). The four cases will be treated in parallel in order to illustrate directly the effects of different controls on eigenstates and on tube stability, even with identical paths in stress space. We begin with the formula (37) and number the four cases as in (3). The results in differential format are (i) do, =(m,a,
+m2a,)d~+(w2dp+dq)/2nrt,
do2 =(m, +2m&,
dc+rdp/t;
(75)
(ii) do, = m, CT,de+dq/2nrt, da2 =(m, +2m2)o, (iii) do, =(m, +2mz)o, da, =(m, +2m2)o,
dc+rdp/t;
(76)
dE+r(dp+dq)/2t, dE+rdp/t;
(77)
(iv) do, = m,o, dE+(dp+dq)/2wt, da, = m,u2 defdp/xrt.
(78)
Equations exactly comparable in structure can be seen in Hill (1991, p. 306). The respective contexts and variables differ significantly, however, because basic quantities in the reference are correlated with a notional family of yield loci which are generated by the contours of a single function @(a,, g2). No such function is invoked in the present analysis, as explained before. The free unit vector m here corresponds formally with grad@/1 grad01 there, while ds and mds correspond with pJ grad@1 ds and p( grad 0) ds (p is a shorthand for @/a grad a). The critical values (40) of the hardening modulus are respectively (i) h = mia, +2m,m,a, (ii) h = m:o, +m,m,a, (iii) h =(m, +2m,)(m,a, (iv) h = m,(m,o,
+m,o,).
+2m:a, f2m:az
; ;
+m2(r2); (79)
Comparable formulae also appear in the reference. The symbol h there, however, is equivalent to (p grad @)2 times the symbol h here. The values of h(de)’ in the two notations are therefore identical.
A general theory of plastic deformation and instability
2055
Eigenstates are characterized also by (41) which imposes constraints on the concurrent rates of change of the two control variables. In the format of the second line of (42) these are (i) dp/ds =
dq/dt: = - @ml + 2m,)nrt ;
(ii) dp/ds = trn, t/r,
dqlde = -
(iii) dp/de = trn, t/r,
dq/dt: = -<(ml +2m2)t/r;
(iv) dp/ds = Sm,xrt,
dq/ds = -[(ml
+ 2m2)wt.
(80)
The associated stress rates at any eigenstate are obtainable from (43) or directly from (74-78). By the latter route the terms in dp and dq which contribute to dg, and do, respectively are simply replaced as they stand by - 5rn2de and [m, ds. There follows (0 d(qi,aJds
= [(mlol +m2c2),(ml
+2m&d+5(-m2,mI);
00 d(cr,,adlde = [m,a,,(m, +2mda21+t(-m2,m,); (iii) d(a,,adlde =(mI+2m,)(a,,~2)+5(-m*,mI); (iv) d(o,,ad/de = ml(aI,~d+5(-mz,m,).
(81)
A stage in the analysis has now been reached which parallels (45). The unit vector 1 is introduced here also and the entries in (81) are split into
(i,m, +12mz) da/da = h
(82)
which is (21) with h given by (79), together with [
(I1
1 (ii)
Vim1 +&m,)5 =
(83)
(iii)
respectively. When 5 # 0 in (80) the local direction of the control path is given by the ratio dpldq. In general the derivatives dp/de and dqlde there are nonzero and have opposite signs. Exceptionally one of the derivatives in (80) may vanish at an eigenstate, namely dp/ds if ml = 0, and dq/ds if m2 = 0 in (ii) or m, + 2mz = 0 in the other three. According to (12) the latter means that the circumferential and through-thickness increments of strain are equal, which in turn would require the stress to be uniaxial (g2 = 0) if the material were fully isotropic. When 5 = 0 in (80) both dp/ds and dq and de vanish and the local direction of the control path depends on the second (or higher) derivatives of p and q. From (83) the local direction of the stress path must now accord with (i) Ml, = (ml +2Q~J(m1~i (4
/2/l,
=h
+m2a2),
+2m2)~2/ml~l,
(iii) and (iv) 12/1, = az/g,,
(84)
2056
R. HILL
analogously to (49). In (iii) and (iv) the conclusion is simply that the local direction must be radial in order to activate any eigenstate. Suppose, more particularly, that the servo-control in each case is operated so that L/&P stays constant with the help of feedback as regards the varying radius. Then by (1) the paths in stress space are always radial, which is to say that 1, and 1, are fixed quantities along each path, while o,l, = 01~1~= a,l, at every stage. Consequently (83) reduces to (85) In cases (iii) and (iv) there, consistently with (84), every radial path is automatically terminated by an active eigenstate (Hill, 1994, pp. 13141315). In cases (i) and (ii), by contrast, “almost all” radial paths are terminated by a latent eigenstate. There are indeed a few situations where < vanishes and an eigenstate is active, but these are quite trivial. In both (i) and (ii) this happens under uniaxial tension G, (P = 0, L a maximum) and when de2 = 0 (tube radius stationary, simultaneous maxima in L, P, and L + xr2P, given that L/lrr2P is constant). In case (i) alone it happens also on the ray r~,/a2 = i (L 3 0, P a maximum), and in case (ii) alone under uniaxial tension g2 (L E - 77r2P,P a maximum). These findings contradict the basic premise in Swift’s (1952) analysis of case (i), namely that the onset of diffuse instability always entails the simultaneous stationarity of L and P. As was customary at that time, Swift modelled strain hardening of the tube material via the Levy-Mises flow rule and a simple equivalence relation based on a singly-infinite family of subsequent Mises ellipses. Restricting the analysis to radial paths in stress space, he then calculated a value of the associated hardening modulus by putting dL = 0 = dP in the rate equations of equilibrium. Finally he asserted that the value was critical for tube stability, and in fact this modulus does agree with (40) here (when the formula is particularized for his constitutive model and control variables). That it must do so is evident because the replacement of (41) at the outset by dp/dE = 0 = dq/de leads directly to active eigenstates in the present sense. On the other hand, Swift failed to notice that in case (i) these are not actually encountered along radial paths except in two special cases (dL = 0 = dP being generally incompatible with a constant ratio of L to xr2P, as said before). All other eigenstates admitted by (41) are latent and such that neither control variable is stationary in case (i). Mellor (1962) assumed the same constitutive model and likewise restricted the analysis to radial paths in stress space. His standpoint can be categorized as case (ii) ; see Mellor (1962), p. 254, first column. The basic premise was that stability would be lost when either P or L + nr2P first attained a maximum. Now d(L + nr’P) = 0 trivially entails dL = 0 and d(nr2P) = 0 simultaneously on any path along which L/xr’P is constant. In effect, then, Mellor calculated separately the strain when each of P and nr2P attained a maximum. It is obvious that the maximum in P occurs first if the tube is expanding, while the maximum in nr2P occurs first if the tube is contracting, and that is what Mellor duly found. From the general standpoint of (41) and (60) here, however, the turning point in one control variable alone has no implication in regard to loss of stability.
A general theory of plastic deformation and instability
2057
An interesting comparison of the respective theories of Swift and Mellor with experimental data has been made by Stout and Hecker (1983). When assessing this, it is important to appreciate that the data points were scaled rather arbitrarily [Stout and Hecker (1983), p. 27, near top of second column]. It is probable that a more satisfactory reconciliation would be achievable by means of the present model, wherein the material parameters are completely free to vary from one path to another. For information about the path-dependent properties of the same material, see Hill et al. (1993).
REFERENCES Davis, E. A. (1945) Yielding and fracture of medium-carbon steel under combined stress. J. Appl. Mech. 12, Al 3-A24. Davis, E. A. and Parker, E. R. (1948) Behaviour of steel under biaxial stress as determined by tests on tubes. J. Appl. Mech. 15,201-209. Hill, R. (1991) A theoretical perspective on in-plane forming of sheet metal. J. Mech. Phys. Solids 39, 295-301. Hill, R. (1994) Classical plasticity: a retrospective view and a new proposal. J. Mech. Phys. Solids 42, 1803-1816. Hill, R., Hecker, S. S. and Stout, M. G. (1993) An investigation of plastic flow and differential work-hardening in orthotropic brass tubes under fluid pressure and axial load. ht. J. Solids and Structures 31,2999-3021. Mellor, P. B. (1962) Tensile instability in thin-walled tubes. J. Mech. Engng Sci. 4, 251-256. Stout, M. G. and Hecker, S. S. (1983) Role of geometry in plastic instability and fracture of tubes and sheet. Mech. Mater. 2, 23-3 1. Stout, M. G., Hecker, S. S. and Bourcier, R. (1983) An evaluation of anisotropic effective stress-strain criteria for the biaxial yield and flow of 2024 aluminum tubes. J. Engng Mater. and Technol. 105,242-249.
Swift, H. W. (1952) Plastic instability under plane stress. J. Mech. Phys. Solids 1, l-18.