Instability and failure of internally pressurized ductile metal cylinders

OO22-5O96/82/030121-34 $03.OO/O Q 1982.Pergamon PressLtd.

J. Mech. Phys. Solids Vol.30, No. 3, pp. 121-154, 1982.

Printed in GreatBritain.

INSTABILITY PRESSURIZED

AND FAILURE OF INTERNALLY DUCTILE METAL CYLINDERS

M. LARSSON, A. NEEDLEMAN,? V. TVERGAARD~ and B. STORAKERS Departmentof Strengthof Materialsand Solid Mechanics. 100 44 Stockholm, Sweden

The Royal Institute of Technology,

(Received 25 June 1981)

ABSTRACT FOR LONG, ductile, thick-walled tubes under internal pressure instabilities and final failure modes are studied experimentally and theoretically. The test specimens are closed-end cylinders made of an aluminum alloy and of pure copper and the experiments have been carried out for a number of different initial external radius to internal radius ratios. The experiments show necking on one side of the tubes at a stage somewhat beyond the maximum internal pressure. All tubes, except for one aluminum alloy tube, failed by shear fracture under decreasing pressure. The aluminum alloy tubes exhibited localized shear deformations in the neck region prior to fracture and also occasionally surface wave instabilities. The numerical investigation is based on an elastic-plastic material model for a solid that develops a vertex on the yield surface, using representations of the uniaxial stress-strain curves found experimentally. In contrast to the simplest flow theory of plasticity this material model predicts shear band instabilities at a realistic level of strain. A rather sharp vertex is used in the material model for the aluminum alloy, while a more blunt vertex is used to characterize copper. The theoretically predicted bifurcation into a necking mode, the cross-sectional shape of the neck, and finally the initiation and growth of shear bands from the highly strained internal surface in the neck region are in good agreement with the experimental observations.

1.

INTRODUCTION

THE STABILITY of long ductile cylindrical tubes under internal pressure has been the subject of several theoretical investigations in recent years. Tubes made of a rigidplastic material have been analyzed by STORAKERS (1971) and by STIFORS and STORAKERS (1973). For a particular class of strain hardening materials these authors find that bifurcation away from the cylindrically symmetric deformation state is excluded prior to the maximum pressure point, whereas after this point such a bifurcation may take place. The range of uniqueness of the cylindrically symmetric mode beyond the point of maximum pressure is of interest when the volume enclosed by the tube is the prescribed quantity. If the internal pressure is prescribed to increase monotonically, stability is lost at the maximum pressure point. CHU (1979) reinvestigated this problem on the basis of elastic-plastic material behavior; mainly because critical modes may be excluded by neglecting elastic deformations. All bifurcation points found in Chu’s study occur after the maximum pressure point and, in the limit of a very thin-walled tube, modes corresponding to all t Division of Engineering, $ Department

Brown University, Providence, RI 02912, U.S.A. of Solid Mechanics, The Technical University of Denmark, 2800 Lyngby, 121

Denmark.

122

M. LARSSON, A. NEEDLEMAN, V. TVERGAARD and B. STOR~KERS

circumferential wave numbers are critical simultaneously. For thicker tubes the critical mode is generally characterized by one circumferential wave, although CHU (1979) found a two wave mode critical for a very high thickness to radius ratio. The bifurcation analyses rely on the general theory of HILL (1958), and the critical modes are determined by numerical solutions through the wall thickness. The experiments to be reported on (see also LARSSON,1978) were carried out in order to study in more detail the instabilities and failure modes of thick-walled closed-end cylinders under internal hydrostatic pressure. Some of the cylinders tested were made of an aluminum alloy and some were made of pure copper. In the tests, carried out for a variety of thickness to radius ratios, the tubes essentially retain cylindrical symmetry until somewhat beyond the maximum pressure. Then circumferentially non-uniform deformation initiates. Thus, the behavior in the tests to be discussed here is in good qualitative agreement with the earlier theoretical bifurcation predictions. The development of the final failure mode in the tube tests exhibited several features which are of interest from a theoretical point of view. The neck shapes observed for the two materials differ considerably. The copper tubes exhibit diffuse necking modes while a short wave pattern of surface waves appears in highly strained parts of the aluminum tubes. Furthermore, the development of localized shear bands in the neck region seems to control the mode of fracture. Theoretical predictions of such features in a solid subject to non-homogeneous straining have been obtained quite recently by TVERGAARD,NEEDLEMANand Lo (198 1) for the analogous case of a plane strain tensile test, and further experience has been gained by TRIANTAFYLLIDISet al. (1982) for plate bending. These investigations were carried out numerically, using a very fine grid to be able to resolve the localized shearing, and the material model applied was the J, corner theory developed by CHRISTOFFERSENand HUTCHINSON (1979). The development of circumferentially non-uniform deformation in internally pressurized cylinders has recently been investigated by TOMITA (1980) using classical smooth yield surface plasticity theory. The computed deformation patterns indicate growth of the most critical mode, which is characterized by one circumferential wave, but no tendency is exhibited for the deformation pattern to develop regions of intense localized shearing. Studies of a homogeneously strained solid show that bifurcation into a localized shear band is first possible when the equations governing incremental equilibrium lose ellipticity, HILL (1962) and RICE (1976). For the classical elastic-plastic solid no such bifurcation is predicted at an achievable strain level, but this result is very sensitive to small deviations from the assumptions of a smooth yield surface and plastic incompressibility, RICE (1976), NEEDLEMANand RICE (1978). Thus, for the J2 corner theory material shear bands initiate at achievable strains; but the subsequent growth of localized straining is quite sensitive to details of the vertex description, HUKHINSON and TVERGAARD (198 1). The critical strain for bifurcation into a surface wave mode is somewhat below that for shear bands (see HUTCHINSON and TVERGAARD, 1980). The numerical investigation for the plane strain tensile test, TVERGAARD, NEEDLEMANand Lo (1981), indicates that shear bands can initiate at the surface, due to the local strain concentrations induced by the growth of surface wave modes. In the present paper the numerical procedure of TVERGAARD et al. (1981) is used to analyze the behavior of the pressurized tubes.

123

Failure of internallypressurizedmetal cylinders METHOD OF ANALYSIS

2.

In the numerical investigation of the deformations, the tubes are assumed to be cylindrical bodies and to remain cylindrical throughout the deformation. Thus, any edge effects observed experimentally are neglected in the analyses. A Lagrangian formulation of the field equations is adopted, using the initial unstressed configuration as reference. A material point is identified by the Cartesian coordinates xi in the reference state, where the initial cylinder axis is chosen as the x3axis. In terms of the displacement components ui on the reference base vectors the Lagrangian strain tensor is given by Vij

=

ft”i.j

+

u,j.i

+

4i”k.j)

(2.1)

where ( ).i denotes covariant differentiation in the reference frame. Here, Latin indices range from 1 to 3. The contravariant components zij of the Kirchhoff stress tensor are related to the corresponding components of the Cauchy stress tensor oij by + = J(Gjg) Here, g and G are the determinants and the current state, respectively. virtual work

(2.2)

of the metric tensors gij and Gij in the reference state Equilibrium is expressed in terms of the principle of

&vij I

@.

dV =

Ti6ui dS

(2.3)

ss

V

where I/ and S are the volume and surface, respectively, in the reference configuration, and T’are the components of the nominal traction vector on the reference base vectors. The boundary conditions express the requirement of zero tractions on the external surface and hydrostatic pressure p on the internal surface T’ = 0,

for

T’ = -pc?n,, Here, the configuration

dependent

{(xl)” +(x2)‘} ‘I2 = b, - At/2 for

((x1)2+(x2)2)112

= a,+At/2.

(2.4) (2.5)

tensor c?‘is given by (see, for example, SEWELL, 1965)

cli’ = teijk&lrnr(gjl +

Uj,l)(gkm

+

Uk,m) (2.6)

where eijk is the alternating tensor and n, is the normal vector, in the reference coordinate system. The initial thickness t is taken to vary with the circumferential angle cp = arccos[x’/{(x’)2 +(x2)‘> 1’2] according to At = g cos 9.

(2.7)

Thus, b, and a, are the initial external and internal radii, respectively, and a thickness inhomogeneity is specified by the amplitude f. The straining in axial direction could be computed by requiring axial equilibrium under the hydrostatic pressure loading. However, since no changes of length were detected in the experiments, the assumption of plane strain (u,“~= uf3 = ~,‘3 = 0) is considered a sufficiently good approximation. Furthermore, with the initial thickness

124

M. LARSON,A. NEEDLEMAN, V. TVERGAARD and B. SIOR~KERS

variation (2.7) the deformations remain symmetric about the xl-axis. Therefore, symmetry conditions are specified for x2 = 0, and only one half of the cylinder is analyzed numerically. The constitutive relations to be used here are those of J, corner theory, developed by CHRISTOFFERSENand HUTCHINSON (1979). This phenomenological theory of plasticity is developed to embody some features of physical models of polycrystalline aggregates, based on single crystal slip (see HUTCHINSON, 1970). The instantaneous moduli for nearly proportional loading are chosen equal to those of J, deformation theory, and for increasing deviation from proportional loading the moduli stiffen monotonically until they coincide with the linear elastic moduli for stress rates directed along or within the corner of the yield surface. With M$, denoting the deformation theory compliances and Aijk, denoting the linear elastic compliances, the plastic compliances of deformation theory are Cijkl = M& - djkl. The yield surface in the neighborhood of the current loading point is taken to be a cone in stress deviator space, with the axis in the direction 1”

= Sij(c,,pqSmnSP4)

where 8 = rij- G’jrij3 is the stress deviator. An angular measure (3of the stress rate direction cos 6, =

and a stress rate potential

l/2

(2.8)

relative to the cone axis is defined by

Cijkl~ij~kf(Cmnpq~mn~P4)1/z

at the vertex is formulated

w = +&jklWl

as

+$f(e)cijk,YijBkl.

Here, T’jdenotes the Jaumann rate of the Kirchhoff convected rate by the expression

(2.10) the strain rate is obtained liij =T

a2w &J aykl

(2.10)

stress tensor, which is related to the

7” = + + Gikrjrqkl + Gjkrilqk,. From the potential

(2.9)

(2.11)

as

yk’= Mijkl(h))?kL.

(2.12)

The transition function f(f3) is unity throughout the total loading range, 0 < 8 < 8,, and is identically zero for 8, < 6 d n, where Qc denotes the angle of the yield surface cone. In the transition regime, 8, < fI d H,, f(e) is taken to decrease monotonically and smoothly from unity to zero in a manner that ensures convexity of the incremental relation. The transition function to be employed here (see also HUTCHINSON and TVERGAARD, 1980 and TVERGAARD, NEEDLEMANand Lo, 1981) is specified by

f(e) = where I(cp) = i(dg/dp)/g,

dd

’

g(cp)Cl + L2((P)l’

(2.13)

and for

= cl-i(~-e,)~e.-e,)137~2:

with 8, = 8, -n/2.

e(q) = cp+ arctan[l(cp)]

for

8, e. G up G 8,

0 < cp 6

(2.14)

Failure of internally

Inverting

pressurized

(2.12) gives the expression

125

metal cylinders

for the Jaumann

rate of the Kirchhoff

7i.i = Ri.ik’(Qkl.

stress (2.15)

In the total loading range, 0 d 8 < B,,, the moduli R are those of a finite strain J, deformation theory. As in previous applications of the J2 corner theory the incremental moduli of a nonlinear elastic solid are employed. The principal axes techniques of HILL (1968) may be used to determine the components Rijklof these moduli on the Eulerian principal axes as Es/E,

Rijkl = 2G, ~ik6jl + ~

6ij6k,

-

for R 1212

-

R 2112

-

R1221

=

-

sijskl

1

’

2 ES/E,-&l-2v,)

s

i=j,

R2121

=

0:

1’

k=l Gsq>

(2.16) q = (cl - .s2)coth(e1 -Ed).

Here, si are the principal logarithmic strains, Sij are the stress deviator components on principal axes, with ee = (3sijsij/2) u2 , 6ij is the Kronecker delta, and E, and E, are the slope and the ratio of stress to strain, respectively, for the uniaxial Kirchhoff stresslogarithmic strain curve. Furthermore, with Young’s modulus E and Poisson’s ratio v, the effective shear modulus is G, = +ES/(l + v,), and the effective Poisson ratio is v, = i + (V -+)E,/E. The components (2.16) on principal Eulerian axes are used to determine the components Rijk’(0) on current deformed coordinates, and substituting (2.11) into (2.15), the incremental constitutive relationship is obtained in the form $J’ = ,r$kl(e)dkl.

(2.17)

The angle BCof the yield surface cone is specified by tan 19,= &

tan /?,,

+--

1][;-

11-l

(2.18)

where p, is taken to be given by tan /?, = - ~,/(0,” -a;),

for

(2.19)

and otherwise /?, = (PC),,,. Here, oy is the initial yield stress in uniaxial tension. The uniaxial Kirchhoff stress-logarithmic strain curve is here represented by an expression of the form

where n, and n, are strain-hardening exponents, and c, and c2 are material parameters. This particular expression is chosen to obtain a reasonably good fit of the measured stress-strain curves for the aluminum alloy and the pure copper used in the tube experiments.

126

M.

LARSSON, A. NEEDLEMAN,

V. TVERGAARD and

B. STOK~ERS

For a perfect elastic-plastic tube bifurcation away from the cylindrically symmetric state has been analyzed by CHU (1979). If internal pressure is prescribed to increase monotonically, stability is lost at the maximum pressure point ; but in cases where the volume enclosed by the tube is the prescribed quantity, the cylindrically symmetric deformation state may continue in a stable way beyond this point. The bifurcation points found by CHU (1979) occur somewhat after the pressure maximum. Both for thin and moderately thick tubes the critical bifurcation mode has one circumferential wave. In particular, for the limit of a very thin-walled tube, the ratio of hoop stress orp to tangent modulus E, at the maximum pressure point is oJE, = 2/3, and bifurcation occurs at crp/Ef = 413. Even for rather thick tubes CHU (1979) finds that 4E,/3 is a good approximation of (6,-a,) at bifurcation, where (-) denotes averages through the thickness. The condition for bifurcation into a shear band under homogeneous plane strain tension is also the criterion for local loss of stability in the most strained part of the pressurized tube. Since the slight elastic compressibility included in the present study has little influence on this criterion, the results obtained by HILL and HUTCHINS~N (1975) for an incompressible solid will be used here. These results are expressed in terms of two instantaneous moduli, p* = E,/3 and p = qE,/3, and in terms of the true tensile stress c, which may be interpreted as the in-plane principal stress difference since the deformations are unaffected by a superposed hydrostatic pressure. For p*/p < 2 bifurcation into a shear band occurs at the elliptic-hyperbolic boundary, with the critical stress (Tand the critical inclination $ between the shear band normal and the tensile direction given by c = 4JCPL*(P -

0-4/l* tan’ $ = 1 + ____ 2/.-a

cl*n

(2.21)

Bifurcation into a surface wave mode on a free surface, HUTCHINS~N and TVERCAARD (1980), is the limiting case of a very short wavelength diffuse mode discussed by HILL and HUTCHINS~N (1975). This instability occurs prior to (2.21), at a stress given by c-r

_=1+a 4P

4P*

JC) 2p--rr __ 2/L+o

(2.22)

.

The relationship (2.22) is also valid under a hydrostatic surface pressure, when 0 denotes the in-plane principal stress difference. The numerical solutions for the pressurized tube are obtained by a linear incremental method. If the current values of the field quantities are taken to express an approximate equilibrium state, the equations governing increments (. ) of the field quantities are obtained by expanding the principle of virtual work (2.3) about this state, using (2.4) and (2.5). To lowest order the incremental equation is

=

s {-pii’-@?‘}n,6u, s

dSs

s pcxi’nrchi dS . 1

(2.23)

Failure of internally

pressurized

127

metal cylinders

p and p are the current hydrostatic pressure and the prescribed pressure increment, respectively, on the internal surface, and both these quantities are zero on the external surface. The correction terms, bracketed in (2.23), are included to prevent drifting of the solution away from the true equilibrium path. A finite element approximation of the displacement increments is employed for the numerical solution of (2.23). The grid consists of quadrilaterals, each built up of four constant strain triangular subelements. Static condensation is used for each quadrilateral to eliminate the nodal degrees of freedom associated with the central node. In the expected neck region a relatively fine grid is used, whereas away from this region the grid size is increased continuously in the circumferential direction. In the formulation (2.23) the pressure increment fi is taken to be prescribed, but this quantity changes sign during the deformation. As in a previous investigation of pressurized spherical shells, TVERGAARD (1976), a mixed finite element-Rayleigh-Ritz method is used to overcome numerical difficulties associated with this phenomenon. Here,

3.

EXPERIMENTAL

METHODS

AND

RESULTS

Five aluminum alloy and four copper closed-end cylinders of different geometries were subjected to expansion by means of internal hydrostatic pressure. The strainhardening characteristics of the two materials employed are depicted in Figs. 1 and 2 where the load (nominal stress)-engineering strain (change in length/initial length, AL/L,) curves obtained in ordinary tensile tests at a strain rate of 10d3 s-r are plotted. The elastic material parameters were found to be given by E = 6.93 x lo4 MPa, v = 0.30 for the aluminum alloy and E = 11.6 x lo4 MPa, v = 0.33 for the copper. The chemical composition of the aluminum alloy by percent weight was Al 97.4, Mg 09, Mn 0.7 and Si 1.0. The copper was pure within 99.90%. Tubular specimens were machined from solid bars having an original diameter of 25 mm. In their final shape, all cylinders were of common length, 190 mm, and external radius, 10 mm. This length to radius ratio was presumed to be sufficient for end effects to be negligible. Five aluminum specimens with internal radii of 5,6,7,8, and 9 mm and

200 I

I

0’

0

0.05

I

I

0.1

0 15

I

0.20

AL/L,

FIG. 1. Load vs engineering

strain, AL/L,,, for the aluminum

alloy

128

M. LARSSON, A. NEEDLEMAN, V. TVERGAARD and B. STCIK~KFRS

-

I

40 I

0

0.125

0.25 AL/L,

FIG. 2. Load vs engineering

I

I

0.375

0 500

strain, AL/L,,

for copper.

four copper specimens with internal radii of 5,6,7, and 8 mm were tested. In all cases the test procedure was repeated with a specimen of identical geometry. In order to restore the as received material condition as closely as possible the copper tubes were annealed at 450°C for half an hour and subsequently air-cooled before testing as were the uniaxial test specimens. In Fig. 3 the geometry of a typical specimen is drawn to scale. Great care was taken, when preparing specimens by drilling and machining centered holes in the solid bars. In no case after the machining procedure, was it possible to detect any deviation from nominal dimensions exceeding 0.01 mm when keeping the specimens intact. The cylinders were subjected to increasing internal pressure applied by means of a hydraulic loading device; with efforts being made to impose loading rates approximately equal to that in the tensile tests. The magnitude of the pressure was determined from the measured resulting force acting on a piston used for load application. At selected instants unloading was performed and the current external diameter and length of the cylinders were measured together with any detectable deviation from axial symmetry. In general no change in length could be detected within a measuring accuracy of 0.05 mm implying a prevailing state of plane strain, any axial strain being less than 3 x lo- 4. A more detailed discussion of the experimental procedure has been given by LARSON (1978). The pressure-expansion curves for aluminum alloy and copper tubes are shown in Figs. 4 and 5, respectively. In each figure, the curves obtained for one of the two specimens tested with a particular initial radius ratio, b,/a,, are displayed. The expansion was determined by dismounting the tubes at selected stages of expansion, and the current external radius measured. The current internal radius, u, was then calculated assuming incompressibility. The measure of expansion employed in Figs. 4 r-__-_-_--_-_-----_-_ L______________-_-__-_

FIG. 3. The test specimen geometry.

Failure of internally

pressurized

129

metal cylinders

160 b,/a,=

1.25

b,/a,= 0’ 0

1 0.025

2.0

b,/a,= 1 I 0.05 0.075

I.11 I I 0.1 0.125

I 0.15

In (a/at) FIG. 4. Pressure

vs average hoop strain at the internal surface, In(&) initial radius ratios.

160

for aluminum

t b,/a,=

0’

alloy tubes with various

I

0

0.1

I

0.2

I

0.3

I

0.4

I

0.5

2.0

I

0.6

In(a/a,) FIG. 5. Pressure

vs average

hoop strain at the internal surface, In(a/n,) for copper radius ratios.

tubes with various

initial

and 5 is In(a/a,). This quantity will be referred to subsequently as the average hoop strain at the inner radius. The observed values of the maximum pressure and the corresponding values of the average hoop strain at the internal radius for each specimen tested are given in Table 1. There is good agreement between the results shown in Table 1 for each of the two copper specimens with the same initial radius ratio, be/a,. There is considerably more variation in the corresponding results for the aluminum alloy tubes. Due to the low hardening properties, the aluminum tubes were not annealed before testing. In retrospect it seems advisable to have done so. When considering the variations in results especially for thicker tubes tested under nominally identical conditions, it appears plausible that material inhomogeneity and anisotropy were induced by the tube forming procedure. From the flatness of the pressure-expansion curves in Figs. 4 and 5 it can be appreciated, however, that the values of the average hoop strain at maximum pressure were indeed difficult to determine accurately.

M. LARSON, A. NEEDLEMAN, V. TVERGAARD and B.

130

TABLE 1. Maximum

STOR~KERS

pressure and average hoop strain at the maximum pressurefbr

each

specimen tested Specimen number Al Al Al Al Al Al Al Al Al Al

1 2 3 4 5 6 7 8 9 10

b,/u, 2.0 2.0 1.67 1.67 1.43 1.43 1.25 I.25 1.11 1.11

lnGhh) 152.4 133.7 94.9 88.1 56.2 66.2 38.1 35.0 15.6 15.0

0.151 0.074 0109 0.063 0.080 0.046 0.070 0.042 0.043 0.027

Specimen number Cu 1 cu2 Cu 3 cu 4 cu5 Cu6 cu 7 Cu 8

h&lo

2.0 2.0 1.67 1.67 1.43 1.43 1.25 1.25

Pin,, MPa 129.9 126.2 94.9 92.4 65-O 66.8 41.2 42.5

ln(a,,&,) 0.37 0.39 @33 0.31 0.30 0.36 0.32 0.27

All tubes were tested to final fracture except for one of each material, for which the tests were deliberately interrupted after a pressure maximum had been passed with the intention of recording the pre-fracture mode of deformation. Figure 6 displays crosssections of deformed aluminum and copper tubes at values of ln(a/a,) about 0.16 and 0.40, respectively. It is particularly interesting lo note that the copper tube in Fig. 6(b) has been deformed into a shape, for which the deviation from circumferential homogeneity is described by a wave number of unity. Detailed measurements proved this to be true within a high degree of accuracy. While the same is also essentially true for the aluminum specimen, this tube has suffered a more severe strain localization as is apparent from Fig. 6(a). Figure 7 depicts an additional phenomenon as exhibited by aluminum tubes in the stages of deformation near fracture and after fracture (also see Fig. S(b)). In the most highly strained region, a string of parallel ridges in the longitudinal direction has evolved. The height of the ridges is approx. 0.05 mm and the wavelength of the patterns is about 0.10 mm. In Fig. 8 the final fracture appearance of an aluminum tube with an initial radius ratio of 1.67 is shown, while Fig. 9 depicts the final fracture appearance for a copper tube with the same initial radius. All tubes, except for one made of aluminum, failed under decreasing pressure due to shear fracture, the surfaces of which formed an angle relative to the radial direction at the inner boundary of approx. 57” for the aluminum alloy and 62” for copper as may be seen in Figs. 8 and 9, respectively. Of more interest in the present context, however, is the pre-fracture deformation mechanism. In order to study this matter in some more detail, several tube samples were subjected to optical microscopy. Several cuts were made orthogonal to the axis of fractured cylinders and the surfaces were ground, polished with diamond paste, and finally chemically etched. Figure 10 shows some results of microscopic observations under oblique illumination. Figure 10(a) shows a section of an aluminum cylinder a bit away from a fractured section. The two slightly curved shear bands are inclined to the internal tube surface by

FIG.6 Deformed

cross-sections of tubes prior to failure. (a) Aluminum alloy with initial radius r atio h,/l~, = 1.67; (h) copper tube wsith initial radius ratio h,/cr, = 1.67. 131

FIG. 7. Surface waves on an aluminum

alloy tube.

FIG. 8. Final fracture

appearance

on an aluminum alloy tube with initial radius ratio b,/a, = 1.67. (a) Crosssectional view ; (b) axial view.

133

9. Final fracture

appearance

on a copper tube with initial radius ratio h,/cr, = 1.67. (a) Cross-section view ; (b) axial view.

134

(4 Cross-sectional

micrograph

of shear bands in an aluminum

alloy tube with initial radius

ratio

I.4 13t(b) intersection ofshear bands in an ~iuminum alloy tube with initial radius ratio f:lo/(to = 1.43; ‘OS? ;-sectional micrograph of ductile fracture in a copper tube of initial radius ratio h,/,

10 = 2%

13s

Failure of internally pressurized metal cylinders

137

approx. 40”. The details of the intersection of the two bands at the external tube surface is shown in Fig. 10(b). A voidage concentration along the bands is observable. The copper behavior was different. Figure IO(c) shows a section of a copper tube with a major crack penetrating more than half way through a neck region. Although there is a certain radial discontinuity at the fractured internal surface, there is no trace of intense shearing ahead of the crack. In no other copper tubes were any shear bands detectable either. 4.

NUMERICAL RESULTS

We first focus on the dependence of the maximum pressure on tube size, for perfect tubes and for tubes with an initial thickness imperfection in the shape of the mode associated with the first bifurcation point. For perfect tubes, neglecting the small effect of elastic compressibility, the stress state at each material point is one of monotonically increasing plane strain tension with a superposed hydrostatic stress. Hence, the maximum pressure attained in a perfect tube is independent of the vertex characteristics. Even for imperfect tubes this remains essentially true since the first bifurcation mode does not involve an abrupt change in loading path and, in any event, the bifurcation mode amplitude grows rather little prior to the attainment of the maximum pressure. Accordingly, the variation of maximum pressure with tube size is governed by the material properties through the uniaxial stress-strain curve via some appropriate yield function. Uniaxial stress-strain curves of the form (2.20) are employed which give a reasonably good fit to the observed uniaxial behaviors of the aluminum alloy and copper materials used in the experiments. A comparison of the computed and observed maximum pressures (and the strains at which the pressure maxima are attained) provides some indication of the appropriateness of the assumption, embodied in the constitutive law, that the material work-hardening in various stress states is correlated by the von Mises effective stress. For one aluminum alloy tube and for one copper tube we also carry out detailed analyses of the evolution of the tube configuration well after the onset of necking. These results do depend on the assumed vertex characteristics. 4.1 Pressure-expansion

results

The uniaxial stress--strain relations of the materials used in the experiments are not well represented by a simple power law. Hence, in the computations carried out here the more complex representation (2.20) is employed. To characterize the aluminum alloy, the material parameters in (2.20) are taken as E = 69.3 x lo3 MPa

Cl = 30.0

n, = 1.18

v = 0.30 C%= 0.022

cY = 66.0 MPa n, = 17.6

(4.1)

while for the copper, we take E = 116.0 x lo3 MPa c1 = 69.0

n, = 1.1

v = 0.33 c2 = 0.03

oY = 35.0 MPa n2 = 4.58.

(4.2)

M. LARSSON,A. NEEDLEMAN,V. TVERCAARD and

138

B. STOR~KEKS

In order to facilitate a direct comparison with the experimental results, the material parameters in (4.1) and (4.2), as well as the results of the computations, are presented in dimensional form. Using the material parameters (4.1) and (4.2) pressure-volume relations were calculated for various size aluminum and copper tubes. Figures 11 and 12 depict the maximum pressure as a function of the ratio of initial outer radius, b,, to initial inner radius, a,, for the aluminum and copper tubes, respectively. In these computations, the initial outer radius of the tubes is considered as fixed at 10 mm and various initial internal radii are specified to give initial radius ratios corresponding to the values employed in the experiments. Calculations were carried out for perfect tubes and for tubes with an imperfection of the form (2.7), with r = 0.05 mm. For the tubes with an initial thickness imperfection, b, is the mean initial outer radius and a, is the mean initial internal radius. The amplitude of the initial imperfection is substantially larger than any plausible value for the tubes employed in the experiments and was chosen to explore the influence of a rather large thickness imperfection. Note also that the imperfection amplitude has been taken to be independent of the tube thickness, so that a larger relative imperfection, f/to, results for thinner tubes. The computations for the imperfect cylinders were carried out using a grid consisting of 8 elements through the thickness (for the aluminum tube with &/a, = 1.11, 4 elements were employed through the thickness) and 48 elements in the circumferential direction, The deformations were constrained to remain symmetric about the xl-axis so that only half the cylinder was analyzed numerically. For the perfect cylinders only 8 elements were used in the circumferential direction. Even for the relatively large imperfection employed, the maximum pressures

ot,,,,‘,,,,‘,“,‘,,,“~,,,l 1 1.25 1.5 1.75

2

2.25

bO/aO FIG. 11. The maximum

0 average

pressure for aluminum alloy tubes as a function of initial radius value obtained for the two specimens used in the experiments, + computed

ratio, /~,/a,; value.

Failure

of internally

pressurized

139

metal cylinders

25

0

1

,.,.‘,,,,‘,,~,‘,,1,1,.,,1 1.75 1.25

1.5

2

2.25

b,/a, FIG. 12. The maximum pressure for copper tubes as a function of initial radius ratio, b,/a,, ; 0 average value obtained for the two specimens used in the experiments, + computed value.

attained for both the aluminum alloy and copper tubes were virtually identical for both the perfect and imperfect cases. For comparison purposes Figs. 11 and 12 display the mean values of the observed maximum pressures for the two specimens with a given initial radius ratio, be/a,, used in the experiments. The agreement between the computed and observed maximum pressures is quite good for the aluminum alloy tubes, with the exception of the tube for which ho/a, = 2.0. On the other hand, the computed maximum pressures for the copper tubes are consistently lO--12% greater than the observed values for all radius ratios considered, indicating a somewhat softer behavior in plane strain than predicted by the von Mises equivalent stress. Figures 13 and 14 show a measure of the enclosed volume per unit length at the maximum pressure as a function of initial radius ratio, b,/a,, for the aluminum alloy and copper tubes, respectively. The enclosed volume per unit cylinder length, V, is given by v=+

+ J aijnj(xi u,)dS

s

(4.3)

where S is the inner surface of the cylinder in the undeformed state, xi are the initial Cartesian coordinates of points on this surface and clii is given by (2.6). We then define the average current internal radius, a, by a = J(V/rd

(4.4)

and the measure of enclosed volume per unit cylinder length employed in Figs. 13 and 14, and subsequently, is ln(u/u,). This quantity represents the average (logarithmic) hoop strain at the inner radius.

M. LARSON, A. NEEDLEMAN,V. TVERGAARD

140

0.16 1'

3 1

8.

I,

z 3 r a,.

I,

2,

and

B. STOR~KERS

- > I,,

8,

s u,

0.06

0.06

FIG.13. The value of the average hoop strain at the internal surface at the maximum pressure, In(a,,Ja,), for aluminum alloy tubes as a function of initial radius ratio, b,/a,; 0 average value obtained for the two connecting values for perfect tubes and specimens used in the experiments ; + computed values with __ _._ connecting the values for tubes with an imperfection of the form (2.7) with c = 005 mm (b, is fixed at 10mm).

0.36

-

0.34

-

2

0.32

-

1

0.30

-

0.26

-

0.26

-

-.

z

FIG.14. The value of the average hoop strain at the internal surface at the maximum pressure, ln(a,,Ja,), for copper tubes as a function of initial radius ratio, b,,/a,; 0 average value obtained for the specimens used in the experiments; + computed value.

Failureof internallypressurizedmetalcylinders

141

As displayed in Fig. 13, the computed values of the average hoop strain at the maximum pressure, ln(a,,Ja,), for the aluminum tubes are somewhat above the observed values, with the exception of the tube with &,/a, = 2.0. The presence of the initial imperfection with amplitude f = O-05 mm lowers the value of ln(a,,Ja,), but even with this rather large imperfection the computed values are still above the observed values. Remembering, however, the difficulties in determining ln(u,,Ju,) precisely, as indicated in Section 3, the discrepancy in Fig. 13 is not believed to be due to the presence of a geometric imperfection. The computed values of ln(u,Ju,) for copper, Fig. 14, are smaller than the observed values. For the copper tubes, the computed average hoop strains at the pressure maximum for the perfect and imperfect tubes coincide on the scale of Fig. 14 and only one curve is shown in this figure. The agreement between the predicted and observed values in Figs. 11-14 is considered fairly good. The remaining discrepancies probably arise because the measured work-hardening rates in uniaxial tension and in cylindrical tube expansion, essentially plane strain tension, are not precisely correlated by the von Mises effective stress and are not due to any deficiencies in representing the uniaxial data with an expression of the form (2.20) using the parameter values (4.1) and (4.2). In fact, these parameter values were chosen to give as good a representation as could be found of the tube expansion data while remaining reasonably faithful to the observed uniaxial stress strain curves.

4.2 Flow localization

results

We now turn attention to detailed analyses of aluminum alloy and copper tubes with an initial radius ratio, &,/a,,, of 1.67 corresponding to tubes with inner radii, a,, of 6 mm and outer radii, b,, of 10 mm. An initial thickness imperfection of the form (2.7) is specified with c = 0.01 mm, which is in the range of the amplitude of thickness imperfection possible in the tubes used in the experiments. In these computations 24 elements were employed through the thickness of the cylinder and 90 elements in the circumferential direction. A variable mesh was used in the circumferential direction, with the mesh in the thinnest part of the cylinder designed to give the most favorable orientation of element diagonals for shear bands as discussed by TVERGAARD, NEEDLEMANand Lo (1981). The undeformed meshes are depicted in Figs. 17 and 21. The aluminum alloy is modelled by a J, corner theory solid with a rather sharp vertex, having a limiting cone angle, (fi,),,,, of 135”. The copper is assumed to have a more blunt limiting cone angle for which (PC),,,,, = 115”. For both materials the angle limiting the total loading range, 8,, is taken as 0, = f3,/2-7r/4 with 8, given by (2.18). For the aluminum alloy material model a tube with bo/uo = 1.67 bifurcates into the diffuse necking mode when ln(u/uJ = 0.126 and the critical strain for bifurcation into the surface wave mode is @168. Bifurcation into the shear band mode occurs at a critical logarithmic strain of 0.214. The critical inclination between the shear band normal and the tensile direction, $, is 5 1.1 O.For the material model representing copper, bifurcation into the diffuse necking mode in a tube with b,,/u, = 1.67 occurs when ln(u/u,) = 0.469, bifurcation into the surface wave mode occurs at the logarithmic strain O-488 and the

142

M. LARSON, A. NEEDLEMAN, V. TVERGAARD and

B. STOR~KERS

60

0

0.025

0.05

0.075

0.1

0.125

0.15

h(a/aa) FIG. 15. Computed curve of pressure versus average hoop strain at the internal surface, In(a/cr,) for an aluminum alloy tube. The initial outer radius, b,, is 10 mm, the initial inner radius, a, is 6 mm and the imtial imperfection is of the form (2.7) with r = 0.01 mm.

0

0.025

0.05

0.075

0.1

0.125

0.15

Wa/a,) FIG. 16. Computed curve of the ratio of the thickness change at the thinnest section, At(O), to the thickness change at the opposite section, At(n), as a function of average hoop strain at the internal surface, In(a/a,), for an aluminum alloy tube. The initial outer radius, b,, is 10 mm, the initial inner radius, aa, is 6 mm and the initial imperfection is of the form (2.7) with r = 0.01 mm.

Failure of internally pressurized metal cylinders

143

logarithmic strain for shear bands is 0.567. In this case the critical band inclination, $, is 604”. Figure 15 shows the curve of pressure versus average logarithmic hoop strain at the inner surface, ln(a/a& for the aluminum material model. The maximum pressure of 85 MPa is attained at ln(a/a,) = 0.093, which agrees well with the corresponding results for a perfect cylinder with b,/a, = 1.67 displayed in Figs. 11 and 13. After the maximum pressure has been attained, the pressure falls rather slowly until diffuse necking occurs. At about ln(a/u,) = 0.120, the pressure begins to fall off much more rapidly. Figure 16 shows the ratio of the change in thickness (current thickness minus initial thickness) at the thinnest section, At(O), to that at the opposite section, At(n) with increasing expansion as measured by ln(u/u,). Since plastic yielding initiates at the thinnest section, thinning occurs more rapidly at this section until the cylinder becomes fully plastic. Once the entire cylinder has yielded, the rate of thinning varies little around the cylinder surface and the ratio At(O)/At(n) actually decays toward unity until the diffuse bifurcation point is approached. Intense local thinning then takes place. The development of the deformation pattern for the aluminum alloy material model is shown in Fig. 17. Depicted in this figure is the finite element mesh at various stages of loading (only the quadrilaterals are shown). For comparison purposes, the undeformed finite element mesh is also displayed. The symmetry about x2 = 0 is enforced by the boundary conditions; only one half of the mesh is employed in the computations. The first stage of deformation depicted in Fig. 17 ln(a/u,) = 0.109, is somewhat beyond the maximum pressure point. No deviation from a cylindrical deformation pattern is visible, although reference to Fig. 16 indicates that some local thinning due to growth of the diffuse necking mode has occurred. At ln(u/u,) = 0.125, the thinning is increasingly localized and a substantial portion of the cylinder has undergone elastic unloading. However, the deviation from a perfectly cylindrical deformation state is barely visible. Shortly after this stage, the highly localized thinning at the thinnest section is evident and flattening of the outer surface near this section takes place (Figs. 17~~). In Fig. 17(e), two shear bands emanating from the local protrusion at the thinnest section are visible. The similarity with the deformed geometry depicted in Fig. 6(a) is evident. Figure 18 gives a more quantitative picture of shear band development. In this figure contours of constant maximum principal logarithmic strain are plotted in the current deformed configuration. Only one half of the cylinder is shown. At the first stage of expansion shown, ln(u/u,) = 0.109, the contour line 0.10 is shifted away from the inner cylinder surface in the diffuse necking region. At ln(u/uJ = 0.125, the critical strain for shear bands has been reached. The effect of shear band initiation on material still in the elliptic regime is reflected in the strain contours. With continued expansion, deformation is almost entirely confined to the necked down region. The fully developed shear band pattern emerges with rather little additional overall tube expansion, as measured by ln(u/u,). Figures 19-22 display the corresponding results obtained for a tube with &,/a, = 1.67 using the copper material model. In Fig. 19 the maximum pressure of 106 MPa is attained at ln(u/u,) = 0.318. The pressure then decreases smoothly until ln(u/u,) is about 0.48. At this point the pressure begins to drop abruptly with increasing expansion. As can be seen by comparing Figs. 19 and 20 this pressure drop begins critical

144

M. LARSSON,A. NEEDLEMAN,V. TVERGAARUand B. STOR~K~RS

UNDEFORMED

CONFIGURATION

(a)

(b) FIG. 17. The initial undeformed finite element mesh and aluminum alloy material model. The initial outer radius, the initial imperfection is of the form (2.7) with e = (c) In(a/a,) = 0.131, (d) In(&)

the deformed mesh at five stages of expansion for the b,, is 10 mm, the initial inner radius, a,. is 6 mm and 0.01 mm. (a) In(n/a,) = 0.109, (b) In(a/a,) = 0.125, = 0,135, (e) ln(a/a,) = 0.139.

somewhat after local thinning initiates and is, in fact, associated with shear band formation. Figure 21 depicts the initial finite element mesh employed in this computation and the deformed mesh at four stages of expansion, while Fig. 22 shows the corresponding contours of constant logarithmic principal strain (in Fig. 22 only half the cylinder is displayed). At the first stage shown in these figures, ln(a/u,) = 0.403, no deviation from a cylindrical deformation pattern is evident even though this configuration is one significantly beyond the maximum pressure point. When ln(a/a,) = 0.474, the strain contours exhibit the pattern associated with the onset of diffuse necking. The last two stages of deformation (c and d in Figs. 21 and 22) depict the course of shear band initiation and growth.

Failure of internally

pressurized

metal cylinders

(e) FIG. 17 (continued).

5.

DISCUSSION

For the ductile metal tubes tested, the deformations remain essentially cylindrically symmetric until after the maximum pressure has been attained. The diffuse necking mode which gives the gross deviation from circumferential homogeneity is characterized by a wave number of unity. After some growth of this mode, failure occurs under decreasing pressure by shear fracture. Particularly, the aluminum alloy tubes exhibit a noticeable localization of strain into shear bands in the necked down region prior to failure, Figs. 6(a) and 10(a), (b). The detailed computations carried out here for the aluminum alloy tube, Figs. 17 and 18, and for the copper tube, Figs. 21 and 22, qualitatively reproduce most of the main features of the observed deformation patterns. These calculations were based on characterizing the material behavior by the phenomenological vertex theory of plasticity, termed J, corner theory, introduced by CHRISTOFWZRSEN and HUTCHINSON (1979).

146

M. LAKSSON,A. NEEDLEMAN,V. TVEKCAARD and B.

STOK~KEKS

In our analysis it is the presence of a vertex on the yield surface at the current loading point that permits the shearing deformation patterns depicted in Figs. 18 and 22 to emerge. The motivation for introducing the phenomenological J, corner theory employed stemmed from the fact that physical models of polycrystalline aggregates based on single crystal slip, e.g. HUTCHINSON (1970), give rise to such a yield surface vertex. Thus, we have presumed that an inherent feature of the plastic flow process, namely the discrete nature of crystalline slip, precipitates shear band initiation and growth in the circumstances analyzed. Other physical mechanisms, for example the 0.05 0.075 0. IO

1

1

(a)

FIG. 18. Contours ofconstant maximum principal logarithmic strain in the deformed configuration (only one half the cylinder is shown) for the aluminum alloy material model. The initial outer radius, b,, is 10 mm, the initial inner radius, co. is 6 mm and the initial imperfection is of the form (2.7) with e= 0.01 mm. (a) ln(a/a,) = 0.109, (b) In(u/a,) = 0.125, (c) In(a/a,) = 0.131, (d) ln(a/a,) = 0.135, (e) ln(a/a,) = 0.139.

Failure of internally

pressurized

metal cylinders

147

.0.15

0130 FIG. 18 (continued).

120

100

80

60

40

20

0 0

0.1

0.2

0.3

0.4

0.5

Ma/a3 FIG. 19. Computed curve of pressure vs average hoop strain at the internal surface, ln(a/a,) for a copper tube,. The initial outer radius, b,, is 10 mm, the initial inner radius, a, is 6 mm and the initial imperfection is of the form (2.7) with ? = 0.01 mm.

148

M. LARSSON,A. NEEDLEMAN, V. TVERCAARD and B. STOR~KEKS 2.5 ~

0

0.1

0.2

0.3

0.4

0.5

Ma/ad

FIG.20. Computed

curve of the ratio of the thickness change at the thinnest section, Ar(O), to the thickness change at the opposite section, At(n), as a function of average circumferential strain at the internal surface, h-@/a,), for a copper tube. The initial outer radius, b,, is 10 mm, the initial inner radius, a,, is 6 mm and the initial imperfection is of the form (2.7) with f = 0.01 mm.

dilational plastic flow induced by the nucleation and growth of microvoids also can promote shear band development. Which mechanism or which combination of mechanisms plays the dominant role is likely to vary from material to material. Even for a given material the dominant mechanism may be stress state dependent as discussed by NEEDLEMAN and RICE (1978). Computations of the behavior of imperfect internally pressurized cylinders analogous to those carried out in this study, but employing classical smooth yield surface plasticity theory have been reported recently by TOMITA (1980). TOMITA (1980) finds that the maximum straining at each radius occurs at the thinnest section, x2 = 0. This is consistent with continued growth of the diffuse necking mode, but no tendency is exhibited for intense shearing to occur obliquely to the radial direction. This lack of localized shearing is not surprising since an analysis which associates shear band initiation with a material instability, reveals that the classical elastic-plastic solid with a smooth yield surface is quite resistant to the localization of deformation into a shear band, RUDNICKI and RICE (1975). We have characterized the aluminum alloy material model by a rather sharp vertex and the copper material model by a rather blunt vertex. Since the aluminum alloy and copper material models have very different uniaxial stress-strain relations, the effect of FIG.21. The initial undeformed finite element mesh and the deformed mesh at four stages of expansion for the copper material model. The initial outer radius, b,, is 10 mm, the initial inner radius, a,, is 6 mm and the initial imperfection is of the form (2.7) with F = @Ol mm. (a) In(a/as) = 0.403, (b) ln(a/a,) = 0.474, (c) ln(a/n,) = 0.483, (d) ln(a/a,) = 0.489.

Failure

of internally

pressurized

metal cylinders

149

150

M. LARSON, A. NEEDLEMAN, V. TVEKGAARD and

B. STOKAK~RS

0.20

(b)

0.25

Ill II 0 25

I

(dl

“““W

o.t30-

FIG 22. Contours of constant maximum principal logarithmic strain in the deformed configuration (only one half the cylinder is shown) for the copper material model. The initial outer radius, b,, is 10 mm, the initial inner radius, a,, is 6 mm and the initial imperfection is of the form (2.7) with g = 0.01 mm. (a) In(a/a,) = 0.403, (b) In(a/a,) = 0.474, (c) In(a/a,) = 0.483, (d) ln(a/a,) = 0.489.

Failure of internally pressurized metal cylinders

151

the vertex characterization on shear band development is not unequivocally exhibited HUTCHINSON and TVERGAARD (1981) and in the present results. However, TRIANTAFYLLIDIS et al. (1982) have studied the effect of vertex characterization on shear band development. In the study of TRIANTAFYLLIDIS et al. (1982), for pure bending of a plane strain strip, the two vertex characterizations employed here, (fi,),,, = 135” and (/_?,)_ = 115”, were used with identical uniaxial stress-strain relations. The more blunt In fact, vertex, (PJmaX = 115”, exhibited significantly slower shear band development. this rather blunt vertex angle was taken here to characterize the copper material model since the tests showed that the copper tubes did not exhibit the pronounced shear band development prior to fracture that the aluminum alloy tubes did, Fig. 10(c). For the aluminum alloy tubes it is perhaps plausible that the stability against flow localization is limited by the plastic flow process itself with the initiation of ductile rupture ensuing in the highly strained shear band. This appears consistent with the observations above and also those of HAHN and ROSENFIELD (1975) on other aluminum alloys. HAHN and ROSENFIELD (1975) noted the occurrence of localized bands of plastic deformation with microcracks often initiating within these deformation bands. Localized shearing prior to any noticeable voidage is also observed in aluminumcopper single crystals, CHANG and ASARO (1981) and in polycrystalline high purity aluminum, PUTTICK(1959). On the other hand, commercially pure copper typically exhibits pronounced microvoid nucleation and growth at the moderate strain levels as encountered in the tube expansion tests, cf. also e.g. PUTTICK(1959). In this material, the weakening effect ofextensive void nucleation and growth, which is not incorporated into the constitutive relation employed here, may play an important role in precipitating localized deformation. It is then plausible that localization and failure are virtually coincident. In this regard, the variation of mean normal stress, $i, obtained from the numerical solution for the copper tube, Figs. 21 and 22, will be commented on briefly. In the early stage of diffuse necking, Fig. 22(b), there is no elevation of mean normal stress in the most highly strained region at the inner boundary. A local elevation in mean normal stress does accompany the shearing deformations exhibited in Figs. 22(c) and (d), so that conditions in this region-increased plastic strains and increased mean normal stress-are favorable for void nucleation and growth. One notable feature of the observed deformation patterns not clearly reproduced by the calculations is the surface wave patterns depicted in Figs. 7 and 8(b). One reason that this feature may not develop in the computations is that the finite element mesh employed could be too coarse to resolve such very short wavelength modes. The surface ripples in Figs. 7 and 8(b) have a wavelength about O-1 mm. The initial mesh spacing shown in Fig. 17 is about 0.095 mm at the inner surface and about 0.158 mm at the outer surface. Since several elements, on the order of &8, are required per half wavelength for adequate resolution, the grid employed is much too crude to resolve the observed ripple pattern. Furthermore, since the surface waves decay rapidly away from the surface, with a decay length of the order of the wavelength, HUTCHINSONand TVERGAARD (1980), the grid is undoubtedly also too crude in the radial direction. However, in Fig. 18(c) (the 0.20 contour) and in Fig. 22(d) (the 0.60 contour), strain contours in the highly strained surface region do exhibit a tendency to oscillate with a wavelength long enough to be resolved by the grid. The computations here were terminated before very

152

M. LARSSON, A. NEEDLEMAN, V. TVERGAARD and B. STORAKEKS

large surface strains developed. TRIANTAFYLLIDIS ef al. (1982) analyzed the pure bending of a plane strip and continued their computations far enough for very large surface strains to develop. Very short wavelength surface ripples, of the order of the mesh spacing, did occur on the compressed surface of one of the strips considered. The rather arbitrary choice of a more blunt vertex, (fi,),,, = 115”, for copper is also related to the fact that no surface waves are observed on the copper tubes, Fig. 9(b). This lack of surface ripples would result from the simplest flow theory of plasticity with a smooth yield surface, see HUTCHINSON and TVERGAARD (1980), which corresponds to the limiting case (fl,),,, + 90” in J2 corner theory. The course of shear band development obtained here for the case of plane strain tube expansion can be contrasted with that encountered by TVERGAARD, NEEDLEMANand Lo (1981) in plane strain tension and TRIANTAFYLLIDIS,NEEDLEMANand TVEROAARI) (1982) in plane strain strip bending using the J, corner theory constitutive law of CHRISTOFFERSENand HUTCHINSON (1979). In the plane strain tension study of TVERGAARDet al. (1981) the effect of various short wavelength thickness imperfections in addition to a diffuse necking mode imperfection on shear band development was analyzed. A variety of shear band patterns were obtained, with the particular pattern depending on the initial thickness imperfection. A similar dependence of the shear band pattern on additional short wavelength surface imperfections probably also occurs for the pressurized tubes. In the plane strain tensile test, the shear bands propagated along nearly straight lines emanating from the local surface strain concentrations induced by the initial thickness imperfection. The deformation gradients in the necked down region into which the shear bands propagated arose primarily from the growth of the diffuse necking mode and were not very large. On the other hand, in pure bending the large prelocalization deformation gradients led to strongly curved shear band patterns, TRIANTAFYLLIDISrt nl. (1982). In tube expansion, significant prelocalization strain gradients exist, but these gradients are not nearly so severe as in the pure bending case. In Figs. 18(b) and (c), the initial direction of shear band growth, as given by the peak of the contour line for 0.20 in Fig. 18(b) and the peak of the contour line for 0.30 in Fig. 18(c) is 50”-52” from the x1axis, in good agreement with the value of 51.1” given by the shear band bifurcation analysis. The initial tendency, as can be seen from the contour line peaks in Fig. 18(c) is for propagation to be along a curve, with the angle tending to increase to about 55”. However, in Figs. 18(d) and (e), the shear band is virtually straight, making an angle of about 55” with the xl-axis. Thus, in this case, where the strain gradient is moderate, the fully developed shear band is straight, despite the initial tendency. We note that at the inner boundary the angle between the shear band and the radial direction is observed to be about 50”, Fig. 10(a). In Fig. 22, for the copper material model, the computation was only carried through the early stages of shear band development and the initial orientation of 57”-58” agrees reasonably well with the orientation of 60.4” given by a shear band bifurcation analysis. As can be seen in Fig. 10(c), the shear fracture occurs essentially straight through the thickness, the angle between the fracture surface and the radial direction being about 60”. TVERGAARD et al. (1981) discussed the importance of an appropriate mesh orientation for resolving shear bands. Here, the undeformed meshes shown in Figs. 17 and 21 were designed to ensure that, in the most highly strained region (where the

1.53

Failureof internallypressurizedmetal cylinders

meshes are finest), the orientation of the element diagonals is along the most favorable angle for shear bands. As in the previous problems considered by TVERGAARD et al. (1981) and TRIANTAFYLLIDIS et al. (1982), the width of the shear bands in an optimally oriented mesh is set by the mesh spacing since there is no material length scale embodied in the constitutive law to set a minimum shear band width. The relationship between the discretized problem solved numerically and the underlying continuum problem has been discussed by TVERGAARD et al. (1981). In one respect, at least, this relationship appears less problematical in the tube expansion problem analyzed here and in the strip bending problem analyzed by TRIANTAFYLLIIXS et ul. (1982). than in the case

of plane

strain

tension.

In plane

strain

tension,

for the J,

corner

theory

solid,

TVERGAARD et al.

(1981) found that internal shear bands formed in the neck. In the discretized problem the separation of the internal bands was set by the mesh spacing due to the lack of a natural length scale. In tube expansion, as in pure bending of a plane strain strip (TRIANTAFYLLIDIS et al., 1982), there is no tendency for such internal bands to form, presumably due to the pronounced deformation gradients. The discussion so far has focussed primarily on the qualitative features of the calculated deformation patterns, which are in good agreement with the general phenomena observed in the tests. The quantitative agreement for some of the pressureexpansion curves is not as satisfactory as we would have liked. As noted previously, the form of the uniaxial stress-strain relation (2.20) with the parameter values (4.1) and (4.2) was chosen as a compromise between an optimum fit to the tensile data and to tube expansion data recorded prior to a pressure maximum. It seems to us most likely that the major reason for any discrepancy is due, at least in part, to the use of the von Mises equivalent stress to correlate the strain-hardening behavior in uniaxial tension and plane strain tension. Implicit in the use of the von Mises equivalent stress is the assumption of plastic isotropy and it is possible that plastic anisotropy may be a factor in this discrepancy. In particular, the underestimation of the maximum pressure for the thickest aluminum tube possibly arises from the strain gradient in conjunction with material inhomogeneity and/or anisotropy.

ACKNOWLEDGEMENTS

The work of A.N. and, in part, the work of V.T. was supported by the U.S. National Science Foundation through Grant ENG76-16421. The computations reported on here were carried out on the Brown University, Division of Engineering, VAX-l l/780 computer. The acquisition of this computer was made possible by grants from the U.S. National Science Foundation (Grant ENG78-19378), the General Electric Foundation and the Digital Equipment Corporation. We are indebted to Mr. A. Thuvander of the Swedish Institute for Metal Research, Stockholm for his able assistance in preparing micrographs.

REFERENCES CHANG, Y. W. and ASARO, R. J. CHRISTOFFERSEN. J. and

1981 1979

Acta Met. 29, 247. J. Mech. Phvs. Solids 27,465.

HUTCHINSON,J. W. CHU, C.-C.

1979

J. appl. Mech. 46, 889.

154

M.

LARSON,

A.

HAHN,G. T. and ROSENFIELD, A. R. HILL, R. HILL, R. HILL, R. HILL, R. and HUTCHINSON, J. W HUTCHINSON, J. W. HUTCHINSON, J. W. and TVERGAARD, V. HUTCHINSON, J. W. and TVERGAARD, V. LARSSON, M.

NEEDLEMAN,V.

TVERGAAKU and B. STOR/IKERS

1975

Met. Trans. 6A, 653.

1958 1962 1968 1975 1970 1980

J. Mech. Phys. Solids 6,236. Ibid. 10, 1. Ibid. 16, 229. Ibid. 23, 239. Proc. R. Sot. Lond. A319, 247. Int. J. Mech. Sci. 22, 339.

1981

Int. J. Solids Struct.

1978

Instabilities in closed-end thick-walled cylinders subject to internal pressure, Diploma Thesis, Dept. of Strength of Materials and Solid Mechanics, The Royal Inst. of Technology, Stockholm, (in Swedish). Mechanics of Sheet Metal Forming (edited by D. P. KOISTINEN and N.-M. WANG),p. 237. Plenum Press, New York.

17. 451

NEEDLEMAN, A. and RICE, J. R.

1978

PUTTICK.K. E RICK.J. R.

1959 1976

Phil. Maq. 4, 964. I‘heoreticul and Applied Mechurzics,(edited

RUDNICKI,J. W. and RICE, J. R. SEWELL,M. J. STORKKERS, B. STIFORS,H. and STOR~KERS, B.

1975 1965 1971 1973

J. Mech. Phys. Proc. Roy. Sot. J. Mech. Phys. Foundations of

TOMITA,Y. TRIANTAFYLLIDIS, N., NEEDLEMAN, A. and TVERGAARD, V. TVERGAARD, V. TVERGAARD, V., NEEDLEMAN, A. and Lo, K. K.

1980 1982

Lond. A286,402. Solids 19, 339. Plasticity, (edited by A. SAWCZUK), Vol. 1, p. 327, Noordhoff, Leyden. Proc. 4th N.C.P. Symp., p. 87 (in Japanese). Int. J. Solids Struct. 18, 121.

1976 1981

J. Mech. Phys. Solids 24, 291 Ibid. 29. 115.

by W. T. KOITER)(Proc. 14th IUTAM Cong., Delft, The Netherlands, 30 August-4 September 1976) p. 207. North-Holland, Amsterdam. Solids 23, 371.