1-s2.0-0045782593900494-main

Computer Methods in Apphed ~~~h~i~s and Engineering 103 (1993) 273-290 North-Holland

CMA 382

Necking in tensile bars with rectangular cross-section

Viggo Tvergaard

Received 25 September 1992 Revised manuscript received 30 October 1992

Tensile instabilities are studied for an el~ti~-plastic bar with rectangular cross-section, using a full three-dimensional numerical analysis. Most analyses are based on the simplest flow theory of plasticity, but also the effect of yield surface curvature and the effect of ductile failure mechanisms are considered in a few computations. For a tensile bar with a square cross-section, the neck development is rather similar to that in a round bar, whereas for a flat plate strip the final neck development appears as a narrow groove inclined to the cross-section of the bar. A range of cross-section aspect ratios are analysed to study the transition between these two modes of necking.

1. Introduction

Localized necking in a tensile test specimen has been studied by a number of authors with focus on round bars for which an axisymmetric solution describes the neck development. Hutchinson and Miles [l] studied bifurcations from a uniform stress state in a homogeneous round tensile bar, based on Hills [2,3] general theory of uniqueness and bi~rcation in elastic-plastic solids, and found that bifurcation occurs a little after the maximum load point with more delay the higher the thickness to length ratio. In the limit of a very long thin bar this result agrees with the classical criterion of ConsidCre [4), and it also agrees with the experimental observation that a neck becomes clearly visible shortly after the maximum load has been passed. Nume~~al studies of neck development in round bars with initial thickness imperfections or imperfections induced by the end conditions have been carried out by Needleman [5], Norris et al. [6] and Argyris et al. f7].

Final failure in the neck of a tensile test specimen has been studied by the use of various material models that account for failure mechanisms. Thus, for a round bar Tvergaard and Needleman [S] used a model that accounts for the nucleation and growth of micro-voids, and found that the experimentally observed cup-cone fracture is predicted without any prior assumption of the fracture path. For a plane strain tensile test Tvergaard et al. [9] have used J, corner theory to study the development of localized shear bands in the neck, leading to final shear fracture of the tensile test specimen.

Correspondence to: Prof. Viggo Tvergaard, Department of Solid Mechanics, The Technical University of Denmark, DK-2800 Lyngby, Denmark.

00457825/93/$06.00 @ 1993 Elsevier Science Publishers B.V. All rights reserved

274 V. Tvergaard, Necking in tensile bars with rectangular cross-section

For biaxially stretched thin sheets localized necking occurs along a line of zero straining in the plane or along the line of minimum straining (see e.g. [lo-121). In the case of pure uniaxial tension in the sheet this localized neck occurs along a line inclined to the normal of the tensile direction, and the critical bifurcation strain is twice that at the load maximum, where necking occurs in a long thin bar. For a long thin plate strip under uniaxial tension the simple one-dimensional Considere analysis still applies, with necking predicted at the maximum load point, but the necking mode observed experimentally in wide flat bars or strips is that of a narrow groove inclined to the cross-section of the bar, as has been known to testing engineers for more than 50 years (see [13]). In relation to sheet metal formability the different types of necking in uniaxial testing conditions have recently been discussed by Bayoumi and Joshi [14]. An approximate three-dimensional bifurcation analysis has been carried out by Miles [15], leading to an asymptotic formula for the effect of moduli and aspect ratios on the delay of bifurcation beyond the load maximum.

In the present paper a three-dimensional numerical analysis is carried out for a bar with rectangular cross-section to study the necking behaviour. For a square cross-section the neck will be similar to that in a round bar, whereas sheet necking is expected for a high aspect ratio of the cross-section. The transition between these two necking modes is studied by considering bars with a range of aspect ratios of the cross-sections. For dynamic loading conditions with softening material behaviour some similar analyses have been carried out by Zbib and Jubran [16]. The present analyses focus on quasi-static behaviour, and most of the computations are based on J2 flow theory with isotropic hardening; but also the effect of void growth on localization and the effect of kinematic hardening are considered.

2. Specimen geometry and basic equations

The bar to be analysed numerically has the initial length 2A,, the initial width 2B, and the initial thickness 2C,,, with the Cartesian reference coordinate system, xi, placed as shown in Fig. 1. An initial thickness imperfection with amplitude 5 is specified so that the actual initial thickness function is

c1(x1,x2)=co 1-scos$ ) [ I

x = x1 + x2 tan(4,) . Cl

(2.1)

For 5 = 0 necking initiates at a bifurcation point, but practically the same solution is obtained

Fig. 1. Tensile test specimen with rectangular cross-section.

V. Tvergaard, Necking in tensile bars with rectangular cross-section 275

for a very small positive value of 4, with the advantage that the post-bifurcation behaviour can be studied directly. If an angle +r # 0 is chosen in (2.1), the imperfection will also amplify one inclined narrow neck more than that symmetric about the x1-x3-plane.

The solutions considered are in all cases symmetric about the x1-x2-plane with free outer surfaces and zero displacements at the centre of the bar,

u3=0, T=T2=0 atx3=O;

T=O at.x2=+Bo;

T = 0 at x3 = C1(xl, x2) ;

ul=u, T2= T3=0 atxi=A,.

Here, U is the prescribed displacement at both ends, so that

(2.2)

(2.3)

(2.4)

(2.5)

the average logarithmic strain is

about the xi-x3-plane and the E, = ln(l + U/A,).

A few analyses x2-x3-plane,

are carried out for solutions symmetric

ul=O,

u2=o,

In these cases only

T2= T3=0 atx=O;

T'=T3=0 atx2=0.

(2.6)

(2.7)

one eighth of the bar needs to be analysed numerically. In most of the analyses one quarter of the bar is analysed, making use of the symmetry

(2.2) together with 180 rotational symmetry about the x3-axis. Then, the region analysed numerically is that shown in Fig. 2, for x3 a 0. This rotational symmetry is expressed by the following conditions on the skew cut surface in Fig. 2, for any given value of x3 and for q = 5:

u(5) = -J(n) 7 u( S) = -u(q) , u(5) = u3b-/) ;

T(5) = Th) , T2(5) = T2(d , T3(t)= -T3(v). P-8)

Standard J, flow theory with isotropic hardening is used for most of the computations. However, to be able to study the effect of voids and the effect of yield surface curvature a kinematic hardening model for a porous ductile material is used. This material model,

Fig. 2. Geometry of region analysed with boundary conditions (2.8).

suggested by Mear and Hutchinson [17] and extended by Tvergaard [IS] to account for void nucleation, makes use of a family of isotropic/kinematic hardening yield surfaces of the form qtr@, a ii cr,, f) = 0, where f is the current void volume fraction, crii is the average macroscopic Cauchy stress tensor and a denotes the stress components at the centre of the yield surface. The radius or of the yield surface for the matrix material is taken to be given by

CFF = (1 - b)a, -i- bo;, , WV

where oY and aM are the initial yietd stress and the matrix flow stress, respectively, and the parameter b is a constant in the range [0, l]. The constitutive relations are formulated such that for b = 1 they reduce to Gursons [19,20] isotropic hardening model, whereas a pure kinematic hardening model appears for b = 0. For b = 1 and f = 0 the expressions reduce to J2 flow theory.

The approximate yield condition to be used here for the porous solid is of the form

@= $ +2q,f cash -l-(4J)2=0, (2.10) where 2; = o - tyi, ge = (3~~~~2~~~ and s= 6 - G@&ii3. Fur q1 = 1, (2.10) is the expression proposed by Mear and Hutchinson 1171, which coincides with that of Gurson f19] for b = 1. The parameter q1 > 1 has been proposed to bring predictions of the Gurson model at low void volume fractions in better agreement with full numerical analyses for periodic arrays of voids [2l, 221. An approximation introduced in (2.10) by Tvergaard and Needleman [S]l to better represent final failure by void coalescence, is not used here.

A Lagrangian formulation of the field equations is employed in the present paper, with a material point identified by the coordinates xi in the reference configuration. The metric tensors in the current configuration and the reference configuration are denoted G, and gij, respectively, with determinants G and g, and qij denotes the Lagrangian strain tensor. The contravariant components of the Cauchy stress tensor # and the Kirchhoff stress tensor 7ii are related by the expression rii = mui.

The plastic part of the macroscopic strain increment 7j,F and the effective plastic strain increment tiF[I for the matrix material are taken to be related by [B]

(2.11)

Then, using the uniaxial true stress natural strain curve for the matrix material, iz = (1 lE, - 1 lE)d,, an expression for the matrix flow stress increment aM is obtained from (2.11). Furthermore, the change of the void volume fraction during an increment of deformation is taken to be given by

(2.12)

where the first term results from the growth of existing voids, and the two last terms model the increment due to nucleation [23]. Nucleation controlled by the plastic strain EL is represented by taking & > 0 and B? = 0 in (2.121, and nucleation is assumed to follow a normal


distribution, with the mean nucleation strain Ed, the standard deviation s and the volume fraction fN of void nucleating particles [18]. When nucleation is controlled by the maximum normal stress on the particle-matrix interface, the sum a, + a:/3 is used as an approximate measure of this maximum stress, thus taking A! = LX?, and the mean stress for nucleation is denoted a,.

Unrealistic oscillatory stress predictions have been found for kinematic hardening solids subject to large shear strains; but it has been shown [24,25] that these stress oscillations disappear if certain corotational stress rates other than the Jaumann rate are used in the finite strain generalization of the constitutive law. For the ductile porous material model Tvergaard and Van der Giessen [26] have incorporated alternative stress rates involving corotation with the crystal substructure spin (the elastic spin) rather than with the continuum spin. The Jaumann rate 2 of the Cauchy stress and the alternative rate 8i are defined by

gj = 6 + (G kui/ + G iku)rjk, , (2.13)

8 = 2 + (Gikuil _ uikGi)wIl, (2.14)

w; = $pP(Gik$ - 7j;Gl,)ak'. (2.15)

In a macroscopic plasticity theory the separation of continuum spin in an elastic part and a plastic part is not defined, and (2.15) is an assumed constitutive law for the plastic spin, in which the factor pp appears as an additional material function. Results will be shown here for p=O, where (2.13) and (2.14) are identical, as well as for a non-zero value of pp.

The plastic part of the strain rate is taken to be [18,26]

(2.16)

The expressions for H and the tensors my and rn: are not repeated here, but it is noted that plastic yielding initiates when @ = 0 and & > 0 during elastic deformation, and continued plastic loading requires @ = 0 and (1 /H)vz~,~~~ 2 0. The hardening rule, expressing the evolution of the yield surface centre during a plastic increment, is taken to be

Ok1 . -kl a =pa ) /_iz-0, (2.17)

where the value of the parameter @ is determined by the consistency condition, d = 0. The uniaxial true stress logarithmic strain curve for the matrix material, defining the value

of the tangent modulus E,, is taken to be represented by the piecewise power law

&= 0 1 u

E for u S oy ,

u 1N

+a, i ) , for a>~,,

(2.18)

where E is Youngs modulus and N is the strain hardening exponent. Finally, the incremental


constitutive law of the form i = Liljlk, to be used in the principle of virtual work, is derived from the assumution that the total strain rate is the sum of the elastic and plastic parts,

3. Numerical method

In terms of the displacement components ui on the reference base vectors (see Fig. 1) the Lagrangian strain tensor is given by

where ( ),j denotes covariant differentiation in the reference frame. The equilibrium equations governing the stress increments ti, the strain increments Gjj, etc., are obtained by expanding the principle of virtual work about the current state, with the current values of the stresses and strains denoted 7ij and qij, respectively. To lowest order this incremental equilibrium equation is

where V and S are the reference volume and surface of the region analysed, and T are the specified nominal surface tractions. The equilibrium correction terms, bracketed in (3.2), prevent drifting of the solution away from the true equilibrium path.

(a>

Fig. 3. Example of a mesh used for the computations. (a) x3-x-plane. (b) x2 = B,. (c) x2 = -B,.


Approximate solutions are obtained by an incremental finite element method based on (3.2). The displacement increments 6 are approximated in terms of three-dimensional twenty-noded isoparametric elements. The volume integrations in (3.2) are carried out using 2 X 2 x 2 point Gaussian quadrature within each element in most computations; but also more accurate 3 X 3 X 3 point-integration is tried for comparison. As an example, a 20 X 8 x 2 mesh used for some of the computations is illustrated in Fig. 3 by a cross-section parallel to the x1-x2-plane and by two cross-sections parallel to the x1-x3-plane. This mesh corresponds to the boundary conditions (2.8), with B,/C, = 8, A,/B, = 4 and 4ffo = 27.5 (see Fig. 2). The mesh is significantly refined in the central part of the bar, where neck development is expected.

4. Results

The elastic-plastic properties of the material to be analysed are specified by the parameters a;/E = 0.0033, v = 0.3 and N = 0.1. First, computations for the boundary conditions (2.2)-

(a)

b) Fig. 4. Contours of maximum principal logarithmic strain near the x3-?-plane, for specimen with B,IC, = 16 and A,/& = 2. Symmetry about all three coordinate planes assumed. (a) E, = 0.163. (b) E, = 0.197.


(2.7) with triple symmetry are briefly discussed, but the majority of results to be presented are obtained using the boundary conditions (2.2)-(2.5) and (2.8).

The specimen first analysed has the initial geometry ~~/C~ = 16 and A,/& = 2, with the initial imperfection amplitude 5 = 0.001 and Cp, = 0. This specimen is symmetric about all three coordinate planes, so that only one eighth of the specimen needs to be analysed numerically (boundary conditions (2.2)-(2.7)), assuming that the symmetries are retained throughout the computation. The computation is carried out with a 20 x 8 x 2 mesh, using a refined uniform mesh in the region where necking is going to occur. Figure 4 shows contours of maximum principal logarithmic strain on a surface through the integration points closest to the x1-x2-plane. The contours are shown at two stages, E, = 0.163 and F, = 0.197, clearly indicating the development of a localized neck. Due to the assumed symmetries, Fig. 4 represents two localized necks crossing each other at the centre of the bar, where the strains are so large that the sheet has become very thin locally. It is noted that the results in Fig. 4 are analogous to the crossing shear bands in the neck of a plane strain tensile test specimen found by Tvergaard et al. [9], although the present necking problem involves thickness variations normal to the plane of the figure.

The symmetries assumed in Fig. 4 do not allow for a bifurcation into a mode, in which only one of the two localized necks continue to develop; but such behaviour is expected at a rather early stage, with a subsequent more rapid development of the remaining localized neck as a function of the overall strain E, than that found when two localized necks grow simultaneous- ly. Therefore, the alternative boundary conditions (2.8), instead of (2.6) and (2.7), are used in the following computations, illustrated in Figs. 5-12. As in Fig. 4 a small imperfection 5 = 0.001 is used to trigger necking into the diffuse mode, without actually passing through a bifurcation point, and a nonzero angle t;b, = 2.9 is used to induce a small asymmetry about the x1--x3-plane, such that one of the localized necks will be preferred over the other. The value of the angle IjbO in Fig. 2 is chosen such that the localized neck will form along a line rather close to the skew edge of the region analysed. This skew edge allows for a rather narrow region with strong mesh refinement (see Fig. 31, which gives a good resolution of the narrow groove forming at localized necking. The value #0 = 27.5 is chosen in all analyses to be presented here.

Figure 5 shows the development of maximum principal logarithmic strain contours in a specimen characterized by the initial dimensions B,IC, = 8 and A,IB, = 4, with material parameters identical to those considered in Fig. 4. These contours on a surface near the xl-x*-plane are shown at four stages, for E, = 0.108, E, = 0.120, E, = 0.132 and E, = 0.140, respectively. According to the Considere condition the onset of necking in a diffuse mode should occur a little after the maximum load point, which occurs at E, = 0.1 for N = 0.1. The onset of such necking may be defined by the the first occurrence of elastic unloading, which is at E, = 0.096 in the present case, due to the small initial imperfection. Thus, the stage shown in Fig. 5(a) is slightly after the onset of necking, and the strain contour for E = 0.1 shows the development of a completely symmetric neck, in spite of the non-zero values of the angles #r and 4. In Fig. 5(b) the neck has grown more, but is still symmetric, while in Fig. 5(c) an inclined narrow localized neck has developed.

For biaxially stretched sheets in the range of negative strain ratios, K = EJE~, Hill [lo] has given the critical strain and the critical angle of inclination for bifurcation into a localized necking mode

281

(a) (b)

(Cl (d)

Fig. 5. Contours of maximum principal logarithmic strain near the x1-x2-plane, for specimen with B,IC,, = 8 and A,/&, = 4. (a) E, = 0.108. (b) E, = 0.120, (c) E, = 0.132. (d) E, = 0.140.

iv EIH = -

l+K tan Jlw =d=X, for K G 0. (4.1)

Thus, for a uniaxial stress state, K = -0.5 and N = 0.1, the critical values are ~~~ = 0.2 and #u = 35.3. In the case of Fig. 5 this critical strain level is first reached in the diffuse neck region, where the strain field is not uniform as assumed in the derivation of (4.1). In Fig. 5(b) the critical strain level qH is exceeded at the centre of the bar, and it is seen that two crossing necks have started to develop, each inclined about 30 to the transverse direction. Figure 5(c) shows that one of these crossing necks has saturated, while the other one has grown into a narrow localized neck, with the current angle of inclination $ = 27. It is noted that this angle is a little smaller than $u.

The thickness variation corresponding to Figs. 5(b), 5(c) and 5(d), at E, = 0.120, &a = 0.132 and E, = 140, respectively, is illustrated in Fig. 6. The figure shows mesh cross-sections through the surface containing the integration points closest to the outer edge, x2 = B,, showing the projection on the x1--x3-plane. The corresponding initial mesh cross-section is shown in Fig. 3(b). In Fig. 6(a), corresponding to the strain fields in Fig. 5(b), the thickness variation is only barely visible; but the cross-section of the localized neck is clearly seen in Figs. 6(b) and 6(c). The width of the neck is determined by three-dimensional effects, which


(a)

Fig. 5. Deformed meshes near x2 = B,, for specimen with B,iC, = 8 and A,/& = 4. {a) e, = 0,120. fb) F, = 0.132. fc) E, = 0.140.

are fully accounted for in the present analysis, and it is noted in Fig. 6 that the width is of the order of the sheet thickness.

The variation of the average nominal traction T, with the average logarithmic strain E,, corresponding to the solution illustrated in Figs. 5 and 6, is shown by the solid curve in Fig. 7. The maximum load is reached at E, = 0.094, slightly before the onset of necking into the diffuse mode, while rapid luad-decay starts somewhat later, at F, = 0.12, just around the stage

20x8~2 mesh 8 integration

points

27 integrution

\

/ points

i

i 1 20x8~3 mesh

20~12x2 mesh /

0.00 0.05 0.10 0.15 Ea 0.20

Fig. 7. Average nominal traction versus average longitudinal strain, for specimen with B,IC, = 8 and A,IB, = 4. Comparison of four different solutions.


where the localized neck begins to develop. It is clear that localized necking leads to final failure without much additional overall straining.

Some additional computations have been carried out for the same case, to test the accuracy of the numerical solution. The localized neck develops in the fine mesh region, but is not exactly parallel to a row of elements, and these elements are rather long in the transverse direction (see Fig. 3(a)). In Fig. 7 the load versus strain curves for the 20 X 8 X 2 mesh are compared with curves for a finer mesh in the transverse direction (20 x 12 x 2 mesh) or a finer mesh through the thickness (20 x 8 x 3 mesh). The very small difference between these three curves indicates that the 20 x 8 x 2 mesh is sufficient, and this mesh has been used for all other computations. For this mesh a computation has also been carried out using 27 integration points within each element (3 x 3 x 3 point Gauss integration), and Fig. 7 shows that the corresponding T, versus E, curve differs only little from that obtained using 8 integration points in each element.

Figure 8 shows a comparison between the load versus strain curves predicted for different values of the aspect ratio B,IC, of the rectangular cross-section, with A,IB, = 4 in all cases. The solid curve for B,IC, = 8 is that also shown in Fig. 7, and the other aspect ratios considered are 4, 8/3 and 2. In each case the length of the fine mesh region is chosen such that the aspect ratio of the element cross sections is identical to that shown in Figs. 3(b) and 3(c), in the fine mesh region. Figure 9 shows the maximum principal strain contours near the middle plane, at E, =0.151 for B,IC, =4 and at E, =0.158 for B,IC, = 8/3. In Fig. 9(a), for B,IC, = 4, a more narrow neck inclined to the transverse direction by + 2: 24 has developed in the centre of the initial diffuse neck region. In Fig. 9(b), for E&,/C, = 8/3, the neck has not

1.5

T /a Y

0.5 -

8dCO=813

0.0 1 0.00 0.05 0.10 0.15 &a

Fig. 8. Average nominal traction versus average longitudinal strain, for specimen with A,IB, = 4. Comparison of different aspect ratios B,IC, of the cross-section.


- I

7 0.1 (a)

(b)

Fig. 9. Contours of maximum principal logarithmic strain near the x1-x*-plane, for specimen with A,IB, = 4. (a) B,,IC, = 4 at F, = 0.151. (b) B,IC, = 813 at E, = 0.158.

grown into a narrow groove although large strains have been reached, and the slight inclination of the strain field relative to the transverse direction may result from the nonzero angle +i in (2.1) combined with a mesh effect. Thus, the analyses show the transition from a final localized necking mode as that predicted by (4.1) for thin sheets, to a necking mode more like that observed in round bars. For the material parameters considered here the transition occurs in the range of aspect ratios B,IC, between 6 and 3. It is noted that based on test results Nadai [13] reported that necking along an oblique line was observed when the aspect ratio of the cross-section was larger than 6 or 7. In both Figs. 9(a) and 9(b) the width of the final neck is of the order of the plate thickness, as also noted in relation to Figs. 5 and 6. Therefore, since the length to width ratio is fixed in the present computations, A,IB, = 4, a larger part of the bar undergoes plastic deformations the thicker the bar, resulting in a corresponding less rapid load decay as shown in Fig. 8.

The effect of ductile failure in the material by the nucleation and growth of voids has been investigated by a number of computations for the bar with A,/& = 4 and B,IC,, = 8, also considered in Figs. 5-7. The material is taken to have no voids initially, but plastic strain controlled nucleation is assumed with the volume fraction of void nucleating particles


f, = 0.01, the mean strain for nucleation Ed = 0.2 and the standard deviation s = 0.1. The computation has been carried out both for b = 1 (isotropic hardening) and for b = 0 (kinematic hardening) in (2.9). Furthermore, kinematic hardening computations have been carried out for pp = 0, thus neglecting plastic spin according to (2.15), and for pp = 2/u,. The load versus strain curves predicted by these computations are shown in Fig. 10. The solid curve in Fig. 10 is identical to that in Fig. 7, corresponding to no voids (f= 0).

In the analyses accounting for void nucleation and growth, the first occurrence of elastic unloading, marking the onset of necking, is at E, = 0.091, slightly earlier than found for f= 0. At this early stage void nucleation has just started, and the void volume fraction is everywhere smaller than 0.003. Subsequently, the voids nucleate and grow rapidly inside the neck, and Fig. 10 shows that this results in significantly earlier occurrence of the abrupt load decay that marks the formation of a localized, inclined neck. Kinematic hardening may be considered a model of a solid that develops a rounded vertex on the yield surface, and Fig. 10 shows that this gives more rapid load decay in the final stage, where the localized neck develops. The very small difference between the curves for pp = 0 and pp = 2/a, indicate that plastic spin does not play an important role in the necking problem considered here.

Figure 11 shows contours of constant maximum principal logarithmic strain and void volume fraction at E, = 0.122 and T,/u,, = 0.965, for the computation with b = 0 and pp = 2/u, in Fig. 10. It is seen that significant void nucleation and growth has only occurred inside the inclined, localized neck, where the peak value of f is 0.093 so that conditions for material failure by void coalescence are approached. In Fig. 11(a) the contour for E = 0.2 shows a trace of the crossing localized neck that has stopped growing at an early stage.

1.5

T 43 y

1.0

0.5

0.0

b=l

0.00 0.05 0.10 0.15

Fig. 10. Average nominal traction versus average longitudinal strain for specimen with A,/& = 4 and I&/C,, = 8. Comparison of predictions for kinematic hardening (b = 0) or isotropic hardening (b = l), with or without void nucleation.


(a)

Fig. 11. Specimen with A,,/& = 4, B,IC, =8, b = 0, pp = 21~r~.f,=O.O1, ~~=0.2ands=O.l at ~,=0.222. (a}

contours of maximum principal loga~thmic strain near the x1--x*-plane. (b) Contours of constaut void volume fraction near the xl-x2-plane.

To distinguish the effect of a rounded vertex from that of void nucleation and growth, a computation has been carried out for f = 0, b = 0 and pp = 210;. In Fig. 12 the predicted load versus strain curve is compared with two of the curves also shown in Fig. 10. It is seen that the new curve, for f= 0 and b = 0, is closer to that for f = 0 and b = 1 than to that accounting for ductile failure. Thus, in the case considered here the nucleation and growth of voids has a stronger effect on early development of localized necking than that resulting from using kinematic hardening.

5. Discussion

The main issue considered in the present paper is that of the competition between different necking modes in a plate strip under uniaxial tension. For a Iong plate strip, necking according to the Considkre condition occurs just after the maximum load; but the localized neck along an oblique line observed in tests becomes critical at a much larger strain, according to sheet


fN=o.ol ,+,=0.2,s=0.1

0.00 0.05 0.10 0.15 Ekl 0.20

Fig. 12. Average nominal traction versus average longitudinal strain for specimen with A,/& = 4 and L&/C, = 8. Comparison of prediction for b = 0, pp = 2/u, and f = 0 with two curves from Fig. 10.

necking analyses. The present analyses show that necking occurs first in a diffuse mode with a neck length of the order of the specimen width, and that localized necking occurs subsequently in the diffuse neck region as the strains grow large enough. For the first onset of necking the delay beyond the load maximum is mainly a function of the specimen length to width ratio; e.g. in the case of Fig. 4 for A,IB, = 2 the onset of necking is at E, = 0.108 (1.15 times the strain at the load maximum), while in the case of Fig. 5 for A,/B, = 4 the onset of necking is at E, = 0.096 (1.02 times the strain at the load maximum). Clearly, a much smaller length to width ratio, or a sheet with no free edges, would be required to find the localized oblique neck first critical.

The transition between the two necking modes as a function of the cross-section aspect ratio, B,IC,, is illustrated by the load versus the strain curves in Fig. 7, and the corresponding strain fields in Figs. 5 and 8. For I&/C, = 8 the predicted localized neck is fully developed, while for B,IC, = 4 there is still a strong trace of the oblique localized neck, and for B,IC, = 8/3 there is practically no trace left. Thus, the predicted transition is rather gradual, and it may be concluded that this transition takes place in the range of aspect ratios between 6 and 3. This is in rather good agreement with the test results reported by Nadai [13] that necking along an oblique line was observed for aspect ratios larger than 6 or 7. Even though necking in a diffuse mode occurs first, this is hardly visible on the test specimen (see Figs. 5 and 6), where the only dominant feature is the localized neck. However, elastic unloading outside the neck region occurs at a strain corresponding to the value of E, at the first onset of necking into the diffuse mode. Therefore, in Fig. 7 for A,/&, = 4 the value of the average strain E, never reaches the critical strain value 0.2 for the onset of necking into the oblique


localized mode, and for AJB, >i 1 the abrupt load drop in Fig. 7 would shift towards the strain value 0.1 corresponding to the load maximum.

It is noted that the symmetries assumed in the first study reported here, Fig. 4, are similar to those assumed by Tvergaard et al. ]9] in a study of shear band formation in the neck of a plane strain tensile test specimen. Such symmetries have also been assumed by Zbib and Jubran [16] in a transient analysis for softening bars with rectangular cross-section, and two crossing localized necks have been found in some of these dynamic analyses. In the present investigation some calculations with the symmetries assumed in Fig. 4 were first used to study the necking mode transition, but the interaction of the two crossing localized necks led to a neck pattern not observed experimentally, and the transition was predicted at somewhat higher values of B,/C, _ Since only one localized neck grows in reality (due to either bifurcation or imperfection), it was considered more interesting to study the necking mode transition for the boundary conditions (2.8), assuming less symmetry.

Analyses accounting for three-dimensional effects in sheet necking have been carried out previously, to evaluate the predictions obtained by the much simpler plane stress analyses. Thus, for equal biaxial stretching, K = E~IE~ = 1, Needleman and Tvergaard [27] have used an axisymmetric finite element study of a stretched circular plate with a small initial thickness reduction near the edge to study neck development, and Tvergaard [28] has used the same type of studies to compare kinematic hardening predictions with isotropic hardening predictions. These full axisymmetric studies, for thickness imperfections corresponding to 5 = 0.01 and 5 = 0.1, have shown that the onset of necking is significantly delayed when three- dimensional effects are accounted for, and that kinematic hardening gives much earlier localization than isotropic hardening. By contrast, the present studies, Figs. 10 and 12, show much less difference between predictions obtained by kinematic hardening and isotropic hardening. This is closely connected to the predictions of the simple plane-stress model that in equal biaxial stretching, K = 1, localization predictions are strongly sensitive to details of the material model, whereas in uniaxial tension, K = -0.5, localization predictions are rather insensitive (see 1121).

Progressive ductile failure by the nucleation and growth of voids gives material softening, which promotes the onset of localization, as has been found in Fig. 10. Also in the presence of ductile failure rather little difference has been found here between predictions for isotropic hardening and predictions for kinematic hardening, representing the effect of a rounded vertex on the yield surface. Again, this agrees with predictions obtained by the simple plane-stress model [18] for uniaxial tension, K = -0.5. In the studies based on ductile porous material models the fmal failure mode inside the localized neck, by shear localization or void coalescence, could be described in detail, analogous to the cup-cone fracture in the neck of a round tensile test specimen analysed by Tvergaard and Needleman [8]. However, in the present analyses the mesh is not sufficiently refined to resolve the final failure mode.

The strong appearance of a secondary bi~rcation mode is not uncommon in plastic instability problems. Thus, in an elastic-plastic tube under internal pressure the first critical bifurcation leads to the development of a localized axisymmetric bulge, and the experimentally observed failure in a localized bulge on one side of the tube results from a secondary bifurcation [29]. Also, in buckling localization problems [30] the first critical buckling mode is usually periodic, while the experimentally observed localized buckle results from a secondary bifurcation point. In the case of a long thin plate strip under uniaxial tension, the behaviour


for no initial imperfections is clearly indicated by the present results based on assuming a small imperfection. In the absence of imperfections the first critical bifurcation point leads to necking in the diffuse mode. Subsequently, the non-uniformly strained neck region provides enough imperfection so that the two crossing localized necks start to develop without any bifurcation; but a secondary bifurcation results in the saturation of one of these necks, while the other oblique localized neck grows into the failure mode that is clearly visible on a test specimen.

References

[l] J.W. Hutchinson and J.P. Miles, Bifurcation analysis on the onset of necking in an elastic-plastic cylinder under uniaxial tension, J. Mech. Phys. Solids 22 (1974) 61-71.

[2] R. Hill, A general theory of uniqueness and stability in elastic-plastic solids, J. Mech. Phys. Solids 6 (1958) 236-249.

[3] R. Hill, Bifurcation and uniqueness in non-linear mechanics of continua, in: Problems of Continuum Mechanics (SIAM, Philadelphia, PA, 1961) 155-164.

[4] M. Considtre, Annales des Ponts er Chausdes 9 (1885) 574. [5] A. Needleman, A numerical study of necking in circular cylindrical bars, J. Mech. Phys. Solids 20 (1972)

111-127. [6] D.M. Norris, B. Moran, J.K. Scudder and D.F. Quiiiones, A computer simulation of the tension test, J.

Mech. Phys. Solids 26 (1978) 1-19. [7] J.H. Argyris, J. St. Doltsinis, P.M. Pimenta and H. Wiistenberg, Thermomechanical response of solids at high

strains - natural approach, Comput. Methods Appl. Mech. Engrg. 32 (1982) 3-57. [8] V. Tvergaard and A. Needleman, Analysis of the cup-cone fracture in a round tensile bar, Acta Metallurgica

32 (1984) 157-169. [9] V. Tvergaard, A Needleman and K.K. Lo, Flow localization in the plane strain tensile test, J. Mech. Phys.

Solids 29 (1981) 115-142. [lo] R. Hill, On discontinuous plastic states, with special reference to localized necking in thin sheets, J. Mech.

Phys. Solids 1 (1952) 19-30. [ll] S. Storen and J.R. Rice, Localized necking in thin sheets, J. Mech. Phys. Solids 23 (1975) 421-441. [12] V. Tvergaard, Bifurcation and imperfection-sensitivity at necking instabilities, Z. Angew. Math. Mech. 60

(1980) T26-T34. [13] A. Nadai, Theory of flow and fracture of solids (McGraw-Hill, New York, 1950). [14] A.E. Bayoumi and R. Joshi, On the formability/instability of stretch-forming sheet metals, Appl. Mech. Rev.

45 (1992) S154-S164. [15] J.P. Miles, The initiation of necking in rectangular elastic/plastic specimens under uniaxial and biaxial tension,

J. Mech. Phys. Solids 23 (1975) 197-213. [16] H.M. Zbib and J.S. Jubran, Dynamic shear banding: A three-dimensional analysis, Intemat. J. Plasticity

(1992) to appear. [17] M.E. Mear and J.W. Hutchinson, Influence of yield surface curvature on flow localization in dilatant plasticity,

Mechanics of Materials 4 (1985) 395-407. [18] V. Tvergaard, Effect of yield surface curvature and void nucleation on plastic flow localization, J. Mech. Phys.

Solids 35 (1987) 43-60. [19] A.L. Gurson, Continuum theory of ductile rupture by void nucleation and growth - I. Yield criteria and flow

rules for porous ductile media, J. Engrg. Materials Technol. 99 (1977) 2-15. [20] A.L. Gurson, Porous rigid-plastic materials containing rigid inconclusions - Yield function, plastic potential,

and void nucleation, in: D.M.R. Taplin, ed., Proc. Intemat. Conf. Fracture 2A (1977) 357-364. [-rJ V. Tvergaard, Influence of voids on shear band instabilities under plane strain conditions, Intemat. J. Fracture

17 (1981) 389-407.


[22] V. Tvergaard, On localization in ductile materials containing spherical voids, Internat. J. Fracture 18 (1982) 237-252.

[23] A. Needleman and J.R. Rice, Limits to ductility set by plastic flow localization, in: D.P. Koistinen et al., eds., Mechanics of Sheet Metal Forming (Plenum, New York, 1978) 237-267.

[24] Y.F. Dafalias, Corotational rates for kinematic hardening at large plastic deformations, J. Appl. Mech. 50 (1983) 561-565.

[2.5] B. Loret, On the effects of plastic rotation in the finite deformation of anisotropic elastoplastic materials, Mechanics of Materials 2 (1983) 287-304.

{26] V. Tvergaard and E. Van der Giessen, Effect of plastic spin on localization predictions for a porous ductile material, J. Mech. Phys. Solids 39 (1991) 763-781.

[27] A. Needleman and V Tvergaard, Necking of biaxially stretched elastic-plastic circular plates, J. Mech. Phys. Solids 25 (1977) 159-183.

(281 V, Tvergaard, Effect of kinematic hardening on localized necking in biaxially stretched sheets, lntemat. J. Mech. Sci. 20 (1978) 651-658.

[29] V. Tvergaard, Bifurcation in elastic-plastic tubes under internal pressure, European J. Mech. A/Solids 9 (1990) 21-35.

[30] V. Tvergaard and A. Needleman, On the localization of buckling patterns, J. Appl. Mech. 47 (1980) 613-619.

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