Eur. J. Mech. A/Solids 20 (2001) 713–729

2001 Éditions scientifiques et médicales Elsevier SAS. All rights reservedS0997-7538(01)01163-9/FLA

Effects of inertia on dynamic neck formation in tensile bars

Kristina Nilsson∗

Division of Mechanics, Lund University, Box 118, S-221 00 Lund, Sweden

(Received 28 September 2000; revised and accepted 7 May 2001)

Abstract – Neck localization during high rate extension of round bars is analyzed numerically. An axisymmetric problem formulation is given andthe material is described as rate independent and elastic–plastic. The time history of neck development is investigated and effects of geometry andinitial thickness imperfections are visualized. Studies of both weakly and strongly developed necks are performed, revealing the occurrence of multiplenecks in some loading cases. The influence on neck formation from background inertia (lateral inertia) corresponding to the inertia originating fromhomogeneous deformation of the bar is examined somewhat qualitatively by introducing an artificial volume load. Calculations show that in the presentanalysis, background inertia does not have any noticeable influence on the necking pattern unless the effect is artificially magnified by three orders ofmagnitude so that it becomes comparable to the yield stress. 2001 Éditions scientifiques et médicales Elsevier SAS

dynamic / background inertia / neck localization / time development / multiple necking

1. Introduction

Under quasi-static conditions neck localization in tensile bars undergoing uniaxial tension occurs shortlyafter the maximum load has been reached. Using a one-dimensional analysis, Considere (1885) showed thatan instability develops at the maximum load point. Introducing a full bifurcation analysis in the problemformulation, Hutchinson and Miles (1974) showed that localization is delayed due to the stiffening effect of themultiaxial stress state. For long slender bars, this corresponds to necking at tensile strain levels slightly higherthanN , whereN is the strain hardening exponent. Changing the aspect ratio of the specimen by using a shorterbar leads to a delay in localization.

For rate independent solids under quasi-static conditions, bifurcation corresponding to loss of ellipticity ofthe rate equilibrium equations governs the onset of localization (Rice, 1976). Bifurcation is also associated withthe occurrence of vanishing wave speeds; body waves that cease to travel through the specimen and becomestationary. Rate dependent solids do not experience loss of ellipticity and true bifurcations are not found atrealistic stress levels. Initial inhomogeneities might trigger localization behavior even for rate dependent solids,but the critical strain level under quasistatic conditions is higher than that observed using rate independentmaterials (Hutchinson and Neale, 1977).

Han and Tvergaard (1995) used a rate independent elastic–plastic solid in their simulations of ring specimens.A large number of necks developed around the circumference for various imperfections, which is in generalagreement with experimental results obtained when expanding rings using electromagnetic loading. They alsoobserved a delay in neck development when the expansion velocity was increased. In their study of expandingrings, Sørensen and Freund (2000) included the effects of rate sensitivity, void growth and thermal softening.

∗ Correspondence and reprints.E-mail address:kristina.nilsson@mek.lth.se (K. Nilsson).

714 K. Nilsson

They found no strong correlation between any of these effects and the resulting necking pattern. Moreover itwas found that many of the necking patterns formed occurred in essentially the same way; via a nearly periodiccritical mode.

Niordson (1965) used an experimental technique based on electromagnetic loading to obtain nearlyhomogeneous deformation of ring specimens, avoiding wave effects present in uniaxial tension tests that mightcause early localization at the loaded end. After localization was initiated and deformation was no longerhomogeneous multiple necks formed, some of them leading to final fracture. This technique has been used forseveral experiments investigating necking behavior of rate sensitive and rate insensitive materials. Altynovaet al. (1996) studied increased ductility in high rate expansion of rings. They found a relation between thenumber of necks and imposed velocity.

The differences in necking pattern obtained using dynamic loading compared to quasi-static conditions, areconsidered as partly due to material inertia. Even though similar surface imperfections are used, several necksmay develop in the dynamic case at sites that correlate weakly or not at all with the neck sites found underquasi-static conditions. Needleman (1989) used a local reduction in flow strength to investigate dynamic shearband propagation in a rate dependent solid under plane strain conditions. Differences in behavior for hardeningand softening materials were investigated using a rectangular block subject to compression. A significantretardation of shear band development due to inertial effects was found. A one-dimensional analysis of around bar subjected to tension was performed by Tugcu et al. (1990) where the stabilizing effects of inertiawere observed (see also Fressengeas and Molinari (1994)). An elastic–viscoplastic constitutive law was usedto model the material behavior of solids with strain and strain rate hardening characteristics. A limitation of theone-dimensional analysis was that at high velocities radial inertia effects may no longer be negligible. Sørensenand Freund (1998) showed that mode inertia, i.e. the inertia following a rapidly growing incipient neck leadsto a suppression of the modes with the longest wave length. In addition they showed that bifurcation modes areavailable essentially everywhere when the imperfection growth rate as well as the strain levels are sufficientlyhigh. Shenoy and Freund (1999) found that very short wavelength modes of nonuniform deformation aresuppressed by background inertia. In the analysis of Shenoy and Freund (1999) a linear stability approachis used, so that results regarding the onset of a bifurcation seem to rely on the assumption that wave effects andimperfection amplitudes are small compared with the effects of background inertia.

In the present paper the necking pattern formed in a round bar of a ductile metal is studied, using a dynamicaxisymmetric finite element approach to simulate uniaxial tension. The aim is to explore the development oflocalization zones with time and in particular effects of background inertia. In the present context backgroundinertia is defined as the inertia originating from idealized homogeneous deformation of the bar. An artificialvolume load chosen as a pressure whose form is similar to that originating from this homogeneous deformationis introduced to qualitatively study the effects of background inertia. Clearly, perfect homogeneous deformationdoes not take place in the actual cases studied. However, parts of the inertial effects due to the global shapeof the bar are investigated in this way. The full time history of neck formation is visualized using a strain rateparameter depicting both localization zones that die out as well as localization zones that continue to growuntil they eventually cause final failure. The effects of specimen size, aspect ratio, end velocity and surfaceimperfections are studied.

During high rate homogeneous extension of round bars a hydrostatic pressure appears in the material due toinertia (e.g., Graff, 1975). It is this pressure effect resulting from assumed incompressible and homogeneousdeformation that is termed background inertia here. The effects of background inertia in the present case ofdynamic loading of a round bar is studied by introducing an artificial volume force in the form of hydrostatictension. This artificial hydrostatic tension is in several cases selected to be precisely opposite to the hydrostaticpressure (background inertia) that results from rapid homogeneous extension of the bar to explore any possible

Effects of inertia on dynamic neck formation 715

effect of background inertia. Attention is thus directed towards the effect of background inertia and for allsimulations regardless of specimen size, aspect ratio, etc., corresponding simulations have been performedwhere the effects of background inertia have been cancelled out by the artificial pressure applied.

2. Problem formulation and method of analysis

The axisymmetric deformation of a round tensile bar subject to dynamic loading is to be studied. MaterialcoordinatesXi are introduced in the reference configuration which occupies the undeformed region 0� X1 �l0, 0 � X2 � r0 (figure 1) with rotational symmetry around theX1 axis. Only half of the bar is studiednumerically due to mirror symmetry about the mid plane at the center of the bar, containing theX2 axis. Thefull initial length and diameter of the bar are thus denoted 2l0 and 2r0, respectively. Initial sinusoidal surfaceimperfections are introduced in the form:

r0 = r0

(1− ξ cos

[πmX1

l0

]), (1)

whereξ is the dimensionless imperfection amplitude andm is the mode number. The aspect ratio of the bar isdefined as:

α = l0

r0. (2)

The following boundary conditions are used for displacementsui and tractionsT i :

u1 = u1(t), T 2 = 0, X1 = l0; (3)

u1 = 0, T 2 = 0, X1 = 0; (4)

u2 = 0, T 1 = 0, X2 = 0; (5)

T 1 = T 2 = 0, X2 = r0, (6)

where the displacementu1(t) at the end of the bar in the direction of the axis of rotational symmetry isdetermined by an imposed end velocity ramped from zero to a constant maximum value according to:

u1(t) ={v0t

2/2t0, t � t0,v0t0/2+ v0(t − t0), t > t0,

(7)

wheret is the time,t0 is a ramp time andv0 the imposed boundary velocity.

Somewhat idealized inertia is viewed as consisting of two parts only; mode inertia and background inertia,where background inertia is the inertia due to shape and velocity changes associated with homogeneous

Figure 1. Part of a round tensile bar represented by a mesh with 100× 10 quadrilaterals, each consisting of four crossed triangles. The bar has an initiallength and diameter equal to 2l0 and 2r0 respectively. Only half of the bar is studied numerically due to mirror symmetry about the mid plane at the

center of the bar, containing theX2 axis. (X1,X2) are cylindrical coordinates and theX1 axis is the axis of rotational symmetry.

716 K. Nilsson

deformation. The velocity field originating from homogeneous deformation of an incompressible bar referredto the material coordinates of the reference configuration is fort > t0:

V 1(X1,X2,X3, t) = X1v0

l0; (8)

V 2(X1,X2,X3, t) = − v0

2l0

X2[1+ v0

l0

(t − t0

2

)]3/2; (9)

V 3(X1,X2,X3, t) = 0, (10)

taking into account that the boundary velocity is ramped. This velocity field may be used to define thecontribution of background inertia to the total stress field of the axisymmetric bar. Considering deformationdue to the velocity field (8)–(10) a stress fieldσ I caused by background inertia can be determined from thedynamic equilibrium condition. The total stress fieldσ T may be considered as consisting of two parts;σ I andσN :

σ T = σ I + σN, (11)

whereσN does not include contributions from background inertia. Using the stress fieldσ I , the hydrostaticpressure contributionp from background inertia can be determined to be of the form:

p(X2, t

) = ρ03

8

(v0

l0

)2 1[1+ v0

l0

(t − t0

2

)]5/2

(r20 − (

X2)2), t > t0. (12)

The effects of background inertia on neck development may be addressed somewhat qualitatively by imposingan artificial volume load corresponding to−ψp whereψ = 1 means that the pressure effects due to backgroundinertia are cancelled out.Figure 2shows the development of the maximum pressurep with time for a bar withl0 = 1 m, r0 = 0.1 m,v0 = 40 m/s andt0 = 10−5 s.

Figure 2. Time development of the maximum inertial pressurep normalized with the yield stressσy for a bar withl0 = 1 m, r0 = 0.1 m,v0 = 40 m/sandt0 = 10−5 s. The pressurep is due to background inertia.

Effects of inertia on dynamic neck formation 717

The deformation time history of the full length of the tensile bar is visualized using contour plots of anormalized average cross sectional strain rateq:

q = 〈εp〉c〈εp〉t , (13)

where〈 〉c and〈 〉t denotes a cross sectional average and a total average, respectively andεp is the maximumprincipal logarithmic strain rate (as in Sørensen and Freund (2000)). This strain rate parameter is plotted overthe reference length of the bar (X1/ l0) as a function of the overall logarithmic strainε = ln(l/ l0) in thefollowing figures. Values ofq higher than unity and lower than unity indicates a strain rate higher than orlower than, respectively, that of the total average strain rate in the bar.

The balance of momentum is expressed in terms of the principle of virtual work:

∫V

τ ij δηij dV =∫S

T iδui dS −∫V

ρ0∂2ui

∂t2δui dV +

∫V

ψpGij δηij dV, (14)

whereV andS refer to volume and surface of the body, respectively, in the reference configuration. In (14) thevolume force introduced to study the effects of background inertia appears in the artificial volume load termcontaining the pressurep. The covariant components of the Lagrangian strain tensor are denoted byηij and thecontravariant components of Kirchhoff stress are denotedτ ij . The mass density at timet = 0 when the bodyis in the reference configuration isρ0, the variational operator isδ andGij is the metric tensor in the currentconfiguration. The stress and strain tensors in (14) are:

τ ij = (ρ0/ρ)σij ; (15)

ηij = 1

2

(ui,j + uj,i + uk

,iuk,j

), (16)

where ρ denotes current density. Comma represents covariant differentiation with respect to the materialcoordinate of the reference configuration. Nominal traction is denotedT i and ui are the displacementvector components in the reference configuration. The constitutive relations of the elastic–plastic materialare represented by rate independentJ2 flow theory (Hutchinson, 1973). The linear incremental form of theconstitutive response for a nearly incompressible material (τ ij ≈ σ ij ) can be expressed in terms of the non-objective rate of Cauchy stress and the increment of Lagrangian strain:

σ ij = Lijkl ηkl, (17)

with the instantaneous moduli specified by:

Lijkl = E

1+ ν

[1

2

(GikGjl + GilGjk

) + ν

1− 2νGijGkl − α

3

2

E/Et − 1

E/Et − (1− 2ν)/3

sij skl

σ 2e

]

− 1

2

(Gikσ jl +Gjkσ il +Gilσ jk +Gjlσ ik

). (18)

HereE is Young’s modulus,ν is Poisson’s ratio andα is an elastic–plastic switch which equals zero or unitydepending on whether or not yielding occurs. The stress deviator in (18) is written as:

sij = σ ij − 1

3GijσklG

kl, (19)

718 K. Nilsson

and the effective Mises stress is:

σe =√

3

2sij sij . (20)

Under uniaxial tension the tangent modulusEt relates the true stress rate and logarithmic strain rate for anincrement of stress according toσ = Et ε. A piecewise power law is adopted:

ε ={ σ

Efor σ � σy,

σy

E

[σσy

]1/Nfor σ > σy,

(21)

whereσy is the initial yield stress andN is the strain hardening exponent.

The numerical simulations are performed using a finite element approach to the dynamic principle of virtualwork (14). Axisymmetric quadrilateral finite elements consisting of four crossed triangles to avoid locking(Nagtegaal et al., 1974) are used in a Lagrangian framework (seefigure 1). A Newmark time integration schemewith a lumped mass matrixM is applied:

D(t+,t) = D(t) + D(t),t + [(1/2− k2)D(t) + k2D(t+,t)

](,t)2; (22)

D(t+,t) = M−1(−F(t+,t)); (23)

D(t+,t) = D(t) + [(1− k1)D(t) + k1D(t+,t)

],t, (24)

whereD denotes nodal displacement,F denotes nodal forces and (˙) indicates differentiation with respect totime. Explicit time integration with no stiffness matrix computation or iteration is obtained by choosingk2 = 0(Belytschko et al., 1976). The other parameter is chosen to bek1 = 1/2. The use of a lumped mass matrix hasbeen shown to be preferable in the case of explicit time integration (Krieg and Kay, 1973) concerning bothaccuracy and efficiency. The size of the time step,t is limited by element size and the characteristic speed ofpropagation of information. The maximum time step is chosen as:

,t < ζ,de/cd, (25)

where,de denotes the minimum element dimension andζ is a constant taken to be in the range 0< ζ < 1.The dilatational wave speedcd , the highest wave speed in an elastic medium, is:

cd =√

E(1− ν)

ρ(1+ ν)(1− 2ν). (26)

The shear wave speed is:

cr =√

E

2ρ(1+ ν). (27)

3. Results

The material parameters are representative of a structural steel;E = 2.069× 1011 Pa,ν = 0.3 andρ0 =7850 kg/m3. The speeds of dilatation and shear waves are thus 5956 m/s and 3183 m/s respectively. The

Effects of inertia on dynamic neck formation 719

initial yield stress isσy = 374.87× 106 Pa and the piecewise power law hardening exponent isN = 0.1. Theimperfection amplitudes considered areξ = 0, ξ = 0.01 andξ = 0.001, using the mode numbersm = 1,m = 2andm = 4. Four different specimen sizes are used;l0 = 1 m, r0 = 0.1 m andl0 = 0.1 m, r0 = 0.01 m hasan aspect ratio ofα = 10 whereasl0 = 1 m, r0 = 0.01 m andl0 = 0.1 m, r0 = 0.001 m has an aspect ratioof α = 100. The mesh used for the aspect ratioα = 10 consists of 10× 100 quadrilaterals (figure 1) and thecorresponding mesh forα = 100 consists of 10× 1000 quadrilaterals. The imposed end velocities are chosenbelowv0 = 100 m/s since the use of higher velocities in the present analysis leads to necking at the loaded endof the specimen.

In figure 3theq-levels of a bar with dimensionsl0 = 1 m, r0 = 0.1 m and no initial imperfection (ξ = 0) aredisplayed. The results using four different values of the end velocityv0, are shown. The ramp time is chosen ast0 = 10−5 s and no artificial volume load is included (ψ = 0). The deformation history is represented in termsof contours of constantq-levels plotted as a function of the reference positionX1/ l0 and overall logarithmicstrainε = ln(l/ l0). The insert to the right of theq contour plot shows the final state of the bar (scaled) with theshading indicating contours of constant principal logarithmic strainεp.

In figure 3av0 = 20 m/s and neck localization occurs at approximatelyX1/ l0 = 0.55. For symmetry reasonsthis means that there are actually two necking sites along the full length of the bar. An increase of the velocityto v0 = 30 m/s leads to a shifting of the most developed neck towards the center of the bar (figure 3b). There isalso a weak localization closer to the end of the bar, but this localization does not develop into all full-grownneck. At a velocity ofv0 = 40 m/s the neck site moves away from the center of the bar, although there is a

(a) (b)

(c) (d)

Figure 3. Contours of normalized average strain rate (q) as a function of time and distance for a tensile bar withl0 = 1 m, r0 = 0.1 m, t0 = 10−5 s,m = 0 andψ = 0, where the end velocities are: (a)v0 = 20 m/s; (b)v0 = 30 m/s; (c)v0 = 40 m/s; (d)v0 = 50 m/s. Inserts to the right show the final

deformation state (scaled bars) with contours of maximum principal logarithmic strain (εp).

720 K. Nilsson

zone with higher than average strain rate present at the center, as can be seen infigure 3c. For velocities ofv0 = 50 m/s and above, necking occurs at the end of the bar,figure 3d.

At the left-hand side offigures 3a–d, an intensively black region originating fromX1/ l0 = 1.0 indicates rapidpropagation along the bar of information reachingX1/ l0 = 0 at approximatelyε = 0.005 (t ≈ 1.5 × 10−4 s).This corresponds approximately to the time required for a dilatation wave to travel the half-length of theundeformed bar.

An increase of the velocity fromv0 = 20 m/s tov0 = 40 m/s causes an increase in logarithmic overallstrain at the final deformation state fromε = 0.26 to ε = 0.39, but when the velocity reachesv0 = 50 m/sthe overall strain level is reduced toε = 0.29 and further increase of end velocity diminishesε even more.A similar behavior is noticed when studying normalized traction at the end of the bar. Infigure 4normalizedengineering traction is plotted against logarithmic overall strain for the cases a–d infigure 3. Large wave effectsare generated at impact, causing initial oscillation of the normalized traction. The critical overall strain level atthe first appearance of incipient neck localization infigure 4 increases when increasing the end velocity fromv0 = 20 m/s (ε = 0.15) (a) tov0 = 30 m/s (ε = 0.17) (b), but decreases again whenv0 = 40 m/s (ε = 0.14) (c).At velocities v0 = 50 m/s (d) and above, necking occurs at the end of the bar and the critical strain levelcontinues to diminish.

In order to study imperfection sensitivity, bars with varying thickness were introduced.Figure 5depicts thecontours of constantq-levels for a bar with dimensionsl0 = 1 m, r0 = 0.1 m and an initial imperfection ofξ = 0.01,m = 1. The ramp time is again chosen ast0 = 10−5 s and no artificial volume load is included,ψ = 0.In figure 5a, v0 = 20 m/s and the final neck occurs at the thin point at the center of the bar. A similar resultis expected in the quasi-static case because of the initial imperfection. Increasing the velocity tov0 = 30 m/sleads to a shifting in the final neck site towards the end of the bar (figure 5b). Further increase of the velocityto v0 = 40 m/s shows that although there is an incipient neck between the center and the end of the bar, thedominating neck is the one developed at the site of the imperfection. The neck precursor atX1/ l0 = 0.4 diesout at an overall logarithmic strain of approximatelyε = 0.25. The neck formation at the imperfection site ofthe bar is termed stable, whilst the neck precursor atX1/ l0 = 0.4 is termed unstable (compare with (Sørensenand Freund, 2000)). At the velocityv0 = 50 m/s, necking occurs at the end of the bar, and further simulationsreveal that this is the case for all velocities abovev0 = 50 m/s for this particular bar.

Figure 4. Normalized traction plotted against logarithmic overall strain for a tensile bar withl0 = 1 m, r0 = 0.1 m, t0 = 10−5 s, m = 0 andψ = 0,where the end velocities are; (a)v0 = 20 m/s, (b)v0 = 30 m/s, (c)v0 = 40 m/s, (d)v0 = 50 m/s.

Effects of inertia on dynamic neck formation 721

(a) (b)

(c) (d)

Figure 5. Contours of normalized average strain rate (q) as a function of time and distance for a tensile bar withl0 = 1 m, r0 = 0.1 m, t0 = 10−5 s,ξ = 0.01,m = 1 andψ = 0, where the end velocities are: (a)v0 = 20 m/s; (b)v0 = 30 m/s; (c)v0 = 40 m/s; (d)v0 = 50 m/s. Inserts to the right show

the final deformation state (scaled bars) with contours of maximum principal logarithmic strain (εp).

Simulations with velocities betweenv0 = 40 m/s andv0 = 50 m/s did not lead to fully developed neckspositioned between the center and the end of the bar. Thus, the transition of dominating neck site from thecenter to the end of the bar takes place with only a weakly developed neck at an intermediate position.

The results for a bar with the same dimensions (l0 = 1 m,r0 = 0.1 m), but with different initial imperfectionsare shown infigure 6. The imperfection amplitude remains atξ = 0.01, but the mode number has been changedto m = 2 in figures 6a–bcorresponding to thin points atX1/ l0 = 0 andX1/ l0 = 1, andm = 4 in figures 6c–dcorresponding to thin points located atX1/ l0 = 0,X1/ l0 = 0.5 andX1/ l0 = 1.

In figure 6awhere the velocity isv0 = 30 m/s, the most developed neck localization occurs at the center ofthe bar, but in addition a weakly developed neck is formed at the end of the bar. This incipient localization doesnot proceed, and instead it fades away at an early stage. When the velocity is increased tov0 = 40 m/s as infigure 6b, the result is the opposite. Necking occurs at the end of the bar whilst a weakly developed localizationis observed at the center. Further increase of the velocity shows similar behavior with a gradually weakeningof the localization at the center of the bar.

Using this initial imperfection, there is no sign of any incipient localization between the thin points. Studiesof velocities betweenv0 = 30 m/s andv0 = 40 m/s show that the transition between necking at the center of thebar and at the end of the bar is performed with no visible signs of other incipient localization sites. It shouldalso be noted that none of the simulations in the present study show the occurrence of two strongly developednecks simultaneously along the bar with aspect ratioα = 10. Only one neck prevails in all test cases.

722 K. Nilsson

(a) (b)

(c) (d)

Figure 6. Contours of normalized average strain rate (q) as a function of time and distance for a tensile bar withl0 = 1 m, r0 = 0.1 m, t0 = 10−5 s,ξ = 0.01 andψ = 0: (a)m = 2, v0 = 30 m/s; (b)m = 2, v0 = 40 m/s; (c)m = 4, v0 = 30 m/s; (d)m = 4, v0 = 40 m/s. Inserts to the right show the final

deformation state (scaled bars) with contours of maximum principal logarithmic strain (εp).

In figure 6c–dthe mode number has been increased tom = 4. Necking occurs at the center of the bar infigure 6c, where the velocity equalsv0 = 30 m/s. Increase of the end velocity tov0 = 40 m/s leads to a shift innecking site, so that the most developed neck is positioned at the end of the bar. In both of these cases, apartfrom the dominant neck, there are weakly developed necks at the thin points. Studies of the necking patternat velocities betweenv0 = 30 m/s andv0 = 40 m/s reveals that there is a transition interval where the mostdeveloped neck occurs atX1/ l0 = 0.5. As in the case withm = 2 only one of the localization zones is stronglydeveloped.

Simulations using a bar with an imperfection amplitude reduced toξ = 0.001 (l0 = 1 m, r0 = 0.1 m) showan inclination towards formation of a fully developed neck located between the center of the bar and the endof the bar, though the actual necking pattern differs between the three different cases of imperfection (modenumberm = 1, m = 2 andm = 4). Qualitatively similar results are obtained as in the case when using a barwith the same dimensions, but no initial imperfection (ξ = 0) where the necking pattern is rather random.

In general, the computations show that the influence of background inertia originating from homogeneousdeformation is small whereas the extension rate have a large influence on the resulting neck formation.Simulations corresponding to all cases mentioned above, but usingψ = 1 and thereby canceling backgroundinertia, show essentially no deviation from the results presented infigures 3–6.

Figure 7 shows a comparison between the results acquired with no artificial volume load and those withcancellation of background inertia. Infigures 7a–bthe size of the bar isl0 = 1 m, r0 = 0.1 m with animperfection ofm = 1, ξ = 0.01 and a ramp time oft0 = 10−5 s. The end velocity is chosen asv0 = 40 m/s.

Effects of inertia on dynamic neck formation 723

(a) (b)

Figure 7. Contours of normalized average strain rate (q) as a function of time and distance,l0 = 1 m, r0 = 0.1 m, t0 = 10−5 s, ξ = 0.01,m = 1 andv0 = 40 m/s. (a) No artificial volume load,ψ = 0. (b) Cancelled background inertia,ψ = 1. Inserts to the right show the final deformation state (scaled

bars) with contours of maximum principal logarithmic strain (εp).

(a) (b)

(c)

Figure 8. Contours of normalized average strain rate (q) as a function of time and distance for a tensile bar withl0 = 1 m, r0 = 0.01 m, t0 = 10−5 s,ξ = 0.01,ψ = 0 andv0 = 30 m/s, where the mode numbers are: (a)m = 1; (b)m = 2; (c) m = 4. Inserts to the right show the final deformation state

(scaled bars) with contours of maximum principal logarithmic strain (εp).

In figure 7ano artificial volume load is added,ψ = 0. There are essentially no changes in the results obtainedwith cancellation of inertial pressureψ = 1, shown infigure 7b. There are some minor differences in details,but the overall necking pattern is the same.

724 K. Nilsson

Imperfection sensitivity is obvious also in the case of an aspect ratioα = 100 where the dimensions of thebar arel0 = 1 m andr0 = 0.01 m. Infigure 8q-levels of a bar with imperfection amplitudeξ = 0.01 and modenumbersm = 1, m = 2 andm = 4 are shown. The inserts of the deformed bars infigure 8are rescaled forclarity. The ramp timet0 = 10−5 s and the end velocityv0 = 30 m/s were chosen. No artificial volume load isincluded,ψ = 0.

Figure 8ashows the necking pattern for the mode numberm = 1. At a strain level ofε = 0.07 it seems likenecking will occur at the center of the bar (q = 2), but almost immediately after the first sign of this incipientlocalization, another neck precursor forms atX1/ l0 = 0.35. Eventually, the latter forms a strongly developedneck whilst the localization pattern closest to the center of the bar only results in three weakly developed necks.The spacing of the necks is almost periodic, with the dominating neck positioned at a slightly larger distancefrom the other neck precursors.

Changing the mode number tom = 2 results in a split of the necking pattern into one localization at thecenter of the bar and one at the end of the bar (figure 8b). At both thin regions of the bar one strongly and oneweakly developed localization occur, but nevertheless only the localization zone at the center of the bar prevails.Again the symmetry of the neck precursors is evident. The spacing between the growing and the declining neckprecursors is essentially the same in both cases.

Studies of the casem = 4 reveals the formation of one strongly developed neck atX1/ l0 = 0.5. There issome non-uniform deformation present at both the end and the center of the bar, but the localization zone atX1/ l0 = 0.5 is clearly dominant (figure 8c).

It should be noted that for this bar with aspect ratioα = 100, necking occurs at an overall logarithmic strainlevel lower than that observed forα = 10. In figure 8a–bε ≈ 0.13 at the final state of deformation, while infigure 8cε ≈ 0.1, indicating that neck localization is initiated at an overall strain levelε < N .

The effects of a smaller specimen size using the same aspect ratios mentioned above are also considered.For the aspect ratio ofα = 10 a bar withl0 = 0.1 m andr0 = 0.01 m is used. The ramp time is chosen ast0 = 10−5 s and the initial imperfections used areξ = 0, ξ = 0.001 andξ = 0.01, with m = 1. Only minordifferences between the cases using no artificial volume load (ψ = 0) and cancellation of background inertialeffects (ψ = 1) are present in the velocity range 20 m/s< v0 < 70 m/s. The overall necking pattern is the same,suggesting that background inertia plays a minor role compared to initial imperfections, end velocity and waveeffects.

(a) (b)

Figure 9. Contours of normalized average strain rate (q) as a function of time and distance for a tensile bar withm = 1, ξ = 0.01,ψ = 0, v0 = 40 m/sandt0 = 10−5 s, (a)l0 = 1 m,r0 = 0.1 m, (b)l0 = 0.1 m, r0 = 0.01 m. Inserts to the right show the final deformation state (scaled bars) with contours

of maximum principal logarithmic strain (εp).

Effects of inertia on dynamic neck formation 725

Comparison between results usingl0 = 1 m, r0 = 0.1 m andl0 = 0.1 m, r0 = 0.01 m show that the finalnecking sites are the same for the two different specimens, but that there are some noticeable differences in thedevelopment of theq-levels.Figure 9shows the results forl0 = 1 m, r0 = 0.1 m (figure 9a) and l0 = 0.1 m,r0 = 0.01 m (figure 9b) usingm = 1, ξ = 0.01, t0 = 10−5 s, v0 = 40 m/s andψ = 0. Two necks are present; astrongly developed neck atX1/ l0 = 0 and a weakly developed neck atX1/ l0 = 0.4. The development of thelatter neck infigure 9bis not as prominent as infigure 9a. Theq-levels do not reach the same values and theneck ceases to grow at an overall logarithmic strain level of approximatelyε = 0.225 compared toε = 0.25in figure 9a.

Using a bar with dimensionsl0 = 0.1 m andr0 = 0.001 m for the aspect ratioα = 100 reveals the samekind of behavior as described in the case ofα = 10. The end velocity is chosen asv0 = 30 m/s and theinitial imperfections areξ = 0, andξ = 0.001 withm = 1. Again, there are no differences in overall neckingpattern between the results using no artificial volume load (ψ = 0) and the results where background inertiais cancelled (ψ = 1). Comparison between the two different sized specimensl0 = 1 m, r0 = 0.01 m andl0 = 0.1 m,r0 = 0.001 m show that the final necking sites are the same for the two bars. Some local differencesin development ofq-levels are present, but the positions of the more and the less developed necks along the barare approximately the same.

The ramp time is varied to examine the effect of rapid loading, since a rapid ramp time such ast0 = 10−5 smight not permit the homogeneous deformation field leading to (12).Figure 10showsq-levels for a bar with

(a) (b)

(c) (d)

Figure 10. Contours of normalized average strain rate (q) as a function of time and distance for a tensile bar withl0 = 1 m, r0 = 0.01 m,ξ = 0.01,m = 1, ψ = 0 andv0 = 30 m/s, where the ramp times are: (a)t0 = 10−5 s; (b) t0 = 10−4 s; (c) t0 = 10−3 s; (d) t0 = 10−2 s. Inserts to the right show

the final deformation state (scaled bars) with contours of maximum principal logarithmic strain (εp).

726 K. Nilsson

initial dimensionsl0 = 1 m andr0 = 0.01 m where an imperfection ofξ = 0.01 andm = 1 is added. The endvelocity equalsv0 = 30 m/s and the ramp times aret0 = 10−5 s, t0 = 10−4 s, t0 = 10−3 s andt0 = 10−2 s.

Studies of the results using various ramp times show no difference between those obtained with no artificialpressure terms added (ψ = 0) and those where effects of background inertia are cancelled out (ψ = 1).Therefore only the results withψ = 0 are shown infigure 10.

In figure 10athe case witht0 = 10−5 s is shown once more (seefigure 8) for comparison reasons. The mostdeveloped neck is located atX1/ l0 = 0.35 with three weakly developed necks closer to the center of the bar.

Increasing the ramp time tot0 = 10−4 s in figure 10bshows little difference compared to the results infigure 10aregarding the large scale necking pattern. The zone withq = 0.8 aroundX1/ l0 = 0.9 is smaller inthis case and there are some differences inq-levels.

Further increase of the ramp time tot0 = 10−3 s leads to a rather different necking pattern, as can be seenin figure 10c. Six different necks are initialized and two of them form fully developed necks. Not only thenumber of necks has changed, but also the strain levels in the individual necks, so that it is no longer the necksfurthest away from the center of the bar that lead to failure. The neck located atX1/ l0 = 0.5 is the first oneto die out, compared to the neck atX1/ l0 = 0.35 in figure 10bwhich is the only one that continues to grow.The effect of the loading wave is no longer visible as a characteristic line because the accompanyingq-levelsare too small, but an extended plot using a larger number ofq-levels reveals that the initial response is stilldominated by dilatational waves. The nearly periodic pattern of neck precursors infigure 10cclosely resembles

(a) (b)

(c) (d)

Figure 11. Contours of normalized average strain rate (q) as a function of time and distance for a tensile bar withl0 = 1 m, r0 = 0.1 m, ξ = 0.01,m = 1, v0 = 40 m/s andt0 = 10−5 s where (a)ψ = 0 (b)ψ = 10 (c)ψ = 100 (d)ψ = 1000. Inserts to the right show the final deformation state (scaled

bars) with contours of maximum principal logarithmic strain (εp).

Effects of inertia on dynamic neck formation 727

that observed by Sørensen and Freund in their analysis of radially expanding rings (Sørensen and Freund,2000).

Another interesting feature is visualized infigure 10d. Increasing the ramp time tot0 = 10−2 s leads tonecking at two separate locations. The necking pattern at the center of the bar is still present. The number ofinitiated necks has been reduced and once again only one of them lead to final failure, this time the one closestto the center. The other neck is located at the end of the bar which in this case means the thickest part of thebar using the initial imperfection described above (m = 1, ξ = 0.01). The localization of the neck at the loadedend of the bar has been associated with high velocities in the cases described above, here (figure 10d) the onsetof necking occurs at an increased ramp timet0.

Simulations using different ramp times for a bar with aspect ratioα = 10 (l0 = 1 m, r0 = 0.1 m) shows thatlocalization at the end of the bar is promoted in this case also when the ramp time is increased tot0 = 10−2 s.The same result is obtained when using a smaller (l0 = 0.1 m, r0 = 0.01 m) bar with the same aspect ratio(α = 10), but no initial imperfections.

Figure 11displays the results of an increase of the factorψ in order to investigate whether or not backgroundinertia effects the necking pattern at all. The values of the inertia parameter used in the simulations are;ψ = 10,ψ = 100 andψ = 1000. The results for a bar withl0 = 1 m, r0 = 0.1 m, ξ = 0.01, m = 1, v0 = 40 m/s andt0 = 10−5 s are shown. Infigure 11athe results usingψ = 0 are shown for comparison. The overall neckingpattern is the same as infigure 11afor ψ = 10 (figure 11b). The overall logarithmic strainε at the finaldeformation state is slightly increased usingψ = 100, but the necking pattern remains unchanged infigure 11c.This is not the case whenψ = 1000 where the artificial pressure term is of the order of the yield stress.Figure 11ddemonstrates how the nature of the neck sites changes as the intermediate mode, which was formerlysuppressed, now forms a strongly developed neck localization. The neck at the center of the bar is now weaklydeveloped and ceases to grow at an overall logarithmic strain ofε = 0.22.

The required size of the artificial background inertia factor to obtain changes in the overall necking patternis investigated for several cases in order to explore the possibility of a smaller value of this factor in someconfiguration. An increase of the ramp time does not effect the size of the artificial inertia factor at which theoverall necking pattern is altered. Changing the ramp time tot0 = 10−3 using the same specimen geometry asin figure 11promotes the transition of an intermediate mode to a neck site at the center of the bar when theinertia factor reaches the valueψ = 1000, so thatp is of the order of the yield stress. Changing the aspectratio of the specimen toα = 100 by using the radiusr0 = 0.01 m using an artificial volume force factor equalto ψ = 1000 leads to an increase of the overall logarithmic strainε at the final state of deformation whenthe simulation is terminated as a fully developed neck has formed. In this case no transition of necking sitewas observed. The same kind of results are obtained when using a bar of smaller dimension (l0 = 0.1 m,r0 = 0.01 m) with no initial imperfections and an aspect ratioα = 10. The hydrostatic pressurep originatingfrom background inertia in (12) is dependent on specimen geometry, imposed end velocity and time. Thusit might be expected that the artificial volume force factor required to obtain changes in the overall neckingpattern is not the same for different configurations. In this study no changes in overall necking pattern occurredfor values smaller thanψ = 1000.

4. Conclusions

The effects of background inertia, geometry, thickness imperfections, imposed final velocity and ramp timeon neck formation have been investigated for cylindrical bars subjected to rapid extension. The study shows thatbackground inertia does not have any significant influence on the necking pattern in any of the cases analyzed,

728 K. Nilsson

unless the effect is artificially magnified to the magnitude of the yield stress. Instead wave effects, geometryand elastic unloading seems to control the formation of the different necking patterns observed.

There is an obvious imperfection sensitivity present when a thickness imperfection amplitude ofξ = 0.01 isused. At sufficiently low velocities necking occurs at the site of a thin point. A transition of neck site occurswhen the end velocity is increased, leading to necking at the end of the bar at large enough velocities. Whena smaller imperfection amplitude ofξ = 0.001 is used the results resemble those obtained using no thicknessimperfection at all, showing a rather random necking pattern strongly dependent on imposed end velocity.

The effects of dynamic loading on the strain level at localization were also studied. Delay in neck localizationis shown to increase with increasing end velocity for a bar with aspect ratioα = 10 until the point when neckingoccurs at the loaded end of the bar due to wave effects. After this instant, an increase of the end velocity onlyleads to earlier neck localization at the end of the bar.

When comparing two bars with the same aspect ratio, but with different specimen size (l0 = 1 m orl0 = 0.1 m) in the velocity range 20 m/s< v0 < 70 m/s the localization sites of strongly and weakly developednecks, respectively, are essentially the same though some differences in strain rate levels are present. A largeraspect ratio allows a larger number of necks to form. When using bars with aspect ratioα = 100 necklocalization occurs at lower levels of overall logarithmic strain compared with bars where the aspect ratioequalsα = 10.

An increase of ramp time leads to changes in necking pattern, including shifting of strongly and weaklydeveloped localization zones, respectively. The results of the simulations using a bar with aspect ratioα = 100,imposed end velocityv0 = 30 m/s and ramp timet0 = 10−3 s indicates that a nearly periodic necking patternexists in a certain loading range. A similar pattern was observed for the case of radically expanding rings ofa viscoplastic material (Sørensen and Freund, 2000) and in rings of a rate independent material (Han andTvergaard, 1995). The nearly periodic necking pattern seems to be a fundamental feature, which is quiteindependent, at least qualitatively, of rate hardening. Sørensen and Freund (2000) also included the effectsof adiabatic heating and failure by nucleation and void-growth, and the necking pattern appeared to be quiteinsensitive to both of these features.

Although multiple necking sites are present in some of the cases analyzed in this study only a few of themcontinue to grow. This type of behavior seems to be characteristic for all of the simulations conducted here,even in the cases using initial thickness imperfections with a mode numberm> 1 only one strongly developedneck appears. A possible explanation is that as soon as a sufficiently developed (strong) localization occurselastic unloading spreads throughout the specimen prohibiting further development at other neck sites.

Background inertia did not have any significant effect on the necking pattern in any of the cases analyzed.Initial imperfections and wave effects influenced the necking pattern in a more profound way. Not onlythickness imperfections, but also changes in end velocity and ramp time lead to large differences in the neckingpattern. This implies that background inertia is not the primary effect when studying dynamic deformation intensile bars withα � 10.

Acknowledgement

Prof. N.J. Sørensen of the Division of Mechanics, Lund University, S-22100 Sweden is acknowledged foradvice and discussion.

Effects of inertia on dynamic neck formation 729

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