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Effects of inertia on dynamic neck formation in tensile bars
Eur. J. Mech. A/Solids 20 (2001) 713–729 2001 Éditions scientifiques et médicales Elsevier SAS. All rights reserved S0997-7538(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 and the material is described as rate independent and elastic–plastic. The time history of neck development is investigated and effects of geometry and initial thickness imperfections are visualized. Studies of both weakly and strongly developed necks are performed, revealing the occurrence of multiple necks in some loading cases. The influence on neck formation from background inertia (lateral inertia) corresponding to the inertia originating from homogeneous deformation of the bar is examined somewhat qualitatively by introducing an artificial volume load. Calculations show that in the present analysis, background inertia does not have any noticeable influence on the necking pattern unless the effect is artificially magnified by three orders of magnitude 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 shortly after the maximum load has been reached. Using a one-dimensional analysis, Considere (1885) showed that an instability develops at the maximum load point. Introducing a full bifurcation analysis in the problem formulation, Hutchinson and Miles (1974) showed that localization is delayed due to the stiffening effect of the multiaxial stress state. For long slender bars, this corresponds to necking at tensile strain levels slightly higher than N , where N is the strain hardening exponent. Changing the aspect ratio of the specimen by using a shorter bar leads to a delay in localization. For rate independent solids under quasi-static conditions, bifurcation corresponding to loss of ellipticity of the rate equilibrium equations governs the onset of localization (Rice, 1976). Bifurcation is also associated with the occurrence of vanishing wave speeds; body waves that cease to travel through the specimen and become stationary. Rate dependent solids do not experience loss of ellipticity and true bifurcations are not found at realistic 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 independent materials (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 general agreement with experimental results obtained when expanding rings using electromagnetic loading. They also observed a delay in neck development when the expansion velocity was increased. In their study of expanding rings, Sørensen and Freund (2000) included the effects of rate sensitivity, void growth and thermal softening. ∗ Correspondence and reprints.
E-mail address: [email protected] (K. Nilsson).
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They found no strong correlation between any of these effects and the resulting necking pattern. Moreover it was found that many of the necking patterns formed occurred in essentially the same way; via a nearly periodic critical mode. Niordson (1965) used an experimental technique based on electromagnetic loading to obtain nearly homogeneous deformation of ring specimens, avoiding wave effects present in uniaxial tension tests that might cause early localization at the loaded end. After localization was initiated and deformation was no longer homogeneous multiple necks formed, some of them leading to final fracture. This technique has been used for several experiments investigating necking behavior of rate sensitive and rate insensitive materials. Altynova et al. (1996) studied increased ductility in high rate expansion of rings. They found a relation between the number of necks and imposed velocity. The differences in necking pattern obtained using dynamic loading compared to quasi-static conditions, are considered as partly due to material inertia. Even though similar surface imperfections are used, several necks may develop in the dynamic case at sites that correlate weakly or not at all with the neck sites found under quasi-static conditions. Needleman (1989) used a local reduction in flow strength to investigate dynamic shear band propagation in a rate dependent solid under plane strain conditions. Differences in behavior for hardening and softening materials were investigated using a rectangular block subject to compression. A significant retardation of shear band development due to inertial effects was found. A one-dimensional analysis of a round bar subjected to tension was performed by Tuˇgcu et al. (1990) where the stabilizing effects of inertia were observed (see also Fressengeas and Molinari (1994)). An elastic–viscoplastic constitutive law was used to model the material behavior of solids with strain and strain rate hardening characteristics. A limitation of the one-dimensional analysis was that at high velocities radial inertia effects may no longer be negligible. Sørensen and Freund (1998) showed that mode inertia, i.e. the inertia following a rapidly growing incipient neck leads to a suppression of the modes with the longest wave length. In addition they showed that bifurcation modes are available essentially everywhere when the imperfection growth rate as well as the strain levels are sufficiently high. Shenoy and Freund (1999) found that very short wavelength modes of nonuniform deformation are suppressed by background inertia. In the analysis of Shenoy and Freund (1999) a linear stability approach is used, so that results regarding the onset of a bifurcation seem to rely on the assumption that wave effects and imperfection 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 dynamic axisymmetric finite element approach to simulate uniaxial tension. The aim is to explore the development of localization zones with time and in particular effects of background inertia. In the present context background inertia is defined as the inertia originating from idealized homogeneous deformation of the bar. An artificial volume load chosen as a pressure whose form is similar to that originating from this homogeneous deformation is introduced to qualitatively study the effects of background inertia. Clearly, perfect homogeneous deformation does not take place in the actual cases studied. However, parts of the inertial effects due to the global shape of the bar are investigated in this way. The full time history of neck formation is visualized using a strain rate parameter depicting both localization zones that die out as well as localization zones that continue to grow until they eventually cause final failure. The effects of specimen size, aspect ratio, end velocity and surface imperfections are studied. During high rate homogeneous extension of round bars a hydrostatic pressure appears in the material due to inertia (e.g., Graff, 1975). It is this pressure effect resulting from assumed incompressible and homogeneous deformation that is termed background inertia here. The effects of background inertia in the present case of dynamic loading of a round bar is studied by introducing an artificial volume force in the form of hydrostatic tension. This artificial hydrostatic tension is in several cases selected to be precisely opposite to the hydrostatic pressure (background inertia) that results from rapid homogeneous extension of the bar to explore any possible
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effect of background inertia. Attention is thus directed towards the effect of background inertia and for all simulations regardless of specimen size, aspect ratio, etc., corresponding simulations have been performed where 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. Material coordinates X i are introduced in the reference configuration which occupies the undeformed region 0 X 1 l0 , 0 X 2 r0 (figure 1) with rotational symmetry around the X 1 axis. Only half of the bar is studied numerically due to mirror symmetry about the mid plane at the center of the bar, containing the X 2 axis. The full initial length and diameter of the bar are thus denoted 2l0 and 2r0 , respectively. Initial sinusoidal surface imperfections are introduced in the form:
π mX 1 r0 = rˆ0 1 − ξ cos l0
,
(1)
where ξ is the dimensionless imperfection amplitude and m is the mode number. The aspect ratio of the bar is defined as: l0 (2) α= . rˆ0 The following boundary conditions are used for displacements ui and tractions T i : u1 = u1 (t),
T 2 = 0,
X 1 = l0 ;
(3)
u1 = 0,
T = 0,
X = 0;
(4)
u2 = 0,
T = 0,
X = 0;
(5)
X = r0 ,
(6)
2 1
T = T = 0, 1
2
1 2 2
where the displacement u1 (t) at the end of the bar in the direction of the axis of rotational symmetry is determined by an imposed end velocity ramped from zero to a constant maximum value according to:
u1 (t) =
v0 t 2 /2t0 , t t0 , v0 t0 /2 + v0 (t − t0 ), t > t0 ,
(7)
where t is the time, t0 is a ramp time and v0 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 initial length 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 the X2 axis. (X1 , X2 ) are cylindrical coordinates and the X1 axis is the axis of rotational symmetry.
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deformation. The velocity field originating from homogeneous deformation of an incompressible bar referred to the material coordinates of the reference configuration is for t > t0 :
V 1 X1 , X2, X3, t =
V 2 X1 , X2, X3, t = −
X 1 v0 ; l0
v0 2l0 1 +
(8)
X2
v0 t l0
− t20
3/2 ;
(9)
V 3 X 1 , X 2 , X 3 , t = 0,
(10)
taking into account that the boundary velocity is ramped. This velocity field may be used to define the contribution of background inertia to the total stress field of the axisymmetric bar. Considering deformation due to the velocity field (8)–(10) a stress field σ I caused by background inertia can be determined from the dynamic equilibrium condition. The total stress field σ T may be considered as consisting of two parts; σ I and σN: σT = σI + σN,
(11)
where σ N does not include contributions from background inertia. Using the stress field σ I , the hydrostatic pressure contribution p from background inertia can be determined to be of the form:
p X 2 , t = ρ0
3 v0 8 l0
2
1+
1
v0 t l0
−
t0 5/2 2
r02 − X 2
2
,
t > t0 .
(12)
The effects of background inertia on neck development may be addressed somewhat qualitatively by imposing an artificial volume load corresponding to −ψp where ψ = 1 means that the pressure effects due to background inertia are cancelled out. Figure 2 shows the development of the maximum pressure p with time for a bar with l0 = 1 m, r0 = 0.1 m, v0 = 40 m/s and t0 = 10−5 s.
Figure 2. Time development of the maximum inertial pressure p normalized with the yield stress σy for a bar with l0 = 1 m, r0 = 0.1 m, v0 = 40 m/s and t0 = 10−5 s. The pressure p is due to background inertia.
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The deformation time history of the full length of the tensile bar is visualized using contour plots of a normalized average cross sectional strain rate q: q=
ε˙p c , ε˙p t
(13)
where c and t denotes a cross sectional average and a total average, respectively and ε˙p is the maximum principal logarithmic strain rate (as in Sørensen and Freund (2000)). This strain rate parameter is plotted over the reference length of the bar (X 1 / l0 ) as a function of the overall logarithmic strain ε = ln(l/ l0 ) in the following figures. Values of q higher than unity and lower than unity indicates a strain rate higher than or lower 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:
τ δηij dV =
ij
V
T δui dS −
i
S
V
∂ 2 ui ρ0 2 δui dV + ∂t
ψpGij δηij dV ,
(14)
V
where V and S refer to volume and surface of the body, respectively, in the reference configuration. In (14) the volume force introduced to study the effects of background inertia appears in the artificial volume load term containing the pressure p. The covariant components of the Lagrangian strain tensor are denoted by ηij and the contravariant components of Kirchhoff stress are denoted τ ij . The mass density at time t = 0 when the body is in the reference configuration is ρ0 , the variational operator is δ and Gij is the metric tensor in the current configuration. The stress and strain tensors in (14) are: (15) τ ij = (ρ0 /ρ)σ ij ; 1 (16) ηij = ui,j + uj,i + uk,i uk,j , 2 where ρ denotes current density. Comma represents covariant differentiation with respect to the material coordinate of the reference configuration. Nominal traction is denoted T i and ui are the displacement vector components in the reference configuration. The constitutive relations of the elastic–plastic material are represented by rate independent J2 flow theory (Hutchinson, 1973). The linear incremental form of the constitutive response for a nearly incompressible material (τ ij ≈ σ ij ) can be expressed in terms of the nonobjective rate of Cauchy stress and the increment of Lagrangian strain: σ˙ ij = Lij kl η˙ kl ,
(17)
with the instantaneous moduli specified by:
Lij kl =
E/Et − 1 s ij s kl E 1 ik j l ν 3 Gij Gkl − αˆ G G + Gil Gj k + 1+ν 2 1 − 2ν 2 E/Et − (1 − 2ν)/3 σe2 1 − Gik σ j l + Gj k σ il + Gil σ j k + Gj l σ ik . 2
(18)
Here E is Young’s modulus, ν is Poisson’s ratio and αˆ is an elastic–plastic switch which equals zero or unity depending on whether or not yielding occurs. The stress deviator in (18) is written as: 1 s ij = σ ij − Gij σkl Gkl , 3
(19)
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and the effective Mises stress is:
σe =
3 ij s sij . 2
(20)
Under uniaxial tension the tangent modulus Et relates the true stress rate and logarithmic strain rate for an increment of stress according to σ˙ = Et ε˙ . A piecewise power law is adopted:
ε=
for σ σy , for σ > σy ,
σ E σy σ 1/N E σy
(21)
where σy is the initial yield stress and N is the strain hardening exponent. The numerical simulations are performed using a finite element approach to the dynamic principle of virtual work (14). Axisymmetric quadrilateral finite elements consisting of four crossed triangles to avoid locking (Nagtegaal et al., 1974) are used in a Lagrangian framework (see figure 1). A Newmark time integration scheme with a lumped mass matrix M is applied:
˙ (t ),t + (1/2 − k2 )D ¨ (t ) + k2 D ¨ (t +,t ) (,t)2 ; D(t +,t ) = D(t ) + D
¨ (t +,t ) = M−1 −F(t +,t ) ; D
(22) (23)
˙ (t ) + (1 − k1 )D ¨ (t ) + k1 D ¨ (t +,t ) ,t, ˙ (t +,t ) = D D
(24)
where D denotes nodal displacement, F denotes nodal forces and (˙) indicates differentiation with respect to time. Explicit time integration with no stiffness matrix computation or iteration is obtained by choosing k2 = 0 (Belytschko et al., 1976). The other parameter is chosen to be k1 = 1/2. The use of a lumped mass matrix has been shown to be preferable in the case of explicit time integration (Krieg and Kay, 1973) concerning both accuracy and efficiency. The size of the time step ,t is limited by element size and the characteristic speed of propagation of information. The maximum time step is chosen as: ,t < ζ ,de /cd ,
(25)
where ,de denotes the minimum element dimension and ζ is a constant taken to be in the range 0 < ζ < 1. The dilatational wave speed cd , the highest wave speed in an elastic medium, is:
cd =
E(1 − ν) . ρ(1 + ν)(1 − 2ν)
The shear wave speed is:
cr =
E . 2ρ(1 + ν)
(26)
(27)
3. Results The material parameters are representative of a structural steel; E = 2.069 × 1011 Pa, ν = 0.3 and ρ0 = 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
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initial yield stress is σy = 374.87 × 106 Pa and the piecewise power law hardening exponent is N = 0.1. The imperfection amplitudes considered are ξ = 0, ξ = 0.01 and ξ = 0.001, using the mode numbers m = 1, m = 2 and m = 4. Four different specimen sizes are used; l0 = 1 m, r0 = 0.1 m and l0 = 0.1 m, r0 = 0.01 m has an aspect ratio of α = 10 whereas l0 = 1 m, r0 = 0.01 m and l0 = 0.1 m, r0 = 0.001 m has an aspect ratio of α = 100. The mesh used for the aspect ratio α = 10 consists of 10 × 100 quadrilaterals (figure 1) and the corresponding mesh for α = 100 consists of 10 × 1000 quadrilaterals. The imposed end velocities are chosen below v0 = 100 m/s since the use of higher velocities in the present analysis leads to necking at the loaded end of the specimen. In figure 3 the q-levels of a bar with dimensions l0 = 1 m, r0 = 0.1 m and no initial imperfection (ξ = 0) are displayed. The results using four different values of the end velocity v0 , are shown. The ramp time is chosen as t0 = 10−5 s and no artificial volume load is included (ψ = 0). The deformation history is represented in terms of contours of constant q-levels plotted as a function of the reference position X 1 / l0 and overall logarithmic strain ε = ln(l/ l0 ). The insert to the right of the q contour plot shows the final state of the bar (scaled) with the shading indicating contours of constant principal logarithmic strain εp . In figure 3a v0 = 20 m/s and neck localization occurs at approximately X 1 / l0 = 0.55. For symmetry reasons this means that there are actually two necking sites along the full length of the bar. An increase of the velocity to v0 = 30 m/s leads to a shifting of the most developed neck towards the center of the bar (figure 3b). There is also a weak localization closer to the end of the bar, but this localization does not develop into all full-grown neck. At a velocity of v0 = 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 with l0 = 1 m, r0 = 0.1 m, t0 = 10−5 s, m = 0 and ψ = 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 ).
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K. Nilsson
zone with higher than average strain rate present at the center, as can be seen in figure 3c. For velocities of v0 = 50 m/s and above, necking occurs at the end of the bar, figure 3d. At the left-hand side of figures 3a–d, an intensively black region originating from X 1 / l0 = 1.0 indicates rapid propagation along the bar of information reaching X 1 / l0 = 0 at approximately ε = 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 the undeformed bar. An increase of the velocity from v0 = 20 m/s to v0 = 40 m/s causes an increase in logarithmic overall strain at the final deformation state from ε = 0.26 to ε = 0.39, but when the velocity reaches v0 = 50 m/s the overall strain level is reduced to ε = 0.29 and further increase of end velocity diminishes ε even more. A similar behavior is noticed when studying normalized traction at the end of the bar. In figure 4 normalized engineering traction is plotted against logarithmic overall strain for the cases a–d in figure 3. Large wave effects are generated at impact, causing initial oscillation of the normalized traction. The critical overall strain level at the first appearance of incipient neck localization in figure 4 increases when increasing the end velocity from v0 = 20 m/s (ε = 0.15) (a) to v0 = 30 m/s (ε = 0.17) (b), but decreases again when v0 = 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 level continues to diminish. In order to study imperfection sensitivity, bars with varying thickness were introduced. Figure 5 depicts the contours of constant q-levels for a bar with dimensions l0 = 1 m, r0 = 0.1 m and an initial imperfection of ξ = 0.01, m = 1. The ramp time is again chosen as t0 = 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 result is expected in the quasi-static case because of the initial imperfection. Increasing the velocity to v0 = 30 m/s leads to a shifting in the final neck site towards the end of the bar (figure 5b). Further increase of the velocity to v0 = 40 m/s shows that although there is an incipient neck between the center and the end of the bar, the dominating neck is the one developed at the site of the imperfection. The neck precursor at X 1 / l0 = 0.4 dies out at an overall logarithmic strain of approximately ε = 0.25. The neck formation at the imperfection site of the bar is termed stable, whilst the neck precursor at X 1 / l0 = 0.4 is termed unstable (compare with (Sørensen and Freund, 2000)). At the velocity v0 = 50 m/s, necking occurs at the end of the bar, and further simulations reveal that this is the case for all velocities above v0 = 50 m/s for this particular bar.
Figure 4. Normalized traction plotted against logarithmic overall strain for a tensile bar with l0 = 1 m, r0 = 0.1 m, t0 = 10−5 s, m = 0 and ψ = 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
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(a)
(b)
(c)
(d)
Figure 5. Contours of normalized average strain rate (q) as a function of time and distance for a tensile bar with l0 = 1 m, r0 = 0.1 m, t0 = 10−5 s, ξ = 0.01, m = 1 and ψ = 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 between v0 = 40 m/s and v0 = 50 m/s did not lead to fully developed necks positioned between the center and the end of the bar. Thus, the transition of dominating neck site from the center 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 imperfections are shown in figure 6. The imperfection amplitude remains at ξ = 0.01, but the mode number has been changed to m = 2 in figures 6a–b corresponding to thin points at X 1 / l0 = 0 and X 1 / l0 = 1, and m = 4 in figures 6c–d corresponding to thin points located at X 1 / l0 = 0, X 1 / l0 = 0.5 and X 1 / l0 = 1. In figure 6a where the velocity is v0 = 30 m/s, the most developed neck localization occurs at the center of the bar, but in addition a weakly developed neck is formed at the end of the bar. This incipient localization does not proceed, and instead it fades away at an early stage. When the velocity is increased to v0 = 40 m/s as in figure 6b, the result is the opposite. Necking occurs at the end of the bar whilst a weakly developed localization is observed at the center. Further increase of the velocity shows similar behavior with a gradually weakening of the localization at the center of the bar. Using this initial imperfection, there is no sign of any incipient localization between the thin points. Studies of velocities between v0 = 30 m/s and v0 = 40 m/s show that the transition between necking at the center of the bar and at the end of the bar is performed with no visible signs of other incipient localization sites. It should also be noted that none of the simulations in the present study show the occurrence of two strongly developed necks simultaneously along the bar with aspect ratio α = 10. Only one neck prevails in all test cases.
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(a)
(b)
(c)
(d)
Figure 6. Contours of normalized average strain rate (q) as a function of time and distance for a tensile bar with l0 = 1 m, r0 = 0.1 m, t0 = 10−5 s, ξ = 0.01 and ψ = 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–d the mode number has been increased to m = 4. Necking occurs at the center of the bar in figure 6c, where the velocity equals v0 = 30 m/s. Increase of the end velocity to v0 = 40 m/s leads to a shift in necking site, so that the most developed neck is positioned at the end of the bar. In both of these cases, apart from the dominant neck, there are weakly developed necks at the thin points. Studies of the necking pattern at velocities between v0 = 30 m/s and v0 = 40 m/s reveals that there is a transition interval where the most developed neck occurs at X 1 / l0 = 0.5. As in the case with m = 2 only one of the localization zones is strongly developed. Simulations using a bar with an imperfection amplitude reduced to ξ = 0.001 (l0 = 1 m, r0 = 0.1 m) show an inclination towards formation of a fully developed neck located between the center of the bar and the end of the bar, though the actual necking pattern differs between the three different cases of imperfection (mode number m = 1, m = 2 and m = 4). Qualitatively similar results are obtained as in the case when using a bar with 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 homogeneous deformation is small whereas the extension rate have a large influence on the resulting neck formation. Simulations corresponding to all cases mentioned above, but using ψ = 1 and thereby canceling background inertia, show essentially no deviation from the results presented in figures 3–6. Figure 7 shows a comparison between the results acquired with no artificial volume load and those with cancellation of background inertia. In figures 7a–b the size of the bar is l0 = 1 m, r0 = 0.1 m with an imperfection of m = 1, ξ = 0.01 and a ramp time of t0 = 10−5 s. The end velocity is chosen as v0 = 40 m/s.
Effects of inertia on dynamic neck formation
(a)
723
(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 and v0 = 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 with l0 = 1 m, r0 = 0.01 m, t0 = 10−5 s, ξ = 0.01, ψ = 0 and v0 = 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 7a no artificial volume load is added, ψ = 0. There are essentially no changes in the results obtained with cancellation of inertial pressure ψ = 1, shown in figure 7b. There are some minor differences in details, but the overall necking pattern is the same.
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Imperfection sensitivity is obvious also in the case of an aspect ratio α = 100 where the dimensions of the bar are l0 = 1 m and r0 = 0.01 m. In figure 8 q-levels of a bar with imperfection amplitude ξ = 0.01 and mode numbers m = 1, m = 2 and m = 4 are shown. The inserts of the deformed bars in figure 8 are rescaled for clarity. The ramp time t0 = 10−5 s and the end velocity v0 = 30 m/s were chosen. No artificial volume load is included, ψ = 0. Figure 8a shows the necking pattern for the mode number m = 1. At a strain level of ε = 0.07 it seems like necking will occur at the center of the bar (q = 2), but almost immediately after the first sign of this incipient localization, another neck precursor forms at X 1 / l0 = 0.35. Eventually, the latter forms a strongly developed neck 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 distance from the other neck precursors. Changing the mode number to m = 2 results in a split of the necking pattern into one localization at the center of the bar and one at the end of the bar (figure 8b). At both thin regions of the bar one strongly and one weakly 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 neck precursors is essentially the same in both cases. Studies of the case m = 4 reveals the formation of one strongly developed neck at X 1 / l0 = 0.5. There is some non-uniform deformation present at both the end and the center of the bar, but the localization zone at X 1 / l0 = 0.5 is clearly dominant (figure 8c). It should be noted that for this bar with aspect ratio α = 100, necking occurs at an overall logarithmic strain level lower than that observed for α = 10. In figure 8a–b ε ≈ 0.13 at the final state of deformation, while in figure 8c ε ≈ 0.1, indicating that neck localization is initiated at an overall strain level ε < N . The effects of a smaller specimen size using the same aspect ratios mentioned above are also considered. For the aspect ratio of α = 10 a bar with l0 = 0.1 m and r0 = 0.01 m is used. The ramp time is chosen as t0 = 10−5 s and the initial imperfections used are ξ = 0, ξ = 0.001 and ξ = 0.01, with m = 1. Only minor differences between the cases using no artificial volume load (ψ = 0) and cancellation of background inertial effects (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 wave effects.
(a)
(b)
Figure 9. Contours of normalized average strain rate (q) as a function of time and distance for a tensile bar with m = 1, ξ = 0.01, ψ = 0, v0 = 40 m/s and t0 = 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 ).
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Comparison between results using l0 = 1 m, r0 = 0.1 m and l0 = 0.1 m, r0 = 0.01 m show that the final necking sites are the same for the two different specimens, but that there are some noticeable differences in the development of the q-levels. Figure 9 shows the results for l0 = 1 m, r0 = 0.1 m (figure 9a) and l0 = 0.1 m, r0 = 0.01 m (figure 9b) using m = 1, ξ = 0.01, t0 = 10−5 s, v0 = 40 m/s and ψ = 0. Two necks are present; a strongly developed neck at X 1 / l0 = 0 and a weakly developed neck at X 1 / l0 = 0.4. The development of the latter neck in figure 9b is not as prominent as in figure 9a. The q-levels do not reach the same values and the neck ceases to grow at an overall logarithmic strain level of approximately ε = 0.225 compared to ε = 0.25 in figure 9a. Using a bar with dimensions l0 = 0.1 m and r0 = 0.001 m for the aspect ratio α = 100 reveals the same kind of behavior as described in the case of α = 10. The end velocity is chosen as v0 = 30 m/s and the initial imperfections are ξ = 0, and ξ = 0.001 with m = 1. Again, there are no differences in overall necking pattern between the results using no artificial volume load (ψ = 0) and the results where background inertia is cancelled (ψ = 1). Comparison between the two different sized specimens l0 = 1 m, r0 = 0.01 m and l0 = 0.1 m, r0 = 0.001 m show that the final necking sites are the same for the two bars. Some local differences in development of q-levels are present, but the positions of the more and the less developed necks along the bar are approximately the same. The ramp time is varied to examine the effect of rapid loading, since a rapid ramp time such as t0 = 10−5 s might not permit the homogeneous deformation field leading to (12). Figure 10 shows q-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 with l0 = 1 m, r0 = 0.01 m, ξ = 0.01, m = 1, ψ = 0 and v0 = 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 ).
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initial dimensions l0 = 1 m and r0 = 0.01 m where an imperfection of ξ = 0.01 and m = 1 is added. The end velocity equals v0 = 30 m/s and the ramp times are t0 = 10−5 s, t0 = 10−4 s, t0 = 10−3 s and t0 = 10−2 s. Studies of the results using various ramp times show no difference between those obtained with no artificial pressure terms added (ψ = 0) and those where effects of background inertia are cancelled out (ψ = 1). Therefore only the results with ψ = 0 are shown in figure 10. In figure 10a the case with t0 = 10−5 s is shown once more (see figure 8) for comparison reasons. The most developed neck is located at X 1 / l0 = 0.35 with three weakly developed necks closer to the center of the bar. Increasing the ramp time to t0 = 10−4 s in figure 10b shows little difference compared to the results in figure 10a regarding the large scale necking pattern. The zone with q = 0.8 around X 1 / l0 = 0.9 is smaller in this case and there are some differences in q-levels. Further increase of the ramp time to t0 = 10−3 s leads to a rather different necking pattern, as can be seen in figure 10c. Six different necks are initialized and two of them form fully developed necks. Not only the number of necks has changed, but also the strain levels in the individual necks, so that it is no longer the necks furthest away from the center of the bar that lead to failure. The neck located at X 1 / l0 = 0.5 is the first one to die out, compared to the neck at X 1 / l0 = 0.35 in figure 10b which is the only one that continues to grow. The effect of the loading wave is no longer visible as a characteristic line because the accompanying q-levels are too small, but an extended plot using a larger number of q-levels reveals that the initial response is still dominated by dilatational waves. The nearly periodic pattern of neck precursors in figure 10c closely 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 with l0 = 1 m, r0 = 0.1 m, ξ = 0.01, m = 1, v0 = 40 m/s and t0 = 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
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that observed by Sørensen and Freund in their analysis of radially expanding rings (Sørensen and Freund, 2000). Another interesting feature is visualized in figure 10d. Increasing the ramp time to t0 = 10−2 s leads to necking at two separate locations. The necking pattern at the center of the bar is still present. The number of initiated necks has been reduced and once again only one of them lead to final failure, this time the one closest to the center. The other neck is located at the end of the bar which in this case means the thickest part of the bar using the initial imperfection described above (m = 1, ξ = 0.01). The localization of the neck at the loaded end of the bar has been associated with high velocities in the cases described above, here (figure 10d) the onset of necking occurs at an increased ramp time t0 . Simulations using different ramp times for a bar with aspect ratio α = 10 (l0 = 1 m, r0 = 0.1 m) shows that localization at the end of the bar is promoted in this case also when the ramp time is increased to t0 = 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 11 displays the results of an increase of the factor ψ in order to investigate whether or not background inertia effects the necking pattern at all. The values of the inertia parameter used in the simulations are; ψ = 10, ψ = 100 and ψ = 1000. The results for a bar with l0 = 1 m, r0 = 0.1 m, ξ = 0.01, m = 1, v0 = 40 m/s and t0 = 10−5 s are shown. In figure 11a the results using ψ = 0 are shown for comparison. The overall necking pattern is the same as in figure 11a for ψ = 10 (figure 11b). The overall logarithmic strain ε at the final deformation state is slightly increased using ψ = 100, but the necking pattern remains unchanged in figure 11c. This is not the case when ψ = 1000 where the artificial pressure term is of the order of the yield stress. Figure 11d demonstrates how the nature of the neck sites changes as the intermediate mode, which was formerly suppressed, now forms a strongly developed neck localization. The neck at the center of the bar is now weakly developed and ceases to grow at an overall logarithmic strain of ε = 0.22. The required size of the artificial background inertia factor to obtain changes in the overall necking pattern is investigated for several cases in order to explore the possibility of a smaller value of this factor in some configuration. An increase of the ramp time does not effect the size of the artificial inertia factor at which the overall necking pattern is altered. Changing the ramp time to t0 = 10−3 using the same specimen geometry as in figure 11 promotes the transition of an intermediate mode to a neck site at the center of the bar when the inertia factor reaches the value ψ = 1000, so that p is of the order of the yield stress. Changing the aspect ratio of the specimen to α = 100 by using the radius r0 = 0.01 m using an artificial volume force factor equal to ψ = 1000 leads to an increase of the overall logarithmic strain ε at the final state of deformation when the simulation is terminated as a fully developed neck has formed. In this case no transition of necking site was 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 ratio α = 10. The hydrostatic pressure p originating from background inertia in (12) is dependent on specimen geometry, imposed end velocity and time. Thus it might be expected that the artificial volume force factor required to obtain changes in the overall necking pattern is not the same for different configurations. In this study no changes in overall necking pattern occurred for values smaller than ψ = 1000. 4. Conclusions The effects of background inertia, geometry, thickness imperfections, imposed final velocity and ramp time on neck formation have been investigated for cylindrical bars subjected to rapid extension. The study shows that background inertia does not have any significant influence on the necking pattern in any of the cases analyzed,
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unless the effect is artificially magnified to the magnitude of the yield stress. Instead wave effects, geometry and 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 of ξ = 0.01 is used. At sufficiently low velocities necking occurs at the site of a thin point. A transition of neck site occurs when the end velocity is increased, leading to necking at the end of the bar at large enough velocities. When a smaller imperfection amplitude of ξ = 0.001 is used the results resemble those obtained using no thickness imperfection 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 localization is shown to increase with increasing end velocity for a bar with aspect ratio α = 10 until the point when necking occurs at the loaded end of the bar due to wave effects. After this instant, an increase of the end velocity only leads 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 or l0 = 0.1 m) in the velocity range 20 m/s < v0 < 70 m/s the localization sites of strongly and weakly developed necks, respectively, are essentially the same though some differences in strain rate levels are present. A larger aspect ratio allows a larger number of necks to form. When using bars with aspect ratio α = 100 neck localization occurs at lower levels of overall logarithmic strain compared with bars where the aspect ratio equals α = 10. An increase of ramp time leads to changes in necking pattern, including shifting of strongly and weakly developed localization zones, respectively. The results of the simulations using a bar with aspect ratio α = 100, imposed end velocity v0 = 30 m/s and ramp time t0 = 10−3 s indicates that a nearly periodic necking pattern exists in a certain loading range. A similar pattern was observed for the case of radically expanding rings of a viscoplastic material (Sørensen and Freund, 2000) and in rings of a rate independent material (Han and Tvergaard, 1995). The nearly periodic necking pattern seems to be a fundamental feature, which is quite independent, at least qualitatively, of rate hardening. Sørensen and Freund (2000) also included the effects of adiabatic heating and failure by nucleation and void-growth, and the necking pattern appeared to be quite insensitive to both of these features. Although multiple necking sites are present in some of the cases analyzed in this study only a few of them continue 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 number m > 1 only one strongly developed neck appears. A possible explanation is that as soon as a sufficiently developed (strong) localization occurs elastic 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 only thickness imperfections, but also changes in end velocity and ramp time lead to large differences in the necking pattern. This implies that background inertia is not the primary effect when studying dynamic deformation in tensile bars with α 10.
Acknowledgement Prof. N.J. Sørensen of the Division of Mechanics, Lund University, S-22100 Sweden is acknowledged for advice and discussion.
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References Altynova, M., Hu, X., Daehn, G.S., 1996. Increased ductility in high velocity electromagnetic ring expansion. Metallurgical and Materials Transactions 27a, 1837–1844. Belytschko, T., Chiapetta, R.L., Bartel, H.D., 1976. Efficient large scale non-linear transient analysis by finite elements. International Journal of Numerical Methods in Engineering 10, 71–84. Considere, M., 1885. L’Emploi du Fer de l’Acier dans les Constructions, Mémoire No. 34. Annales des Ponts et Chaussées, Paris, pp. 574–775. Fressengeas, C., Molinari, A., 1994. Fragmentation of rapidly streching sheets. Eur. J. Mech. A/Solids 13, 251–268. Graff, K.F., 1975. Wave Motion in Elastic Solids. Clarendon Press, Oxford. Han, J.B., Tvergaard, V., 1995. Effect of inertia on the necking behaviour of ring specimens under rapid radial expansion. Eur. J. Mech. A/Solids 14, 287–307. Hutchinson, J.W., 1973. Finite strain analysis of elastic–plastic solids and structures, in: Hartung, R.F. (Ed.), Numerical Solutions of Nonlinear Structural Problems. ASME, New York, pp. 17–29. Hutchinson, J.W., Miles, J.P., 1974. Bifurcation analysis of the onset of necking in an elastic/plastic cylinder under uniaxial tension. J. Mech. Phys. Solids 22, 61–71. Hutchinson, J.W., Neale, K.W., 1977. Influence of strain rate sensitivity on necking under uniaxial tension. Acta Met. 25, 839–846. Krieg, R.D., Kay, S.W., 1973. Transient shell response by numerical time integration. International Journal of Numerical Methods in Engineering 7, 273–286. Nagtegaal, J., Parks, D.M., Rice, J.R., 1974. On the numerical accurate finite element solutions in the fully plastic range. Computer Method in Applied Mechanics and Engineering 4, 153–177. Needleman, A., 1989. Dynamic shear band development in plane strain. Journal of Applied Mechanics 56 (1). Niordson, F.I., 1965. A unit for testing materials at high strain rates. Experimental Mechanics 5, 29–32. Rice J.R., 1976. The localization of plastic deformation. Theoretical and Applied Mechanics, 207–220. Shenoy, V.B., Freund, L.B., 1999. Necking bifurcations during high strain rate extension. J. Mech. Phys. Solids 47, 2209–2233. Sørensen, N.J., Freund, L.B., 1998. Dynamic bifurcation during high-rate planar extension of a rectangular block. Eur. J. Mech. A/Solids 17, 709– 724. Sørensen, N.J., Freund, L.B., 2000. Unstable neck formation in a ductile ring subjected to impulsive radial loading. Int. J. Sol. Struct. 2265–2283. Tuˇgcu, P., Neale, K.W., Lahoud, A.E., 1990. Inertial effects on necking in tension. Int. J. Solids Structures 26 (11), 1275–1285.
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 and the material is described as rate independent and elastic–plastic. The time history of neck development is investigated and effects of geometry and initial thickness imperfections are visualized. Studies of both weakly and strongly developed necks are performed, revealing the occurrence of multiple necks in some loading cases. The influence on neck formation from background inertia (lateral inertia) corresponding to the inertia originating from homogeneous deformation of the bar is examined somewhat qualitatively by introducing an artificial volume load. Calculations show that in the present analysis, background inertia does not have any noticeable influence on the necking pattern unless the effect is artificially magnified by three orders of magnitude 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 shortly after the maximum load has been reached. Using a one-dimensional analysis, Considere (1885) showed that an instability develops at the maximum load point. Introducing a full bifurcation analysis in the problem formulation, Hutchinson and Miles (1974) showed that localization is delayed due to the stiffening effect of the multiaxial stress state. For long slender bars, this corresponds to necking at tensile strain levels slightly higher than N , where N is the strain hardening exponent. Changing the aspect ratio of the specimen by using a shorter bar leads to a delay in localization. For rate independent solids under quasi-static conditions, bifurcation corresponding to loss of ellipticity of the rate equilibrium equations governs the onset of localization (Rice, 1976). Bifurcation is also associated with the occurrence of vanishing wave speeds; body waves that cease to travel through the specimen and become stationary. Rate dependent solids do not experience loss of ellipticity and true bifurcations are not found at realistic 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 independent materials (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 general agreement with experimental results obtained when expanding rings using electromagnetic loading. They also observed a delay in neck development when the expansion velocity was increased. In their study of expanding rings, Sørensen and Freund (2000) included the effects of rate sensitivity, void growth and thermal softening. ∗ Correspondence and reprints.
E-mail address: [email protected] (K. Nilsson).
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They found no strong correlation between any of these effects and the resulting necking pattern. Moreover it was found that many of the necking patterns formed occurred in essentially the same way; via a nearly periodic critical mode. Niordson (1965) used an experimental technique based on electromagnetic loading to obtain nearly homogeneous deformation of ring specimens, avoiding wave effects present in uniaxial tension tests that might cause early localization at the loaded end. After localization was initiated and deformation was no longer homogeneous multiple necks formed, some of them leading to final fracture. This technique has been used for several experiments investigating necking behavior of rate sensitive and rate insensitive materials. Altynova et al. (1996) studied increased ductility in high rate expansion of rings. They found a relation between the number of necks and imposed velocity. The differences in necking pattern obtained using dynamic loading compared to quasi-static conditions, are considered as partly due to material inertia. Even though similar surface imperfections are used, several necks may develop in the dynamic case at sites that correlate weakly or not at all with the neck sites found under quasi-static conditions. Needleman (1989) used a local reduction in flow strength to investigate dynamic shear band propagation in a rate dependent solid under plane strain conditions. Differences in behavior for hardening and softening materials were investigated using a rectangular block subject to compression. A significant retardation of shear band development due to inertial effects was found. A one-dimensional analysis of a round bar subjected to tension was performed by Tuˇgcu et al. (1990) where the stabilizing effects of inertia were observed (see also Fressengeas and Molinari (1994)). An elastic–viscoplastic constitutive law was used to model the material behavior of solids with strain and strain rate hardening characteristics. A limitation of the one-dimensional analysis was that at high velocities radial inertia effects may no longer be negligible. Sørensen and Freund (1998) showed that mode inertia, i.e. the inertia following a rapidly growing incipient neck leads to a suppression of the modes with the longest wave length. In addition they showed that bifurcation modes are available essentially everywhere when the imperfection growth rate as well as the strain levels are sufficiently high. Shenoy and Freund (1999) found that very short wavelength modes of nonuniform deformation are suppressed by background inertia. In the analysis of Shenoy and Freund (1999) a linear stability approach is used, so that results regarding the onset of a bifurcation seem to rely on the assumption that wave effects and imperfection 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 dynamic axisymmetric finite element approach to simulate uniaxial tension. The aim is to explore the development of localization zones with time and in particular effects of background inertia. In the present context background inertia is defined as the inertia originating from idealized homogeneous deformation of the bar. An artificial volume load chosen as a pressure whose form is similar to that originating from this homogeneous deformation is introduced to qualitatively study the effects of background inertia. Clearly, perfect homogeneous deformation does not take place in the actual cases studied. However, parts of the inertial effects due to the global shape of the bar are investigated in this way. The full time history of neck formation is visualized using a strain rate parameter depicting both localization zones that die out as well as localization zones that continue to grow until they eventually cause final failure. The effects of specimen size, aspect ratio, end velocity and surface imperfections are studied. During high rate homogeneous extension of round bars a hydrostatic pressure appears in the material due to inertia (e.g., Graff, 1975). It is this pressure effect resulting from assumed incompressible and homogeneous deformation that is termed background inertia here. The effects of background inertia in the present case of dynamic loading of a round bar is studied by introducing an artificial volume force in the form of hydrostatic tension. This artificial hydrostatic tension is in several cases selected to be precisely opposite to the hydrostatic pressure (background inertia) that results from rapid homogeneous extension of the bar to explore any possible
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effect of background inertia. Attention is thus directed towards the effect of background inertia and for all simulations regardless of specimen size, aspect ratio, etc., corresponding simulations have been performed where 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. Material coordinates X i are introduced in the reference configuration which occupies the undeformed region 0 X 1 l0 , 0 X 2 r0 (figure 1) with rotational symmetry around the X 1 axis. Only half of the bar is studied numerically due to mirror symmetry about the mid plane at the center of the bar, containing the X 2 axis. The full initial length and diameter of the bar are thus denoted 2l0 and 2r0 , respectively. Initial sinusoidal surface imperfections are introduced in the form:
π mX 1 r0 = rˆ0 1 − ξ cos l0
,
(1)
where ξ is the dimensionless imperfection amplitude and m is the mode number. The aspect ratio of the bar is defined as: l0 (2) α= . rˆ0 The following boundary conditions are used for displacements ui and tractions T i : u1 = u1 (t),
T 2 = 0,
X 1 = l0 ;
(3)
u1 = 0,
T = 0,
X = 0;
(4)
u2 = 0,
T = 0,
X = 0;
(5)
X = r0 ,
(6)
2 1
T = T = 0, 1
2
1 2 2
where the displacement u1 (t) at the end of the bar in the direction of the axis of rotational symmetry is determined by an imposed end velocity ramped from zero to a constant maximum value according to:
u1 (t) =
v0 t 2 /2t0 , t t0 , v0 t0 /2 + v0 (t − t0 ), t > t0 ,
(7)
where t is the time, t0 is a ramp time and v0 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 initial length 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 the X2 axis. (X1 , X2 ) are cylindrical coordinates and the X1 axis is the axis of rotational symmetry.
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deformation. The velocity field originating from homogeneous deformation of an incompressible bar referred to the material coordinates of the reference configuration is for t > t0 :
V 1 X1 , X2, X3, t =
V 2 X1 , X2, X3, t = −
X 1 v0 ; l0
v0 2l0 1 +
(8)
X2
v0 t l0
− t20
3/2 ;
(9)
V 3 X 1 , X 2 , X 3 , t = 0,
(10)
taking into account that the boundary velocity is ramped. This velocity field may be used to define the contribution of background inertia to the total stress field of the axisymmetric bar. Considering deformation due to the velocity field (8)–(10) a stress field σ I caused by background inertia can be determined from the dynamic equilibrium condition. The total stress field σ T may be considered as consisting of two parts; σ I and σN: σT = σI + σN,
(11)
where σ N does not include contributions from background inertia. Using the stress field σ I , the hydrostatic pressure contribution p from background inertia can be determined to be of the form:
p X 2 , t = ρ0
3 v0 8 l0
2
1+
1
v0 t l0
−
t0 5/2 2
r02 − X 2
2
,
t > t0 .
(12)
The effects of background inertia on neck development may be addressed somewhat qualitatively by imposing an artificial volume load corresponding to −ψp where ψ = 1 means that the pressure effects due to background inertia are cancelled out. Figure 2 shows the development of the maximum pressure p with time for a bar with l0 = 1 m, r0 = 0.1 m, v0 = 40 m/s and t0 = 10−5 s.
Figure 2. Time development of the maximum inertial pressure p normalized with the yield stress σy for a bar with l0 = 1 m, r0 = 0.1 m, v0 = 40 m/s and t0 = 10−5 s. The pressure p is due to background inertia.
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The deformation time history of the full length of the tensile bar is visualized using contour plots of a normalized average cross sectional strain rate q: q=
ε˙p c , ε˙p t
(13)
where c and t denotes a cross sectional average and a total average, respectively and ε˙p is the maximum principal logarithmic strain rate (as in Sørensen and Freund (2000)). This strain rate parameter is plotted over the reference length of the bar (X 1 / l0 ) as a function of the overall logarithmic strain ε = ln(l/ l0 ) in the following figures. Values of q higher than unity and lower than unity indicates a strain rate higher than or lower 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:
τ δηij dV =
ij
V
T δui dS −
i
S
V
∂ 2 ui ρ0 2 δui dV + ∂t
ψpGij δηij dV ,
(14)
V
where V and S refer to volume and surface of the body, respectively, in the reference configuration. In (14) the volume force introduced to study the effects of background inertia appears in the artificial volume load term containing the pressure p. The covariant components of the Lagrangian strain tensor are denoted by ηij and the contravariant components of Kirchhoff stress are denoted τ ij . The mass density at time t = 0 when the body is in the reference configuration is ρ0 , the variational operator is δ and Gij is the metric tensor in the current configuration. The stress and strain tensors in (14) are: (15) τ ij = (ρ0 /ρ)σ ij ; 1 (16) ηij = ui,j + uj,i + uk,i uk,j , 2 where ρ denotes current density. Comma represents covariant differentiation with respect to the material coordinate of the reference configuration. Nominal traction is denoted T i and ui are the displacement vector components in the reference configuration. The constitutive relations of the elastic–plastic material are represented by rate independent J2 flow theory (Hutchinson, 1973). The linear incremental form of the constitutive response for a nearly incompressible material (τ ij ≈ σ ij ) can be expressed in terms of the nonobjective rate of Cauchy stress and the increment of Lagrangian strain: σ˙ ij = Lij kl η˙ kl ,
(17)
with the instantaneous moduli specified by:
Lij kl =
E/Et − 1 s ij s kl E 1 ik j l ν 3 Gij Gkl − αˆ G G + Gil Gj k + 1+ν 2 1 − 2ν 2 E/Et − (1 − 2ν)/3 σe2 1 − Gik σ j l + Gj k σ il + Gil σ j k + Gj l σ ik . 2
(18)
Here E is Young’s modulus, ν is Poisson’s ratio and αˆ is an elastic–plastic switch which equals zero or unity depending on whether or not yielding occurs. The stress deviator in (18) is written as: 1 s ij = σ ij − Gij σkl Gkl , 3
(19)
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and the effective Mises stress is:
σe =
3 ij s sij . 2
(20)
Under uniaxial tension the tangent modulus Et relates the true stress rate and logarithmic strain rate for an increment of stress according to σ˙ = Et ε˙ . A piecewise power law is adopted:
ε=
for σ σy , for σ > σy ,
σ E σy σ 1/N E σy
(21)
where σy is the initial yield stress and N is the strain hardening exponent. The numerical simulations are performed using a finite element approach to the dynamic principle of virtual work (14). Axisymmetric quadrilateral finite elements consisting of four crossed triangles to avoid locking (Nagtegaal et al., 1974) are used in a Lagrangian framework (see figure 1). A Newmark time integration scheme with a lumped mass matrix M is applied:
˙ (t ),t + (1/2 − k2 )D ¨ (t ) + k2 D ¨ (t +,t ) (,t)2 ; D(t +,t ) = D(t ) + D
¨ (t +,t ) = M−1 −F(t +,t ) ; D
(22) (23)
˙ (t ) + (1 − k1 )D ¨ (t ) + k1 D ¨ (t +,t ) ,t, ˙ (t +,t ) = D D
(24)
where D denotes nodal displacement, F denotes nodal forces and (˙) indicates differentiation with respect to time. Explicit time integration with no stiffness matrix computation or iteration is obtained by choosing k2 = 0 (Belytschko et al., 1976). The other parameter is chosen to be k1 = 1/2. The use of a lumped mass matrix has been shown to be preferable in the case of explicit time integration (Krieg and Kay, 1973) concerning both accuracy and efficiency. The size of the time step ,t is limited by element size and the characteristic speed of propagation of information. The maximum time step is chosen as: ,t < ζ ,de /cd ,
(25)
where ,de denotes the minimum element dimension and ζ is a constant taken to be in the range 0 < ζ < 1. The dilatational wave speed cd , the highest wave speed in an elastic medium, is:
cd =
E(1 − ν) . ρ(1 + ν)(1 − 2ν)
The shear wave speed is:
cr =
E . 2ρ(1 + ν)
(26)
(27)
3. Results The material parameters are representative of a structural steel; E = 2.069 × 1011 Pa, ν = 0.3 and ρ0 = 7850 kg/m3 . The speeds of dilatation and shear waves are thus 5956 m/s and 3183 m/s respectively. The
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initial yield stress is σy = 374.87 × 106 Pa and the piecewise power law hardening exponent is N = 0.1. The imperfection amplitudes considered are ξ = 0, ξ = 0.01 and ξ = 0.001, using the mode numbers m = 1, m = 2 and m = 4. Four different specimen sizes are used; l0 = 1 m, r0 = 0.1 m and l0 = 0.1 m, r0 = 0.01 m has an aspect ratio of α = 10 whereas l0 = 1 m, r0 = 0.01 m and l0 = 0.1 m, r0 = 0.001 m has an aspect ratio of α = 100. The mesh used for the aspect ratio α = 10 consists of 10 × 100 quadrilaterals (figure 1) and the corresponding mesh for α = 100 consists of 10 × 1000 quadrilaterals. The imposed end velocities are chosen below v0 = 100 m/s since the use of higher velocities in the present analysis leads to necking at the loaded end of the specimen. In figure 3 the q-levels of a bar with dimensions l0 = 1 m, r0 = 0.1 m and no initial imperfection (ξ = 0) are displayed. The results using four different values of the end velocity v0 , are shown. The ramp time is chosen as t0 = 10−5 s and no artificial volume load is included (ψ = 0). The deformation history is represented in terms of contours of constant q-levels plotted as a function of the reference position X 1 / l0 and overall logarithmic strain ε = ln(l/ l0 ). The insert to the right of the q contour plot shows the final state of the bar (scaled) with the shading indicating contours of constant principal logarithmic strain εp . In figure 3a v0 = 20 m/s and neck localization occurs at approximately X 1 / l0 = 0.55. For symmetry reasons this means that there are actually two necking sites along the full length of the bar. An increase of the velocity to v0 = 30 m/s leads to a shifting of the most developed neck towards the center of the bar (figure 3b). There is also a weak localization closer to the end of the bar, but this localization does not develop into all full-grown neck. At a velocity of v0 = 40 m/s the neck site moves away from the center of the bar, although there is a
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Figure 3. Contours of normalized average strain rate (q) as a function of time and distance for a tensile bar with l0 = 1 m, r0 = 0.1 m, t0 = 10−5 s, m = 0 and ψ = 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 ).
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zone with higher than average strain rate present at the center, as can be seen in figure 3c. For velocities of v0 = 50 m/s and above, necking occurs at the end of the bar, figure 3d. At the left-hand side of figures 3a–d, an intensively black region originating from X 1 / l0 = 1.0 indicates rapid propagation along the bar of information reaching X 1 / l0 = 0 at approximately ε = 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 the undeformed bar. An increase of the velocity from v0 = 20 m/s to v0 = 40 m/s causes an increase in logarithmic overall strain at the final deformation state from ε = 0.26 to ε = 0.39, but when the velocity reaches v0 = 50 m/s the overall strain level is reduced to ε = 0.29 and further increase of end velocity diminishes ε even more. A similar behavior is noticed when studying normalized traction at the end of the bar. In figure 4 normalized engineering traction is plotted against logarithmic overall strain for the cases a–d in figure 3. Large wave effects are generated at impact, causing initial oscillation of the normalized traction. The critical overall strain level at the first appearance of incipient neck localization in figure 4 increases when increasing the end velocity from v0 = 20 m/s (ε = 0.15) (a) to v0 = 30 m/s (ε = 0.17) (b), but decreases again when v0 = 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 level continues to diminish. In order to study imperfection sensitivity, bars with varying thickness were introduced. Figure 5 depicts the contours of constant q-levels for a bar with dimensions l0 = 1 m, r0 = 0.1 m and an initial imperfection of ξ = 0.01, m = 1. The ramp time is again chosen as t0 = 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 result is expected in the quasi-static case because of the initial imperfection. Increasing the velocity to v0 = 30 m/s leads to a shifting in the final neck site towards the end of the bar (figure 5b). Further increase of the velocity to v0 = 40 m/s shows that although there is an incipient neck between the center and the end of the bar, the dominating neck is the one developed at the site of the imperfection. The neck precursor at X 1 / l0 = 0.4 dies out at an overall logarithmic strain of approximately ε = 0.25. The neck formation at the imperfection site of the bar is termed stable, whilst the neck precursor at X 1 / l0 = 0.4 is termed unstable (compare with (Sørensen and Freund, 2000)). At the velocity v0 = 50 m/s, necking occurs at the end of the bar, and further simulations reveal that this is the case for all velocities above v0 = 50 m/s for this particular bar.
Figure 4. Normalized traction plotted against logarithmic overall strain for a tensile bar with l0 = 1 m, r0 = 0.1 m, t0 = 10−5 s, m = 0 and ψ = 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.
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Figure 5. Contours of normalized average strain rate (q) as a function of time and distance for a tensile bar with l0 = 1 m, r0 = 0.1 m, t0 = 10−5 s, ξ = 0.01, m = 1 and ψ = 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 between v0 = 40 m/s and v0 = 50 m/s did not lead to fully developed necks positioned between the center and the end of the bar. Thus, the transition of dominating neck site from the center 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 imperfections are shown in figure 6. The imperfection amplitude remains at ξ = 0.01, but the mode number has been changed to m = 2 in figures 6a–b corresponding to thin points at X 1 / l0 = 0 and X 1 / l0 = 1, and m = 4 in figures 6c–d corresponding to thin points located at X 1 / l0 = 0, X 1 / l0 = 0.5 and X 1 / l0 = 1. In figure 6a where the velocity is v0 = 30 m/s, the most developed neck localization occurs at the center of the bar, but in addition a weakly developed neck is formed at the end of the bar. This incipient localization does not proceed, and instead it fades away at an early stage. When the velocity is increased to v0 = 40 m/s as in figure 6b, the result is the opposite. Necking occurs at the end of the bar whilst a weakly developed localization is observed at the center. Further increase of the velocity shows similar behavior with a gradually weakening of the localization at the center of the bar. Using this initial imperfection, there is no sign of any incipient localization between the thin points. Studies of velocities between v0 = 30 m/s and v0 = 40 m/s show that the transition between necking at the center of the bar and at the end of the bar is performed with no visible signs of other incipient localization sites. It should also be noted that none of the simulations in the present study show the occurrence of two strongly developed necks simultaneously along the bar with aspect ratio α = 10. Only one neck prevails in all test cases.
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Figure 6. Contours of normalized average strain rate (q) as a function of time and distance for a tensile bar with l0 = 1 m, r0 = 0.1 m, t0 = 10−5 s, ξ = 0.01 and ψ = 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–d the mode number has been increased to m = 4. Necking occurs at the center of the bar in figure 6c, where the velocity equals v0 = 30 m/s. Increase of the end velocity to v0 = 40 m/s leads to a shift in necking site, so that the most developed neck is positioned at the end of the bar. In both of these cases, apart from the dominant neck, there are weakly developed necks at the thin points. Studies of the necking pattern at velocities between v0 = 30 m/s and v0 = 40 m/s reveals that there is a transition interval where the most developed neck occurs at X 1 / l0 = 0.5. As in the case with m = 2 only one of the localization zones is strongly developed. Simulations using a bar with an imperfection amplitude reduced to ξ = 0.001 (l0 = 1 m, r0 = 0.1 m) show an inclination towards formation of a fully developed neck located between the center of the bar and the end of the bar, though the actual necking pattern differs between the three different cases of imperfection (mode number m = 1, m = 2 and m = 4). Qualitatively similar results are obtained as in the case when using a bar with 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 homogeneous deformation is small whereas the extension rate have a large influence on the resulting neck formation. Simulations corresponding to all cases mentioned above, but using ψ = 1 and thereby canceling background inertia, show essentially no deviation from the results presented in figures 3–6. Figure 7 shows a comparison between the results acquired with no artificial volume load and those with cancellation of background inertia. In figures 7a–b the size of the bar is l0 = 1 m, r0 = 0.1 m with an imperfection of m = 1, ξ = 0.01 and a ramp time of t0 = 10−5 s. The end velocity is chosen as v0 = 40 m/s.
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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 and v0 = 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 ).
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(c) Figure 8. Contours of normalized average strain rate (q) as a function of time and distance for a tensile bar with l0 = 1 m, r0 = 0.01 m, t0 = 10−5 s, ξ = 0.01, ψ = 0 and v0 = 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 7a no artificial volume load is added, ψ = 0. There are essentially no changes in the results obtained with cancellation of inertial pressure ψ = 1, shown in figure 7b. There are some minor differences in details, but the overall necking pattern is the same.
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Imperfection sensitivity is obvious also in the case of an aspect ratio α = 100 where the dimensions of the bar are l0 = 1 m and r0 = 0.01 m. In figure 8 q-levels of a bar with imperfection amplitude ξ = 0.01 and mode numbers m = 1, m = 2 and m = 4 are shown. The inserts of the deformed bars in figure 8 are rescaled for clarity. The ramp time t0 = 10−5 s and the end velocity v0 = 30 m/s were chosen. No artificial volume load is included, ψ = 0. Figure 8a shows the necking pattern for the mode number m = 1. At a strain level of ε = 0.07 it seems like necking will occur at the center of the bar (q = 2), but almost immediately after the first sign of this incipient localization, another neck precursor forms at X 1 / l0 = 0.35. Eventually, the latter forms a strongly developed neck 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 distance from the other neck precursors. Changing the mode number to m = 2 results in a split of the necking pattern into one localization at the center of the bar and one at the end of the bar (figure 8b). At both thin regions of the bar one strongly and one weakly 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 neck precursors is essentially the same in both cases. Studies of the case m = 4 reveals the formation of one strongly developed neck at X 1 / l0 = 0.5. There is some non-uniform deformation present at both the end and the center of the bar, but the localization zone at X 1 / l0 = 0.5 is clearly dominant (figure 8c). It should be noted that for this bar with aspect ratio α = 100, necking occurs at an overall logarithmic strain level lower than that observed for α = 10. In figure 8a–b ε ≈ 0.13 at the final state of deformation, while in figure 8c ε ≈ 0.1, indicating that neck localization is initiated at an overall strain level ε < N . The effects of a smaller specimen size using the same aspect ratios mentioned above are also considered. For the aspect ratio of α = 10 a bar with l0 = 0.1 m and r0 = 0.01 m is used. The ramp time is chosen as t0 = 10−5 s and the initial imperfections used are ξ = 0, ξ = 0.001 and ξ = 0.01, with m = 1. Only minor differences between the cases using no artificial volume load (ψ = 0) and cancellation of background inertial effects (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 wave effects.
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Figure 9. Contours of normalized average strain rate (q) as a function of time and distance for a tensile bar with m = 1, ξ = 0.01, ψ = 0, v0 = 40 m/s and t0 = 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 ).
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Comparison between results using l0 = 1 m, r0 = 0.1 m and l0 = 0.1 m, r0 = 0.01 m show that the final necking sites are the same for the two different specimens, but that there are some noticeable differences in the development of the q-levels. Figure 9 shows the results for l0 = 1 m, r0 = 0.1 m (figure 9a) and l0 = 0.1 m, r0 = 0.01 m (figure 9b) using m = 1, ξ = 0.01, t0 = 10−5 s, v0 = 40 m/s and ψ = 0. Two necks are present; a strongly developed neck at X 1 / l0 = 0 and a weakly developed neck at X 1 / l0 = 0.4. The development of the latter neck in figure 9b is not as prominent as in figure 9a. The q-levels do not reach the same values and the neck ceases to grow at an overall logarithmic strain level of approximately ε = 0.225 compared to ε = 0.25 in figure 9a. Using a bar with dimensions l0 = 0.1 m and r0 = 0.001 m for the aspect ratio α = 100 reveals the same kind of behavior as described in the case of α = 10. The end velocity is chosen as v0 = 30 m/s and the initial imperfections are ξ = 0, and ξ = 0.001 with m = 1. Again, there are no differences in overall necking pattern between the results using no artificial volume load (ψ = 0) and the results where background inertia is cancelled (ψ = 1). Comparison between the two different sized specimens l0 = 1 m, r0 = 0.01 m and l0 = 0.1 m, r0 = 0.001 m show that the final necking sites are the same for the two bars. Some local differences in development of q-levels are present, but the positions of the more and the less developed necks along the bar are approximately the same. The ramp time is varied to examine the effect of rapid loading, since a rapid ramp time such as t0 = 10−5 s might not permit the homogeneous deformation field leading to (12). Figure 10 shows q-levels for a bar with
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Figure 10. Contours of normalized average strain rate (q) as a function of time and distance for a tensile bar with l0 = 1 m, r0 = 0.01 m, ξ = 0.01, m = 1, ψ = 0 and v0 = 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 ).
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initial dimensions l0 = 1 m and r0 = 0.01 m where an imperfection of ξ = 0.01 and m = 1 is added. The end velocity equals v0 = 30 m/s and the ramp times are t0 = 10−5 s, t0 = 10−4 s, t0 = 10−3 s and t0 = 10−2 s. Studies of the results using various ramp times show no difference between those obtained with no artificial pressure terms added (ψ = 0) and those where effects of background inertia are cancelled out (ψ = 1). Therefore only the results with ψ = 0 are shown in figure 10. In figure 10a the case with t0 = 10−5 s is shown once more (see figure 8) for comparison reasons. The most developed neck is located at X 1 / l0 = 0.35 with three weakly developed necks closer to the center of the bar. Increasing the ramp time to t0 = 10−4 s in figure 10b shows little difference compared to the results in figure 10a regarding the large scale necking pattern. The zone with q = 0.8 around X 1 / l0 = 0.9 is smaller in this case and there are some differences in q-levels. Further increase of the ramp time to t0 = 10−3 s leads to a rather different necking pattern, as can be seen in figure 10c. Six different necks are initialized and two of them form fully developed necks. Not only the number of necks has changed, but also the strain levels in the individual necks, so that it is no longer the necks furthest away from the center of the bar that lead to failure. The neck located at X 1 / l0 = 0.5 is the first one to die out, compared to the neck at X 1 / l0 = 0.35 in figure 10b which is the only one that continues to grow. The effect of the loading wave is no longer visible as a characteristic line because the accompanying q-levels are too small, but an extended plot using a larger number of q-levels reveals that the initial response is still dominated by dilatational waves. The nearly periodic pattern of neck precursors in figure 10c closely resembles
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Figure 11. Contours of normalized average strain rate (q) as a function of time and distance for a tensile bar with l0 = 1 m, r0 = 0.1 m, ξ = 0.01, m = 1, v0 = 40 m/s and t0 = 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 ).
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that observed by Sørensen and Freund in their analysis of radially expanding rings (Sørensen and Freund, 2000). Another interesting feature is visualized in figure 10d. Increasing the ramp time to t0 = 10−2 s leads to necking at two separate locations. The necking pattern at the center of the bar is still present. The number of initiated necks has been reduced and once again only one of them lead to final failure, this time the one closest to the center. The other neck is located at the end of the bar which in this case means the thickest part of the bar using the initial imperfection described above (m = 1, ξ = 0.01). The localization of the neck at the loaded end of the bar has been associated with high velocities in the cases described above, here (figure 10d) the onset of necking occurs at an increased ramp time t0 . Simulations using different ramp times for a bar with aspect ratio α = 10 (l0 = 1 m, r0 = 0.1 m) shows that localization at the end of the bar is promoted in this case also when the ramp time is increased to t0 = 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 11 displays the results of an increase of the factor ψ in order to investigate whether or not background inertia effects the necking pattern at all. The values of the inertia parameter used in the simulations are; ψ = 10, ψ = 100 and ψ = 1000. The results for a bar with l0 = 1 m, r0 = 0.1 m, ξ = 0.01, m = 1, v0 = 40 m/s and t0 = 10−5 s are shown. In figure 11a the results using ψ = 0 are shown for comparison. The overall necking pattern is the same as in figure 11a for ψ = 10 (figure 11b). The overall logarithmic strain ε at the final deformation state is slightly increased using ψ = 100, but the necking pattern remains unchanged in figure 11c. This is not the case when ψ = 1000 where the artificial pressure term is of the order of the yield stress. Figure 11d demonstrates how the nature of the neck sites changes as the intermediate mode, which was formerly suppressed, now forms a strongly developed neck localization. The neck at the center of the bar is now weakly developed and ceases to grow at an overall logarithmic strain of ε = 0.22. The required size of the artificial background inertia factor to obtain changes in the overall necking pattern is investigated for several cases in order to explore the possibility of a smaller value of this factor in some configuration. An increase of the ramp time does not effect the size of the artificial inertia factor at which the overall necking pattern is altered. Changing the ramp time to t0 = 10−3 using the same specimen geometry as in figure 11 promotes the transition of an intermediate mode to a neck site at the center of the bar when the inertia factor reaches the value ψ = 1000, so that p is of the order of the yield stress. Changing the aspect ratio of the specimen to α = 100 by using the radius r0 = 0.01 m using an artificial volume force factor equal to ψ = 1000 leads to an increase of the overall logarithmic strain ε at the final state of deformation when the simulation is terminated as a fully developed neck has formed. In this case no transition of necking site was 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 ratio α = 10. The hydrostatic pressure p originating from background inertia in (12) is dependent on specimen geometry, imposed end velocity and time. Thus it might be expected that the artificial volume force factor required to obtain changes in the overall necking pattern is not the same for different configurations. In this study no changes in overall necking pattern occurred for values smaller than ψ = 1000. 4. Conclusions The effects of background inertia, geometry, thickness imperfections, imposed final velocity and ramp time on neck formation have been investigated for cylindrical bars subjected to rapid extension. The study shows that background inertia does not have any significant influence on the necking pattern in any of the cases analyzed,
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unless the effect is artificially magnified to the magnitude of the yield stress. Instead wave effects, geometry and 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 of ξ = 0.01 is used. At sufficiently low velocities necking occurs at the site of a thin point. A transition of neck site occurs when the end velocity is increased, leading to necking at the end of the bar at large enough velocities. When a smaller imperfection amplitude of ξ = 0.001 is used the results resemble those obtained using no thickness imperfection 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 localization is shown to increase with increasing end velocity for a bar with aspect ratio α = 10 until the point when necking occurs at the loaded end of the bar due to wave effects. After this instant, an increase of the end velocity only leads 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 or l0 = 0.1 m) in the velocity range 20 m/s < v0 < 70 m/s the localization sites of strongly and weakly developed necks, respectively, are essentially the same though some differences in strain rate levels are present. A larger aspect ratio allows a larger number of necks to form. When using bars with aspect ratio α = 100 neck localization occurs at lower levels of overall logarithmic strain compared with bars where the aspect ratio equals α = 10. An increase of ramp time leads to changes in necking pattern, including shifting of strongly and weakly developed localization zones, respectively. The results of the simulations using a bar with aspect ratio α = 100, imposed end velocity v0 = 30 m/s and ramp time t0 = 10−3 s indicates that a nearly periodic necking pattern exists in a certain loading range. A similar pattern was observed for the case of radically expanding rings of a viscoplastic material (Sørensen and Freund, 2000) and in rings of a rate independent material (Han and Tvergaard, 1995). The nearly periodic necking pattern seems to be a fundamental feature, which is quite independent, at least qualitatively, of rate hardening. Sørensen and Freund (2000) also included the effects of adiabatic heating and failure by nucleation and void-growth, and the necking pattern appeared to be quite insensitive to both of these features. Although multiple necking sites are present in some of the cases analyzed in this study only a few of them continue 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 number m > 1 only one strongly developed neck appears. A possible explanation is that as soon as a sufficiently developed (strong) localization occurs elastic 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 only thickness imperfections, but also changes in end velocity and ramp time lead to large differences in the necking pattern. This implies that background inertia is not the primary effect when studying dynamic deformation in tensile bars with α 10.
Acknowledgement Prof. N.J. Sørensen of the Division of Mechanics, Lund University, S-22100 Sweden is acknowledged for advice and discussion.
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References Altynova, M., Hu, X., Daehn, G.S., 1996. Increased ductility in high velocity electromagnetic ring expansion. Metallurgical and Materials Transactions 27a, 1837–1844. Belytschko, T., Chiapetta, R.L., Bartel, H.D., 1976. Efficient large scale non-linear transient analysis by finite elements. International Journal of Numerical Methods in Engineering 10, 71–84. Considere, M., 1885. L’Emploi du Fer de l’Acier dans les Constructions, Mémoire No. 34. Annales des Ponts et Chaussées, Paris, pp. 574–775. Fressengeas, C., Molinari, A., 1994. Fragmentation of rapidly streching sheets. Eur. J. Mech. A/Solids 13, 251–268. Graff, K.F., 1975. Wave Motion in Elastic Solids. Clarendon Press, Oxford. Han, J.B., Tvergaard, V., 1995. Effect of inertia on the necking behaviour of ring specimens under rapid radial expansion. Eur. J. Mech. A/Solids 14, 287–307. Hutchinson, J.W., 1973. Finite strain analysis of elastic–plastic solids and structures, in: Hartung, R.F. (Ed.), Numerical Solutions of Nonlinear Structural Problems. ASME, New York, pp. 17–29. Hutchinson, J.W., Miles, J.P., 1974. Bifurcation analysis of the onset of necking in an elastic/plastic cylinder under uniaxial tension. J. Mech. Phys. Solids 22, 61–71. Hutchinson, J.W., Neale, K.W., 1977. Influence of strain rate sensitivity on necking under uniaxial tension. Acta Met. 25, 839–846. Krieg, R.D., Kay, S.W., 1973. Transient shell response by numerical time integration. International Journal of Numerical Methods in Engineering 7, 273–286. Nagtegaal, J., Parks, D.M., Rice, J.R., 1974. On the numerical accurate finite element solutions in the fully plastic range. Computer Method in Applied Mechanics and Engineering 4, 153–177. Needleman, A., 1989. Dynamic shear band development in plane strain. Journal of Applied Mechanics 56 (1). Niordson, F.I., 1965. A unit for testing materials at high strain rates. Experimental Mechanics 5, 29–32. Rice J.R., 1976. The localization of plastic deformation. Theoretical and Applied Mechanics, 207–220. Shenoy, V.B., Freund, L.B., 1999. Necking bifurcations during high strain rate extension. J. Mech. Phys. Solids 47, 2209–2233. Sørensen, N.J., Freund, L.B., 1998. Dynamic bifurcation during high-rate planar extension of a rectangular block. Eur. J. Mech. A/Solids 17, 709– 724. Sørensen, N.J., Freund, L.B., 2000. Unstable neck formation in a ductile ring subjected to impulsive radial loading. Int. J. Sol. Struct. 2265–2283. Tuˇgcu, P., Neale, K.W., Lahoud, A.E., 1990. Inertial effects on necking in tension. Int. J. Solids Structures 26 (11), 1275–1285.