Study of the development of plastic instabilities during tests on metallic plates biaxially loaded in their plane, in tension or in compression

– Plastic instabilities develop during tension and compression tests on metallic plates biaxially loaded in their plane. They limit the acceptable deformation levels during sheet forming. Carrying out a Linear Stability Analysis, we study the onset of their development. We calculate the growth rate of small symmetrical and antisymmetrical defects with respect to the median plane of the plate, periodic along the loading directions, and we determine the dominant mode. This 3 D model applies to dynamic tests whatever the thickness. It retrieves classical results for thin plates statically loaded in tension. Plane tension and compression tests are two particular 2 D cases of this model. In plane strain tension on ductile non viscous materials, we retrieve that the ﬁrst instabilities, which are also long wavelength necking ones, arise very little time before the applied force is maximum; this is consistent with the experimental observations of Consid`ere. As time goes by, antisymmetric modes with shorter wavelengths compete with the symmetric ones.


Introduction
Plastic instabilities develop during tension and compression tests on metallic plates biaxially loaded in their plane.They limit the acceptable deformation levels during sheet forming.Many publications have dealt with the onset and the development of necking in tension, since the pioneering works of Considère (1885) [1].Many of them deal with thin plates statically loaded [2][3][4][5][6][7][8][9].
In order to study the onset of the development of these instabilities, we carry out a Linear Stability Analysis [7,8,[10][11][12][13][14][15].We consider a plate dynamically loaded with constant velocities applied at its edges (our model applies whatever the thickness) (see Fig. 1), and we calculate the growth-rate θ of small perturbations δ x of the material trajectories of the mean homogeneous flow of the perfect plate (cf.[15], Chap.7) 1 , that are representative of the instabilities.We suppose that they grow exponentially a Corresponding author: dominique.jouve@cea.fr 1 For an incompressible material, the projection on the x1and x2-loading directions of the velocity of each material particle remains constant with time.The velocity gradients along these directions are uniform at each time (but they evolve with time).
Fig. 1.Plate biaxially loaded in tension in its plane, with constant velocities applied at its edges.
(δ x = e θt × − → F (space variables)).They are supposed to be periodic along the x 1 -and x 2 -loading directions (we denote respectively by λ 1 and λ 2 the wavelength along the x 1 -and x 2 -axes, and by γ i = 2π/λ i (i = 1, 2) the Article published by EDP Sciences denotes the wavelength associated with the dominant mode in x1-direction.4. Ȧ denotes Lagrangian derivative of A, for any quantity A. corresponding wavenumbers).They are symmetric or antisymmetric with respect to the median plane of the plate (cf.Fig. 2).We draw maps θ(γ 1 , γ 2 ), and we deduce the dominant mode, i.e. the most unstable pair of wavelengths (λ 2 ), and thus we identify the sites where plastic deformation begins to localize.
The material is a metal supposed to be homogeneous, isotropic, incompressible, elastoviscoplastic.Its yield strength Y depends on plastic strain ε p , absolute temperature T and plastic strain-rate εp .Its shear modulus G depends on T .The material is supposed to satisfy the Von Mises plasticity criterion (S : S = 2Y 2 /3, S denoting the deviatoric part of Cauchy stress tensor Σ), and the normality flow rule (plastic strain rate tensor Its evolution is supposed to be adiabatic, and plastic work to be fully converted into heat.Damage, thermal expansion and heat conduction are neglected.
Linear stability analysis solves the evolution equations of first order perturbations of the mean homogeneous ground flow of the perfect plate: volume and energy conservation equations, lagrangian momentum equations, flow rule, and we write that plastic strain increases with time, so we have simultaneously: S : S = 2Y 2 /3 and S : Ṡ = 2Y Ẏ /3.
We have developed a 3D Linear Stability Analysis: it applies to dynamic tests, whatever the thickness of the plate.In tension, when inertial effects are negligible, and for sufficient viscous effects, the wavelength of the most unstable defects is large compared to thickness, and we retrieve the growth-rate of necking plastic instabilities calculated by Dudzinski and Molinari in their Compte Rendu à l'Académie des Sciences in 1988 [7,8,16,17], in the framework of generalized plane stress theory [18,19].
In this paper, first in Section 2 we search for the dominant mode for different loadings; then we deal with plane tension and compression tests in Section 3 (the dimension of the plate is infinite along one loading direction).

Main results of our 3D model 2.1 Searching for plastic strain localization lines
We suppose that the perturbation δ x of material trajectories is periodic along the x 1 -and x 2 -loading directions, and that the end faces normal to the loading directions remain plane over time.Thus we have (cf.[15],  Chap.18), in Lagrangian coordinates x 0i (i = 1, 2, 3):

Antisymmetric defects
with, setting : In the plastic strain localization zones, perturbation δx 3 is extremum (see Fig. 2) (we set: In view of the form given to δ x, in the planes at x 03 = ±L 03 , plastic strain concentrates along straight lines, defined by the following equations: The wave vector γ = γ 1 − → e 1 + γ 2 − → e 2 is normal to the localization lines "+".The angle between the x 2 -axis and these lines is equal to (see Fig. 3): (for localization lines "-", this angle equals −ψ).
Table 1.Properties of the material.
We illustrate the search for the dominant mode and the plastic strain localization zones for different loadings and for a given material.

Material
Given a (fictitious) metal, whose physical properties are: 1. constant mass density: ρ = 5000 kg.m −3 ; 2. its yield strength Y obeys a constitutive law in the form proposed by Johnson and Cook [20], and, as in the model of Steinberg-Cochran-Guinan [21], we suppose that the ratio Y/G is independent of temperature.Thus we have: The coefficients we have chosen in relations ( 6) and (7) are given in Table 1; they are representative of the behaviour of usual metals (cf.[20], Table 1).In particular, with the value chosen for G 0 , the order of magnitude of the ratio Y/G for metals, i.e. one percent, is satisfied; 3. constant isochoric heat capacity:

Dominant mode and plastic strain localization lines
We carry out tension and compression tests on plates made with the material of Section 2.2, whose initial thickness equals 2L 03 = 2 cm.By convention, x 1 -direction is major principal stress direction (|Σ 11 | ≥ |Σ 22 |).The initial velocity gradient along x 1 -axis, D 11 = V 01 /L 01 , equals 10 s −1 in absolute value.Two tension (or compression) tests along x 1 -axis differ only in the value of the velocity gradient ratio α = D 22 /D 11 .
We carry out the linear stability analysis at initial time t 0 = 0: then plastic strain ε p equals zero, and temperature is supposed to be equal to 305 K.
We draw the map θ(γ 1 , γ 2 ), and we identify the dominant mode, i.e. the pair of wave numbers (γ 2 ) hav-ing the largest growth-rate.The associated wave vector − → e 2 in the plane of the loading directions is normal to the thinnest lines, where plastic strain concentrates during the linear phase of the development of necking.These are zero rate extension lines (along these lines, we have: D tt = ∂v t /∂x t = 0, t denoting the tangent direction).They are inclined at Hill's angle with respect to minor principal stress direction x 2 [2]: The dominant mode associated with these symmetric modes is all the less unstable that the absolute value of α increases (see Figs. 4a, 5a and 5b).Its orientation is given by Hill's angle (cf.Fig. 4c).

Loadings between uniaxial tension (α = −0.5) and simple shear (α = −1).
Let us examine Figure 5a once again.From uniaxial tension along x 1 -axis (α = −0.5) to simple shear 3), we see that unstable antisymmetric modes overtake symmetric modes.The wavelength along x 1 -axis of the dominant mode associated with these antisymmetric modes is infinite (γ = ∞), and plastic strain preferably concentrates along lines parallel to x 1 -axis; the wavelength λ growth-rate θ (d) all the larger) that we get nearer to simple shear (cf.Figs.4a and 4b).
Tension simultaneously along x 1 -and x 2 -axes (Fig. 5b) The most unstable defects are symmetric with respect to the median plane of the plate: these are necks.For α < 1, the wavelength along minor principal stress direction x 2 associated with the dominant mode is infinite (γ = ∞), and plastic strain concentrates along lines parallel to x 2 -axis.Plane tension (α = 0) is the most unstable loading condition2 .Getting from plane tension (α = 0) to balanced stretching (α = 1), the (symmetric) dominant mode becomes less and less unstable, and the associated wavelength λ (d) 1 larger and larger (cf.Fig. 4b).For balanced stretching, the wave vector appears in the equations of the model only in its norm γ = γ 2 1 + γ 2 2 , and the iso-θ curves on the map θ(γ 1 , γ 2 ) are circle quarters centred at the origin, and all orientations for localization lines are equiprobable.
Finally, in the neighbourhood of plane tension, there exist also unstable antisymmetric modes (cf.Figs.4a, 4b, and 5a, 5b).Their wavelength is comparable to the thick-ness of the plate, and is shorter than the one of the unstable symmetric modes.Due to viscous effects, they do not dominate the symmetric modes.

Compression along
x 1 -axis (Figs. 6 and 7) The most unstable defects are antisymmetric with respect to the median plane of the plate.For α < 1, plastic strain concentrates preferably along lines parallel to minor principal stress direction x 2 .
The flow is all the less unstable (and the localization lines all the more spaced out) that we get farther from plane strain3 .For balanced compression (α = 1), as for balanced stretching, the wave vector γ = γ 1 − → e 1 + γ 2 − → e 2 appears in the equations of the model only in its norm γ 2 1 + γ 2 2 , and the iso-θ curves in the (γ 1 , γ 2 )-plane are circle quarters centred at the origin; all orientations of localization lines are equiprobable.

Competition between symmetric and antisymmetric modes
For plates loaded in tension (for velocity gradient ratio between −0.5 and 1) or in compression along major principal stress direction x 1 , plane strain (D 22 = 0: the dimension of the plate along x 2 -direction is infinite) is the most unstable loading condition.It has been widely investigated in the literature [12][13][14][22][23][24].For us, it is a particular case of our general 3D model.Then we observe a competition between symmetric and antisymmetric modes.This competition is all the more important that viscous effects are lower.The θ(γ 1 ) curve is made up of a succession of branches, associated with symmetric and antisymmetric modes, alternatively.The first branch, in the field of the longest wavelengths, is associated with symmetric modes in tension, and with antisymmetric modes in compression [25].This competition has been revealed in the past, notably by Hill and Hutchinson in tension [26], and by Young in compression [27].For very low viscous effects, the dominant mode is not always symmetric in tension, and antisymmetric in compression (cf.Fig. 8).

First instabilities in tension
In plane strain tension on non viscous ductile metals, the first instabilities, which are also long wavelength necking ones, arise very little time before the applied force is maximum: this result is consistent with the experimental observations of Considère [1].Then, the growth-rate associated with the dominant mode increases, whereas the associated wavelength decreases (the number of necks increases).The instability develops clearly once the ratio θ/ εp has reached a certain threshold level (typically 10) (cf.Fig. 9) [15,28].

Shear bands
In plane strain tension and compression, in the absence of viscous effects, our linear stability analysis shows  that, when the effect of work-hardening on the evolution of yield strength Y (ε p , T ) no longer sufficiently prevails over the effect of thermal softening, symmetric and antisymmetric defects having arbitrarily short wavelength develop with an infinite growth-rate [15,16].Such instabilities develop as soon as the relative variation of Y for a plastic strain increment δε p satisfies the following inequality (we set: Nevertheless, we have not been able to predict analytically the spatial dependence of such perturbations.We simply see that this condition for the absence of a minimal wavelength below which shorter wavelengths are all stable (cutting wavelength) coincides with the condition for the instantaneous onset of infinitely dense networks of shear bands inclined at 45 • with respect to the loading axis, shown by the bifurcation analysis of Hill and Hutchinson in tension [26], and the one of Young in compression [27] for static tests [15,16,29].When the condition ( 9) is satisfied, we see the onset these networks in numerical simulations, all the more rapidly and densely that the mesh is refined.Due to the absence of a physical length (and time) scale in the problem, simulations are always mesh-sensitive (see Fig. 10).The transition from an elastic perfectly plastic constitutive law (Y = constant) to a "sufficiently" viscous Norton's law [30] (Y ∝ εm p , with: m ≥ 0.05) reintroduces a cutting wavelength [10][11][12][13][14].

Conclusion and future works
In this paper, we have shown results obtained carrying out a rigorous 3D linear stability analysis of the de-velopment of plastic instabilities during tension and compression tests on metallic plates biaxially loaded in their plane.The material is supposed to satisfy the plasticity criterion of Von Mises, and the normality flow rule.
We are undertaking to generalize our model to more complex materials, accounting for damage [31,32] and anisotropy effects [33], studying the influence of the shape of the yield surface [5,34], and even texture [35,36].In the limiting case of static tests on thin plates, in the absence of damage effects and for an orthotropic material obeying Hill's plasticity criterion (1948) [33], we will have to retrieve previous analytical results published in 1991 by Dudzinski and Molinari [8].

Fig. 2 .
Fig. 2. Symmetric and antisymmetric defects with respect to the median plane of the plate, periodic along the x1-and x2-loading directions.

- 1 Fig. 10 .
Fig. 10.Onset of a shear band network during a plane tension test on a material having constant yield strength Y (then inequality (9) is well satisfied), in lieu of a long wavelength multimodal perturbation of the material velocity field introduced at initial time t0 = 0 of the simulation, all the more rapidly that the mesh is refined.Initial dimensions of the plate: 2L01 = 20 cm; 2L03 = 2 cm -Stretching velocity V01 = 1 m.s −1 -Mass density ρ = 5000 kg.m −3 -yield strength Y = 1 GPa -shear modulus G = 100 GPa -Two simulations of the test have been carried out, with initially square elements, with sides 100 μm long for the finest mesh, and 1 mm long for the coarsest one[15].We compare the plastic strain rate εp in the two simulations at time t = 40 μs.