M Xavier

Sébastien Mercier

C Czarnota

S El Mai

D Jouve

J L L Dequiedt

A Molinari

S El Mai

Extension of 1D linear stability analysis

Keywords: Dynamic necking, Bridgman correction factor, Instability, Stability analysis, Ring, Plate

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Introduction

Fragmentation of ductile metals subject to dynamic expansion is a topic of interest for both civil and defense industries. The characterization of the fragment size distribution is one of the main concern since the kinetic energy is related to the size and velocity of fragments. The fracture process in ductile materials may be triggered by instabilities within the plastic flow that result in necking i.e. the occurrence of local thinnings where the plastic deformation is localized. When the structure expands at high-strain rates, multiple necks develop until failure resulting in multiple fragments.

Numerous experimental works using electromagnetical or explosive loadings have been conducted to understand the process of localization; various geometries were studied such as cylinders (see [START_REF] Wesenberg | Dynamic fracture of 6061-T6 aluminium cylinders[END_REF], [START_REF] Olive | Rupture behaviour of metals in explosive expansion[END_REF], [START_REF] Fyfe | Dynamic pre-strain and inertia effects on the fracture of metals[END_REF] or [START_REF] Goto | Investigation of the fracture and fragmentation of explosively driven rings and cylinders[END_REF]), rings (see [START_REF] Niordson | A unit for testing materials at high strain rates[END_REF], [START_REF] Llorca | Expanding ring test: numerical simulation -application to the analysis of experimental data[END_REF], [START_REF] Zhang | On the dynamics of necking and fragmentation-I. Real-time and post-mortem observations in Al 6061[END_REF]) or hemispherical shells (see [START_REF] Mercier | Multiple necking during the dynamic expansion of hemispherical metallic shells, from experiments to modelling[END_REF]). From all experiments, it can be infered that the fracture process is divided in three steps (see [START_REF] Zhang | On the dynamics of necking and fragmentation-I. Real-time and post-mortem observations in Al 6061[END_REF] for high-quality images): homogeneous deformation, development of local thinnings, fracture at some neck sites.

Analytical models have been developed in the last half-century to predict the fragment size distribution observed in experimental works. Pioneering work of [START_REF] Mott | Fragmentation of shell cases[END_REF] used a probabilistic approach and accounted for the stress release wave propagation within a metallic ring. While instantaneous fracture was postulated in Mott theory, [START_REF] Grady | Fragmentation of solids under impulsive stress loading[END_REF] enriched it by accounting for the fracture energy dissipation and obtained results in agreement with experimental observations of [START_REF] Wesenberg | Dynamic fracture of 6061-T6 aluminium cylinders[END_REF].

Bifurcation and linear stability analyses have been widely used in the last decades to tackle the issue of predicting necking by focusing on the early stages of the deformation. One may refer to [START_REF] Hill | Bifurcation phenomena in the plane tension test[END_REF] and [START_REF] Hutchinson | Influence of strain rate sensitivity on necking under uniaxial tension[END_REF] for quasi-static conditions. Under dynamic conditions, Fressengeas andMolinari (1985, 1994), [START_REF] Shenoy | Necking bifurcations during high strain rate extension[END_REF] developed a multidimensional linear stability analysis (modal analysis). Inertia is shown to damp long wavelengths while viscosity and multiaxiality of the flow tend to extinguish short wavelengths. At each step of the deformation process, an intermediate wavelength having the largest growth rate thus emerges and was supposed to characterize the mean neck spacing observed in experiments. [START_REF] Mercier | Predictions of bifurcation and instabilities during dynamic extension[END_REF] and [START_REF] Mercier | Analysis of multiple necking in rings under rapid radial expansion[END_REF] respectively applied linear stability analyses to the case of cylindrical bars and tubes. Using this method, [START_REF] Jouve | Etude analytique de l'instabilité plastique de striction pour une plaque sollicitée en traction biaxiale[END_REF] retrieved the average fragment size of the experimental work of [START_REF] Olive | Rupture behaviour of metals in explosive expansion[END_REF] on the dynamic expansion of steel cylinder. [START_REF] Mercier | Multiple necking during the dynamic expansion of hemispherical metallic shells, from experiments to modelling[END_REF] analyzed the expansion of a hemispherical shell using a plate theory. In addition, [START_REF] Jouve | Analytic study of the onset of plastic necking instabilities during biaxial tension tests on metallic plates[END_REF] studied instabilities in metallic plates under dynamic biaxial loading. A 1D formalism was also proposed in the literature where the stress multiaxiality in the neck region is accounted for by adopting the Bridgman correction factor, see for instance [START_REF] Zhou | An elastic-visco-plastic analysis of ductile expanding ring[END_REF] for the dynamic stretching of round bar. Note that the neck development in biaxial loading was also investigated in [START_REF] Zaera | Collective behavior and spacing of necks in ductile plates subjected to dynamic biaxial loading[END_REF] by considering linear stability analysis using the Bridgman correction factor [START_REF] Bridgman | Studies in large plastic flow and fracture[END_REF]. However, most of the methods are based on a frozen coefficient theory that implicitly supposes that the time evolution of the perturbation is much faster than the one of the homogeneous solution. This strong assumption is not verified for very high strain rates. [START_REF] Xavier | Extension of linear stability analysis for the dynamic stretching of plates: Spatio-temporal evolution of the perturbation[END_REF] developed a linear stability formalism without adopting the frozen coefficient theory. Within a 2D theoretical framework, they observed that while growth rates obtained with the frozen coefficient theory are retrieved with the new development in the late stage of the deformation process, they are highly different in the early stage. This induces strong discrepancies in terms of perturbation amplitudes. Recently, [START_REF] Jacques | An analytical model for necking strains in stretched plates under dynamic biaxial loading[END_REF] proposed a two-zone model for the dynamic biaxial stretching of a plate adopting also the Bridgman correction factor. Dynamic forming limit curves obtained with this analytical approach compared well with finite element calculations. It is also interesting to mention the approaches developed by [START_REF] Audoly | Analysis of necking based on a one-dimensional model[END_REF] and [START_REF] Audoly | One-dimensional modeling of necking in rate-dependent materials[END_REF] who proposed a new 1D methodology to analyze the necking of round bars and sheets in quasi-static conditions. Interestingly, they were able to find a new analytical expression of the Bridgman correction factor. As in all works of the literature, the neck retardation is influenced by positive strain hardening, strain-rate dependence and also by non-uniform stress distribution in the neck.

The aim of the paper is to revisit the development proposed in [START_REF] Xavier | Extension of linear stability analysis for the dynamic stretching of plates: Spatio-temporal evolution of the perturbation[END_REF] by considering a 1D approach with introduction of a Bridgman correction factor. In this paper, the necking process is studied during the stretching of a rectangular plate. It will be seen in Appendix B that the theory is easily extended to describe the dynamic extension of a cylindrical rod, which is assumed in the literature to be representative of the ring expansion. In Section 2, the 1D theoretical framework based on the formalism of [START_REF] Zhou | An elastic-visco-plastic analysis of ductile expanding ring[END_REF] is derived. The system of equations governing the evolution of perturbations is established. The frozen coefficient theory is not used and the time evolution of the perturbation amplitudes is governed by two differential equations: one equation for the perturbed accumulated plastic strain and one for the thermal perturbation. In Section 3, the multiple necking of a plate made of copper is analyzed. The material is exhibiting strain hardening, strain rate sensitivity and thermal softening. The difference between the proposed development and the 1D classical linear stability analysis based on the frozen coefficient theory (1D-CLSA) is revealed. The influence of the modeling of multiaxiality of the plastic flow is analyzed by comparing results obtained with the present model, referred to as 1D-XLSA and the 2D-XLSA model of [START_REF] Xavier | Extension of linear stability analysis for the dynamic stretching of plates: Spatio-temporal evolution of the perturbation[END_REF]. The role of the initial material state is also discussed.

Modeling

Consider a plate of initial half-length L 0 , initial thickness h 0 and infinite width, see Fig. 1. The initial thickness can be non uniform so that one has the possibility to explore the situation where h 0 varies along the longitudinal axis: h 0 (X). The plate is submitted to plane strain conditions by imposing a constant velocity V 0 to the surfaces X = ±L 0 :

v x (±L 0 , t) = ± ε0 L 0 where ε0 = V 0 L 0 (1)
where ε0 represents the nominal strain rate. In the following, we assume that at any material point in the plate, the initial velocity field is : v x (X, t = 0) = ε0 X. In this paper, a 1D approach will be presented, so the index x referring to the component or the direction will be omitted next.

The material is assumed thermo-viscoplastic. During plastic straining, the flow stress σ Y is described as the sum of two terms: a rate insensitive thermoplastic yield stress σ p Y (ε p , T ) and a viscous overstress σ v Y (ε p , εp , T ):

σ Y = σ p Y + σ v Y = f (ε p , εp , T ) , ( 2
)
where T is the absolute temperature. εp is the equivalent plastic strain rate. Since elasticity is neglected in the present approach, one has, within the von Mises theory framework, εp = 2 3 D : D 1 2 , with D being the strain rate tensor and (:) being the double contraction for second order tensors. ε p = t 0 εp dτ + ε 0 p is the accumulated plastic strain with ε 0 p being a pre-strain parameter. We propose to develop a 1D Lagrangian approach for the prediction of multiple necking. As a consequence, any mechanical or thermal field G, solution of the problem, is only related to the Lagrangian coordinate X and

h 0 V 0 V 0 X Y 2L 0 h V 0 V 0 X Y 2L a) b)
Figure 1: Schematic representation of the plate, of initial half-length L0 and uniform initial thickness h0 at a) initial time t = 0 and b) current time t. Velocities ±V0 are applied to the extremities X = ±L0. In the figure, the situation of a configuration without geometrical defects is presented. time t : G(X, t). In the following, dependencies relatively to X and t will be omitted for the sake of conciseness.

The field equations are recalled below. More details can be found in [START_REF] Fressengeas | Inertia and thermal effects on the localization of plastic flow[END_REF], [START_REF] Zhou | An elastic-visco-plastic analysis of ductile expanding ring[END_REF] or Rodríguez-Martínez et al. (2013a).

The conservation of linear momentum in the longitudinal direction is written as:

ρ 0 h 0 ∂v ∂t = ∂(hσ) ∂X (3)
where ρ 0 is the initial mass density of the material. Incompressibility is supposed i.e ρ 0 is constant, and assumed uniform. v is the current velocity in the longitudinal direction, h 0 (resp. h) the initial (resp. current) plate thickness and σ the current longitudinal Cauchy stress (at time t).

From the definition of the true strain ε:

ε = ln ∂x ∂X , (4)
the continuity equation leads to:

∂v ∂X = εe ε (5)
where ε is the longitudinal strain rate and x the current position. During plane strain stretching, the thickness of the plate decreases. Elasticity being neglected and owing to volume preserving of the plastic deformation, one gets:

h = h 0 e -ε (6)
Under plane strain stretching, and since the material is assumed rigid plastic, the equivalent plastic strain rate εp is related to the longitudinal strain rate ε through:

εp = 2 √ 3 ε (7)
By time integration of Eq.( 7), we have:

ε p -ε 0 p = 2 √ 3 ε (8)
To account for the multidimensional aspect of the stress field within the neck region, the 1D approach is complemented by considering the Bridgman correction for plane strain, see [START_REF] Bridgman | Studies in large plastic flow and fracture[END_REF]. Therefore the true stress σ is replaced by an average axial stress accross the plate thickness defined as:

σ = 2σ Y √ 3 B(Φ) (9)
where B(Φ) is the Bridgman correction factor. More details on this correction factor are presented in Appendix C.

Φ is a parameter depending on the thickness profile and given by:

Φ = h 8 ∂ 2 h ∂x 2 (10)
When the faces (Y = ±h 0 /2) of the plate remain planar (homogeneous deformation of the plate with initial uniform thickness as displayed in Fig. 1), Φ = 0 and B(Φ) = 1. In that case, the true stress is simply σ = 2σ Y √ 3 .

For limited amplitude of necks, the term B(Φ) can be approximated by its first order Taylor expansion:

B(Φ) = 1 + 2Φ 3 + o(Φ 2 ) ( 11
)
The plate is subjected to high strain rate loading such that adiabatic conditions prevail with no heat conduction inside. Thus, heat equation reduces to:

ρ 0 C Ṫ = β T Q σ Y εp (12)
where β T Q , the Taylor-Quinney coefficient, represents the proportion of plastic work converted into heat. In the present work, β T Q is assumed constant. Nevertheless, the present approach can be extended without difficulties to account for the evolution of β T Q during the deformation process, as it has been discussed in the literature, see [START_REF] Mason | On the strain and strain rate dependence of the fraction of plastic work converted to heat: an experimental study using high speed infrared detectors and the kolsky bar[END_REF], [START_REF] Rittel | On the conversion of plastic work to heat during high strain rate deformation of glassy polymers[END_REF], [START_REF] Longère | Evaluation of the inelastic heat fraction in the context of microstructure-supported dynamic plasticity modelling[END_REF] or [START_REF] Stainier | Study and validation of a variational theory of thermo-mechanical coupling in finite visco-plasticity[END_REF]. C is the heat capacity of the material assumed constant in this work. Note that heat conduction might have effects for short wavelength perturbations. The characteristic time for conduction τ cond related to the Fourier dimensionless number, is linked to the characteristic size of the domain and the diffusivity α. In our approach, we can assume that the shortest wavelength for necks is of the order of the plate thickness. So, one gets τ cond ≈ h 2 0 /α. For a plate thickness of 1 mm and for a material representative of copper, τ cond ≈ 0.008 s. The assumption of adiabatic condition during multiple necking process seems to be valid for a nominal strain rate over 1000 s -1 , condition fulfilled in the present paper.

When the plate has initially a uniform thickness (h 0 (X) = h b 0 ), a homogeneous solution as presented in Fig. 1 may exist. The corresponding background solution G b , satisfying Eqs (3)-( 12) with boundary conditions Eq.(1), can be expressed as follows:

x b = (1 + ε0 t)X ε b = ln(1 + ε0 t) > 0 h b = h b 0 1 + ε0 t εb p = 2 √ 3 ε0 1 + ε0 t Φ b = 0 σ b = 2σ b Y √ 3 (13)
In the present work, at time t = 0, the plate thickness is non uniform. An initial thickness profile is prescribed h(X, t = 0) = h 0 (X). We assume that the initial thickness is the sum of the homogeneous background thickness and a small perturbation: h 0 (X) = h b 0 + δh 0 (X), such that h b 0 =< h 0 (X) > represents the mean value of the initial thickness over the whole length of the specimen. In that case, the average value of the perturbation is vanishing < δh 0 (X) >= 0. Similarly, an initial pre-strain profile ε 0 p (X) = ε 0 p b + δε 0 p (X) is also prescribed. This field is somehow induced by the thermo-mechanical treatment, related to the processing route or the presence of microstructural defects. Note also that as the initial longitudinal velocity field is v(X, t = 0) = ε0 X, the initial longitudinal strain rate ε0 (X) is uniform, equal to ε0 . Necessarily, the initial perturbations in longitudinal strain rate and equivalent plastic strain rate are vanishing: δ ε(X, t = 0) = δ εp (X, t = 0) = 0. Finally, an initial temperature profile is prescribed: T 0 (X) = T b 0 + δT 0 (X). Therefore, due to these initial pre-defined inhomogeneities in the plate, all other fields of the problem will be also initially heterogeneous. At t = 0, the plate has not been stretched yet, meaning that the longitudinal strain is ε(X, t = 0) = 0. Therefore, ε b (X, t = 0) = 0 and δε(X, t = 0) = 0.

During the deformation process, any field G will be the sum of the homogeneous solution G b and a time evolving perturbation δG(X, t):

v = v b + δv ε = ε b + δε ε p = ε b p + δε p h = h b + δh σ = σ b + δσ σ Y = σ b Y + δσ Y ε = εb + δ ε Φ = Φ b + δΦ T = T b + δT εp = εb p + δ εp (14)
Substituting relations ( 14) into the governing equations ( 3)-( 12) and adopting the homogeneous solution defined in Eq.( 13) provide a set of equations for the perturbed fields:

∂δv ∂X = e ε b εb δε + δ ε δh = -h b 0 e -ε b δε + δh 0 e -ε b δσ = 2δσ Y √ 3 + 4 3 √ 3 σ b Y δΦ ρ 0 h b 0 ∂δv ∂t = h b ∂δσ ∂X + σ b ∂δh ∂X δΦ = h b exp(-2ε b ) 8 ∂ 2 δh ∂X 2 δ εp = 2 √ 3 δ ε δε p = 2 √ 3 δε + δε 0 p δσ Y = σ b Y ,εp δε p + σ b Y , εp δ εp + σ b Y ,T δT ρ 0 Cδ Ṫ = β T Q ( εb p δσ Y + σ b Y δ εp ) (15)
where σ b Y ,ω denotes the partial derivative of σ b Y with respect to the variable ω (ω being ε p , εp or T ). A modal analysis is carried out. As in [START_REF] Xavier | Extension of linear stability analysis for the dynamic stretching of plates: Spatio-temporal evolution of the perturbation[END_REF], any perturbation field δG is searched of the form δG(X, t) = cos (γX) Ĝ(γ, t), (except for the perturbation of the velocity whose form is δv = sin(γX)v(γ, t), see Eq.3) where γ is the longitudinal wavenumber of the perturbation. The prescribed mathematical description of perturbation fields is to the author's point of view compatible with the studied configuration where a material defect is imposed. Indeed, the plate has a non uniform thickness h 0 (X), of sinusoidal shape with longitudinal wavenumber γ. In the literature, some recent contributions (see [START_REF] Ravi-Chandar | Dynamic stability of a bar under high loading rate: Response to local perturbations[END_REF] or Vaz-Romero et al. ( 2017)) have analyzed the dynamic stretching of a homogeneous configuration where a localized defect in velocity and strain rate was introduced. In that specific case, the propagation of the defect in the sample prior to a phase where well defined necks occur was observed. By adopting the above form δG(X, t), we are not able to analyze the propagation of such localized defect. This is not our goal. It has been checked by finite element calculations, that the stretching of a plate with a non uniform thickness (pattern with a single wavenumber γ) does not lead to propagation phenomena of the material defect.

To satisfy the boundary condition Eq.(1) (i.e. δv(X = ±L 0 , t) = 0), the wavenumber γ is related to the initial length of the plate by: γL 0 = kπ (k is an integer representing the initial number of necks along the X direction). Functions Ĝ depend only on time and wavenumber γ, and may have positive or negative value. Their absolute values represent the true amplitude of each perturbation field. Nevertheless, in the following Ĝ will be named amplitudes. From now on, their dependencies on γ will be omitted for the sake of conciseness. In our analysis, the time dependency of Ĝ is not enforced to be of exponential form (as in [START_REF] Zhou | An elastic-visco-plastic analysis of ductile expanding ring[END_REF] or Rodríguez-Martínez et al. (2013a)). In that sense, the proposed methodology is similar to the one recently proposed by [START_REF] Xavier | Extension of linear stability analysis for the dynamic stretching of plates: Spatio-temporal evolution of the perturbation[END_REF]. Nevertheless, contrary to [START_REF] Xavier | Extension of linear stability analysis for the dynamic stretching of plates: Spatio-temporal evolution of the perturbation[END_REF] in which a 2D approach is derived, the present work is restricted to the development of a 1D approach.

The ten perturbed quantities (δv, δε, δ ε, δε p , δ εp , δh, δσ, δσ Y , δΦ and δT ) or equivalently the associated amplitudes (v, ε, ε, εp , εp , ĥ, σ, σY , Φ and T ) are related to each other owing to the nine relationships of Eqs (15). In this set of equations, the terms δε 0 p and δh 0 (and the associated amplitudes ε0 p and ĥ0 ) are initial prescribed quantities.

Next, as in [START_REF] Xavier | Extension of linear stability analysis for the dynamic stretching of plates: Spatio-temporal evolution of the perturbation[END_REF], a mechanical perturbation (here represented by the perturbation in accumulated plastic strain εp ) and a thermal perturbation (here T ) are selected to be representative of the perturbation. Note that another choice could have been possible for the mechanical perturbation but the present choice is the one that leads to the simplest derivation. After adequate combinations, the set of Eqs (15) reduces to only two ordinary differential equations coupling the perturbation amplitude εp of the accumulated plastic strain and the perturbation amplitude T of the temperature:

a 1 εp + a 2 εp + a 3 εp + a 4 T = s 0 Ṫ = b 1 T + b 2 εp + b 3 εp (16)
Coefficients a i and b i in Eq.( 16) are only time dependent, related to the homogeneous solution:

a 1 (t) = ρ 0 1 + ε0 t 2 √ 3 b 1 (t) = β T Q 2 ε0 ρ 0 C √ 3(1 + ε0 t) σ b Y ,T a 2 (t) = ρ 0 √ 3 ε0 + 2γ 2 √ 3(1 + ε0 t) σ b Y , εp b 2 (t) = β T Q ρ 0 C ( 2 ε0 √ 3(1 + ε0 t) σ b Y , εp + σ b Y ) a 3 (t) = 2γ 2 √ 3(1 + ε0 t) σ b Y ,εp + (h b 0 ) 2 σ b Y 12(1 + ε0 t) 5 γ 4 - γ 2 σ b Y 1 + ε0 t b 3 (t) = β T Q 2 ε0 ρ 0 C √ 3(1 + ε0 t) σ b Y ,εp a 4 (t) = 2γ 2 √ 3(1 + ε0 t) σ b Y ,T (17)
s 0 is a time dependent coefficient that depends also on the prescribed amplitudes ĥ0 and ε0 p :

s 0 = ĥ0 h b 0 + √ 3 2 ε0 p (h b 0 ) 2 γ 4 σ b Y 6 √ 3(1 + ε0 t) 5 - 2σ b Y γ 2 √ 3(1 + ε0 t) (18)
The system ( 16) is composed respectively of a second order and of a first order ordinary differential equations for εp and T . Thus, initial conditions εp (t = 0), εp (t = 0) and T (t = 0) are required. However, as stated previously, due to the initial conditions on the imposed strain rate field, necessarily, εp (t = 0) = 0.

The system ( 16) is solved using a finite difference scheme with a splitting strategy in which each differential equation is solved successively within a timestep. A convergence study revealed that predictions are converged for sufficiently small timesteps. Remember that one of the main interest of the proposed work is that time evolutions of perturbation amplitudes are direct outcomes of the model.

As already mentioned, in the linear stability analysis of the literature with Bridgman correction factor as in [START_REF] Zhou | An elastic-visco-plastic analysis of ductile expanding ring[END_REF] or in Rodríguez-Martínez et al. (2013a) (in the case of a cylindrical bar) or [START_REF] Zaera | Collective behavior and spacing of necks in ductile plates subjected to dynamic biaxial loading[END_REF] (in the case of biaxial stretching of a plate), a frozen coefficient theory is used and Ĝ(t) is assumed of exponential form: Ĝ(t) = Ĝo e ηt with η being the instantaneous growth rate of the perturbation and Ĝo the initial amplitude.

In that case, all fields have the same growth rate. The governing equation for the instantaneous growth rate η is developed in Appendix A. By performing the linear stability analysis at various consecutive times, the amplitude of any perturbation field Ĝ is estimated by the relation Ĝ = Ĝ0 exp( t t1 ηdτ ), where t 1 is the first time during the deformation process where the growth rate η becomes positive, see [START_REF] Petit | Breakup of copper shaped-charge jets: Experiments, numerical simulations, and analytical modeling[END_REF], El Maï (2014) or [START_REF] Vaz-Romero | Multiple necking pattern in nonlinear elastic bars subjected to dynamic stretching: The role of defects and inertia[END_REF]. Implicitly, we assume in this paper that before t 1 , the perturbation is not evolving. Of course, this is a crude assumption and it could be of interest to investigate the response of the classical linear stability analysis in a configuration where non positive growth rate is captured. This is not the subject of the present paper and this work is currently under investigation. With this type of approach, the relative amplitudes (defined as the ratio between the current and the initial amplitudes) of all fields are identical.

With the proposed extended Bridgman analysis, the amplitudes εp and T are natural outcomes of the model.

As seen in [START_REF] Xavier | Extension of linear stability analysis for the dynamic stretching of plates: Spatio-temporal evolution of the perturbation[END_REF], there is no simple link between them except their combined interaction inherited from the thermo-mechanical coupling and described by Eqs ( 16). As a consequence, their respective growth rates are not identical. We define the instantaneous growth rate of the mechanical perturbation η εp by:

η εp (γ, t) = εp εp (19)
which is depending on time t and the wavenumber γ. In the following, the dependency upon t and γ will be omitted for the sake of conciseness. As in [START_REF] Xavier | Extension of linear stability analysis for the dynamic stretching of plates: Spatio-temporal evolution of the perturbation[END_REF], the instantaneous growth rate of the thermal perturbation η T (γ, t) = Ṫ T could have been also introduced. Nevertheless, a comparison between the mechanical and thermal growth rate leads to trends similar to those observed in [START_REF] Xavier | Extension of linear stability analysis for the dynamic stretching of plates: Spatio-temporal evolution of the perturbation[END_REF]. So, results concerning the thermal growth rate evolution are not shown in the present work.

Results

In the present work, the initial thickness is not uniform. The thickness perturbation will be of sinusoidal shape: δh 0 (X) = cos(γX) ĥ0 . ĥ0 is the amplitude of the thickness perturbation.The plate has the following dimensions: L 0 = 180.8 mm, h b 0 = 2.4 mm and ĥ0 = 8 µm. The initial longitudinal strain rate ε0 is = 2300 s -1 , .i.e. εX (X, t = 0) = ε0 .

The material, representative of copper, has a thermo-viscoplastic behavior modeled by a power law, see [START_REF] Mercier | Multiple necking during the dynamic expansion of hemispherical metallic shells, from experiments to modelling[END_REF]:

σ Y = Kε n p εp m T -q (20)
Values for the hardening coefficient n, the strain rate sensitivity parameter m and the thermal softening parameter q are taken from [START_REF] Mercier | Multiple necking during the dynamic expansion of hemispherical metallic shells, from experiments to modelling[END_REF] and given in Table (1). For the calculations, the value As mentioned in the previous section, initial conditions for the temperature and pre-strain are compulsory.

β T Q = 1 is selected. ρ 0 ( kg/m 3 ) C(J/kg/K) K(I.S.) n m q β T Q T 0 (K) ε 0 p b 8930
Owing to the initial longitudinal velocity field Eq. ( 1), the perturbation in strain rate has initially a zero amplitude: εp (t = 0) = 0. In the following of the paper, the initial temperature field and pre-strain are non uniform: T 0 (X) =

T b 0 + δT 0 (X) and ε 0 p (X) = ε 0 p b + δε 0 p (X).
As for the plate thickness, initial perturbations in temperature and accumulated plastic strain (named also pre-strain at t = 0) are of sinusoidal shape: δT 0 (X) = cos(γX) T0 and δε 0 p (X) = cos(γX)ε 0 p . Throughout the paper, we will consider for illustrative purpose that T (t = 0) = T0 = 4 K and εp (t = 0) = ε0 p (to be defined). In addition, as in [START_REF] Xavier | Extension of linear stability analysis for the dynamic stretching of plates: Spatio-temporal evolution of the perturbation[END_REF], we enforce a link between the initial perturbed quantities for the thickness and the plastic strain such that,

δε 0 p (X) = - 2 √ 3 δh 0 (X) h b 0 (21)
From Eq.( 21) and the prescribed value of ĥ0 = 8 µm, we found: ε0 p = -3.85.10 -3 . In fact, by considering the above relationship, one observes from Eq.( 18) that s 0 = 0. As a consequence, the set of differential equations ( 16) is slightly simplified. Nevertheless, the complete theory has been derived without such a link and no relationship is necessary between ĥ0 and ε0 p . In addition, the adopted initial conditions are consistent with the ones used in [START_REF] Xavier | Extension of linear stability analysis for the dynamic stretching of plates: Spatio-temporal evolution of the perturbation[END_REF] where a similar relation between the perturbation of the plate thickness and the perturbation of the pre-strain was considered.

To provide an evaluation of the traction force needed to stretch the plate dynamically, we propose to display its evolution for the background solution (determined with the condition of uniform thickness h b 0 ). In that case, from Eq. ( 13), the plastic strain within the plate is :

ε b p = ε 0 p b + 2 √ 3 ln(1 + ε0 t).
The temperature evolution is governed by Eq. ( 12). As a consequence, the force F b applied at the lateral surfaces X = ±L 0 (per unit width) is defined as:

F b = σ b h b = 2 √ 3 σ b Y h b 0 e -ε b (22)
Of course, when a perturbation is introducted, the force will be changed, mostly after the maximum of the force is reached.

Fig. 2 displays the time evolution of the normalized force F b /F max for a plate of uniform thickness under nominal strain rate ε0 = 2300 s -1 . For the parameters adopted in the present paper (see Table 1), the force first increases and reaches its maximum value at around t = 50 µs. It will be checked that for non homogeneous plates as considered next, the amplification of the perturbation is mainly observed for time larger than 50 µs.

Growth rate and amplitude estimations

In this section, the predictions of the evolution of the perturbation are derived with our new model. In that case, the time evolution of the perturbation amplitude does not have a predefined exponential form as in the 1D-CLSA theory, and is evolving according to a system of ordinary differential equations. We propose thus to compare results 22). Plane strain conditions are prescribed. Thermo-mechanical coupling is accounted for. The material is thermoviscoplastic, with a powerlaw flow stress, see Eq.( 20). Material parameters are listed in Table 1. The nominal strain rate is 2300 s -1 . The plate is homogeneous with uniform thickness.

obtained from the 1D-XLSA and 1D-CLSA approaches. As mentioned in [START_REF] Fressengeas | Fragmentation of rapidly stretching sheets[END_REF], [START_REF] Shenoy | Necking bifurcations during high strain rate extension[END_REF] or [START_REF] Mercier | Predictions of bifurcation and instabilities during dynamic extension[END_REF], an intermediate wavenumber γ a for which the growth rate is maximum is selected due to the combined effect of inertia, viscosity and stress triaxiality, see Fig. 3a). A second dominant wavenumber γ b is observed for the amplitude in Fig. 3b). The two dominant modes (defined via a criterion based on the critical growth rate or amplitude) are different: γ a = 2.2/h b 0 and γ b = 1.54/h b 0 . This result, already observed in El Maï (2014), may have some influence on the estimated number of necks since in the literature, either a criterion based on a critical normalized growth rate (for instance in [START_REF] Shenoy | Necking bifurcations during high strain rate extension[END_REF] Fig. 3a) also shows that growth rates obtained at t = 200 µs adopting the proposed approach derived from Eq.( 16) or the linear stability analysis based on the frozen coefficient assumption (1D-CLSA, see Appendix A) are similar at least for wavenumbers 0.3/h b 0 ≤ γ ≤ 2.4/h b 0 . For perturbation modes with wavenumber close to γ I = 0.13/h b 0 , growth rates predicted by the 1D-XLSA model are large as displayed in Fig. 3a) with the dashed vertical asymptotic line. As discussed in [START_REF] Xavier | Extension of linear stability analysis for the dynamic stretching of plates: Spatio-temporal evolution of the perturbation[END_REF], the large value of the growth rate defined in Eq.( 19) is due to the fact that the amplitude of the perturbation vanishes for γ I , see Fig. 3b). Fig. 3b) shows that amplitudes are clearly different from one model to another, while both models lead to the same dominant wavenumber, i.e γ b = 1.54/h b 0 . Indeed, with the proposed approach, the normalized amplitude for the dominant mode is close to 270 while with the 1D-CLSA, the normalized amplitude is around 58, more than four times smaller. Remember that at time t = 200 µs, growth rates predicted by the two approaches are identical for the dominant mode γ a , see The zoomed part for small wavenumbers in b) illustrates the fact that amplitude tends to zero for the wavenumber γI . The material is thermoviscoplastic, with a powerlaw flow stress, see Eq.( 20). Material parameters are listed in Table 1. The nominal strain rate is ε0 = 2300 s -1 . Initial thickness is h b 0 = 2.4 mm. Initial amplitude of the thickness perturbation is ĥ0 = 8 µm.

Fig. 3a). This amplitude difference is solely due to history effect, so large differences in growth rate may exist at the early stage of the deformation process. This result in terms of amplitude was already stated in Xavier et al.

(2020) when comparing 2D linear stability approaches. 0 , shown to be the dominant wavenumber (in terms of amplitude) at t = 200 µs. Fig. 4a) shows that with the proposed 1D-XLSA model, the growth rate of the perturbation is initially negative. Therefore, the perturbation amplitude first decreases as observed on Fig. 4b). At around t I = 12 µs, the amplitude is almost approaching zero. As the amplitude is close to zero and the growth rate εp has a finite value, the corresponding normalized growth rate η εp (t) defined in Eq.( 19) reaches a very large value. For t > 12 µs, η εp (t) is always positive. So the amplitude of the perturbation increases. The initial amplitude is retrieved at t = 22.5 µs. This amplitude increase occurs when the force is still increasing in Fig. 2. An important outcome from our model is therefore the capability to predict the perturbation evolution at any deformation stage. It has been checked (results not presented here) that, for all wavenumbers, the perturbation amplitude is varying during the phase of stable flow (here t < 50 µs). For some wavenumbers, contrary to the situation presented here for γ b = 1.54/h b 0 , the perturbation amplitude becomes larger than the initial amplitude beyond t > 50 µs. When the 1D-CLSA model is adopted, it is seen that the normalized growth rate has limited value up to t = 50 µs. So far, in the literature, the 1D-CLSA model has been used to investigate multiple necking mainly when the plastic flow is unstable, for which η/ ε0 >> 1 (see [START_REF] Jouve | Analytic study of plastic necking instabilities during plane tension tests[END_REF] for further details). As already mentioned, a specific analysis of the response of the classical linear stability analysis for stable or slightly unstable flows will be conducted in the near future. Meanwhile, in this paper, the amplitude of perturbation associated to the 1D-CLSA model will be assumed frozen, roughly up to t = 50 µs. For t = 100 µs, the normalized growth rate η/ ε0 predicted based on the 1D-CLSA approach is around 10 while with the proposed approach, the value η εp / ε0 is close to 12.8. At still t = 100 µs, the relative amplitude for the proposed approach is 6.47 and only 1.64 with the 1D-CLSA theory.

Later at t = 200 µs, both growth rates η εp and η coincide. Nevertheless, in terms of amplitude, a difference still exists; the amplitude predicted by the proposed model being 4.4 times larger than the one obtained with the 1D-CLSA model. This comparison shows clearly the importance of the early deformation stage, which may strongly influence the perturbation evolution. For predicting capability of the number of necks and most important, for the time scenario of fragmentation, this difference in amplitude may be important. Indeed, if we consider that necks start to trigger intense localization process when the normalized amplitude of the defect has reached a ratio of 30 (arbitrary value taken here only for illustrative purpose), Fig. 4b) shows that this stage will be initiated at t = 147 µs for the present model while it is delayed up to t = 182 µs with the 1D-CLSA based on the frozen theory coefficient. 20). Material parameters are listed in Table 1. The nominal strain rate is 2300s -1 . Initial thickness is h b 0 = 2.4 mm. Initial amplitude of the thickness perturbation is ĥ0 = 8 µm.

Role of multiaxiality modeling and of initial plastic strain perturbations

By construction, spatial heterogeneity of initial conditions may lead to difference in perturbation evolution. We represents the perturbation in the transversal direction and depends on the mode γ. In the rest of the paper, the dependence of Ĝ2D-XLSA on γ will be omitted for the sake of notation conciseness. Initial conditions for the perturbation not only consist in providing the initial amplitude of Ĝ2D-XLSA but also in defining the spatial variation of the initial perturbation along the transversal direction. In [START_REF] Xavier | Extension of linear stability analysis for the dynamic stretching of plates: Spatio-temporal evolution of the perturbation[END_REF], the evolution of each perturbation related to mechanical variables is associated to a stream function ψ 2D-XLSA (X, Y, t) whose through-thickness variation is ψ2D-XLSA (Y, t). The through-thickness variation of the perturbed temperature is T 2D-XLSA (Y, t). In particular, the through-thickness variation of the amplitude of the accumulated plastic strain perturbation is associated with the stream function as shown below:

ε2D-XLSA p (Y, t) = 2γ √ 3 ∂ ψ2D-XLSA ∂Y ( 23
)
For the 2D-XLSA model, initial conditions are specified by prescribing the initial shape of three quantities: ψ2D-XLSA (Y, t = 0), ψ2D-XLSA (Y, t = 0) and T 2D-XLSA (Y, t = 0). To be consistent with the present work, we select ψ2D-XLSA (Y, t = 0) = 0 (no initial perturbation of the strain rate) and T 2D-XLSA (Y, t = 0) = 4 K.

In the first test, we defined initial condition IC1 by considering the specific form of the stream function:

ψ2D-XLSA (Y, t = 0) = √ 3ε 0 p 2γ Y = - ĥ0 2γ 2Y h b 0
. It is clear from Eq.( 23) that IC1 case leads to an homogeneous perturbation amplitude of the accumulated plastic strain across the thickness: ε2D-XLSA p (Y, t = 0) = εo p , see Fig. 5. This value is the one adopted as the initial amplitude of the plastic strain perturbation in the 1D-XLSA model, see previous results. Therefore, a fair comparison between the proposed model and the 2D-XLSA approach with IC1 can be made. A second initial condition, IC2, is proposed with a pronounced fluctuation along the through thickness direction: ψ2D-XLSA (Y, t = 0) = -ĥ0 2γ sin(ωπ2Y /h b 0 ) sin (ωπ) with ω = 4 3 for IC2. As a consequence, the initial accumulated plastic strain perturbation for IC2 is oscillatory in the plate thickness, see Fig. 5, leading to initial material heterogeneity. Fig. 6 presents the evolution of the normalized amplitude of the plastic strain perturbation as a function of the normalized wavenumber γh b 0 at t = 200 µs. First, the dominant wavenumber predicted by the 2D-XLSA model is slightly affected by the adopted initial conditions. Its value is around γ 1.54/h b 0 , and is consistent with the one obtained from the 1D-XLSA, see Fig. 3b). Nevertheless, a difference in terms of amplitude exists.

In that case, the difference in amplitude from the 1D-XLSA and 2D-XLSA with IC1 condition, is due to a difference in modeling strategy for the stress triaxiality which develops in the neck. In 1D-XLSA, the Bridgman correction factor is introduced while for the 2D-XLSA theory, the multi-dimensional aspect of the stress-field is by construction present in the model. A larger difference is observed with IC2. Indeed, the initial condition generates an oscillatory plastic strain in the cross-section, see Fig. 5. The material heterogeneity will induced an additional stress heterogeneity in the neck region. It could be important to evaluate how the initial material heterogeneity affects predictions for other material constitutive responses. Preliminary calculations have been carried out (results not presented here) and it has been observed that for a viscoplastic material (behavior without strain hardening and thermal softening, other material parameters being kept), no difference in growth rate prediction was observed between IC1 and IC2 for the 2D-XLSA method. Differences in predictions based on the 1D-XLSA and 2D-XLSA methods were present for large wavenumbers. For small wavenumbers, both models led to consistent results. A deeper analysis is needed and will be done in a future work.

It is also interesting to notice that for small wavenumbers, the proposed 1D-XLSA model and the 2D-XLSA approach with different initial conditions lead to the same results (at least for γh b 0 < 1). Indeed, it is known from the literature (see for instance [START_REF] Fressengeas | Fragmentation of rapidly stretching sheets[END_REF]) that the damping of the amplitude of perturbation mode with small wavenumber is mostly due to inertia. This inertia effect is present in all models. On the contrary, for perturbations with a large wavenumber, a clear difference exists between all modeling routes considered in the work. It is also known from [START_REF] Fressengeas | Fragmentation of rapidly stretching sheets[END_REF] that modes with a large wavenumber are damped mostly because of the multi-dimensionality of the flow inside the neck together with the viscoplastic properties of the material. With IC2 condition, the heterogeneity of the stress field in the neck is enlarged because of the initial material heterogeneity induced by the heteregeneous plastic strain field. This situation provides a clear understanding of the respective role of stress triaxiality in the neck region and also on the role of the material heterogeneity.

-1 -0.5 0 0.5 1 0 0.02 0.01

-0.02 -0.01 ε0 p 2Y /h b 0 ε2D-XLSA p (Y, t = 0)

IC1 IC2

Figure 5: Spatial variation of the initial perturbation of the accumulated plastic strain along the transversal direction. The two initial conditions IC1 and IC2 are displayed. Those initial conditions for the plastic strain are induced by the stream function ψ2D-XLSA , see Eq.( 23). Initial thickness is h b 0 = 2.4 mm. Initial amplitude of the thickness perturbation is ĥ0 = 8 µm. Initial amplitude of the perturbation for IC1 is ε0 p = -0.00385. Fig. 7 presents the time evolution of a) the normalized amplitude |ε p /ε 0 p | and b) the normalized growth rate η εp defined in Eq.( 19) of the perturbed plastic strain for the wavenumber γ 2 = 2.5/h b 0 . The evolution of the perturbation is highly dependent on the initial conditions. Results obtained by the 1D-XLSA are close to those obtained with the 2D-XLSA model and initial condition IC1. In that configuration, the amplitude decreases, vanishes for t = 8.7 µs and then increases. Normalized amplitudes derived with initial condition IC2 vanishe later, at t = 69 µs. Such a difference in the early stage of the deformation is transfered to later stage. Fig. 7b) depicts the time evolution of the normalized mechanical growth rate η εp ε0 (see Eq.19) for γ 2 = 2.5/h b 0 . For the 2D-XLSA model, the mechanical growth rate is defined in [START_REF] Xavier | Extension of linear stability analysis for the dynamic stretching of plates: Spatio-temporal evolution of the perturbation[END_REF] A comparison between predictions of the 2D-XLSA model with various initial conditions and of the 1D-XLSA is performed. The material is thermoviscoplastic, with a powerlaw flow stress, see Eq.( 20). Material parameters are listed in Table 1. The nominal strain rate is ε0 = 2300 s -1 . Initial thickness is h b 0 = 2.4 mm. Initial amplitude of the thickness perturbation is ĥ0 = 8 µm. 19) of the perturbed plastic strain as a function of time. The wavenumber considered is γ2 = 2.5/h b 0 . The predictions of the 2D-XLSA model with the two initial conditions are compared to the 1D-XLSA. The material is thermoviscoplastic, with a powerlaw flow stress, see Eq.( 20). Material parameters are listed in Table 1. The nominal strain rate is ε0 = 2300 s -1 . Initial thickness is h b 0 = 2.4 mm. Initial amplitude of the thickness perturbation is ĥ0 = 8 µm.

in Eq.( 19) when the amplitude is close to zero. It is seen that the time duration of the deformation process with negative growth rate is varying from t = 8.5 µs with the proposed 1D-XLSA model to t = 69 µs for the 2D-XLSA model. During this stage, the normalized amplitude decreases, as displayed in Fig. 7 a). Later, the growth rate remains always positive and the corresponding amplitude of the plastic strain perturbation is strictly increasing.

Only for t > 150 µs, growth rates are coinciding for both models and all initial conditions. An interesting outcome of the paper is to highlight how the way multiaxiality effects are introduced in the model affects the perturbation evolution for large wavenumbers. This is in agreement with [START_REF] Fressengeas | Fragmentation of rapidly stretching sheets[END_REF]. To provide a measure of the linearity of the solution at a given time, a polynomial interpolation of function ψ2D-XLSA is performed: ψ2D-XLSA (Y, t) = ∞ n=0 q 2n+1 (t)Y 2n+1 . Note that all coefficients related to an even order are null as the function is antisymmetric with respect to Y so as to be representative of necking mode. q 1 is the coefficient in the polynomial expansion representative of the linear shape of ψ2D-XLSA . Fig. 9 displays the absolute value of the normalized coefficient q 2n+1 q 1 for n in the range [0, 7] at different times a) t = 125 µs, b) t = 200 µs. Two wavenumbers γ 1 = 0.82/h b 0 and γ 2 = 2.5/h b 0 are considered. With the initial condition IC2, it is found (results not presented here) that q 3 is the dominant coefficient at t = 0 with q 3 ≈ 3q 1 . At t = 125 µs, Fig9a)

shows that ratios |q 2n+1 /q 1 | have been reduced for n ≥ 1 meaning that the stream function amplitude is less nonlinear. This is particularly visible for the perturbation with the small wavenumber γ 1 for which the coefficient q 1 is the dominant one. In other words, for this small wavenumber, the time duration when the effect of the modeling of multiaxiality and of the influence of the initial condition are significant, is short and the proposed 1D-XLSA model provides consistent results when compared to a multidimensional approach as the 2D-XLSA method. In the early deformation process, for the perturbation mode of large wavenumber γ 2 , the stream function remains highly non-linear at t = 125 µs, meaning that the perturbed plastic strain presents a strong fluctuation inside the plate thickness. Therefore, discrepancies in terms of amplitude and growth rate are expected, as observed in the early stage of the deformation process between the 2D-XLSA and the proposed 1D-XLSA models. At t = 200 µs, it is seen in Fig. 9b) that the stream function presents an almost linear fluctuation within the plate thickness for both wavenumbers as q 1 is clearly the dominant coefficient. Therefore, the proposed 1D-XLSA model and the 2D-XLSA approach capture similar growth rates, see Fig. 20). Material parameters are listed in Table 1. The nominal strain rate is ε0 = 2300 s -1 . Initial thickness is h b 0 = 2.4 mm. Initial amplitude of the thickness perturbation is ĥ0 = 8 µm. 20). Material parameters are listed in Table 1. The nominal strain rate is ε0 = 2300 s -1 . Initial thickness is h b 0 = 2.4 mm. Initial amplitude of the thickness perturbation is ĥ0 = 8 µm.

Conclusion

A new linear stability analysis of a plate under plane strain loading is proposed to model multiple necking. A onedimensional framework is used and multiaxiality of the stress field within the neck region is taken into account with a Bridgman correction factor. Contrary to models in the literature, except in [START_REF] Xavier | Extension of linear stability analysis for the dynamic stretching of plates: Spatio-temporal evolution of the perturbation[END_REF], the assumption of exponential growth of the perturbation, only valid when the time scale separation prevails, is no longer used. The thermo-mechanical problem is analyzed which leads finally to a system of two time differential equations coupling a mechanical perturbation and a thermal perturbation. The theory is exemplified by considering the stretching of a plate made of a thermo-viscoplastic material with strain hardening.

The importance of the modeling of multiaxiality and the role of the initial material configuration have been addressed. Indeed, while in the proposed 1D model the Bridgman correction factor is used, the 2D-XLSA model of [START_REF] Xavier | Extension of linear stability analysis for the dynamic stretching of plates: Spatio-temporal evolution of the perturbation[END_REF] for which the stress field within the plate is really multi-dimensional, is also considered.

It is shown that for perturbations with a large wavenumber, a discrepancy in amplitude is observed mainly due to difference in growth rates at the beginning of the stretching. For perturbations with a small wavenumber, both approaches provide similar amplitude evolution. It is therefore seen that the 1D-XLSA framework provides consistent results when compared to the 2D-XLSA model. It could be also of interest to perform an extensive comparison for various constitutive material responses. Note that the 1D-XLSA model of the present work is much simpler when compared to 2D-XLSA (also from a numerical point of view).

One of the main features of the proposed model is that the evolution of the perturbation can be captured at any deformation stage (during stable or unstable flow). This capability appears as an important achievement when compared to 1D linear stability analyses of the literature. Indeed, classical linear stability analyses based on frozen coefficient theory (as often adopted in the literature) are usually used to analyze the deformation stage during unstable flow, i.e. when positive growth rate for the perturbation is encountered (often late in the deformation process). Thus, during stable flow, we have assumed in the CLSA that the perturbation is frozen (no amplitude evolution is assumed). It is clear that, it could be of interest to further analyze with the classical linear stability analysis the evolution of perturbations during stable flow.

Comparisons with results obtained with linear stability analysis based on the frozen coefficient theory are performed. While growth rates of the two models (1D-CLSA and 1D-XLSA) are similar in the late deformation stages, large differences in terms of amplitude are found due to discrepancies inherited from the early deformation stage. This finding is valid for perturbations with any wavenumber. In addition, various initial conditions can be treated with the 1D-XLSA approach, while up to now, the 1D-CLSA method was not considering this possibility.

In a future work, finite element calculations will be conducted to compare predictions in terms of amplitude and spacing of necks. The goal will be to investigate how the 1D-XLSA model can help in predicting fragmentation of structural elements.

A 1D Linear stability analysis with frozen coefficient assumption

The governing equations for the perturbed quantities are given by the set (15) in which the initial conditions δh 0 and δε 0 p are not considered, see [START_REF] Zhou | An elastic-visco-plastic analysis of ductile expanding ring[END_REF] or [START_REF] Zaera | Collective behavior and spacing of necks in ductile plates subjected to dynamic biaxial loading[END_REF]. In the frozen coefficient theory, perturbations are searched under the form δG(X, t) = Ĝ0 cos(γX) exp(ηt) with η the growth rate of the perturbation and Ĝ0 an amplitude factor, while δv is searched as δv(X, t) = v0 sin(γX) exp(ηt). When adopting the frozen coefficient theory, it is assumed that the time scale associated with the evolution of the perturbation is much smaller than the one associated to the homogeneous solution.

At each time t of the deformation process, a perturbation is superimposed to the background homogeneous solution. So, the expression of the perturbed quantities are substituted in the set of equations ( 15). After some rearrangement, and by selecting the perturbed plastic strain and the perturbed temperature as the two main variables: δε p (X, t) = ε0 p cos(γX) exp(ηt) and δT (X, t) = T 0 cos(γX) exp(ηt), one obtains from the set of Eqs (15):

a 1 η 2 + a 2 η + a 3 ε0 p + a 4 T 0 = 0 η T 0 = b 1 T 0 + b 2 η ε0 p + b 3 ε0 p (24)
Coefficients in Eq.( 24) are the same as those provided in Eq.( 17) The combination of the two relations (24) provides:

a 1 η 2 + a 2 η + a 3 + a 4 b 2 η + b 3 η -b 1 = 0 (25)
After simple manipulation, Eq.( 25) leads to a third order polynomial equation whose solution is η. The largest positive root of Eq.( 25) is usually adopted as the instantaneous growth rate of the perturbation. It is used to represent the time evolution of the perturbation when the Bridgman approximation is considered in the linear stability analysis, see [START_REF] Zhou | An elastic-visco-plastic analysis of ductile expanding ring[END_REF]), Rodríguez-Martínez et al. (2013a).

B 1D Linear stability analysis of ductile expanding ring

Fragmentation of rings has been widely discussed in the literature and studied analytically with the use of the linear stability analysis of the dynamic extension of a round bar, see [START_REF] Jeanclaude | Dynamic necking of rods at high strain rates[END_REF], [START_REF] Mercier | Predictions of bifurcation and instabilities during dynamic extension[END_REF], [START_REF] Zhou | An elastic-visco-plastic analysis of ductile expanding ring[END_REF]), El Maï et al. (2014), Rodríguez-Martínez et al. (2013a). We propose to use the method of the present paper to extend the work of [START_REF] Zhou | An elastic-visco-plastic analysis of ductile expanding ring[END_REF] without considering the frozen coefficient theory. The linear stability analysis of a round bar, of initial cross-section A 0 (A 0 = πr 2 0 ) is developed through a Lagrangian 1D analysis. In the present approach, we investigate the possibility of having an initial imperfection for the radius. Therefore, it is assumed that r 0 is possibly evolving along the length of the bar: r 0 (X) = r b 0 + δr 0 (X) and the average of the perturbation is zero : < δr 0 (X) >= 0. At time t = 0, the bar is strained under uniform nominal strain-rate ε0 = V 0 /L 0 . V 0 is the constant traction velocity and L 0 the initial half-length of the bar.

The Lagrangian coordinate is X (-L 0 ≤ X ≤ L 0 ). At time t, we define the following quantities: x the Eulerian coordinate, v the longitudinal velocity, σ the true stress, ε = ln( ∂x ∂X ) the true strain and A = πr 2 the current section. All those variables are only related to t and X. The multidimensional aspect of the stress field within the neck region is accounted for by replacing the true stress σ by an average stress complemented by the Bridgman correction factor B(Φ) defined in [START_REF] Bridgman | Studies in large plastic flow and fracture[END_REF], see also Appendix C. Governing equations detailed in [START_REF] Zhou | An elastic-visco-plastic analysis of ductile expanding ring[END_REF] are briefly recalled:

ρ 0 A 0 ∂v ∂t = ∂Aσ ∂X ∂x ∂X = e ε ∂v ∂X = εe ε A = A 0 e -ε σ = 1 + 1 Φ ln(1 + Φ)σ Y Φ = 1 2 r ∂ 2 r ∂x 2 σ Y = σ Y (ε p , εp , T ) ρ 0 C Ṫ = β T Q σ Y εp ε p -ε 0 p = ε (26)
with C the thermal capacity supposed constant, β T Q the Taylor-Quinney coefficient, supposed constant, and ρ 0 the mass density. The material is assumed incompressible. ε 0 p is a pre-strain parameter. The homogeneous solution G b , representative of the deformation mode without any imperfections defined from Eqs (26) can be found in [START_REF] Zhou | An elastic-visco-plastic analysis of ductile expanding ring[END_REF]:

x b = (1 + ε0 t)X ε b = ln(1 + ε0 t) > 0 Φ b = 0 A b = A b 0 e -ε b εb p = ε0 1 + ε0 t (27)
where A b 0 = πr b 0 is the initial cross-section, without imperfection. In the 1D linear stability analysis of the literature, as in [START_REF] Zhou | An elastic-visco-plastic analysis of ductile expanding ring[END_REF], at each time step, a perturbation is added to the homogeneous solution. With the present approach, the perturbation is introduced at the initial time and its evolution is deduced consistently:

v = v b + δv ε = ε b + δε ε p = ε p b + δε p A = A b + δA σ = σ b + δσ σ Y = σ Y b + δσ Y ε = εb + δ ε Φ = Φ b + δΦ T = T b + δT (28)
Substitution of Eq.( 28) in the governing equations ( 26) and using the homogeneous solution defined in Eqs

with σ b Y ,ω denoting the partial derivative of σ b Y with respect to the variable ω (ω being ε p , εp or T ). As for the stretching of the plate developed in the main part of the paper, perturbation fields δG are searched of the form δG = cos(γX) Ĝ(γ, t) (except for the perturbation of the velocity whose form is δv = sin(γX)v(γ, t), see Eq.3) with γ = kπ L0 being the longitudinal wavenumber of the perturbation. After adequate combination, the set of Eqs (29) simplifies into two ordinary differential equations coupling the main unknowns εp and T :

c 1 εp + c 2 εp + c 3 εp + c 4 T = f 0 Ṫ = d 1 T + d 2 εp + d 3 εp (30)
The coefficients c i and d i in Eq.( 30) are only time dependent, related to the homogeneous solution: f 0 is a time dependent coefficient that depends on the prescribed values of the amplitude Â0 of the perturbation of the initial cross-section and the amplitude ε0 p of the perturbation of the pre-strain parameter:

c 1 (t) = ρ 0 (1 + ε0 t) c 3 (t) = σ b Y A 0 b γ 4 8π(1 + ε0 t) 4 + σ b Y ,εp γ 2 1 + ε0 t - σ b Y γ 2 1 + ε0 t c 2 (t) = 2ρ 0 ε0 + γ 2 σ b Y , εp 1 + ε0 t c 4 (t) = γ 2 σ b Y ,T 1 + ε0 t d 1 (t) = β T Q ε0 ρ 0 C(1 + ε0 t) σ b Y ,T d 3 (t) = β T Q ε0 ρ 0 C(1 + ε0 t) σ b Y ,εp d 2 (t) = β T Q ρ 0 C ( ε0 1 + ε0 t σ b Y , εp + σ b Y ) (31)
f 0 = -ε0 p + Â0 A b 0 γ 2 σ b Y (1 + ε0 t) - A b 0 γ 4 σ b Y 8π(1 + ε0 t) 4 (32)
The system (30) has to be completed with initial conditions on εp (t = 0), εp (t = 0) = 0 (since initial uniform strain rate is prescribed) and T (t = 0). As for plane strain, when the frozen coefficient theory is adopted, the system (29), in which initial conditions δε 0 p and δA 0 are neglected, leads to a third order polynomial equation for the growth rate η. This equation can be found elsewhere, [START_REF] Zhou | An elastic-visco-plastic analysis of ductile expanding ring[END_REF]

C Bridgman correction factor calculation

In this appendix, the expression of the Bridgman correction factor for plane strain conditions is briefly discussed, see [START_REF] Bridgman | Studies in large plastic flow and fracture[END_REF] for more details.

Let us consider a half-plate with a non-uniform thickness h(x), see Fig. 10. The radius of curvature in the neck region is r N > 0 while in the area where the plate is thicker than in the neck region, the surface profile is concave with a radius of curvature r T < 0.

In the original paper of [START_REF] Bridgman | Studies in large plastic flow and fracture[END_REF], a simplified analysis of the stress field at the neck has been proposed.

From geometrical considerations on the shape of the neck region and of the isostatic lines (assuming a circle shape), the author found that the longitudinal stress at the neck is fluctuating in the cross-section according to the following expression:

σ(x) = 2 σ Y √ 3 1 + ln 1 + 1 4 h r N 1 -4 y 2 h 2 (33)
with σ Y being the flow stress.

Noticing that Φ = h 4r N , and by integration along the thickness of the plate at the neck, the expression of the

Figure 2 :

 2 Figure 2: Time evolution of the normalized force F b /Fmax registered at X = ±L0 by unit width, see Eq.(22). Plane strain conditions are prescribed. Thermo-mechanical coupling is accounted for. The material is thermoviscoplastic, with a powerlaw flow stress, see Eq.(20). Material parameters are listed in Table1. The nominal strain rate is 2300 s -1 . The plate is homogeneous with uniform thickness.

Fig. 3

 3 Fig. 3 represents a) the normalized growth rate η εp / ε0 and b) the normalized amplitude | εp εo p | as a function of the normalized wavenumber γh b 0 at t = 200 µs.As mentioned in[START_REF] Fressengeas | Fragmentation of rapidly stretching sheets[END_REF],[START_REF] Shenoy | Necking bifurcations during high strain rate extension[END_REF] or[START_REF] Mercier | Predictions of bifurcation and instabilities during dynamic extension[END_REF], an intermediate wavenumber γ a for which the growth rate is

  ) or on a critical normalized amplitude (for instance in ElMaï et al. (2014) or[START_REF] Vaz-Romero | Multiple necking pattern in nonlinear elastic bars subjected to dynamic stretching: The role of defects and inertia[END_REF]) can be found. In[START_REF] Vaz-Romero | Multiple necking pattern in nonlinear elastic bars subjected to dynamic stretching: The role of defects and inertia[END_REF], the case of a non-linear elastic rod under dynamic stretching was analyzed by finite element calculations and by using the 1D-CLSA model. The authors have shown that the criterion based on the amplitude was relevant to reproduce the neck spacing observed in calculations. Further works are necessary to confirm this result.

Figure 3

 3 Figure 3: a) Normalized growth rate η εp / ε0 and b) normalized amplitude | εp εo p | as a function of the normalized wavenumber γh b 0 at t = 200 µs.The zoomed part for small wavenumbers in b) illustrates the fact that amplitude tends to zero for the wavenumber γI . The material is thermoviscoplastic, with a powerlaw flow stress, see Eq.(20). Material parameters are listed in Table1. The nominal strain rate is ε0 = 2300 s -1 . Initial thickness is h b 0 = 2.4 mm. Initial amplitude of the thickness perturbation is ĥ0 = 8 µm.

Fig. 4

 4 Fig.4 presents the time evolution of a) the normalized growth rate η εp / ε0 and b) the normalized amplitude | εp εo p | for γ b = 1.54/h b0 , shown to be the dominant wavenumber (in terms of amplitude) at t = 200 µs. Fig.4a) shows that with the proposed 1D-XLSA model, the growth rate of the perturbation is initially negative. Therefore,

FurtherFigure 4 :

 4 Figure 4: Time evolution of a) the normalized growth rate η εp / ε0 and b) the normalized amplitude | εp εo p | for the 1D-XLSA model and for the 1D-CLSA model. Results are provided for the specific wavenumber γ b . The vertical dashed line observed for time tI is due to the observation that the predicted amplitude for the perturbation vanishes. The material is thermoviscoplastic, with a powerlaw flow stress, see Eq.(20). Material parameters are listed in Table1. The nominal strain rate is 2300s -1 . Initial thickness is h b 0 = 2.4 mm. Initial amplitude of the thickness perturbation is ĥ0 = 8 µm.

  propose in the following to investigate this aspect via a comparison between predictions of the proposed 1D-XLSA model, where the through thickness dependency of fields is disregarded, and of the 2D-XLSA model where this dependency is kept. In brief (more details can be found in[START_REF] Xavier | Extension of linear stability analysis for the dynamic stretching of plates: Spatio-temporal evolution of the perturbation[END_REF]), any perturbation fields in the 2D-XLSA model can be written as δG 2D-XLSA (X, Y, t) = F (γX) Ĝ2D-XLSA (γ, Y, t) where Y is the Lagrangian coordinate in the transversal direction. The function F (γX) is of sinusoidal shape and Ĝ2D-XLSA (γ, Y, t)

Figure 6 :

 6 Figure 6: Normalized amplitude | εp εo p | as a function of the normalized wavenumber γh b 0 at t = 200 µs.A comparison between predictions of the 2D-XLSA model with various initial conditions and of the 1D-XLSA is performed. The material is thermoviscoplastic, with a powerlaw flow stress, see Eq.(20). Material parameters are listed in Table1. The nominal strain rate is ε0 = 2300 s -1 . Initial thickness is h b 0 = 2.4 mm. Initial amplitude of the thickness perturbation is ĥ0 = 8 µm.

Figure 7 :

 7 Figure 7: Evolution of a) the normalized amplitude |εp/ε 0 p | and b) the normalized growth rate η εp defined in Eq.(19) of the perturbed plastic strain as a function of time. The wavenumber considered is γ2 = 2.5/h b 0 . The predictions of the 2D-XLSA model with the two initial conditions are compared to the 1D-XLSA. The material is thermoviscoplastic, with a powerlaw flow stress, see Eq.(20). Material parameters are listed in Table1. The nominal strain rate is ε0 = 2300 s -1 . Initial thickness is h b 0 = 2.4 mm. Initial amplitude of the thickness perturbation is ĥ0 = 8 µm.

Fig. 8

 8 Fig.8 displays the through-thickness variation of the stream function ψ2D-XLSA (Y, t) computed by the 2D-XLSA model at a) t = 125 µs and b) t = 200 µs. The two wavenumbers γ 1 = 0.82/h b 0 and γ 2 = 2.5/h b 0 are considered. Only the initial condition IC2 is adopted as it generates an initial fluctuation of the plastic strain perturbation in the plate thickness. At t = 125 µs, the spatial variation in the Y direction of the stream function is non-linear, leading to a heterogeneous perturbation of the plastic deformation in the thickness of the plate. Therefore, difference between the 1D-XLSA and 2D-XLSA can be expected. At t = 200 µs i.e. in the late deformation stage, the spatial variation of the stream function amplitude becomes almost linear for both wavenumbers. Thus, the 2D-XLSA model provides results similar to those obtained with the proposed 1D model in terms of growth rate, see Fig.7b).

Figure 8 :

 8 Figure 8: Spatial variation in the plate thickness of the stream function computed by the 2D-XLSA model for a) t = 125 µs and b) t = 200 µs. Wavenumbers γ1 = 0.82/h b 0 and γ2 = 2.5/h b 0 are considered. Initial condition IC2 is used, see Fig.5. The material is thermoviscoplastic, with a powerlaw flow stress, see Eq.(20). Material parameters are listed in Table1. The nominal strain rate is ε0 = 2300 s -1 . Initial thickness is h b 0 = 2.4 mm. Initial amplitude of the thickness perturbation is ĥ0 = 8 µm.

Figure 9 :

 9 Figure 9: Values of the normalized coefficients q 2n+1 q 1 | of the polynomial interpolation of ψ2D-XLSA for n in the range [0, 7]. Interpolation is performed at a) t = 125 µs and b) t = 200 µs. Two perturbation modes with wavenumber γ1 = 0.82/h b 0 and γ2 = 2.5/h b 0 are considered. Initial condition IC2 of Fig.5 is selected. The material is thermoviscoplastic, with a powerlaw flow stress, see Eq.(20). Material parameters are listed in Table1. The nominal strain rate is ε0 = 2300 s -1 . Initial thickness is h b 0 = 2.4 mm. Initial amplitude of the thickness perturbation is ĥ0 = 8 µm.

(

  27) provide the governing equations for the perturbed fields:∂δv ∂X = e ε b εb δε + δ ε δA = -A b δε + δA 0 e -ε b δσ = δσ Y δ ε δσ Y = σ b Y ,εp δε p + σ b Y , εp δ εp + σ b Y ,T δT ρ 0 Cδ Ṫ = β T Q ( εb p δσ Y + σ b Y δ εp )

Figure 10 :

 10 Figure 10: Schematic representation of the plate during deformation. The thickness profile is non-uniform.

  , Rodríguez-Martínez et al. (2013b), Rodríguez-Martínez et al. (2013a) or Vaz-Romero et al. (2017).

Table 1 :

 1 Material parameters for copper taken from[START_REF] Mercier | Multiple necking during the dynamic expansion of hemispherical metallic shells, from experiments to modelling[END_REF]. The flow stress is described by a powerlaw, see Eq.(20).

	383	1.075 10 9 0.35 0.0098 0.111	1	300	0.256

Bridgman correction factor is obtained as in [START_REF] Bridgman | Studies in large plastic flow and fracture[END_REF]:

The expression is valid for Φ > 0. Thus, the Bridgman correction factor obtained in Eq.( 34) for plane strain cannot be used when the profile is concave. Let us analyse now the stress field where the plate is thicker than in the neck region, labelled T in Fig. 10. The center of curvature of the external contour c T is located on the other side of the axis of the plate. The radius of curvature is r T < 0. By applying the same strategy as in [START_REF] Bridgman | Studies in large plastic flow and fracture[END_REF] for the region where the plate is locally the thickest, one can easily show that the expression for the longitudinal stress is still given by Eq.( 33), replacing r N by r T . Note that for the concave case, the parameter Φ = h 4r T is negative. By integration as in Eq. ( 34), the novel expression of the Bridgman correction factor in the concave region where the plate is the thickest is:

In the present paper, for limited sine perturbation of the plate thickness, one has r N >> h and -r T >> h. As a consequence, Φ remains close to zero. The Taylor expansion of Eqs ( 34) and ( 35) leads to the expression (11):

The last relation is valid at the neck and in the area where the plate is thicker than in the neck region. In the present paper, for limited amplitude of the perturbation, we assume that the formula (36) is extended by continuity so that the Taylor expansion of the Bridgman correction factor given by Eq. ( 36) is adopted for the entire plate, see for instance [START_REF] Zhou | An elastic-visco-plastic analysis of ductile expanding ring[END_REF].