# 1D linear stability analysis based on an adjusted correction factor for dynamic expansion: Application to plates and rods

U. Houire, Sébastien Mercier, C. Czarnota, M. Xavier, S. El Maï

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U. Houire, Sébastien Mercier, C. Czarnota, M. Xavier, S. El Maï. 1D linear stability analysis based on an adjusted correction factor for dynamic expansion: Application to plates and rods. Journal of the Mechanics and Physics of Solids, 2026, 208, pp.106487. 10.1016/j.jmps.2025.106487. hal-05445083

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# 1D linear stability analysis based on an adjusted correction factor for dynamic expansion: application to plates and rods

U. Houire$^{a,b,c}$, S. Mercier$^{b,*}$, C. Czarnota$^b$, M. Xavier$^a$, S. El Maïc

$^a$CEA, DAM, DIF, 91297 Arpajon - FRANCE

$^b$Université de Lorraine, CNRS, Arts et Métiers Institute of Technology, LEM3, F-57000, Metz - FRANCE

$^c$CEA DAM, Gramat, 46500 Gramat - FRANCE

## Abstract

This work investigates the onset and development of plastic strain localization during the dynamic expansion of metallic shells. The multiple necking and fragmentation scenario are here viewed as originating from the development of geometrical perturbations (i.e., surface roughness), whose time evolution plays a critical role for the late free flight of fragments. Based on the extended 1DXLSA (One-Dimensional eXtended Linear Stability Analysis) model of Xavier et al. (2021), and using the 2DXLSA of Xavier et al. (2020) and FEM calculations as references, we propose an adjustment of the stress approximation in the neck section to better capture the onset of multiple necking in cylindrical (plate) and ring (round bar) geometries. A modified Bridgman correction factor is then introduced, which highlights the limitations of the previous 1DXLSA study. A good agreement in terms of

<footnote>* Corresponding author</footnote>

<footnote>Email address: sebastien.mercier@univ-lorraine.fr (S. Mercier)</footnote>
time evolution of the perturbations is obtained between Finite element sim-
ulations, two-dimensional linear stability approach and the new 1D model.

This is a post-peer-review, pre-copyedit version of an article published in
Journal of the Mechanics and Physics of Solids. The final authenticated ver-
sion is available online at: https://doi.org/10.1016/j.jmps.2025.106487

***Keywords:*** Dynamic necking, Bridgman correction factor, Unit cell calculations, Multimodal surface profile, Stability analysis, Ring expansion, Plate extension

**1. Introduction**

Fragmentation of thin metallic structures subjected to high strain-rate
expansion is a phenomenon of significant importance in both civil and defense
applications. In ductile metals, the resulting fragments originate from local
thinning regions, commonly referred to as necks. Some of them are arrested,
while others lead to fracture, Zhang & Ravi-Chandar (2006). Accurately
predicting not only the number of fragments but also the temporal evolution
of necking remains a key objective for understanding and controlling dynamic
failure.

Numerous facilities have been developed in research laboratories to con-
duct thin structure expansions in order to understand the process of localiza-
tion. Three configurations have been commonly investigated: the dynamic
expansion of thin-walled tubes (see Wesenberg & Sagartz (1977); Olive et al. (1979); Fyfe & Rajendran (1980); Goto et al. (2008)), hemispherical shells (see Campbell et al. (2007); Mercier et al. (2010); Bolis et al. (2013); Callahan et al. (2023)), and rings, driven by electromagnetic or explosive loading. Among these, the ring expansion test remains the most widely adopted configuration due to its simplicity and its ability to produce well-controlled radial kinematics (see Niordson (1965); Llorca & Juanicotena (1997); Zhang & Ravi-Chandar (2006)). More recent studies have developed impact-driven expansion tests for tubes (Nieto-Fuentes et al., 2023a), rings (Nieto-Fuentes et al. (2023b); Gant et al. (2024)), and biaxial plate bulging (Simpson et al. (2024)). In most of the works, high speed cameras have been used to record all stages leading to fragmentation: homogeneous deformation, development of local thinnings, fracture at some necking sites. So, a precise time evolution of the free surface up to failure can be captured.

In parallel with experimental efforts, analytical and statistical models have been developed to explain fragment-size distributions. Classical statistical fragmentation theories originate from the pioneering works of Mott (1947) and Grady (1981), which considered randomly distributed failure sites and instantaneous stress release upon fracture. These models successfully captured the bell-shaped fragment size distribution observed in expanding thin cylinders, as in the experiments of Wesenberg & Sagartz (1977). More recently, Dequiedt (2025) refined the Mott assumption by incorporating non-instantaneous stress release, dissipative effects, and defect interactions. This
work showed that while the classical assumptions remain valid for low defect energies, interactions between defect activation and release waves may significantly modify the predicted fragment-size distribution.

The present work adopts a complementary point of view. The multiple necking and fragmentation scenario for ductile thin metallic casings is viewed as the consequence of the development of plate thickness fluctuations originating from initial surface roughness. Several modeling routes have been proposed in the literature to investigate how a thickness defect contributes to dynamic strain localization. Extending the classic two-zone model introduced by Marciniak & Kuczyński (1967) for the forming limit diagram, Jacques (2020) included the contribution of inertia and the Bridgman formulation. The author was able to characterize the necking strain as of function of the perturbation wavenumber. The predictions obtained with this non-linear approach compared well with finite element calculations. In our work, to model the evolution of the local shell thickness, we propose to adopt linear stability analyses as this technique is well recognized in the literature for being capable of explaining the incipient multiple necking observed in dynamic expansion of thin structures. Under dynamic conditions, multidimensional linear stability analyses conducted by Fressengeas & Molinari (1985, 1994); Shenoy & Freund (1999) revealed the essential role of inertia, which damps long wavelength modes, while viscosity and stress multiaxiality extinguish short wavelengths perturbations. The interplay between these two effects leads to the selection of an intermediate wavelength corresponding to the
characteristic neck spacing observed in experiments. Among others, Shenoy
& Freund (1999); Mercier & Molinari (2003, 2004); Jouve (2013) applied
such multidimensional linear stability analyses to the dynamic expansion of
rods and tubes to describe multiple necking patterns. The predictions of
Jouve (2010) showed excellent agreement with the experimental fragment
sizes reported by Olive et al. (1979) during the dynamic expansion of steel
cylinders.

Nevertheless, it has to be noticed that multidimensional linear stabil-
ity analyses require substantial analytical developments. Simplified one-
dimensional (1D) formalisms were therefore introduced to retain the essential
physical mechanisms while facilitating analytical treatment. One can refer
for instance to Zhou et al. (2006) for the dynamic stretching of round bars
using the Brigdman correction factor. As the model derivation is signifi-
cantly simpler while still providing accurate predictions of the average neck
spacing, numerous contributions have adopted or extended this approach.
Predictions of 1D stability analyses were compared to Finite Element (FE)
results for rods (Vaz-Romero et al. (2017)) but also for sheets under plain
strain (Rodríguez-Martínez et al. (2017); Zaera et al. (2015); N’souglo et al.
(2020)).

In most linear stability analyses published in the literature, the frozen
coefficient theory is adopted. The underlying assumption is that the time
evolution of the perturbation is much faster than that of the homogeneous so-
lution, also referred to as the background solution or background flow in the
following. This assumption, however, becomes questionable at high strain rates. Therefore, the eXtended Linear Stability Analysis (XLSA) framework was introduced, accounting for the time-dependent coupling between the background and perturbation fields. Recent developments include the multidimensional XLSA for plane-strain sheet extension proposed by Xavier (2019) and the one-dimensional formulation incorporating Bridgman correction factors for rods and plates introduced by Xavier et al. (2021).

A common feature of all 1D models is that the stress multiaxiality in the neck region is accounted for by adopting a correction factor. This factor scales the uniaxial stress to capture the increase in mean axial stress associated with the triaxial state induced by necking. The origin of this correction can be traced back to the seminal work of Bridgman (1952), who derived an analytical expression assuming a circular neck profile, and uniform equivalent strain and stress in the neck section. Alternative formulations were later proposed (Davidenkov & Spiridonova (1947); Siebel & Schwaigere (1948); Chen (1978)). It is also interesting to mention the recent approaches developed by Audoly & Hutchinson (2016) for non-linear elastic materials and, by extension, for rate-independent elastic-plastic materials. They proposed a new methodology to analyze the necking of round bars and sheets under quasi-static conditions and derived an analytical expression for the correction factor. For round bars, the expression proposed by Siebel & Schwaigere (1948) was retrieved by Audoly & Hutchinson (2016). Adopting this model to analyze post necking behavior, Beltramo et al. (2023) combined the experiments
tal neck profile with finite element simulations to identify the elastic-plastic
response of copper. Their study also examined the role of the hardening
exponent on the neck profile. Later, Audoly & Hutchinson (2019) extended
their one-dimensional approach to rate-dependent materials (powerlaw for
the flow stress accounting for strain hardening and strain rate sensitivity).
Nevertheless, as stated by the authors, the necking analysis for this type of
materials was developed under the assumption of slow straining, meaning
that inertial effects were not considered. Most of the contributions in the
literature were dedicated to necking in a round bar, except for Bridgman
(1952) and Audoly & Hutchinson (2019). All correction factors depend on
the dimensionless ratio *a*/R, where *a* and *R* denote the neck radius and cur-
vature radius, respectively. None of these expressions were derived under
dynamic loading conditions.

Since no analytical solution exists even for idealized cases such as perfectly
plastic materials, FE simulations have been widely used to investigate the
development of necking, the associated evolution of triaxiality, and their
relation to the Bridgman factor, mostly under uniaxial tension. The first
simulations were proposed by Needleman (1972). By modeling the neck
development in a round bar, the author showed that if the radii of the bar
and the curvature, together with the axial force are known quantities (as
captured here using FE simulations), the Bridgman (1952) formula is in good
agreement with the results of the full-field calculation in the early stages of
necking, but fails to capture the actual triaxiality in the later stages.
As the Brigdman correction factor was originally derived for perfectly plastic materials, many studies have investigated its efficiency for various constitutive models, including those accounting for strain hardening. Tu et al. (2020) evaluated the relative error in plastic strain associated with different correction factors for three types of plastic behavior: perfectly plastic, linear, and nonlinear hardening. Chen et al. (2019) evaluated the accuracy of the Bridgman (1952); Siebel & Schwaigere (1948); Chen (1978) models based on FE simulations and proposed to introduce a dependence on the hardening exponent. Most of the papers aimed at extending the classical correction factor, mostly based on the ratio *a/R*. However, since this ratio is difficult to measure precisely in experiments, Mirone (2004) proposed an alternative approach and described the correction factor as a polynomial function of the reduced post-necking strain. La Rosa et al. (2003) showed that this mathematical formulation provided good predictions of the post-necking response for a wide range of metallic materials, provided that the necking strain is known a priori. For dynamic conditions, implementing this modeling route may be challenging due to the stabilizing effect of inertia, and therefore, it will not be adopted in our work. Brünig (1998) further highlighted the influence of surface imperfections on the onset of necking, showing that the amplitude of the defect controls the strain at localization. More recently, Kumar et al. (2024) investigated the combined influence of inertia, surface roughness, and porosity on dynamic necking in additively manufactured plates, while El Maï et al. (2014) analyzed similar effects using linear stability theory.
The present work aims to evaluate the accuracy of the predictions of the XLSA compared to those from FE calculations. This comparison will concern both the selection of the dominant mode (as often presented in the literature) but also the time evolution of the free surface, up to unloading. Underlying limitations will also be discussed. It is expected by the authors that the proposed development, validated by comparison with the finite element method, will be considered as a potential alternative solution for predicting the spatial distribution of neck spacing in real configurations.

The document is organized as follows. In a first part, the configuration of interest is presented. Here, a metallic sheet is stretched at high strain rate. The free surface has some surface roughness depicted in the form of a white noise. The predictions of the corresponding FE model and the two XLSA are compared. It will be seen that the 2DXLSA approach of Xavier et al. (2020) reproduces the FE results accurately. In contrast, the 1DXLSA approach of Xavier et al. (2021) fails to predict the profile evolution up to necking. In the second part, unit cell calculations are performed to understand the origin of this discrepancy. Based on the findings, a new proposition for the correction factor is presented. Finally, it is demonstrated that the new 1DXLSA is able to recover the surface profile obtained by the 2DXLSA or FE simulations, up to the occurrence of unloading.

## 2. Dynamic extension of a plate

### 2.1. *Multimodal study*
In this work, we focus on the configuration illustrated in Fig. 1. A plate of initial length $2L_{01} = 2$ m and initial average thickness $h_0^b = 10$ mm is dynamically stretched by applying constant velocities $\pm V_0$ at both ends ($X_1 = \pm L_{01}$), while the lateral surfaces are stress-free. A material point of the plate is identified by its Lagrangian coordinates ($X_1, X_3$) and its Eulerian coordinates ($x_1, x_3$). Note that our selected reference configuration is thicker than usual configurations adopted in the literature (of the order of 2 mm for the cylindrical geometry of Olive et al. (1979), of the order of 1 mm for ring expansion).

The nominal longitudinal strain rate is $\dot{\epsilon}_0 = V_0/L_{01} = 2000$ s$^{-1}$. A plane-strain condition is assumed, implying an infinite width in the transverse direction 2.

The initial plate thickness is not uniform and a symmetric geometric imperfection is introduced in the plate configuration. At time $t = 0$, a perturbation is superimposed on the mean thickness profile. The perturbed thickness, at a longitudinal position $X_1$ varying from $-L_{01}$ to $L_{01}$, is given by:

$$h_0(X_1) = h_0^b - \delta h_0(X_1) = h_0^b - \sum_{p=1}^{p_{\max}} \hat{h}_0 \cos(\gamma_p X_1 + \varphi_p^0) \quad (1)$$

where $\hat{h}_0$ is the initial amplitude of all perturbation mode indices (a white noise is considered). Equation (1) therefore represents a truncated Fourier series up to mode $p_{\max}$. The term $\varphi_p^0, p = 1...p_{\max}$, defines the initial phase as
an independent random variable uniformly distributed in $[0, 2\pi]$. The plate length $2L_{01}$ defines the fundamental period, and the wavenumber of mode $p$ is:

$$ \gamma_p = \frac{p\pi}{L_{01}} \qquad (2) $$

This representation of the initial geometry follows the classical description of random surface profiles proposed by Nayak (1971).

Owing to the symmetry of the plate with respect to the plane $X_3 = 0$ (see Fig. 1), only symmetric modes develop during deformation, see Jouve (2014). The transverse coordinate of the top free surface (representative of half the thickness) is defined as $z(X_1, t)$. At $t = 0$, $z(X_1, t = 0) = \frac{h_0(X_1)}{2}$, $h_0(X_1)$ given by Eq. (1). The current plate thickness is $h(X_1, t) = 2z(X_1, t)$.

<figure><img src="image_2.png" /><figcaption>Figure 1: Schematic of the studied configuration. A plate of initial length $2L_{01}$ and mean thickness $h_0^b$ is stretched at high strain rate by applying constant velocities at both ends. The lateral surfaces are stress-free.</figcaption></figure>

Experimental and numerical studies of dynamic fragmentation showed that, for ductile materials, the neck spacing distribution has a bell shape,
with a dominant wavelength. The minimum neck spacing is typically of the order of the initial bar diameter or plate thickness, depending on the configuration of interest. The parameter $p_{max}$ in Eq. (1) sets the maximum number of modes considered. As discussed in El Maï et al. (2022), $p_{max}$ must be large enough to include all wavelengths that can contribute to the necking pattern. Indeed, an overly small $p_{max}$ would exclude modes with wavelengths comparable to the observable neck spacing, thus compromising the description of the phenomenon. For a plate configuration, a minimum distance between necks of the order of the mean thickness $h_0^b$ leads to the following conditions for $p_{max}$:

$$p_{max} \geq \frac{2L_{01}}{h_0^b} \qquad (3)$$

In our study, considering the adopted initial plate length and thickness, we set $p_{max} = 200$, which satisfies Eq.(3).

It will be shown next that, for this configuration, the dominant mode index $p_{dom}$ is much smaller than $p_{max}$ (indeed $p_{dom} < 100$ for $\dot{\epsilon}_0 = 2000 \text{ s}^{-1}$).

The initial surface roughness of the top profile is evaluated as:

$$R_a^0 = \langle |z(X_1, 0) - \langle z(X_1, 0) \rangle| \rangle = \frac{\hat{h}_0}{4L_{01}} \int_{-L_{01}}^{L_{01}} \left| \sum_{p=1}^{p_{max}} \cos(\gamma_p X_1 + \varphi_p^0) \right| dX_1 \quad (4)$$

where the mean operator $\langle \cdot \rangle = \frac{1}{2L_{01}} \int_{-L_{01}}^{L_{01}} \cdot dX_1$ has been used. For the reference configuration defined previously, the initial value of the surface
$$ \sigma_Y = \sigma_0 (1 + \varepsilon_p^n) \qquad (8) $$

where $\varepsilon_p = \int d_{eq}^p dt$ is the accumulated plastic strain. Material constants $\sigma_0$, $n$ and elastic parameters are listed in Table 1. The material considered here is representative of a generic ductile metal with a large mass density (identical to that of copper) to promote inertial effects. In section 4.2, it will be seen that our approach remains valid for various sets of material parameters ($\sigma_0$ varying in the range (0.250 – 1 GPa), $n$ in the range (0.05 – 0.25) and $\rho$ in the range (2700 – 16000 kg/m³). Such a range of material properties covers a broad diversity of metallic materials (assuming elastic-plastic behavior).

While the theoretical 1D and 2D XLSA formulations neglect elasticity, the finite-element (FE) simulations include it explicitly. Note that elastic effects have been considered in some stability analyses (Jouve (2014, 2015)). However, since LSA does not address unloading phenomena, omitting elasticity does not affect the description of the pre-localization stage.

<table><caption>Table 1: Material parameters adopted for the reference configuration. The flow stress is given by Eq. (8). Elastic properties are required only for finite element analyses.</caption><thead><tr><th>ρ₀ (kg/m³)</th><th>σ₀ (GPa)</th><th>n</th><th>E (GPa)</th><th>ν</th></tr></thead><tbody><tr><td>8930</td><td>1</td><td>0.25</td><td>150</td><td>0.33</td></tr></tbody></table>

The objective in this section is to compare surface profiles predicted by finite element (FE) simulations (used as reference solution), the 2DXLSA model (Xavier et al. (2020)), and the 1DXLSA model (Xavier et al. (2021))
for the above configuration. We impose an initial plastic strain $\epsilon_p^0$ of 0.15, ensuring that at $t = 0$, the Considère criterion is met and the plastic flow is unstable. For the adopted hardening law in Eq. 8, the Considère strain under plane-strain conditions is approximately 0.1.

FE simulations are carried out using ABAQUS/Explicit. Due to the symmetry of the problem illustrated in Fig. 1, only the upper half of the plate is modeled. The mesh contains 7202 elements along the longitudinal direction and 9 elements through the thickness. So a total of 64818 elements is used for all subsequent multimodal simulations. A mesh-sensitivity analysis confirmed that this discretization provides mesh-independent results. Four-node, bilinear, plane-strain elements with reduced integration and hourglass control (CPE4R in ABAQUS) were adopted. In addition, the yield stress expression given in Eq. (8) has been implemented in ABAQUS using the VUHARD user subroutine.

To minimize wave propagation within the specimen before necking localization, it is important to initialize the mechanical fields in the FEM (see Xue et al. (2008); Zaera et al. (2015)). As the surface roughness is limited ($R_a^0 = 8 \ \mu\text{m}$) while $h_0^b = 10 \ \text{mm}$, we applied the velocity and stress fields representative of a perfectly uniform extension at the nominal strain rate $\dot{\epsilon}_0$. Under plane-strain conditions, the background flow has been detailed in previous works (see Fressengeas & Molinari (1994); Shenoy & Freund (1999); Xavier et al. (2020); Kumar et al. (2024) for FEM). The initial velocities are defined as:
$$v_1(X_1, X_3, 0) = X_1 \dot{\epsilon}_0 \qquad v_2(X_1, X_3, 0) = 0 \qquad v_3(X_1, X_3, 0) = -X_3 \dot{\epsilon}_0 \quad (9)$$

In addition, the stress tensor is initialized at all points in the domain with Cauchy stress tensor components given by:

$$\begin{aligned}
\sigma_{11}(X_1, X_3, 0) &= \frac{2\sigma_Y^0}{\sqrt{3}} - P^I(X_3, 0) \\
\sigma_{22}(X_1, X_3, 0) &= \frac{\sigma_Y^0}{\sqrt{3}} - P^I(X_3, 0) \\
\sigma_{33}(X_1, X_3, 0) &= -P^I(X_3, 0)
\end{aligned}
\qquad (10)$$

where $\sigma_Y^0$ is the initial yield stress and $P^I(X_3, 0)$ the initial inertial pressure expressed as:

$$P^I(X_3, 0) = \rho \dot{\epsilon}_0^2 \left( \left( \frac{h_0^b}{2} \right)^2 - X_3^2 \right) \qquad (11)$$

For more details on the derivation of the inertial pressure, the reader may refer to Shenoy & Freund (1999). As a uniform pre-strain $\epsilon_p^0$ is initially imposed, the initial yield stress $\sigma_Y^0$ in Eq. (10) is obtained from Eq. (8) after substituting $\epsilon_p$ by $\epsilon_p^0$: $\sigma_Y^0 = \sigma_0(1 + \epsilon_p^{0n})$.

By imposing the stress field defined in Eqs. (10)–(11), we ensure that the initial stress state nearly satisfies the yield condition, leading to a homogeneous plastic state from the beginning of the FE simulation. Consequently, wave effects are expected to be minimal until the occurrence of unloading.

Although the nominal strain rate ($\dot{\epsilon}_0 = 2000 \text{ s}^{-1}$) is relatively low in com-
parison with other studies conducted in dynamic regimes, the plate thickness is 10 mm in our study whereas the ring diameter or cylinder wall thickness in experiments is typically around 1 – 2 mm. This results in an overall inertia contribution comparable to that found in other studies.

Following the works of Walsh (1984); Zhou & Wierzbicki (1996); Zaera et al. (2015); Rodríguez-Martínez et al. (2017), the influence of inertia during dynamic necking can be quantified by the following dimensionless parameter:

$$I = h_0^b \dot{\epsilon}_0 \sqrt{\frac{\rho}{\sigma_Y^0}} \qquad (12)$$

For the reference case analyzed in this study ($h_0^b = 10$ mm) where material parameters are listed in Table 1 together with a pre-strain of 0.15 and an imposed strain rate $\dot{\epsilon}_0 = 2000 \text{ s}^{-1}$, one obtains $I = 0.047$. Although explosive fragmentation experiments typically involve much higher strain rates ($10^5 \text{ s}^{-1}$) and consequently larger inertial parameters, our values of $I$ remains comparable to those reported for ring and cylinder tests using electromagnetic or mechanical expansion techniques. For instance, in Rodríguez-Martínez et al. (2017), the dimensionless value is $I = 0.08$, whereas for Grady & Olsen (2003), Zhou et al. (2006), it is smaller with $I = 0.018$.

Fig. 2a presents the top free surface profile for selected times in the range [0, 80] µs, obtained by FE simulations. Obviously, the thickness decreases with time due to stretching. In addition, the amplitude of surface fluctuations (peak-to-trough) clearly increases over time. The particular position
$X_1 = -67$ mm is indicated in Fig. 2a by a black arrow. This corresponds to the first place where the thickness remains constant, marking the onset of localization of plastic deformation. Fig. 2b presents the time evolution of the thickness $h = 2z$ at the location $X_1 = -67$ mm up to $t = 110$ µs. The plateau value observed between $t = 65$ µs and $t = 110$ µs further indicates elastic unloading. Accordingly, for a meaningful analysis of the predictions, the comparison between the three modeling routes (FEM, 2DXLSA, 1DXLSA) should be restricted to the time interval preceding the onset of elastic unloading in the FEM simulation, i.e., up to $t = 65$ µs.

The predictive capabilities of the two XLSA methods are next evaluated. For that purpose, the free surface profiles predicted by the 2DXLSA model and the 1DXLSA model are superimposed to the numerical ones on Figs. 3a–3b at $t = 30$ µs, and Figs. 3c–3d at $t = 65$ µs, when the plastic strain rate vanishes for the first time in the FE model, see Fig. 2. Figs. 3a and 3c show that the surface profiles predicted by the 2DXLSA approach are in good agreement with the FE results, in terms of both amplitude and shape. This is the first confirmation in the literature that the extended linear stability approach proposed in Xavier et al. (2020) is accurate in predicting the time evolution of a multimodal perturbation up to the onset of necking. In contrast, Figs. 3b and 3d reveal notable differences between the simulated plate profiles and the ones predicted by the 1DXLSA model. While the overall shape of the profile is preserved, the amplitudes are significantly larger with the 1DXLSA model. This suggests that the 1DXLSA model overestimates
<figure><img src="image_3.png" /><figcaption>Figure 2: Time evolution of (a) the top free-surface profile and (b) the local thickness at $X_1 = -67$ mm from FE simulations. Elastic unloading first occurs at $t = 65$ µs. The nominal strain rate is 2000 s⁻¹. Material parameters are given in Table 1. The initial plate length is $2L_{01} = 2$ m and the mean initial thickness is $h_0^b = 10$ mm. The initial surface perturbation follows Eq. (1) with $\hat{h}_0 = 2$ µm, $p_{max} = 200$.</figcaption></figure>
the growth rate of certain modes compared to the finite element simulations.

To summarize, the predictive capability of 2DXLSA is confirmed during all
time period up to *t* = 65 µs. The free surface profiles captured by FE and
2DXLSA coincide in Fig. 3a at *t* = 30 µs. On the contrary, the discrepancy
already exists with the 1DXLSA predictions at *t* = 30 µs, see Fig. 3b.

<figure><img src="image_4.png" /><figcaption>Figure 3: Evolution of the top free-surface profile between FE and (a,c) the 2DXLSA model of Xavier et al. (2020), and (b,d) the 1DXLSA model of Xavier et al. (2021). The time of analysis is 30 µs in (a,b) and 65 µs in (c,d). The nominal strain rate is 2000 s⁻¹. Material parameters are given in Table 1. The multimodal perturbation is a white noise with pmax = 200 and ĥ₀ = 2 µm.</figcaption></figure>
To better understand these differences, a discrete Fast Fourier Transform (FFT) is applied to the free-surface profiles, such as those shown in Fig. 3. The discrete surface height $z(X_1)$ is sampled at $N$ equally spaced positions $X_n = L_{01}(-1+2n/N)$ with $n \in [0, N-1]$. The corresponding half thickness is $z(X_n)$. The FFT allows to characterize the amplitudes and phases of modes associated with equally spaced frequencies $\gamma_p = \frac{\pi p}{L_{01}}$. Next, we focus only on the amplitude, defined as follows:

$$S(\gamma_p) = \frac{4}{N} \left| \sum_{n=0}^{N-1} z(X_n) e^{-i2\pi pn/N} \right| \quad (13)$$

<figure><img src="image_5.png" /><figcaption>Figure 4: Amplitude spectra of the surface profiles obtained at $t = 65 \mu s$ and shown in Figs. 3c and 3d: comparison between FE, 2DXLSA, and 1DXLSA. Mode indices $p = 30$ and $p = 100$ are highlighted and examined in Section 2.2. The nominal strain rate is $2000 \ s^{-1}$. Material parameters are given in Table 1. The multimodal perturbation is a white noise with $p_{max} = 200$ and $\hat{h}_0 = 2 \mu m$.</figcaption></figure>

Figure 4 compares, for each modeling approach, the mode amplitudes
as a function of the dimensionless wavenumber $\gamma_p h_0^b$, at $t = 65$ µs. All three approaches yield bell-shaped spectra with a maximum reached for an intermediate wavenumber. The dominant mode is consistent across models, with $\gamma_{pdom} h_0^b = 2.23$ for 1DXLSA and 2.01 for 2DXLSA and FE, in agreement with Fig. 3. This dominant mode is typically associated with the mean neck spacing and is known to be well captured by stability analyses (see Fressengeas & Molinari (1985); Shenoy & Freund (1999); Mercier & Molinari (2004); Rodríguez-Martínez et al. (2017)).

As explained in Fressengeas & Molinari (1985, 1994), the selection of an intermediate wavenumber mode results from the interplay between inertia, which damps perturbations with small wavenumber (hence the superposition of the three curves in Fig. 4) and multiaxiality of the mechanical field within the neck which inhibits the development of perturbation with large wavenumber. From Fig. 4, the damping of the amplitude for intermediate to large wavenumber modes is insufficient in the 1DXLSA approach. Figure 4 also demonstrates that the amplitude of perturbations associated with the most unstable modes grows more rapidly when using the 1DXLSA model compared to the finite element simulations. As a consequence, this first multimodal calculation suggests that the discrepancy likely stems from the treatment of multiaxial effects within the 1DXLSA framework. The 2DXLSA, which embeds multiaxiality in its bidimensional formulation, does not suffer from this limitation.

When a precise description of the free surface evolution over time is re-
quired, the current 1DXLSA approach cannot be used to accurately describe
neck development. To address this issue, we next propose to conduct unit cell
calculations to evaluate the true multiaxial stress state which develops within
the neck. This will allow us to examine whether the 1DXLSA approach can
reproduce it, and to propose some extension for a better consideration of
multiaxiality.

## 2.2. Periodic unit cell

This section analyzes the discrepancies between the 1DXLSA and the two
reference routes (2DXLSA and finite element simulations), with emphasis
on the implementation of multiaxial effects. Following Xue et al. (2008),
we perform FE calculations on a periodic unit cell containing a sinusoidal
geometric imperfection.

The constitutive parameters are identical to those used in the previous
section (Table 1). The initial average thickness $h_0^b = 10$ mm is also pre-
served. Various lengths $L_{01}/p$ of the periodic unit cell are selected so that
the wavenumber of the imperfection $\gamma_p = \frac{p\pi}{L_{01}}$ is varied by changing the value
of the mode index $p$. To preserve identical nominal strain rate $\dot{\epsilon}_0$, the velocity
applied at both ends of the unit cell is $\pm V_0/p$.

The initial thickness is not uniform, $h_0(X_1) = h_0^b - \hat{h}_0^{UC} \cos(\gamma_p X_1)$ with
$\hat{h}_0^{UC} = 2 \ \mu\text{m}$. The position of the top surface profile at $t=0$ is $z(X_1, t=0) = h_0(X_1)/2$.

Figure 5 illustrates the adopted configuration. Only one quarter of the
domain is meshed and the following symmetry boundary conditions have been implemented: $v_1(X_1 = 0, X_3, t) = 0$, $v_3(X_1, X_3 = 0, t) = 0$. The initial velocity and stress fields are still given by Eqs. (9) and (10). Note that the whole plate of length $2L_{01}$ is obtained by merging $p$ unit cells using periodicity.

Finite element simulations are performed for perturbation mode indices $p$ varying in the interval $1 \le p \le 200$.

The finite element mesh is designed with a fixed initial thickness-to-length aspect ratio of two to ensure numerical accuracy and avoid excessive element distortions during dynamic extension. To achieve an appropriate discretization, the half thickness is divided into 29 elements, while the number of elements along the half length of the unit cell $L_{01}/p$ is adjusted. As dynamic extension proceeds, the aspect ratio naturally decreases.

Because of the development of the neck, the plastic strain and stress are heterogeneous through the thickness. We propose to follow these quantities at two locations: the peak section ($X_1 = L_{01}/p$), where the thickness is maximum, and the valley section ($X_1 = 0$), where it is minimum, see Fig. 5. In addition to local fields at these two sections, we consider through-thickness averages of the quantities of interest, denoted as $\overline{\varepsilon}_p$ for accumulated plastic strain and $\Sigma_{11}$ for the longitudinal stress, and defined as:

$$ \overline{\varepsilon}_p(X_1, t) = \frac{1}{h_0(X_1)} \int_{-h_0(X_1)/2}^{h_0(X_1)/2} \varepsilon_p(X_1, X_3, t) \, dX_3 \quad (14) \\ \Sigma_{11}(X_1, t) = \frac{1}{h_0(X_1)} \int_{-h_0(X_1)/2}^{h_0(X_1)/2} \sigma_{11}(X_1, X_3, t) \, dX_3 $$
<figure><img src="image_6.png" /><figcaption>Figure 5: Finite element model of 1/4 of a unit cell of length $2L_{01}/p$, average thickness $h_0^b$. The imperfection amplitude, $\hat{h}_0^{UC}$, is enhanced for better visualization. The velocity applied at the end of the unit cell is adjusted depending on the perturbation mode index $p$.</figcaption></figure>

where $X_1 = 0$ (respectively $X_1 = L_{01}/p$) when the valley (respectively peak) section is considered.

We analyze two distinct configurations:

* The first one with the mode index $p = 30$, characterized by a dimensionless wavenumber $\gamma_p h_0^b \simeq 0.94$. In Fig. 4, it was shown that all the three models are able to predict accurately the amplitude of this mode.

* The second one with $p = 100$, corresponding to $\gamma_p h_0^b \simeq 3.14$. It was clear in Fig. 4 that the 1DXLSA approach is unable to accurately predict the time evolution of this perturbation mode.

Figure 6 shows the evolution of selected features for the configuration
with mode index $p = 30$. Figure 6a presents the time evolution of the average plastic strain $\overline{\varepsilon}_p$ in the peak and valley sections of the unit cell, as obtained from finite element simulations. The localization time $t_{\text{loc}}$ is defined as the instant when $\overline{\dot{\varepsilon}}_p$ at the peak section reaches a maximum, i.e. when $\overline{\dot{\varepsilon}}_p < \xi$ with $\xi = 10^{-1} \text{s}^{-1}$. At this moment, localized necking appears and rapidly develops, concentrating plastic strain in the valley region of the unit cell (cf. Xue et al. (2008); Marciniak & Kuczyński (1967)). As in experiments, unloading elastic waves, known as Mott waves, also emerge and propagate within the numerical unit cell. Consequently, plastic strain stops evolving in the peak region after $t_{\text{loc}}$. In our study, with $\xi = 10^{-1} \text{s}^{-1}$, $t_{\text{loc}}$ is identified as 210 µs. Nevertheless, note from Fig. 6 that the two curves for $\overline{\varepsilon}_p$ separates well before localization, as early as $t = 125 \text{ µs}$ (for mode index $p = 30$).

Figure 6b compares the time evolution of the mean plastic strain at the peak section, denoted $\overline{\varepsilon}_p^{\text{peak}}$, for the FE, 1DXLSA, and 2DXLSA approaches. Up to $t \simeq 210 \text{ µs}$, the three predictions are almost indistinguishable. Recall that linear stability theories are not able to model the unloading stage (elasticity is neglected), the XLSA curves therefore decrease after reaching their maximum, which indicates the end of their validity range.

As a consequence, in the following, the validity domain of the two XLSA approaches is limited in time. The upper limit of the validity time domain is estimated as the moment when the mean plastic strain $\overline{\varepsilon}_p$ at peak reaches its maximum. Recall that for mode index $p = 30$, the three models lead to the same localisation time: $t_{\text{loc}} \simeq 210 \text{ µs}$.
Figure 6c highlights the consistency of the perturbation development obtained with the 1DXLSA and 2DXLSA models, by displaying the normalized top free surface $z/\langle z \rangle$ at $t = 200~\mu s \le t_{loc}$. Note that the surface profile is plotted in the Lagrangian coordinate system for $-L_{01} \le X_1 \le L_{01}$, using the symmetry condition at $X_1 = 0$. The free surface profiles are almost superimposed, confirming the validity of the 1D and 2D XLSA predictions, for perturbation mode index $p = 30$.

Finally, Fig. 6d shows the distribution of the accumulated plastic strain, $\epsilon_p$, normalized by the background solution, $\epsilon_p^b$. The results are obtained from finite element simulations at $t = 200~\mu s$. We observe that the accumulated plastic strain remains homogeneous through the thickness direction, confirming the unidirectional characteristic of the flow. This observation explains why the 1DXLSA approach is sufficient to reproduce FEM results for mode index $p = 30$.

We now examine in Fig. 7 the case of a perturbation with mode index $p = 100$. As shown in Fig. 4, noticeable differences arise between the 1D and 2D XLSA predictions for this higher wavenumber mode.

Figure 7a presents the time evolution of the mean plastic strain in the peak and valley sections, as obtained from the FE simulations. At early times, when the perturbation amplitude remains small, the two curves coincide, indicating a quasi-homogeneous response up to $t \simeq 100~\mu s$. Beyond this stage, the predictions of the strain fields progressively diverge. The plastic strain rate in the peak region decreases and eventually vanishes at
<figure><img src="image_7.png" /><figcaption>Figure 6: Results for a perturbation mode with index $p = 30$ ($\gamma_p h_0^b = 0.94$). (a) Time evolution of the mean plastic strain $\overline{\epsilon}_p$ in the peak and valley sections of the unit cell captured by FEM. (b) $\overline{\epsilon}_p$ in the peak section predicted by the 1DXLSA, 2DXLSA and FEM approaches. (c) Normalized top free surface $z/\langle z \rangle$ at $t = 200$ µs. Comparison of predictions from the 1DXLSA, 2DXLSA, and FEM approaches. (d) Map of the normalized plastic strain $\epsilon_p/\epsilon_p^b$ in the unit cell at $t = 200$ µs obtained by FEM. Material parameters are given in Table 1. The geometrical and loading conditions are those of Fig. 5 with $L_{01} = 1$ m, $p = 30$, $h_0^b = 10$ mm, $\hat{h}_0^{UC} = 2$ µm, and $\dot{\epsilon}_0 = 2000$ s$^{-1}$.</figcaption></figure>
$t_{\text{loc}} = 148$ µs, marking the onset of elastic unloading. Simultaneously, the plastic strain in the valley section increases sharply, leading to plastic localization within the neck. Similar evolution was already depicted in Fig. 6a for mode index $p = 30$.

Figure 7b compares the time evolution of the mean plastic strain at the peak section, denoted $\bar{\varepsilon}_p^{\text{peak}}$, for the FE, 1DXLSA, and 2DXLSA approaches. The localization times predicted by 2DXLSA (maximum $\bar{\varepsilon}_p^{\text{peak}}$) and FE (onset of elastic unloading) are consistent with $t_{\text{loc}}(p = 100) = 148$ µs. For the 1DXLSA model, the localization occurs earlier ($t = 110$ µs) as the 1DXLSA model overestimates the growth rate of the perturbation, see Fig. 4.

Although the evolution of $\bar{\varepsilon}_p^{\text{peak}}$ remains comparable for all models up to about $t = 100$ µs as indicated in Fig. 7b, it is interesting to note that the normalized surface profile captured by the 1DXLSA approach already differs. As shown in Fig. 8a, the 1DXLSA overestimates the perturbation amplitude at $t = 80$ µs, and the discrepancy becomes more pronounced at $t = 115$ µs (Fig. 8b). The free surface profile predicted by 2DXLSA still coincides with the FEM result, while the perturbation amplitude is three times larger with the 1DXLSA.

Figure 9a shows the map of the normalized plastic strain field $\varepsilon_p/\varepsilon_p^b$ at $t = 115$ µs, as obtained from FE simulation. Unlike for mode index $p = 30$, heterogeneities in plastic strain $\varepsilon_p$ appear along the thickness direction. Clearly, a 1D modeling may fail to capture this heterogeneity.

It is now useful to clarify how multiaxial effects are accounted for in the
<figure><img src="image_8.png" /><figcaption>Figure 7: Results for a perturbation mode with index $p = 100$ ($\gamma_p h_0^b = 3.14$). (a) Evolution of the mean plastic strain $\bar{\epsilon}_p$ in the peak and valley sections of the unit cell obtained by FEM. (b) Evolution of $\bar{\epsilon}_p$ in the peak section. Comparison of the predictions from the 1DXLSA, 2DXLSA, and FEM approaches. Material parameters are given in Table 1. The geometrical and loading conditions are those of Fig. 5 with $L_{01} = 1$ m, $p = 100$, $h_0^b = 10$ mm, $\hat{h}_0^{UC} = 2$ µm, and $\dot{\epsilon}_0 = 2000$ s$^{-1}$.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 8: Normalized top free surface profile, $z/\langle z \rangle$, at (a) $t = 80$ µs, (b) $t = 115$ µs. The predictions from the three modeling routes are compared. Material parameters are given in Table 1. The geometrical and loading conditions are those of Fig. 5 with $L_{01} = 1$ m, $p = 100$, $h_0^b = 10$ mm, $\hat{h}_0^{UC} = 2$ µm, and $\dot{\epsilon}_0 = 2000$ s$^{-1}$.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Figure 9: Results for a perturbation mode with index $p = 100$ ($\gamma_p h_0^b = 3.14$) (a) Map of the normalized plastic strain $\epsilon_p/\epsilon_p^b$ in the unit cell at $t = 115$ µs obtained by FEM. (b) Normalized longitudinal stress $\sigma_{11}/\Sigma_{11}^b$ across the thickness in the valley section at $t = 115$ µs given by Eq. (15), 2DXLSA and FEM. $\Sigma_{11}^b$ is the longitudinal stress component prevailing during static homogeneous plate extension. Material parameters are given in Table 1. The geometrical and loading conditions are those of Fig. 5 with $L_{01} = 1$ m, $p = 100$, $h_0^b = 10$ mm, $\hat{h}_0^{UC} = 2$ µm, and $\dot{\epsilon}_0 = 2000$ s$^{-1}$.</figcaption></figure>
1DXLSA of Xavier et al. (2021) and recall the main analytical development on which this approach is based. Several models have been proposed in the literature to reduce the multidimensional flow in the neck region to a 1D formalism. For instance, as the neck develops, Bridgman (1952) proposed to apply a correction for the deviation of the stress distribution from uniformity. Assuming that the strain is uniform across the neck and that the stress is the sum of a constant contribution and a hydrostatic term varying along the cross-section, he proposed an explicit expression for the longitudinal stress component $\sigma_{11}(x_3)$ within the neck section:

$$ \sigma_{11}(x_3) = \sum_{11}^{b} \left( 1 + \ln \left( 1 + \phi \left( 1 - \left[ \frac{2x_3}{h} \right]^2 \right) \right) \right) \quad (15) $$

where $x_3$ is the current position within the cross-section and $\phi = \frac{h}{4r}$ is the ratio of the current plate thickness $h$ to the current radius of curvature $r$ all taken at the valley position (i.e. for $X_1 = 0$ and $-\frac{h}{2} \le x_3 \le \frac{h}{2}$, see Fig. 5). Time dependence has been omitted for $\phi, h$ and $\sum_{11}^b$ for sake of clarity. $\sum_{11}^b$ corresponds to the static longitudinal stress component existing in a configuration without any imperfection, at the same time. The use of the static contribution, $\sum_{11}^b$, is valid as the inertial pressure is limited in our approach. As for 1DXLSA, 1D linear stability analysis (see Zhou et al. (2006); Rodríguez-Martínez et al. (2017)) commonly adopts this approach to evaluate the effective stress in the perturbed configuration. Bridgman (1952) mentioned in his book that the neck geometry parameters, $h$ and $r$, could
be measured from experiments. For the present work, these quantities are evaluated at each time of the deformation process based on the FE top free surface profile, as the one illustrated, for instance, for $p = 100$ at $t = 115$ µs in Fig. 8b. Figure 9b compares the normalized stress estimated using Bridgman's approximation (Eq. (15)) to the axial stress $\sigma_{11}$ captured by FEM and the one provided by the 2DXLSA model. The comparison is performed at $t = 115$ µs. It is seen that the stress profile across the full plate thickness obtained in FEM and 2DXLSA approaches almost coincide. In contrast, except in the vicinity of the outer surface (i.e., for $x_3 = \pm h/2$, $\sigma_{11}/\Sigma_{11}^b = 1$, see Eq. (15)), the Bridgman's model overestimates the normalized longitudinal stress in the thickness. Note that for mode index $p = 30$, the longitudinal stress was almost homogeneous in the valley section ($X_1 = 0$) and well captured by all three approaches (results not presented for the sake of brevity).

The finding has important consequences for the 1DXLSA approach which adopts the Bridgman correction factor derived from Eq. (15). Indeed, as presented in Xavier et al. (2021), the mean longitudinal stress in a cross-section, $\Sigma_{11}$, is related to the longitudinal stress $\Sigma_{11}^b$ existing in a homogeneous plate under static conditions through the Bridgman (1952) correction factor $B(\phi)$:

$$ \Sigma_{11} = \Sigma_{11}^b B(\phi) \qquad (16) $$

with $\Sigma_{11}^b = \frac{2}{\sqrt{3}}\sigma_Y$. The term $B(\phi)$ is given by:
$$B(\phi) = \frac{1}{h} \int_{-h/2}^{h/2} \frac{\sigma_{11}(x_3)}{\Sigma_{11}^b} dx_3 = \sqrt{1 + \frac{1}{\phi} \ln(1 + 2\phi + 2\sqrt{\phi(1 + \phi)})} - 1 \quad (17)$$

with $\sigma_{11}(x_3)$ given by Eq. (15).

Calculations on unit cells for two different modes ($p = 30$ and $p = 100$) help identifying a possible reason for the discrepancies observed in the development of a large wavenumber perturbation. It is important to remember that the combination of the 1D LSA with the Bridgman correction factor has proved its efficiency in predicting the mean neck spacing (Zhou et al. (2006); Rodríguez-Martínez et al. (2021)). Nevertheless, predicting the precise time evolution of a sinusoidal perturbation over a wide range of wavenumbers is lacking.

Next, to enhance the relevance of the 1DXLSA model for intermediate to large wavenumber modes, modifications to the treatment of multiaxiality are proposed and analyzed. The aim is to introduce a correction that better captures the stabilizing effect of multiaxiality, thereby reducing the prediction discrepancies between the 1DXLSA approach and both the 2DXLSA model and numerical simulations.

### 3. Improving the multiaxiality correction in the 1DXLSA model

The 1DXLSA approach of Xavier et al. (2021) adopts the formalism proposed by Bridgman (1952) to mimic the stress multiaxiality which develops
within the neck. It appears in the 1DXLSA development of Xavier et al. (2021) that the correction factor is only introduced in the model via the first-order Taylor expansion of $B(\phi)$:

$$B(\phi) = 1 + \frac{2}{3}\phi + o(\phi) \qquad (18)$$

As shown in the previous sections, the 1DXLSA approach is not capable of precisely predicting the free surface evolution for a perturbation with intermediate mode index $p$. The discrepancy is attributed to the difficulty in properly capturing the stress multiaxiality with the adopted correction factor. The idea is therefore to heuristically extend the approach by considering a modified first order approximation:

$$B(\phi) = 1 + K\phi + o(\phi) \qquad (19)$$

where $K$ is a parameter to be defined. With the present proposition, the main equations of the 1DXLSA method are slightly modified and are presented in Appendix A.1.

To first understand the influence of the parameter $K$ on the free surface profile, a parametric study is carried out by varying $K$ in the range 2/3 (value derived from the Bridgman approach) to 1.1. Figure 10 shows the normalized free surface position at the peak section ($X_1 = L_{01}/p$, see Fig. 5) of the unit cell, $\frac{z_{peak}}{\langle z \rangle}$, as a function of the dimensionless wavenumber $\gamma_p h_0^b$, at time $t = 115$ µs. Predictions of the 2DXLSA serve as reference as a
very good agreement has been observed between the 2DXLSA and the FEM approaches. Predictions with the 1DXLSA model based on various values for the parameter *K* are superimposed in Fig. 10. It can be observed that the dominant mode slightly decreases with an increase of the parameter *K*. For small wavenumbers $γ_p h_0^b ≤ 1$, all curves coincide. Indeed, the damping of short-wavenumber perturbations is mostly controlled by inertia (see Fressengeas & Molinari (1985), Mercier & Molinari (2003)). Thus, the adopted value of the parameter *K* has no significant impact in the development of such perturbations. For intermediate to large wavenumbers (here $γ_p h_0^b ∈ [1.5, 4]$), the effect of the *K* value is important. A key observation from this figure is that the plate thickness at the peak section decreases as the parameter *K* increases. Unfortunately, using a single value for all wavenumbers is not suitable to reproduce the 2DXLSA results, as seen in Fig. 10. Indeed, the parameter *K* must be wavenumber-dependent and should also decrease with $γ_p$. Therefore, it is necessary to define an optimal value, $K_{opt}$, that makes the predictions of the free surface by the new 1DXLSA as close as possible to those obtained with the 2DXLSA or, equivalently, with the FEM, throughout the deformation process.
<figure><img src="image_11.png" /><figcaption>Figure 10: Normalized top free surface position at the peak section, $z^{peak}/\langle z \rangle$, at $t = 115$ µs predicted by the 1DXLSA model with different K values (as introduced in Eq. (19)). Comparison with the 2DXLSA model and finite element simulations. Material parameters are given in Table 1. The geometrical and loading conditions are those of Fig. 5 with $L_{01} = 1$ m, varying $p$, $h_0^b = 10$ mm, $\hat{h}_0^{UC} = 2$ µm, and $\dot{\epsilon}_0 = 2000$ s$^{-1}$.</figcaption></figure>

The first question to answer is whether the coefficient K in Eq. (19) is constant or time-dependent for a given configuration. For each perturbation mode characterized by the wavenumber $\gamma_p$, the correction coefficient $K_{\text{opt}}(\gamma_p, t)$ is identified so that the new 1DXLSA reproduces as closely as possible the 2DXLSA response. At each time increment, the value of $K_{\text{opt}}$ is determined by minimizing the instantaneous difference between the amplitude perturbation at the valley section ($X_1 = 0$) predicted by the two approaches. This identification procedure is repeated at successive times, defining the time dependence of $K_{\text{opt}}(\gamma_p, t)$. Note that the reference plate thickness at the valley section could be obtained also from finite element cal-
culations. Since the 2DXLSA and FEM approaches provide the same results, the 2DXLSA model offers more straightforward access to the field of interest. It is therefore adopted here as the reference solution.

Figure 11a presents the variation of $K_{opt}$ over time for mode indices in the range [30, 100]. Modes with an index $p < 30$ are not considered in the present analysis, since it was shown in Fig. 10 that the parameter $K$ has a negligible influence on the surface profile in this range. Figure 11a shows that $K_{opt}$ depends on time. Interestingly, over a quite long period of time, $K_{opt}(\gamma_p, t)$ reaches a plateau, which remains nearly constant up to time at localization, $t_{loc}$. Since the plate thickness perturbation is initially small and due to inertia, strain localization occurs at a late stage of the deformation process. Accordingly, $K_{opt}$ can be defined as:

$$K_{opt}(\gamma_p) = K_{opt}(\gamma_p, t_{loc}) \qquad (20)$$

without inducing significant consequences on the growth of the perturbations. For clarity, Fig. 11a highlights, for each mode, the localization time and the corresponding $K_{opt}$ value using black circles.

Additionally, since the new 1DXLSA approach relies on the first-order expansion relationship (19), it is necessary to ensure that the parameter $\phi = \frac{h}{4r}$ remains small. Figure 11b presents the time evolution of $\phi$ for all $\gamma_p$ values considered in Fig. 11a. As expected, the coefficient $\phi$ increases with time, and is larger at the localization time, when $\gamma_p$ becomes larger.
Nevertheless, the maximum value of $\phi$ remains below 0.01, which confirms the validity of the Taylor expansion in the LSA.

Similar calculations are conducted with a larger initial amplitude $\hat{h}_0^{UC} = 8~\mu\text{m}$ of the unit-cell thickness perturbation. Figure 12a shows the dependence of $K_{\text{opt}}$ from Eq. (20) with respect to the wavenumber (or mode index), for the two values of the initial amplitude $\hat{h}_0^{UC}$. Clearly, the dependence to the wavenumber of $K_{\text{opt}}$ is nearly identical in both cases ($\hat{h}_0^{UC} = 2~\mu\text{m}$ or $\hat{h}_0^{UC} = 8~\mu\text{m}$). This result suggests that for reasonably small initial perturbation amplitudes (as compared to the initial plate thickness $h_0^b$), a fixed value for the coefficient $K_{\text{opt}}$ can be adopted for a given $\gamma_p$. Finally, Fig. 12a shows that, for the considered geometry and material, the $K_{\text{opt}}$ parameter is a linearly decreasing function of the dimensionless wavenumber $\gamma_p h_0^b$, up to $\gamma_p h_0^b = 2.83$ (or $p = 90$). It is noticeable that the transition step occurs at a wavenumber which corresponds to the right-hand side foot of the bell shape in Fig. 10. For larger mode indices, assuming a constant value of $K_{\text{opt}}$ appears reasonable. Based on these calculations, the following expression is proposed for the master curve of $K_{\text{opt}}$:

$$ K_{\text{opt}} = \begin{cases} -0.1317\gamma_p h_0^b + 1.2161 & \text{if } \gamma_p h_0^b \le 2.83 \\ 0.844 & \text{if } \gamma_p h_0^b \ge 2.83 \end{cases} \qquad (21) $$

It is clearly observed that, for all perturbation modes, the required $K_{\text{opt}}$ value is larger than 2/3, corresponding to the one predicted by the first-
<figure><img src="image_12.png" /><figcaption>Figure 11: Time evolution for various perturbations of (a) the parameter $K_{opt}$ (b) the parameter $\phi = \frac{h}{4r}$, evaluated based on the 2DXLSA approach. The black circle on each curve designates the localization time $t_{loc}$. Material parameters are given in Table 1. The geometrical and loading conditions are those of Fig. 5 with $L_{01} = 1$ m, $p$ in the range [30; 100], $h_0^b = 10$ mm, $\hat{h}_0^{UC} = 2$ µm, and $\dot{\varepsilon}_0 = 2000$ s$^{-1}$.</figcaption></figure>
order Taylor expansion of the Bridgman correction factor. As highlighted in Fressengeas & Molinari (1985, 1994), the multiaxial character of the stress state in the neck inhibits the development of high-wavenumber perturbations. Enhancing this multiaxial character may therefore help delaying the onset of unstable modes as suggested by the results of Fig. 10.

The $K_{opt}$ parameter has been calibrated by considering the thickness of the unit cell at the valley section. An alternative identification procedure can be proposed based on the stress profile through the neck cross-section. In this approach, for each wavenumber $\gamma_p$, the parameter $K_{opt}$ is obtained by minimizing the absolute difference between the average longitudinal stress determined using the 2DXLSA model, and the corresponding one obtained from the 1DXLSA model. Figure 12b compares the values of $K_{opt}$ identified from the stress correlation (green triangles), with those obtained from the thickness calibration shown in Fig. 12a. Both methods yield consistent results, with only limited differences. Calculations are performed with $\hat{h}_0^{UC} = 2 \mu m$.

Note that the dynamic extension of the bar configuration is discussed in Appendix A.2, together with the derivation of the $K_{opt}$ parameter for this configuration. It appears that the correction factor does not strongly depend, in this case, on the perturbation mode. Considering $K_{opt} = 1$ for the dynamic extension of a round bar, larger than $K = 1/2$ as deduced from the Bridgman's analysis, is consequently a reasonable assumption.

A first validation of the modification based on Eq. (21) in the 1DXLSA is
carried out by revisiting the results of Fig. 9b. Assuming that $B(\phi) = 1+K\phi$, we propose a heuristic formulation for the stress distribution through the plate thickness as an extension of the formulation proposed by Bridgman (see Eq. (15)). The following expression, similar to Eq. (15) when $K = 2/3$, is proposed:

$$ \sigma_{11}(x_3) = \sum_{11}^{b} \left( 1 + \ln \left( 1 + \frac{3}{2} K \phi \left( 1 - 4 \frac{x_3^2}{h^2} \right) \right) \right) \quad (22) $$

As for Eq. (15), time dependence has been omitted for $\phi, h$ and $\sum_{11}^{b}$ for the sake of clarity. In addition, with this expression, the new correction factor becomes:

$$ B(\phi) = \sqrt{1 + \frac{2}{3K\phi}} \ln \left( 1 + 3K\phi + 2\sqrt{\frac{3}{2}K\phi\left(1 + \frac{3}{2}K\phi\right)} \right) - 1 \quad (23) $$

For the mode index $p = 30$, the stress distribution in the plate thickness is quite homogeneous as $\phi$ is very small. So $\sigma_{11} \simeq \sum_{11}^{b}$. For mode index $p = 100$, the original Bridgman expression Eq. (15) has proved to overestimate the longitudinal stress in the valley region (neck), see Fig. 9b. With the proposed modification, it is seen in Fig. 13a that Eq. (22) yields results consistent with those captured with the 2DXLSA or FE models. Indeed, the axial stress gradient along the thickness is reproduced more accurately. Recall that Eq. (22) considers $\sigma_{11}(x_3 = \pm h/2)/\sum_{11}^{b} = 1$ at the free surfaces of the plate. The assumption of the Bridgman model is not verified by the finite element results. One can notice nevertheless an underestimation of the
<figure><img src="image_13.png" /><figcaption>Figure 12: $K_{opt}$ values derived for use in the new 1DXLSA model as a function of the dimensionless wavenumber a) Identification based on the plate thickness at the neck. As the $K_{opt}$ parameter is quite insensitive to the defect amplitude $\hat{h}_0^{UC}$, a master curve can be defined (see solid black line). The default value $K = 2/3$ leads to an underestimation of the multiaxial stress effect in the neck section. b) Identification based on the average longitudinal stress in the neck region. The solid line (Eq. (21)), identified from the thickness correlation method, provides an accurate representation of the correction factor for both identification approaches. The geometrical and loading conditions are those of Fig. 5 with $L_{01} = 1$ m, $p$ in the range [30; 100], $h_0^b = 10$ mm, $\hat{h}_0^{UC} = 2$ µm, and $\dot{\epsilon}_0 = 2000$ s$^{-1}$.</figcaption></figure>
longitudinal stress.

While the present approach remains empirical in nature and relies on a calibration, a valuable next step would be to revisit the analytical derivation originally proposed by Bridgman (1952) and to establish a theoretical basis for the expression (22). This aspect could be addressed in future work.

Figure 13b displays the time evolution of the mean plastic strain $\overline{\varepsilon}_p^{\text{peak}}$ in the peak section. It is observed that the predictions of the new 1DXLSA model are now aligned with those of the finite element simulations up to $t_{\text{loc}}$, confirming the relevance of the improvement.

Next, the new 1DXLSA capability is tested by tracking the perturbation development up to $t_{\text{loc}}$ for mode indices in the range [25, 160]. Figure 14 shows the localization time $t_{\text{loc}}$ as a function of the dimensionless wavenumber of the perturbation evaluated from the three approaches: FEM, 1DXLSA of Xavier et al. (2021), new 1DXLSA. Note that 2DXLSA predicts similar results as FEM. It was previously observed that the 1DXLSA model tends to predict earlier localization when the studied mode was associated with a dimensionless wavenumber $\gamma_p h_0^b > 1.0$. With the new 1DXLSA model, $t_{\text{loc}}$ times are now faithfully reproduced for all considered perturbation modes, ensuring better consistency with predictions of reference models (2DXLSA or FEM).

Figure 15 displays both the normalized position of the top free surface at peak $z^{\text{peak}}/\langle z \rangle$ and the corresponding normalized average longitudinal stress $\Sigma_{11}/\Sigma_{11}^b$ in the valley of the unit cell, evaluated at $t = 115$ µs for the differ-
<figure><img src="image_14.png" /><figcaption>Figure 13: Assessment of the new approach for the perturbation mode index $p = 100$ ($\gamma_p h_0^b = 3.14$) in terms of: (a) Normalized longitudinal stress $\sigma_{11}/\Sigma_{11}^b$ at $t = 115$ µs within the thickness at the valley ($X_1 = 0$) of the unit cell. The curve obtained from Eq. (22) is plotted for $K = 0.844$ (b) Evolution of mean plastic strain at peak $\bar{\varepsilon}_p^{peak}$. Comparison of predictions from the 1DXLSA, new 1DXLSA, and FE results. Material parameters are given in Table 1. The geometrical and loading conditions are those of Fig. 5 with $L_{01} = 1$ m, $p = 100$, $h_0^b = 10$ mm, $\hat{h}_0^{UC} = 2$ µm, and $\dot{\varepsilon}_0 = 2000$ s$^{-1}$.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Figure 14: Localization time $t_{\text{loc}}$ as a function of dimensionless wavenumber. The predicted localization time $t_{\text{loc}}$ with the new 1DXLSA model aligns with the computed ones given by 2DXLSA or FEM. Material parameters are given in Table 1. The geometrical and loading conditions are those of Fig. 5 with $L_{01} = 1$ m, $p$ in the range [30; 200], $h_0^b = 10$ mm, $\hat{h}_0^{\text{UC}} = 2$ µm, and $\dot{\varepsilon}_0 = 2000$ s$^{-1}$.</figcaption></figure>
ent models with respect to the normalized wavenumber $\gamma_p h_0^b$. A significant improvement is noted with the new 1DXLSA. Predictions now closely match those of the 2DXLSA model and finite element simulations, thus validating the benefit of the proposed correction.

The new 1DXLSA model has been developed in a monomodal context using unit cell calculations enabling the calibration of the optimal correction parameter $K_{opt}$ for each perturbation mode. The next step is to assess its relevance in the multimodal context presented in Section 2.1.

# 4. Validation and parametric study of the new 1DXLSA model in a multimodal configuration

This section examines the multimodal configuration of the plate ($2L_0 = 2$ m, $h_0^0 = 10$ mm) introduced in Section 2.1 to validate the predictions of the new 1DXLSA model, with the initial surface profile defined by Eq. 1. The analysis is first carried out for the reference material and strain rate, while varying the initial surface roughness. In a second step, a parametric study is conducted to analyze how material parameters and loading rate affect the model predictions. The effect of the pre-strain parameter is also discussed.

## 4.1. Validation of the model and influence of the initial surface roughness

The material parameters are still presented in Table 1, and the simulations are conducted at a reference nominal strain rate $\dot{\epsilon}_0 = 2000 \text{ s}^{-1}$.

Figure 16a compares the top free surface predicted by finite element simulations and by the new 1DXLSA model at $t = 65 \mu\text{s}$, for the surface roughness
<figure><img src="image_16.png" /><figcaption>Figure 15: Evaluation of the predictive capability of the proposed approach. (a) Normalized position of the top surface $z^{peak}/\langle z \rangle$ at peak section versus dimensionless wavenumber $\gamma_p h_0^b$ and (b) Normalized mean longitudinal stress $\Sigma_{11}/\Sigma_{11}^b$ at valley section for $t = 115$ µs versus dimensionless wavenumber $\gamma_p h_0^b$. Comparison of 1DXLSA, new 1DXLSA, and FEM (or 2DXLSA). Material parameters are given in Table 1. The geometrical and loading conditions are those of Fig. 5 with $L_{01} = 1$ m, varying $p$, $h_0^b = 10$ mm, $\hat{h}_0^{UC} = 2$ µm, and $\dot{\epsilon}_0 = 2000$ s$^{-1}$.</figcaption></figure>
defined by $\hat{h}_0 = 2~\mu\text{m}$, $p_{\text{max}} = 200$. The two profiles are nearly superimposed, demonstrating the excellent consistency between both approaches. This agreement is further supported by the amplitude spectra presented in Fig. 16b, which compares the results obtained from FEM, the original 1DXLSA model, and the new 1DXLSA model. The improvement achieved with the present model is particularly noticeable for perturbations with dimensionless wavenumber $\gamma_p h_0^b > 1.0$, as the new model shows much better agreement with numerical simulations.

The close agreement is not limited to the late stage of the deformation process, just before localization occurs. Figure 17 compares the surface profiles predicted by the 2DXLSA, 1DXLSA, and the new 1DXLSA models at different times ($t = 15, 30, 45$ and $60~\mu\text{s}$) prior to the first unloading, which occurs at $65~\mu\text{s}$ in the FEM calculation. FEM results are not shown, as they almost coincide with those obtained from the 2DXLSA model. It clearly appears that the new 1DXLSA model provides a much better agreement with the full two-dimensional approach at all selected times than the original 1DXLSA of Xavier et al. (2021). Nevertheless, the quantitative agreement is slightly better at later times than at the early stages of the dynamic extension. As shown in Fig. 11a, $K_{\text{opt}}$ depends on time and was identified at the onset of localization (i.e. for relatively large time, see Eq. (20)). For early times, the value of the correction factor should therefore be larger than $K_{\text{opt}}$, which explains the small discrepancies observed between the 2DXLSA and the new 1DXLSA models at early times. More importantly, Fig. 17
<figure><img src="image_17.png" /><figcaption>Figure 16: Validation of the new 1DXLSA based on a white noise perturbation of the free surface. (a) Top free surface of the plate and (b) amplitude spectra of the surface profile. Predictions, based on the new 1DXLSA, 1DXLSA, 2DXLSA and FEM, are shown at $t = 65$ µs. The nominal strain rate is 2000 s⁻¹. Material parameters are given in Table 1. The multimodal perturbation is a white noise with $p_{max} = 200$ and $\hat{h}_0 = 2$ µm.</figcaption></figure>
shows that discrepancies with the classical 1DXLSA model arise well before
unloading begins.

<figure><img src="image_18.png" /><figcaption>Figure 17: Comparison of the top free surface profiles, at different times, predicted by the 2DXLSA, 1DXLSA, and new 1DXLSA models. The nominal strain rate is 2000 s⁻¹. Material parameters are given in Table 1. The multimodal perturbation is a white noise with pmax = 200 and ĥ₀ = 2 µm.</figcaption></figure>

Further validation is pursued by considering additional configurations,
varying the amplitude $\hat{h}_0$ and the maximum mode index $p_{max}$. Figure 18a
presents the top free surface predicted by the FEM and by the new 1DXLSA
model at $t = 73$ µs, obtained with $p_{max} = 100$ and $\hat{h}_0 = 2$ µm. Beyond
this time, unloading waves propagate in the FE simulation, making further
comparison with XLSA unreliable. At $t = 73$ µs, predictions obtained with
the new 1DXLSA and the FEM are similar. The amplitude spectra derived
from the free surface profiles of Fig. 18a are shown in Fig. 18b. Excellent
agreement is observed in terms of amplitude between the new 1DXLSA model
and the FEM, confirming the robustness of the proposed model within a
multimodal framework.

The comparison is extended in Figs. 18c and 18d, obtained for $p_{max} = 100$, and a larger initial perturbation amplitude $\hat{h}_{0}^{UC} = 8$ µm, at the localization time $t_{loc} = 37.5$ µs. Once again, an excellent agreement is observed between the predictions of the new 1DXLSA model and those from the FEM.

These results demonstrate that identifying $K_{opt}$ from unit cell calculations
is sufficient for modeling the response of a plate with multimodal perturba-
tions under dynamic extension. Further validations of this procedure for
other materials (different constitutive responses) and configurations (geom-
etry and loading rate) could be considered in future work. To pave the way
for an extended use of the proposed modeling, the following section presents
an analysis of the sensitivity of the model predictions to material parame-
ters and loading rates, using $K_{opt}$ identified from the reference configuration
which led to Eq. (21).
<figure><img src="image_19.png" /><figcaption>Figure 18: Multimodal perturbation during the dynamic extension of a plate. Predictions are given by the new 1DXLSA, 2DXLSA, and FEM. (a,c) Free surface profiles. (b,d) Corresponding amplitude spectra. For (a,b) $p_{max} = 100$, $\hat{h}_0 = 2$ µm and $t = 73$ µs. For (c,d) $p_{max} = 100$, $\hat{h}_0 = 8$ µm and $t = 37.5$ µs. The nominal stress rate is 2000 s⁻¹. Material and loading conditions are those used in Fig. 5. For clarity, the surface profile predicted by the original 1DXLSA is not presented. Only the corresponding amplitude spectrum is depicted in (b,d).</figcaption></figure>
## 4.2. Influence of material parameters, strain rate, and pre-strain on the model predictions

### 4.2.1. Effect of material parameters

This subsection examines, under the same nominal strain rate $\dot{\varepsilon}_0 = 2000 \text{ s}^{-1}$, the influence of material parameters, namely the reference stress $\sigma_0$, the strain-hardening parameter $n$, and the mass density $\rho$, on the new 1DXLSA predictions.

First, the stress parameter $\sigma_0$ (which takes a large value in the reference configuration) is reduced by a factor of four. Note that the initial yield stress, $\sigma_Y^0 = \sigma_0 (1 + (\varepsilon_p^0)^n)$, thus is reduced by the same factor, while the inertial parameter $I$ is doubled, see Eq. (12). The comparison between the different models is presented in Fig. 19a for the top free surface and the corresponding amplitude spectrum in Fig. 19b at $t = 128 \mu\text{s}$, i.e. at the onset of localization. Localization occurs later than for the reference material, as the reduced yield stress enhances inertial effects, which in turn delays the development of instabilities. Only small discrepancies are observed and the new 1DXLSA model reproduces the 2D FEM results with good accuracy.

A similar, however slightly more pronounced, discrepancy is observed in Figs. 19c and 19d, when the strain hardening parameter is reduced from $n = 0.25$ to 0.1. In this case, the inertial parameter is only slightly modified, decreasing from $I = 0.047$ to 0.044.

The effect of the mass density is now illustrated in Fig. 20 for $\rho = 2700 \text{ kg/m}^3$ (mass density of aluminum) and $\rho = 16000 \text{ kg/m}^3$ (mass den-
<figure><img src="image_20.png" /><figcaption>Figure 19: Predictive capabilities of the new 1DXLSA model, using $K_{opt}$ definition given by Eq. (21). The constitutive parameters ($\sigma_0$ and $n$) are varied, with all other parameters taken from Table 1. (a-c) Top free surface profiles and (b-d) corresponding amplitude spectra. Results are shown for the new 1DXLSA, the original 1DXLSA and reference models. In all cases, the nominal strain rate is 2000 s$^{-1}$, and the comparison is conducted at the onset of localization. The multimodal perturbation is a white noise with $p_{max}$ = 200 and $\hat{h}_0$ = 2 $\mu$m. As for Fig. 18, only the surface profiles based on FEM and new 1DXLSA are presented, while amplitude spectra are depicted for all modeling routes.</figcaption></figure>
sity of tantalum). These two cases lead to values of the inertial parameter $I = 0.026$ and $I = 0.063$, respectively, as compared to $I = 0.047$ for the reference case.

For the low-density case ($\rho = 2700 \text{ kg/m}^3$), a smaller perturbation amplitude ($\hat{h}_0 = 0.1 \text{ µm}$) was adopted in order to delay the localization time, which occurs very rapidly when using the reference amplitude. Since the influence of the initial perturbation amplitude was found to be negligible (see Section 4.1), this modification does not affect the validity of the comparison. Consequently, in Fig. 20b, the localization time ($t_{\text{loc}} = 73 \text{ µs}$) appears longer than that obtained for the reference case ($\rho = 8930 \text{ kg/m}^3$, see Fig. 16b). By contrast, in Fig. 20d ($t_{\text{loc}} = 90 \text{ µs}$), which considers the same initial perturbation amplitude, increasing the mass density amplifies inertial effects, which in turn reduce the growth rate of the perturbations associated with each wavenumber in the multimodal surface profile. In both cases (lower and higher inertia regimes), the new 1DXLSA model, using $K_{\text{opt}}$ given by Eq. (21), reproduces the 2D reference calculation with very good agreement, although a small discrepancy is observed at low mass density. Again, the original 1DXLSA model, although accurately predicting the dominant mode, fails to properly capture the amplitude of the perturbation growth observed in the full 2D model.

### 4.2.2. Influence of the strain rate

In this section, the influence of the strain rate on the model predic-
<figure><img src="image_21.png" /><figcaption>Figure 20: Predictive capabilities of the new 1DXLSA model, using $K_{opt}$ given by Eq. (21). The mass density $\rho$ is varied, with all other parameters taken from Table 1. (a-c) Top free surface profiles and (b-d) corresponding amplitude spectra. Results are shown for the new 1DXLSA, the original 1DXLSA and reference models. In all cases, the nominal strain rate is $2000 \text{ s}^{-1}$, and the comparison is conducted at the onset of localization. The multimodal perturbation is a white noise with $p_{max} = 200$ and $\hat{h}_0 = 0.1 \text{ µm}$ for (a,b) and $2 \text{ µm}$ for (c,d). As for Fig. 18, only the surface profiles based on FEM and new 1DXLSA are presented, while amplitude spectra are depicted for all modeling routes.</figcaption></figure>
tions is examined, considering the reference material parameters listed in Table 1 and an initial plastic strain of $\epsilon_p^0 = 0.15$. Three nominal strain rates are considered: $\dot{\epsilon}_0 = 1000 \text{ s}^{-1}$ (half the reference strain rate), $6000 \text{ s}^{-1}$, and $10000 \text{ s}^{-1}$, corresponding to inertial parameter values ranging from 0.024 up to 0.23. Although explosive fragmentation experiments typically involve much higher strain rates ($> 10^5 \text{ s}^{-1}$) and consequently larger inertial parameters, the present values of $I$ remain comparable, and even exceed for $\dot{\epsilon}_0 = 10000 \text{ s}^{-1}$ those calculated for typical ring and cylinder expansion tests using electromagnetic or mechanical loading techniques. Indeed, as already mentioned, for dynamic ring expansions using magnetic techniques, a value of $I$ of 0.02 can be inferred from the experiments of Grady & Olsen (2003) on U-6Nb alloy at $10000 \text{ s}^{-1}$. For impact-driven ring expansions, $I \simeq 0.04$ in the work of Gant et al. (2024) on copper alloys, and values up to $I \simeq 0.07$ can be estimated from the experiments of Nieto-Fuentes et al. (2023b) on porous AlSi10Mg at strain rates approaching $15000 \text{ s}^{-1}$. Note that impact-driven tests on cylinders allowed to reach values of $I$ close to 0.2 at strain rates around $23000 \text{ s}^{-1}$.

Figure 21 presents the comparative results obtained for the three strain rates. The predictive capability of the new 1DXLSA model is preserved when the strain rate is reduced, showing an excellent agreement with the reference 2D result (FEM or 2DXLSA) for both the top surface profile (Fig. 21a) and the corresponding amplitude spectrum (Fig. 21b). Although the agreement remains satisfactory for $\dot{\epsilon}_0 = 6000 \text{ s}^{-1}$, (Figs. 21c and 21d) and even
for 10000 s⁻¹ (Figs. 21e and 21f), it is gradually deteriorating as the strain
rate increases. The main discrepancies appear for short wavelengths (large
wavenumbers). As revealed in all previous analyses, the original 1DXLSA
model deviates further from the reference 2D result (FEM or 2DXLSA).

4.2.3. *Influence of the pre-strain*

In all previous calculations, a large initial plastic strain $\epsilon_p^0 = 0.15$ was
adopted to ensure that, at $t = 0$, the Considère criterion is satisfied and the
plastic flow is initially unstable.

One of the main outcomes of the extended approaches introduced by
Xavier et al. (2020) for 2D configurations and derived in Xavier et al. (2021)
for 1D configurations is that the evolution of initial perturbations can be
analyzed during the early stage of deformation, i.e., before the Considère
criterion is met. This stage is usually disregarded in classical linear stability
analyses, since they do not provide positive perturbation growth rates prior
to the Considère strain.

The case of an initially stable flow is therefore analyzed here by con-
sidering a nominal strain rate of $\dot{\epsilon}_0 = 2000 \text{ s}^{-1}$ and the reference material
parameters from Table 1, with a vanishing initial plastic strain ($\epsilon_p^0 = 0$).
Comparison of the predictions obtained from the FEM, 2DXLSA, 1DXLSA,
and new 1DXLSA models (using $K_{\text{opt}}$ from Eq. (21)) are shown in Fig. 22.

Apart from the spurious oscillations observed in the FEM results, an
excellent agreement is found between the 2DXLSA approach and the FEM
<figure><img src="image_22.png" /><figcaption>**Figure 21:** Influence of the nominal strain rate on the predictive capabilities of the new 1DXLSA model, using $K_{opt}$ given by Eq. (21). Materials parameters are given in Tab. 1. (a-c-e) Top free surfaces and (b-d-f) corresponding amplitude spectra for $\dot{\epsilon}_0 = 1000 \text{ s}^{-1}$ (a-b), $6000 \text{ s}^{-1}$ (c-d), and $10000 \text{ s}^{-1}$ (e-f). Results are shown for the new 1DXLSA, the original 1DXLSA and reference models. In all cases, the comparison is conducted at the onset of localization. The multimodal perturbation is a white noise with $p_{max} = 200$ and $\hat{h}_0 = 2 \mu\text{m}$.</figcaption></figure>
calculations, confirming the ability of the extended approach (which does
not rely on the frozen-coefficient approximation) to describe the evolution
of perturbations from an initially stable flow. Moreover, the new 1DXLSA
model also reproduces the reference results (FEM or equivalently 2DXLSA)
with close agreement, both in terms of surface profile (Fig. 22a) and corre-
sponding amplitude spectrum (Fig. 22b). The original 1DXLSA model still
overestimates the growth rate, see Fig. 22b.

<figure><img src="image_23.png" /><figcaption>Figure 22: Comparison between FEM, 2DXLSA, new 1DXLSA, and 1DXLSA of (a) the top free surface profiles and (b) the corresponding amplitude spectra at the time of localization. The flow is initially stable ($\epsilon_p^0 = 0$). The nominal strain rate is 2000 s$^{-1}$. Material parameters are given in Table 1. The multimodal perturbation is a white noise with $p_{max} = 200$ and $\hat{h}_0 = 2$ µm.</figcaption></figure>

From the above analysis, it can be concluded that the new 1DXLSA model
reproduces the 2D reference calculations with good accuracy, with deviations
that remain moderate. In contrast, the large discrepancies observed with
the original 1DXLSA model of Xavier et al. (2021) confirm that the stress
heterogeneity within the neck region cannot be properly captured without
including a correction factor into the original 1DXLSA model.

Results for the bar configuration are reported in Appendix A.2 for the
reference case only. The detailed parametric influence for this configuration
will be the subject of a forthcoming work.

The $K_{\text{opt}}$ parameter may require further adjustment to better account
for the influence of the strain rate and material parameters of the adopted
constitutive law, as well as other factors yet to be explored, such as strain-
rate and thermal dependencies, which will be the subject of future work.

**5. Conclusion**

This work provides a framework for modeling multiple necking during
the dynamic expansion of metallic cylindrical shells (or plates) and rings
(or bars), through a modification of the 1DXLSA model of Xavier et al.
(2021). By benchmarking its predictions against finite element simulations
and the more comprehensive 2DXLSA model, we have identified limitations
of the original 1D approach of Xavier et al. (2021). While the 1DXLSA model
effectively captures the dominant wavelength, it systematically overestimates
the rate of development for large-wavenumber perturbations. This is due to
limitations in the estimation of the correction factor which is introduced to
capture the stress multiaxiality within the neck region.

The 2DXLSA model, by contrast, shows excellent agreement with numer-
ical simulations across all investigated configurations, confirming its robust-
ness and justifying its use as a reference model for correcting the 1DXLSA
approach while retaining its computational efficiency.

To improve the accuracy of the 1DXLSA, the first-order Taylor expansion of the Bridgman (1952) correction factor is replaced by a similar first-order expansion containing an adjustable coefficient *K*.

From FE unit cell calculations, it is shown that, for metallic plates, the optimal value of the coefficient *K* depends on the perturbation wavenumber and evolves during the deformation process. A mathematical expression deduced from FEM is proposed in the present work. By incorporating this modification in the 1DXLSA framework, the predictive capability of the 1D model is improved and compares well to both reference predictions (FEM or, equivalently, 2DXLSA).

The proposed correction methodology was thoroughly validated by considering both unit-cell (monomodal) and multimodal configurations for plates under plane strain extension. It was also validated for the dynamic extension of round bars, as demonstrated in the Appendix A.2. These results highlight the relevance of the new 1DXLSA model for capturing dynamic necking.

Furthermore, the influence of the initial perturbation amplitude on the value of the adjustable parameter *K* is found to be minimal, assuming that the amplitude remains within a reasonable range close to that used for calibration. The parametric analysis performed in this study further demonstrates that the optimal coefficient $K_{opt}$ exhibits a moderate sensitivity to the material parameters of the considered constitutive model and strain rates. However, the results suggest that additional refinement of $K_{opt}$ could further
Acknowledgment

This research is supported by the LRC FRAGDYN, a research laboratory under agreement between CEA DAM, ISAE-SUPERO, the Université de Lorraine, the École des Mines de Saint-Etienne and the CNRS.

Appendix A. Governing equations of the new 1DXLSA model

Appendix A.1. *Dynamic stretching of a plate under plane strain assumption*

This appendix outlines the modifications introduced to the 1DXLSA model of Xavier et al. (2021) in order to improve the representation of multiaxial stress effects during dynamic extension. These changes are based on a revised first-order Taylor expansion of the Bridgman correction factor, incorporating an adjustable coefficient K, see Eq. (19).

The derivation follows exactly the framework of the original 1DXLSA model, as presented in detail in Xavier et al. (2021). The present section highlights only the modifications introduced in replacing Eq. (18) by Eq. (19).

The true stress $\sigma$ (longitudinal component $\sigma_{11}$ in fact) is replaced by an average axial stress across the plate thickness defined as $\sigma = \frac{2\sigma_y}{\sqrt{3}}B(\phi)$. Accordingly, with Eq. (19), the perturbed stress field $\delta\sigma$ becomes:

$$ \delta\sigma = \frac{2}{\sqrt{3}}\delta\sigma_Y + \frac{2}{\sqrt{3}}K\sigma_Y^b\delta\phi \quad (A.1) $$

This is the only modification in the set of Eq. (15) of Xavier et al. (2021). The system governing the perturbation amplitudes of the plastic strain $\dot{\epsilon}_p$
and the temperature $\hat{T}$ remains otherwise unchanged:

$$ 
\begin{aligned}
a_1 \hat{\varepsilon}_p + a_2 \hat{\varepsilon}_p + a_3 \hat{\varepsilon}_p + a_4 \hat{T} &= s_0 \\
\hat{T} &= b_1 \hat{T} + b_2 \hat{\varepsilon}_p + b_3 \hat{\varepsilon}_p
\end{aligned} 
\qquad (A.2) $$

except for the coefficient $a_3$:

$$ a_3 = \frac{2\gamma^2}{\sqrt{3}(1 + \dot{\varepsilon}_0 t)} \sigma_{Y, \varepsilon_p}^b + \frac{(h_0^b)^2 \sigma_Y^b K}{8(1 + \dot{\varepsilon}_0 t)^5} \gamma^4 - \frac{\gamma^2 \sigma_Y^b}{1 + \dot{\varepsilon}_0 t} \qquad (A.3) $$

and the source term $s_0$:

$$ s_0 = \left( \frac{\hat{h}_0}{h_0^b} + \frac{\sqrt{3}}{2} \hat{\varepsilon}_p^0 \right) \left[ \frac{(h_0^b)^2 \gamma^4 \sigma_Y^b K}{4\sqrt{3}(1 + \dot{\varepsilon}_0 t)^5} - \frac{2\sigma_Y^b \gamma^2}{\sqrt{3}(1 + \dot{\varepsilon}_0 t)} \right] \qquad (A.4) $$

Expressions for $a_1, a_2, a_4, b_1, b_2, b_3$ can be found in Xavier et al. (2021). The results presented in the whole set of figures 3 to 22 are obtained as solutions of Eqs. (A.2).

## Appendix A.2. Dynamic extension of a round bar

The goal is to extend the 1DXLSA analysis proposed in Xavier et al. (2021) for round bar (of initial length $2L_{01}$, initial radius $R_0^b$ and initial area $A_0^b$), in a similar manner as proposed in this work for a plate.

The first order Taylor expansion of the Bridgman correction factor ($B(\phi) = 1 + \phi/2$) is replaced by the following expression: $B(\phi) = 1 + K\phi$ with $K$ being an adjustable parameter.

Following the Bridgman approach, the stress within the round bar cross-section at neck ($X_1 = 0$) becomes:
$$ \sigma_{11}(r) = \Sigma_{11}^{b} \left[ 1 + \ln \left( 1 + 2K\phi \left( 1 - \frac{r^2}{r_0^2} \right) \right) \right] \quad (A.5) $$

where $r_0$ is the current radius in the neck section. $\Sigma_{11}^b$ still represents the static longitudinal stress in a homogeneous round bar subject to uniaxial tension at a nominal strain rate $\dot{\epsilon}_0$.

The modified correction factor becomes:

$$ B(\phi) = \left(1 + \frac{1}{2K\phi}\right) \ln(1 + 2K\phi) \quad (A.6) $$

As for the plate configuration, the only change concerns the evaluation of the perturbed longitudinal stress field:

$$ \delta\sigma_{11} = \delta\sigma_Y + K\sigma_Y^b\delta\phi \quad (A.7) $$

Following the same derivation as in Xavier et al. (2021) the coupled system for $\hat{\epsilon}_p$ and $\hat{T}$ takes the form:

$$ \begin{aligned} c_1 \ddot{\hat{\epsilon}}_p + c_2 \dot{\hat{\epsilon}}_p + c_3 \hat{\epsilon}_p + c_4 \hat{T} &= f_0 \\ \dot{\hat{T}} &= d_1 \hat{T} + d_2 \dot{\hat{\epsilon}}_p + d_3 \hat{\epsilon}_p \end{aligned} \quad (A.8) $$

The modified coefficient $c_3$ is:

$$ c_3 = \frac{\sigma_Y^b A_0^b \gamma^4 K}{4\pi(1 + \dot{\epsilon}_0 t)^4} + \frac{\sigma_{Y,\epsilon_p}^b \gamma^2}{1 + \dot{\epsilon}_0 t} - \frac{\sigma_Y^b \gamma^2}{1 + \dot{\epsilon}_0 t} \quad (A.9) $$

and the source term $f_0$ becomes:
$$f_0 = - \left( \hat{\varepsilon}_p^0 + \frac{\hat{A}_0}{A_0^b} \right) \left[ \frac{\gamma^2 \sigma_Y^b}{1 + \dot{\varepsilon}_0 t} - \frac{A_0^b \gamma^4 \sigma_Y^b K}{4\pi (1 + \dot{\varepsilon}_0 t)^4} \right] \quad (A.10)$$

Expressions for $c_1, c_2, c_4, d_1, d_2, d_3$ can be found in Xavier et al. (2021). Next, the determination of the adjustable parameter $K$ is performed based on unit cell calculations, following the methodology proposed by Xue et al. (2008) and as presented previously for a plate configuration. Note that the multidimensional XLSA approach for the round bar extension has not been developed. So the reference calculation will be captured using only FEM. The material parameters are those of Section 2.2.

For the present case, the average initial diameter of the bar is fixed: $D_0^b = 2R_0^b = 10$ mm. The initial length of the round bar is set to $2L_{01} = 2$ m, and the wavelength of the imperfection is varied by selecting unit cells of length $2L_{01}/p$, where $p \in \mathbb{N}$. The associated wavenumber is defined as $\gamma_p = \frac{p\pi}{L_{01}}$.

A constant nominal strain rate $\dot{\varepsilon}_0$ is applied to the unit cell so that the axial velocity applied at the ends of each unit cell is adjusted to $\pm V_0/p$, with $V_0 = 2000$ m/s.

The free surface of the unit cell is perturbed as for the plate configuration so the initial diameter is:

$$D_0(X_1) = D_0^b - \hat{D}_0^{UC} \cos(\gamma_p X_1) \quad (A.11)$$

with $\hat{D}_0^{UC} = 2$ µm. The radial position of the free surface at time $t = 0$ is
thus defined by $R_0(X_1) = D(X_1, t = 0)/2$.

Figure A1a presents the normalized current radius of the round bar at the peak section for various values of $K$ in the range [0.5, 1.1]. Adopting a value of 0.5 (value derived from the first order Taylor expansion of the Bridgman coefficient $B(\phi)$ for round bar) leads to a slightly larger dominant mode index, as compared to the FEM result. Furthermore, amplitudes of the perturbations are not consistent when compared to the ones given by FEM, see Fig. A1a. For small wavenumber perturbation, multiaxial effects are negligible and the value of $K$ has no significant influence on the current radius of the round bar.

It is clearly seen that as the value of $K$ increases, the normalized radius is smaller, meaning that the development of the instability is slowed down. This is a clear illustration of the stabilizing role of the multiaxiality of the stress field within the neck region.

From multiple unit cell calculations, the optimal value for the adjustable coefficient $K$ is displayed in Fig. A1b. As discussed previously, the evolution of perturbations in the small wavenumber regime is controlled mainly by inertia. For the adopted configuration and loading, this regime ends when $\gamma_p D_0^b = 1.8$ (see shaded part in Fig. A1b).

In the intermediate to large wavenumber regime, the correction becomes essential. An optimal value $K_{opt}$ for each perturbation mode has been captured, which minimizes the difference in the perturbation amplitude captured by the new 1DXLSA model and the reference solution (FEM), see black dots
in Fig. A1b.

From the FE results, it is seen in Fig. A1b that the variation of $K_{\text{opt}}$ is more limited than for the plate configuration. $K_{\text{opt}}$ is varying in the range [0.9, 1.2]. This value is quite different from the one derived when adopting the Bridgman theory ($K = 1/2$). As for plate, a mathematical expression can be proposed to describe the variation of $K_{\text{opt}}$ with the dimensionless wavenumber. The best fit curve is given by $K_{\text{opt}}(\gamma_p) = 3.45 \times 10^{-10}\gamma_p^4 - 3.29 \times 10^{-7}\gamma_p^3 + 1.13 \times 10^{-4}\gamma_p^2 - 1.57 \times 10^{-2}\gamma_p + 1.67$. Nevertheless, due to limited variations of $K_{\text{opt}}$ in Fig. A1b, adopting a unique value $K_{\text{opt}} = 1$ is a reasonable approximation.

The predictive performance of the new 1DXLSA model (with $K = 1$) is illustrated in Fig. A2a, which shows the localization time as a function of the dimensionless wavenumber. The results demonstrate the improved accuracy of the new model, particularly for perturbations with intermediate to large wavenumber where multiaxial effects play a significant role in the development of the multiple necking pattern.

Finally, a round bar configuration with a multimodal perturbation is stretched at a nominal strain rate $\dot{\epsilon}_0 = 2000 \text{ s}^{-1}$. The perturbation is similar to the one given in Eq. (1), replacing $h_0$ by $D_0$. Fig. A2b displays the current radius of the bar at the localization time $t = 35 \ \mu\text{s}$. The good agreement between finite element simulations and the predictions of the new 1DXLSA model confirms the accuracy of the proposed correction (with a constant value of $K = 1$). Additional simulations were performed still with
<figure><img src="image_24.png" /><figcaption>Figure A1: Calibration of parameter K for round bar. (a) Normalized current radius at peak of the unit-cell $R^{peak}/\langle R \rangle$ at $t = 80~\mu s$ predicted by the new 1DXLSA model with different K values. Comparison with finite element simulations. (b) Optimal value of the adjustable coefficient $K_{opt}$ as a function of the perturbation mode for a round bar configuration ($D_0^b = 10~\text{mm}$, $L_{01} = 1~\text{m}$). The nominal strain rate is $2000~s^{-1}$. Material parameters are given in Table 1.</figcaption></figure>
a multimodal perturbation and an initial amplitude four times larger. The predicted surface profiles remained in excellent agreement with FEM results (results not shown here). These observations demonstrate that preserving the parameter value $K_{opt} = 1$, identified from small-amplitude perturbations ($\hat{D}_0^{UC} = 2 \ \mu\text{m}$), remains valid even when a larger initial imperfection ($\hat{D}_0^{UC} = 8 \ \mu\text{m}$) is considered. As for the plate configuration, a validation of the strategy for additional configurations could be done in a future work.
<figure><img src="image_25.png" /></figure>

Figure A2: Validation of the new 1DXLSA approach (with $K_{opt} = 1$) for round bar. Comparison of predictions based on the 1DXLSA, the new 1DXLSA, and finite element results: (a) Localization time as a function of the dimensionless wavenumber based on unit cell calculations. (b) Current radius of the round bar at $t = 35~\mu s$ with a multimodal perturbation. A white noise defined by $p_{max} = 200$ and $\hat{D}_0^{UC} = 2~\mu m$ serves as initial perturbation of the round bar free surface. The nominal strain rate is $2000~s^{-1}$. Material parameters are given in Table 1.
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