Experimental characterization of
aluminum/polymer/aluminum sandwich structures
under various loading rates and temperatures:
Establishing a constitutive relationship for LDPE

Amine Bendarma, Alexis Rusinek, Christophe Czarnota, T. Jankowiak, Richard Bernier, T. Lodygowski

► To cite this version:

Amine Bendarma, Alexis Rusinek, Christophe Czarnota, T. Jankowiak, Richard Bernier, et al.. Experimental characterization of aluminum/polymer/aluminum sandwich structures under various loading rates and temperatures: Establishing a constitutive relationship for LDPE. Composite Structures, 2025, 351, pp.118616. 10.1016/j.compstruct.2024.118616. hal-05494005

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# Experimental characterization of aluminum/polymer/aluminum sandwich structures under various loading rates and temperatures: Establishing a constitutive relationship for LDPE☆

A. Bendarma<sup>a,b,c</sup>, A. Rusinek<sup>d,e,f</sup>, C. Czarnota<sup>d</sup>, T. Jankowiak<sup>c</sup>, R. Bernier<sup>d</sup>, T. Lodygowski<sup>c</sup>

## ARTICLE INFO

### Keywords:

- Alucobond
- Composite sandwich structures
- Strain rate sensitivity

- Temperature sensitivity
- Mechanical behavior

- LDPE (Low-Density Polyethylene)

## ABSTRACT

Composite sandwich structures, for example Alucobond, have appeared as auspicious materials in a diversity of engineering applications due to their lightweight, high strength to weight ratio and thermal insulation properties. Analyzing their mechanical behavior in different loading rates is critical to optimizing their performance. This study inspects the mechanical response of Alucobond sandwich composite structures under dynamic and quasi-static compression loadings. Dynamic compression tests, carried out at high strain rates, and quasi-static compression tests, carried out at lower strain rates and temperatures, were used to predict the behavior of the material under different loading conditions. In addition, the mechanical properties of the low-density polyethylene (LDPE) core layer were characterized under dynamic and quasi-static loading conditions, with varying strain rates and temperatures, leading to the establishment of a constitutive relationship for the LDPE material. The comparative study of Alucobond and LDPE highlights the influence of material composition and loading conditions on the mechanical behavior of the composite sandwich structures.

## 1. Introduction

The subject of the presented research are the properties of a composite material called alucobond, which is used in many industries due to its lightness, high strength-to-weight ratio and thermal insulation properties [1]. Encompassed of two layers of aluminum with a core layer of low-density polyethylene (LDPE), Alucobond shows potential applications in the construction, automotive, and aerospace sectors [2].

In the domain of behavior evaluation, compression tests are crucial for understanding how composite sandwich structures react to various loading conditions [3]. Dynamic and quasi static compression tests conducted at different strain rates, offer valuable insights into how the material responds to varying loading rates [4].

The importance of considering both dynamic and quasi-static loading conditions when characterizing the mechanical properties of composite sandwich structures has been highlighted by numerous

studies. For instance, complete testing methodologies have been established to be necessary by Smith et al. [5], as they presented that the mechanical behavior of aluminum foam core sandwich structures varied significantly under dynamic and quasi-static loading.

When metals undergo dynamic testing, they experience strain rates and brief loading periods leading to distinct material responses and failure patterns [6]. This type of testing is crucial, in the field of materials science and engineering as it helps predict how metals will behave in situations like high-speed impacts, explosions, and ballistic events [7]. Knowing how metals respond to loads is essential for improving the safety and reliability of parts used in various industries such, as aerospace, defense, automotive, and industrial sectors.

In the years there have been notable advancements, in experimental methods for studying the dynamic behavior of metallic materials. By using tools such as high-speed cameras and laser interferometry, researchers can now observe the details of how metals deform and fail

<footnote><sup>a</sup> Arts et Métiers Campus de Rabat Technopolis, Rocade de Rabat-Salé, 11100 Sala Al Jadida, Morocco</footnote>

<footnote><sup>b</sup> Universiapolis, Technical University of Agadir, Laboratory for Sustainable Innovation and Applied Research, Bab Al Madina, BP 8143, Qr Tlila, 80000 Agadir, Morocco</footnote>

<footnote><sup>c</sup> Poznan University of Technology, Institute of Structural Analysis, Piotrowo 5, 60-965 Poznan, Poland</footnote>

<footnote><sup>d</sup> Université de Lorraine, CNRS UMR 7239, Arts et Métiers Sciences et Technologie, LEM3, F-57000 Metz, France</footnote>

<footnote>* The paper is dedicated to our friend and colleague Prof. Tomasz Sadowski.</footnote>

<footnote>* Corresponding author.
E-mail address: alexis.rusinek@univ-lorraine.fr (A. Rusinek).</footnote>
under dynamic loads. Additionally, numerical simulations like finite element analysis play a key role in modeling and forecasting how metals react to dynamic situations [8–20].

Polymer mechanics research has focused on how low-density polyethylene (LDPE) polymers respond to changes in strain rate due to their nature and molecular structure [21]. The mechanical properties of LDPE, such as ductility, stiffness and strength are known to vary with strain rates [22]. Recent studies by Lamnii et al [23], and Wang et al. [5] have shed light on the dynamic behavior of LDPE. Lamnii et al [23] have delved into the influence of temperature on LDPE, revealing how it affects material properties and deformation mechanisms. A deep understanding of LDPEs sensitivity to strain rate and temperature is essential, for modeling and designing structures that incorporate this polymer.

In the field of metal mechanics, aluminum alloys have attracted a lot of attention, for their thermal and strain rate sensitivities [24–27]. Barlat et al [26], and Zhang et al [27] have examined how aluminum alloys change under applied stress demonstrating that the movement of defects and the evolution of the material’s structure play crucial roles in its response. Additionally, Bendarma et al [28] have investigated the thermal sensitivity under various strain rates of aluminum alloys, Fig. 1.

In Bendarma et al. [29], researchers have been exploring the effect of strain rates and temperatures, on aluminum and LDPE materials. The ballistic performance of Alucobond has also been examined, highlighting the importance of understanding how constituent materials behave under extreme conditions to predict the overall behavior of the composite under loads.

The behavior of aluminum changes with initial temperatures. During dynamic compression, at a strain rate of approximatively equals to2500 s⁻¹, the stress at a strain level of 0.2 decreases by about 20 % when the temperature increases from 20 °C (room temperature, Fig. 1a) to 170 °C (Fig. 1b).

LDPE also shows a marked effect of temperature, under static and dynamic loading, as illustrated in Fig. 2. Using the data reported in Fig. 2, the temperature sensitivity $\nu = \partial\sigma/\partial T|_{\varepsilon,\dot{\varepsilon}}$ may be defined. As an illustration, under static loading, the average temperature sensitivity is close to −0.41 MPa/°C for any strain level reported, whereas under dynamic loading, this sensitivity is estimated as −0.13 MPa/°C.

The paper examines how the Alucobond composite structure behaves under different loading conditions, both static and dynamic, during compression and tension. To gain a clearer understanding, it also looks at how, when isolated, LDPE and aluminum behave. This information helps to predict the real behavior of Alucobond and assists in designing similar structures for future works.

<figure><img src="image_3.png" /><figcaption>Fig. 2. True stress versus temperature for a strain level of 7.5 % and 20 % under quasi-static and dynamic loading, [29].</figcaption></figure>

## 2. Experimental conditions and techniques

### 2.1. Static tensile and compression test description

To conduct tensile and compression tests on Alucobond and LDPE samples, a Zwick Roell universal testing machine was employed with a maximum load capacity of 100 kN. Additionally, the machine was equipped with a climate chamber, allowing precise temperature control within a range of -80 °C to +150 °C (Fig. 1). This thermal control capability is essential for conducting tests at extreme temperatures and evaluating the mechanical behavior of materials under varied environmental conditions. The combination of high load capacity and precise thermal control makes the Zwick Roell machine a reliable and versatile tool for characterizing the mechanical properties of composite materials, such as Alucobond and LDPE, in this study, a constant loading rate is applied to the specimen up to failure for a strain rate varying from 10⁻⁴ s⁻¹ to 10⁻¹ s⁻¹. For high strain rates, a Split Hopkinson Pressure Bars (SHPB) system has been used to conduct experiments (see forthcoming section 2.2). Fig. 3 shows a schematic of the composite panel from which specimens have been machined. Various sample geometries have been considered, depending on the type of loading, see Fig. 4.

<figure><img src="image_4.png" /><figcaption>Fig. 1. Experimental dynamic compression results using the SHPB system, (a) SHPB curves at room temperature and various strain rates, (b) SHPB curves at 170 °C.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 3. Scheme of 3 layers used for the Alucobond [30].</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4. Specimens geometry for quasi-static tests: Tensile test specimen used for Alucobond and LDPE characterization with a) $L_0$ = 40 mm and b) $L_0$ = 20 mm. Compression test specimen for c) Alucobond and d) LDPE.</figcaption></figure>

## 2.2. Dynamic compression tests

At high deformation rates, the most widely used technique to study the dynamic properties of materials relies on the Split Hopkinson Pressure Bar (SHPB). This method was initially developed by Kolsky [31] and has undergone several improvements. Its principle is based on the theory of elastic waves propagation in bars, and allows the dynamic characterization of materials for strain rates from $5 \cdot 10^2 s^{-1}$ to $10^4 s^{-1}$ in

compression. The principle of Hopkinson bars is well-known and has been the topic of numerous publications [28,32–37].

The SHPB system used for this study is illustrated in Fig. 5. Specimens are sandwiched between two identical bars of high tensile strength, generally close to 2 GPa. The first and second bars are referred as the “Input bar” and “Output bar” respectively. The input bar is impacted by a blunt cylindrical impactor to generate an initial elastic wave which will be at the origin of the dynamic loading of the specimen.

To perform the tests for different initial temperatures, a specific furnace has been designed to achieve and maintain a uniformly distributed temperature using an electronic controller, Fig. 6. Thus, a constant volume is heated to an initial temperature $T_0$. This volume is larger than the chamber volume where the sample is sandwiched between the two Hopkinson bars, Fig. 6-c. To maintain a constant temperature in the sample chamber, a fan is used to evacuate the hot air flow as needed.

# 3. Analysis of the tensile behavior

## 3.1. Alucobond

In this section the characteristics of the Alucobond material are determined by conducting tensile tests at different strain rates. The tests involved Alucobond samples with gauge lengths $L_0$ of 20 mm and 40 mm under strain rates of 0.001, 0.01 and $0.1 s^{-1}$, see Fig. 4a and 4b. Fig. 7 presents the results, highlighting key findings such as fracture in samples with a 20 mm and 40 mm initial gauge length and a consistent failure pattern across different strain rates and shapes. This section aims to elucidate the behavior of Alucobond, providing insights into its structural strength, failure mechanisms, and suitability for various engineering applications.

Fig. 7b and 7c show fractures samples at the end of tests conducted under various low strain rates. Notably, early rupture is observed for specimens with an initial gauge length of 20 mm due to faster plastic

<figure><img src="image_7.png" /><figcaption>Fig. 5. Schematic description of the Hopkinson Bar device [33,34].</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Fig. 6. a - Split Hopkinson Pressure Bars used in dynamic tests with a furnace; b - Description of the internal part; c - Description of the airflow, [28,37].</figcaption></figure>

localization along the active part of the specimen, as discussed in [39]. This could be due to the failure occurring near the edge of the sample (Fig. 7c) rather than in the main section, as observed in samples with an initial length of 40 mm (Fig. 7b). It is interesting to note that the failure occurs first in aluminum for each test performed not depending on the geometry used.

## 3.2. LDPE tensile tests

In this section, the mechanical properties of low-density polyethylene (LDPE) material are examined through a series of tensile tests conducted at room temperature with a strain rate ranging from 0.001 to 0.2 s⁻¹. In this section, the initial length of the specimen is L₀ = 20 mm.

The true stress – true strain curves shown in Fig. 8a for LDPE reveal the material’s strain rate dependence, as observed for polymers. This behavior is linked to the nature of LDPE, where higher strain rates reduce the time available for polymer chains to align with the loading direction, resulting in a strength increased. Conversely, a decrease in the fracture strain is observed at large strain rates which can also be attributed to restricted movement within the polymer chains, leading to premature failure. The observed strain rate sensitivity highlights the importance of considering strain rate effects when evaluating polymer properties particularly in situations where loading rates may vary.

# 4. Analysis of the compression behavior

## 4.1. Alucobond composite material

### 4.1.1. Quasi-Static compression tests

Quasi-static compression tests were conducted on Alucobond using two different strain rates of 0.001 and 0.01 s⁻¹ at room temperature. Fig. 9 illustrates the corresponding true-stress true-strain curves, along with snapshots of some samples at the end of test. The results are portrayed up to a true strain of ~ 0.5.

A notable observation made on the samples is the non-uniform deformation behavior between the layers of LDPE and aluminum, Fig. 9. Clearly, LDPE exhibits greater ductility compared to aluminum.

The diameter of the Alucobond sample at the end of the test carried out at a strain rate of 0.01 s⁻¹ is 11.32 mm, whereas for the test conducted at a lower strain rate of 0.001 s⁻¹, the diameter is 13.63 mm. The greater ductility of LDPE is a characteristic feature of polymers, which typically exhibit higher levels of deformation before failure compared to metals. As a thermoplastic polymer, LDPE undergoes significant plastic deformation before reaching its ultimate failure point. In contrast, aluminum, being a metal, tends to fracture with a limited accumulated plastic strain. The uneven deformation between materials observed in compression, at various strain rates in the quasi-static regime, underscores the nature of the Alucobond composite, where each
<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 7. a) Tensile testing curves of Alucobond at room temperature and different strain rates (0.001, 0.01, and 0.1 s⁻¹). Fractured specimens b) L₀ = 40 mm and c) L₀ = 20 mm.</figcaption></figure>

component contributes uniquely to the overall mechanical response.

### 4.1.2. *Dynamic compression tests*

The dynamic behavior of Alucobond was characterized using the Split Hopkinson Pressure Bar (SHPB) system. The experiments were conducted at strain rates of 2137, 2142 2590, 2595, 3242, 3313, and 3362 s⁻¹ and room temperature. The findings indicated that the composite structure, made of aluminum and LDPE, showed sensitivity to strain rate.

In the compression curves obtained (as shown in Fig. 10), it can be seen that the behavior of Alucobond varies with the applied strain rate. Generally, an increase in strain rate results in greater stiffness and strength of the material. This phenomenon can be attributed to several factors, including increased confinement and reduced time for plastic deformation to develop. Furthermore, the strain rate sensitivity can be related to the viscoelastic nature of LDPE, which constitutes the core layer of the composite structure. At higher strain rates, the polymer chains in LDPE have less time to reorganize, leading to stress increase.

In order to provide a synthetic representation of the dynamic

compression, the representative behavior of Alucobond is proposed in Fig. 11 where the results obtained at a strain rate of 3313, 2595 and 2142 s⁻¹ have been selected. Results of Fig. 11 demonstrate that the fracture pattern varies with the strain rate level, revealing an increase in the number of radial necks and fragments observed on the aluminum surface as the strain rate rises. This phenomenon can be attributed to inertia effects that become more pronounced at higher strain rates, leading to more accumulated plastic strain at fracture. This is similar to what is observed in the dynamic fragmentation of rings and cylinders [38,39].

### 4.1.3. *Comparison of dynamic and quasi-static tests*

To gain a better understanding about the behavior of Alucobond under dynamic and quasi-static compression loadings, Fig. 12 presents a compilation of results, in terms of stress versus strain responses at varying strain rates.

In Fig. 12, it is observed that when the strain rates are lower, such as under slow loading conditions, the material can deform significantly before failure (which occurs at very large strain not reported in the
<figure><img src="image_11.png" /><figcaption>Fig. 8. a) Tensile testing curves of LDPE at room temperature and different strain rates of 0.001, 0.01, 0.1, and 0.2 s⁻¹, b) Fracture samples of initial gauge length L₀ = 20 mm.</figcaption></figure>

graph where the static responses are plotted up to a strain of ~ 0.5). Conversely, at higher strain rates typical of fast dynamic loading, the stress–strain curves display a rapid increase in stress indicating that the material becomes more resistant. This sensitivity to strain rate emphasizes the importance of accounting for loading rate effects when

designing structures with Alucobond.

## 4.2. LDPE compression tests

### 4.2.1. Quasi-Static compression tests

Quasi-static compression experiments were conducted on LDPE at room temperature, using two different strain rates (0.01 and 0.001 s⁻¹). From Fig. 13, showing stress–strain curves, one observed that, similarly to the dynamic compression tests, quasi-static tests revealed a strain rate dependency.

At 0.001 s⁻¹ the material exhibited greater stiffness and strength compared to when tested at 0.01 s⁻¹. This indicates that LDPEs mechanical behavior is influenced by how deformation's applied during these tests. The materials response to strain rate in these conditions may be due to factors like changes or viscoelastic properties, which vary with loading rates. Recognizing LDPEs sensitivity to strain rate in compression tests emphasizes the need to account for loading rate effects when designing structures using LDPE.

### 4.2.2. Dynamic compression tests

Dynamic compression experiments were performed on LDPE using the Split Hopkinson Pressure Bar (SHPB) system, at room temperature across three strain rates (2483, 2919 and 3953 s⁻¹).

It is important to note that LDPE is an amorphous thermoplastic polymer, meaning it lacks a regular crystalline structure, unlike metals such as Face Centered Cubic (FCC) or Body Centered Cubic (BCC). LDPE has a random molecular arrangement, which significantly influences its mechanical properties. While LDPE differs structurally from crystalline materials, its mechanical characteristics—including strength, stiffness, and ductility—are influenced by factors such as molecular weight, chain interactions, and the use of additives. Fig. 14 illustrates that the polymer is strain rate sensitive under compressive loading. The strain at failure and the stress both increase as the strain rate increases.

### 4.2.3. Comparison between dynamic and quasi-static tests

This section discusses how the mechanical behavior of LDPE varies from quasi-static to dynamic compression tests, highlighting noticeable differences in strain rate sensitivity.

As demonstrated in Fig. 15, LDPE shows distinct compression behaviors depending on the rate at which deformation is applied. During dynamic compression tests, LDPE demonstrated sensitivity to strain rate maintaining a consistent response across various loading rates. However, during static compression tests performed at lower strain rates (0.001 s⁻¹) LDPE displayed a more pronounced sensitivity to strain rate,

<figure><img src="image_12.png" /><figcaption>Fig. 9. Quasi-static compression tests of Alucobond at room temperature and different strain rates (0.001 and 0.01 s⁻¹).</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Fig. 10. Dynamic compression testing of Alucobond using the Split Hopkinson Pressure Bar (SHPB) system. Tests were conducted at room temperature and different strain rates.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 11. Dynamic compression testing of Alucobond with different initial strain rates, 3313, 2595, and 2142 s⁻¹.</figcaption></figure>

resulting in noticeable differences in the yield strength. Possible reasons for this sensitivity include polymer chain relaxation and thermal effects.

### 4.3. Comparison between Alucobond and LDPE behaviors

#### 4.3.1. Quasi-Static loading

This section presents the behavior of Alucobond and LDPE when subjected to static loadings. The results of compression tests carried out at room temperature and two strain rates (0.001 and 0.01 s⁻¹) are shown in Fig. 16.

Alucobond demonstrated a larger rigidity and toughness in contrast to LDPE, which can be linked to its structure consisting of aluminum layers and an LDPE core. The strain rate sensitivity and the hardening behavior are similar for both materials up to a strain level of ~ 0.35, corresponding to an increase of the hardening index after an inflection point.

<figure><img src="image_15.png" /><figcaption>Fig. 12. Comparison of the mechanical behavior of Alucobond under dynamic and static loading conditions.</figcaption></figure>

#### 4.3.2. Dynamic loading and material behavior comparison

In this section a comparative analysis studies the dynamic mechanical behavior of aluminum, LDPE, and Alucobond at ambient temperature. Dynamic compression tests were conducted using strain rates of 3313, 2585, and 2140 s⁻¹ for Alucobond, 2483, 2919, and 3953 s⁻¹ for LDPE, and 2358, 28 40 and 3725 s⁻¹ for aluminum. The modulus of the composite, E<sub>Comp</sub>, has been defined as follows, Eq. (1), using a series composite description (iso-stress). The value calculated is approximately E<sub>Comp</sub> = 399MPa.

$$E_{\text{Comp}} = \frac{E_{\text{Al}} \cdot E_{\text{LDPE}}(e_{\text{Al}} + e_{\text{LDPE}})}{e_{\text{Al}} \cdot E_{\text{LDPE}} + e_{\text{LDPE}} \cdot E_{\text{Al}}} \quad (1)$$

where E<sub>Al</sub> and E<sub>LDPE</sub> are the Young's moduli of the aluminum and LDPE, respectively. e<sub>Al</sub> and e<sub>LDPE</sub> are the thicknesses of aluminum and LDPE layers making up the composite material. For our experimental configuration, see Fig. 3, e<sub>Al</sub> = 1mm(0.5 mm on each side of the LDPE core) and e<sub>LDPE</sub> = 3mm.
<figure><img src="image_16.png" /><figcaption>Fig. 13. Quasi-Static compression test of LDPE at different strain rates, 0.01 and 0.001 s⁻¹.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 14. Dynamic Compression Testing of LDPE Using the Split Hopkinson Pressure Bar (SHPB) Technique.</figcaption></figure>

From Fig. 17, which displays the stress strain curves for both materials under dynamic loading conditions, it is observed that incorporating aluminum on each side of the polymer LDPE, as is the case with Alucobond, induces an average stress increase of 35 %. The yield strength of LDPE under dynamic loading is approximately 30 MPa. Considering Alucobond, the yield stress is about 40 MPa showing a difference approximately of 10 MPa compared to LDPE.

### 4.3.3. Strain rate and temperature sensitivities

An important aspect of understanding the mechanical behavior of LDPE is its sensitivity to strain rate, which is influential at different stress levels. Based on the data depicted in Fig. 18, LDPEs reliance on strain rate can be estimated by examining its behavior near the point of yielding (at a strain of 0.1) and when it reaches an accumulated strain of 0.3.

By examining where the elastic response intersects with the hardening response (shown in Figs. 15 and 17), the material's sensitivity to strain rate near its initial yield point was evaluated. Additionally, LDPE's behavior at an accumulated strain of 0.3 was analyzed to

<figure><img src="image_18.png" /><figcaption>Fig. 15. Comparison of LDPE's Response in Dynamic and Quasi-Static Compression Tests.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Fig. 16. Comparative Analysis of Alucobond and LDPE under Quasi-Static Loading Conditions.</figcaption></figure>

understand how its mechanical properties change as deformation increases and how this change is influenced by the strain rate.

The thermal dependency is illustrated in Fig. 18 which displays strain-strain curves at a strain rate of 2700 s⁻¹ across temperatures ranging from 20 °C (room temperature) to 80 °C. It is evident that the composite structure, which was previously shown to depend on strain rate, also strongly depends on temperature. Specifically, there is a difference of ~ 10 MPa in stress at a strain of 0.1, and ~ 27 MPa at a strain level of 0.3, when the results obtained at 20 °C (room temperature) and 80 °C are examined.

### 4.3.4. LDPE constitutive relation modeling

An essential aspect of understanding the mechanical behavior of LDPE is its sensitivity to strain rate, which plays a significant role in various loading conditions. Based on experiments, a model is proposed here to define the strain rate sensitivity of the LDPE as well as its hardening behavior. Recently, a model and experiments have been published in [41] with the Mises equivalent flow stress expressed as:
<figure><img src="image_20.png" /><figcaption>Fig. 17. Comparative analysis of the dynamic mechanical behavior of Alucobond, LDPE, and aluminum at room temperature and strain rates.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Fig. 18. Temperature dependence of Alucobond.</figcaption></figure>

$$ \sigma(\epsilon, \dot{\epsilon}) = \sigma_0 \left[ 1 + \left( \frac{\dot{\epsilon}}{D} \right)^{1/p} \right] \qquad (2) $$

where $\sigma_0$ is the initial yield stress, $\dot{\epsilon}$ the Huber-Mises equivalent plastic strain rate, and $D$ and $p$ are the parameters proposed by Cowper-Symonds to describe strain rate sensitivity.

Using Eq. (2), Xu et al. [3] identified the coefficients $D$ and $p$ from SHPB tests and $\sigma_0$ has been determined from a quasi-static test at an initial strain rate of 0.001 s⁻¹, assuming a perfectly plastic material. In the present work, the same material was used and a hardening was observed under quasi-static and dynamic loadings, Fig. 15. For this reason, a constant hardening may not be used to describe the LDPE layer making up our composite. The proposed model relies on a coupling between the Johnson-Cook model for the hardening and the Cowper-Symonds modified law to describe the strain rate sensitivity, Eq. (3).

$$ \sigma(\epsilon, \dot{\epsilon}) = (A + B\epsilon^n) \left[ 1 + \left( \frac{\dot{\epsilon} - \dot{\epsilon}_0}{D} \right)^{1/p} \right] \qquad (3) $$

where $\langle \cdot \rangle$ is the Macaulay bracket, and $\epsilon = \int_0^t \dot{\epsilon}(\tau)d\tau$ is the accumulated

plastic strain, $\dot{\epsilon}_0$ being a reference strain rate. As in the Johnson-Cook model [40], A stands for the initial yield stress at a strain rate $\dot{\epsilon} = \dot{\epsilon}_0$, $B$ and $n$ are the parameters defining the strain hardening while $D$ and $p$ are defined similarly as in Eq. (2).

In the following Table 1, the constants are reported for Eq.2 and 3 corresponding to a LDPE with hardening (present work) or idealized as a perfectly plastic material as proposed by Xu et al. [41].

In the following curve, Fig. 19, a comparison is reported in terms of behavior and strain rate sensitivity. It is clear the LDPE material shows hardening with an increase in plastic strain under both quasi-static and dynamic loading conditions. However, the strain rate sensitivity obtained from our experimental study is lower compared to that reported by Xu et al. [41]. In general, a distinct material formulation is observed. In our case due to hardening increase and strain rate sensitivity, the material exhibits multiplicative behavior that induces an interplay between hardening and strain rate sensitivity, as described in Eq. 4-b. In contrast, concerning the behavior reported by Xu et al. [41], the material is idealized as perfectly plastic and described by Eq. 4-b.

$$ \sigma(\dot{\epsilon}) = \sigma_0 * f(\dot{\epsilon}) \qquad (4-a) $$

$$ \sigma(\epsilon, \dot{\epsilon}) = g(\epsilon) * f(\dot{\epsilon}) \qquad (4-b) $$

where $g(\epsilon)$ defines the hardening function, $\sigma_0$ the constant yield stress and $f(\dot{\epsilon})$ is the strain rate dependent function.

A comparison is reported in the following picture comparing our experiments with Eq. (3) and the data published in [41]. It is clear that a perfectly plastic model behavior is not correct and misestimates the real material behavior. In Fig. 19-a, the hardening obtained using Eq. (3) with parameters of Table 1, it is demonstrated that the overall response agrees with experiments of Xu et al. [41]. Finally, the hardening observed from our experiments is the same as with the previous work [41]. It is also important to note that the Young modulus generally reported in the literature for LDPE 200 ≤ E ≤ 300 MPa is well restituted by our experiments, yielding a value of 300 MPa.

Thus, using the same material name, LDPE, there exists a notable difference in terms of behavior but also concerning the strain rate sensitivity. However, it is clear, based on the data reported in [41] under dynamic loading, that the material cannot be defined as perfectly plastic, Fig. 19-a. The higher stress level observed in [41] may be influenced by several factors, such as the friction effect at the bar-specimen interface [35], which can induce overstress effects.

## 5. Conclusion

The dynamic and static compression tests conducted on Alucobond to investigate its mechanical properties provided informative experimental results on the behavior of this composite material under varying loading conditions. Through the tests, Alucobond displayed high sensitivity to strain rate, demonstrating increased stiffness and strength at higher strain rates. The non-uniform deformation observed between LDPE and aluminum in static compression test highlighted LDPE’s greater ductility compared to aluminum. This difference in deformation behavior of LDPE and aluminum illustrated distinct deformation

<table><caption>Table 1
Constants used with Eq.1 and 2 for LDPE.</caption><thead><tr><th>Constants</th><th>Present work with hardening</th><th>Modeling parameters from Xu et al. [41]</th></tr></thead><tbody><tr><td>E (MPa)</td><td>300</td><td></td></tr><tr><td>σ<sub>0</sub> (MPa)</td><td></td><td>15</td></tr><tr><td>A (MPa)</td><td>8</td><td></td></tr><tr><td>B (MPa)</td><td>22</td><td></td></tr><tr><td>n (–)</td><td>0.4</td><td></td></tr><tr><td>ε̇<sub>0</sub> (1/s)</td><td>0.001</td><td></td></tr><tr><td>D (1/s)</td><td>1500</td><td>104</td></tr><tr><td>p (–)</td><td>6</td><td>3,72</td></tr></tbody></table>
<figure><img src="image_22.png" /><figcaption>Fig. 19. a- Comparison between the constitutive relation and experiments in terms of hardening, for various strain rates; b- Definition of the strain rate sensitivity, comparison between experiments and modeling, our work and Xu et al. [41].</figcaption></figure>

mechanisms between a polymer and metal.

During static compression tests, Alucobond showed higher stiffness and strength than LDPE, as evidenced by its Young's modulus and yield strength values. Conversely, LDPE showed higher deformation capability due to its viscoelastic properties. This comparison emphasized how these materials respond differently to loading indicating their suitability for applications depending on the needs, for stiffness,

strength and deformation behavior.

In dynamic compression tests, both Alucobond and LDPE showed limited strain rate sensitivity. Alucobond demonstrated consistent behavior regardless of strain rate, whereas LDPE exhibited larger flow stresses with increasing strain rates, indicating higher sensitivity to these effects. This comparison provided insights into how these materials behave under different loading conditions. Additionally, a constitutive relationship has been formulated for LDPE. When combined with an appropriate model for aluminum, this relationship could effectively describe the behavior of the composite material under diverse loading conditions, including dynamic events.

Combining the results of dynamic and static compression tests allows for a direct comparison, highlighting variations in the mechanical behavior of Alucobond under dynamic and static loading conditions.

In summary, the comprehensive analysis of the mechanical properties of Alucobond has provided a better understanding of its behavior under different loading conditions in terms of strain rates and temperatures. This study highlights the importance of considering factors such as strain rate sensitivity and the distinct deformation mechanism of its constituent materials in the design and engineering of applications utilizing this material.

### CRediT authorship contribution statement

A. Bendarma: Writing – review & editing, Writing – original draft, Visualization, Validation, Data curation, Conceptualization. A. Rusinnek: Visualization, Validation, Supervision, Formal analysis, Data curation, Conceptualization. C. Czarnota: Validation, Supervision, Methodology, Formal analysis. T. Jankowiak: Visualization, Validation, Supervision, Conceptualization. R. Bernier: Formal analysis, Data curation. T. Lodygowski: Supervision, Formal analysis, Data curation, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Data availability

The data that has been used is confidential.

### Acknowledgement

We would like to thank the experimental resources of the LEM3 Rheology and Rupture Dynamics platform, labeled INFRA+ (University of Lorraine - CNRS UMR 7239)

## Appendix A

The projectile launched with an initial velocity $V_0$ impacts the incoming bar and generates an incident elastic wave $\varepsilon_l$ which propagates in the bar with a celerity $C_0$ [28]:

$$C_0 = \sqrt{\frac{E}{\rho}} \qquad (A-1)$$

$E$ and $\rho$ denote respectively the Young's modulus and the density of the SHPB bars. Thus, the intensity of the incident stress is defined by [28]:

$$\sigma_l = \frac{1}{2} \rho C_0 V_0 = E \varepsilon_l \qquad (A-2)$$

The characteristic equations associated with a one-dimensional elastic wave propagation in the bar makes it possible to express the particle velocities at the two interfaces [28]:
$$ \begin{cases} v_e(t) = C_0(\epsilon_I(t) - \epsilon_R(t)) \\ v_s(t) = C_0\epsilon_T(t) \end{cases} \qquad (A-3) $$

The average rate of axial strain in the specimen is given by:

$$ \dot{\epsilon}_n = \frac{v_e(t) - v_s(t)}{l_0} = \frac{C_0}{l_0}(\epsilon_I - \epsilon_R - \epsilon_T) \qquad (A-4) $$

where $l_0$ denotes the initial length of the specimen. Since the displacement and force are known it is possible to calculate stress and strain at the two specimen/bar interfaces:

$$ \left\{ \begin{aligned} F_{\text{input}}(t) &= A_0 E |\epsilon_I(t) + \epsilon_R(t)| \\ F_{\text{output}}(t) &= A_0 E \epsilon_T(t) \end{aligned} \right. \qquad (A-5) $$

The nominal average axial stress in the specimen is expressed as follows [28]:

$$ \left\{ \begin{aligned} \sigma(t) &= \frac{F_{\text{input}}(t) + F_{\text{output}}(t)}{2A_s} \\ \sigma(t) &= \frac{A_0 E [\epsilon_I(t) + \epsilon_R(t) + \epsilon_T(t)]}{2A_s} \end{aligned} \right. \qquad (A-6) $$

where $A_0$ is the cross-section of the bar and $A_s$ is the cross-section of the specimen. The stress is supposed to be balanced if the boundary conditions are frictionless, the stress in the specimen is also uniaxial.

In other words, assuming that the stress and strain are homogeneous and the stress is uniaxial in the specimen, as well as the elastic wave propagation is one-dimensional (1D) without dispersion in the bars [28].

Once the force equilibrium is achieved ($F_{\text{input}} = F_{\text{output}}$), the equation, reduces to:

$$ \epsilon_I(t) + \epsilon_R(t) = \epsilon_T(t) \qquad (A-7) $$

## References

[1] Khan T, Acar V, Aydin MR, Hülagü B, Akbulut H, Seydibeyoğlu MÖ. A review on recent advances in sandwich structures based on polyurethane foam cores. Polym Compos 2020;41(6):2355-400.

[2] Harhash M, Gilbert RR, Hartmann S, Palkowski H. Experimental characterization, analytical and numerical investigations of metal/polymer/metal sandwich composites-Part 1: Deep drawing. Compos Struct 2018;202:1308-21.

[3] Patekar V, Kale K. State of the art review on mechanical properties of sandwich composite structures. Polym Compos 2022;43(9):5820-30.

[4] Qu J, Ju D, Gao S, Chen J. Research on the dynamic mechanical properties of polymethacrylimide foam sandwich structure. Compos Struct 2018;204:22-30.

[5] Yu JL, Wang X, Wei ZG, Wang EH. Deformation and failure mechanism of dynamically loaded sandwich beams with aluminum-foam core. Int J Impact Eng 2003;28(3):331-47.

[6] Meyers MA. Dynamic behavior of materials. John wiley & sons; 1994.

[7] Fan Z, Fang X, Ji S. Microstructure and mechanical proper-ties of rhoe-diecast (RDC) aluminium alloys. Mater Sci Eng: A 2005;412(1-2):298-306.

[8] Wu HC, Yip MC. Strain rate and strain rate history effects on the dynamic behavior of metallic materials. Int J Solids Struct 1980;16(6):515-36.

[9] Lambert, JP, Jonas, GH. Towards standardization in termi-nal ballistics testing: velocity representation. Army ballistic research lab aberdeen proving ground MD; 1976.

[10] Garouge SE, Tarfaoui M, Hassoon OH, Minor HE, Bendarma A. Effect of stacking sequence on the mechanical performance of the composite structure under slamming impact. Mater Today: Proc 2022;52:29-39.

[11] Jankowiak T, Rusinek A, Bendarma A. Protocol to define material behaviour and failure strain level at low and high strain rates based on compression test. J Theore Appl Mech 2018;56(2):471-81.

[12] Bendarma A, Jankowiak T, Rusinek A, Lodygowski T, Klosak M. Perforation tests of aluminum alloy specimens for a wide range of temperatures using high-performance thermal chamber-experimental and numerical analysis. In: IOP Conference Series: Materi-als Science and Engineering (Vol. 491, No. 1, p. 012027). IOP Publishing; 2019, March.

[13] Jia B, Rusinek A, Bahi S, Bernier R, Pesci R, Bendarma A. Perforation behavior of 304 stainless steel plates at various temperatures. J Dynamic Behavior Mater 2019; 5:416-31.

[14] Bendarma A, Gourgue H, Jankowiak T, Rusinek A, Kardellass S, Klosak M. Perforation tests of composite structure specimens at wide range of temperatures and strain rates-experimental analy-sis. Mater Today: Proc 2020;24:7-10.

[15] Akhzouz H, El Minor H, Tatane M, Bendarma A. Physical characterization of bio-composite CEB stabilized with Argan nut shell and cement. Mater Today: Proc 2021;36:107-14.

[16] Bendarma A, Rusinek A, Jankowiak T, Lodygowski T, Jia B. Experimental analysis of the aluminum alloy sheet subjected to impact and perforation process. Mater Today: Proc 2021;36:88-93.

[17] Garouge SE, Tarfaoui M, Minor HE, Bendarma A. The improvement of the physical and mechanical properties of CNTs based composite material. Mater Today: Proc 2022;52:64-70.

[18] Jia B, Rusinek A, Pesci R, Bernier R, Bahi S, Bendarma A, et al. Simple shear behavior and constitutive modeling of 304 stainless steel over a wide range of strain rates and tempera-tures. Int J Impact Eng 2021;154:103896.

[19] Bendarma A, Garouge SE, Rusinek A, Akhzouz H, Kardellass S, Bouslikhane S, et al. The behaviour of aluminum al-loy 1050 sheet subjected to impact and perforation process: Experi-mental and numerical approaches. Mater Today: Proc 2022;52: 40-4.

[20] Klosak M, Jankowiak T, Rusinek A, Bendarma A, Sielicki PW, Lodygowski T. Mechanical properties of brass under impact and perforation tests for a wide range of temperatures: experimental and nu-merical approach. Materials 2020;13(24): 5821.

[21] Dasari A, Misra RDK. On the strain rate sensitivity of high density polyethylene and polypropylenes. Mater Sci Eng A 2003;358(1-2):356-71.

[22] Qi Z, Hu N, Li G, Zeng D, Su X. Constitutive modeling for the elastic-viscoplastic behavior of high density polyethylene under cyclic loading. Int J Plast 2019;113: 125-44.

[23] Lamnii H, Abdelaziz MN, Ayoub G, Colin X, Maschke U. Experimental investigation and modeling attempt on the effects of ultraviolet aging on the fatigue behavior of an LDPE semi-crystalline polymer. Int J Fatigue 2021;142:105952.

[24] Bendarma A, Jankowiak T, Łodygowski T, Rusinek A, Klósak M. Experimental and numerical analysis of the aluminum alloy AW5005 behavior subjected to tension and perforation under dynamic loading. J Theor Appl Mech 2017;55(4):1219-33.

[25] Pandya KS, Roth CC, Mohr D. Strain rate and temperature dependent fracture of aluminum alloy 7075: Experiments and neural network modeling. Int J Plast 2020; 135:102788.

[26] Barlat F, Glazov MV, Brem JC, Lege DJ. A simple model for dislocation behavior, strain and strain rate hardening evolution in deforming aluminum alloys. Int J Plast 2002;18(7):919-39.

[27] Zhang W, Chen X, Zhuo B, Li P, He L. Effect of strain rate and temperature on dynamic mechanical behavior and microstructure evolution of ultra-high strength aluminum alloy. Mater Sci Eng A 2018;730:336-44.

[28] Bendarma A, Jankowiak T, Rusinek A, Lodygowski T, Jia B, Mi-guélez MH, et al. Dynamic behavior of aluminum al-loy Aw 5005 undergoing interfacial friction and specimen configuration in split Hopkinson pressure bar system at high strain rates and tempera-tures. Materials 2020;13(20):4614.

[29] Bendarma A, Jankowiak T, Rusinek A, Lodygowski T, Czarnota C, Bernier R, et al. Experimental and numerical analysis of aluminum-polyethylene composite structure subjected to tension and perforation under dynamic loading for a wide range of temperatures. J Dynamic Behavior Mater 2023:1-24.
[30] ALUCOBOND, 2017. [Online] Available at: https://www.transport_industry.com/fr/alucobond.html.

[31] Kolsky H. An investigation of the mechanical properties of materials at very high rates of loading. Proc Phys Soc. Section B, 62(11), 676; 1949.

[32] Rusinek A, Rodríguez-Martínez JA, Arias A, Klepaczko JR, López-Puente J. Influence of conical projectile diameter on perpendicular impact of thin steel plate. Eng Fract Mech 2008;75(10):2946-67.

[33] Lindholm US. Some experiments with the split hopkinson pressure bar*. J Mech Phys Solids 1964;12(5):317-35.

[34] Chen WW, Song B. Split Hopkinson (Kolsky) bar: design, testing and applications. Springer Science & Business Media; 2010.

[35] Jankowiak T, Rusinek A, Lodygowski T. Validation of the Klepacz-ko-Malinowski model for friction correction and recommendations on Split Hopkinson Pressure Bar. Finite Elem Anal Des 2011;47(10):1191-208.

[36] Lennon AM, Ramesh KT. A technique for measuring the dynamic behavior of materials at high temperatures. Int J Plasticity 1998;14(12):1279-92.

[37] Rusinek A, Bernier R, Boumbimba RM, Klosak M, Jankowiak T, Voyiadjis GZ. New devices to capture the temperature effect under dynamic compression and impact perforation of polymers, application to PMMA. Polym Test 2018;65:1-9.

[38] Wood P, Rusinek A, Platek P, Janiszewski J, Sienkiewicz J, Gunputh UF, et al. High strain rate effect on tensile ductility and fracture of AM fabricated Inconel 718 with voided microstructures. Mater Des 2021;208:109908.

[39] Rusinek A, Zaera R. Finite element simulation of steel ring fragmentation under radial expansion. Int J Impact Eng 2007;34(4):799-822.

[40] Johnson GR. A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures. In: Proceedings of the 7th International Symposium on Ballistics, The Hague, Netherlands, 1983.

[41] Xu MM, Huang GY, Feng SS, Qin XY, McShane GJ, Stronge WJ. Perforation resistance of aluminum/polyethylene sandwich structure. Mater Des 2016;100: 92-101.