<figure><img src="image_1.png" /></figure>

HAL
open science

# Impact of pre-softening on rubber adhesion

Pierre-Yves Corbel, Julien Jumel

► To cite this version:

Pierre-Yves Corbel, Julien Jumel. Impact of pre-softening on rubber adhesion. Composites Part B: Engineering, 2025, 303, pp.112534. 10.1016/j.compositesb.2025.112534. hal-05379648

HAL Id: hal-05379648

https://hal.science/hal-05379648v1

Submitted on 24 Nov 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_3.png" /></figure>

# Impact of pre-softening on rubber adhesion

Pierre-Yves Corbel<sup>D*</sup>, Julien Jumel

Institut de Recherche Dupuy de Lome, UMR CNRS 6027, ENSTA Bretagne, 2 rue Francois Verny, Brest, 29200, France

<figure><img src="image_4.png" /></figure>

## ARTICLE INFO

**Keywords:**

Adhesion

Mullins effect

Rubber composite

RCAIT

## ABSTRACT

The effect of stress softening, namely the Mullins effect, on elastomers' mechanical properties such as fatigue, strength, or toughness is widely studied, but mainly on bulk materials. The present contribution presents experimental evidence that applying a pre-softening load may significantly alter the critical strain energy release rate (SERR), which controls the interface adhesion between a soft solid and a rigid substrate. Here, we focus on the adhesion of a natural rubber matrix to a single metal wire, characterized using the Rubber Cord Adhesion Inflation Test. This protocol allows for the observation of a stable steady-state crack propagation under an axisymmetric configuration. An experimental protocol is proposed that utilizes a pre-load sequence to achieve a controlled softening of the rubber sheath surrounding the wire. Quantification is carried out through numerical simulations and subsequent material characterization. The adhesion results indicate that the pre-loading sequence reduces the specimen's overall resistance to decohesion. A more detailed analysis reveals that the critical SERR value is also lower. These findings provide new insights into the role of energy dissipation, both within the bulk material and at the interface level, in influencing the interface failure between an elastomer matrix and a rigid reinforcement.

## 1. Introduction

The mechanical performances of rubber composite materials and laminates are closely linked to that of the interface between their constituent at all scales. Toughness, delamination, and fracture mechanisms are controlled not only by the cohesive failure of the matrix and the reinforcements but also by the decohesion between the matrix and reinforcements, which in some cases favors the occurrence of bridging, bifurcation or blunting mechanisms [1]. However, on a more macroscopic scale, fracture processes are often simulated using cohesive modeling for part and component design [2-7]. This phenomenological approach introduces a decoupling between the interface's fracture properties and the surrounding medium's behavior. While this hypothesis deserves specific discussion [8], it is also necessary to assess whether these two local effects can be identified and distinguished from the analysis of crack initiation and propagation tests, which are macroscopic.

We are particularly interested in the characterization of adhesion in rubber composites using fibers, wires, and cords of different materials (polyamide, steel, etc.) as reinforcements [7,9,10]. These materials are used in various applications such as car tires, transmission belts [11-13], anti-vibration components, or air springs [14]. In service, these materials are exposed to various environmental conditions [13], and to complex mechanical stresses [11,12,15] (unsteady, multi-axial, etc.)

so that the matrix, reinforcements, and interfaces are in return also subjected to complex stresses before decohesion is observed. The conditions leading to decohesion between layers or between reinforcement and matrix are typically analyzed using basic or standardized tests (such as peeling tests, H-tests, and single lap shear tests) [7,9,12,14-16]. These methods primarily provide a general overview and quantification of the fracture behavior of specimens. As a result, accurately quantifying the specific local mechanisms and interactions that drive the interface separation is still challenging. Few tests are available to evaluate the adhesion between a single reinforcement fiber and a matrix, and even fewer for a rubber matrix. The majority of tests are derived from pull-out type tests [17,18] but may differ in terms of specimen geometry, loading conditions and/or test fixtures. The behavior of metal/elastomer interfaces is sometimes studied using analogous planar and more classical peel test configurations [19-21]. However, these are never strictly representative of real configurations. A peel test with a cord is possible if the material allows sufficient bending, for instance, an aramid cord [13].

In this context, the Rubber Cord Adhesion Inflation Test (RCAIT) was specifically developed to characterize the adhesion between a unit reinforcement and an elastomer matrix [22] to address the lack of alternative test configurations. This test, which is described in detail below, is similar to a tubular blister test [21,23,24] and consists in

<footnote>* Corresponding author.
E-mail address: pierre.corbel@ensta-bretagne.org (P.-Y. Corbel).</footnote>
debonding the rubber from the wire by injecting a pressurized fluid. This test configuration allows for the propagation of debonding under self-similar and stable conditions, and can be adapted to apply more complex loading conditions which would be more representative to in-service ones. To the best of our knowledge, no studies have been conducted today to investigate the effect of pre-loads on the debonding performance of rubber-wire specimens, nor on soft to rigid adhesive systems in general. The particular geometry of the RCAIT specimens allows us here to apply pre-loading sequences to the rubber before the crack propagation test without damaging the interface to study their effects on the crack propagation conditions. The primary objective of this contribution is to observe and analyze the impact of pre-softening on adhesive performance.

Indeed, to obtain reinforced rubber with optimum mechanical performance, it is not only necessary to carefully tune the adhesion between the matrix and the reinforcement through a suitable surface treatment or coating but also to use an appropriate rubber formulation [25,26]. Fillers of different natures, sizes, and shapes are generally added to the elastomer matrix to improve the strength, stiffness, toughness, or wear resistance [27]. Another important consequence of filler addition is the development of a rate-independent hysteretic response when a load-unload sequence is applied, known as stress softening or Mullins effect [28–30]. Other soft materials can exhibit a similar phenomenon, like hydrogels [31]. The elastomer behavior deviates from strict elasticity; when subsequent loads are applied, the stress versus elongation evolution follows that defined during the unloading path following the previous maximum applied load. If the stress exceeds the last maximum stress, the stress versus elongation evolution path returns to the curve corresponding to a virgin material. The physical origin of the Mullins effect is still debated in the literature. It is attributed to matrix-filler debonding, chain breakage, and interaction between bounded and free rubber or molecular slippage in connecting filaments. Softening can be partially recovered after a given period, accelerated by temperature or solvent swelling [30]. Finally, softening phenomena can lead to directional anisotropy, making the complete modeling of the phenomenon complex [30,32]. In addition to the large dissipation due to the first cycle softening, other hysteretic responses of smaller amplitude may be observed due to other phenomena such as material viscosity or strain-induced crystallization [30]. The consequence of a cycle softening on the rubber is a significant decrease in fracture toughness [31,33,34].

These effects are extensively studied on bulk rubber as softening procedures can be easily implemented on bulk specimens before any mechanical characterization tests, including crack propagation ones [31,34,35]. As mentioned above, the implementation of such a softening procedure before adhesion testing using standard peel test protocol [21] or more original ones is not reported, although it can drastically modify the measured adhesion performances of the system. The planar geometry and the presence of a rigid substrate make it difficult to apply a softening load sequence to the rubber layer. The RCAIT is used below to overcome these difficulties and to evaluate the effect of a softening sequence applied to the specimen on the resulting measured critical strain energy release rate (SERR) of the interface. We first demonstrate experimentally the effect of pre-softening on adhesion performance, with the description of the RCAIT method, pre-softening method, and the results. Then a finite element analysis evaluates the accommodated energy field during the pre-softening step. Those results are then incorporated to the energy balance analysis, which allows us to calculate the critical SERR of the samples after the pre-softening procedure. The remaining discrepancy between the experimental and theoretical/numerical results is finally discussed. The larger decrease in adhesion performance observed experimentally compared to the predicted ones could be explained with similar arguments that for the fracture in bulk rubber. For engineering purposes, the present work gives new insight for more reliable characterization and design of rubber-reinforced composite.

## 2. Experimental evidence of pre-softening influence on adhesion

This section present the RCAIT procedure, which is used to evaluate the adhesion between rubber and a single metal wire. It also introduces the pre-loading sequence, which is applied for pre-softening the rubber sheath prior to the crack propagation test. The RCAIT results on as-received and pre-loaded specimens demonstrate a significant reduction in adhesive performance for increasing pre-load intensity.

### 2.1. Rubber-wire adhesion test protocol

The RCAIT protocol [22] used to measure the critical SERR of a rubber-metal wire bonded specimen is illustrated in Fig. 1. The specimen is obtained by curing a cylindrical sheath of uncured rubber around a single steel wire using a special mould which ensures correct alignment of the wire with the axis of the cylindrical sheath. The exact composition of the rubber is not disclosed by our industrial partner. It consists of natural rubber, with 60 phr of carbon black and 7 phr of sulfur, which allows for cross-linking during the curing process. The cold-drawn steel wire is coated with a thin brass layer, approximately 0.05 to 0.1 µm thick, before undergoing the final wet-drawing process. The strong bond between the rubber and the metal wire is due to the formation of a covalent bond between the sulfur contained in the rubber and the copper from the brass coating combined with the growth of a dendritic structure at the interface [26]. An anti-adhesive tape is applied along one side of the wire to prevent the formation of such a covalent bond and create an artificial initial decohesion of 30 mm, leaving 70 mm of bonded interface. The pre-cracked side of the specimen is also terminated with a shoulder to facilitate clamping in the test fixtures (on left in Fig. 1a).

During RCAIT, the crack onset and propagation are achieved by injecting a pressurized fluid between the rubber sheath and the wire (see Fig. 1b). A clamp is then required to secure the specimen to the high-pressure hydraulic circuit and to ensure fluid tightness. Before crack propagation is observed, the rubber sheath first inflates. However, inflation of a hyperelastic tube is known to cause structural instability/localized deformation, also known as bulging [36] which could lead to failure of the rubber sheath rather than crack propagation. To prevent such an unexpected failure mode, the specimen is then placed in a confinement which limits diametrical expansion. A silicone lubricant is applied to the tube and specimen surfaces to achieve an almost frictionless contact condition between the confinement and the rubber sheath, allowing free longitudinal expansion [37]. During the test, the fluid is injected at a constant volume injection rate using a syringe pump system, the pressure is measured with the sensor connected to the hydraulic circuit. A typical test result is divided into two sequences (see Fig. 2). First, the initial inflation regime is observed, where the pressure versus strain and pressure versus injected volume evolution are mainly representative of the mechanical response of a thick-walled, confined rubber tube inflated by internal pressure. Then a constant pressure plateau is observed, corresponding to a steady-state crack propagation regime. Finally, as the crack tip approaches the end of the specimen, unstable failure with complete decohesion is observed. Further details of the experimental protocol and critical SERR evaluation can be found in [37].

From a macroscopic point of view, the RCAIT is a cylindrical version of the confined blister test [24]. In this sense, it belongs to the family of peel tests conventionally used to characterize the adhesion between a soft material and a rigid substrate. The values obtained are expressed in peel force per unit length, which can be equivalently expressed in terms of energy per unit area. This critical energy restitution rate therefore represents the work provided to the system to create a unit crack surface. In the case of soft materials, a significant proportion of the energy supplied to the system is stored in the deformable sheath, without participating to the fracture process, and must therefore be extracted from the balance by additional measurements or modeling. As
<figure><img src="image_5.png" /><figcaption>**Fig. 1.** Geometry and dimensions of the Rubber Cord Adhesion Inflation Test in (a) initial and (b) deformed configuration. The pre-softened region in the rubber sheath is represented with a darker shading. Two typical tensile test responses are associated with the reference and softened region.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Fig. 2.** Experimental evolution of the pressure with the injected volume during a RCAIT on a reference sample.</figcaption></figure>

mentioned in the introduction, a key challenge in analyzing adhesion tests on the rubber materials studied in this work is that significant energy dissipation occurs due to the Mullins effect during the initial deformation path. This dissipation occurs independently of any fracture process and far from the fracture process zone.

When evaluating the adhesion between rubber and a rigid substrate, it is important to consider the relative contribution of energy dissipation due to Mullins damage to the total energy required for crack initiation. While the fundamental quantities and phenomena involved in the energy balance remain, in principle, independent of the test method, their specific contributions do, in fact, depend on the material properties and deformation pathways during crack propagation. Consequently, these contributions can be influenced by the test conditions, including specimen geometry and loading conditions. For example, in a peel test, the peeled section is not only subjected to increasing tension but also to a bending load/unload cycle, so that local unloading can be expected [21]. Conversely, the RCAIT test and the confined blister test impose a monotonic strain evolution on the deformable section, preventing material unloading throughout the test [38]. As a result, the stress-stretch response can be effectively modeled using a simple hyperelastic law, identified from tensile test data. In service, rubber can be subjected to significant stresses and strains regardless of the presence of a crack, resulting in material softening and significant energy consumption even before a crack initiates or propagates. A major limitation of peel tests is their inability to apply a preconditioning loading sequence to the rubber, as the deformation is constrained by the high rigidity of the substrates. The particular geometry of the specimens used for RCAIT allows the rubber to be deformed while limiting the stresses applied to the interface. The protocol presented

below is therefore used to quantify the effect of pre-softening on the decohesion conditions of a wire/rubber interface.

## 2.2. Pre-softening method

Applying uniaxial tension to the specimen along the direction of the reinforcement would not allow sufficient deformation to be applied to the rubber, comparable to that measured during the actual test, as the stiffness of the reinforcement limits that of the rubber in the bonded area. Such a loading condition would only accommodate the rubber along the pre-cracked length without modifying the system in the bonded one. Diametrical compression tests are a standard method for determining the strength of brittle materials (see Brazilian tests [39]). These tests produce a heterogeneous stress and strain field across the specimen section and are not usually used to characterize materials such as elastomers. However, several studies have been carried out on configurations similar to the one studied here, including mechanical analyses of O-ring components [40], biological applications [41], and compression tests on spheres in the context of granular media modeling [42,43]. While applying diametrical compression cycles could be an interesting method to accommodate the rubber sheath on the length that adheres to the wire reinforcement, they would also lead to highly heterogeneous rubber softening, which would break the axisymmetric nature of the specimen and the test.

To achieve partial restoration of the specimen's axisymmetry, the diametrical compression sequences are repeated in the same zone, but with a change of the loading direction. This is achieved by rotating the specimen around the cord's axis between each sequence. By employing this method, the accumulation of Mullin's effect in different directions results in a more homogeneous rubber softening along the circumference (see 3b). A testing machine is equipped with compression plates on which an additive manufactured part is mounted to radially distribute 3 RCAIT specimens (see Fig. 3a). This allows the load applied to the plates to be balanced and centered so as not to damage the load cell. Furthermore, the specimen cross-section was observed throughout the loading and unloading sequences using a camera placed in front of the specimen. The load cell on this machine remains limited to 5 kN, which limits the softening that can be applied to the rubber when the full bonded length of the 3 specimens is subjected to diametrical compression. In addition, to enhance the pre-softening level, some specimens are repositioned on the tool to compress a shorter length $l_c$ placed at the bonded end. Fig. 3c illustrates the result of such a procedure in the case of an RCAIT specimen that has already undergone an initial pre-softening sequence over a length
<figure><img src="image_7.png" /><figcaption>Fig. 3. Successive diametrical compression protocol for RCAIT samples with experimental setup in (a) with 3 samples on the compression plateau, the axial view of the successive diametrical compression in the initial and deformed (drawn transparent) configuration in (b), the evolution of the force with displacement during a compression step in (c), and the sequence of rotations θ sequence applied to the RCAIT samples in (d).</figcaption></figure>

of 50 mm. The result shown here then corresponds to a second pre-softening sequence during which diametrical compression is applied over a length of 25 mm. This example illustrates the presence of the anticipated Mullins effect, as evidenced by a pronounced hysteretic response observed during unloading following the initial loading (black circle). Less pronounced hysteretic cycles are also observed during the subsequent cycles (white triangle), which are close to the initial un-loading. Stabilized cycles are observed starting from the second applied cycle, indicating additional dissipative mechanisms, such as material viscosity, and potential friction between the plates and the specimen. Regardless of the cause, the shift in mechanical response during the diametrical compression sequence demonstrates the effectiveness of this procedure in conditioning the rubber before performing the RCAIT. A first estimate of the energy dissipated can also be obtained from the force versus crosshead displacement response recorded during these pre-softening cycles. However, this estimate needs to be confirmed by more accurate analyses due to potential experimental errors, squeezing displacement uncertainties, edge effects, and friction.

As mentioned above, applying a loading sequence along a single diameter breaks the axisymmetric nature of the test. Therefore, this procedure is repeated in four orientations θ of the specimen in the sequence [0; π/2; π/4; 3π/4] as illustrated in Fig. 3b and Fig. 3d. It is essential to follow this cycle sequence precisely, rather than using 45° increments, to ensure that the Mullins damage distribution resulting from the preceding loads and induced by the current one remains symmetrical with respect to the applied load direction. This point will be further explained using numerical simulations. The pre-softening tests are conducted here by imposing loading paths with an imposed transverse velocity (1 mm/min) and by reversing the loading direction when the machine capacity is reached (i.e. 5kN).

## 2.3. Results of adhesion on pre-softened samples

Once the pre-softening procedure is complete, the adhesion between the wire and the softened rubber is evaluated using the RCAIT [22]. The resulting inflation curves for the nominal and the two pre-softening levels are presented in Fig. 4.

Following the critical inflation pressure that governs the onset of crack propagation, a pressure plateau is observed corresponding to a steady-state crack growth regime. The black curves represent the results obtained on the reference specimen which were not subjected to any pre-softening cycles. The second specimen (red line) was subjected to a cyclic pre-load over a 50 mm length (33 N/mm). The third specimen (blue line) was also subjected to the same loading sequence, followed by an additional one over a 25 mm length (66 N/mm) to enhance the softening of the rubber. Two distinct successive plateaus are then observed on the P(ΔV) evolutions for these specimens as the crack propagates along the pre-accommodated lengths. The drop in pressure

<figure><img src="image_8.png" /><figcaption>Fig. 4. Experimental results of RCAIT compares samples in reference state and two levels of softening, on the pressure–volume evolution (left) and on the observed critical pressure plateau with applied pre-softening compression level (right).</figcaption></figure>

plateau is indicated with a vertical dotted line and marked with arrows. These observations indicate that the pre-softening procedure effectively reduced (at least partially) the Mullins dissipation during the crack propagation sequence, since the energy required to inject the fluid and then to extend the rubber/cord cohesion is reduced. Consequently, the global energy dissipated and resulting specimen critical SERR should also be reduced [38]. However, the rubber sheath stiffness is modified by the pre-softening cycle and the subsequent energy balance analysis must account for it to evaluate correctly the stored energy of the pre-softened samples [33,34].

# 3. Numerical evaluation of pre-softening distribution

The diametrical compression procedure generates complex stress and strain fields across the specimen's cross-section, resulting in a heterogeneous distribution of softening within the rubber sheath. To quantitatively assess the softening induced by the four successive cyclic compression sequences, finite element simulations are performed.

## 3.1. Rubber constitutive behavior

Before performing any finite element simulations, an appropriate model must be identified to accurately capture the mechanical behavior of the rubber under cyclic loading conditions. The model should, at a minimum, replicate the pre-softening observed during the first unloading sequence caused by the Mullins effect. The stress versus elongation evolution recorded during a uniaxial tension test is fitted with a hyperelastic model based on a strain energy potential proposed by Khajehsaeid et al. [44], and defined by

$$ \tilde{W} = A \left[ \frac{1}{a} \exp(a(I_1 - 3)) + b(I_1 - 2)(1 - \ln(I_1 - 2)) - \frac{1}{a} - b \right], \quad (1) $$
<figure><img src="image_9.png" /><figcaption>Fig. 5. Experimental evolution of nominal stress against stretch in tensile load (strain rate at 1%/s) compared to the analytic solution with identified material parameters.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 6. Superposition of the finite element discretization on the experimental sample picture in initial (a) and deformed (b) configuration during the diametrical compression at $F/l_c = 66$ N/mm with the applied boundary conditions in (c).</figcaption></figure>

<caption>**Table 1**</caption>

Parameters of the hyperelastic part (A, a, b) and the hysteretic part (r, m, β) of the mechanical behavior model.

<table><thead><tr><th>A</th><th>a</th><th>b</th><th>r</th><th>m</th><th>β</th></tr></thead><tbody><tr><td>1.14 MPa</td><td>-0.96</td><td>-0.91</td><td>1.73</td><td>0.63</td><td>0.10</td></tr></tbody></table>

where $I_1$ is the first invariant of the left Cauchy-Green deformation tensor $I_1 = \text{tr}(\mathbf{B})$. The Cauchy-Green tensor is related to the transformation gradient tensor with $\mathbf{B} = \mathbf{F}\mathbf{F}^T$. The potential requires the identification of 3 parameters including the modulus A. Parameters a and b respectively control the behavior at low elongation and the stiffening that occurs as elongation increases.

The Mullins effect, characterized by rate-independent hysteresis during the initial loading cycle, is indicative of material softening in rubber. To accurately capture this phenomenon, a range of models of varying complexity have been proposed in the literature [30]. The simple pseudo-elastic model proposed by Ogden and Roxburgh [45] is readily available in commercial finite element software like Abaqus. This model, chosen for the present work, modifies the hyperelastic strain energy potential $\tilde{W}$ by a factor $\eta$, which depends on the maximum strain energy encountered during the loading history. The modified strain energy potential is then expressed as

$$W(I_1, \eta) = \eta \tilde{W}(I_1) + \phi(\eta). \qquad (2)$$

The scaling parameter $\eta$ remains equal to 1 during initial loading and decreases to $\eta_{min}$ during unloading. The original formulation [45] for $\eta$ is

$$\eta(W_{max}, \tilde{W}) = 1 - \frac{1}{r} \text{erf} \left( \frac{W_{max} - \tilde{W}}{m + \beta W_{max}} \right), \qquad (3)$$

where m, β, and r are material parameters and $W_{max}$ is the maximum strain energy experienced by the material. The relation between dissipated energy $\phi(\eta)$ and strain energies is defined by

$$\phi(\eta) = \frac{m + \beta W_{max}}{r\sqrt{\pi}} \left[ \exp\left(-\frac{W_{max} - \tilde{W}}{m + \beta W_{max}}\right)^2 - 1 \right] + (1 - \eta)W_{max}. \qquad (4)$$

To identify the material parameter values, a tensile test was performed on a dumbbell-shaped rubber specimen at a constant strain rate of 1%/s. Three increasing load-unload cycles were followed by a final cycle to failure. For identification purposes, unloading paths were rescaled to eliminate residual strain similarly to the work of Ogden and Roxburgh [45]. Fig. 5 compares experimental data with model predictions using the identified parameters, listed in Table 1.

## 3.2. Finite element modeling of diametrical compression procedure

Finite element simulations were performed to gain insight into the energy dissipation and softening of the rubber sheath during four successive cyclic diametrical compression tests. While a full 3D finite element model would be ideal for capturing the complex strain, stress, and damage fields, the computational cost is prohibitive. Simplified 2D simulations were therefore used. The experimental data presented in Fig. 4 show abrupt pressure drops during crack propagation tests on pre-loaded specimens, indicating a relatively short transition zone between undamaged and damaged material. This observation supports the use of 2D simulations, which can provide valuable insights into the overall material behavior with a limited computational cost.

The specimen cross-section is divided into two distinct regions: a rigid wire with a radius of 0.65 mm and a rubber sheath with an outer radius of 4.575 mm. The rubber sheath is discretized using hexahedral elements with quadratic interpolation, full integration, and a hybrid pressure-displacement formulation. To accurately represent the loading conditions along most of the specimen length between the compression plates, plane strain conditions are imposed by constraining the displacement of the element nodes in the z-direction (see Fig. 6c). The decision to use hexahedral elements rather than simpler 2D elements is motivated by the subsequent energy analysis, which is more straightforward with volumetric elements. Fig. 6a and b show the undeformed and deformed geometries together with the finite element mesh, superimposed with the experimental observation of the deformed specimen.

The Exp-Ln hyperelastic model [44], calibrated with the material parameters identified in Section 3.1, is implemented in Abaqus through the UHYPER user subroutine. The Ogden-Roxburgh model [45] readily available in Abaqus is used to consider the effects of Mullins damage dissipation. Assuming a perfect bond between the rubber sheath and the wire, kinematic coupling enforces displacement continuity between the two components.

To simplify the numerical evaluation of the loading sequences, the axial displacement of the wire is constrained to zero throughout the simulation. The angular orientation of the entire geometry is adjusted between each step to accurately mimic the experimental protocol. The experimental loading conditions of the cross-section are reproduced by placing the mesh between two rigid solid planes. A surface contact condition with a Coulomb friction coefficient of $\mu = 0.3$ is used to model the interaction between the rubber and the compression plates. In order to ensure symmetrical deformation, an opposite displacement is imposed on the top and bottom plates. The maximum relative displacement between the two plates is set to match the experimental measurements obtained with the camera.

## 3.3. Results of finite elements

Fig. 7 presents the evolution of the accumulated dissipated energy density across the specimen cross-section after each of the four diametrical compression steps (0, $\pi/2$, $\pi/4$, $3\pi/4$) at the two compression level (33 and 66 N/mm). The initial application of a diametrical compression loading along a single direction results in highly localized energy dissipation and subsequent softening, mainly along the vertical and horizontal diameters of the cross-section.

After the application of four compression cycles, the energy dissipation becomes more evenly distributed in the vicinity of the wire-rubber interface, where damage and crack propagation are expected
<figure><img src="image_11.png" /><figcaption>Fig. 7. Map of the dissipated energy by softening $\phi(\eta_{min})$ (mJ/mm$^3$) on a section of the sample at the end of each compression step for $F/l_c$ = 33 N/mm (top) and $F/l_c$ = 66 N/mm (bottom).</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 8. Map of the maximum principal strain on a sample section at the peak of each compression step $F/l_c$ = 66 N/mm in the undeformed configuration.</figcaption></figure>

to occur during the RCAIT. However, lobe-like patterns persist at a distance of approximately $3R_{wire}$ from the interface. Significant but non-axisymmetric energy dissipation and softening due to the discontinuous pre-loading process is then still observed with an angular periodicity of $\pi/4$. However, during the RCAIT, a rapid decrease in rubber deformation in the radial direction from the rubber/wire interface is predicted. Consequently, the circumferential variation in softening, which is more pronounced farther away from the interface, is not expected to significantly affect the axisymmetric nature of crack propagation at the crack tip scale. Global energy dissipation is also expected to be mainly influenced by the radial evolution of the average circumferential dissipated energy density. Finally, the results presented in Fig. 7 show that the overall distribution of dissipated energy remains relatively consistent for both loading cases. While material nonlinearity and geometric effects, such as boundary conditions and large deformations, do not significantly alter the spatial distribution of softening, they do influence the overall intensity of the softening. These results are then used to consider the global effect of the pre-loading sequence in the subsequent RCAIT analysis.

The diametrical compression steps could lead to the formation of micro-cracks or even the debonding between the rubber matrix and the wire. This would also strongly contribute to the reduction in crack propagation of the specimen. However, experimental observations and numerical results suggest that such damage is very unlikely. The numerical results presented in Fig. 4 illustrate the deformation levels within the matrix for each accommodation step at the maximum load level. The maximum principal strain observed across the specimen cross-section under the most severe loading condition ($F/l_c$ = 66 N/mm) is approximately 1.15 (see Fig. 8), corresponding to an elongation ratio of $\lambda$ = 3.15. This value remains significantly lower than the ultimate elongation at break ($\lambda_{rupt}$ = 4), as determined from tensile tests on dumbbell specimens. Therefore, it is unlikely that crack initiation occurs in the matrix during the pre-softening diametrical compression

sequence. Additionally, cohesive failure is observed in all cases due to strong bonding between the rubber and the cords thanks to the brass coating. There is no evidence of adhesive/interfacial failure on the specimens that were subjected to softening loading. Finally, the most severe loading conditions during the diametrical compression sequences are expected on the edge of the bonded length, where the plane strain condition is relieved and shear stress is present along the rubber cord interface. No evidence of crack onset has been found from camera observation or that would modify the crack onset regime during the initial inflation sequence of the RCAIT. Therefore, we can conclude that the drop in critical pressure during crack propagation can only be attributed to the softening of the rubber sheath, aside from any other potential damage mechanisms.

## 4. Calculation of critical SERR after pre-softening

The RCAIT method allows the determination of the critical SERR that governs the decohesion of the rubber sheath from the wire by considering a global energy balance and an infinitesimal extension of the crack. Accurate quantification of the energy delivered by the injection of the pressurized fluid, which results in the inflation of the rubber sheath is critical. A semi-analytical model was previously developed to describe the inflation of a thick-walled, homogeneous, hyperelastic tube, to compute the energy balance and determine the SERR [22]. The softening induced by the preloading sequence was evaluated numerically using finite element simulations in the previous section. Rather than modifying the semi-analytical model [22], we use the finite element model again to compute the energy needed to inflate the presoftened rubber sheath as it would be experienced during RCAIT. It is essential to accurately update the mechanical behavior of the rubber sheath, particularly accounting for the effects of the diametrical compression accommodation sequences, to determine the specimen's critical SERR. While the critical fluid pressure driving
<figure><img src="image_13.png" /><figcaption>Fig. 9. Principle of energy balance analysis of Rivlin and Thomas [47] between a section of solid ahead of the crack (adhering to the cord) and one behind (under inflation). The stored energy variation $\delta E$ corresponds to the sheath deformation and the exterior work variation $\delta W_{ext}$ corresponds to the pressurized fluid injection. The dark shaded area on the tube section represents the softening obtained with the cyclic diametrical compression protocol.</figcaption></figure>

crack propagation serves as a measurable macroscopic quantity, it is inherently dependent on both the specimen geometry and the stress-stretch response of the material surrounding the crack path. Therefore, ensuring a proper characterization of the rubber's mechanical response is crucial for evaluating whether the critical SERR can be considered an intrinsic material property.

## 4.1. Energy balance analysis

In the framework of the elastic fracture mechanics of Griffith [46], the SERR $G$ relates the energy dissipated divided by the crack surface increases. It can be obtained by balancing the variation of the stored energy, $E$, with the work provided to the system, $W_{ext}$, through the application of external loads and then divided with the variation of cracked surface $S$. The variation of both energies can be evaluated with the energy balance analysis of Rivlin and Thomas [47]. The balance between the bonded section and a cracked and fully inflated section after an infinitesimal crack length increment $\delta a$ is obtained with the relation:

$$G = \frac{\delta W_{ext} - \delta E}{2\pi R_{wire} \delta a}, \qquad (5)$$

as illustrated in Fig. 9. The term $\delta W_{ext}$ represents the additional energy required to inject the pressurized fluid, which induces interfacial decohesion along an infinitesimal length $\delta a$. Note that the surface created by the crack advance is a tubular surface, with a radius equal to the one of the wire $R_{wire}$. It is calculated with the equation:

$$\delta W_{ext} = P\pi(r_{int}^2 - R_{wire}^2)\lambda_z\delta a = P(s_{int} - \pi R_{wire}^2)\lambda_z\delta a, \qquad (6)$$

where $r_{int}$ represents the deformed inner radius of the rubber sheath. It is equivalent to $s_{int} = \pi r_{int}^2$ if the sheath deformation is axisymmetric. The extension of the cracked portion of the specimen is greater than $\delta a$ owing to the longitudinal stretch $\lambda_z$, which serves as a scaling factor. In the case of hyperelastic rubber behavior, the stored energy increment $\delta E$ in the rubber sheath can be determined using the following relation:

$$\delta E = 2\pi\lambda_z\delta a \int_0^{2\pi} \int_{r_{int}}^{r_{ext}} W(r, \theta)r \, dr \, d\theta, \qquad (7)$$

where $W$ represents the hyperelastic strain potential used to derive the stress-strain behavior, while $r_{ext}$ is the external radius of the rubber sheath in its inflated state.

Applying Eq. (7) to pre-loaded specimens becomes complex due to the non-uniform development of Mullins softening over the cross section and the global reduction in stiffness. This requires replacing of the function $W$ with the modified expression introduced in Eq. (2), but also taking into account the non-axisymmetric nature of the rubber

sheath expansion through numerical simulations. During the propagation of the decohesion, a steady-state regime is observed, characterized by constant pressure and no unloading. This justifies the use of Eq. (5) to approximate the energy balance condition driving infinitesimal decohesion growth, given that only direct stress-strain load paths are involved in the description of the rubber sheath expansion. The critical SERR $G_C$ is computed with Eq. (5) at the critical pressure during crack propagation (see right axis of Fig. 4). However, the interpretation of $G_C$ requires clarification. The dissipated energy is global, making it difficult to distinguish between the non-recoverable energy dissipated during rubber sheath expansion and the energy dissipated locally near the crack tip due to damage and other non-linear processes driving the extension of the reinforcement-matrix decohesion. However, when considering the direct load path, the evaluation of the energy required to inflate the rubber sheath naturally includes both reversible potential energy and dissipated one. The main difficulty is then to anticipate the effect of presoftening on the inflation of the rubber sheath from the simulations when it cannot be determined experimentally because the rubber is bonded to the cord, although it is necessary for consistent energy balance analysis.

The definition of $\delta W_{ext}$ remains unchanged; however, the work required to inflate the rubber sheath at a given pressure now depends not only on the fluid pressure but also on the intensity of the load applied during the pre-softening process. Whilst semi-analytical models have been used previously to evaluate the total energy required for rubber inflation [22], they are unsuitable for radially and circumferentially varying material stiffness. However, as shown in Fig. 7, the pre-softening procedure induces rubber softening that is not purely axisymmetric. Therefore, numerical simulation is preferred for determining $\delta E$ as it offers greater accuracy and flexibility compared to the approximations inherent in semi-analytical approaches.

## 4.2. Finite element computation of the modified stored energy

To update Eq. (7) and to take into account the stiffness degradation due to the pre-softening sequence, the $\delta E$ value is then determined numerically using the finite element model described in Section 3.2. After simulating the pre-softening sequence, the boundary and loading conditions are modified to determine the evolution of $P$ versus $\delta V$ during the inflation of a unit length of the thick rubber sheath under conditions representative of those observed experimentally. In particular, the inner radius is decoupled from the wire by replacing the kinematic coupling with a uniform pressure distribution. A single layer of elements is considered along the z-direction with a length of 1 mm, corresponding to $\delta a$ in Eq. (7). On one side, the z-displacement is set to zero while the elements are free to expand in the radial direction. On the other side surface elements are used to enforce the same longitudinal displacement of all nodes using kinematic coupling but again allowing free radial expansion. An axial force $F_z = P_{int} \cdot s_{int}$ is uniformly distributed to the surface elements which is representative of the closed-end boundary condition. The non-axisymmetric nature of the rubber behavior due to the circumferential Mullins damage distribution has a small but visible effect on the radial expansion of the tube. Slight circumferential variation of the inner wall radius can be seen in Fig. 9. These variations are taken into account in the evaluation of $s_{int}$. Finally, the confinement of the specimen with the lubricated glass tube is modeled by using a surface contact condition between the outer wall of the rubber tube and a solid cylinder surface representing the inner wall of the confinement tube. Frictionless contact is used here as it has been experimentally validated previously [37].

Simulations are performed for both the reference case of a non-softened specimen and specimens subjected to pre-softening sequences for two different load levels. The softening and energy dissipation distributions resulting from the preloading sequence simulations are used as the initial state for the inflation regime simulation. The resulting normalized variations of the inner volume with the inflation pressure
<figure><img src="image_14.png" /><figcaption>**Fig. 10.** Comparison of the evolution of normalized inner volume variation with inflation pressure for reference and softened sheaths at two $F/l_c$ compression levels. The theory for the reference state is from derived from Kane et al. [22]. The annotations a,b,c and d corresponds to the loading states of the dissipation maps of Fig. 11.</figcaption></figure>

for the different diametrical compression steps are shown in Fig. 10. The results obtained with the numerical simulation for the reference (without pre-softening) configuration are in close agreement with those obtained using the semi-analytical resolution [22]. The effect of the pre-softening process is observable for pressures below ca. 6 MPa where the stiffness decrease is noticeable. The reduction in stiffness becomes more pronounced with increasing compression force during the diametrical compression, with the most compliant response being observed at 66 N/mm. Beyond approximately 6 MPa, the reference response is restored (5.5 MPa for $F/l_c$ = 33 N/mm, 6.5 MPa for $F/l_c$ = 66 N/mm), which means that above this pressure value, the majority of the rubber sheath has reached or exceeded an energy density greater than that experienced during the pre-softening protocol. This is confirmed by the evolution of the distribution of the energy dissipated by Mullins across the rubber tube section, shown in Fig. 11 for 4 inflation pressure values (i.e. 0, 3, 5 and 7 MPa) and $F/l_c$ = 33N/mm. These color maps are presented in the undeformed state for clarity. The normalized volume and pressure corresponding to each dissipation map of Fig. 11 is reported in Fig. 10 with the filled symbol and annotation a,b,c and d. It can be seen that the damage distribution due to the pre-softening is gradually overpassed by that due to inflation. In the inflation regime, the higher energy density is located at the inner wall position where the elongations are maximum. Up to about 7 MPa inflation pressure, the damage field from the pre-softening sequences is still visible but only by the presence of the lobes marking the diametrical load directions which are progressively covered by the newly created axisymmetric energy front distribution.

The linear energy required to inflate a unit length of the rubber sheath at a given pressure is shown in the inset of Fig. 10. Unlike semi-analytical hyperelastic models, this approach allows the total energy to be decomposed into recoverable potential energy and energy dissipated through Mullins damage. Fig. 10 presents the evolution of the linear energy $\delta E/\delta a$ as a function of pressure $P$ for the three cases studied previously: a virgin specimen and specimens preloaded with transverse loads of respectively $F/l_c$ = 33 N/mm and $F/l_c$ = 66 N/mm. The difference between these curves may seem difficult to discern and primarily noticeable at the early stage of inflation (i.e. $P_{int} < 5$MPa) when the local energy density due to the inflation remains below the level applied during the pre-softening process, and the effects of rubber softening are still evident. As expected, softened specimens require less energy to reach a given pressure, with the energy reduction increasing as the load applied during the pre-softening process rises. While the effects of pre-softening are no longer apparent on the inflation curve beyond the critical pressure — where the local energy density surpasses that encountered during the pre-softening sequence — they remain evident in the energy evolution curve due to the cumulative nature of

stored energy. In other words, the energy dissipated during the pre-softening step is permanently “erased” from the stored energy creating an offset in the energy curve. Beyond the critical pressure, the curves are parallel, indicating that the rubber has recovered its nominal behavior. The offset along the $\delta E/\delta a$ axis represents the energy dissipated during the pre-softening process.

The information required to calculate the SERR, $G$, using Eq. (5) for various pressure values are obtained from the inflation curves and the results of Abaqus simulations. To avoid any assumptions regarding the axisymmetric nature of the deformation, the inner surface $s_{int}$, is directly computed from the radial positions of the nodes of the inner wall in the deformed state. This leads to a more rigorous evaluation of $\delta W_{ext}$. As anticipated, deviations from fully axisymmetric deformation decrease as the inflation pressure increases due to the recovery of reference rubber behavior. Additionally, Abaqus provides a direct method to determine the energy required to inflate the rubber sheath by using the variable ALLIE.

### 4.3. Results of the finite element energy balance analysis

Finally, the predicted evolutions of the SERR as a function of fluid pressure, obtained from the previous analysis, is presented in Fig. 12 for the three test cases: nominal (unaccommodated - black markers), preload of 33 N/mm (red markers) and preload of 66 N/mm (blue markers). The SERR evolution for the nominal case was also evaluated using the semi-analytical solution presented by Kane et al. [22], assuming hyperelastic behavior. The theoretical prediction (solid black curve) confirms the accuracy of the numerical simulations, despite their inherent numerical approximations and complexity due to complex boundary conditions, geometry, and material non-linearities. The $G(P)$ calibration curves are very similar and again show parallel evolution above the critical pressure when the rubber sheath returns to its original behavior. For a given pressure value, the global specimen SERR increases with increasing applied load during the pre-softening process. This is consistent with the reduction in rubber sheath stiffness and the expected softening effect. Therefore, for the same failure condition at the rubber-wire interface scale, it is expected that the sole pre-softening procedure will result in a drastic reduction in the critical inflation pressure. Contrary to the common interpretation of adhesion tests, especially those dealing with rubber-wire adhesion and more generally peel tests, the critical load value (here the inflation pressure) does not exclusively reflect the interface performance, and the energy dissipation is not localized only in the vicinity of the interface. The evaluation of the energy dissipation due to the interface separation process should be carefully quantified in the total, which is a common limitation of such tests. Therefore, the critical fluid pressure that drives the expansion of decohesion should be considered as a global quantity that characterizes the overall performance of the specimen, but can also vary significantly for the same interface quality depending on the mechanical properties of the rubber sheath. An accurate evaluation of $G_c$ then requires a precise recalibration of the curve from which the critical SERR is determined from the measured critical pressure value.

To accurately determine the specimen critical SERR, $G_c$, from the measured pressure values for the different specimen types, the $G(P)$ calibration curves must be updated unless the $G_c$ values of the preloaded specimens would be underestimated. Using the calibration curves obtained from the Abaqus simulations, it is found that $G_c$ decreases with increasing preload intensity, regardless of the bulk dissipation in the rubber sheath. Specifically, a significant reduction of approximately 20% and 35% in $G_c$ was observed for a preload of 33 N/mm and a preload of 66 N/mm, respectively, highlighting the impact of the specimen softening on the overall adhesive performance of the specimen as evidenced by the decrease in injected fluid pressure but also on the fracture resistance of the rubber/wire interface, which is rigorously evaluated here apart by reevaluating of the energy required to inflate the rubber sheath. It should be noted that without recalibration of
<figure><img src="image_15.png" /><figcaption>**Fig. 11.** Map of the dissipated energy density by softening $\phi(\eta_{min})$ (mJ/mm$^3$) on a section of the sample during the inflation step for $F/l_c$ = 33 N/mm in the undeformed configuration.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>**Fig. 12.** Comparison of the evolution of the SERR with inflation pressure for reference and softened sheaths at two $F/l_c$ compression levels. The inset is the variation of critical SERR normalized by the critical SERR of the reference with the Mullins effect dissipation during the diametrical compression protocol. The theory for the reference state is from Kane et al. [22].</figcaption></figure>

the $G(P)$ curve to account for specimen softening, the reduction in $G_c$ would have been overestimated, resulting in apparent reductions of 30% and 55% for a preload 33 N/mm, and a preload 66 N/mm, respectively.

## 5. Discussion

The RCAIT is proposed to characterize the critical SERR, which quantifies the adhesive performance of a rubber material bonded to a metal wire. A key advantage of the RCAIT is its ability to ensure a steady-state crack propagation regime over the entire crack propagation length, which facilitates the establishment of a rigorous energy balance analysis. In this context, a method is proposed to investigate the effect of rubber softening before testing by applying four successive diametrical compression loads, trying to maintain the axisymmetric condition inherent to the RCAIT. The stress-softening from the pre-softening step decreases the sheath stiffness and this decrease is accounted for in the energy balance analysis thanks to a finite element model. After a rigorous update of the experimental data reduction framework to take into account the modification of the rubber behavior due to the application of pre-accommodation cycles, it results that a visible reduction of the critical SERR is observed with increasing amplitude of the preload cycles, this effect being distinct from the reduction of the total energy required to propagate the crack along the specimen. It should be noted that this effect is controlled by the rubber close to the interface with the wire, as cohesive failure was observed in all cases. This decrease can be interpreted with the mechanical behavior of the rubber matrix, where an already pre-softened material point will dissipate less energy during the loading-unloading path (if the strain energy viewed during the pre-loading sequence is locally overpassed by the strain energy viewed during the crack propagation) as the crack pass-by as demonstrated by the work of Qi et al. [33].

A comprehensive interpretation of the reduction in $G_c$ will require a more detailed analysis of the various phenomena driving the interface

decohesion and quantifying the energy dissipation contributions associated with these mechanisms, but focusing on the near interface region. Among these analyses, the most important is to describe more precisely the load path followed by the rubber at different radius positions during the steady-state crack propagation regime. While the direct load path studied in Section 4 provides useful insight, it is not fully representative of the one that should be experienced during the crack propagation regime [38]. This discrepancy is particularly significant in the vicinity of the interface crack tip region, where more complex load paths and stress distributions are likely to occur. The presence of a highly dissipative medium to carry the load to the interface complicates the evaluation of the local properties of the interface from macroscopic testing and analysis. Indeed, accurate modeling of the fracture process zone is required for more rigorous evaluation of the local stress/strain paths during crack propagation, consistent evaluation of the dissipated energies, and accurate identification of the contribution of individual specimens and component properties to the resistance to crack propagation. The distribution of softening is heterogeneous in the sheath (see Fig. 7) and the interaction between the crack tip and a gradient of damage is still an open question. These findings highlight the need for more comprehensive and local analyses of the interface failure process, with a particular emphasis on modeling the irreversible phenomena that occur in the vicinity of the crack tip, as carried out on bulk material [31,33,34]. The particular configuration of a soft dissipative solid bonded to a rigid substrate increases the complexity, and the effect of mode mixity can interplay with the Mullins effect: previous work showed that the mode mixity varies with the critical pressure and can modify the global dissipation [48].

The engineering implication of the present article for elastomer composites is to gain insight into the reason leading to the reduction in global fracture resistance and adhesion performance between components after the application of accommodation cycles. Two scales and contributions should now be considered separately, the whole matrix scale and the near interface region. For instance, Valentin et al. [13] reports a decrease of peel strength between rubber and polyamide reinforcement after a fatigue loading as a transmission belt application (out-of-plane loading of a rubber-polyamide fiber ply and bending). The sharp evolution between the virgin specimen (peel force of 3.2 N/mm) and a specimen with 2 million cycles (peel force 2.8 N/mm) could be explained, based on the present study, by the softening of the rubber matrix. A significant decrease (20% and 35%) of the critical SERR of the interface between the reinforcement and the rubber matrix is observed with our pre-softening protocol (see Fig. 12). Besides, most rubber composite fatigue and aging studies are done on rubber composite and completed by single reinforcement adhesion tests [7,12,14]. However, the present study suggests that rigorous analysis is needed to accurately determine $G_c$ and more generally, to evaluate the mechanical performance of the interface since the evolution of the surrounding matrix due to complex and/or preload can significantly alter the response of the whole specimen at constant adhesive performance. Such improved analysis is relevant when testing configurations which are used on car tires or transmission belts which are subjected to cyclic loading conditions. In addition, the sequential diametrical compression introduced to soften the rubber can be applied or adapted to specimens other than those used for RCAIT, such as single reinforcement pull-off specimens.
## 6. Conclusion

The RCAIT procedure was carried out on specimens in their reference condition and specimens subjected to two different levels of matrix pre-loading. The results revealed a significant reduction in the critical pressure required for matrix-reinforcement interface decohesion as the preload level was increased. Finite element analysis was subsequently used to evaluate the softening of the rubber sheath numerically and the energy dissipation during preloading. The Ogden-Roxburgh model was used here to describe the Mullins effect and the hysteretic response observed during the loading and unloading cycles. This model also calculates the energy required to inflate the rubber tube during RCAIT, which is essential for energy balance analysis and determining critical SERR. The energy balance analysis emphasizes that the critical pressure which drive the crack propagation accounts for the total dissipative effects occurring throughout the specimen during the crack propagation. To evaluate the $G_C$ value which represent more intrinsically to properties of the rubber-wire interface/interphase the energy balance analysis should be updated to consider the softening of the rubber sheath behavior. After appropriate calibration and accounting for the stiffness decrease by the Mullins effect in the rubber bulk, $G_C$ values are obtained which exhibit small but significant reduction of $G_C$ due to application of accommodation cycles.

### CRediT authorship contribution statement

**Pierre-Yves Corbel:** Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Julien Jumel:** Writing – review & editing, Supervision, Resources, Project administration, Funding acquisition.

### Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: P.-Y. Corbel reports equipment, drugs, or supplies was provided by Michelin Technology Center in Ladoux. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgments

The authors wish to acknowledge Dr. T. Rey of MFP Michelin for industrial support. P.-Y. Corbel acknowledges the Region Bretagne (ARED grant) and the ENSTA Bretagne for the PhD funding. All authors have read and approved the manuscript.

### Data availability

Data will be made available on request.

### References

[1] Rao S, Daniel IM, Gdoutos EE. Mechanical properties and failure behavior of cord/rubber composites. Appl Compos Mater 2004;11(6):353-75, Number: 6.

[2] Meng Q, Chang M. Interfacial crack propagation between a rigid fiber and a hyperelastic elastomer: Experiments and modeling. Int J Solids Struct 2020;188:189-141-54.

[3] Liechti KM, Wu. Mixed-mode J-D. Time-dependent rubber/metal debonding. J Mech Phys Solids 2001;49(5):1039-72, Number: 5.

[4] Ning S, Su B, Liu S, Zhou T, He S, Wu J, et al. Failure behaviors of cord-rubber composite materials under mixed mode I/II loading: Experimental and numerical simulation. Mater Today Commun 2024;41:110349.

[5] Shi C, Zhao Y, Zhu X, Wan X, Li B. Study on the shear failure mechanism of vulcanized rubber-metal bonding interface based on cohesive zone model. Int J Adhes Adhes 2024;134:103807.

[6] Chen L, Gu B. Predicting fatigue damage in interphase of short fiber reinforced rubber composites under transverse load. Polym Compos 2018;39(5):1455-67.

[7] Pisaneschi G, Brugo TM, Gosseddu P, Scalet G, Zucchelli A. Enhanced interfacial adhesion in shape memory alloy hybrid composites via an elastomeric interface: An experimental and numerical investigation. Compos Part B: Eng 2024;286:111785.

[8] Creton C, Cicotti M. Fracture and adhesion of soft materials: a review. Rep Progr Phys 2016;79(4):046601.

[9] Huang W, Li Y, Zhao H, Wang W, Yu B, Ning N, et al. A new eco-friendly dipping system for PA66 fiber cords/rubber composites with strong interfacial adhesion and good fatigue stability. Compos Part B: Eng 2023;253:110541.

[10] Rey T, Razan F, Robin E, Faure S, Le Cam J-B, Chagnon G, et al. Mechanical characterization and comparison of different NiTi/silicone rubber interfaces. Int J Adhes Adhes 2014;48:67-74.

[11] Tonatto MLP, Forte MMC, Amico SC. Compressive-tensile fatigue behavior of cords/rubber composites. Polym Test 2017;61:185-90.

[12] Luo G, Guo J, Zhang C, Yang X. Life prediction of cord/rubber laminates under multiaxial fatigue. Int J Fatigue 2023;174:107733.

[13] Valantin C, Lacroix F, Deffarges M-P, Morcel J, Ait Hocine N. Interfacial damage on fatigue-loaded textile-rubber composites. J Appl Polym Sci 2015;132(4).

[14] Eitzen AV, Flamm M, Steinweger T, Weltin U. On the lifetime prediction of rolling lobe air springs. Eng Fail Anal 2018;94:313-26.

[15] Popa CM, Gebhardt C, Raje N, Steenwyk B, Kaliske M. Investigation of cord-rubber composite durability by the material force method. Eng Fract Mech 2020;229:106909.

[16] Jamshidi M, Afshar F, Mohammadi N, Pourmahdian S. Study on cord/rubber interface at elevated temperatures by H-pull test method. Appl Surf Sci 2005;249(1):208-15, Number: 1.

[17] Standard test method for adhesion between tire bead wire and rubber. 2014, Issue: D1871-04(2014).

[18] Standard test method for adhesion between steel tire cords and rubber. 2014, Issue: D2229-10(2014).

[19] Cook JW, Edge S, Packham DE. The adhesion of natural rubber to steel and the use of the peel test to study its nature. Int J Adhes Adhes 1997;17(4):333-7, Number: 4.

[20] Neggers J, Hoefnagels JPM, van der Sluis O, Sedaghat O, Geers MGD. Analysis of the dissipative mechanisms in metal-elastomer interfaces. Eng Fract Mech 2015;149:412-24.

[21] Bartlett MD, Case SW, Kinloch AJ, Dillard DA. Peel tests for quantifying adhesion and toughness: A review. Prog Mater Sci 2023;137:101086.

[22] Kane K, Jumel J, Lallet F, Mbiakop-Ngassa A, Vacherand J-M, Shanahan MER. A novel inflation adhesion test for elastomeric matrix / steel cord. Int J Solids Struct 2019;160:40-50.

[23] Dannenberg EM. The effects of surface chemical interactions on the properties of filler-reinforced rubbers. Rubber Chem Technol 1975;48(3):410-44.

[24] Chang Y-S, Lai Y-H, Dillard DA. The constrained blister—A nearly constant strain energy release rate test for adhesives. J Adhes 1989;27(4):197-211.

[25] Crowther BG. Handbook of rubber bonding. Smithers Rapra Publishing; 2001.

[26] van Ooij WJ, Harakuni PB, Buytaert G. Adhesion of steel tire cord to rubber. Rubber Chem Technol 2009;82(3):315-39, Number: 3.

[27] Vilgis TA, Heinrich G, Klüppel M. Reinforcement of polymer nano-composites: theory, experiments and applications. Cambridge: Cambridge University Press; 2009.

[28] Bouasse H, Carrière Z. Sur les courbes de traction du caoutchouc vulcanisé. Ann de la Faculté Des Sci de l' Université de Toulouse Pour Les Sci Mathématiques Les Sci Phys 1903;5(3):257-83, Number: 3.

[29] Mullins L. Effect of stretching on the properties of rubber. Rubber Chem Technol 1948;21(2):281-300, Number: 2.

[30] Diani J, Fayolle B, Gilormini P. A review on the Mullins effect. Eur Polym J 2009;45(3):601-12, Number: 3.

[31] Zhang T, Lin S, Yuk H, Zhao X. Predicting fracture energies and crack-tip fields of soft tough materials. Extrem Mech Lett 2015;4:1-8.

[32] Machado G, Chagnon G, Favier D. Induced anisotropy by the Mullins effect in filled silicone rubber. Mech Mater 2012;50:70-80.

[33] Qi Y, Caillard J, Long R. Fracture toughness of soft materials with rate-independent hysteresis. J Mech Phys Solids 2018;118:341-64.

[34] Roucou D, Diani J, Brieu M, Colombo D. Experimental identification of fracture toughness of a carbon black-filled styrene butadiene rubber undergoing energy dissipation by Mullins softening. Mech Mater 2020;151:103645.

[35] Diani J, Brieu M, Batzler K, Zerlauth P. Effect of the Mullins softening on mode I fracture of carbon-black filled rubbers. Int J Fract 2015;194(1):11-8, Number: 1.

[36] Mallock A. Note on the instability of india-rubber tubes and balloons when distended by fluid pressure. Proc R Soc Lond 1891;49(296-301):458-63.

[37] Corbel P-Y, Jumel J. Rubber cord adhesion inflation test: Influence of envelope/confinement friction. Eng Fract Mech 2024;298:109919.

[38] Corbel P-Y, Jumel J. Simulation of fluid driven crack propagation along metal/elastomer interface - application to the numerical analysis of the rubber cord adhesion inflation test. Mech Mater 2023;185:104774.
[39] Guo H, Aziz NI, Schmidt LC. Rock fracture-toughness determination by the Brazilian test. Eng Geol 1993;33(3):177-88.

[40] George AF, Strozzi A, Rich JI. Stress fields in a compressed unconstrained elastomeric o-ring seal and a comparison of computer predictions and experimental results. Tribol Int 1987;20(5):237-47.

[41] Fraser S, Barberio CG, Chaudhry T, Power DM, Tan S, Lawless BM, et al. The effect of injurious compression on the elastic, hyper-elastic and visco-elastic properties of porcine peripheral nerves. J Mech Behav Biomed Mater 2021;121:104624.

[42] Shima S, Tatara Y, Iio M, Shu C, Lucero J. Large deformations of a rubber sphere under diametral compression : Part 2 : Experiments on many rubber materials and comparisons of theories with experiments. JSME Int J Ser A, Mech Mater Eng 1993;36:197-205.

[43] James JD, Ludwick JM, Wheeler ML, Oyen ML. Compressive failure of hydrogel spheres. J Mater Res 2020;35(10):1227-35.

[44] Khajehsaeid H, Arghavani J, Naghdabadi R. A hyperelastic constitutive model for rubber-like materials. Eur J Mech A Solids 2013;38:144-51.

[45] Ogden RW, Roxburgh DG. A pseudo-elastic model for the Mullins effect in filled rubber. Proc R. Soc Lond A 1999;455(1988):2861-77, Number: 1988.

[46] Griffith AA, Taylor GI. The phenomena of rupture and flow in solids. Philos Trans Royal Soc London Series A Contain Pap Math Phys Character 1921;221(582-593):163-98, Number: 582-593 Publisher: Royal Society.

[47] Rivlin RS, Thomas AG. Rupture of rubber. I. characteristic energy for tearing. J Polym Sci 1953;10(3):291-318.

[48] Corbel P-Y, Jumel J. Characterization of the interface fracture energy dependency on mixed mode fracture between rigid fiber and soft matrix. J Mech Phys Solids 2024;190:105745.