# Towards a new methodology to characterize the fracture energies of the woven composite/copper interface in mode I and mode II: Application to printed circuit boards

Charaf-Eddine Ziouani, Gautier Girard, Sébastien Mercier, François Lechleiter

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Charaf-Eddine Ziouani, Gautier Girard, Sébastien Mercier, François Lechleiter. Towards a new methodology to characterize the fracture energies of the woven composite/copper interface in mode I and mode II: Application to printed circuit boards. Engineering Fracture Mechanics, 2025, 333, pp.111819. 10.1016/j.engfracmech.2025.111819. hal-05442160

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# Towards a new methodology to characterize the fracture energies of the woven composite / copper interface in mode I and mode II: Application to printed circuit boards

**Charaf-Eddine ZIOUANI**¹, **Gautier GIRARD**¹, **Sébastien MERCIER**¹, and **François LECHLEITER**²

¹Université de Lorraine - CNRS - Arts et Métiers Institute of Technology, LEM3, F-57000, Metz, France

²CIMULEC, Z.I Les Jonquières, 57365 Ennery, France

## Abstract

The continuous trend toward miniaturization in electronic devices has stimulated the development of a new generation of printed circuit boards (PCBs) with embedded components. Throughout their lifespan, PCBs are subjected to thermal loads generated by heat from active components or the surrounding environment. In particular, mismatches in thermal expansion coefficients between materials are a leading cause of thermal stresses, often resulting in layer delamination, either between insulating substrates or at the copper-substrate interface. Traditionally, the peel test has been the dominant method for evaluating interfacial energy within PCBs, offering an estimate of the interface energy based on the IPC standard. During peeling, the copper layer often undergoes significant plastic deformation, complicating the precise determination of the fracture energy. Thus, achieving an accurate assessment of the mechanical response at the interface remains a challenging task. To overcome these limitations, we have designed a new specimen and adapted the Double Cantilever Beam (DCB) and End Notched Flexure (ENF) tests to the PCB context where layer thickness is significantly constrained (with copper layers ranging from 17 to 70 µm). Prior to experimentation, simulations demonstrate that, unlike the peel test, the DCB and ENF configurations exhibit minimal plastic dissipation. One of the main outcomes of the work is that a precise description of the plastic behavior of copper is not necessary to determine accurate estimations of the critical strain energy release rates in mode I and mode II. Furthermore, a notable advantage of these methods is their ability to maintain a controlled fracture mode, whereas the peel test inherently involves a mixed-mode (I and II) fracture process. The synergy between finite element analysis and experimental testing provides critical insights about the framework of application of the methods.

This is a post-peer-review, pre-copyedit version of an article published in Engineering Fracture Mechanics. The final authenticated version is available online at: DOI: https://doi.org/10.1016/j.engfracmech.2025.111819

## 1 Introduction

Hybrid and electric vehicles have emerged as pivotal solutions to address the challenges posed by climate change. This shift, from thermal to electrical vehicles, has driven the printed circuit board (PCB) industry to advance its expertise, achieving higher levels of performance and reliability to meet the demands of this evolving market. A PCB primarily consists of two key materials: copper and composite laminate. Copper serves as the conductive medium, while the composite laminate, as an electrical insulator, provides mechanical strength to the assembly.
Nomenclature

3P-ENF Three-Point End Notched Flexure

ΔP Force drop

δ Crack Shear Displacement

δn, δt Normal opening displacement, Shear sliding displacement

ṗ, p Effective plastic strain rate, Accumulated plastic strain

λ Crack shear displacement compliance

σ, σc Normal stress at the interface, Critical normal stress for the cohesive zone model

σ0 Copper initial yield stress

τ, τc Tangential stress at the interface, Critical tangential stress for the cohesive zone model

s Deviator of the Cauchy stress tensor

X Backstress tensor

θ Rotation of loading point in DCB test

a, aᵢ Current crack Length, Crack Length after initiation phase

a0DCB, a03P-ENF Initial Crack Length in DCB specimen, Initial Crack Length in 3P-ENF specimen

C, γ Kinematic hardening parameters

C<sub>L</sub> Loading line compliance

CBT Classical Beam Theory

CZM Cohesive Zone Model

DCB, ADCB Double Cantilever Beam, Asymmetric Double Cantilever Beam

DIC Digital Image Correlation

E₁₁, E₂₂, ν₁₂, G₁₂, Elastic moduli of the substrate

E<sub>Cu</sub>, ν<sub>Cu</sub> Elastic moduli of copper

FEM Finite Element Method

G<sub>Ic</sub>, G<sub>IIc</sub> Critical strain energy release rate in mode I, Critical strain energy release rate in mode II

G<sub>I</sub>, G<sub>II</sub> Strain energy release rate in mode I, Strain energy release rate in mode II

I Second moment of the cross sectional area in the crack-free portion.

J J-integral

K<sub>nn</sub>, K<sub>ll</sub> Interface stiffnesses in the cohesive zone model

L, b, h Length, width and half thickness of the specimen

P Force

PCB Printed Circuit Board

PVF, PTFE Polyvinyl Fluoride, Polytetrafluoroethylene

Q, b Isotropic hardening parameters

R(p) Isotropic hardening variable

s Half the support span in 3P-ENF test

SERR Strain Energy Release Rate

t Time

ν Displacement of the loading point in DCB test and 3P-ENF test
PCBs are prone to three primary failure mechanisms: copper trace fracture mostly due to low cycle fatigue, and delamination of the interface between copper trace and the substrate, or cohesive fracture within the substrate itself. These failures stem from the mismatch in the thermal expansion coefficients of the materials, generating stresses within the PCB structure during thermal fluctuations, such as those encountered during the reflow soldering process or during the actual lifetime of the product. To ensure reliability, PCBs undergo a rigorous qualification process. They must withstand a specified number of thermal cycles within a specified temperature range (e.g. -55°C to 125°C).

To address the delamination issue, many experimental configurations have been proposed in the literature. The Double Cantilever Beam (DCB) test, initially proposed by [1], is a widely used experimental method in fracture mechanics to characterize the toughness in mode I (opening mode), particularly for adhesive joints, composites, and other bonded structures. Due to its effectiveness, the DCB test has become a standard for several scientific organizations, including ASTM [2], JIS, and others. The test generates a load-displacement curve from which the Strain Energy Release Rate (SERR) can be calculated. However, discrepancies have been observed between theoretical predictions and experimental results, leading to the development, by Williams et al. [3, 4, 5] among others, of various methods to establish accurate correlations.

In addition to Mode I, Mode II, commonly referred to as in-plane shear mode presents usually a higher associated energy compared to Mode I. Mode II fracture energy is typically evaluated using experimental tests [6] such as the Three-Point End-Notched Flexure (3P-ENF), Four-Point End-Notched Flexure (4P-ENF) and End-Loaded Split (ELS) tests, as outlined by Davies et al. [7]. The 3P-ENF test involves applying a three-point bending load to a pre-notched specimen, promoting crack propagation from the notch. By recording the applied displacement and corresponding force, critical parameters such as fracture energy can be evaluated. Among Mode II fracture characterization methods, the 3P-ENF test offers unique advantages, but also faces certain limitations compared to alternatives such as the ELS and 4P-ENF tests. The ELS test, for example, applies a load directly to the end of the specimen, offering a simpler and more direct approach to measure crack propagation in Mode II. However, it requires precise control over load application at the specimen extremeties, which can pose challenges in execution. In contrast, the 4P-ENF test adopts a four-point bending configuration to achieve a more uniform load distribution, reducing parasitic stresses near the load application points. This often results in more accurate measurements for certain materials, though it introduces greater complexity in test setup and implementation. For all these tests, a force-displacement curve is generated, from which the fracture energy can be calculated. However, theoretical predictions based on the classical beam theory often deviate slightly from experimental results. Corrections and alternative approaches have been proposed to address these discrepancies. For example, in their work, Yoshihara et al. [6] introduced six methods to optimize the analysis and interpretation of the 3P-ENF test results.

Many fracture processes result in a combination of these two modes (e.g., in the peel test), making essential to understand the interaction between the modes I and II. The peel test aims to characterize the interface between a ductile film and a rigid substrate [8, 9, 10]. As copper develops large plasticity during peeling, the analysis is quite complex [11]. Indeed, the interface fracture energy can be estimated based on energy balance, considering both the external work and plastic dissipation. Other configurations derived from the peel test have been proposed in the literature. Without being exhaustive, one can cite the Mandrel Peel test to assess the fracture energy for metal-composite interface [12] or the Climbing Drum Peel test proposed in the work of Daghia et al. [13] to determine the fracture toughness of composite laminates. Finite Element (FEM) simulations are also employed for the analysis of the peel test [14, 15] where the interface response is modeled within the framework of the Cohesive Zone Model (CZM). The fracture energies $G_{Ic}$ and $G_{IIc}$ were calibrated instead of being independently determined
through dedicated tests, as proposed in the present work. According to the CZM, crack behavior is governed by a traction-separation law [16] that links the stresses at the crack tip to the displacement of the crack lips. These stresses are bounded by the so-called critical stress threshold, denoted as $\sigma_c$ for Mode I loading and $\tau_c$ for Mode II loading. An interesting method to derive the traction-separation law was proposed by [17] and [18]. The approach consists in plotting the evolution of the J-integral as a function of the opening displacement of the interface in the case of a DCB test (or the tangential displacement in mode II loading conditions), and then approximating this evolution using a suitable analytical function. The corresponding stress-displacement (traction-separation) curve can then be obtained by derivation of this function. Another key concept in the Cohesive Zone Model is the Fracture Process Zone (FPZ), which refers to the localized region ahead of a crack tip where material degradation, micro-cracking, and inelastic processes occur prior to full crack propagation. In the context of the CZM, it represents the area over which cohesive tractions are transmitted and energy dissipation due to fracture takes place. The FPZ thus governs the transition from undamaged to fully separated material (or interface) and controls the evolution of the crack according to initiation and propagation criteria. This implies that fracture does not occur instantaneously, but rather results from a sequence of damage events triggered progressively. Lot of works focused on assessing the length of the FPZ, Turon et al. summed these contributions in their work [19] and provided guidelines to choose the interface stiffness, the critical stress and the element length in the simulations with their respective effect on the load-displacement curve.

In most of the contribution of the literature, symmetric DCB or ENF samples are considered with arms made of the same materials [20] and [21]. Some works investigated asymmetric configurations to determine the fracture toughness [22] as well as bi-material specimens [23] for multi-material assembly.

Concerning works in electronics and apart from peel test, Calabretta et al. [24] adopted Asymmetric DCB (ADCB) and Four Point Bending to characterize the adhesion of a 0.25 mm thick copper layer on a 2 mm thick resin plate. In the work of Xiao et al. [25], a mixed-mode setup is employed for the same goal. Nevertheless, in both approaches, the thickness of the copper layer is not consistent with the one present in PCBs. Thus, adopting the identified interface properties may induce errors while simulating PCB structure using FEM.

While the focus of our work is made on the copper-composite interface in PCBs, it has to be noted that this interface is also present in lithium-ion batteries [26]. Moreover, the interface characterization methodology used in the present work can be extended to different ductile film/substrate interfaces (photovoltaic modules [27], food packaging [28]) to assess the fracture energy.

The objective of this paper is to develop a new sample configuration valid for both mode I and mode II, where plasticity is limited during the delamination process so the classical beam theory (CBT) in elasticity remains valid to characterize the fracture energy in both modes. The paper is organized as follows: A series of DCB and 3P-ENF tests were conducted on specimens containing the composite substrate/copper interface. To initiate the crack at the right interface, a specific sample with a dedicated manufacturing procedure was designed. A particular attention has been paid to manufacture the specimen with materials (laminates, copper foil) and processes used in the PCB industry. The experimental results were analyzed to calculate the fracture energy in both modes. Additionally, numerical simulations were performed to validate our experimental observations and to gain further insights into the quantification of plastic dissipation rate which is shown to remain limited compared to the interfacial damage work rate. From the limited plastic dissipation, it is shown that the derivation procedure of the fracture energy in both modes can be carried out using classical beam theory. Another main outcome is that a precise characterization of the plastic behavior is not necessary to the evaluation of $G_{Ic}$ and $G_{IIc}$.
# 2 Theoretical analysis

## 2.1 Mode I Energy release rate

As discussed in section 1, the double cantilever beam (DCB) test is widely adopted in the literature to characterize the critical strain energy release rate, $G_{Ic}$, in mode I (see figure 1a). In this work, unlike classical DCB specimen, a 35 µm copper film is introduced at the center of the specimen (see figure 1b). More details on the adopted configuration will be provided in section 3. The DCB specimen is then subjected to a tensile loading. The force $P$, the displacement $v$ and the rotation $\theta$ at the loading pins are recorded during the test, see section 3.

a)

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

b)

<figure><img src="image_3.png" /><figcaption>Figure 1: Schematic view of a) classical DCB configuration. b) DCB configuration with the copper/composite interface. The PVF (Polyvinyl Fluoride) film is introduced to initiate the crack at a distance $a_{0,DCB}$ from the loading pins.</figcaption></figure>

With such information, at least two modeling routes have been proposed in the literature to determine the fracture energy and recalled next. The first approach is based on the Classical Beam Theory and Williams's correction [3, 4, 5, 29]. As a consequence, the strain energy release rate in mode I is written:

$$G_I = \frac{3P^2(bC_L)^{\frac{2}{3}}}{E_{11}^{\frac{1}{3}}b^2h} \quad (1)$$

where $C_L = \frac{v}{P}$ is the loading-line compliance, $b$ the width of the specimen, $h$ half the thickness and $E_{11}$ the longitudinal Young's modulus.

The second route is based on the J integral [30, 31] which quantifies the energy required to propagate a crack in a specimen subject to external forces. It acts as a measure of the critical energy necessary for the crack to extend when no plasticity, or dissipative phenomenon (except the interface fracture) occurs. Mathematically, one can express the J-integral, when the crack propagates along the x-axis, as:

$$J = \int_{\Gamma} (Wdy - T \frac{du}{dx} ds) \quad (2)$$
where W is the strain energy density, T the external traction forces applied, u the displacement vector and Γ the integration contour, ds is an element of arc length along Γ. As the J integral is path independent, Paris et al. [32] adopted a contour of integration which corresponds to the outer boundary of the DCB specimen. Finally, they showed that the measurement of the applied load per unit width and the relative rotation between the loading points, are only needed for the evaluation of the J-integral:

$$J = \frac{P}{b}(\theta_u - \theta_l) \qquad (3)$$

where $\theta_u$ (respectively $\theta_l$) is the rotation of the upper (respectively lower) loading point. Note that Eq 3 has been adopted by Gorman et al. [17] to determine the mode-I traction-separation law of an adhesively bonded double cantilever beam. Högberg et al. [33] used the mixed mode double cantilever beam. By evaluating the J integral also on the outer contour of the specimen, the authors were also able to capture the response of the interface in mixed mode loading conditions. When the applied force is perpendicular to the specimen plane, the classical DCB configuration was retrieved and the J-integral value is given by Eq 3 as expected.

In this work, both strategies will be adopted to capture $G_{Ic}$. Indeed, equation 1 will be used to compute the interface fracture energy in opening mode. The $G_{Ic}$ value will be confirmed by the evaluation of the J-integral. Note that the evaluation of $G_{Ic}$ based on Eq 1 and Eq 3 does not need any evaluation of the current crack length, only the load-line compliance or rotations are used. Furthermore, the bottom arm is thicker than the top arm, with the film at the external part. So an asymmetric theory could have been used instead. No significant difference was found for the $G_{Ic}$ when adopting such a theory. This aspect is not integrated in this paper work. As plasticity may develop in the copper film, the validation of the present approach based on the elastic beam theory will be discussed.

## 2.2 Mode II Energy release rate

The ENF is used to analyze crack propagation in pure mode II. The specimen used in the DCB test is subject here to a three point bending. During the test, the force P, the loading pin displacement v and the Crack Shear Displacement (CSD) δ are recorded, see figure 2b.
<figure><img src="image_4.png" /><figcaption>Figure 2: ENF specimen with copper layer inserted under three-points bending: a) Initial configuration with still a PVF film. b) Deformed configuration. As the crack advances from right to left, the Crack Shear Displacement (CSD, δ) is recorded.</figcaption></figure>

From these three informations (P, v and δ), numerous works in the literature have analyzed this test and proposed a methodology to capture the critical strain energy release rate $G_{IIc}$ [3, 6] assuming pure elastic response of the constituents. Following Yoshihara et al [6] and Kageyama et al [34], the loading-line compliance $C_L$ and the CSD compliance $\lambda$ are expressed as:

$$C_L = \frac{v}{P} = \frac{2L^3 + 3(a + \chi h)^3}{12E_{11}I} \quad (4)$$

$$\lambda = \frac{\delta}{P} = \frac{12(a + \chi h)^2}{E_{11}bh^2} \quad (5)$$

where $E_{11}$ is the Young's modulus in the longitudinal direction, b the specimen width, h the half thickness, a the crack length. $\chi$ is the correction factor proposed by Williams [3] and I is the second moment of the cross section in the crack-free portion. From [34], the strain energy release rate in the in-plane shear mode (mode II), $G_{II}$, can be written:

$$G_{II} = \frac{P^2}{2b} \frac{dC_L}{da} = \frac{3P^2\lambda}{8bh} \quad (6)$$

In this work, equation (6) will be used to access the mode II fracture energy. As for mode I, the current crack length is not needed to provide an estimate of $G_{IIc}$.

Based on this theoretical background, the energy release rates in both Mode I and Mode II will be quantified in the following sections for the copper/composite interface which exists in PCBs. As already mentioned, equations (1) to (6) are developed for elastic materials. The challenge is to validate their applications even with a ductile sheet within the structure which may trigger plasticity during delamination.
# 3 Strategy for the characterization of the copper - composite interface

This section provides a detailed overview of the experimental protocol, including the manufacturing process of the new specimen, its geometry, and the mechanical properties of the materials used. The experimental tests in modes I and II will then be described.

## 3.1 Sample development

As reviewed in the introduction, the characterization of the copper-composite interface is of fundamental interest to predict the lifespan of all electronic devices. One possibility is to perform an Asymmetric Double Cantilever Beam (ADCB) [24] with a thick copper layer. Nevertheless, this configuration with thick layers does not correspond to the one present in PCB. Therefore, a tailored specimen was engineered by integrating a 35 µm copper foil, a common thickness encountered in the PCB industry, into the core of a standard DCB or ENF configuration. The idea is to develop a single sample valid for both tests. The first step of the manufacturing process consists in an assembly of woven pre-impregnated (B-stage) and laminate woven composite layers together. After hot pressing, a thick FR-4 substrate is obtained. Next a pre-preg layer and the copper film are stacked, then assembled by hot pressing with the prepared FR-4 substrate, see figure 3a. Note that the thickness of the FR-4 substrate with the pre-preg layer is around 1.6 mm. To initiate delamination at the copper-composite interface, a PolyVinyl Fluoride (PVF) film is partially inserted over the top surface of the copper foil. An additional FR-4 block is then placed over the assembly, using a pre-preg layer that covers both the PVF film and the copper foil. During hot pressing, the resin of the pre-preg cures and adheres to the thick FR-4 part (top part of the sample) and the copper film, see figure 3b.

<figure><img src="image_5.png" /><figcaption>Figure 3: Manufacturing process of the DCB (and 3P-ENF) specimen with copper foil (not to scale): a) Integration of the 35 µm copper foil to the bottom part. b) Assembly of the bottom arm, the PVF film, a second pre-preg layer and the top arm leading to the sample depicted in fig 1b and 2. The total thickness of the specimen ranges from 3.2mm to 3.4mm.</figcaption></figure>

Note that the whole specimen is an assembly of more than 30 layers in total. It has been seen from preliminary tests that even with a PVF film, the crack propagation may initiate along different interfaces. Therefore, the manufacturing procedure was updated. To facilitate crack initiation at the right interface, a spray of Polytetrafluoroethylene (PTFE) was added ahead of the PVF layer over a distance of 10 mm approximately. Note that the samples were all prepared at CIMULEC company, ensuring that regular process and materials used in the PCB industry are adopted.

Although the same sample will be used for DCB and ENF tests, the initial crack length ($a_0$) varies for both configurations as the loading conditions are different (see figures 1 and 2). The nominal crack length $a_0$ and the geometric data are reported in
<table><caption>table 1.</caption><thead><tr><th>L [mm]</th><th>2h [mm]</th><th>b [mm]</th><th>a<sub>0,DCB</sub> [mm]</th><th>a<sub>0,ENF</sub> [mm]</th><th>2s [mm]</th></tr></thead><tbody><tr><td>150</td><td>3.2-3.4</td><td>25</td><td>62</td><td>30.5</td><td>80</td></tr></tbody></table>

<caption>Table 1: Geometrical features for the tested specimens in DCB and 3P-ENF configurations</caption>

In the present work, the two thick FR-4 arms are made of fiber glass fabric based composite (Figure 3a). The mechanical behavior of the composite is considered elastic orthotropic. The in-plane elastic moduli are obtained through tensile tests conducted at various angles (0, 45°, 90°), see Table 2. The mechanical behavior in the out-of-plane direction is determined via numerical simulations at the level of the composite microstructure, based on the works of Girard et al. [35, 36]. The two layers surrounding copper (see figure 3b) are made of different pre-preg materials (referred to hereafter as Mat-1, a fiberglass epoxy composite with a high glass transition temperature, and Mat-2, a hydrocarbon/ceramic bondply). The elastic response of both Mat-1 and Mat-2 has been characterized with the same methodology as that adopted for FR-4, see Table 2. It is seen that a 2–3 GPa difference in the Young’s modulus $E_{11}$ is found when comparing the three materials. Due to the thinness of the pre-preg layer (around 100 µm when compared to the 1.6 mm thick arm), its presence in the sample does not affect the overall stiffness. Indeed, when comparing the bending stiffness of the 1.6 mm FR-4 arm to that of the 1.5 mm FR-4 and 100 µm pre-preg layer, only a 1.5% difference in bending stiffness was observed between the two configurations. As a consequence, the adopted finite element model will consider that both arms are made only of FR-4 material. This assumption is relevant since the Young’s modulus $E_{11}$ of the three materials are equivalent and the pre-preg layer is thin.

<table><caption>Table 2: Mechanical properties of the FR4, Mat-1 and Mat-2 materials identified based on uniaxial testing at 0°, 45° and 90°. Note that all the materials adopted in the present work are commonly used in the PCB industry.</caption><thead><tr><th>Material</th><th>E<sub>11</sub> [GPa]</th><th>E<sub>22</sub> [GPa]</th><th>ν<sub>12</sub></th><th>G<sub>12</sub> [GPa]</th></tr></thead><tbody><tr><td>FR-4</td><td>22,7</td><td>20,3</td><td>0.168</td><td>5,77</td></tr><tr><td>Mat-1</td><td>20.3</td><td>19,3</td><td>0.17</td><td>5,77</td></tr><tr><td>Mat-2</td><td>25.3</td><td>23.7</td><td>0.27</td><td>4.89</td></tr></tbody></table>

## 3.2 Experimental protocol for DCB and 3P-ENF tests

Figure 4 shows the experimental setup for the DCB test. Tests are conducted on a MTS Criterion C41.103 with a force sensor of 500 N. The DCB specimen is attached to the testing machine by designed hinges inserted in the grips of the machine (Figures 4a and 4b). A screw clamps the specimen to the hinges. A 10 mm/min crosshead velocity is applied to the upper grip while keeping the lower one fixed. The displacement of the crosshead, the force and the rotations at the loading points are recorded continuously. From the displacement $v$ and the force $P$, the loading line compliance is calculated leading to the evaluation $G_{Ic}$ by Eq 1. The rotations are assessed using the Digital Image Correlation (DIC) technique at points shown on Figure 4b. Another method to confirm the fracture energy measurements is to use the value of the J-integral, which can be computed from the force at the loading points and the respective rotations, see Eq 3. It is important to remember that, with the lack of any dissipative phenomenon (except the damage and fracture of the interface), the value of the J-integral is equal to the strain energy release rate [30, 31]. Note that for DCB as well as 3P-ENF tests, a short initiation phase is prescribed before
the full propagation stage.

<figure><img src="image_6.png" /><figcaption>Figure 4: DCB test configuration: a) The specimen is clamped to the MTS testing machine using hinges. b) Rotations analyzed by DIC. The colored layers are the areas where the rotations are extracted. c) Evolution of loading pin rotations over time (dotted line for the upper point and solid line for the lower point). The initiation phase is operated from 0 s to 80 s and the propagation phase extends from 80 s to the end.</figcaption></figure>

Figure 4c shows the evolution of the rotation at the loading points with respect to time. One can easily distinguish, in figure 4c, that the initiation phase is operated from 0 s to 80 s where the initial crack length $a_0$ is modified to what will be denoted next $a_i$. In fact, the objective during the initiation phase is to propagate the crack, through the PVF film and the PTFE spray region. This means that after the initiation phase, the crack length increases from $a_0$ to $a_i$, with $a_0 < a_i$. The value of $a_i$ may vary from sample to sample depending on the exact positioning and the advance of the crack during the initiation phase.

The propagation phase is operated from 80 s to the end.

<figure><img src="image_7.png" /><figcaption>Figure 5: 3P-ENF test configuration: a) The specimen is loaded under 3 point bending. b) Crack Shear Displacement (CSD) analyzed by DIC. The colored layers are the areas where the CSD is extracted. c) Zoom to define precisely the CSD</figcaption></figure>

Figure 5a shows the 3P-ENF experimental setup. Tests are conducted on an INSTRON E3000 with a force sensor of 5 kN. The sample is subject to three point bending. The support span 2s is equal to 80 mm, the diameter of the rollers is 5 mm, see figure 2 for the test configuration. The loading pin moved vertically at a velocity of 10 mm/min. Both the applied load and the displacement of the loading pin are recorded. Additionally, a camera is used to monitor the crack shear displacement $\delta$. It
captures images of the extremity of the specimen (figure 5c) so as to deduce the relative slip between the arms of the specimen (figure 5b). The measurement was done using the DIC software VIC-2D. The colored region represents the area where DIC is applied.

# 4 Results

## 4.1 DCB test

To initiate propagation, a preliminary loading-unloading step, named initiation phase in sections above, is first imposed. It has been checked that the crack has propagated only of few millimeters from $a_0$ to $a_i$. Secondly, the DCB is loaded at 10 mm/min and the crack propagates along the specimen. Only this monotonic loading phase is studied to characterize the interface. After testing, the fracture surfaces of both parts of the specimens have been observed carefully. Figure 6 shows that, effectively, the crack has propagated along the copper - composite interface. This is representative of an adhesive delamination as expected.

<figure><img src="image_8.png" /><figcaption>Figure 6: View of the fracture surfaces for the DCB configuration with Mat-1 as pre-preg.</figcaption></figure>

Figure 7 presents the force-displacement curves. It is clearly seen that after a linear response, the force reaches a maximum. At this stage, the crack has already started to propagate. Subsequently, the force necessary for further crack propagation reduces as the crack length increases. One can observe a good repeatability of the results. The average maximum load for Mat-1 is 12.21 N ± 0.05N and 9.98 ± 0.3N for Mat-2. The slope of the force-displacement curve at the origin, corresponding to the initial stiffness of the assembly, is smaller for Mat-2 configuration. As the design is similar (almost the same thickness), this means that the crack length after the initiation phase for Mat-2 is larger than for Mat-1. In our approach, we combine FE simulations and experiments. Therefore, the crack length $a_i$ will not be determined experimentally but by calibration with FE. It is clear that another strategy could have been proposed. In [21] (Equation 1 of the paper), the compliance $C_L = v/P$ in mode I (see figure 1) can be related to the equivalent crack tip $a_e$. Knowing the evolution of the compliance $C_L$, the current crack length $a_e$ can be deduced and, therefore, the initial crack length $a_i$. We have observed (results not present here) that the methodology was able to confirm the FE value of the initial crack length $a_i$ but a more precise value was captured by FE.
<figure><img src="image_9.png" /><figcaption>Figure 7: Experimental force displacement curves for 2 different pre-preg (Mat-1 and Mat-2)-copper interfaces. The loading velocity of the DCB test is 10 mm/min. The maximum force for Mat-1 is larger than for Mat-2.</figcaption></figure>

Using equation (1), the mode I strain energy release rate is evaluated and plotted with respect to time, as shown in figure 8. The curves of figure 8 exhibit two distinct phases. From 0 to 100 s for Mat-1 (respectively 0 to 60s for Mat-2), $G_I$ increases gradually, representative of a transient regime. From 100 to 130s for Mat-1 (respectively 60s to 130s for Mat-2), the strain energy release rate has reached a plateau value, which enables to define the critical strain energy release rate $G_{Ic}$, assuming that the interface delamination and crack propagation are the only dissipative phenomena expected. The corresponding values for configuration with Mat-1 and configuration with Mat-2 are respectively: $G_{Ic}^{Mat-1} = 145 \pm 3N/m$ and $G_{Ic}^{Mat-2} = 80 \pm 2.5N/m$, see Table 3. The critical energy release rate for Mat-2 is 55 % smaller than the one for Mat-1.

<figure><img src="image_10.png" /><figcaption>Figure 8: Evolution of the strain energy release rate with respect to time for the two pre-pregs Mat-1 and Mat-2 during the DCB test. The loading velocity is 10 mm/min.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Figure 9: Time evolution of: a) the rotation difference between the upper and the lower loading points. b) the value of J-integral captured from equation 3. Results for both configurations (Mat-1 and Mat-2) are presented. The loading velocity during the DCB test is 10 mm/min.</figcaption></figure>

As mentioned in section 3, the rotation of the DCB arms is captured by DIC, see figure 4b. Based on the data, displayed in figure 9a, and adopting equation (3), the J-integral value is evaluated. Figure 9b shows that after a transient regime, a plateau can be defined. This plateau value is the combination of the decreasing force at late time ($t \ge 60s$ in figure 7) and the increasing rotation in fig 9a. For $t \ge 60s$ (respectively $t \ge 80s$) the product $P(\theta_u - \theta_l)$ is almost constant for Mat-2 (respectively for Mat-1). The corresponding plateau values for configurations with Mat-1 and with Mat-2 are respectively: $J^{Mat-1} = 130 \pm 4.3N/m$ and $J^{Mat-2} = 75 \pm 2N/m$, see Table 3. Assuming no plastic dissipation, the plateau value of the J-integral corresponds to the critical energy release rate in mode I. Interestingly, comparing the plateau values in figures 8, 9b and Table 3, one observes that both methods, based on the force-displacement (Eq 1) and the J-integral (Eq 3), provide consistent results. This is a first evaluation showing that plastic dissipation in copper during crack propagation is probably limited.

<table><caption>Table 3: Values of the critical energy release rate and of the J-integral at plateau for both configurations used. The loading velocity is 10 mm/min</caption><thead><tr><td>Pre-preg material</td><td>Maximum Force (N)</td><td>G<sub>Ic</sub> (N/m)</td><td>J (N/m)</td></tr></thead><tbody><tr><td>Mat-1</td><td>12.21 ± 0.05</td><td>145 ± 3</td><td>130 ± 4.3</td></tr><tr><td>Mat-2</td><td>9.8 ± 0.3</td><td>80 ± 2.5</td><td>75 ± 2</td></tr></tbody></table>

To summarize, figures 8, 9b and Table 3 confirm that the two considered interfaces have different fracture energies. The fracture energy of the configuration with Mat-1 is evaluated around 145 N/m while the fracture energy is close to 80 N/m for the second configuration. It is interesting to note that between both configurations, only the pre-preg material is different so the selection of materials for specific properties (insulation, hyper frequency, thermal conductivity...) modifies to a large extent the interface behavior.
## 4.2 3P-ENF tests

The same experimental approach is adopted for Mode II with the 3P-ENF test. The test is performed via a two step loading. Specifically, the specimen is first loaded until a noticeable force drop $\Delta P = 2$N occurs, indicating the onset of crack propagation. The system is then unloaded by removing the contact with the central loading pin. Secondly, the displacement of the central pin is imposed at a constant velocity of 10 mm/min until the crack propagates along the interface. The load - displacement curves of the second phase are plotted in figures 10a and 10b for both configurations.

<figure><img src="image_12.png" /><figcaption>Figure 10: Experimental 3P-ENF force displacement curves for a) Mat-1 b) Mat-2. The results shown here correspond to the propagation phase. The loading velocity is 10 mm/min.</figcaption></figure>

Figure 10 show that the force initially rises linearly up to a certain critical force (the crack is not evolving yet). As the displacement of the loading pin further increases, a slight decrease in force is then observed due to crack propagation. Once the crack extends beneath the loading pin, the force reaches a minimum. As the crack advances further, the force increases again. One can see, in fig10b, that with configuration involving Mat-2, the two curves look like what is expected for 3P-ENF tests and as described previously. Nevertheless, the force drops is of a limited amplitude. For the test Mat1-S1, a force drop of 20N is measured. For samples 2 and 3, the force drop can not be noticed. This response is observed because the initiation phase propagated the crack unstably further than expected (up to beneath the loading pin). This is confirmed by analyzing the slope at the origin of the force-displacement curve which is smaller for samples 2 and 3 meaning that $a_{iENF}^{Mat1-S2} > a_{iENF}^{Mat1-S1}$. The curves show a similar response in the last segment.

A digital camera is recording all tests. As samples are prepared with a speckle of black dots, the DIC technique enables to measure the Crack Shear Displacement (CSD) observed on the tip of the specimen, see figure 5.

The critical strain energy release rate in mode II for Mat1-S1 is computed using equation 6 along the period where the crack advances up to underneath the loading pin. The evolution of $G_{II}$ over time, not presented here, follows a trend similar to that observed in mode I. Particularly, a stabilization into a plateau is observed, which defines the critical strain energy release rate ($G_{IIc}^{Mat-1} = 270N/m$). For the Copper/Mat-1 configuration, tests S2 and S3 turned out to be unstable. Therefore, the identification of $G_{IIc}$ was based solely on test S1. The correlation with tests S2 and S3 was subsequently achieved through a calibration procedure of $a_i$. For the Copper/Mat-2 configuration, the force drop was not marked enough to capture a proper
$G_{IIc}$ identification following Eq 6. As a result, the correlation between the experiments and the simulations was performed by FE calibration. The value of $G_{IIc}$ is much higher for the interface involving Mat-2. Values are provided in Table 4.

<table><caption>Table 4: Experimental values of the maximum force, the critical values of CSD and SERR for both configurations (Mat-1 and Mat-2) obtained from 3P-ENF tests. $δ_c$ refers to CSD at maximum force. The velocity of the central pin is 10 mm/min. For Mat-1, the $G_{IIc}$ was determined using Eq 6 and confirmed by FE simulations. For Mat-2, $G_{IIc}$ was determined by calibration of FE simulations to reproduce precisely experimental force-displacement curves.</caption><thead><tr><td>Configuration</td><td>Maximum Force (N)</td><td>δ<sub>c</sub> (mm)</td><td>G<sub>IIc</sub> (N/m)</td></tr></thead><tbody><tr><td>Material-1</td><td>169</td><td>0.157 ±5 × 10<sup>-3</sup></td><td>270 ±3</td></tr><tr><td>Material-2</td><td>217</td><td>0.247 ±4.1 × 10<sup>-3</sup></td><td>533</td></tr></tbody></table>

The interface fracture energies in Modes I and II have been measured for two distinct configurations combining experiments and theoretical methods based on the classical beam theory (i.e assuming no inelastic dissipation). It is observed that: $G_{Ic}$(Mat-1) > $G_{Ic}$(Mat-2) and $G_{IIc}$(Mat-1) < $G_{IIc}$(Mat-2). A discrepancy arose when comparing the fracture energies $G_{Ic}$ and $G_{IIc}$ to the values reported by the manufacturer. Indeed, data-sheets typically provide the peel strength to copper as defined in the IPC-TM-650 2.4.8C Standard using the peel test. The given values tend to be significantly higher, see Appendix A. This difference is primarily attributed to the significant contribution of plastic dissipation during peel tests. Indeed, according to the IPC-TM-650 2.4.8C standard, the interface fracture energy is determined from the plateau value of the peel force divided by the width of the peel specimen without considering the role of plastic dissipation.

To complement the results of Table 4, finite element (FE) simulations will be conducted next to validate the experimental outcomes, and to offer a deeper understanding on how the interfaces delaminate. A crucial point will be the validation of the assumption of limited plastic dissipation, during delamination in DCB and ENF configurations which allows to use analytical expressions (Eqs 1 and 6) developed for classical DCB and ENF specimens with the CBT framework (elastic response).

## 4.3 Numerical Model

To analyze experiments and observations in terms of fracture energy values and to provide further insights about the interface behavior, finite element (FE) simulations of the two configurations (DCB and 3P-ENF) have been carried out using the commercial software ABAQUS and the CZM framework. As the specimen is slightly asymmetric, the CZM must handle the potential mixed mode loading at the interface at only the upper interface between copper and the composite substrate. Cohesive contact was used for the DCB test while cohesive elements were implemented for the 3P-ENF test at the upper interface between copper and composite which is subject to delamination (The lower copper/composite interface is not supposed to delaminate. So no cohesive elements are introduced at this interface). It is important to mention that the PVF film does not adhere to any constituent. In the literature [37], both approaches — using cohesive contact and cohesive elements — are considered equivalent for modeling interfacial fracture, provided that the same traction-separation law and material parameters are adopted. Therefore, the use of one or the other formulation does not introduce any significant difference in the predicted response. To confirm this, FE simulations of DCB test were conducted and showed that the force-displacement curves derived from both approaches (cohesive contact and cohesive elements) are perfectly superimposed, demonstrating the equivalence of the two modeling strategies in terms of mechanical response and energy release rate.

A bilinear traction separation law is adopted for both modes. The behavior of the cohesive zone is first linear with $σ =$
$K_{nn}$, $\delta_n$ and $\tau = K_{tt}$, $\delta_t$ where $\sigma$ and $\tau$ are respectively the normal and tangential components of the traction forces. $\delta_n$ and $\delta_t$ are respectively the normal and shear displacements. The interface stiffnesses were chosen to be very large ($K_{nn} = K_{tt} = 10^6$ N/mm³). Crack initiation occurs when the following quadratic condition is satisfied:

$$ \left( \frac{\langle \sigma \rangle}{\sigma_c} \right)^2 + \left( \frac{\tau}{\tau_c} \right)^2 = 1 \qquad (7) $$

where $\langle x \rangle$ are the Macaulay brackets ($\langle x \rangle = 0$ if $x < 0$ and $x$ if $x > 0$). Meanwhile, propagation occurs when the second criterion is met:

$$ \frac{G_I}{G_{Ic}} + \frac{G_{II}}{G_{IIc}} = 1 \qquad (8) $$

with $G_I = \int \sigma d\delta_n$ and $G_{II} = \int \tau d\delta_t$. $G_{Ic}$ and $G_{IIc}$ are respectively the critical energy release rates in Mode I and Mode II. The elastic response of the FR-4 material is orthotropic. Its behavior was already presented in section 2. Copper foil characterization was performed in Girard et al. [38]. Based on their work, copper has an elastic-plastic response. The elastic response is governed by the isotropic Hooke's law, while the plastic behavior is described by a combined kinematic and isotropic hardening following the model of Chaboche [39]. A $J_2$ flow law is adopted to link the plastic strain rate tensor $d^p$ to the deviator of the Cauchy stress tensor $s$ and the backstress tensor $X$ by:

$$ d^p = \frac{3}{2} \dot{p} \frac{\mathbf{s} - \mathbf{X}}{J_2(\mathbf{s} - \mathbf{X})} \qquad (9) $$

where $J_2(\mathbf{s} - \mathbf{X}) = \sqrt{\frac{3}{2}(\mathbf{s} - \mathbf{X}) : (\mathbf{s} - \mathbf{X})}$ is the effective stress. The effective plastic strain rate $\dot{p}$ equals to $\dot{p} = \sqrt{\frac{2}{3}d^p : d^p}$. The yield surface is defined by:

$$ F = J_2(\mathbf{s} - \mathbf{X}) - R(p) - \sigma_0 \qquad (10) $$

where $\sigma_0$ is the initial yield stress, $p = \int \dot{p} dt$ the accumulated plastic strain. The evolution law for the isotropic hardening variable $R(p)$ follows the first order differential equation:

$$ \dot{R} = b(Q - R)\dot{p}, \qquad (11) $$

where $b$ and $Q$ are material parameters. As in Girard et al. [38], the kinematic hardening variable is the sum of 3 components $\mathbf{X} = \sum_{i=1}^3 \mathbf{X}_i$. The evolution law of each component of the backstress is given by:

$$ \dot{\mathbf{X}}_i = \frac{2}{3} C_i d^p - \gamma_i \mathbf{X} \dot{p}, \qquad (12) $$

where $C_i$ and $\gamma_i$ are material parameters as well. Table 5 gives the values of the identified mechanical properties used to model the behavior of the copper film.

<table><caption>Table 5: Mechanical properties of the copper film identified in LEM3 based on the methodology presented in Girard et al. [38]. Note that a different copper grade is studied in this work leading to different material parameters.</caption><thead><tr><th>E<sub>Cu</sub> [MPa]</th><th>ν<sub>Cu</sub></th><th>σ<sub>0</sub> [MPa]</th><th>Q [MPa]</th><th>b</th><th>C<sub>1</sub> [MPa]</th><th>γ<sub>1</sub></th><th>C<sub>2</sub> [MPa]</th><th>γ<sub>2</sub></th><th>C<sub>3</sub> [MPa]</th><th>γ<sub>3</sub></th></tr></thead><tbody><tr><td>95000</td><td>0.44</td><td>30.01</td><td>56.8</td><td>396</td><td>244100</td><td>2236</td><td>9123</td><td>200.3</td><td>2647</td><td>26.84</td></tr></tbody></table>
The same geometry used in the DCB and 3P-ENF experiments was modeled in the finite element software ABAQUS, following the geometrical specifications provided in Table 1. The only unknown geometrical parameter was the crack length after the initiation phase, denoted $a_i$, which was calibrated through a series of numerical simulations. Note that the initial crack length $a_i$ does not affect the portion of the curve corresponding to the force drop with increasing displacement (as illustrated in Figure 11b). A mesh sensitivity study was carried out to ensure the accuracy and stability of the numerical results.

For the DCB simulations, an element length of 0.09 mm was used along the specimen (for both copper and composite). Each composite layer was discretized with 14 elements through its thickness, while the copper layer thickness was meshed with 15 elements. The model contains thus 23,338 CPE4R elements.

A constant displacement rate of 10 mm/min was applied in the DCB configuration through reference points kinematically coupled to the specimen, allowing an accurate representation of the hinges and their motion.

For the 3P-ENF simulations, the cohesive elements have a thickness of 0.001 mm. The copper layer was discretized along the length with an element size of 0.1 mm (so as the cohesive layer), while the composite was meshed with a 0.2 mm element size. The copper layer was meshed with 15 elements through its thickness, and each composite layer with 9 elements. Thus the number of elements is 36,630.

In the 3P-ENF model, three analytical rigid surfaces were used to represent the two supports and the central loading pin. Normal hard contact and frictionless tangential contact were defined between these rigid bodies and the specimen. A constant displacement rate of 10 mm/min was applied to the central loading pin in the 3P-ENF configuration.

## 4.4 FEM validation

The finite element model described in section 4.3 is adopted to reproduce the DCB experimental results. $G_{Ic}$ and $G_{IIc}$ have already been identified in section 4.1. Therefore for mode I configuration, the only unknown parameters are the critical stress $\sigma_c$ in the crack initiation criterion (Eq.7) and the crack length $a_i$. By construction, the PVF film is positioned precisely so that $a_{0_{DCB}}$ from figure 1 is given by the PCB manufacturer. Nevertheless, the first loading-unloading step expands this crack length. The parameters $\sigma_c$ and $a_i$ were calibrated to reproduce accurately the experimental load-displacement curve.

### 4.4.1 Effect of $\sigma_c$ and $a_i$ on the force displacement curve

Figure 11 shows a parametric analysis of the effect of the critical stress $\sigma_c$ and of the initial crack length $a_i$ on the force-displacement curve during a DCB test representative of Mat-1.
<figure><img src="image_13.png" /><figcaption>Figure 11: Effect on the DCB force-displacement curve of: a) the critical stress $\sigma_c$ and b) the initial crack length $a_i$. The DCB configuration of Fig 1b is adopted. The loading velocity is 10 mm/min. Note that $G_{Ic}$ and $G_{IIc}$ adopted in the present calculations are representative of Mat-1. Values have been identified in section 4, see Table 3 and Table 4. $\tau_c$ was set equal to $\sigma_c$.</figcaption></figure>

In these calculations, the value of $\tau_c$ has not any significant impact on the force-displacement curve, when a reasonable value is adopted. In that sense, $\tau_c$ is set equal to $\sigma_c$ unless otherwise stated.

It is shown in fig.11a that increasing $\sigma_c$ modifies the force-displacement curve in the vicinity of the maximum force. As the critical stress becomes larger, the maximum force is enhanced. It is also shown that for $\sigma_c > 3MPa$, the change in maximum force remains limited for the configuration adopted here. From figure 11a, it is clear that it will not be possible to propose a precise value of $\sigma_c$. This result was expected since detecting such an effect through experiments proves to be difficult. Note also that this parameter has no influence on the initial slope nor on the late response during the delamination process. With a DCB experiment, at best, a lower bond can be estimated. As mentioned in the literature [16, 40], the DCB test does not allow for the direct identification of the critical stress. Therefore, the value provided here is an estimation intended to facilitate correlation with the experimental results. The real value is probably larger and could be identified in another independent test, such as the peel test.

Figure 11b shows the effect of the crack length $a_i$ on the force-displacement curve. As expected, a larger crack length $a_i$ leads to a decrease of the initial slope, and of the maximum force. Figure 11b shows that a 2mm difference in $a_i$ has a noticeable impact on the initial slope of the force-displacement curve. This result confirms our choice to estimate the $a_i$ value by calibration with FE and comparison to experimental cuvres. No effect of $a_i$ is expected during the crack propagation stage. From these parametric studies, it is expected that DCB test simulations will provide a clear definition of $a_i$ and a lower estimate for $\sigma_c$.

### 4.4.2 Finite element simulations of DCB and 3P-ENF tests

The calibration of $a_i$, $\sigma_c$ and $\tau_c$ has been performed to mimic the experiments.
<figure><img src="image_14.png" /><figcaption>Figure 12: Comparison of the numerical DCB force displacement curves and of the experimental results: a) Configuration with Mat-1 simulated with $a_i = 62\,mm$ and $\sigma_c = 3MPa$. b) Configuration with Mat-2 simulated with $a_i = 70\,mm$ and $\sigma_c = 3MPa$. The loading displacement is 10 mm/min. For the present calculations, values of $G_{Ic}$ and $G_{IIc}$ are those reported in Table 3 and Table 4. $\tau_c$ was set to 3 MPa.</figcaption></figure>

Figures 12a and 12b show that the DCB numerical model can accurately predict the interface delamination process, as the numerical force-displacement curves for both configurations overlap the experimental ones during phases of elastic response and crack propagation. As discussed about figure 11a, the only limitation is the precise determination of the critical stress $\sigma_c$. In fact, the critical stress $\sigma_c$ was estimated by calibrating its value to ensure that the numerical maximum force matches the experimental results. The adopted values $\sigma_c^{Mat-1} = \sigma_c^{Mat-2} = 3MPa$ are seen as a low estimate but not a defined value (for the present interface model).

Concerning the ENF configuration, two parameters need also to be calibrated: the crack length $a_i$ and the critical stress $\tau_c$. Since these quantities affect the force-displacement curves in a similar way to what was seen for the DCB configuration, the parametric analyses are not presented here. Figure 13a displays two experimental force-displacement curves for Mat-1 configuration. The ENF-Mat1-S1 presents the classical response already seen in literature [20]. From this curve, the $G_{IIc}$ value of 270 N/m has been determined. Adopting this value of $G_{IIc}$ and by fixing $a_i = 33\,mm$ and $\tau_c = 20MPa$ in the finite element model, the initial slope and the force drop are accurately reproduced for Mat1-S1. For the other tests Mat1-S2 and S3, the curves do not present a local maximum. In addition, the initial slope of ENF-Mat1-S2 curve (S3 as well) is lower, meaning that $a_i$ is larger. Taking $G_{IIc} = 270\,N/m$ and $\tau_c = 20MPa$, the calibration of $a_i$ leads to a value of $a_i = 41\,mm$. Therefore, this value confirms the previous comments on figure 10 that the crack tip is below the loading pin for Mat1-S2. For Mat1-S1, a distance of approximately 7 mm between the initial position of the crack tip and the loading pin exists before the propagation phase.

Concerning Mat-2 configuration, the two tests present similar force-displacement response. As mentioned in section 4.2, the $G_{IIc}$ for Mat-2 is not determined from Eq 6 but by calibration. By adopting $G_{IIc} = 530\,N/m$ and $a_i = 33\,mm$, the predicted force-displacement curves by FEM coincide with those of Mat2-S1 and Mat2-S2 tests. While the $G_{IIc}$ for Mat-2 is twice larger than for Mat-1, it is seen that adopting also a critical shear stress of 20 MPa provides accurate description of all experiments. It should be emphasized that the 3P-ENF test is not suited for accurately determining the critical stress. Consequently the
values of the shear critical stress adopted here $\tau_c^{Mat-1} = \tau_c^{Mat-2} = 20MPa$ provide only an estimate of the shear critical stress and presumably a lower estimate. Table 6 summarizes the cohesive parameters adopted for both configurations. It should be noted that in the DCB simulations of Fig 12, $\tau_c$ was set equal to $\sigma_c$, and $G_{IIc}$ was assigned the identified mode II fracture energy. It has been checked that when adopting the parameters of the CZM given in Table 6, the predictions given in figure 12 are unchanged. In the 3P-ENF simulations, $\sigma_c$ and $G_{IIc}$ were set to values given in Table 6.

<figure><img src="image_15.png" /><figcaption>Figure 13: Comparison of the numerical and experimental ENF force displacement curves: a) simulation of the Mat-1 configuration with two different crack lengths $a_i = 33$ mm and $a_i = 41$ mm. b) simulation of the Mat-2 configuration with $a_i = 33$ mm.</figcaption></figure>

<table><caption>Table 6: Cohesive parameters of the two interfaces in both modes. The fracture energies $G_{Ic}$ and $G_{IIc}$ have been determined in section 3 (except $G_{IIc}$ of Mat-2). The critical stresses $\sigma_c$ and $\tau_c$ are calibrated in this section. As the DCB and ENF tests are not strongly sensitive to these parameters, the values reported in Table 6 must be considered as lower bonds.</caption><thead><tr><th>Pre-preg</th><th>σ<sub>c</sub> (MPa)</th><th>G<sub>Ic</sub> (N/m)</th><th>τ<sub>c</sub> (MPa)</th><th>G<sub>IIc</sub> (N/m)</th></tr></thead><tbody><tr><td>Mat-1</td><td>3</td><td>145</td><td>20</td><td>270</td></tr><tr><td>Mat-2</td><td>3</td><td>80</td><td>20</td><td>533</td></tr></tbody></table>

# 5 Discussion

The aim of this work is to identify the fracture energies in modes I and II of the Copper-Composite interface present in PCBs. At this point, experimental tests were conducted and validated by FE simulations. It is interesting to note that, under mode I loading conditions, the copper-composite interface with Mat-1 demonstrates greater toughness than the one with Mat-2. However, the opposite trend is observed under mode II loading, see Table 6. This highlights the strong sensitivity of the copper-pre-preg interface behavior to the mode mixity ratio as the same assembly was used for all tests. Moreover, this insight can guide the selection of pre-pregs for specific PCB regions that must withstand particular stress conditions during use.
The identification procedure was carried out under two assumptions:

(i) The interface is subjected to Mode I in DCB and Mode II in ENF

(ii) Delamination occurs without any plastic dissipation

Next, FEM simulations of the DCB configuration will serve to validate the two previous statements.

<figure><img src="image_16.png" /><figcaption>Figure 14: Interfacial normal and shear stresses along the interface involving Mat-1: a) DCB test where the crack tip is at position $a = a_i + 20mm = 82mm$ and b) 3P-ENF test when the crack tip is at position $a = a_i + 4mm = 37mm$. $\sigma_c$, $G_{Ic}$, $\tau_c$ and $G_{IIc}$ were set to the values shown in Table 6. The origin of the distance along the interface corresponds to initial position of the crack tip ($a_i$).</figcaption></figure>

Figures 14a and 14b show the opening and shear stresses along the interface for DCB and 3P-ENF with Mat-1 configuration respectively. The crack has propagated almost 20 mm in Fig 14a, so the current crack tip position is $a = a_i + 20mm = 82mm$ (i.e the upper hinge has reached a displacement of $v = 1.8mm$). In Fig 14b, the normal and shear stresses along the interface are captured at the beginning of the propagation phase where $a = a_i + 4mm$ (i.e the loading pin has reached a vertical displacement of $v = 1.8mm$). As shown clearly in Figure 14a, the shear stress remains limited compared to the opening (normal) stress. In contrast, Figure 14b shows that the shear stress is significantly higher than the normal stress. Then, it is clear that the slight asymmetry of the specimen due to the copper film inserted in the sample does not affect strongly the mode mixity. Remember that for DCB or 3P-ENF samples with two identical arms, pure mode I or pure mode II is prevailing. Such a result validates the assumption (i) for both configurations.

The second goal of adopting DCB and ENF tests instead of the peel test, a standard in the PCB industry, is to avoid the predominant effect of plasticity. Indeed the large plastic dissipation in the copper foil during peeling makes the identification of the interface properties quite complex [14, 15]. In that sense, the goal of our work is to identify precisely $G_{Ic}$ and $G_{IIc}$ with a method that does not trigger as much plasticity in copper as in the peel test. To answer that question, we propose to check whether plasticity development, when the delamination process between the FR4 and the copper film occurs, contributes to some extent to the balance of power. In that perspective, the balance of power is illustrated in figure 15. During the simulations, the net input of work rate due to external forces is redistributed among three contributions:
1. Elastic energy stored within the arms,

2. Interfacial damage energy driving crack propagation, and

3. Plastic dissipation within the copper film.

The resulting power balance equation can be written as:

$$P_{\text{ext}} = P_{\text{elas}} + P_{\text{interface}} + P_{\text{plas}} \quad (13)$$

where $P_{\text{ext}}$ is the rate of external work, $P_{\text{elas}}$ is the rate of elastic energy stored in the arms, $P_{\text{interface}}$ represents the power associated with interfacial crack propagation, and $P_{\text{plas}}$ denotes the plastic power dissipated within the copper film—both at the crack tip and during arm bending. This formulation enables a detailed analysis of the instantaneous contributions of the different mechanisms: elastic storage, plastic dissipation, and interfacial damage throughout crack propagation, offering a comprehensive temporal understanding of the fracture process.

<figure><img src="image_17.png" /><figcaption>Figure 15: Rate of energies collected for Mat-1 configuration: a) for the whole DCB model. b) for the whole 3P-ENF model. $\sigma_c$, $G_{Ic}$, $\tau_c$ and $G_{IIc}$ were set to the values identified in Table 6. The geometrical configurations are depicted in figure 1.</figcaption></figure>

The rate of external work (solid line) closely follows the load-displacement curve of the DCB and 3P-ENF tests. This is expected, since the external work rate is calculated as the product of the reaction force $P$ and the loading velocity ($\dot{v} = 10mm/min$), which remains constant in this case. The curves exhibit two distinct stages. In the first stage, which starts at $t = 0s$, most of the external work rate is stored as elastic strain energy (dotted line). The second stage starts just before the peak load, when crack propagation initiates and damage develops. At this point, a significant part of the external work rate is used to extend crack (dashed line), while the remaining part continues to be stored elastically in the specimen arms. For DCB or 3P-ENF, the work rate due to plasticity (solid dotted line) remains small. It is noteworthy that the rate of energy dissipated through interfacial damage is considerably greater than that one dissipated through plasticity in mode I but also in mode II configurations.

To understand how plasticity develops within the copper film, during DCB, it is interesting to present the time evolution of the plastic strain in copper (close to the interface) at different positions: $x = a_i - 49.7mm$, $x = a_i - 26mm$, $x = a_i - 12.75mm$,
$x = a_i - 5\text{mm}$, $x = a_i$, $x = a_i + 7\text{mm}$ and $x = a_i + 15.6\text{mm}$. The first four points ($x < a_i$) are placed at the rear of the initial crack tip position, the three others ($x \ge a_i$) are placed ahead of the initial crack tip so copper at that particular position will experience the delamination process. Remember that for DCB test with Mat-1 configuration, the delamination process starts at $t \approx 65s$, see figure 7 or 15.

<figure><img src="image_18.png" /><figcaption>Figure 16: Time evolution of the accumulated plastic strain within the copper film for the Mat-1 configuration. $\sigma_c$, $G_{Ic}$, $\tau_c$ and $G_{IIc}$ were set to the values identified in Table 6. The geometrical configurations is depicted in figure 1.</figcaption></figure>

In figure 16, for the four points positioned at the rear of the crack tip, the accumulated plastic strain increases with time up to $t \approx 65s$ and remains constant hereafter. This behavior is due to the bending of the DCB arms as the displacement $v$ increases. Plasticity develops up to the moment when the crack starts its propagation. Then, the four positions are facing unloading and the accumulated plastic strain is frozen.

For the three positions at and ahead of the crack tip ($x = a_i$, $x = a_i + 7\text{mm}$ and $x = a_i + 15.6\text{mm}$), the maximum accumulated plastic strain is almost identical. This value is governed by the bending of the arm and the curvature at the crack tip. Indeed, we have observed (result not presented here) that the local curvature at the time of delamination is similar for any position ahead of the crack tip. With this figure, we clearly observed that plasticity is mostly accumulated due to bending of the arms and that no significant (if any) plasticity is generated at the crack tip.

It is also important to note the very limited level of plastic strain ($p = 1.2 \times 10^{-3}$) due to the small curvature of the thick substrate arm. For peel tests [14] and [15], the radius of curvature is very small and the plastic strain cumulated in the copper film is consequently large. The climbing drum peel test offers, in that perspective, an intermediate configuration [13].

The investigation, carried out in figure 15, was further extended to account for practical configurations commonly observed in PCBs. In this industry, three standard copper foil thicknesses are typically employed: 17.5 µm, 35 µm, and 70 µm. These three thicknesses were therefore considered and simulated using three distinct cohesive interface behaviors: $G_{Ic} = 145$ N/m and $\sigma_c = 3$ MPa, $G_{Ic} = 300$ N/m and $\sigma_c = 6$ MPa and $G_{Ic} = 300$ N/m and $\sigma_c = 3$ MPa. Figure 17 shows a comparison of the rate
of exterior work, elastic, plastic and interface damage energies, still for DCB tests. In this figure, each column corresponds to the same thickness, while each row represents the same interface behavior.

<figure><img src="image_19.png" /><figcaption>Figure 17: Rate of energies collected for the whole DCB model with Mat-1 configuration for the 9 selected configurations: Three copper film thicknesses ($h_{Cu} = 17.5\mu m$, $35\mu m$ and $70\mu m$) and three interface responses ($G_{Ic} = 145N/m$, $\sigma_c = 3MPa$, $G_{Ic} = 300N/m$, $\sigma_c = 3MPa$ and $G_{Ic} = 300N/m$, $\sigma_c = 6MPa$). Each column corresponds to the same thickness, while each row represents the same interface behavior. $\tau_c$ and $G_{IIc}$ were set to the values identified in Table 6.</figcaption></figure>

It is interesting to observe that for all configurations in Fig 17, the energy rate spent to damage the interface is significantly dominant compared to the energy rate dissipated through plasticity regardless the copper foil thickness ($\frac{P_{plas}}{P_{interface}} < 3\%$ for $h_{cu} = 17.5\mu m$ and $35\mu m$ and $< 5\%$ for $70\mu m$). It should be mentioned that even by selecting $\sigma_c = 20MPa$, the plastic dissipation was checked to stay limited as well (results not presented here). From figure 17, it is clear that even with a ductile film within a DCB specimen, plasticity development is limited, validating assumption (ii). Then, for the DCB configuration with a thin metal foil studied in this work, adopting the classical beam theory to capture the fracture energy is sound.

A direct outcome of the present study is that the precise description of the copper plastic behavior is not compulsory to capture the fracture energy in mode I and mode II. This is a major advantage of the design, see Appendix A for more details. In the PCB industry, for various applications, the copper foil thickness may vary from $17.5\mu m$ up to $70\mu m$. Depending on the selected pre-preg material, the $G_{Ic}$ and $\sigma_c$ values may vary as well. It is important to check whether the mode I assumption is still valid for a thicker copper foil.
<figure><img src="image_20.png" /><figcaption>Figure 18: Time evolution of the normal and shear stresses for the 9 selected configurations: Three copper film thicknesses ($h_{Cu} = 17.5 \mu m$, $35 \mu m$ and $70 \mu m$) and three interface responses ($G_{Ic} = 145 N/m$, $\sigma_c = 3MPa$, $G_{Ic} = 300 N/m$, $\sigma_c = 3MPa$ and $G_{Ic} = 300 N/m$, $\sigma_c = 6MPa$). Each column corresponds to the same thickness, while each row represents the same interface behavior. $\tau_c$ and $G_{IIc}$ were set to the values identified in Table 6.</figcaption></figure>

One can observe, in all configurations of Fig 18, that the opening stress remains dominant compared to the shear stress. However, the contribution of the latter tends to increase with the copper foil thickness, which is expected, as the configuration gradually shifts toward an Asymmetrical Double Cantilever Beam (ADCB). Note that all the stresses are extracted from the element positioned at a distance of 3 mm ahead of the initial crack tip. This validates the mode I assumption (i).

Focusing on the behavior ($G_{Ic} = 300 N/m$, $\sigma_c = 3 MPa$), the stress tends to last for a relative long period, almost twice that of ($G_{Ic} = 300 N/m$, $\sigma_c = 6 MPa$). This observation can be explained by the distance of separation $\delta_f$. Specifically, for a fixed fracture energy, a lower critical stress results in a larger separation ($\delta_f = \frac{1}{2} \frac{G_{Ic}}{\sigma_c}$) compared to a configuration with higher critical stress ($G_c = 300 N/m$, $\sigma_c = 6 MPa$). This implies that a greater displacement of the arms is required, leading to a longer time before the stress vanishes within the cohesive zone element. Comparing the two configurations — ($G_{Ic} = 145 N/m$, $\sigma_c = 3 MPa$) and ($G_{Ic} = 300 N/m$, $\sigma_c = 6 MPa$) — it is observed that a larger critical stress tends to delay the onset of damage. This is because reaching the critical stress requires a larger displacement in the first configuration compared to the second, thereby causing the initiation of damage to occur later in time.

# 6 Conclusion

A new experimental method is proposed to identify the critical strain energy release rates in mode I and II for the woven composite /copper interface in PCBs. The classical DCB and 3P-ENF tests are adapted. For that purpose, a specific sample with a copper foil in its core has been proposed. In the present work, DCB and 3P-ENF tests are performed for two different
interfaces encountered in PCBs. Next, the classical analytical methods are used to estimate the fracture energies in Mode I and Mode II. Numerical modeling has been carried out to validate the findings. It is shown that the plastic work rate in the copper film remains limited relative to the interface damage work rate necessary to extend the crack in the copper-composite interface, even for tougher interfaces and thicker copper films. Simulations also confirm the predominant mode I and mode II stressing of the interface in DCB and 3P-ENF setups respectively. Therefore, the proposed development addresses the limitations of the peel test, current standard of the PCB industry, by minimizing the impact of plasticity. In peel test, a precise characterization of the plastic response of copper is compulsory to estimate the effective interface energy from the peel force. With the present work, the fracture energy in mode I and mode II can be obtained without a precise definition of the plastic behavior. As for classical DCB and ENF test, the determination of the critical stress is not possible. This remains a topic for a future work. The proposed method is arguably not more difficult to perform and analyze, while predictions are physically robust.

## Acknowledgment

The authors acknowledge the financial support of Agence Nationale de la Recherche (ANR) through the project EMICI ANR-21-CE08-0007. The authors also thank the experimental facilities MécaRhéo from LEM3 (Université de Lorraine - CNRS UMR 7239).

## Appendix A : Role of the plastic behavior of copper in peel and DCB test

The proposed method aims to overcome the limitations of the peel test, which is commonly used as a standard for evaluating the fracture strength of the copper/composite interface in PCBs. In particular, it seeks to avoid the plastic dissipation that occurs in the film due to bending and reverse bending during peeling of the copper foil. In this appendix, a comparison between the interface fracture energy obtained from the conventional peel test and that determined using the proposed methodology in this paper is presented.

The interface energy can be obtained in a peel test of a ductile film on a rigid substrate using the following formula, [8] :

$$ \Gamma = \frac{F}{b}(1 - \cos\theta) - \Psi \qquad (14) $$

Where F is the peel force, b is the film width, θ is the peel angle and Ψ refers to the work expenditure mostly due to plastic bending in the ductile film. Ψ depends on the plastic behavior of the film. Some expressions can be found in the literature for various constitutive relationships: Kim and Kim [8] (Perfectly plastic), Kim and Aravas [9] (Power law), Kinloch et al.[10] (Bilinear response), Simlissi et al.[14] (Voce law) and Girard et al.[15] (Kinematic hardening).

From FE simulations, it has been observed that the delamination process occurs under mixed mode (see Simlissi et al. [14]). Girard et al. [15] showed, based on FE simulations that the ratio of the work expenditure over peeling force can be quite large (30 % for a high interface fracture energy, Γ = 1055N/m).

Simlissi et al. [14] compared the predicted interface fracture energy for various plastic behaviors, assuming that the peeling force and the curvature are known quantities. From the energy balance, they showed that adopting an elastic perfectly plastic
response or a Voce hardening law may lead to difference in interface strength up to 40 %, for the configuration considered in their work. As the elastic-plastic response is not always known (and can be challenging to identify), the peel test seems not to be a dedicated route for a precise characterization of the interface behavior.

To illustrate this, peel tests for the configuration Copper/Mat-2 are conducted using a MTS machine equipped with a 90° peel fixture and a moving table. The machine has a maximum displacement capacity of 1000 mm and a maximum crosshead velocity of 50 mm/s. During the peel tests, the crosshead and table move at the same velocity (0.5 mm/s) to preserve the 90° angle, as they are connected by an inextensible cable. The film is initially under zero load prior to peeling. The peel force is recorded using a 50N load cell, see figure 19. It should be noted that, for each configuration, a steady-state peeling regime is reached after a few millimeters of crosshead displacement. The copper film thickness is 35 µm with a 10 mm width. It is bonded to a Mat-2 substrate (length of 200 mm and a width of 55mm).

Figure 19 presents the evolution of the peel force with respect to time for a 90° peel angle. Three tests are conducted and the resulting force is around 13 N. It is worth noting that, according to the IPC-TM-650 2.4.8C standard, the interfacial fracture energy is directly calculated from the measured peel force, without any consideration of the plastic dissipation occurring in the copper film, leading to a value of $\Gamma = 1300$ N/m. From the copper behavior adopted in the present work, curvature measurement, Girard et al. analysis [15] enable to evaluate the interfacial fracture energy which is significantly lower: 257 N/m. Moreover, during peel tests, crack propagation occurs under an uncontrolled mixed-mode fracture, which makes it difficult to accurately characterize the mode related interfacial properties.

<figure><img src="image_21.png" /><figcaption>Figure 19: Time evolution of the peel force of the configuration copper/Mat2 subjected to a 90° peel test. The loading velocity is 0.5mm/s</figcaption></figure>
<table><caption>Table 7: Values of the peel force, the interface strength Γ (calculated based on the work of [28]), and the interface energies in mode I and mode II for the Copper/Mat-2 configuration, see Table 6.</caption><thead><tr><th>Interface</th><th>Peel force (N)</th><th>Γ (N/m)</th><th>G<sub>Ic</sub> (N/m)</th><th>G<sub>IIc</sub> (N/m)</th></tr></thead><tbody><tr><td>Copper / Mat-2</td><td>12.97 ± 0.25</td><td>257</td><td>80</td><td>530</td></tr></tbody></table>

One can observe that the interface fracture energy assessed from the peel test (mixed mode) lies between the mode I and the mode II fracture energies evaluated from DCB and 3P-ENF tests, see Table 7.

On the other side, DCB test simulations were carried out for two other copper behaviors: Pure elastic and elastic-plastic with an isotropic hardening. The resulting curves are then compared to the DCB curve in the present work while the copper foil was assumed to present a combined hardening, see figure 20.

<figure><img src="image_22.png" /><figcaption>Figure 20: Effect of the plastic behavior of the copper layer on the DCB test outputs: a) Force-displacement curve. b) The evolution of the strain energy release rate. The cohesive parameters are the ones relative to the copper/Mat-1 configuration (see Table 6).</figcaption></figure>

It can be observed, in figure 20a, that the selected plastic behavior of copper slightly influences the region prior to crack propagation, as the bending stress induces minor plastic deformation in the copper layer during this phase. During crack propagation, however, the plasticity of copper has no effect on the strain energy release rate, as all curves overlap (see figure 20 b).

Unlike peel test, the consistent predictions of $G_{Ic}$ for various plastic responses, allow to disregard the characterization of the plastic behavior of the copper film when assessing the interface fracture energy in PCBs with the framework described in this manuscript.

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