# Estimation of interlaminar shear strength of CF / PEKK composites by full-field data-based numerical simulations of short beam shear tests

Juan Pablo Marquez Costa, R. Arquier, G. Lesimple, Jean-Baptiste Marijon, Bruno Fayolle, Ilias Iliopoulos, Gilles Regnier, Guillaume Miquelard-Garnier

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Juan Pablo Marquez Costa, R. Arquier, G. Lesimple, Jean-Baptiste Marijon, Bruno Fayolle, et al. Estimation of interlaminar shear strength of CF / PEKK composites by full-field data-based numerical simulations of short beam shear tests. Polymer Composites, 2025, 46 (S3), 10.1002/pc.29959. hal-05468374

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# Estimation of interlaminar shear strength of CF/PEKK composites by full-field data-based numerical simulations of short beam shear tests

J. P. Márquez Costa ID

R. Arquier

G. Lesimple

J.-B. Marijon ID

B. Fayolle ID

I. Iliopoulos ID

G. Regnier ID

G. Miquelard-Garnier ID

Laboratoire PIMM, Arts et Métiers
Sciences et Technologies, CNRS-UMR
8006, CNAM, Paris, France

**Correspondence**

J. P. Márquez Costa, Laboratoire PIMM,
Arts et Métiers Sciences et Technologies,
CNRS-UMR 8006, CNAM, 151 boulevard
de l'Hôpital, F-75013 Paris, France.
Email: juan-pablo.marquez_costa@
ensam.eu

**Funding information**

Projets Structurants Pour la
Compétitivité (PSPC)

**Abstract**

A hybrid methodology is proposed to estimate the Interlaminar Shear Strength (ILSS) of Carbon Fiber and Polyetherketoneketone (CF/PEKK) composites from Short Beam Shear (SBS) tests. Digital Image Correlation (DIC) is used to capture the apparent shear behavior during loading as well as to detect the first sign of diffuse damage, characteristic of thermoplastic behavior. Simulations based on Finite Element Method (FEM) considering the orthotropic ply behavior allowed us to estimate the shear stress fields and finally, to identify the ILSS using a DIC-based criterion. The methodology is applied to two types of manufactured laminates prepared with a non-optimal and an optimal manufacturing cycle, to assess the impact of consolidation on macroscopic shear behavior. The estimated ILSS values (62.3 and 70.2 MPa for each manufacturing cycle, respectively) are compared with those obtained from the standard ASTM D2344 (88.1 and 95.7 MPa, respectively) whose limitations are discussed, particularly the out-of-plane shear behavior hypothesis.

**Highlights**

* ILSS obtained via dialog between DIC full-field data and FEM-based simulations.

* Significant differences between SBS strength and ILSS for thermoplastic composites.

* SBS strength is not representative of ILSS for thermoplastic composites.

* ILSS is strongly correlated to the composite microstructure at the interplay.

* ILSS is impacted by nonlinear shear behavior.

**KEYWORDS**

carbon fiber reinforced thermoplastic, composites, finite element analysis, full-field data, interlaminar shear strength, short beam test

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

© 2025 The Author(s). *Polymer Composites* published by Wiley Periodicals LLC on behalf of Society of Plastics Engineers.
# 1 INTRODUCTION

In the aerospace field, high-performance thermoplastic composites reinforced by continuous carbon fibers (CF) are increasingly competing with thermoset-based ones.¹⁻⁴ They have the advantage of being potentially welded and recycled.³⁻⁶ Among the existing thermoplastics (TP), the polyaryletherketone (PAEK) family is the most suitable for aerospace applications as it combines high chemical and oxidative resistance with good thermomechanical and impact properties.⁷⁻¹⁰ TP composites are, similarly to thermoset (TS) ones, individual 200 µm-thick plies with unidirectional fibers (also named prepregs) that are manually or automatically stacked to form a laminate with the desired thickness. The laminate with n plies, either unidirectional (UD) [0]n, cross-ply [0/90]n, or quasi-isotropic [0/45/90/-45]sn (s for symmetric laminate) then needs to be consolidated to give the material the required mechanical properties. Although historically processed in energy-intensive devices such as press or autoclave, studies in the last decade have focused on in situ consolidation through Automated Fiber Placement (AFP) or vacuum bag only (VBO).¹,³,⁴

The manufacturing qualification of composites involves multiple levels of characterization tests to ensure the performance of the components. Due to their inherent anisotropic behavior, basic standard tests are required to identify specific mechanical properties,¹¹ which are then used for sizing of more complex parts by experimental and numerical design.¹² Predicting damage and fracture is a crucial part of the design process to avoid premature failure of parts. This requires extensive experimental campaigns to be compared with advanced modeling (e.g., based on Finite Element Method; FEM).¹³⁻¹⁶ Concerning stratified laminates, interply delamination is a critical failure mechanism. It consists of the separation of two adjacent plies due to external loads (e.g., a mechanical, thermal or environmental source).¹⁷ Consequently, the interlaminar mechanical properties, such as crack onset strength and crack propagation, remain crucial to correctly describe the failure of composites. These properties depend on the material behavior and microstructure, but especially within the interfacial region and its vicinity, which in this case corresponds to the resin-rich zone between two adjacent plies.¹⁰ Dávila et al.¹⁸ studied the dispersion of the interface thickness for UD CF laminates embedded in a TS matrix after consolidation in autoclave. They showed through microscopic investigation that the interface thickness can vary between 4% and 9% of the ply thickness. Other authors highlighted the importance of the manufacturing cycle, notably the effect of curing¹⁹ and pressure²⁰ on void content and interlaminar properties for CF-TS composites. This issue has been less studied in the case of CF-TP composites, as studies focused on the healing

mechanism during the cooling phase,²¹ which affects directly the crystallinity and the interface adhesion, or the effect of pressure,²²,²³ leading to improved interlaminar properties in both UD and cross-ply configurations. Arquier et al. showed an increase in SBS strength with the consolidation time for VBO, related to a slow interply reorganization.¹⁰

Consequently, both the ply orientation and the manufacturing cycle can lead to different microstructures at the interface, which in turn impact the mechanical properties of the interply. To quantify it, different experimental tests are found in standards²⁴⁻²⁷ to characterize delamination, considering both the crack onset and crack propagation depending on the loading mode. The interface strength is classically defined as the stress necessary to create a finite crack in the interface, which could, in a second time, propagate at the interface plane.

It is well known that in the case of CF-TP, material can exhibit complex mechanical responses, notably highlighted in both transverse and shear behaviors, leading to viscous and/or plastic effects for rate-dependent loadings and high stress versus strain solicitation respectively.²⁸,²⁹

Gerritzen et al.³⁰ proposed to reproduce the shear material mechanical response of CF-TP composites using an apparent elastic shear behavior. Zhu et al.¹⁴ observed a softening behavior related to viscous and plastic effects with increasing temperature, coupled with a matrix damage tolerance mechanism. Laurin et al.¹⁶ correlated FE simulations based on the Onera Progressive Failure Model (OPFM), concluding that matrix plasticity should be considered in the model. Choupin et al.⁸ highlighted a strong plastic-dependent PEKK behavior leading to a diffuse damage mechanism. Kovačević et al.³¹ suggested a coupling between plasticity and diffuse damage for CF-TP composites under shear loadings using off-axis tensile tests. The estimate of the crack onset strength remains especially important when shear loadings are involved for two main reasons: (i) the complex shear behavior of the composite ply and (ii) the nature of the diffuse damage, when slip by shear forces makes it difficult to monitor using experimental techniques (e.g., optical tools, as proposed in Ref. [32,33]).

The ASTM D2344 standard²⁵ proposes an experimental protocol based on the Short Beam Shear (SBS) test. It promotes interlaminar failure in laminates by maximizing out-of-plane shear and minimizing bending stresses.³⁴ The SBS strength is calculated as follows:

$$SBS = \frac{3 F_{max}}{4 w e} \qquad (1)$$

with $F_{max}$ the maximum load applied, w the specimen width, and e the specimen thickness.
According to the standard, the validity of SBS strength as interlaminar shear strength (ILSS) depends on the material behavior. The equivalence requires several assumptions: (i) the laminate must be symmetric and its properties balanced with respect to the longitudinal axis of the sample, (ii) the mechanical properties are assumed to be elastic, and (iii) the failure mode involves shear as the dominant applied loading.

As mentioned before, the complex elasto-visco-plastic and damageable behaviors of thermoplastic polymer matrix composites make it impossible to use Equation 1 for the ILSS estimation. Instead of that, the ILSS refers to the shear stress reached at the first sign of damage, assumed to be the consequence of underlayer physical phenomena such as dissipative mechanisms including coupled viscous, plastic, and diffuse damage. Consequently, the orthotropic composite behavior must be considered, including a *minima* non-elastic shear behavior for a more realistic estimation of ILSS.

Makeev and co-workers³²,³³ have proposed inverse methods based on full-field data from DIC and numerical methods to assess the shear stress / strain mechanical response for composite laminates. They proposed fitting experimental data by imposing a Ramberg-Osgood constitutive law for macroscopic shear elasto-plastic behavior. This behavior is compared to the shear behavior obtained with V-notched beam specimens for epoxy matrix and carbon fiber laminates. They concluded that a good agreement between the results from SBS tests and V-notched beam tests can be achieved. However, this approach was validated for brittle composites based on epoxy matrix. The following questions arise: (i) how to develop a suitable approach based on experimental and numerical data to analyze the shear response of thermoplastic composites? (ii) how to take advantage of such hybrid methodology to identify the first sign of damage?

In consequence, we propose to use the SBS test to quantify the interlaminar shear strength (ILSS) of UD thermoplastic laminates by a different approach. To do so, experimental SBS tests equipped with Digital Image Correlation (DIC) are carried out and confronted with FEM-based simulations. The comparison between experimental data and numerical simulations allows us to overcome the main hypotheses of the standard to get a realistic value of ILSS. The distinguishing feature of the proposed approach lies in the convenience of deducing the mechanical shear behavior using experimental data and without knowing a priori the constitutive behavior. Consequently, the identification of complex models based on highly accurate constitutive laws which require convenient standard measurements could be overcome, thus avoiding cumbersome laboratory tests.

2 | MATERIALS AND METHODS

This section first presents the composite material used and the manufacturing procedure. Then, a brief description of the characterization of in-plane shear behavior is presented. Finally, the experimental setup for Short Beam Shear (SBS) test as well as the test protocol is described.

**2.1 | Material and sample preparation**

CF/PEKK UD tapes are kindly provided by Hexcel Reinforcements (France). These tapes contain 34 wt% of PEKK Kepstan 7002 (Arkema) having a terepthaloyl (para) and isopthaloyl (meta) isomers (T/I) ratio of 70/30, and 66 wt% (corresponding to about 60 vol%) of unsized AS7 carbon fibers.³⁵ The glass transition temperature (*T*<sub>g</sub>) of the PEKK matrix is 162 °C and its melting temperature (*T*<sub>m</sub>) is 332 °C.³⁶ 24-ply laminates are manufactured by stacking 150 × 150 mm² UD tapes according to the desired layup sequence ([0]₂₄ for the presented study). Out-of-autoclave consolidation of the layup is performed using the VBO process.⁴ Laminates are sealed under a polymeric bag in which vacuum is applied. Consolidation thus occurs at 1 bar of pressure under temperature in an oven. The applied thermal cycle consists in a constant heating ramp (7 °C/min) up to a temperature plateau higher than the PEKK *T*<sub>m</sub> during a fixed time, followed by a constant cooling ramp (−7 °C/min). In the present study, 2 cycles were compared depending on the final temperature reached and the duration of the isothermal step: a “non-optimal” manufacturing cycle with a plateau at 375 °C for 50 min, and an “optimal” one with a plateau at 390 °C for 90 min, named in the following N-OMC and OMC respectively. For each manufacturing condition, optical micrography probes how time and temperature have a direct link with consolidation and the material microstructure, which can be characterized by analyzing on multiple images the void content in the ply volume (φ) and the fiber volume fraction at interfaces between plies (*V*<sub>f</sub><sup>int</sup>), as discussed by Arquier et al.¹⁰ Figure 1 shows through typical images of the microstructure that OMC leads to a smaller φ (white boxes), as well as smoother and thinner interfaces (higher *V*<sub>f</sub><sup>int</sup>) in comparison to N-OMC.

**2.2 | Experimental testing**

2.2.1 | In-plane shear characterization

In order to establish the shear behavior of the composite,
both in-plane and out-of-plane shear characterizations
<figure><img src="image_2.png" /><figcaption>**FIGURE 1** Micrograph of the UD consolidated composite's cross-section. (left) N-OMC with $V_f^{int} \approx 0.5$ and $\phi \approx 0.002$. (right) OMC with $V_f^{int} \approx 0.58$ and $\phi \approx 0.0005$.</figcaption></figure>

are possible.³⁷⁻⁴² Regarding elementary mechanisms, both mechanical behaviors are the consequence of the shear behavior of the matrix with respect to fibers, as proposed in Ref. [43]. In the case of UD composites, the ply can be assumed to be transverse isotropic, which leads to an equivalence between in-plane and out-of-plane shear behaviors.⁴⁴ Some authors discussed this equivalence.³⁷,⁴⁵,⁴⁶ Tsai et al.³⁷ demonstrated this experimentally for graphite/epoxy composites, confirming the validity of the assumption of transverse isotropy. Marquez Costa et al.⁴⁵,⁴⁶ assumed transverse isotropic behavior of a CF-TP composites to study the shear behavior at the interface under mode II loading. Under this assumption, the complexity of out-of-plane shear tests for laminates can be overcome. Sinclair et al.⁴³ demonstrated that the 10-deg off-axis tensile test is an accurate and convenient test method for the characterization of in-plane shear behavior of unidirectional composites, significantly reducing residual stresses in comparison with classical ±45-deg specimens. Then, experimental off-axis (10°) tensile tests on CF/PEKK UD samples have been carried out to obtain the experimental evolution of the secant in-plane shear modulus during loading.

### 2.2.2 | Setup and protocol of SBS test

SBS test is performed to promote interlaminar crack onset according to ASTM D2344 standard.²⁵ The specimens tested are cut from 150 × 150 mm² panels using a diamond wire saw/diamond saw from Escil. UD laminates are used to monitor the longitudinal out-of-plane shear strength. The SBS tests are performed on an Instron 5581 machine using a 50 kN cell force at a speed of 1 mm/min. The experimental setup is composed of two lower rollers (supports) of diameter 3 mm and an upper

load roller of diameter 6 mm (Figure 2). The specimens' geometry is proportional to the sample thickness (e) as follows: the length of the sample is equal to 6e, the width to 2e, and the span between supports to 4e, according to the ASTM D2344 standard.²⁵ Note that this setup can support a maximum load of 10 kN. The global coordinate system for experimental measuring (X-Y-Z) as well as the local coordinate system related to the ply orientation (1-longitudinal, 2-transverse aux fibers and 3-out-of-plane) are shown in Figure 2.

The test is instrumented with Digital Image Correlation (DIC), using a Manta G-917B camera with a telecentric lens and a LED panel (Figure 2). A random black and white speckle pattern is applied on one edge (plane XZ) of the samples using spray paint, allowing the deformation to be monitored. The images are captured every 200 ms and recorded using VIC-snap software (KiloNewton). The DIC is performed using VIC2D software (KiloNewton) with a step size and subset size of 12 and 27 pixels, respectively.⁴⁷ The SBS strength is then calculated from Equation 1. Classically, the results are averaged over at least five specimens.

Two experimental campaigns have been carried out for UD samples. The first one concerns the N-OMC, while the second one concerns the OMC. Samples' dimensions are presented in Table 1.

## 3 | HYBRID METHODOLOGY FOR ILSS ESTIMATION

This section presents the modeling of the SBS test based on FEM and numerical simulations carried out to estimate stress vs. strain evolution within the laminate. Then, a hybrid strategy based on full-field displacement and strain data is presented for the quantification of the ILSS.
**FIGURE 2** Test setup and schematic of the short beam shear test, showing the measuring coordinate system (X-Y-Z) and material coordinate system (1-2-3) corresponding to the ply orientation.

<figure><img src="image_3.png" /><figcaption>TABLE 1 Sample dimensions.</figcaption></figure>

<table><thead><tr><th colspan="3">N-OMC</th><th colspan="3">OMC</th></tr><tr><th>Sample</th><th>e [mm]</th><th>w [mm]</th><th>Sample</th><th>e [mm]</th><th>w [mm]</th></tr></thead><tbody><tr><td>1</td><td>4.8</td><td>9.65</td><td>1</td><td>4.6</td><td>9.9</td></tr><tr><td>2</td><td>4.8</td><td>9.8</td><td>2</td><td>4.6</td><td>9.8</td></tr><tr><td>3</td><td>4.7</td><td>9.55</td><td>3</td><td>4.6</td><td>9.8</td></tr><tr><td>4</td><td>4.7</td><td>10.25</td><td>4</td><td>4.6</td><td>10</td></tr><tr><td>5</td><td>4.7</td><td>9</td><td>5</td><td>4.6</td><td>9.5</td></tr></tbody></table>

**3.1 | Modeling of SBS test**

SBS test has been modeled by finite elements in Abaqus. The stacking sequence is explicitly modeled considering the ply thickness in the out-of-plane direction. No damage is assumed in the SBS model. The ply behavior is described by its constitutive law, generally as follows:

$$ \boldsymbol{\sigma} = \mathbf{C} : (\boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}^{\text{th}} - \boldsymbol{\varepsilon}^{\text{ve}} - \boldsymbol{\varepsilon}^{\text{p}} - \boldsymbol{\varepsilon}^{\text{in}}) \quad (2) $$

where $\boldsymbol{\sigma}$ is the stress 2nd order tensor, $\boldsymbol{\varepsilon}$ is the strain 2nd order tensor, $\boldsymbol{\varepsilon}^{\text{th}}$ is the thermal strain 2nd order tensor, $\boldsymbol{\varepsilon}^{\text{ve}}$ is the viscous strain 2nd order tensor, $\boldsymbol{\varepsilon}^{\text{p}}$ is the plastic strain 2nd order tensor, $\boldsymbol{\varepsilon}^{\text{in}}$ is the inelastic strain 2nd order tensor and $\mathbf{C}$ is the stiffness 4th order tensor. In the following, we consider: (i) $\boldsymbol{\varepsilon}^{\text{th}} = \mathbf{0}$ at the ply scale because of UD stratification, (ii) $\boldsymbol{\varepsilon}^{\text{ve}} = \mathbf{0}$ because of negligible rate effects at the loading range of interest as explained in deeper detail in section 3.2.2., (iii) $\boldsymbol{\varepsilon}^{\text{p}}$ is highlighted in SBS test on the shear ply behavior as shown in detail in section 3.2.1. and (iv) $\boldsymbol{\varepsilon}^{\text{in}} = \mathbf{0}$ before any damage evolution. In a first approach, the ply behavior can be assumed as elastic orthotropic. Later on, apparent elastic shear behavior related to nonlinear sources (e.g., plasticity) will

be considered through the secant shear modulus as described in section 3.2.1. Table 2 shows the mechanical properties of the constituent (fiber and matrix).$^{34,36}$ Longitudinal, transverse and in-plane shear modulus as well as Poisson's ratios have been analytically calculated using the mixing laws given by Equations 3–7.$^{11}$ Out-of-plane behavior is assumed to be equivalent to in-plane behavior under the hypothesis of transverse isotropy. The load roller and the support rollers are considered as rigid bodies because of their higher stiffness compared to the macroscopic response of the laminate.

$$ E_1 = E_f V_f + E_m (1 - V_f) \quad (3) $$

$$ \nu_{12} = \nu_f V_f + \nu_m (1 - V_f) \approx \nu_{13} \quad (4) $$

$$ G_{12} = G_m \frac{G_{fl}(1+V_f) + G_m(1-V_f)}{G_{fl}(1-V_f) + G_m(1+V_f)} \approx G_{13} \approx G_{23} \quad (5) $$

$$ E_2 = \frac{2}{\frac{1}{2K_L} + \frac{1}{2G_{23}} + 2 \frac{\nu_{12}^2}{E_1}} \approx E_3 \quad (6) $$

$$ \nu_{23} = \frac{E_2}{2G_{23}} - 1 \quad (7) $$

where $V_f = 0.55$, $E_f = 243$ GPa,$^{34}$ $E_m = 3.8$ GPa,$^{35}$ $E_{ft} = 20$ GPa,$^{48}$ $G_{fl} = 9.7$ GPa,$^{48,49}$ $G_{ft} = 6.3$ GPa,$^{48,49}$ $\nu_f = 0.27^{51}$ and $\nu_m = 0.35$ are the fiber volume fraction, the fiber longitudinal modulus, the matrix modulus, the transverse fiber modulus, the fiber longitudinal shear modulus, the fiber transverse shear modulus and the fiber and matrix Poisson ratios respectively.

Surface-to-surface contact is considered between the rollers and the sample as proposed in.$^{52}$ To define this contact, interaction properties have been taken from literature, that is, normal hard contact and a friction coefficient of 0.3 for
<table><caption>TABLE 2 Mechanical properties assuming elastic orthotropic ply behavior.</caption><thead><tr><th>E<sub>1</sub> [GPa]</th><th>E<sub>2</sub> [GPa]</th><th>E<sub>3</sub> [GPa]</th><th>ν<sub>12</sub> [-]</th><th>ν<sub>13</sub> [-]</th><th>ν<sub>23</sub> [-]</th><th>G<sub>12</sub> [GPa]</th><th>G<sub>13</sub> [GPa]</th><th>G<sub>23</sub> [GPa]</th></tr></thead><tbody><tr><td>135.4</td><td>8.2</td><td>8.2</td><td>0.31</td><td>0.31</td><td>0.48</td><td>3.4</td><td>3.4</td><td>3.4</td></tr></tbody></table>

Note: Under the hypothesis of transverse isotropy, only 5 constants are required.

<figure><img src="image_4.png" /><figcaption>FIGURE 3 Evolution of out-of-plane shear stress for 3D simulation as a function of the Y coordinates for 3 different X coordinates: X = 0 (applied load) x = d/4 (half distance to the support) and x = d/2 (support). Due to symmetry only one quarter of the sample is considered for the simulation.</figcaption></figure>

tangential contact.⁵³ Boundary conditions are defined in accordance with the experimental protocol. Support rollers are fixed (zero displacements and rotations in all directions X, Y, Z), while an imposed displacement on the load roller in the Z direction allows simulating the sample loading. Additionally, XZ and YZ symmetries are applied due to the symmetric load case during the SBS test.

According to García-Rodríguez et al.,⁵² 3D modeling is mandatory to consider edge effects in the transverse Y-direction. Figure 3 shows the evolution of out-of-plane shear stress for 3D simulation as a function of the Y coordinates for 3 different X coordinates. It is clear that the maximal shear stress appears at the edge. For this reason, the information given by DIC at the edge of the samples could be compared to numerical calculations. This configuration will be exploited in the next section.

## 3.2 | Hybrid approach

The ILSS will be estimated at the first sign of diffuse damage (crack onset) by direct analysis of the experimental full-field data and the correlated numerical stress estimation. Figure 4 shows a flowchart of the hybrid methodology for estimating the ILSS. For each step, one simulation is conducted by applying the corresponding displacement on the load roller. The out-of-plane shear strain field ($ε_{xz}$) is extracted from the respective DIC map and projected on the integration points of the FEM-based simulation. Then, as proposed in the next

section, apparent linear shear behavior will be computed, assuming a dependency of $G_{12}$ (and consequently on $G_{13}$) on shear strain $ε_{12}$ (and consequently on $ε_{13}$), as measured experimentally by off-axis tensile testing. Finally, stress fields will be computed for further analysis of $τ_{13}$.

This procedure can be repeated to reproduce the loading during the SBS test. Finally, by computing the reaction force at each load step, the simulated macroscopic response force versus displacement can be obtained for comparison with experimental data.

### 3.2.1 | Data-based elasto-visco-plastic shear behavior

Experimental off-axis (10°) tensile tests of CF/PEKK UD samples have been carried out to obtain the experimental in-plane shear modulus evolution during loading. The following expressions are used to convert global stress and strain (XY in the measuring coordinate system) into in-plane shear stress and strain (12 in the ply coordinate system):

$$τ_{12} = σ_X \sinθ\cosθ \quad (8)$$

$$ε_{12} = (ε_{YY} - ε_{XX})\sin(2θ) + ε_{XY}\cos(2θ) \quad (9)$$

where $σ_X$ refers to the longitudinal stress applied to the sample, $θ$ is the off-axis angle and $ε_{XX}$, $ε_{YY}$ and $ε_{XY}$ are the longitudinal, transverse and in-plane shear strains.
<figure><img src="image_5.png" /><figcaption>FIGURE 4 Flowchart for estimating shear stress fields based on a hybrid approach combining DIC data and FEM-based simulations.</figcaption></figure>

Figure 5 (left) shows the evolution of the shear stress $\tau_{12}$ as a function of the shear strain $\epsilon_{12}$ during loading for 5 tests as well as the average shear behavior. It is possible to deduce the apparent shear modulus as a function of the shear strain, defined as the secant modulus as follows:

$$G_{12}^{\text{secant}} = \frac{\tau_{12} - \tau_{12}(0)}{\epsilon_{12} - \epsilon_{12}(0)} \quad (10)$$

where $\tau_{12}(0)$ and $\epsilon_{12}(0)$ are the initial shear stress and strain.

Figure 5 (right) shows the evolution of the secant shear modulus from Equation 10 considering the average shear behavior of Figure 5 (left). This evolution will be used as an input for the computation of numerical FEM-based simulations.

### 3.2.2 | Full-field data-based SBS test

Figure 6 shows the macroscopic curve force versus displacement during the SBS test, while Figure 7 shows the

loading scenario by local DIC at four different points of interest up to failure. First, the macroscopic response evolves non-linearly as a consequence of the initial contact between the load roller and the sample.⁵² Then, the macroscopic response evolves linearly, which is also observed by DIC (from map 211 to map 311) where displacement fields evolve linearly with the force. Then, a nonlinear response is observed both on the macroscopic response and the local displacement fields. The first signs of diffuse damage later appear, and can be identified where displacement field gradients change (map 361). At this point, load continues to increase up to the maximum applied load (map 491) due to the tensile and compression mechanical states in the plies combined with the shear state at the interface. Finally, crack onset propagates, leading to failure.

Damage crack onset is detected by comparing the longitudinal displacement fields on equidistant lines with respect to the applied load. Figure 8 shows the evolution of the mean displacement values measured by DIC on both the right (x = d/4) and left (x = -d/4) sides of the specimen, where crack onset is expected to appear according to
<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>FIGURE 5 (left) Evolution of the shear stress vs. shear strain and (right) evolution of the apparent shear modulus versus shear strain obtained from off-axis tensile tests.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>FIGURE 6 Evolution of normal load as a function of normal displacement during SBS test for a UD composite from Instron machine (raw data) and from DIC, with the corresponding linear regression (dashed orange line).</figcaption></figure>

numerical simulations. The zoomed region (continuous line) shows the difference between the two sides mean displacement ($\Delta U_{mean}$). A threshold (dotted line) of 0.5 px has been retained, which corresponds to the value at which $\Delta U_{mean}$ evolves at a higher rate. The intersection between this threshold and $\Delta U_{mean}$ defines the damage threshold.

To validate this criterion, the local shear strain $\epsilon_{xz}$ in the vicinity of the crack onset has been analyzed. It seems to evolve linearly with time before the first sign of damage and reaches a maximum value around 3% (red dot), as shown in Figure 9 (left). This is the consequence of the experimental displacement rate of 1 mm/min, which leads to quasi-static conditions (constant shear strain rate before damage), as shown in Figure 9 (right). Just after, $\epsilon_{xz}$ evolves nonlinearly and the shear strain rate becomes

noisy. The transition point is in good agreement with the damage threshold based on DIC.

The apparent effects of the complex shear elasto-visco-plastic ply behavior can be reproduced by an elastic shear behavior whose apparent shear modulus can be updated for each step using Equation 10. The secant shear modulus is defined as a function of the shear strain level in agreement with the characteristic shear behavior of ply as presented in section 3.2.1. This hypothesis neglects underlying mechanisms (e.g., viscosity, plasticity). Thus, even if it is only valid in the current testing conditions of SBS tests, it is sufficient to estimate a shear stress level up to ILSS.

## 4 | RESULTS AND DISCUSSION

This section presents first the calculation of SBS strength from macroscopic SBS tests. Then, computed ILSS values are presented following the methodology described in section 3. Finally, a discussion that focuses on the advantages and limitations of the proposed method is presented.

### 4.1 | SBS strength with standard

Figure 10 shows the macroscopic response force versus displacement during the SBS test for both N-OMC and OMC samples.

It is observed that force-displacement curves of the OMC samples display less dispersion than those consolidated with the N-OMC ones, which is especially noticeable for the maximum load. From previous data and using Equation 1, the SBS strength can be calculated. Figure 11 shows the mean value and the standard deviation for both
i)

<figure><img src="image_9.png" /></figure>

ii)

<figure><img src="image_10.png" /></figure>

iii)

<figure><img src="image_11.png" /></figure>

iv)

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /><figcaption>FIGURE 7 Loading scenario during SBS test for a UD composite based on DIC. Evolution of longitudinal and transverse displacement fields from DIC (i) in the linear regime (map 211), (ii) in the last nonlinear regime (map 311), (iii) at detected ILSS (map 361) and (iv) at maximal load (map 491).</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>FIGURE 8 Evolution of the crack onset criterion based on the mean longitudinal displacement measured by DIC. The inset shows the difference between the right (x = d/4) and left (x = -d/4) sides of the specimen (ΔU<sub>mean</sub>) and the threshold (dotted line) of 0.5 px.</figcaption></figure>

manufacturing conditions. A higher mean value for the OMC samples is observed (95.7 MPa) compared to the N-OMC ones (88.1 MPa). It can also be observed that the standard deviation is higher for the latest, but remains small in both cases. Due to the good reproducibility of the SBS test, only test 1 of each configuration is presented in the following paragraph.

## 4.2 | Numerical estimation of the ILSS based on DIC data

The methodology presented in Section 3.2 is applied for one representative case of both OMC and N-OMC. We recall that, according to our definition of ILSS in Section 3.2, ILSS values are expected to differ from SBS ones presented in Figure 11, related to the peak of force, and which assumes linear elastic behavior. To emphasize
<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /><figcaption>FIGURE 9 (right) Evolution during SBS test of: (right) shear strain (orange line) and normal load (blue) and (left) shear strain rate.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>FIGURE 10 Applied force as a function of the displacement during SBS test on [0]₂₄ CF/PEKK samples, OOA-consolidated following: (left) N-OMC and (right) OMC.</figcaption></figure>

this, Figure 12 shows the comparison for each case between the macroscopic experimental response (continuous blue line) and the numerical response with: (i) linear elastic shear behavior and (ii) nonlinear elastic shear behavior by the hybrid approach. The mechanical response in linear elasticity is clearly overestimated (dashed orange line). Regarding the proposed approach, the numerical values computed for specific maps are shown as red dots. These predicted values of force are compared to the experimental data and absolute errors are presented in Tables 3 and 4 for each case.

A very good correlation is observed in both cases up to ILSS. Consequently, it is possible to analyze the computed stress fields, at least up to ILSS as detected by DIC. Figure 13 shows the shear stress field on the edge of the sample, whose ILSS value has been highlighted in blue,

corresponding to the maximal τ₁₃ value in the vicinity where crack onset is expected to appear.

We notice again a slightly smaller value of ILSS for the N-OMC sample (62.3 MPa) compared to the OMC one, whose value is 70.2 MPa. This is a direct consequence of two main factors: (i) the shear behavior is stiffer, and (ii) the damage onset appears later when consolidation of the composite is improved.

## 4.3 | Discussion about SBS strength and ILSS

Both the SBS strength and the ILSS show an improved strength of the interface in the interply region when the manufacturing cycle is optimized, correlated with the microstructure shown in Figure 1. Temperature and time
play a crucial role in the process mechanisms (Darcy versus squeeze flow, shear-banding, volatile diffusion at the ply and healing at the interface),⁴ leading to a smooth

<figure><img src="image_21.png" /><figcaption>FIGURE 11 SBS strength values of [0]₂₄ CF/PEKK samples, OOA-consolidated following: (left) N-OMC and (right) OMC.</figcaption></figure>

<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /><figcaption>FIGURE 12 Comparison between numerical and experimental force vs. displacement curves. First crack onset detected: (left) at map = 236 for N-OMC and (right) at map = 361 for OMC.</figcaption></figure>

<table><caption>TABLE 3 Absolute errors in force between numerical simulation and experimental curve for N-OMC and OMC.</caption><thead><tr><th>Map [–]</th><th>61</th><th>111</th><th>161</th><th>211</th><th>236</th><th>261</th><th>311</th><th>361</th><th>411</th><th>461</th></tr></thead><tbody><tr><td>Absolute error [N] N-OMC</td><td>260.9</td><td>41.5</td><td>110.9</td><td>138.1</td><td>164.3</td><td>193.5</td><td>263.0</td><td>–</td><td>–</td><td>–</td></tr><tr><td>Absolute error [N] OMC</td><td>205.3</td><td>77.7</td><td>274.3</td><td>354.8</td><td>–</td><td>324.3</td><td>151.3</td><td>60.9</td><td>226.4</td><td>135.6</td></tr></tbody></table>

<table><caption>TABLE 4 Summary of experimental and computed results for N-OMC and OMC samples.</caption><thead><tr><th rowspan="2"></th><th colspan="3">Last-point of macroscopic linear regime</th><th colspan="3">Detected ILSS</th><th colspan="3">Maximal load</th></tr><tr><th>Map [–]</th><th>τ<sub>13</sub> [MPa]</th><th>ε<sub>xz</sub> [%]</th><th>Map [–]</th><th>τ<sub>13</sub> [MPa]</th><th>ε<sub>xz</sub> [%]</th><th>Map [–]</th><th>τ<sub>SBS</sub> [MPa]</th><th>ε<sub>xz</sub> [%]</th></tr></thead><tbody><tr><td>N-OMC</td><td>211</td><td>53.7</td><td>1.2</td><td>236</td><td>62.3</td><td>1.6</td><td>311</td><td>88.1</td><td>–</td></tr><tr><td>OMC</td><td>311</td><td>61.0</td><td>2.0</td><td>361</td><td>70.2</td><td>3.1</td><td>491</td><td>95.7</td><td>–</td></tr></tbody></table>

interface between plies, a lower void content¹⁰ and consequently a stiffer local behavior.

Since SBS strength is computed from the macroscopic response, its calculation using Equation 1 neglects any sources of nonlinearity, leading to overestimated strengths that cannot be considered as ILSS for thermoplastic composites. To overcome this, the hybrid approach offers the following advantages: (i) it accounts for the apparent effect of nonlinearities (e.g., plasticity) on the apparent shear mechanical response, which directly influences interlaminar mechanical properties, (ii) through multi-step elastic full-field data-based FEM simulations, the complex apparent nonlinear evolution of the out-of-plane shear behavior of CF/PEKK is revealed. It results in local softening of shear stresses and explains the discrepancy between ILSS and SBS values, and (iii) a simple criterion based on DIC displacement measurements is employed. It enables the detection of ILSS in good agreement with the evolution of shear strain in the vicinity of the expected crack onset.
<figure><img src="image_24.png" /><figcaption>**FIGURE 13** Numerical out-of-plane shear stress field $\tau_{13}$ at: (top) map = 236 for N-OMC and (bottom) map = 361 for OMC. Highlighted value corresponding to estimated ILSS: (top) 62.3 MPa and (bottom) 70.2 MPa.</figcaption></figure>

5 | CONCLUSION

A hybrid methodology based on full-field data from DIC and FEM simulations has been presented to estimate the value of ILSS from SBS tests for two manufacturing conditions of thermoplastic CF/PEKK composite materials. The compared conditions consisted of: (i) a non-optimal cycle where the composite has been consolidated at 375 °C for 50 min and (ii) an optimal cycle at 390 °C for 90 min. The DIC instrumentation of SBS tests has revealed that first damage onset appears earlier than the force peak. This is a consequence of the complex PEKK/CF behavior, notably in the shear direction, for which an elastic response is not accurate enough. For this reason, the SBS strength as described in the standard ASTM D2344 is not valid to estimate a quantitative value of ILSS. However, the presented methodology has been able to predict the stress field up to crack onset (assumed to appear with first sign of damage), and then, up to ILSS. For this, the apparent shear behavior has been used as an input for FEM-based simulations, while strain fields from DIC have been used to locally update the apparent shear modulus during loading. Table 4 presents a recompilation of experimental and computed

values for N-OMC and OMC at the following points of interest: last point of the macroscopic linear regime, detected ILSS, and maximal load.

ILSS values of 62.3 and 70.2 MPa have been found for the N-OMC and OMC, respectively. These values are far from the SBS strengths (88.1 and 95.7 MPa respectively). In fact, two reasons explain this gap: (i) the crack onset for ILSS appears earlier than the peak force, and (ii) the shear behavior evolves non-linearly with the shear strain, leading to smoother shear stress fields during loading with respect to a purely elastic behavior as assumed in Equation 1. The differences between both cases can be related to the microstructure of the interface. To confirm its impact, future work should be focused on the characterization of shear behavior and failure for both manufacturing conditions.

Based on previous results, the present approach can be refined and further validated in two directions: (i) identifying a more accurate constitutive law and (ii) extending the approach to different loading conditions. For the first aspect, viscous effects should be considered initially, followed by plasticity and damage for both the ply behavior and the interface. For the second aspect, an extended set of experimental data is required to account
for rate-dependent and nonlinear shear behavior. In both cases, complementary tests such as [±45]ns tensile tests or V-notched tests would be valuable to compare in-plane and out-of-plane shear behavior and to reassert the assumption of transverse isotropic behavior. Moreover, instrumentation complementary to optical tools (e.g., acoustic emission) should help to better identify the crack onset and confirm the proposed technique based on the local displacement threshold described in Section 3.2 during SBS tests. Moreover, full-field displacement errors between experimental (DIC) and numerical data (FEM) should be compared for a best accurate prediction of ILSS.

Other laminate configurations (e.g., cross-ply, quasi-isotropic) shall be analyzed to deepen the understanding of not only the relationship between manufacturing and interface microstructure, but also the impact of residual stresses on interlaminar properties. Finally, crack propagation tests and modeling with an explicit interface description should allow to highlight the interface toughness as proposed by Tijs et al.⁵⁴ The effect of an optimal manufacturing cycle could be evidenced in the improvement not only of the interface strength, but also of the critical energy release rate, looking for structural applications.

## ACKNOWLEDGMENTS

This work was conducted under the framework of HAICoPAS, a PSPC project (projet de recherche et développement structurant pour la compétitivité, project number: PSPC.AAP-7.0_HAICoPAS) granted by Bpifrance, and of the Industrial Chair Arkema (Arkema/CNRS-ENSAM-Cnam). The authors would like to thank Arkema, Hexcel, and more specifically Lucien Fiore, Nicolas Cadorin, Michel Glotin, Jean-Paul Moulin, Henri-Alexandre Cayzac, Jérôme Pascal, Sylvie Tencé-Girault, and Yves Deyrail for fruitful discussions.

## DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## ORCID

J. P. Márquez Costa † https://orcid.org/0000-0002-1618-2625

J.-B. Marijon † https://orcid.org/0000-0003-4358-0159

B. Fayolle † https://orcid.org/0000-0002-9578-1692

I. Iliopoulos † https://orcid.org/0000-0002-9368-2906

G. Regnier † https://orcid.org/0000-0002-1543-1837

G. Miquelard-Garnier † https://orcid.org/0000-0002-0251-8941

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**SUPPORTING INFORMATION**

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**How to cite this article:** Márquez Costa JP, Arquier R, Lesimple G, et al. Estimation of interlaminar shear strength of CF/PEKK composites by full-field data-based numerical simulations of short beam shear tests. *Polym Compos.* 2025;1-15. doi:10.1002/pc.29959