# Experimental and theoretical investigation of the low velocity impact response of CFRP composite beams

Omar Baabdullah, Ramzi Othman, U.A. Khashaba

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Omar Baabdullah, Ramzi Othman, U.A. Khashaba. Experimental and theoretical investigation of the low velocity impact response of CFRP composite beams. Results in engineering, 2025, 28, pp.107128. 10.1016/j.rineng.2025.107128. hal-05502761

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Submitted on 10 Feb 2026

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Research paper

# Experimental and theoretical investigation of the low velocity impact response of CFRP composite beams

Omar Baabdullah<sup>a,d</sup>, Ramzi Othman<sup>a,b,*</sup> ◆ U.A. Khashaba<sup>a,c</sup>

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

## ARTICLE INFO

### Keywords:

Low-velocity impact (LVI)

Carbon fiber reinforced polymer (CFRP)

composite

Energy absorption

Damage

## ABSTRACT

This paper presents an investigation into the low-velocity impact (LVI) behavior of carbon fiber reinforced polymer (CFRP) composite beams. The main goal is to explore the experimental and theoretical responses of woven CFRP composites under LVI loading. To simplify the typically complex geometries of composite structures, this study focuses on beam configurations rather than plates. The effects of impact energy on damage characteristics, peak load, and energy absorption are comprehensively analyzed. The force and energy curves exhibit three distinct behaviors across varying impact levels. Elastic behavior is evident at impact energies of 2, 5, and 7.5 J. The damage threshold in terms of force, displacement, and energy for the composite beam is demonstrated at intermediate impact levels of 10, 12.5, and 15 J. At the highest impact levels of 17.5 and 20 J, the specimen undergoes complete collapse, as indicated by a sharp drop in the force-time curves, open-loop force-displacement curves, and a continuous increase in the energy-displacement curves without a descending path. The absorbed energy by the composite beam demonstrates distinct behaviors across the low, intermediate, and high impact energy ranges. A damage-based theoretical model has been proposed that effectively captures the key characteristics of the beam's impact response. In addition to the damage mechanism, the model reveals the presence of an auxiliary energy absorption mechanism that operates concurrently. Notably, this secondary mechanism persists across all levels of impact energy, including those within the lowest range. The findings provide valuable insights for enhancing simulation accuracy for impact-resistant composite structures.

## 1. Introduction

Carbon fiber reinforced polymers (CFRPs) combine high stiffness-to-weight ratio, tensile strength, corrosion resistance, and fatigue performance, making them widely used in aerospace, automotive, marine, wind energy, and sporting goods applications [1,2]. Low-velocity impact (LVI) incidents, such as tool drops, gravel strikes, or debris contact, are nearly unavoidable during the manufacturing and service life of CFRP products.

Impact performance and damage tolerance are crucial parameters for CFRP structures, especially in service environments prone to accidental drop-weight or foreign object impact. Low-velocity impacts may result in barely visible damage such as matrix cracking and interlaminar delamination, which can lead to significant reductions (up to 50–75 %) in post-impact compressive strength [3]. This behavior is further

influenced by environmental factors such as moisture and temperature, as shown in recent studies on compression-after-impact (CAI) performance in CFRPs exposed to distilled water immersion and thermomechanical impact conditions [3]. Furthermore, scarf adhesive joints modified with various nanofillers have shown notable improvements in impact stiffness, energy absorption capacity, residual strength, and fatigue resistance, making them highly promising for repairing CFRP structures exposed to thermomechanical loading conditions [4–6]. These insights emphasize the importance of evaluating CFRPs not only in ideal conditions but also under realistic environmental and mechanical loading scenarios. Wang et al. [7] emphasized the significant influence of stacking sequences on the low-velocity impact behavior of CFRP laminates. Their findings indicate that symmetric stacking configurations deliver superior impact responses compared to asymmetric arrangements, characterized by higher peak impact forces, increased

<footnote><sup>a</sup> Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia</footnote>

<footnote><sup>b</sup> Ecole Centrale de Nantes, Institut de Recherche en Génie Civil et Mécanique, UMR CNRS 6183, BP 92101, 44321 Nantes cedex 3, France</footnote>

<footnote><sup>c</sup> Mechanical Design and Production Engineering Department, Faculty of Engineering, Zagazig University, Zagazig 44519, Egypt</footnote>

<footnote><sup>d</sup> Mechanical Engineering Department, College of Engineering and Architecture, Umm Al-Qura University, P.O. Box 5555, Makkah 24381, Saudi Arabia</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: ramzi.othman@ec-nantes.fr (R. Othman).</footnote>
rebound energies, smaller damage areas, reduced internal damage, and lower damage volumes under equivalent impact conditions. These advantages are mainly due to the more uniform stress distribution through the laminate thickness in symmetric configurations, effectively reducing interlaminar shear stresses and delaying the progression of delamination.

In the literature, several strategies have been investigated to improve the impact resistance of composites. These approaches include designing hybrid or sandwich composites, modifying the stacking sequence and number of plies, or integrating natural fibers, nanofillers, and other reinforcements [8]. For instance, studies on hybrid composites demonstrate enhanced peak load and energy absorption, though these benefits often come with a slight reduction in stiffness compared to pure composites [9,10]. Additionally, research consistently shows that both maximum contact force and damage tend to increase with higher impact energy levels [10,11]. Wang et al. [12] specifically explored the effects of stacking sequences in flax and carbon fabric composites under low-velocity impact (LVI). Their findings revealed that using carbon fiber layers as exterior reinforcements improved energy absorption capabilities. Studying also the effects of flax and carbon stacking sequences, Masud and Mubashar [13] reported improved impact performance when flax fibers were symmetrically distributed within the composite. However, a smaller damage area was observed with an asymmetrical distribution of flax-reinforced layers. Through machine-learning assisted analysis, Hiremath et al. [14] found that hybrid carbon-Kevlar composites outperform both plain carbon and Kevlar laminates in terms of impact resistance during their comparative study of LVI response.

Research also highlights the effects of alternative reinforcement materials and stacking configurations. Hiremath et al. [15] showed that variations in stacking sequences can result in distinct damage patterns and failure mechanisms. Li et al. [16] investigated how the lay-up sequence affects the low-velocity impact response of GFRP laminates by conducting double-point impact and compression-after-impact tests. Kazemi et al. [17] found that a novel hybrid thermoplastic composite using ultra-high molecular weight polyethylene (UHMWPE) fibers showed reduced structural degradation and increased energy absorption. Thermoplastic laminates, in general, exhibit greater ductility and lower contact force relative to thermosetting laminates [17,18]. For example, Zamani et al. [19] observed that Kevlar surface laminates improved stiffness and impact mitigation, resulting in reduced contact time and smaller damage areas on laminate surfaces. Additionally, Azimpour-Shishevan and Mohtadi-Bonab [20] reported that multi-walled carbon nanotubes (MWCNT) can improve matrix toughness and fiber adhesion, thereby enhancing the overall impact resistance. Alshehri et al. [21] further noted that incorporating helical MWCNT increased peak force and impact resistance, boosting composite performance.

Comparative studies suggest that glass fibers can exhibit better impact performance than carbon fibers. Arranging carbon layers between glass layers has shown to protect the carbon fiber and enhance resistance to damage [22-24]. Grasso and Xu [24] explored the delamination threshold load (DTL) of glass/carbon fiber composites and observed no abrupt load drop or slope change, suggesting that further research on these characteristics is needed. Siddiqui et al. [25] highlighted that the impact resistance of natural fiber composites can be optimized by increasing the natural fiber content up to an optimal level. In another study, Demir and Ekici [26] examined the structural response of various natural fabric reinforcements, emphasizing the importance of increasing composite thickness to withstand impacts.

The influence of fiber orientation has also been studied extensively. Lin and Waas [27] found that a higher number of 45° plies resulted in smaller impact damage regions at the same energy level, while the damage pattern varied with different stacking sequences. Conversely, Kaware et al. [28] found that 0°/90° layups resulted in greater interlaminar damage under LVI. Pei et al. [29] showed that woven carbon

fibers with a 30° off-axis orientation performed best among different fiber angles, though strength decreased as the angle increased. However, Sun et al. [30] reported that off-axis orientations of 0° and 90° produced greater maximum displacement, with absorbed energies remaining consistent across different off-axis angles. Wu et al. [31] discovered that finely dispersed resin pockets and misaligned yarns reduce damage progression and enhance material performance, despite differences in permanent damage and deformation extent.

Researchers have also examined the role of impactor geometry. Studies show that composites experience greater maximum forces and reduced displacements with larger impactor sizes, while round-nosed impactors yield longer contact times, greater deformation, and increased delamination [32,33]. Nassir et al. [34] reported that multiple low-energy impacts allowed for greater energy absorption compared to a single high-energy impact necessary for perforating the composite. Ko et al. [35] observed that polymer fibers can exhibit similar or even greater specific energy absorption compared to carbon fibers despite their lower strength. In analyzing LVI damage, Balasubramani et al. [36] successfully identified, visualized, localized, and predicted damages in GFRP through experimental and numerical analyses. Environmental factors such as humidity, temperature, and aging also impact the response of composites to impact, as reviewed by Fernandes et al. [37]. For example, Gunoz et al. [38] found that seawater and hydrochloric acid significantly degraded the mechanical performance of glass and carbon fiber composites. Wang et al. [39] noted that hybrid composites with carbon fiber surfaces showed reduced water absorption and diffusion rates compared to flax-only composites.

This paper investigates the effect of impact energy on the low-velocity impact behavior of woven CFRP composites at room temperature, employing a theoretical model to predict the experimental LVI results. Importantly, while much of the existing literature focuses on composite plates, this study examines composite beams, which were selected in this study instead of plates in order to simplify both the geometry and the analysis. This approach allows for easier simulation and offers insights that may be beneficial for future modeling efforts. Moreover, beam geometry outperforms plate geometry by facilitating better damage tracking during testing [40,41]. The simplification primarily stems from the fact that beams can be effectively modeled using one-dimensional theories such as Euler-Bernoulli theory, while plates require more complex two-dimensional formulations. This simplification leads to governing equations that are easier to handle analytically and numerically. In finite element modeling, beams also offer the advantage of significantly fewer degrees of freedom compared to plates, resulting in lower computational costs and easier interpretation of results. Additionally, numerous real-world applications utilize composite laminates in structural elements like wind turbine blades, robotic manipulators, and drive shafts, which are often modeled as beams under transverse loading. These components may encounter impact events, whether during installation, routine maintenance, or operational use [42]. However, we acknowledge that this simplification can limit the direct applicability of the results to more complex geometries, including plates and shells.

While large number of works were interested in the LVI response of composite plates, much fewer works investigated the LVI response of composite beams. Several researchers [4,40,41,43-45] have demonstrated that beam tests offer clear advantages over plate configurations for studying low-velocity impact (LVI) in composites, owing to their simpler boundary conditions and enhanced visibility of internal damage progression. Additionally, advanced techniques such as ultra-high-speed imaging and digital image correlation can be employed to capture real-time sequences of damage initiation and propagation. Bozkurt et al. [40,41] performed detailed experimental and numerical investigations on cross-ply CFRP beams, revealing a distinct failure sequence—starting with diagonal matrix cracking in the 90° plies, followed by delamination at adjacent interfaces. Their high-fidelity finite element simulations, incorporating LaRC05 and cohesive zone modeling, successfully
reproduced experimental strain fields and crack propagation dynamics, highlighting the critical role of accurately modeled boundary conditions.

Hallett [43] conducted three-point bending tests on cross-ply CFRP beams, showing how variations in impact velocity and mass influence failure modes and threshold energies. Their tests revealed that impacts conducted at different velocities and masses (but similar energies) influenced the failure mode and threshold velocities for damage initiation significantly. Thorsson et al. [44] examined matrix crack-delamination interactions in sandwich composite beams, demonstrating failure initiation in the core and its progression through the interface using digital image correlation technique and a smeared-crack FE model. Similarly, Topac et al. [45] conducted both experimental and numerical investigations on the damage evolution in unidirectional CFRP composite beams subjected to low-velocity impacts. Using ultra-high-speed imaging combined with digital image correlation, they successfully captured the progression of diagonal matrix cracks followed by delamination at ply interfaces. Their finite element simulations closely matched the experimental results.

Rotem [46] investigated the effect of material type by comparing a carbon/epoxy laminate with a less brittle glass/epoxy laminate. The beams were initially subjected to LVI tests performed on a drop-weight tower. Subsequently, their residual mechanical properties were assessed through static three-point bending tests. Chenghong et al. [47] investigated impact damage modes using fractography while dealing with glass/epoxy, basalt epoxy and aramid/epoxy composite beams. Santiuste et al. [42] characterized the post-impact bending strength of glass/epoxy composite beams. They reported that the residual strength decreases as the absorbed energy increases. Additionally, wider composites beams show higher normalized residual strength than beams with reduced width. Boroujerdy and Kiani [48] investigated the dynamic response of laminated composite beams impacted by multiple projectiles, and accounting for thermal effects. A model based on first-order shear deformation theory, Hertzian contact, and the Ritz method was developed and solved using the Runge-Kutta approach. Recently, Fand and Wang [49] examined the LVI response of a carbon nanotube-reinforced composite beam with an auxetic (negative Poisson's ratio) design. A modified Hertz model and Reddy beam theory were used to derive governing equations, which were solved via perturbation methods.

This study has to main objectives. First, this work aims at experimentally characterizing the low-velocity impact (LVI) response of CFRP beams to establish benchmark data for future modelling. Second, this investigation aims to develop a simplified mass-spring model accounting for damage to simulate LVI behavior under varying impact energies. The paper outlines the experimental methodology and results in Section 2, presents the theoretical model and its validation in Section 3, and concludes with key findings.

## 2. Experimental investigation

### 2.1. Materials

Specimens were cut from carbon fibre reinforced epoxy (CFRE) composite laminates. The laminates were fabricated by CHN Carbon Fiber Technology Co. from 25 layers of T300-3k woven carbon fiber fabric (200 g/m²) and YPH-120-23 A/B epoxy matrix using prepreg technique. The mechanical properties of the CFRE have been investigated in previous work [50-53] and are shown in Table 1. The fiber volume fraction ($V_f$) of 61.8 % ± 0.2 % was determined experimentally using the ignition technique according to ASTM D3171-99. The composite laminates were provided as sheets of 500 × 500 × 5 mm³. Subsequently, beams were cut to dimensions of 125 × 24 × 5 mm³. The dimensions of the specimens used are shown in Fig. 1. The beam specimens were cut from the laminate using an abrasive waterjet technique to achieve precise dimensions of 125 × 24 × 5 mm³, as shown in Fig. 1.

<table><caption>**Table 1**
The mechanical properties of CFRP [50-53].</caption><thead><tr><td>Properties</td><td>Value</td></tr></thead><tbody><tr><td>Tensile strength, σ<sub>t</sub> (MPa)</td><td>895.28</td></tr><tr><td>Tensile modulus, E<sub>t</sub> (GPa)</td><td>81.66</td></tr><tr><td>Poisson's ratio, ν<sub>12</sub></td><td>0.052</td></tr><tr><td>Shear strength, τ (MPa)</td><td>145.41</td></tr><tr><td>Shear modulus, G (GPa)</td><td>6.94</td></tr><tr><td>Compressive strength, σ<sub>c</sub> (MPa)</td><td>633.3</td></tr><tr><td>Compressive modulus, E<sub>c</sub> (GPa)</td><td>80.9</td></tr></tbody></table>

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Dimensions of the impacted CFRP specimen.</figcaption></figure>

Appropriate cutting parameters for CFRP composites were selected through the machine software to minimize kerf damage, delamination, and eliminate matrix burning at the specimen edges, providing advantages over other traditional cutting methods [4,54].

### 2.2. Low-velocity impact tests

Low-velocity impact experiments were conducted using a drop-weight testing machine (model CEAST 9340), as depicted in Fig. 2a. The tests were performed at room temperature, following the guidelines specified by ASTM D7136. The upper surfaces of the beam specimens were carefully marked and positioned at the center of a steel plate with a rectangular window of 125 × 75 mm², resulting in a span length of 75 mm. As evident from Fig. 2b, the beam specimen was securely clamped within 25 mm from each end using two parallel steel strips, each fitted with two evenly spaced clamps, establishing a robust clamped-clamped boundary condition that prevented specimen sliding or rotation during the impact event. These adaptations ensured consistent impact loading and reliable energy absorption measurement while maintaining the core principles of ASTM D7136, such as using a hemispherical striker (16 mm diameter) and similar impact energy levels. It should be noted that beam specimens were selected instead of standard plates to simplify the modeling, analysis, and enhance damage tracking during testing. This approach has also been adopted in several previous studies [4,40,41,

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** (a) CEAST9340 drop-weight impact tester, and (b) Clamping the specimen inside conditioning thermal chamber.</figcaption></figure>
43–45], which demonstrated the suitability of beam geometries for capturing LVI behavior in composite structures.

The impact tests were performed at seven different energy levels: 2 J, 5 J, 7.5 J, 12.5 J, 15 J, 17.5 J, and 20 J. The total mass of the impactor used in the tests was 3.132 kg. It is possible to achieve different impact energies either by varying the impact velocity or by changing the drop mass. In this study, we specifically investigated the effect of varying the impact energy by adjusting the impact velocity while keeping the drop mass constant. Alternatively, varying the drop mass while maintaining a constant impact velocity is likely to produce different results. To ensure repeatability, three tests were performed for each energy level. A representative test is shown when analyzing the time evolution of force or energy. However, average values are reported for key parameters such as absorbed energy and maximum force.

The software integrated with the machine automatically calculated and recorded the impactor’s drop height and velocity during the tests. However, these values can also be manually derived using fundamental equations for energy and motion, as outlined below:

$$h = E_i/mg \qquad (1)$$

$$v_i = \sqrt{\frac{2E_i}{m}} = \sqrt{2gh} \qquad (2)$$

where h represents the drop height of the impactor, E_i is the impact energy (1/2 mv_i^2 = mgh), v_i is the impact velocity, g is the gravity acceleration (g = 9.81 m/s^2), and m is the total mass of the impactor. Eqs. (1) and (2) express the conversion of the impactor’s initial potential energy into kinetic energy. As the impactor falls from height h, its potential energy (mgh) is gradually and entirely converted into kinetic energy (1/2 mv_i^2), ensuring precise control of impact velocity and energy in each test. The specimens were impact tested using a 16 mm diameter hemispherical impactor. The impact machine was equipped with a pneumatic anti-rebound system designed to prevent secondary impacts on undamaged specimens.

The impactor was equipped with a strain-gauge-based force sensor, allowing for the time variation of the impactor-specimen contact force to be recorded during each test. The measured force represents the reaction force applied by the specimen onto the impactor. As the specimen interacts with the impactor, it first decelerates the impactor and then accelerates it back upwards. Based on Newton’s Second Law of Motion and the recorded contact force (F(t)), the acceleration of the impactor (a(t)), can be determined as follows [55]:

$$a(t) = g - \frac{F(t)}{m} \qquad (4)$$

where g, is the gravity acceleration. Once the acceleration a(t) is determined, the impactor’s velocity v(t) can be computed by performing a time-integration of the acceleration. After obtaining the velocity, the displacement u(t) can also be calculated by integrating the velocity over time as follows [55]:

$$v(t) = v_i - \int_0^t a(\tau) d\tau \qquad (5)$$

$$u(t) = \int_0^t v(\tau) d\tau \qquad (6)$$

The initial displacement is set to zero at the moment of impact. These calculations provide a complete description of the impactor’s motion during the test, allowing for a detailed analysis of the impact event. The impactor’s energy is as important as its motion characteristics. The mechanical power of the contact force is given by:

$$P(t) = F(t)v(t), \qquad (7)$$

The energy of the impactor can be calculated using time integration as follows:

$$E(t) = \int_0^t F(\tau)v(\tau) d\tau \qquad (8)$$

It is clear from Eqs. (4) to (8) that all calculated parameters—such as displacement, velocity, and energy—are derived from the force measurement and the initial impact velocity. Therefore, any error in the calibration of the force measurement can lead to significant inaccuracies in the computed displacement, velocity, and energy. To minimize the effects of potential miscalibration, the drop-weight impact machine was calibrated using the technique developed in [5]. A reference elastic test was performed prior to testing the composite beams. Composite plates, similar to those used in [53], were subjected to impacts at an energy level of 5 J. The force transducer was calibrated based on the assumption of an elastic impact.

As the impactor comes into contact with the specimen, it decelerates to a complete stop. Then, it is accelerated upwards. Once the impactor’s velocity exceeds the beam velocity, the impactor bounces off and loses contact with the specimen. The impactor’s energy after it bounces off (when the force returns to zero) is known as the residual or rebound energy (E_r). Subtracting the residual energy from the impact energy gives the energy absorbed by the specimen (E_a). Thus, the absorbed energy is regarded as the ultimate value of the energy that is transmitted from the impactor to the composite beam:

$$E_a = E_i - E_r \qquad (9)$$

The absorbed energy ratio (ρ) was then computed using the formula:

$$\rho = \frac{E_i - E_r}{E_i} \times 100 = \frac{E_a}{E_i} \times 100 \qquad (10)$$

Furthermore, the peak force (F_max) and maximum deformation (u_max) are additionally estimated for each test. In this study, particular attention is also given to identifying and analyzing the damage threshold force, damage threshold energy, and damage threshold displacement, which correspond to the respective values of force, energy, and displacement at the onset of a significant damage.

# 3. Results and discussion on the experimental observation

## 3.1. Post-impact observations

Fig. 3 presents images of several representative impacted specimens. The composite beam impacted at 5 J shows almost no visible damage, except for a small circular imprint approximately 2 mm in diameter on the front surface. A similar imprint is also observed at 7.5 J, which may be attributed to localized matrix plastification or surface fracture of a few fibers. The side view does not reveal any noticeable alterations, indicating that any damage is confined to the impact zone.

The composite beam impacted at 12.5 J, Fig. 3a and b, is representative of the behavior observed at both 10 J and 15 J. The top (front) view shows a zigzag failure line running across the width of the beam, indicating damage propagation from one side to the other. The side view reveals clear evidence of surface matrix and fiber fracture, along with delamination between the composite layers. However, this damage appears more concentrated in the front and rear layers, while the middle layers remain largely unaffected. The enlarged side view in Fig. 3c indicates that fracture damage was more pronounced on the bottom surface of the composite compared to the top. This asymmetry is explained by the bending stress distribution, where the bottom surface was subjected to maximum tensile stresses while the top surface was under compression. When the applied tensile stresses on the bottom side
<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Images of representative impacted specimens: (a) top view, (b) side view, and (c) zoom of the side view on the failure zone.</figcaption></figure>

exceeded the composite's tensile strength, crack initiation occurred, followed by rapid propagation driven by high strain energy release rates in the lower plies. The progression of these cracks was further facilitated by interlaminar failure, resulting in extensive delamination.

At 17.5 J, the composite beam exhibits complete fracture, splitting into two separate parts. The failure results from extensive fiber and matrix fracture, accompanied by delamination on both sides of the fracture zone. Unlike lower impact energies, the front, middle, and rear layers all show significant damage, including matrix cracking, fiber breakage, and interlaminar separation. Similar behavior was also observed at 20 J, confirming a consistent failure mode at higher impact energies.

## 3.2. Force vs time curves

Fig. 4 displays the force-time curves for composite beams at various impact energy levels, revealing three distinct behaviors. Fig. 4a illustrates the curves for impact energies between 2 and 7.5 J, showing the first behavior. It is obvious from Fig. 4a that the force-time curves follow a half-sine wave pattern, which is characteristic of impacts that cause

negligible damage and exhibit nearly elastic behavior [55]. In these cases, the interaction between the dropped mass and the composite beam resembles a mass-spring system. This interaction begins upon contact and corresponds to approximately half the oscillation period, resulting in the observed half-sine curves. The force increases steadily over time, reaches a maximum, and then nearly-symmetrically decreases to zero. While ideal symmetry would indicate a perfectly elastic response, slight asymmetry is expected due to minor energy losses, which likely arise from localized damage in the contact area or the viscoelastic nature of the polymer matrix within the composite. Despite these losses, the system behaves predominantly elastically.

The peak force for these impact energies is consistently reached at around 2.2 ms, which corresponds to a quarter oscillation period of the mass-spring system formed by the dropped mass and the impacted beam. Since the dropped mass remains constant for all impact energies, the oscillation period is primarily influenced by the overall stiffness of the composite beam. Using the dropped mass of 3.132 kg and a quarter oscillation period (T/4) of 2.2 ms, the overall stiffness of the composite beam (k) can be estimated at 1.596 MN/m ($T = 2\pi\sqrt{m/k}$). This preliminary stiffness value provides an important insight into the structural
<figure><img src="image_6.png" /><figcaption>Fig. 4. Illustrative samples of force time relationship under different energy levels: (a) 2 J-7.5 J, (b) 10 J-15 J, and (c) 17.5 J-20 J.</figcaption></figure>

integrity of the beam, suggesting that its mechanical response under low-energy impacts is mainly governed by its inherent elasticity.

It is evident from Fig. 4a that the maximum force increases with the impact energy. The peak force is reached when all the kinetic energy of the dropped mass is converted into deformation energy within the impacted beam. As the impact energy increases, more energy is absorbed as deformation energy, leading to larger displacements or deflections. These larger displacements correspond to higher force levels since the stiffness of the composite beam is not expected to be significantly affected by the varying impact energy. Thus, the relationship between impact energy and force reflects the beam's capacity to deform elastically under increasing loads.

The second behavior is observed for impact energies ranging between 10 and 15 J (Fig. 4b). Initially, during the ascending region, the force-time curves resemble those recorded at lower impact energies, with the force rising to a peak load. However, during the descending region, unlike lower-energy impacts, the peak load is followed by a sharp drop. The peak loads at 10 J, 12.5 J, and 15 J occur earlier than those at lower energies and are immediately followed by a steep decline, indicating significant damage to the composite beam. Despite this, the force does not fall to zero, suggesting that while the beam is damaged, it has not completely lost its stiffness and still offers some resistance to the falling mass. The peak force, in this case, does not represent the full conversion of the mass's kinetic energy into deformation energy but rather marks the damage threshold force (the point where damage begins).

After this sharp drop, the force-time curve takes on a sine-wave pattern, distinct from the pre-threshold portion. This indicates that although the beam's stiffness is reduced due to damage, it is not entirely lost. This is further supported by the longer time it takes for the force to return to zero. For impact energies between 2 and 7.5 J, it takes around

4.5 ms for the force to peak and then drop to zero. At impact energies of 10 J, 12.5 J, and 15 J, this process extends to approximately 5 ms, 6 ms, and 6.75 ms, respectively. The longer duration corresponds to a longer oscillation period, reflecting a reduction in the beam's stiffness.

Fig. 4c depicts the third behavior, associated with impact energies of 17.5 and 20 J. It is evident from Fig. 4c that the force rises rapidly, reaching a peak known as the damage threshold force, marking the point at which the material can no longer sustain the load without damage. Following this peak, a sharp drop in force signifies the onset of damage, occurring both within the ply plane and between adjacent plies. At this stage, the damage threshold energy and corresponding deflection can be determined. As long as the impact energy remains below this threshold, the beam retains its structural integrity. However, once the threshold is exceeded and damage initiates, high-frequency noise is often detected, likely caused by energy released from fiber breakage that propagates rapidly through the material.

The final stage of the impact response is characterized by a steep drop in force, indicating catastrophic failure of the beam. This suggests the material has absorbed the maximum energy it can withstand, leading to total structural failure. Interestingly, after collapse, the force rises again between 2 and 8 ms, despite the beam's full collapse. This secondary increase in force is due to friction between the collapsed beam fragments and the dropped mass.

To summarize, the three energy intervals presented in Fig. 4(a), 4(b), and 4(c) correspond to distinct damage regimes observed during impact. At low impact energies (2–7.5 J), as shown in Fig. 4(a), the response is predominantly elastic with minimal damage, resulting in a nearly parabolic load-displacement curve. In contrast, Fig. 4(b) captures the transition region (10–15 J), where the impact energy is sufficient to initiate noticeable damage. This leads to a sudden drop in load, indicating the onset of internal failure, though the structure retains partial
load-carrying capacity. Finally, Fig. 4(c) illustrates the behavior at high energy levels, where the impact exceeds the structural failure threshold. In this regime, the specimen undergoes severe damage and collapse, as evidenced by the significant reduction in load. These observations clearly demonstrate the evolution of damage and failure mechanisms with increasing impact energy.

Regardless of the impact energy range, force-time curves consistently exhibit undulations. Khashaba and Othman [53], as well as Othman et al. [56], attributed such oscillations to the inertia of the testing machine, which is a well-known effect in dynamic testing systems. Moreover, the standard processing of low-velocity impact (LVI) experiments often neglects the influence of the bar-shaped geometry of the drop mass. To mitigate these oscillations, wave analysis methods have been proposed [58,59] for drop-bar testing devices. Anuse et al. [57] suggested these fluctuations also arise from noise and applied filtering techniques to smooth the force signals. However, high-fidelity numerical simulations replicated these undulations [40], linking them to damage creation and propagation within impacted specimens. It is most likely that both machine inertia and damage mechanisms contribute to the oscillations. Thus, unfiltered force-time measurements are considered in this study.

### 3.3. Force vs displacement curves

Fig. 5 illustrates representative examples of the force-displacement (beam deflection) curves for different impact energies, highlighting three distinct behaviors. Fig. 5a displays the force-displacement curves for impact energies ranging from 2 to 7.5 J. It is evident from Fig. 5a that at lower impact energies (2–7.5 J), the force increases almost linearly with displacement during the ascending curve until it reaches a peak (Fig. 5a). After this peak, the force decreases in a similarly linear pattern

during the descending curve. During the initial phase of the ascending curve, the slope of the force-displacement remains consistent for all three impact energies, reflecting the beam's overall stiffness. This linear behavior indicates that the beam responds elastically to the impact up to a certain point. However, the decreasing phase of the force-displacement curve deviates from the increasing phase, indicating that the energy absorbed during the impact is not fully recoverable. This slight discrepancy suggests that some of the impact energy is dissipated, either through matrix viscosity or due to localized damage in the impact zone. Despite this, the absorbed energy appears relatively minor, as the ascending and descending portions of the force-displacement curve are still closely aligned.

As the impact energy increases, both the peak force and the maximum displacement increase correspondingly. The peak force occurs at the point of maximum displacement, indicating that this is the moment when the kinetic energy of the dropped mass is entirely transferred to the beam as deformation energy. At this peak, the downward motion of the mass comes to a stop, and the beam begins to accelerate the mass upward, marking for the beam the transition from energy absorption portion to energy restitution portion. This behavior suggests that the beam can store and release energy effectively, although some energy is dissipated due to internal viscosity or minor damage mechanisms within the material.

Fig. 5b shows illustrative samples of the force-displacement curves for impact energies in the range of 10–15 J. The curves in Fig. 5b reveals several distinct phases that characterize the response of the impacted composite beam. Initially, the force increases linearly with the beam's deflection, indicating an elastic response. The slope of this initial linear portion is almost identical for all three impact energies, reflecting the beam's stiffness in the early stages of impact. This elastic behavior continues until the force reaches a peak, marking the onset of damage

<figure><img src="image_7.png" /><figcaption>Fig. 5. Illustrative samples of force displacement relationship under different energy levels: (a) 2 J-7.5 J, (b) 10 J-15 J, and (c) 17.5 J-20 J.</figcaption></figure>
within the material. At this point, the force no longer increases, indi-
cating the transition from elastic deformation to the damage phase.

This peak force, known as the damage threshold force, is different
from what is observed at lower impact energies, such as those below 7.5
J. At lower energies, Fig. 5a, the peak force coincides with the maximum
displacement, indicating that the beam's structural integrity is still
intact. However, for impact energies between 10 and 15 J, the peak force
occurs earlier in the deflection process and does not correspond to
maximum displacement. Instead, it marks the initiation of significant
internal damage within the composite beam, as the material begins to
lose its load-bearing capacity.

Once the damage threshold force is exceeded, the force undergoes a
sharp drop, which becomes more pronounced with increasing impact
energy. The most significant drop is observed at 15 J, followed by 12.5 J
and 10 J. The steeper the drop, the greater the extent of damage inflicted
on the beam, indicating more severe material failure at higher impact
energies. Despite this sharp drop, the force does not fall to zero, indi-
cating that the damaged beam remains structurally capable of with-
standing further deformation.

Following the sharp drop in force after peak loading, a secondary rise
in the force signal is observed as the beam continues to deflect. This
stage corresponds to the remaining transfer of energy from the impactor
to the specimen, culminating at the point of maximum displacement
where the impact energy is fully absorbed. Beyond this point, the beam
begins to recover and restores part of the stored elastic energy, propel-
ling the impactor upwards as the structure attempts to return to its
original shape, resulting in the characteristic closed-loop shape of the
force-displacement curve. Notably, the contact force drops to zero while
the beam has not yet returned to its undeformed state, indicating a loss
of contact between the beam and the impactor. This phenomenon arises
due to partial energy dissipation through internal mechanisms such as
material damping, matrix plasticity, and damage. Consequently, the
mass continues its upward motion due to its inertia, producing a nearly
zero force response during the unloading phase.

Fig. 8c shows the curves for higher impact energies of 17.5 J and 20
J. It is clear from Fig. 5c that the force-deflection curves also exhibit an
initial linear portion. In this phase, the force increases almost linearly
with the beam’s deflection until reaching a maximum corresponding to
the damage threshold force. Unlike lower energy impacts, the peak force
in these cases does not coincide with the beam’s maximum deflection.
Instead, it marks the point where significant internal damage begins
within the composite material.

After the peak force is reached, a sharp drop follows, signifying the
onset of damage. This initial sharp drop is then followed by a brief
oscillatory rise in force. The beam is damaged but has not yet fully
collapsed, and the force oscillation may be interpreted as a sign of the
damage propagating through the material. During this stage, the com-
posite structure is still absorbing energy, but it is gradually losing its
capacity to resist the impact. After this short oscillatory increase, the
force rapidly declines to zero, indicating the complete collapse of the
specimen and resulting in an open-loop force-displacement curve. At
this point, the structural integrity of the beam is entirely compromised.
However, even after the beam has collapsed, its fragments may still offer
some frictional resistance to the falling mass. This resistance could
explain the slight fluctuation in force after the collapse, where a minor
increase is observed before the force drops again.

At higher impact energies, the displacement continues to increase
throughout the process, suggesting that the energy from the falling mass
is fully transferred to the beam without being restituted. This contrasts
with lower and intermediate impact energies, where a portion of the
impact energy may be returned to the falling mass. This continuous
increase in displacement indicates that the beam is undergoing signifi-
cant deformation until it can no longer withstand the impact and breaks
apart completely.

3.4. Energy vs displacement curves

Fig. 6 depicts examples of the energy-displacement curves for different impact energy levels, highlighting three distinct behaviors. Fig. 6a displays the energy-displacement curves for impact energies ranging from 2 to 7.5 J. At low energies, the curves exhibit a parabolic shape, indicating elastic behavior. However, the descending paths are slightly offset from the ascending ones, confirming minor energy absorption, as previously noted Sections 3.1 and 3.2. The descending paths are marginally higher and intersect the energy axis, showing the energy absorbed ($E_a$) by the composite beam. The ascending path represents the phase where the energy is transferred from the falling mass to the beam (impact energy, $E_i$), while the descending path reflects the phase of restitution of energy from the beam back to the mass (rebound energy, $E_r$), Fig. 6a. In a purely elastic response, the two paths would overlap, with the descending path returning to zero when the displacement does. The energy value when the displacement returns to zero corresponds to the energy absorbed by the beam or the energy non-restituted to the mass. Correspondingly, the energy absorbed by the beam is 0.48 J, 1.29 J, and 2.03 J for impact energies of 2 J, 5 J, and 7.5 J, respectively.

Fig. 6b shows illustrative samples of the energy-displacement curves for impact energies in the range of 10–15 J. At intermediate impact energies, the energy-displacement curves exhibit characteristics similar to those observed at lower energies. Initially, the energy starts at zero and increases with displacement until it matches the impact energy, indicating the complete transfer of kinetic energy from the falling mass to the composite beam. Following this, the energy decreases to a certain asymptotic value, which represents the energy absorbed by the beam during the impact. Thus, the energy does not vanish when the deflection returns to zero, suggesting that not all energy is restituted to the falling mass by the beam. The absorbed energies for impact levels of 10 J, 12.5 J, and 15 J are recorded as 4.84 J, 9.94 J, and 13.61 J, respectively.

A key distinction between low and intermediate impact energies is evident in the behavior of the curves. At low energy, the curves remain predominantly parabolic until reaching maximum deflection. In contrast, at intermediate energies, while the curves start parabolic, they display a noticeable change in slope at specific deflection and energy values. This shift occurs at approximately 3.57 mm, 4.05 mm, and 4.32 mm for impact energies of 10 J, 12.5 J, and 15 J, respectively. The change in slope signifies the initiation of substantial damage within the specimen. At this critical point, the displacement corresponds to the damage threshold displacement, and the energy represents the damage threshold energy.

Fig. 6c shows the energy-displacement curves for higher impact en-
ergies of 17.5 J and 20 J. At the highest impact energies (17.5 J and 20
J), the curves initially follow a parabolic pattern like that of low and
intermediate energies. However, they also exhibit a change in slope,
indicating the initiation of substantial damage within the beam, as seen
with intermediate energies. This change occurs at displacements of 3.95
mm and 4.24 mm for impact energies of 17.5 J and 20 J, respectively.
Another change in slope occurs at a deflection of 5 mm, marking the
complete collapse of the beam. Beyond this point, the beam offers no
resistance to the falling mass, except for some friction caused by the
beam’s fragments. The energy continues to increase with displacement,
as it is not able to rebound the falling mass. In other words, the collapsed
beam cannot stop the falling mass or alter its motion upward, so the
descending path in the energy-displacement curve is not observed.

3.5. Some key parameters

3.5.1. Maximum force

The maximum force, or peak force, is highly sensitive to impact
energy, as shown in Fig. 7. In general, the peak force rises with
increasing impact energy (Fig. 7a), primarily due to the corresponding
increase in impact velocity [11,60]. At low-impact energies, the rate of
change is significant but diminishes as the impact energy increases.
<figure><img src="image_8.png" /><figcaption>Fig. 6. Samples of energy-displacement relationship under different impact energy levels: (a) 2 J-7.5 J, (b) 10 J-15 J, and (c) 17.5 J-20 J.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Fig. 7. Effect of the impact energy on the maximum force: (a) Maximum force vs. impact energy, and (b) Maximum force vs square root of impact energy.</figcaption></figure>

Specifically, for impact energies below 7.5 J, the peak force is very sensitive, reaching its maximum at the point of maximum deflection, Fig. 7a, when all the kinetic energy from the falling mass is transferred to the beam. For impact energies above 10 J, however, the peak force corresponds to the damage threshold force, beyond which substantial damage is initiated in the beam. In this range, the peak force becomes less sensitive to further increases in impact energy.

Fig. 7b shows the variation of peak force as a function of the square root of the impact energy. According to Hooke's law, force (F) is proportional to displacement (x), expressed as $F = kx$. Additionally, in purely elastic behavior, the impact energy (E) is related to displacement as $E_i = (1/2) kx^2$. Consequently, the peak force is proportional to the square root of the impact energy, represented as $F_{max.elas} \propto \sqrt{E_i}$. This explains the linear relationship observed between the peak force and the square root of the impact energy at low impact energies, as shown in Fig. 10b. The estimated elastic stiffness is approximately 1.5 MN/m, which closely aligns with the overall elastic stiffness (k) of 1.596 MN/m determined in Section 3.1. However, for impact energies greater than 10 J, the curve deviates from the linear elastic behavior, indicating the onset of damage in the specimen as the impact energy increases. The extent of damage becomes more significant as the curve deviates further from elastic behavior.

### 3.5.2. Maximum and damage threshold displacements

This section examines the effect of impact energy on the beam's maximum deflection and damage threshold displacement. Fig. 8a shows the experimental results for these two parameters as a function of impact energy. The damage threshold displacement is presented for impact energies above 10 J, as no significant damage is observed for lower impact energies. It is evident from Fig. 8a that as the impact energy increases, the damage threshold displacement rises at a modest rate. At

an impact energy of 10 J, the damage threshold displacement is 3.54 mm, rising to 4.25 mm at 20 J, yielding an increase rate of 71 µm/J.

The maximum displacement increases as the impact energy rises. In the low impact energy range (2–7.5 J), the rate of increase in maximum displacement decreases as impact energy increases. In this range, the behavior is nearly linear elastic, meaning the maximum displacement proportional to the square root of the impact energy, ($u_{max.elas} \propto \sqrt{E_i}$), as clarified in Section 3.4.1. At intermediate impact energies (10–15 J), the rate of increase in maximum displacement remains constant relative to impact energy, leading to a nearly linear increase in maximum deflection as energy rises. In this range, the impact energies exceed the composite beam's energy tolerance, inducing significant damage in the specimen, which reduces stiffness and results in a higher maximum displacement, as shown in Fig. 8a and 8b. At the highest impact energy range (17.5–20 J), the maximum displacement increases almost linearly with impact energy but at a higher rate than in the intermediate range. For the highest energies, impacts lead to the complete collapse of the beams. After collapse, only the fragments of the beam offer some resistance to the falling mass through friction, allowing significantly larger deflections to occur.

In Fig. 8b, the maximum deflection is plotted against the square root of the impact energy, with experimental results compared to purely elastic behavior, assuming the beam's elastic stiffness to be 1.45 MN/m. In the low impact energy range, the experimental points align with the elastic behavior, indicating that little or no damage occurs in the composite beam. In the intermediate range, the experimental points begin to deviate from elastic behavior, suggesting that some damage occurs within the specimen. Finally, for the highest impact energies, the experimental points deviate significantly from elastic behavior, corresponding to the complete collapse of the specimen caused by the impacts.

<figure><img src="image_10.png" /><figcaption>Fig. 8. (a) Effect of the impact energy on the maximum deflection and the damage threshold displacement, and (b) Variation of maximum deflection with the square root of impact energy.</figcaption></figure>
### 3.5.3. Absorbed and damage threshold energy

This section investigates the behavior of the energy absorbed by the composite beam and the damage threshold energy, as illustrated in Fig. 9. Fig. 9a presents the variation of absorbed energy in relation to the impact energy. Overall, the absorbed energy increases with increasing impact energy. Indeed, it is widely observed that an increase in impact energy leads to a corresponding increase in the extent of damage within the impacted specimen [10]. However, once again, the variation in absorbed energy suggests three distinct behaviors corresponding to low, intermediate, and high impact energy ranges.

At low impact energies (2–7.5 J), the absorbed energy increases almost linearly with impact energy, exhibiting a relatively low slope. It is important to recall that no substantial damage is observed within this range of impact energies, suggesting that the energy absorbed is primarily due to local or contact damage, or possibly the viscous behavior of the composite matrix. In the intermediate impact energy range (10–15 J), the absorbed energy continues to increase linearly with impact energy but with a steeper slope. In this range, significant damage to the composite beam is detected, which justifies the greater increase in absorbed energy compared to the low-energy range. At higher impact energies (17.5–20 J), the absorbed energy remains nearly constant ≈ 13.9 J. This plateau indicates that the composite beam has reached its saturation point, corresponding to the maximum energy it can absorb. It is important to clarify that the absorbed energy is measured at the point when the impact force drops to zero. As such, any energy absorbed due to friction after the specimen collapses is excluded from this measurement.

Additionally, Fig. 9 shows a piecewise linear interpolation (represented by lines) superimposed on the experimental results (represented by blue dots). The intersections of the linear interpolations mark the

transitions between the three energy ranges. Specifically, Fig. 9a indicates that the transition between the low and intermediate energy ranges occurs at an impact energy of 8.4 J, while the transition between the intermediate and high energy ranges occurs at an impact energy of 15.05 J. The absorbed energy by the composite beam demonstrates distinct behaviors across the low, intermediate, and high impact energy ranges, with clear transition points, as illustrated in Fig. 9a.

In addition to absorbed energy, another key energy parameter of interest is the damage threshold energy, which represents the minimum energy required to initiate substantial damage within the composite beam. The findings reported by Katunin et al. [61] indicate that damage mechanisms, including delamination, matrix cracking, and fiber fracture, are triggered only after exceeding a specific threshold of impact energy. Fig. 9b shows the variation of damage threshold energy as a function of impact energy. For impact energies up to 7.5 J, no significant damage is observed, indicating that the threshold energy is only relevant for impact energies of 10 J and above. Fig. 9b also demonstrates that, overall, the damage threshold energy increases with impact energy. This trend suggests that as impact energy rises, progressively more energy is required to reach the point where substantial damage occurs. This increase can be attributed to the energy absorption mechanisms of the composite beam, including matrix microcracking, delamination, and fiber pull-out, which exhibit greater resistance as the strain rate increases at higher impact energies, where the impact velocity also rises. Similar behavior was reported by Khashaba [62,63]. Their findings on the low-velocity impact (LVI) behavior of woven GFRP composites indicated that the impact bending stiffness increases with rising impact energy. This is attributed to higher strain rates, which reduce the molecular mobility of polymer chains. Consequently, the material exhibits more brittle behavior, resulting in a higher damage threshold force.

<figure><img src="image_11.png" /><figcaption>Fig. 9. (a) Effect of the impact energy on absorbed energy, (b) Effect of the impact energy on damage threshold energy, and (c) Absorbed energy split between local and structural absorption.</figcaption></figure>
These statements were validated in the present study through the observed increase in the damage threshold energy of the CFRP composite beams with increasing the impact energy.

To identify the transition between low and intermediate impact energy ranges, a linear interpolation of the damage threshold energy variation is shown in Fig. 9b (red line). The black line in the same figure represents the impact energy (line “x = y”). When the red line is above the black line, the damage threshold energy exceeds the impact energy, indicating that no significant damage should occur, which characterizes the low-impact energy regime. Conversely, when the red line is below the black line, the impact energy surpasses the damage threshold energy, meaning the energy transferred to the specimen is sufficient to cause damage, placing it in the intermediate to high impact energy regime. The intersection of these lines marks the transition between these regimes, occurring at an impact energy of 8.9 J. This value aligns closely with the transition point determined from Fig. 9a, which was 8.4 J, reinforcing the consistency of the observed damage threshold energy across different analyses.

Regarding energy absorption, Fig. 9b presents the total energy absorbed by the impact beam, without distinguishing between the energy absorbed locally in the contact region and the energy associated with structural bending deformation. Fig. 9c attempts to separate these two contributions. In the lower impact energy range, no substantial structural damage is observed, suggesting that energy absorption is primarily due to local contact deformation at the point of impact. Within this range, the absorbed energy increases linearly with impact energy. To estimate local or contact energy absorption beyond the low-impact energy range, a linear extrapolation was applied. The locally absorbed energy due to contact deformation is shown by the red squares across the entire impact energy range in Fig. 9c. The difference between the total absorbed energy and the local contact energy provides the energy absorbed due to structural bending deformation, which is represented by magenta triangles in Fig. 9c. Interestingly, the structural bending energy absorption is absent at low impact energies, becomes more significant in the intermediate impact range, and then decreases at higher impact energies. The separation between energy absorption modes offers valuable insights into the composite beam’s response to varying impact intensities. At lower energies, local deformation predominates, while structural mechanisms become more significant as the impact energy rises.

Fig. 9(c) also compares the rebound energy with the absorbed energy across the different impact energy ranges. In the low-impact energy regime, the response is dominated by the rebound energy, with only a small fraction of the impact energy absorbed; most of the energy is restituted to the dropped mass. In this interval, both absorbed and rebound energies increase with impact energy. In the intermediate range, however, the trend changes: rebound energy decreases while absorbed energy rises significantly. This shift reflects the onset of more severe damage in the CFRP beam, which reduces its ability to push the dropped mass upward, thereby restituting less energy. At the highest impact energies, the beam collapses completely, losing its capacity to resist the impacting mass, and the rebound energy drops to zero.

# 4. Results and discussion on the theoretical investigation

## 4.1. Theoretical model

In this section, a simple theoretical model to account for damage in the system is developed. A simplified schematic of the drop-weight experiment is shown in Fig. 10. The falling mass is considered as a particle with mass m, initially raised to a height h. Upon release, this mass impacts a double-clamped beam characterized by mass m_b, stiffness modulus E_b, and area moment of inertia I_b. After free fall, the mass impacts the composite beam at an initial speed $v_i = \sqrt{2gh}$.

When the falling mass contacts the composite beam, the low-velocity

<figure><img src="image_12.png" /><figcaption>Fig. 10. Schematic of the impact process.</figcaption></figure>

impact scenario can be modeled using a two-degree-of-freedom mass-spring system (Fig. 11a). In this model, m represents the mass of the falling object, m_b is the beam’s mass, k_c denotes the local stiffness due to contact, k_b represents the global bending stiffness of the beam, δ is the beam’s deflection, and u is the displacement of the mass. Given that the beam’s mass is significantly smaller than that of the impactor (m_b << m), it can be neglected, reducing the system to a one-degree-of-freedom mass-spring model, as shown in Fig. 11b [64].

The contact and bending springs are arranged in series, allowing them to be represented by a single equivalent spring with stiffness k, as illustrated in Fig. 11c, where:

$$k = \frac{k_c k_b}{k_c + k_b} \qquad (11)$$

Applying Newton’s Second Law, the displacement of the falling mass follows the differential equation for a classic one-degree-of-freedom free vibration system:

$$ma + ku = 0. \qquad (12)$$

where a is the acceleration of the mass, given by $a = \frac{d^2u}{dt^2}$. The initial conditions are taken at the instant the mass reaches the beam. Therefore, the initial displacement $u_0$ is set to zero ($u_0 = u(t=0) = 0$), marking the starting point of the mass’s displacement. The initial velocity, $v_0$, is equal to the impact velocity $v_i$, which is $(v_0 = v(t=0) = v_i = \sqrt{2gh})$.

To account for damage occurring in the composite beam, a piecewise damage variation with respect to displacement is considered here. Shabani et al. [65] proposed a piecewise fiber damage evolution model under tensile and compressive loading. In the context of our mass-spring model, damage is treated as the cumulative effect of multiple damage

<figure><img src="image_13.png" /><figcaption>Fig. 11. Mass-spring representation of the low-velocity impact problem (a) Considering the beam’s mass, (b) Neglecting the specimen’s mass, (c) Using an equivalent stiffness for the two springs in series.</figcaption></figure>
mechanisms, including matrix cracking, delamination, fiber fracture, and fiber kinking. This holistic approach ensures that the damage contributions from all potential failure modes are accounted for within the model. It is illustrated in Fig. 12. Damage is assumed to be zero until a critical displacement, $u_{thr}$, is reached [66]. This damage threshold displacement marks the point at which substantial damage begins within the specimen. For low displacements ($0 \le u < u_{thr}$), no damage is considered, although some energy absorption is recorded in this range. This absorbed energy may be due to localized damage in the contact/impact area or the viscoelastic behavior of the composite's polymeric matrix. In both cases, the damage is negligible as it does not significantly affect the overall behavior of the composite beam.

Damage initiation is assumed to be instantaneous, as indicated by a sharp drop in the force-time curves (Fig. 4). The initial damage is represented by $d_i$. Once damage begins, it remains constant for displacements between $u_{thr}$ and $u_{icp}$, where $u_{icp}$ is the displacement at which the beam's collapse process starts. Although slight increases in damage may occur within this range, any variation is neglected for simplicity. As displacement surpasses $u_{icp}$, complete failure of the specimen begins, and damage increases significantly until total collapse at ($u = u_{cp}$), where $u_{cp}$ is the displacement corresponding to full specimen collapse, or also the final failure equivalent displacement [66].

The damage model depicted in Fig. 12 reflects the three phases of deformation observed during low-velocity impact on composite beams: the undamaged deformation phase, the damage initiation phase, and the collapse phase. Moreover, this damage model represents a cumulative effect that encompasses various damage mechanisms, such as matrix cracking, delamination, and fiber fracture [53,61].

The formal expression for damage progression is provided by the following piecewise function:

$$d(u) = \begin{cases} 0 & (0 \le u \le u_{thr}) \\ d_i & (u_{thr} \le u < u_{icp}) \\ d_i + \frac{1 - d_i}{u_{cp} - u_{icp}} u & (u_{icp} \le u \le u_{cp}) \end{cases} \quad (13)$$

This function captures the damage evolution from undamaged to complete failure in three stages, with initial, gradual, and final phases of damage.

As damage increases, it proportionally reduces the specimen's overall stiffness [61]. Consequently, the evolution of stiffness in relation to displacement can be represented as follows:

$$k = k_u(1 - d). \quad (14)$$

where $k_u$ denotes the initial stiffness of the undamaged beam.

Since stiffness depends on displacement, an iterative numerical

scheme is necessary to solve the displacement-dependent differential Eq. (12). Here, time is divided into small increments of length $\Delta t$, enabling calculation of displacement, velocity, acceleration, force, and energy at discrete times ($t_j = j\Delta t$), where $j$ ranges from 0 to the maximum iteration count $N$.

The iterative process begins with the initial conditions of the free-vibration problem:

$$u_0 = 0. \quad (15a)$$

$$v_0 = v_i. \quad (15b)$$

$$a_0 = 0. \quad (15c)$$

To calculate displacement, velocity, and acceleration at a general step $j$, sufficiently short step length $\Delta t$ is assumed. This allows us to use the displacement of the previous step to update the damage at each step $j$:

$$d_j = d(u_{j-1}). \quad (16)$$

With the damage determined at step $j$, the stiffness can be updated using Eq. (14):

$$k_j = k_u(1 - d_j). \quad (17)$$

Having calculated the specimen's stiffness, the acceleration at step $j$ is then determined from Eq. (12):

$$a_j = -k_j u_j / m. \quad (18)$$

With acceleration known, the velocity and displacement at step $j$ are computed through numerical integration:

$$v_j = v_{j-1} + a_j \Delta t. \quad (19)$$

and

$$u_j = u_{j-1} + v_j \Delta t. \quad (20)$$

These Eqs. (18 to 20) provide the acceleration, velocity, and displacement for each step $j$ in the iterative process. This scheme repeats until the maximum number of iterations $N$ is reached, ensuring that each phase of the damage evolution is captured. Once the displacement progression is complete, the force, power, and energy are calculated using Eqs. (4), (7), and (8), respectively, to gain insights into the overall mechanical response and energy absorption characteristics of the composite specimen. This detailed iterative approach enables accurate modeling of the specimen's dynamic response, considering the progressive damage and its impact on stiffness.

## 4.2. Results and discussion on the comparison of developed theoretical model and experimental data

The experimental and theoretical force-displacement curves are shown in Fig. 13. In the low-impact energy range (2 to 7.5 J), the model accurately captures the loading phase of the curves (Fig. 13a, 13b, and 13c). As the model assumes an elastic response, the theoretical unloading curve overlaps with the loading curve. However, the experimental unloading curves fall slightly below the loading curves, indicating some energy dissipation. Additionally, the experimental curves display oscillations not observed in the model-generated curves, indeed, such undulations are commonly attributed either to the inertia of the testing machine [53,54] or to the propagation of damage mechanisms induced by impact. The simplified model proposed in this study does not incorporate the geometry of the drop mass and, as a result, cannot capture the effects related to machine inertia. Furthermore, the model does not aim to distinguish between individual damage mechanisms; instead, it represents the overall damage evolution within the composite beam.

In the intermediate impact energy range (10 to 15 J), the model also

<figure><img src="image_14.png" /><figcaption>Fig. 12. Piecewise damage evolution with displacement.</figcaption></figure>
<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /></figure>

<figure><img src="image_22.png" /><figcaption>Fig. 13. Comparison between the experimental and theoretical force-displacement curves (a) 2 J, (b) 5 J, (c) 7.5 J, (d) 10 J, (e) 12.5 J, (f) 15 J, (g) 17.5 J and (h) 20 J.</figcaption></figure>
accurately represents the loading phase (Fig. 13d, 13e, and 13f). Here, the theoretical unloading curve lies slightly below the loading curve, suggesting some energy absorption. Similarly, the experimental unloading curves fall below the loading curves, though at a greater offset than the theoretical predictions, indicating that the damage model may underestimate the energy absorbed by the composite beam during impact. As in the low-energy range, oscillations appear in the experimental curves but are absent from the theoretical results, since the model does not account for the drop mass geometry or differentiate between individual damage mechanisms.

At the highest impact energy levels (17.5 and 20 J), the model curves align well with the experimental data (Fig. 13g and h), aside from some oscillations. At this energy level, unloading behavior cannot be observed, as the specimen collapses and cannot reverse the displacement of the impacting mass. Here, the displacement only increases. The simulation concludes when the damage parameter reaches 1, and the beam's stiffness drops to zero.

To obtain the theoretical curves in Fig. 13, five damage model parameters—$k_u$, $d_i$, $u_{thr}$, $u_{icp}$, and $u_{cp}$—were adjusted to best fit the experimental data. The resulting parameter values across all impact energies are listed in Table 2. The stiffness $k_u$ of the undamaged beam falls between 1.42 and 1.56 MN/m, with an average of 1.476 MN/m, suggesting that this parameter is not particularly sensitive to impact velocity or energy. These results align well with previous findings using alternative methods. For instance, in Section 2.2.1.1, the undamaged stiffness was estimated at 1.596 MN/m using the quarter-period time, and in Section 2.2.3.1, it was estimated at 1.5 MN/m using peak force values. Maximum deflection analysis in Section 2.2.3.2 yielded a stiffness estimate of 1.46 MN/m, further supporting these findings.

In contrast to $k_u$, the initial damage parameter $d_i$—representing the primary substantial damage in the composite beam—shows greater sensitivity to impact energy/velocity, increasing with higher impact energies. Notably, the increase in $d_i$ is more pronounced between 10 and 12.5 J. This behavior is also observed for $u_{thr}$, the damage threshold displacement. For these two parameters, which characterize damage initiation, it appears that the process is sensitive to impact energy, with higher energy leading to more initial damage and a later onset of damage.

Examining the collapse phase, parameters $u_{icp}$ and $u_{cp}$ show limited sensitivity to impact velocity or energy; their values remain similar across different impact velocities. However, due to the limited number of impact energy levels tested, clear conclusions regarding these parameters are challenging.

The elastic-damage model introduced in Section 3.1 performs well, capturing key experimental characteristics. However, the model underestimates the energy absorbed by the composite beam, likely due to an additional energy absorption mechanism beyond damage mechanism. The theoretical model accounts primarily for damage-related energy absorption, while experimental data suggest that some energy may be absorbed through viscous effects or other mechanisms in addition to the damage mechanism.

Fig. 14 compares the energy absorbed by the specimen during impact, as measured experimentally and predicted theoretically. Overlaid on this comparison is the structural damage energy absorbed, as calculated in Section 2.2.3.3. The structural damage energy was estimated by subtracting the extrapolated local or contact energy absorption from the total energy. In the low and intermediate impact energy

<table><caption>**Table 2**
Identified constants for the damage model.</caption><thead><tr><th></th><th>2</th><th>5</th><th>7.5</th><th>10</th><th>12.5</th><th>15</th><th>17.5</th><th>20</th></tr></thead><tbody><tr><td>E<sub>i</sub> (J)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>k<sub>u</sub> (MN/m)</td><td>1.46</td><td>1.42</td><td>1.49</td><td>1.42</td><td>1.53</td><td>1.56</td><td>1.46</td><td>1.47</td></tr><tr><td>d<sub>i</sub></td><td>-</td><td>-</td><td>-</td><td>0.30</td><td>0.55</td><td>0.63</td><td>0.60</td><td>0.58</td></tr><tr><td>u<sub>thr</sub> (mm)</td><td>-</td><td>-</td><td>-</td><td>3.52</td><td>3.78</td><td>3.89</td><td>3.95</td><td>4.07</td></tr><tr><td>u<sub>icp</sub> (mm)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>4.78</td><td>4.70</td></tr><tr><td>u<sub>cp</sub> (mm)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>5.00</td><td>4.95</td></tr></tbody></table>

<figure><img src="image_23.png" /><figcaption>**Fig. 14.** Comparison between the experimental and theoretical energy absorption.</figcaption></figure>

ranges, the theoretical energy absorption prediction aligns more closely with the energy absorbed by structural damage, supporting the notion that damage is just one mechanism of energy absorption, with other mechanisms coexisting. At the highest impact energy range, the theoretical energy absorption corresponds to the total energy absorbed rather than only the structural damage. This is primarily because, at these high energy levels, no unloading occurs, making it challenging to distinguish damage-based from non-damage-based energy absorption.

The auxiliary energy absorption mechanism can be attributed to local damage, as evidenced by the circular impactor imprints observed on the front surface of the composite beams at 5 and 7.5 J, as discussed in Section 2.1. These imprints likely result from localized plastification of the matrix or fracture of a few surface fibers. However, the damage remains confined to the impact zone and primarily affects only the front surface of the composite beam. In addition to local damage, matrix viscosity also contributes to the auxiliary energy absorption. Epoxy, as a polymeric matrix, exhibits strain-rate sensitivity due to its inherent viscosity [67], and its mechanical behavior is more accurately described by a complex Young's modulus that incorporates both elastic and viscous components [68,69]. The current model described in Section 4.1 represents the composite beam using an elastic stiffness. Future work should enhance this model by incorporating both stiffness and damping parameters to account for the viscoelastic nature of the composite beam and better represent its response under impact loading.

## 5. Conclusion

This study examined the low-velocity impact (LVI) behavior of carbon fiber reinforced polymer (CFRP) composite beams through a combination of experimental testing and theoretical modeling. The results revealed three distinct behavioral regimes based on impact energy levels: an elastic regime (2–7.5 J), an intermediate damage regime (10–15 J), and a complete collapse regime (17.5–20 J). Load-displacement and energy absorption profiles clearly demonstrated that increasing impact energy leads to progressive damage and structural degradation. Damage indicators such as the initiation threshold $d_i$ and threshold displacement $u_{thr}$ were found to be sensitive to impact energy, while the stiffness parameter $k_u$, representative of the undamaged material state, remained nearly constant. The proposed damage-based theoretical model effectively captured essential aspects of the beam's impact response, including initial stiffness and the evolution of damage. Additionally, the model results point to the contribution of secondary energy dissipation mechanisms—such as matrix plasticity or viscous effects—which persist even at lower energy levels and are not solely attributable to material damage. These findings underscore the complexity of impact phenomena in CFRP composites and suggest
avenues for further refinement of predictive models. By focusing on beam geometries rather than plates, this work offers a simplified yet effective approach for simulating and analyzing composite impact behavior.

## CRediT authorship contribution statement

**Omar Baabdullah:** Writing – original draft, Visualization, Validation, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.
**Ramzi Othman:** Writing – original draft, Validation, Investigation, Formal analysis, Conceptualization.
**U.A. Khashaba:** Writing – review & editing, Writing – original draft, Supervision, Methodology, Investigation, Data curation, Conceptualization.

## Declaration of competing interest

The authors declare that there are no conflicts of interest related to this manuscript. No financial, personal, or professional relationships influenced the work reported in the submitted manuscript.

## Data availability

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

## References

[1] M.S. Ul Abrar, K.F.N. Ezaz, M.J. Hasan, R.I. Pranto, T.A. Alvy, M.Z Hossain, Speed-dependent impact analysis on a car bumper structure using various materials, Results Eng. 21 (2024) 101927.

[2] R.M. Abdalrahman, S. Rostam, H.D. Lafta, Bettered strength and fracture behavior of plain weave carbon fiber reinforced epoxy laminate with open hole under the influence of hole shape/size and strain rate, Results Eng. 25 (2025) 103906.

[3] R. Othman, U.A. Khashaba, K.H. Almitani, Compression after impact of CFRP composites immersed in distilled water, Int. J. Crash. 26 (2021) 490–500.

[4] U.A. Khashaba, R. Othman, I.M.R. Najjar, Experimental analysis of composite scarf adhesive joints modified with multiwalled carbon nanotubes under bending and thermomechanical impact loads, Proc. Inst. Mech. Eng. Part L 234 (2020) 48–64.

[5] R. Othman, U.A. Khashaba, Calibration of the low-velocity impact experiments, Polym. Compos. (2025) 1–22, https://doi.org/10.1002/pc.29896.

[6] U.A. Khashaba, Dynamic analysis of scarf adhesive joints in CFRP composites modified with $Al_2O_3$-nanoparticles under fatigue loading at different temperatures, Compos. Part A 143 (2021) 106277.

[7] M. Wang, Z. Pan, Z. Ying, Z. Wu, Symmetric and asymmetric intercalation effect on the low-velocity impact behavior of carbon/Kevlar hybrid woven laminates, Compos. Struct. 297 (2022) 115919.

[8] K. Kaware, M. Kotambkar, Low velocity impact response and influence of parameters to improve the damage resistance of composite structures/materials: a critical review, Int. J. Crash. 27 (2022) 1232–1256.

[9] Y. Fang, D. Sheng, Z. Lin, P. Fei, Study of low-velocity impact behavior of hybrid Fiber-reinforced metal laminates, Polymers 16 (2) (2024) 173.

[10] Y. Li, Y. Jin, X. Chang, Y. Shang, D. Cai, On low-velocity impact response and compression after impact of hybrid woven composite laminates, Coatings 14 (8) (2024) 986.

[11] U. Gezer, B. Demir, Y. Kepir, A. Günöz, M. Kara, A numerical study on the low-velocity impact response of hybrid composite materials, Turk. J. Eng. 7 (4) (2023) 314–321.

[12] A. Wang, X. Wang, G. Xian, The influence of stacking sequence on the low-velocity impact response and damping behavior of carbon and flax fabric reinforced hybrid composites, Polym. Test. 104 (2021) 107384.

[13] M. Masud, A. Mubashar, Investigating stacking configuration and Fiber hybridization effects on low-velocity impact behavior in twill woven carbon/flax composites, MATEC Web Conf. 398 (2024) 01009.

[14] S. Hiremath, Y. Zhang, L. Zhang, T.W. Kim, Machine learning approach to evaluating impact behavior in fabric-laminated composite materials, Res. Eng. 23 (2024) 102576.

[15] S. Hiremath, Y. Zhang, T.W. Kim, Low-velocity impact behavior in multi-layered structures and hybrid composites via sandwich stacking techniques, Polym. Comos. 45 (2024) 7439–7454.

[16] H. Li, K. Liu, Z. Tao, Z. Yu, L. Ye, W. Xiao, Effect of stacking sequence on the double-point low velocity impact response and compression-after-impact behavior of glass fiber-reinforced composite laminates, Polym. Compos. (2025), https://doi.org/10.1002/pc.29504 in press.

[17] M.E. Kazemi, L. Shanmugam, A. Dadashi, M. Shakouri, D. Lu, Z. Du, Y. Hu, J. Wang, W. Zhang, L. Yang, J. Yang, Investigating the roles of Fiber, resin, and stacking sequence on the low-velocity impact response of novel hybrid thermoplastic, Comos. B 207 (2021) 108554.

[18] L. Han, H. Qi, J. Yang, F. Chu, C. Lin, P. Liu, Q. Zhang, Study on the low-velocity impact response and damage mechanisms of thermoplastic composites, Polymers 16 (6) (2024) 791.

[19] M.R. Zamani, Influence of kevlar in carbon Fiber hybrid composite laminate under low-velocity impact, Power Syst. Technol. 48 (2) (2024) 1672–1685.

[20] F. Azimpour-Shishevan, M.A Mohtadi-Bonab, An experimental study on the effects of introducing carbon nanotube on low velocity impact behavior of carbon/aramid Fiber reinforced intra-ply hybrid composites, J. Text Inst. 114 (8) (2023) 1167–1176.

[21] A.H. Alshehri, A. Alamry, S.S.R. Koloor, B. Alzahrani, A. Arockiarajan, Investigating low velocity impact and compression after impact behaviors of carbon Fiber/epoxy composites reinforced with helical multiwalled carbon nanotubes, In press, J. Eng. Res. (2024), https://doi.org/10.1016/j.jer.2024.07.017.

[22] H. Laaouidi, M. Tarfaoui, M. Nachtane, M. Trihi, Energy absorption characteristics in hybrid composite materials for marine applications under impact loading: case of tidal current turbine, Int. J. Energy Res. 45 (4) (2021) 5894–5911.

[23] A. Saylik, Ş. Temiz, Low-speed impact behavior of Fiber-reinforced polymer-based glass, carbon, and glass/carbon hybrid composites, Mater. Test 64 (6) (2022) 820–831, https://doi.org/10.1515/mt-2021-2179.

[24] M. Grasso, Y. Xu, Threshold identification and damage characterization of woven GF/CF composites under low-velocity impact, J. Compos. Sci. 6 (2022) 305.

[25] M.A.S. Siddiqui, M.S. Rabbi, S. Dewanjee, Low-velocity impact response of natural Fiber reinforced composites: a comprehensive review on influential parameters, Compos. Part C 12 (2023) 100422.

[26] M. Demir, R. Ekici, Investigation of low-velocity impact behaviors of polymer composites reinforced with different natural Fiber fabrics, Polym. Compos. 45 (6) (2024) 4928–4946, https://doi.org/10.1002/pc.28099.

[27] S. Lin, A.M. Waas, The effect of stacking sequence on the LVI damage of laminated composites; experiments and analysis, Compos. A 145 (2021) 106377.

[28] K. Kaware, M. Kotambkar, A. Sontakkey, N. Talekar, Finite element analysis of CFRP composite under low velocity impact to improve the impact strength, Int. J. Crashworthiness 28 (6) (2023) 717–731.

[29] N. Pei, Q. Wu, L. Long, Comparison of off-axis angle on the low velocity impact behaviors for carbon-Fiber-reinforced polymer laminates, J. Compos. Mater. (16) (2022) 2483–2494.

[30] J. Sun, Y. Dai, L. Huang, D. Zhang, J. Zhao, Micromechanisms and characterization of low-velocity impact damage in 3D woven composites, Materials 15 (19) (2022) 6636.

[31] Wu H., Li X., Yan K., Yuan M., Huang C., Zhang Q. Experimental and simulation study on the failure mechanism of weaving structures on 3d att woven composites under low-velocity impact. Available at SSRN 2024. doi:10.2139/ssrn.4902130.

[32] M.I. Özsoy, Impactor diameter and ply number effects on the impact behavior of carbon Fiber composite laminates, Gazi Univ. J. Sci. Part C 10 (3) (2022) 439–454.

[33] Y. Ding, J. Liu, Z.E.C. Hall, R.A. Brooks, H. Liu, A.J. Kinloch, J.P. Dear, Damage and energy absorption behavior of composite laminates under impact loading using different impactor geometries, Compos. Struct. 321 (2023) 117259.

[34] N.A. Nassir, R.S. Birch, W.J. Cantwell, Z.W. Guan, The response of glass fibre reinforced PEKK laminates subjected to single and multiple impact loading, Polym. Test. 131 (2024) 108323.

[35] S. Ko, Y. Qiao, T.J. Roosendaal, A.C. Denny, J.L. Ramos, W. Kuang, K.W. Shahwan, K.L. Simmons, Low-Velocity Repeated Impact Behaviors of Polymer Fiber Reinforced Plastics (PFRPs), Pacific Northwest National Laboratory (PNNL), Richland, WA (United States), 2023. PNNL-SA-186407.

[36] K. Balasubramaniam, D. Ziaja, M. Jurek, P. Fiborek, P. Malinowski, Experimental and numerical analysis of multiple low-velocity impact damages in a glass fibered composite structure, Materials 14 (23) (2021) 7268.

[37] O. Fernandes, J. Dutta, Y. Pai, Effect of various factors and gyrothermal ageing environment on the low velocity impact response of fibre reinforced polymer composites- a comprehensive review, Cogent. Eng. 10 (1) (2023) 2247228.

[38] A. Gunoz, M. Kur, M. Kara, Impact response of hybrid composites under varying environmental conditions: a comparative study, Polym. Compos. 45 (13) (2024) 12481–12494.

[39] A. Wang, X. Wang, G. Xian, Mechanical, low-velocity impact, and hydrothermal aging properties of flax/carbon hybrid composite plates, Polym. Test. 90 (2020) 106759.

[40] O.A. Batmaz, M.O. Bozkurt, E. Gurses, D. Coker, High-fidelity simulations of low-velocity impact induced matrix cracking and dynamic delamination progression in CFRP beams, Compos. A 177 (2024) 107960.

[41] M.O. Bozkurt, D. Coker, In-situ investigation of dynamic failure in [0$_5$/90$_3$]$_6$ CFRP beams under quasi-static and low-velocity impact loadings, Int. J. Solids Struct. 217–218 (2021) 134–154.

[42] C. Santiuste, S. Sanchez-Saez, E. Barbero, Residual flexural strength after low-velocity impact in glass/polyester composite beams, Compos. Struct. 92 (2010) 25–30.

[43] S.R. Hallett, Three-point beam impact tests on T300/914 carbon-fibre composites, Compos. Sci. Technol. 60 (2000) 115–124, https://doi.org/10.1016/S0266-3538 (99)00099-8.

[44] S.I. Thorsson, J. Xie, J. Marek, A.M. Waas, Matrix crack interacting with a delamination in an impacted sandwich composite beam, Eng. Fract. Mech. 163 (2016) 476–486.

[45] O.T. Topac, B. Gozluklu, E. Gurses, D. Coker, Experimental and computational study of the damage process in CFRP composite beams under low-velocity impact, Comos. A 92 (2017) 167–182.

[46] A. Rotem, Residual flexural strength of FRP composite specimens subjected to transverse impact loading, Sampe J. 24 (1988) 19–25.
[47] H. Chenghong, L. Yubin, Z. Zuoguang, S. Zhijie, Impact damage modes and residual flexural properties of composite beam, J. Reinf. Plast. Compos. 27 (2008) 1163–1175.

[48] M.S. Boroujerdy, Y. Kiani, Low velocity impact analysis of composite laminated beams subjected to multiple impacts in thermal field, Z. Angew. Math. Mech. 96 (2016) 843–856.

[49] Y. Fan, Y. Wang, The effect of negative Poisson's ratio on the low-velocity impact response of an auxetic nanocomposite laminate beam, Int. J. Mech. Mater. Des. 17 (2021) 153–169.

[50] U.A. Khashaba, A.A. Aljinaidi, M.A. Hamed, Analysis of adhesively bonded CFRE composite scarf joints modified with MWCNTs, Compos. A 71 (2015) 59–71.

[51] U.A. Khashaba, A.A. Aljinaidi, M.A. Hamed, Development of CFRE composite scarf adhesive joints with SiC and Al2O3 nanoparticle, Compos. Struct. 128 (2015) 415–427.

[52] U.A. Khashaba, A.I. Khdair, Open hole compressive elastic and strength analysis of CFRE composites for aerospace applications, Aerosp. Sci. Technol. 60 (2017) 96–107.

[53] U.A. Khashaba, R. Othman, Low-velocity impact of woven CFRE composites under different temperature levels, Int. J. Impact. Eng. 108 (2017) 191–204.

[54] U.A. Khashaba, Impact of standard and unexpected bolt-hole-washer tolerances on the performance of pinned and torqued joints in CFRP composites, Alex. Eng. J. 81 (2023) 290–303.

[55] S.Z.M. Daud, J. Lim, M. Amir, S.W. Kim, Enhancing impact energy absorption in composite sandwich structures through synergistic smart material integration, Results Eng. 21 (2024) 101902.

[56] R. Othman, P. Guegan, G. Challita, F. Pasco, D. LeBreton, A modified servo-hydraulic machine for testing at intermediate strain rates, Int. J. Impact. Eng. 36 (2009) 460–467.

[57] V.S. Anuse, K. Shankar, R. Velmurugan, S.K. Ha, Compression-after-impact analysis of carbon fiber reinforced composite laminate with different ply orientation sequences, Int. J. Impact. Eng. 167 (2022) 104277.

[58] J.C. Petiteau, R. Othman, P. Guégan, H. Le Sourne, E. Verron, A drop-bar setup for the compressive testing of rubber-like materials in the intermediate strain rate range, Strain. 50 (2014) 555–562.

[59] B. Song, B. Sanborn, J. Heister, R. Everett, T. Martinez, G. Groves, E. Johnson, D. Kenney, M. Knight, M. Spletzer, Development of “Dropkinson” bar for intermediate strain-rate testing, Dyn. Behav. Mater. 1 (2018) 81–83, https://doi.org/10.1007/978-3-319-95089-1_12.

[60] R.L. Caldwell, M.J. Donough, A.W. Phillips, N.A. St John, B.G Prusty, Dynamic response of composite materials subjected to low velocity impacts in water: a numerical study, Compos. Commun. 44 (2023) 101775.

[61] A. Katunin, W. Danek, A. Wronkowicz, K. Dragan, Methodology of residual strength prediction of composite structures with low-velocity impact damage based on NDT inspections and numerical-experimental CAI testing, Int. J. Impact. Eng. 181 (2023) 104762.

[62] U.A. Khashaba, A comprehensive analysis of low velocity impact response of [0/±45/90]s thin woven GFRP composites at room temperature, Compos. Struct. 339 (2024) 118160.

[63] U.A. Khashaba, A comprehensive study of low-velocity impact behavior and damage evaluation of angle-ply thin woven GFRP composites under elevated temperatures, Compos. Sci. Technol. 252 (2024) 110608.

[64] Q. Zhu, Z. Yu, A modified spring-mass model of low-velocity-impact on composite laminate including shearing deformation at delaminated region and reduction on membrane stiffness, Int. J. Impact. Eng. 185 (2024) 104847.

[65] P. Shabani, L. Li, J. Laliberte, Low-velocity impact (LVI) and compression after impact (CAI) of Double-double composite laminates, Compos. Struct. 351 (2025) 118615.

[66] W. Liu, H. Liu, Z. Cheng, H. He, Q. Kan, G. Kang, Prediction of low-velocity impact responses for bio-inspired helicoidal laminates based on machine learning, Int. J. Impact. Eng. 195 (2025) 105144.

[67] G. Challita, R. Othman, K. Khalil, Compression and shear behavior of epoxy SA 80 bulk adhesive over wide ranges of strain rate, J. Polym. Eng. 36 (2016) 165–171.

[68] M.N Bussac, P. Collet, G. Gary, R. Othman, An optimisation method for separating and rebuilding one-dimensional dispersive waves from multi-point measurements. Application to elastic or viscoelastic bars, J. Mech. Phys. Solids 50 (2002) 321–349.

[69] R. Othman, On the use of complex Young's modulus while processing polymeric Kolsky-Hopkinson bars' experiments, Int. J. Impact. Eng. 73 (2014) 123–134.