Mechanical Analysis of Corrugated Cardboard Subjected to Shear Stresses

Abstract

Corrugated cardboard is widely used for packaging due to its sturdiness and lightweight properties. However, its ability to withstand lateral forces during handling, transportation, and storage is critical for maintaining its structural integrity. In this study, the problem of understanding and quantifying the strain experienced by corrugated cardboard boxes under shear stresses at various points in their lifecycle is addressed. Specifically, a precise methodology for testing shear stresses was developed and validated, enhancing the understanding of the material’s behaviour. Several key problems are addressed in this research, including the quantification of the amount of strain experienced by cardboard boxes from a warehouse to the final delivery, the development of a reliable testing methodology to measure shear stresses, and the understanding of the mechanical behaviour and structural integrity of corrugated cardboard under these conditions. It is anticipated that the findings will improve the design and durability of corrugated cardboard packaging, ensuring better performance during transportation and storage. This research contributes to the optimisation of logistics, the improvement of packaging quality, and the support of sustainable development by enhancing the reuse and recyclability of corrugated cardboard.

1. Introduction

Corrugated cardboard plays a critical role in the packaging industry. It provides security and transportation easily through an adaptable and economical medium. Cardboard, in a universally dynamic marketplace like this, has helped in becoming part of the solution for the carrying of goods, especially for moderate to fragile items. Over time, corrugated cardboard has been transformed in a phenomenal way from an elementary carrier of packages to a strategic component in protection and logistic solutions.

This is a very environmentally friendly, readily recyclable ingredient, not only making its presence inevitable in the packaging world, but also becoming a favourite of any manufacturer or consumer concerned with sustainable development. The unique characteristics of this material, in terms of mechanical strength, durability, and flexibility, have made it widely applicable in almost all industries. The mechanical strength and durability of corrugated cardboard depend mainly on its direction of orientation with respect to the direction of the corrugation. Precisely for this reason, cardboard boxes show their corrugations on a vertical orientation. This design has intended to optimise the load-bearing capacity of cardboard packages.

The structural integrity of corrugated cardboard, particularly its strength, is mainly oriented along the direction of its corrugations. This characteristic has led to a design preference for vertically oriented corrugations in cardboard boxes, optimising their load-bearing capacity. Corrugated cardboard also can absorb impacts. The present study aims to describe the tangential stresses experienced by corrugated cardboard, especially at the top and bottom facets of boxes when subjected to stacking forces during transport.

In practise, tangential stresses may not always be visibly evident on the surface of corrugated cardboard boxes, but they act throughout the whole lifecycle of the package. Corrugated cardboard boxes are very useful tools for the storage, transportation, and handling of almost all products, even if their weight is quite high. It is incredibly deceptive how much they offer in strength, considering the generally fragile nature of the material. This research was focused on the study and measurement of tangential stresses occurring in all phases of the use of packages of corrugated cardboard, starting from the handling inside a warehouse and continuing with the loading and unloading operations from trucks, up to the delivery to the final consumer. The main purpose of the research consists of the development and validation of very accurate methodology for testing tangential stresses that influence the structure of corrugated cardboard. This way, a gateway to find the solutions for the optimisation of performance and hence the durability of corrugated cardboard boxes can be opened. Consequently, the efficiency of logistics, improvement of packing, and transportation quality should take a big step forward. From a strength point of view, the real strength of corrugated cardboard is in fact parallel to the corrugations. For many corrugated cardboard applications, this characteristic often introduces the necessity to prioritise the vertical position of the corrugations when designing cardboard boxes to ensure their maximum load-bearing capacity. In this way, the packaging in corrugated cardboard not only offers remarkable impact and compression protection, but also ensures that the contents housed in the boxes are delivered to users completely safely. This paper addresses such tangential stresses, which are particularly located at the top and bottom capacities of the boxes when these are subjected to a piling load during its transport. The strength in the horizontal direction of the corrugations and the influence that these tangential stresses exercise on them was deemed to be important for evaluation. Packaging paper is a basic material in a wide array of sectors, from food to fragile protection. All these functions depend on the mechanical properties of the paper to be fulfilled in a proper way. Tensile strength is one of these properties, which has stimulated the interest of researchers and industry people. The scientific context of the study of tangential stresses in the structure of corrugated cardboard and its role within the field of mechanical engineering and packaging is intended to be addressed. One of the most important parameters that accounts for the real application performance in packaging paper is tensile strength. Additionally, in-plane mechanical properties are defined by the machine direction (MD) and cross-direction (CD) and are base properties of the material in its capability to resist those types of tensile or torsion forces encountered during its usage. And at the same time, the z-direction out-of-plane mechanical properties are also of major importance, dependent on the external actions or pressures which they will be able to resist. These actions will be exerted over the paper during its functional life, especially in the stacking process, which is the process when the paper is stacked or piled, that is, overlapped exactly and discreetly; this is one of the main characteristics of the tensile stresses used in the design or improvement of packaging products.

The object of this work regarding the corrugated cardboard-type CO3 cardboard is a means of analysing this material. CO3 has two main different directions, namely “Machine Direction” (MD-x) and “Cross-Direction” (CD-y). Sections normal to the CD-y orientation show large stiffness with respect to normal stresses Ny and bending moments My, as indicated in Figure 1, which is an important material characteristic.

Figure 1 illustrates the geometric parameters of corrugated cardboard. As the mechanical properties of the core are always determined experimentally in its flat state, it is necessary to convert the constitutive law from its local reference in tilted 1, 2, and 3 directions (where 1 represents the tangential direction, 2 represents the y-direction, and 3 represents the normal direction) into the global reference of the cardboard’s XYZ plane. The angle, θ, can be expressed in terms of x. In this part of the work, the focus is placed on the structural properties of a composite material consisting of laminated sheets and a corrugated cardboard core.

1.1. Global and Analitycal Models

Previous research is divided into three distinct categories, which are analytical models, numerical models, and experimental models.

One of the first studies related to the problem of tangential stresses is from Libove C and Hubka R [1], who in a paper on the elastic behaviour of material in corrugated-core structures, describe transverse shear stiffness in the direction along and perpendicular to the direction along the corrugation. They observed that transverse shear stiffness in the direction perpendicular to the direction along the corrugation is much smaller when the deformation of the liners was taken into consideration. An analytical development is presented in [2] from 1992, where an extension of the membrane analogy method was used to calculate the torsional rigidity of a corrugated board. Some of the most important contributions to the study of these problems are from T. Nordstrand and co-workers. Additionally, the papers [3] and [4] represent an important theoretical contribution dealing with composite structures. Then, there is Nordstrand’s PhD thesis [5], in which he presents non-linear constitutive models for the prediction of the strength of corrugated board structures.

Other papers were focused on the elasto-plastic comportment of the paper and composite structure, such as Stenberg (2001), (2003) [6,7]. Regarding a physical explanation of the plastic behaviour, Seth and Page (1983) [8] show that the plastic behaviour of the structure is due to the plasticity of the fibres, and the effect of bond breakage is small compared to the influence of mechanical properties of the material.

A significant contribution comes from Aboura and al. [9], who developed an analytical homogenisation model based on the theory of laminated plates, comparing the obtained results with numerical and experimental data. Buannic [10] proposed a homogenisation theory based on the asymptotic expansion method, presenting a finite element calculation of the efficient behaviour properties. There are also studies about the circular economy and the recycling of cardboard products. In [11], authors show how different fractions of per- and poly-fluoroalkyl (PFAS) are to be restricted in paper and board materials.

1.2. Experimental and Numerical Analysis

Many papers have studied the structural mechanical behaviour of a whole box made of corrugated cardboard. In these studies, experimental results and FEM modelling have been shown to determine the complex twisting behaviour of a composite material in the classical box configuration. Studies such as those of Isaksson, P., Hägglund (2005) use the FEM for modelling and structure analysis to observe how a corrugated board panel deforms during severe mechanical loading [12].

In papers such as [13] (McKee et al., 1963), boxes made of corrugated cardboard with knife-cut perforations were analysed. Also, a numerical analysis (Matlab, FEM) based on a series of top-to-bottom compression strength tests of boxes were conducted in order to determine their experimental ultimate loads [13]. Also, in the paper of Tobi Fadiji et al. (2019) [14], the authors aimed to study the creep behaviour of three corrugated paperboard packages at two environmental conditions under constant load for a 12 h duration. The paperboard boxes were designed for a control package without vent holes, as well as standard vented and multi-vented packages. There is various research regarding the comportment of corrugated boxes. In [15], the authors contribute to research into the behaviour of the material of corrugated cardboard under extreme radiation, with valuable implications for fire safety and hazard assessment (2024). In [16], a FEM study is presented concerning a simplified FEM bending model for corrugated boards (2023).

1.3. Shear Comportment

Regarding the analysis of tangential stresses in the composite material structure of corrugated cardboard, there are studies such as [17], from Hämäläinen et al. (2017), that propose the SCT tester as a vehicle to indirectly measure the transverse shear modulus of thin paper grades. This paper is based on the idea that the standardised SCT compression test also has the effect of tangential stress on the composite structure. Hämäläinen et al. (2017) use this test as an experiment for estimating of the transverse shear modulus, which subsequently will be used in FE simulations of the SCT.

Experimental studies have been conducted for quasi-static axial crushing of single-wall corrugated paperboard, Jing Bai et al. (2019) [18], susceptibility to impact damage of apples inside ventilated corrugated paperboard packages, Tobi Fadiji et al. (2016) [19], lock-in infrared thermography for the evaluation of the structural performance of corrugated paperboard structures, N. Navaranjana, R. Jonesb [20], and thermally buffered corrugated packaging for preserving the postharvest freshness of mushrooms, Suman Singh et al. (2018) [21]. In the direction of the present article, i.e., for the study of the material behaviour of corrugated cardboard under shear stresses, important contributions have been made by Nordstrand, T., Carlsson, L.A. [3] (1994), Altenbach [22] (2000), Hernández-Pérez et al. [23] (2014), Avilés et al. [24], and Mrówczyński et al. [25] (2023). In the paper of Nordstrand and Carlsson [3], the authors perform a calculation to determine the upper limit of the transverse shear modulus in the CD of the corrugation, assuming that the liners are rigid sheets. Furthermore, the authors perform a quantification of the effective shear modulus influenced by the deformation of the liners, which is also compared with the FEM results. In the paper [22], Altenbach et al. show that while there is a consensus on the calculation results for the determination of bending stiffness, tension/compression, in-plane shear, and even torsional stiffness for sandwich and laminated plate models, large differences occur in the determination of transverse stiffness. The authors propose an approach that leads to the solution of a Sturm–Liouville problem, which would be a computational solution for both laminated and sandwich plates. The authors Hernández-Pérez et al. [23] deal with the twist stiffness of single- and double-wall corrugated cardboard. The first-order shear deformation (FOSD) theory is used as a numerical method and FEA is also used for comparison, but in both analyses, a homogenised core is used. According to the FOSD analysis, the torsional stiffness depends linearly on the transverse shear moduli of the web core along both main directions of the core. Similarly, the same method is used for double-walled boards. It is concluded that the FOSD method is less demanding than an FEA. In paper [24], Avilés et al. also deal with the torsional stiffness of sandwich plates. The authors start from the idea that the compliance of sandwich panels with a low density/modulus core is not adequately predicted by many previous studies. The authors propose to perform a measurement to extract the shear moduli of monolithic, composite, and sandwich panels, and then propose to use these results to determine the torsional stiffness. In paper [25], published in MDPI, Mrówczyński et al. deal with the dynamic behaviour of corrugated packaging. They present a simplified method to determine the mechanical strength of cardboard boxes subjected to dynamic transport stresses.

The authors propose a working algorithm design for three calculation steps, comprising a static analysis of the compressive strength of the packaging, a random vibration analysis in the frequency range used to determine the resonance frequencies for the packaging, and a dynamic analysis of the packaging loaded with the previously calculated resonance frequencies. The numerical model calculation is complemented with results and data obtained from measurements. There is also the possibility to verify the results of the numerical simulation of compression tests for several box geometries. The results show the advantages of using a simplified calculation method.

2. Materials and Methods

2.1. Materials

The preparation of specimens for testing tangential stresses comprised cutting samples of material of dimensions (length × width × height) of 70 × 20 × H mm, dimensions sufficient to accommodate for a wavelength of corrugation to be analysed, as seen in Figure 2. The dimension denoted as “H”, the height of the sample, is dependent on the cardboard type analysed.

The materials used for samples are two configurations. Six samples, noted as “ORIGINAL” to 11E, follow a single-wall board model (as shown in Figure 2) and one sample, noted as 35EB, is a double-wall board model. The dimensions of the samples are shown in Table 1.

Concerning the mechanical properties of materials, the in-plane elastic properties of the samples used in the experiment are considered by Young’s modulus, Poisson’s ratio, and shear modulus. The values obtained by Baum (1985) [26] and Harrysson [27] will be used. Young’s modulus and Poisson’s ratio can be directly determined. This is shown in [27]. The shear modulus cannot be that easily measured or established directly and has been a subject of discussion in many papers. However, with the methodology and results now presented by Stenberg, 2023, the study will also consider in-plane data to be known. On the other hand, the study also uses Young’s modulus obtained from a uniaxial test in 45 directions by Stenberg (2003) [6]. Consequently, the in-plane shear modulus, G12, can be calculated using a coordinate transformation. The elastic constant related to the out-of-plane direction, i.e., E₃₃, G₁₃, and G₂₃, and Poisson’s ratios ν₁₃ and ν₂₃, are also taken from Stenberg (2003) [6].

With the values obtained by Baum (1985) [26] and also used by Harrsson et al. [27]., the elastic properties are given as E₁ = 3050 MPa; E₂ = 1172 MPa; G₁₂ = 720 MPa; and ν₁₂ = 0:3.

As it has been shown, there are not many specialised studies dealing with shearing tests. In the article [9], the authors, according to the ASTM 273-61 standard, perform a shear test. In this test for the studied specimen, the two parts, the lower and upper liner, are bonded to the loading blocks with a two-component epoxy glue adhesive (araldite), which cures for 24 h at room temperature. Then, it is conditioned at 23 C and 50% RH for 24 h. Basically, the loading is only tangential. The shear moduli are determined according to the ASTM C273-61 recommendation. The values obtained are given for pure shear stress separately for the skin (liner) and corrugated part. Thus, the values are G12 (skin) = 2052.76 MPa and G12 (corrugated) = 267.83 MPa.

Due to the fact that in this paper, the numerical results agree very well with the experimental ones and are also in accordance with the ASTM 273-61 standard, the results will be used in the numerical modelling as a basis for the values of the mechanical properties determined for the geometrical dimensions of our specimens.

2.2. The Testing Rig

Experimental setups were established using the CETR (Bruker, Billerica, MA, USA) UMT-2 stand, a platform capable of measuring force and positioning at low speeds. The CETR UMT-2 Micro tribometer is extensively utilised in both academic and industrial sectors. It features the capability to accurately measure forces ranging from 1 mN to 2000 N, offering resolutions from 10 N to 100 mN thanks to its modular sensors. A visual representation of the experimental configuration is shown in Figure 3 below.

When additional components are added to the base system to perform a specific type of tribological study for tangential stresses, the configuration of the test stand can be described as follows: The vertical positioning subassembly is equipped with a motorised system and is equipped with a specialised sensor to provide precise information about its position. This sensor monitors and transmits data about the loading force acting on a specific specimen, in this case, a corrugated cardboard specimen of type CO3.

The 2D force sensor assembly can be installed in two ways, either directly on the lateral positioning stage or on an extension block that is mounted on the lateral positioning stage itself. This sensor has the ability to detect two important types of forces, the vertical or loading force and the lateral friction force. The information provided by this sensor is used to provide feedback within the vertical positioning system to either maintain a specific force constant or to monitor and control the force in real-time, depending on the requirements.

2.3. Specimen Preparation, Testing Methodology, and Results

The preparation of specimens for testing tangential stresses comprised cutting samples of material of dimensions of 70 × 20 × H mm, dimensions sufficient to accommodate for the wavelength of corrugation to be analysed. A custom fixing device was designed in order to ensure proper analysis of the cardboard samples, as seen in Figure 4, below.

The device that has been designed is composed of multiple components, each having a well-defined role in the testing process. At position 1, there is a sturdy support plate, serving as a stable base for the entire device. The plate provides the necessary support and is manufactured from S325 steel that has been laser cut. In position 2, there is another plate, which attaches to position 1 to form a lower subassembly. It is also made from S325 steel using the same manufacturing technology. Inside this subassembly, at position 3, there a plexiglass piece with a thickness of 6 mm. This element is important, as it is used in both the lower and upper subassemblies of the device. The primary function of these pieces is to serve as grips for the testing device. To ensure optimal adhesion during tensile strength tests, a special abrasive paper is applied to these pieces.

At positions 4 and 5, there are two components made of S325 steel, forming a subassembly. These are securely joined using M5 screws at position 6. In this subassembly, a 6 mm thick plexiglass disc (referred to as position 7) is affixed using 3M double-sided adhesive tape before being permanently bonded to the piece at position 5. In this process, one of the pieces from position 3 is bonded to this disc using a fast-setting adhesive called Interbond, creating a subassembly that cannot be disassembled through detachment. The component at position 8 is crafted from a 36 mm diameter round bar, precision-machined on a lathe. The M5 × 20 mm screws, positioned at number 9, play a crucial role in the assembly of the working platform of the testing stand. The setup for testing can be seen in Figure 5.

3. Results

Table 2 presents the results obtained from testing 5 samples of each type of cardboard subjected to tangential stress determinations. To ensure concrete and representative results, the highest values recorded for each type of cardboard were used in determining the final outcomes. This approach was chosen to optimally reflect the material’s performance under maximum stress conditions and to provide an accurate assessment of the cardboard’s behaviour under tangential loading.

Firstly, the influence of the stand mounting option, whether by bonding with superglue or by abrasive adhesion, was evaluated. Thus, the same sample, “ORIGINAL”, was tested in both mounting variants for tangential stress in the stand.

It can be seen from Figure 6 that there is a difference in the results for the same load, Fz = 100 N, and the same material, “ORIGINAL”. In up to about 2 mm of movement distance in the x-direction of the load Fz, the results look identical with respect to the tangential effect. After this point, quite large differences start to appear, and at the end of the stroke, at 8 mm, the difference is almost 50%. The maximum effect is given by bonding with superglue. In the same experiment, the friction force was also measured for the same mounting options. It can be seen from Figure 7 that the influence of the mounting on the variation in the friction force is largely consistent with the results for the variation in the Fx force determined previously. The equivalent friction coefficient was calculated to be the ratio between the normal and the tangential forces measured. The use of an adhesive, although better at transmitting the tangential force from the test rig through the cardboard and back to the test rig, had a major disadvantage. The glue seeped through the cardboard because of the capillary effect and artificially increased the rigidity of the cardboard. This seepage compromised the integrity of the material by altering its natural mechanical properties, leading to an inaccurate representation of the material’s true behaviour under shear stress. Therefore, the mounting method using an abrasive solution was selected, as it ensured a more accurate and reliable measurement of the corrugated cardboard’s natural response to tangential forces without altering its inherent properties.

In the second experiment, the variation in the tangential force was determined for different increasing values of the applied normal force. This study was carried out for a single specimen, i.e., for the specimen marked “ORIGINAL”. This specimen corresponds to the corrugated cardboard type “ORIGINAL”. The tangential force variation can be observed in Figure 8. It presents a visualisation of the relationship between the applied normal force (Fz) and the corresponding tangential force (Fx) experienced by a specimen of corrugated cardboard under testing. The x-axis represents the displacement in millimetres (mm), while the y-axis quantifies the tangential force in Newtons (N).

The graph presents five different curves corresponding to the application of five various normal forces, namely 100 N, 75 N, 50 N, 25 N, and 10 N. These five normal force levels are represented as Fz = 10, Fz = 25, Fz = 50, Fz = 75, and Fz = 100. When displacement starts from 0 mm, there is a rapid decrease in the tangential force—the gradient of the force applied by resistance against the movement. The tangential force gradient corresponding to the increment in the normal force will be linearly proportional. In the initial segments of the curves, the shear force increases proportionally with the normal force, indicating a linear relationship. This is evident from the initial slopes of the curves, which show a consistent rise in shear force with increasing normal force, reflecting the material’s elastic behaviour.

However, as the displacement progresses, the curves begin to diverge, suggesting that other factors start to influence the shear force, leading to a more complex, non-linear relationship. The reviewer’s observation that the initial parts of the curves coincide implies that the material exhibits similar initial shear responses across different normal forces, but this does not mean the shear force is independent of the normal force. Instead, it highlights the elastic region where the material’s response is linear before transitioning into non-linear behaviour due to factors such as material deformation and slippage.

The direction of the tangential force against the movement is represented by the negative value. This is easily observed in the curve for the highest normal force application, equal to 100 N. Along the range of displacement, each curve flattens, as though stipulating that the increment in tangential force gradually becomes stabilised. The flattening of the curve is generally observed, applying lower normal forces as seen in the curve for Fz = 10, which has the least magnitude change in the tangential force between both ends of the displacement. On average, the greater the normal force applied is, the greater the magnitude of tangential force. The curve for Fz = 10 has the smallest magnitude of tangential force, which is near the x-axis, indicating little to no resistance to the tangential stress. In contrast, the greatest magnitude of tangential force is denoted in the curve applied with the greatest normal force, which is Fz = 100, indicating strong resistance to tangential movement. The trend readily observable in the graph is the consistency of the curves’ behaviour along the range of applied normal forces. It is seen that a linear relationship may be assumed between the normal force and the initial tangential force reaction. This is followed by the non-linear behaviour of the material after the measured force shows a local maximum in its pattern; this may suggest a complex response to the stress applied.

As seen in Figure 8, the experimental results reveal that up to a deformation of 2 mm, the corrugated cardboard exhibits elastic deformation behaviour. During this phase, the shear force increases proportionally with the applied deformation, indicating an efficient redistribution of stresses within the corrugated structure. The cardboard resists the tangential stresses through the flexion and compression of the corrugations, reflecting the material’s ability to elastically return to its original shape once the forces are released.

However, after exceeding 2 mm of deformation, the cardboard enters a plastic deformation phase where the shear force stabilises. In this stage, the material loses its elasticity, and the corrugations gradually collapse, leading to permanent deformation. Specifically, the corrugations no longer respond by flexing but begin to flatten out, losing their capacity to absorb additional tangential stresses. As a result, the shear force remains relatively constant because the material can no longer resist the applied loads effectively, and further deformation occurs without a proportional increase in shear force.

Structurally, the corrugations in the cardboard undergo two distinct phases. Initially, under the action of tangential forces, the corrugations deform elastically, contributing to shear resistance through flexion and compression. However, beyond the 2 mm deformation point, the corrugations plastically deform and start to flatten. At this stage, the corrugations’ ability to distribute stresses is compromised, and the internal structure of the cardboard begins to fail. The applied tangential forces are no longer effectively distributed through the compression of the corrugation, resulting in localised permanent deformation.

Moreover, the abrasive surface used to ensure adhesion between the material and the testing rig plays a significant role during the initial stages of the test. This adhesion minimises slippage and allows for an efficient transfer of tangential forces onto the corrugated cardboard. However, once the corrugations flatten beyond 2 mm, the degree of adhesion becomes less relevant, as deformation no longer occurs through corrugation flexion, but rather through the sliding of the indenter over the material’s surface. This suggests that frictional forces between the surfaces reach a stabilisation point, and further cardboard deformation does not result in a proportional increase in shear force.

In conclusion, the behaviour of the corrugated cardboard under tangential stresses transitions from an elastic phase, where the corrugation actively resists through flexion and compression, to a plastic phase, where the corrugation collapses and progressively flattens. After 2 mm of deformation, the material is no longer capable of sustaining additional stresses, and the shear forces reach a state of stabilisation. This indicates structural failure of the corrugation, explaining the behaviour observed in the experimental data. Next, the variation in the tangential force for the chosen mounting and for a single normally applied load, Fz = 100 N, for all specimen variants, “Original” to 35EB, was investigated.

Figure 9 presents the deformation results of the corrugated cardboard samples under a normal load of 100 N. The figure illustrates the differences in deformation across various samples, providing a visual comparison of their structural integrity under identical loading conditions, with the samples after testing shown on the left side and the samples before testing on the right side. The observed deformations highlight the varying stiffness and load-bearing capacities of the different board configurations. Specifically, samples with greater thickness and more pronounced corrugation patterns demonstrate less deformation, indicating higher resistance to compressive forces.

Figure 10 shows the deformation test results of the samples under tangential loads for a normal force load Fz = 100 N. This graduation of the deformation allows for a more detailed viewing of the behaviour of the samples and the deformation transition that occurs through them. What can be said about the general behaviour of the samples is that the first elastic behaviour of the material corresponds with a displacement between (0 and 1.8 mm). It can also be observed that the point of inflection is taking place in all samples in this zone of behaviour at a small deformation distance (0.7–0.9 mm). This can be interpreted by the fact that initially, there is a zone of elastic deformation (a quasi-static deformation) and then the tangential displacement is triggered.

A second zone of behaviour is that of strong deformation; in other words, non-linear behaviour on the curves of the samples—a zone in which the bonding between the fluting and the liners is also stressed. A third zone is that of plastic behaviour, i.e., the deformation of the corrugation until its position in the sandwich package is stable and follows what is practically a plateau behaviour. The tangential load graph has a similar appearance to a stick–slip phenomenon, this being in accordance with the quality and texture of the liner surface.

The sample with the smallest dimensions, h_tot = 1.530 mm and wave 2p = 3.500 mm, is the sample 11E. It is obvious that for this one, the transition from elastic to plastic behaviour is very fast and for a minimum tangential load (Fx = 46 N). The C3211,2B specimen has a dimensional structure with a total thickness h_tot almost double that of 11E and double the wavelength, 2p = 6 mm. It is evident that the elastic behaviour is more pronounced for this sample, taking into account that the upper liner thickness t₂ is higher by 30%. At the same time, the transition zone to the plastic deformation is much smoother, though the force value for the shear deformation tangential force is approximately the same.

The other samples have rather the same behaviour, but the value of the tangential force is much higher. The common height value is between h_tot = (3–4) mm. Note the behaviour of the 28,5C sample having a wavelength of 2p = 3.6 mm, so it is smaller than the others in this group, which means it has the smallest tangential force value. The double-face 35EB sample behaviour is also very interesting, as the tangential force value at deformation is about the same value as the samples in this group. The elastic behaviour is much improved, also due to the double sandwich effect, meaning double corrugation, even if the size h_tot is still about equal to that of the other specimens. It is also of interest to note the value after which the maximum value of the tangential force is reached, after which the specimen enters a sliding motion, with the friction coefficient practically tending to the unit value.

The distribution of the maximum values of the tangential force for the all the specimens is shown in Figure 11. Also noted is the tangential displacement distance at which these values occur. The displacement is an important parameter because it directly relates to the deformation behaviour of the corrugated cardboard under tangential stresses, allowing for a detailed observation of the cardboard’s ability to resist shear forces.

In Table 3, the analysis of the measurement results reveals that the datasets 11E, 21C, and 35EB exhibit high consistency and stability. These datasets have low standard deviations and coefficients of variation, indicating minimal measurement errors and highly precise readings. In contrast, dataset 28,5C shows moderate variation, with a slightly higher standard deviation and an increased coefficient of variation, suggesting greater variability between measurements but still within acceptable limits. Dataset 24,5C stands out due to its significant standard deviation and high coefficient of variation, pointing to considerable variability and more pronounced measurement errors, which may be attributed to inconsistencies or differences between specimens. Dataset 21C demonstrates moderate variation, with measurement errors that are higher than those for the original sample and C3211,B but lower compared to 24,5C. Lastly, dataset 11E, despite having a small standard deviation, shows a high coefficient of variation, indicating considerable relative variation in the context of small values. This suggests that although the absolute measurements are fairly consistent, the relative errors are more significant. Overall, these observations provide a comprehensive view of the precision and reliability of measurements across different datasets, highlighting variations in measurement consistency and potential sources of error.

4. Model Validation and Finite Element Analysis

The mechanical behaviour of corrugated cardboard under tangential stresses was investigated in this study.

In order to validate the experimental study carried out in this work, a finite element analysis was performed using Ansys code. The analysis was performed for a single specimen, namely the “Original” specimen, which was subjected to different applied forces. For the numerical study, Hex 20-type finite elements were used, i.e., Solid 186 elements, which allow for the study of Ux, Uy, and Uz displacements. The behaviour was studied for the isotropic elastic model.

The material properties for the skin (liner) and cardboard (fluting) are as described in Section 2.1. Figure 12 shows that the deformation is approximately equal in size to that obtained in the experimental study. From Figure 8, it can be seen that at the same force, the tangential displacement of the specimen in its upper part is approximately linear up to a value of about 2.2 mm. In the numerical model, this displacement, due to the simplifying assumptions used, is obtained at a value of 1.856 mm, from which, as can be seen from the figure below, a “collapse” of the model occurs, and the elastic deformation range is exceeded.

After validating the model, an evaluation of the stress states was performed using finite element analysis. Figure 13 illustrates the tangential stress state for the given data and applied loads from the first moment of applied tangential force forth. The analysis reveals that the maximum tangential stress observed is 6.1 MPa. However, within the corrugated layer, the stress distribution shows a notable green region, representing approximately 2.7136 MPa.

The stress values within the corrugated structure vary, with the observed range extending from 1.3568 MPa to 4.0704 MPa. This range indicates that while the maximum stress is relatively high, the majority of the corrugated layer experiences lower stress levels. The green zone, which represents 2.7136 MPa, is significant, as it highlights areas of relatively lower but still substantial stress within the corrugated structure.

The detailed stress distribution is critical for understanding the material’s behaviour under tangential loads, as it helps identify regions where stress concentrations might occur and where material performance is crucial. By analysing these stress values, it can be better understood how the corrugated structure responds to loading and the design can be optimised to improve its structural integrity and performance.

The von Mises stress state analysis was conducted to evaluate the structural behaviour of the specimen under normal loading, with the results presented in Figure 14. The maximum observed von Mises stress value was 18.441 MPa, indicating the highest level of stress the material is subjected to. In contrast, the yellow region in the figure, located after the bonding area and representing a stress of approximately 12.294 MPa, indicates a significant stress level but one that is lower than the theoretical maximum. This value was derived from the finite element analysis (FEA) performed in the study, which simulated the material’s response under the applied loads. The 12.294 MPa stress represents an area of high stress within the corrugated structure, yet it remains below the peak stress values observed elsewhere in the model. It is important to note that this value was not directly compared to experimental results but rather serves as an intermediate stress level within the simulation’s stress distribution, providing insight into the material’s behaviour beyond the bonding area.

To better understand the significance of this value, the stress value was calculated to be 12.294 MPa, which represents approximately 66.6% of the maximum value of 18.441 MPa. This highlights that while the stress in the yellow region is considerable and approaches the material’s performance limits, it does not reach the maximum theoretical stress.

This interpretation suggests that the yellow region represents an area of high stress but not critical, which is essential for identifying potential failure points and making necessary design adjustments. Comparing these results with experimental data is crucial for validating the model and ensuring that simulations accurately reflect the material’s real behaviour. This detailed analysis contributes to optimising the corrugated cardboard, ensuring it maintains structural integrity under varying loading and usage conditions.

5. Discussion

The mechanical behaviour of corrugated cardboard under tangential stresses was investigated in this study, providing valuable insights into its structural integrity and performance during transportation and storage. Several key observations were made as follows: the mounting technique evidently had great influence over the tangential stress response of the corrugated cardboard. The data analysis shows that the use of abrasive material as a mounting technique gives repeatable and reliable results in contrast to binding with superglue. This is probably due to better engagement and less slip from the abrasive surface, which create a condition that keeps the specimen steady throughout the testing. This again shows the importance of using proper mounting techniques with regard to mechanical testing so that the results are reliable and repeatable.

The linear relation with an increase in the normal force was then established, followed by a non-linear complex interaction at higher forces. The behaviour of this result would suggest corrugated cardboard should show a predictable response under low to moderate normal forces, representing possibly an important feature for the packaging design to withstand standard loading conditions. The response of the material to higher forces is much more complex with the interaction of compressive, shearing, and adhesive forces in the corrugated structure. This fact highlights the need for detailed testing and modelling so that a prediction of cardboard performance under varying load conditions is correctly made.

Three distinct regions of deformation behaviour were then identified, an initial elastic region, a transition zone, and a plastic region. During the initial elastic region, the material will deform proportionally to the applied force but without permanent deformation. Through the transition zone, the bond between the fluting and liners is stressed and non-linear behaviour starts. Finally, in the plastic region, the material deforms permanently until reaching a stable configuration. Such an understanding of the mechanical limits of corrugated cardboard and different stages of deformation shall be crucial in designing packaging that will absorb the impacts and stresses without failure.

It was concluded that corrugated cardboard exhibits significantly greater stiffness in the direction of the corrugations. The worst-case loading scenario was investigated, focusing on the direction perpendicular to the corrugations. By examining this less favourable orientation, the material’s mechanical properties and strength endurance under challenging conditions were rigorously assessed, providing critical insights into its performance limits. The specimens with different corrugation patterns and thicknesses underwent different responses to the applied tangential stresses. For example, the smallest sample underwent a very rapid transition from elastic to plastic behaviour, showing that the resistance to tangential forces was small. The corresponding transition was much smoother for the thicker samples with more pronounced corrugation patterns and was also more enhanced, elastic, and more resistant to tangential forces. In other words, the results obtained underline the need for design modifications toward challenging performance requirements.

Future work details must focus on the dynamic behaviour of corrugated cardboard under changing environmental conditions, like the cycles of humidity and temperature, which compromise their mechanical properties. Further studying the long-term effects of cycles of loading and unloading may furnish an understanding of corrugated cardboard fatigue behaviour. Such advanced modelling techniques, like finite element analysis (FEA), are also applied to simulate complex loading scenarios and predict the performance of cardboard packaging in the service state.

6. Conclusions

Some important and experiment-related key findings are as follows: the tangential stress response of the corrugated cardboard is highly dependent on the mounting method and the applied normal force. The more consistent results obtained show that mounting with abrasive adhesion is better than those obtained with bonding with superglue. The normal force versus tangential force has a linear trend at the beginning but shifted to a non-linear complex interaction for higher forces.

The study established deformation behaviour with three different regions, an initial elastic region, a transition zone, and a plastic region. The results show that the design of corrugated board packages must pay attention to the direction of the corrugation and the cardboard’s special mechanical properties. The results prove that strength and durability of the corrugated cardboard are highly orientation dependent in a mechanical sense. Cardboard thickness and patterns of the corrugation are also very influencing factors in the mechanical performance. Design solutions showed sensitivity in the toroidal direction; in other words, they were different for some patterns and thicknesses, allowing for optimisation.

In conclusion, the work gives insight into the mechanical behaviour of corrugated cardboard under tangential stresses and provides a robust methodology for future studies. The findings should assist in the design of new packaging solutions for better protection and improved durability of the selected goods during transport. Future research in this area should investigate dynamic loading and the influences of environmental conditions in further enhancing the reliability of corrugated cardboard packaging solutions.

Author Contributions

Methodology, F.I.B. and M.B.; Software, D.A.P. and M.S.; Formal analysis, A.C.; Data curation, S.C., D.A.P. and M.S.; Writing—review & editing, A.C.; Funding acquisition, D.A.P. and M.S. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Data are contained within the article.

Acknowledgments

We gratefully thank the PubArt—Program for their support of the publication of scientific articles and the communications of the National University of Science and Technology, Politehnica, Bucharest.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. The main directions of a CO3 corrugated cardboard.

Figure 2. The main dimensions of a CO3 corrugated cardboard sample.

Figure 3. CETR (Bruker) UMT-2 testing rig.

Figure 4. Corrugated cardboard sample gripping device.

Figure 5. The operation mode of the clamping system.

Figure 6. Tangential force as a function of sample mounting.

Figure 7. Friction coefficient as a function of sample mounting.

Figure 8. Tangential force Fx variation in “ORIGINAL” sample with normal load Fz applied.

Figure 9. Samples undeformed and deformed after normal load Fz = 100 N.

Figure 10. Tangential force at normal load Fz = 100 N, v = 0.5 mm/s.

Figure 11. Distribution of maximum values of tangential force function of length displacements.

Figure 12. Deformation results for “Original” specimen at Fx = 100 N.

Figure 13. Shear stress state.

Figure 14. Von Mises stress state.

Table 1

Main dimensions of the samples used in the experiment.

Samples	t1	t2	tc	htotal	h	hc	2p	nr 2p
	(mm)	(mm)	(mm)	(mm)		(mm)	(mm)	-
Original	0.300	0.200	0.170	3.000	2.850	2.500	6.000	3.000
C3211,2B	0.130	0.130	0.150	2.800	2.740	2.540	6.000	3.000
28,5C	0.241	0.241	0.169	4.000	3.880	3.520	7.500	3.000
24,5C	0.200	0.170	0.200	3.800	3.700	3.430	7.500	3.000
21C	0.160	0.160	0.130	3.700	3.620	3.380	7.000	3.000
11E	0.120	0.100	0.170	1.530	1.470	1.310	3.500	3.000
35EB	0.170	0.170	0.130	4.000	3.830	3.660	3.500	5.000
	0.170		0.150				6.000	3.000

Table 2

Results.

Sample	Test 1 Displacement [mm]	Test 2 Displacement [mm]	Test 3 Displacement [mm]	Test 4 Displacement [mm]	Test 5 Displacement [mm]	Sustained Tangential Force [N]
Original	7.011	7.024	7.067	7.051	7.074	$\approx 57$
C3211,2B	6.954	6.994	7.022	7.011	7.016	$\approx 86$
28,5C	7.005	7.000	6.897	6.983	7.002	$\approx 72$
24,5C	6.898	6.862	6.995	6.988	6.692	$\approx 77$
21C	6.098	6.298	6.228	6.128	6.082	$\approx 76$
11E	1.899	1.832	1.940	1.917	1.967	$\approx 49$
35EB	6.911	6.954	6.991	6.889	6.996	$\approx 79$

Table 3

Statistical analysis of test measurements.

Specimen n = 5	Mean (x)	Standard Deviation (s)	Coefficient of Variation (ν)	Minimum (min)	Maximum (max)
Original	7.0456 mm	0.026 mm	0.37%	7.001 mm	$7.074$ mm
C3211,2B	7.0006 mm	0.028 mm	0.40%	6.954 mm	$7.022$ mm
28,5C	6.9774 mm	0.043 mm	0.61%	6.897 mm	7.050 mm
24,5C	6.8870 mm	0.126 mm	1.83%	6.692 mm	$6.995$ mm
21C	6.1660 mm	0.089 mm	1.44%	6.082 mm	$6.298$ mm
11E	1.9110 mm	0.053 mm	2.77%	1.832 mm	$1.967$ mm
35EB	6.9482 mm	0.039 mm	0.56%	6.889 mm	6.996 mm

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