*Article* **Investigation of the Effect of Pallet Top-Deck Stiffness on Corrugated Box Compression Strength as a Function of Multiple Unit Load Design Variables**

**Saewhan Kim <sup>1</sup> , Laszlo Horvath 1, \* , Jennifer D. Russell <sup>1</sup> and Jonghun Park 2**


**Abstract:** Unit loads consisting of a pallet, packages, and a product securement system are the dominant way of shipping products across the United States. The most common packaging types used in unit loads are corrugated boxes. Due to the great stresses created during unit load stacking, accurately predicting the compression strength of corrugated boxes is critical to preventing unit load failure. Although many variables affect the compression strength of corrugated boxes, recently, it was found that changing the pallet's top deck stiffness can significantly affect compression strength. However, there is still a lack of understanding of how these different factors influence this phenomenon. This study investigated the effect of pallet's top-deck stiffness on corrugated box compression strength as a function of initial top deck thickness, pallet wood species, box size, and board grade. The amount of increase in top deck thickness needed to lower the board grade of corrugated boxes by one level from the initial unit load scenario was determined using PDS™. The benefits of increasing top deck thickness diminish as the initial top deck thickness increases due to less severe pallet deflection from the start. The benefits were more pronounced as higher board grade boxes were initially used, and as smaller-sized boxes were used due to the heavier weights of these unit loads. Therefore, supposing that a company uses lower stiffness pallets or heavy corrugated boxes for their unit loads, this study suggests that they will find more opportunities to optimize their unit loads by increasing their pallet's top deck thickness.

**Keywords:** corrugated box; compression strength; pallet; unit load; unit load optimization

## **1. Introduction**

Historically, the distribution packaging industry has adapted the method of unitizing single, multiple, or bulk products on a solid platform to make the handling, storing, and transporting of these products easier [1]. This arrangement is called a unit load. In today's supply chains, 80% of products are moved in unit load form [2]. The most common base platform for unit loads is a pallet. Pallets can be made of different materials such as wood, plastic, paper, or metal. Among these materials, wood is by far the most commonly used to manufacture pallets. Wood is the material of choice for over 90% of companies that use pallets in their supply chains in the United States [3]. Furthermore, approximately 804 million new and recycled wood pallets were manufactured in 2016 [4]. Just as wood pallets have become one of the essential elements of a unit load, corrugated boxes also play a crucial role. Corrugated boxes are the most used primary and secondary packaging; in fact, 72% of unit loads are built using corrugated boxes [3].

When designing a unit load, accurately predicting corrugated box compression strength is crucial to avoid package failure from the vertical compression forces during distribution and storage. Therefore, numerous studies have investigated the factors that

**Citation:** Kim, S.; Horvath, L.; Russell, J.D.; Park, J. Investigation of the Effect of Pallet Top-Deck Stiffness on Corrugated Box Compression Strength as a Function of Multiple Unit Load Design Variables. *Materials* **2021**, *14*, 6613. https://doi.org/ 10.3390/ma14216613

Academic Editor: Tomasz Garbowski

Received: 4 October 2021 Accepted: 30 October 2021 Published: 3 November 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

affect the compression strength of corrugated boxes, including material properties [5–10], manufacturing methods [6,11–16], environmental condition factors [6,8,12,17], and the palletization factor [18–26].

Wood pallet characteristics, such as pallet gaps and pallet overhang, have been included among the main palletization factors that affect box compression strength. In relatively recent years, researchers have endeavored to correlate pallet top-deck stiffness to corrugated box compression strength. Baker [19] and Phanthanousy [24] examined the relationships between the differences in stress concentrations and box compression strength. However, their studies were inconclusive. Phanthanousy found that the stiffness of the pallet's top deck has no notable effect on box compression strength when the wood pallet is designed with deck board gaps.

Meanwhile, Baker [19] found that pallet top-deck stiffness significantly affects box compression strength when the wood pallet is designed with no deck board gaps. Their studies only evaluated situations in which all corners of the boxes were symmetrically supported. However, in many cases, the top deck board of a wood pallet deforms by the weight of the top load and creates asymmetric support conditions for the loaded products. Baker [19] highlighted that asymmetrically supported corrugated boxes are a prevalent condition in most unit loads, and his research found that asymmetric support can decrease corrugated box compression strength by as much as 15%.

In 2020, Quesenberry et al. [25] further investigated the effect of wood pallet top-deck stiffness on corrugated box compression strength when box corners are asymmetrically supported. They concluded that a stiffer top deck board could increase the compression strength of asymmetrically supported corrugated boxes up to 37% when the unit loads are double-stacked on the floor [25]. They also discovered that the effect of pallet top-deck stiffness on box compression strength could be utilized to lower the cost of a unit load by decreasing the required board grade of corrugated boxes and increasing the pallet's top deck thickness. However, the experimental design utilized by Quesenberry only focused on a limited number of variables. Furthermore, the pallet design utilized for his experimental unit load consisted of a single wood species and singular moisture content.

Additionally, the corrugated boxes were made of a single board grade, two flute sizes, and two box sizes. In practice, many wood species with varying moisture content are available for pallet manufacturing; meanwhile, corrugated boxes are produced in multiple board grades and sizes. Nevertheless, there is an absence of studies investigating how these variations may change the effect of top deck stiffness on corrugated box compression strength.

Therefore, the objective of this current paper is to investigate the effect of pallet topdeck stiffness on the compression strength of asymmetrically supported corrugated boxes as a function of currently under-studied variables, including initial top deck thickness, pallet wood species, box size, and board grade.

#### **2. Materials and Methods**

This study consisted of two main sections: validation of the analytical pallet design software and unit load scenario analysis.

#### *2.1. Software Validation*

The commercially available pallet design software Pallet Design System™ (PDS™) v. 6.2, created by NWPCA (National Wooden Pallet & Container Association, Alexandria, VA, USA) was utilized to replace numerous physical experiments in this study. The box performance data predicted by PDS™ and that Quesenberry et al. [25] found were compared to confirm that the software reproduced the results from the experiment.

#### 2.1.1. Corrugated Box Description for Validation

The same designs of corrugated boxes used by Quesenberry et al. [25] were used to build the unit load model in PDS™ for predictive software validation. Specified parameters from Quesenberry et al. [25] included: Regular Slotted Container (RSC) style with two different external dimensions (length × width × height) 406.4 mm × 247.7 mm × 254 mm and 609.6 mm × 247.7 mm × 254 mm. Unit loads were built with four layers of boxes, and the configuration of boxes was either 3 boxes × 4 boxes (length × width) or 2 boxes × 4 boxes. Both sizes of boxes were built with nominal 0.57 kg/mm Edge Crush Test (ECT) value B-flute and C-flute corrugated board.

#### 2.1.2. Pallet Description for Validation

Quesenberry et al. [25] simulated a 1219.2 mm × 1016 mm GMA™ (Grocery Manufacturers Association) style pallet by using a custom-built, quarter-section pallet for testing purposes. For software validation, a full-sized 1219.2 mm × 1016 mm stringer class, double face, non-reversible, partial four-way, unidirectional bottom, flush, GMA™ style pallet was modeled in PDS™ (see Figure 1). The pallet consisted of three stringers, seven top deck boards, five bottom deck boards, and two fasteners per joint. The stringers were 1219.2 mm long, 31.8 mm wide, and 88.9 mm high. The top and bottom deck boards were 1016 mm long and 88.9 mm wide. The four top deck board thicknesses studied were: 9.5 mm, 12.7 mm, 15.9 mm, and 19.1 mm. All top deck boards were equally spaced 99.6 mm apart. Lead bottom deck boards were spaced 292.1 mm away from the interior bottom deck boards, and the interior bottom deck boards were spaced 95.3 mm apart. Number 1 & better (premium & better), kiln-dried, Spruce–Pine–Fir (SPF) lumber was used for all pallet components.

**Figure 1.** Picture of GMA pallet used for software validation (image generated using PDS™).

2.1.3. Comparison of Box Load Factor and Box Compression Strength Factor

During software validation, the box load factors computed by PDS™ and the box compression strength factor derived from the thesis of Quesenberry [27] were compared. The box load factor is the ratio of the weights on worst loaded box edges to the load if it were evenly distributed. Meanwhile, the box compression strength factor is a new term developed by the authors and is defined as the ratio of the box compression strength when box corners are symmetrically supported on rigid supports to the box compression strength when its corners are asymmetrically supported on an actual pallet. Both the box load factor and the box compression strength factor ultimately provide information about the compression performance of the corrugated box.

Process of Computing Box Load Factor

Box load factors were computed using PDS™ following the steps described below. *Step 1:* Built a unit load in PDS™ using boxes and pallets previously described in Sections 2.1.1 and 2.1.2.

*Step 2:* Set the top deck board thickness to the lowest level (9.5 mm).

*Step 3:* Set the weight in the box to the load that will just fail the boxes (box safety factor of one) when the support condition is a single floor stack.

*Step 4*: Report current box load factor when support condition is single floor stack.

*Step 5*: Increased the top deck board thickness to the following levels (12.7 mm, 15.9 mm, and 19.1 mm).

*Step 6*: Repeat *steps 3* and *4* for each level of top deck board thickness.

*Step 7:* Repeat the process for two flute sizes (B and C flute) and two box sizes.

Process of Calculating Box Compression Strength Factor

The box compression strength factor from Quesenberry's study was calculated using Equation (1):

$$\text{CSF} = \frac{\text{SCS}\_{\text{avg}}}{\text{ACC}\_{\text{avg}}} \tag{1}$$

where:

*CSF* = Box compression strength factor.

*SCSavg* = Average box compression strength when box corners are symmetrically supported on a rigid platform.

*ACSavg* = Average box compression strength when box corners are asymmetrically supported on the actual pallet.

The unit load scenarios used to calculate the box compression strength factors were varied by two flute sizes, two box sizes, and four thickness levels.

#### Statistical Analysis

The independent *t*-test was conducted to see whether the difference between box load factors from PDS™ and box compression strength factors from the experiment were statistically significant or not. To confirm the normality assumption of the independent *t*-test, we also ran the Shapiro–Wilk test for each group separately. The similarities between the box performance data from PDS™ and the experiment were also assessed using the Pearson correlation coefficient. The Pearson correlation coefficient is a way to investigate linear dependence between two variables. The measured correlation coefficient (*r*) ranges between −1 and +1. When the *r*-value is −1, it indicates a strong negative correlation, while +1 indicates a strong positive correlation, and 0 means no relation. Both statistical analyses were conducted at a significance level of 0.05. The analyses were done using SAS JMP Pro 15® software (SAS Enterprises, Raleigh, NC, USA).

#### *2.2. Unit Load Scenario Analysis*

The concept of a unit load cost optimization method that allows for corrugated boxes with decreased board grades by increasing the pallet's top deck thickness was adopted from Quesenberry et al. [25] to modify each unit load scenario. In other words, the analysis was done by determining how much the top deck thickness needed to increase to lower the corrugated board grade by one level from the initial unit load scenario's specific deck board thickness and board grade. A total of 234 unit load scenarios were designed with varying factors for investigation.

#### 2.2.1. Corrugated Box Description for Unit Load Scenario Analysis

Three sizes of RSC-style corrugated boxes were investigated to explore the effect of different box sizes. Three box sizes were chosen that would cover the entire top surface of the 1219.2 mm × 1016 mm pallet and create asymmetrically supported corners. The external dimensions were 203.2 mm × 304.8 mm × 254 mm (small box), 406.4 mm × 254 mm × 254 mm (medium box), and 609.6 mm × 337.8 mm × 254 mm (large box). The boxes were organized in 4 × 5, 3 × 4, and 2 × 3 arrays for small, medium, and large boxes, respectively. Four layers of boxes were used for each unit load. Unit load configurations using the different box sizes are depicted in Figure 2. The boxes were built with two different flute sizes: single-wall C-flute and double-wall BC-flute. The

C-flute and BC-flute corrugated boards were made of commonly manufactured board grades for each flute size. C-flute boards were modeled with nominal 0.52 kg/mm., 0.57 kg/mm, 0.71 kg/mm, and 0.79 kg/mm ECT. BC-flute boards were modeled with nominal 0.86 kg/mm, 0.91 kg/mm, 1.09 kg/mm, and 1.27 kg/mm ECT.

**Figure 2.** Image of investigated unit load configurations (image generated using PDS™). (**a**) Unit load with small boxes, (**b**) unit load with medium boxes, and (**c**) unit load with large boxes.

### 2.2.2. Pallet Description for Unit Load Scenario Analysis

For the unit load scenario analysis, the most common size of GMA™ style wood pallet was used. The 1219.2 mm × 1016 mm GMA™ style stringer class wooden pallet is the most commonly used pallet design in North America [28,29]. The specifications were 1219.2 mm × 1016 mm, stringer class, double face, non-reversible, partial four-way, unidirectional bottom, flush, GMA™ style pallet (see Figure 3). The pallet design had three stringers, two lead top deck boards, five interior top deck boards, five bottom deck boards, two fasteners per joint on the interior top deck boards and for all bottom deck board connections, and three fasteners per joint on the lead top deck boards. The pallet design utilized for the unit load scenario analysis (Figure 3) had 50.8 mm wider lead top deck boards than the pallet design used for the software validation process (Figure 1). The spacing between top deck boards has also changed accordingly. The stringers were 1219.2 mm long, 31.8 mm wide, and 88.9 mm high. The interior top deck boards and bottom deck boards were 1016 mm long and 88.9 mm wide. The lead top deck boards were 1016 mm long and 139.7 mm wide. The bottom deck boards were 9.5 mm thick. All top deck boards were spaced 82.6 mm apart. The lead bottom deck boards were spaced 292.1 mm apart from interior bottom deck boards, and interior bottom deck boards were spaced 95.3 mm apart from each other. Number 1 & better (premium & better) grade lumber was used for all pallet components.

**Figure 3.** Picture of GMA pallet used for analysis (image generated using PDS™).

Initial top deck thicknesses were varied by four levels to explore which changes in deck board thicknesses would be required to reduce by one level the initial board grade specified for the corrugated boxes. The investigated initial top deck thickness levels were 9.5 mm, 12.7 mm, 15.9 mm, and 19.1 mm. However, unit load scenarios built with kilndried southern yellow pine (KD SYP) pallets were designed with 11.1 mm top deck boards, and this thickness was increased to 17.5 mm for the optimized design. This limitation was due to the availability of raw material sizes; only the 11.1 mm and 17.5 mm dimensions could be manufactured effectively.

Wood species used for pallet construction were also varied. The wood species commonly used for pallet construction in the southeastern United States were selected. Selected wood materials were: green, high-density hardwood (Grn HD HW); green, low-density hardwood (Grn LD HW); green, southern yellow pine (Grn SYP); and kiln-dried, southern yellow pine (KD SYP). Green lumber contained 25% or greater moisture content, and kiln-dried lumber had a maximum of 19% moisture content.

#### 2.2.3. Variable Factors

Several factors of the unit load were varied to identify the characteristics that could change the effect of the pallet's top deck board stiffness on box compression strength. The factors evaluated were initial top deck board thickness, pallet wood species, box size, and corrugated board grade. The variable factors that relate to pallets are listed in Table 1, and Table 2 contains the variable factors relating to the boxes.


**Table 1.** Summary table of variable factors related to pallets.

**Table 2.** Summary table of variable factors related to corrugated boxes.


#### 2.2.4. Analysis Method

Measurement of Top Deck Thickness Increase

The unit load cost optimization method adopted from Quesenberry et al. [25] was investigated by varying the factors introduced in Section 2.2.3. The change in top deck thickness required to reduce by one level the corrugated board grade used, without downgrading box performance, was measured. This analysis was done with the unit load in the double floor stacked condition. A box safety factor of 3 was selected for the unit load design to comply with the requirements of the ISTA 3E testing standard [30].

Required steps in the analysis were as follows:

*Step 1:* Construct the unit load in PDS™.

*Step 2:* Set pallet material as one of the listed wood species (i.e., green high-density hardwood). *Step 3:* Set the top deck board as the lowest initial top deck board thickness (9.5 mm). In the case of KD SYP, always set the initial top deck thickness as 11.1 mm.

*Step 4:* Set corrugated boxes as the higher ECT values in the selected range of board grade (i.e., Choose 0.57 kg/mm if the range was decreasing from 0.57 kg/mm to 0.52 kg/mm).

*Step 5:* Determine the weight of the box that works to create a box safety factor of three for the double floor stacking condition.

*Step 6:* Create a new unit load with the corrugated boxes made of lower ECT value from the selected range of corrugated board grade and apply the weight determined in *step 5* (i.e., Select 0.52 kg/mm if the range was 0.57 kg/mm to 0.52 kg/mm).

*Step 7:* Continuously increase the top deck thickness by 1.6 mm until the unit load again reaches the safety factor of three for safe operation. In the case of KD SYP, always increase the top deck thickness to 17.5 mm.

*Step 8:* Report the total increase in the top deck board thickness required to achieve the required safety factor of three.

*Step 9:* Repeat *step 1* to *step 8* after changing the pallet wood species.

*Step 10:* Repeat from *step 1* to *step 9* after increasing the initial top deck stiffness level.

*Step 11:* Repeat from *step 1* to *step 10* after changing the range of board grade (i.e., changing from a range of 0.57–0.52 kg/mm to a range of 0.71—0.57 kg/mm).

*Step 12:* Repeat from *step 1* to *step 11* after changing the size of corrugated boxes (i.e., changing from a small to a medium size box).

#### Unit Load Scenario Classification System

The amount that the top deck thickness increased was categorized as one of three grades to make it easier to identify which scenarios had smaller or larger increases in top deck thickness: less than 12.7 mm (grade 1), 12.7 mm to 25.4 mm (grade 2), and beyond 25.4 mm increase (grade 3). For better visualization, a color-coding system was also applied; green for grade 1, yellow for grade 2, and red for grade 3. Grade 1 scenarios were considered as cases with high potential to apply the unit load optimization process. Grade 2 scenarios were considered cases that may be possible to apply the optimization method depending on the manufacturer's circumstances. Because pallets made of deck boards thicker than 25.4 mm are unprecedented; grade 3 scenarios were considered unrealistic unit load designs.

#### **3. Results and Discussion**

#### *3.1. Software Validation Results*

Measurement of the box load factors and box compression strength factors on varied top deck thicknesses, box sizes, and flute sizes are presented in Table 3. The comparison of box load factors and box compression strength factors is plotted in Figure 4. It was observed that PDS™ tends to overestimate the effect of top deck stiffness when compared to the experiment results. However, the independent *t*-test showed that the difference between PDS™ and the experiment was not statistically significant (*t*(25) = −0.85, *p*-value = 0.40). The Shapiro-Wilk test confirmed that the normality assumptions were met (PDS: W = 0.927, *p*-value = 0.216; Quesenberry: W = 0.919, *p*-value = 0.160). Furthermore, the Pearson correlation coefficient revealed a strong positive correlation, *r* = 0.911 (*p*-value < 0.0001), between box load factors from PDS™ and box compression strength factors from experiment results. In other words, the PDS™ and Quesenberry's [27] experiments had a similar pattern.


**Table 3.** Summary table of box load and compression strength factors.

−

**Figure 4.** Comparison of box load factors and box compression strength factors of each type of boxes in response to pallet top-deck thickness. (**a**) Small C-flute box scenarios, (**b**) shows large C-flute box scenarios, (**c**) shows small B-flute box scenarios, and (**d**) shows large B-flute box scenarios.

#### *3.2. Unit Load Scenario Analysis Results*

Tables 4 and 5 report the amount of top deck board thickness increase required to reduce the corrugated board grade by one level as a function of starting top deck thickness, wood species, initial board grade, and box sizes for the unit loads consisting of C-flute boxes and BC-flute boxes, respectively. A streamlined grading system has been applied, as described in Section 2.2.4, for better visualization and identification of the level of top deck thickness increase. The top deck thickness increase for grade 3 scenarios was reported as N/A (not applicable) because adding an extra inch of thickness to a pallet deck board is highly cost-prohibitive.


**Table 4.** The amount of top deck board thickness required to optimize unit loads consisting of C-flute boxes.

Grade 1: less than 12.7 mm (green), Grade 2: 12.7 mm to 25.4 mm (yellow), Grade 3: beyond 25.4 mm increase (red). Note: Grn HD HW: green high-density hardwood, Grn LD HW: green low-density hardwood, Grn SYP: green southern yellow pine.

**Table 5.** The amount of top deck board thickness required to optimize unit loads consisting of BC-flute boxes.


Grade 1: less than 12.7 mm (green), Grade 2: 12.7 mm to 25.4 mm (yellow), Grade 3: beyond 25.4 mm increase (red). Note: Grn HD HW: green high-density hardwood, Grn LD HW: green low-density hardwood, Grn SYP: green southern yellow pine.

> Tables 6 and 7 present the KD SYP scenarios' amount of top deck board thickness increase required to reduce the corrugated board grade by one level as a function of the different factors for the unit loads built using C-flute and BC-flute boxes, respectively.


**Table 6.** The amount of top deck board thickness required to optimize unit loads consisting of KD SYP pallet and C-flute boxes.

Note: The deckboard thickness sizes available for kiln-dried southern yellow pine (KD SYP) were limited because the available raw material size only allows the cost-effective production of 11.1 mm and 17.5 mm deckboard thicknesses.

**Table 7.** The amount of top deck board thickness required to optimize unit loads consisting of KD SYP pallet and BC-flute boxes.


Note: The deckboard thickness sizes available for kiln-dried southern yellow pine (KD SYP) were limited because the available raw material size only allows the cost-effective production of 11.1 mm and 17.5 mm deckboard thicknesses.

> To investigate how different factors such as the initial top deck board thickness, pallet wood species, box size, and board grade effect the feasibility of optimizing the strength of the corrugated boxes by changing the stiffness of the pallets, researchers looked at the changes in the proportions of different grade scenarios in response to each variable factor.

> Figure 5 shows how the proportions of various grade scenarios changed when different initial top deck thicknesses were used for the pallet design. As the initial top deck thickness increased, there was a significant reduction in the proportion of grade 1 scenarios. These are the scenarios where it is highly feasible to reduce the corrugated board grade with a reasonable amount of top deck thickness change. The proportion of grade 1 scenarios started from 78% with 9.5 mm initial top deck thickness and decreased to 50%, 24%, and 4% when the initial top deck thickness was 12.7 mm, 15.9 mm, and 19.1 mm, respectively. Correlatingly, the ratio of grade 3 scenarios was almost inversely proportional to the ratio of grade 1 scenarios as the initial top deck thickness increased. The proportion of grade 3 increased from 17% to 31%, 70%, and 91% when the initial top deck thickness was 9.5 mm, 12.7 mm, 15.9 mm, and 19.1 mm, respectively. Unlike other grade scenarios, no consistent trend was found in the proportion of grade 2 scenarios.

> Figure 6 shows the changes in the proportions of the various unit load scenario grades when different wood species were used to build the pallets. The percent of different grade levels were similar for the scenarios using green low-density hardwood and green SYP with around 40% grade 1, 10% grade 2, and 50% grade 3. KD SYP scenarios behaved differently than the other wood species scenarios. They had a much lower number of feasible scenarios than the others. Grade 1 scenarios of KD SYP accounted for only 28%, while grade 1 scenarios of green lumber accounted for between 35–40%. The reduction of feasible scenarios might be attributable to the high stiffness of the KD SYP species. A highly stiff top deck will not bend enough to make a difference in board grade when top deck thickness changes. In addition, the results could have been affected by the limited

availability of various KD SYP thicknesses. KD SYP lumber required a larger jump in top deck thicknesses than the 1.59 mm increases used with green species.

**Figure 5.** Changes in the proportions of the different grade scenarios in response to the initial top deck thickness for green wood scenarios.

**Figure 6.** Changes in the proportions of the different grade scenarios in response to the pallet wood species. Note: KD SYP: kiln-dried southern yellow pine, Grn HD HW: green high-density hardwood, Grn LD HW: green low-density hardwood, Grn SYP: green southern yellow pine.

Furthermore, the proportion of the grade 1 scenarios for Grn HD HW was slightly lower (35%) than the other green lumber scenarios (40–42%). Since Grn HD HW does not have a limit on the level of top deck thickness increase, this could provide further evidence that the stiffness of the material affects the feasibility of the design scenario. Overall, the results indicate that the feasibility of using increased deck board thickness to lower the corrugated boxboard grade decreases when species with higher material stiffness are initially used to construct the pallets.

Similar trends in the proportional changes of different unit load scenario grades were observed from the initial top deck thickness effect and the pallet wood species effect. Both results indicated a significant reduction in the potential to decrease board grade by increasing top deck stiffness when the pallet was initially designed with stiffer pallet wood

material. In other words, this unit load optimization method is more effective when the unit load is initially designed using lower stiffness pallets.

Figure 7 displays changes in the proportions of different grade scenarios as a function of the range of board grade reduction. It was discovered that for the scenarios where the ECT change is greater between the consecutive board grade levels, the proportion of grade 1 scenarios decreases, and the ratio of grade 3 considerably increases. The ratio of grade 1 scenarios ranged between 41% and 82% for the cases with 0.05 kg/mm to 0.08 kg/mm ECT reduction. On the other hand, the proportion of grade 1 scenarios ranged only between 8% and 28% when it required 0.14 kg/mm to 0.18 kg/mm ECT value reduction. These results also show that the higher the initial board grade is, the more opportunities there are to reduce the board material with minor changes to top deck thickness. For instance, the proportion of grade 1 scenarios significantly increased from 41% to 49% and 82% when the board grade reduction range was 0.57–0.52 kg/mm, 0.79–0.71 kg/mm, and 0.91–0.86 kg/mm ECT, respectively. In this analysis, higher board grade also meant that the boxes supported more weight than lower board grade boxes. It indicates that the effect was more prominent for scenarios that had greater unit load weight because having more weight in the boxes causes more bending to the deck boards, which increases stress concentrations on the boxes.

**Figure 7.** Changes in the proportions of the different grade scenarios in response to the range of board grade reduction.

Figure 8 shows changes in the proportions of various unit load scenario grades for the three different box sizes. The proportion of the grade 1 scenarios decreased from 57% to 39% and 21% and the proportion of the grade 3 scenario increased from 38% to 47% and 72% as package size increased from small to medium to large boxes. There was no consistent trend with the proportion of grade 2 scenarios. The results indicated that the feasibility is greater to reduce the corrugated board grade by increasing the thickness of top deck boards for unit loads consisting of small-sized boxes rather than larger ones. Similar to the board grade effect, this trend could be explained by weight differences per unit load. Although each small box held a lighter weight than the medium and large boxes in this analysis, the small box scenarios contained much heavier weight as a whole unit load than the scenarios with larger-sized boxes because these unit loads required more of the small boxes to create the same size load.

Overall, it was found that all investigated variable factors had an observable influence on the feasibility of using an increase in pallet top-deck stiffness to lower the board grade of the corrugated boxes. Unit load scenarios to which it was more feasible to apply the unit load cost optimization method were observed as the initial unit load was designed with less stiff pallet top-deck boards; either thinner top deck boards or lower density wood species. For box-related variables, unit loads of smaller-sized boxes, unit loads with a smaller range of board grade reduction, and unit loads with higher initial board grades all created more favorable situations on which to apply the unit load optimization method due to the heavier weight of these unit loads.

#### **4. Limitations and Assumptions**


#### **5. Conclusions**

The key findings of this study were as follows:


Therefore, this study suggests that companies that use low stiffness pallets or have unit loads of heavy boxes could have more opportunities to optimize their unit loads by increasing the top deck thickness of their pallets.

The study also revealed that changing the top deck board stiffness cannot be done without considering the effects of other factors such as initial top deck board thickness, pallet wood species, box size, and board grade. Therefore, the unit load optimization process that reduces corrugated board grade by increasing top deck stiffness needs to be a holistic process.

The next phase of the project will focus on investigating whether the increase in pallet top-deck stiffness and the resulting reduction in corrugated boxboard grade can create an environmentally beneficial scenario.

**Author Contributions:** S.K. and L.H. developed the research methodology, S.K. conducted the modeling and the analysis, S.K. developed the original draft, L.H., J.P. and J.D.R. reviewed and edited the paper, L.H. finalized the paper. This research project was conducted under the supervision of L.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Industrial Affiliate Program of the Center for Packaging and Unit Load Design at Virginia Tech.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Acknowledgments:** The PDS™ software package used for the data analysis provided by the National Wooden Pallet and Container Association.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Tomasz Garbowski 1 , Anna Knitter-Pi ˛atkowska 2, \* and Aleksander Marek 3**


**Abstract:** The standard edge crush test (ECT) allows the determination of the crushing strength of the corrugated cardboard. Unfortunately, this test cannot be used to estimate the compressive stiffness, which is an equally important parameter. This is because any attempt to determine this parameter using current lab equipment quickly ends in a fiasco. The biggest obstacle is obtaining a reliable measurement of displacements and strains in the corrugated cardboard sample. In this paper, we present a method that not only allows for the reliable identification of the stiffness in the loaded direction of orthotropy in the corrugated board sample, but also the full orthotropic material stiffness matrix. The proposed method uses two samples: (a) traditional, cut crosswise to the wave direction of the corrugated core, and (b) cut at an angle of 45 ◦ . Additionally, in both cases, an optical system with digital image correlation (DIC) was used to measure the displacements and strains on the outer surfaces of samples. The use of a non-contact measuring system allowed us to avoid using the measurement of displacements from the crosshead, which is burdened with a large error. Apart from the new experimental configuration, the article also proposes a simple algorithm to quickly characterize all sought stiffness parameters. The obtained results are finally compared with the results obtained in the homogenization procedure of the cross-section of the corrugated board. The results were consistent in both cases.

**Keywords:** corrugated cardboard; edge crush test; orthotropic elasticity; digital image correlation; compressive stiffness

#### **1. Introduction**

The increasing consumer demands and absorptive power of the merchant market in today's world, resulting in the need to pack, store and securely ship more and more various goods, in addition to growing ecological awareness, have led to the increasing interest of manufacturers in cardboard packaging. This fact, in turn, has triggered the inevitable, continuous, and intensive development of numerous corrugated cardboard testing techniques over the last decades.

Assessing the load-bearing capacity of corrugated cardboard products is crucial for their proper design, production final usage, and re-use processes. It is important to emphasize here that corrugated cardboard comprises a few layers, and thus can be called a sandwich structure. Its mechanical properties are directly related to two characteristic in-plane directions of orthotropy, i.e., a machine direction (MD) that is perpendicular to the main axis of the fluting and parallel to the paperboard fiber alignment, and a cross direction (CD), which is parallel to the fluting.

Numerous approaches to sandwich element strength determination, including for corrugated cardboard, can be found in the literature. Analytical methods, starting already in the 1950s, were presented, e.g., in [1–5], whereas numerical methods can be found in [6–11],

**Citation:** Garbowski, T.; Knitter-Pi ˛atkowska, A.; Marek, A. New Edge Crush Test Configuration Enhanced with Full-Field Strain Measurements. *Materials* **2021**, *14*, 5768. https://doi.org/10.3390/ ma14195768

Academic Editor: Aniello Riccio

Received: 31 August 2021 Accepted: 29 September 2021 Published: 2 October 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

and analytical-numerical techniques in [12–16]. Analytical calculations of the edge crush resistance of cellular paperboard, both in MD and CD, based on the paperboard's geometric parameters and the mechanical properties of the materials used for its production, was discussed by Kmita-Fudalej et al. [17]. Park et al. [18] investigated the edgewise compression behavior of corrugated paperboard while applying the finite element method (FEM) as well as experimental analysis, i.e., load vs. displacement plots, edge crush tests (ECT) and failure mechanisms. In recent years, methods of artificial intelligence, including artificial neural networks, have become widespread to predict the strength of composite materials, e.g., sandwich structures as presented by Wong et al. [19].

While executing numerical simulations in examining corrugated cardboard, the comprehensive knowledge of each layer's material properties is necessary. By reason of the anisotropy of the paper-based materials, this is a demanding task. In such a case a good solution is to implement a method called homogenization. This approach efficiently allows us to simplify multi-layer models into single-layered model, described by the effective properties of the composite [9,10,20]. The application of this technique has the benefits of significant savings in computation time while maintaining the accuracy of the results. Hohe [21] presented the strain energy approach as being applicable to sandwich panels for homogenization and proposed an equivalence of a representative element of the heterogeneous and homogenized elements for this purpose.

Another option, in addition to analytical or numerical analysis, for the estimation of corrugated board strength is to carry out measurements from an experiment. Physical testing is very common in the paper industry, and a number of typical tests have been developed to unify the process of the characterization of corrugated cardboard mechanical properties. The aforementioned ECT is used to evaluate the compressive strength, the load during this examination is applied perpendicularly to the axis of the flutes. In the bending test (BNT), four-point bending is executed, two supports are at the bottom of the cardboard whereas two equal forces act on the sample from the opposite side. The shear stiffness test (SST) involves twisting the cardboard cross-section by applying a pair of forces to opposite corners while the other two remain supported. In the torsional stiffness test (TST) the cardboard sample is twisted in both directions. The box compressive test (BCT) is conducted to examine the load bearing capacity of the whole cardboard box [12–14,22]. The bursting and humidity tests should also be mentioned here.

Since ECT is standardized, four different methods have been described, i.e., the edge-clamping method [23], the neck-down method [24], the rectangular test specimen method [24–26] and the edge-reinforced method [27,28]. One of the major characteristics which differentiates these tests is the shape of the samples. To assemble the measurements from the outer surfaces of the specimen during the examination, video extensometry can be employed. Such a procedure is based on the measurement of the relative distances between pairs of points traced across images captured at different load values [15]. This is a method comparable to, yet simpler than digital image correlation (DIC) which, as full-field non-contact optical measurement method, is gaining more popularity in the field of experimental mechanics since it ensures very high accuracy of data acquisition. Hägglund et al. applied DIC while examining thickness changes during the ECT of damaged and undamaged panels made of corrugated paperboard [29]. The implementation of DIC for the investigation of the strain and stress fields of paperboard panels subjected to BCT and analysis of their post-buckling behavior was discussed by Viguié et al. in [30–32]. A distortional hardening plasticity model for paperboard was presented by Borgqvist et al. [33], who introduced a yield surface characterized by multiple hardening variables attained from simple uniaxial tests. The comparison between the results acquired from the model and the experimental results received while using DIC were demonstrated as well. Combined compression and bending tests of paperboards and laminates for liquid containers while applying DIC were executed by Cocchetti et al. [34,35], who identified the material parameters of anisotropic elastic-plastic material models of foils. For this purpose, inverse analysis was employed while processing the results received from both the experiment

and the numerical FEM simulations. DIC and the virtual fields method (VFM) for the recognition of general anisotropy parameters of a filter paper and a paperboard have been discussed by Considine [36]. Åslund et al. applied the detailed FEM for the investigation of the corrugated sandwich panel failure mechanism while performing the ECT and compared the results with the measurements obtained with the use of DIC [37]. Zappa et al. studied the inflation of the paperboard composites which are used in the packaging of beverages while applying DIC [38]. Paperboard boxes with ventilation holes subjected to a compression load were investigated using DIC by Fadiji et al. [39]. of the corrugated sandwich panel failure mechanism while performing the ECT and compared the results with the measurements obtained with the use of DIC [37]. Zappa et al. studied the inflation of the paperboard composites which are used in the packaging of beverages while applying DIC [38]. Paperboard boxes with ventilation holes subjected to a compression load were investigated using DIC by Fadiji et al. [39]. It should be pointed out that in a large part of the above-mentioned studies, 3-ply corrugated cardboard specimens were tested. In this study, 5-ply double-wall corrugated

analysis was employed while processing the results received from both the experiment and the numerical FEM simulations. DIC and the virtual fields method (VFM) for the recognition of general anisotropy parameters of a filter paper and a paperboard have been discussed by Considine [36]. Åslund et al. applied the detailed FEM for the investigation

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It should be pointed out that in a large part of the above-mentioned studies, 3-ply corrugated cardboard specimens were tested. In this study, 5-ply double-wall corrugated cardboard samples were examined. While performing ECT, an optical system with digital image correlation (DIC) is used to determine the displacements on the outer surface of the specimen. The proposed method uses two types of samples, i.e., traditional, cut crosswise to the direction of the wave direction of the corrugated core, and a novel procedure involving a cut at an angle of 45◦ . Such an approach not only allows for the reliable identification of the stiffness in one direction of orthotropy, but also for the measurement of the full material stiffness matrix, i.e., 4 independent parameters. The obtained results were verified by the results acquired in the homogenization procedure of the cross-section of the corrugated board. As proven, in both cases, the outcomes were very consistent. cardboard samples were examined. While performing ECT, an optical system with digital image correlation (DIC) is used to determine the displacements on the outer surface of the specimen. The proposed method uses two types of samples, i.e., traditional, cut crosswise to the direction of the wave direction of the corrugated core, and a novel procedure involving a cut at an angle of 45°. Such an approach not only allows for the reliable identification of the stiffness in one direction of orthotropy, but also for the measurement of the full material stiffness matrix, i.e., 4 independent parameters. The obtained results were verified by the results acquired in the homogenization procedure of the cross-section of the corrugated board. As proven, in both cases, the outcomes were very consistent. **2. Materials and Methods** 

#### **2. Materials and Methods** *2.1. Corrugated Cardboard*

#### *2.1. Corrugated Cardboard*

of Table 1.

In the current study, a 5-ply corrugated cardboard marked as EB-650 was used. The top liner is made of white, coated, recycled cardboard TLWC with a grammage of 140 g/m<sup>2</sup> . The cross-section has two corrugated layers: (a) low flute (E wave) and (b) high flute (B wave). Both the wavy layers and the flat layer between them, forming the mid liner, are made of lightweight WB cardboard, also recycled, with a grammage of 100 g/m<sup>2</sup> . As a bottom liner, again the white recycled test liner with a grammage of 120 g/m<sup>2</sup> is used. The geometry of the cross-section of the corrugated board and the configuration of the respective layers are shown in Figure 1, where 5 samples are placed one on top of the other. In the current study, a 5-ply corrugated cardboard marked as EB-650 was used. The top liner is made of white, coated, recycled cardboard TLWC with a grammage of 140 g/m<sup>2</sup> . The cross-section has two corrugated layers: (a) low flute (E wave) and (b) high flute (B wave). Both the wavy layers and the flat layer between them, forming the mid liner, are made of lightweight WB cardboard, also recycled, with a grammage of 100 g/m<sup>2</sup> bottom liner, again the white recycled test liner with a grammage of 120 g/m<sup>2</sup> is used. The geometry of the cross-section of the corrugated board and the configuration of the respective layers are shown in Figure 1, where 5 samples are placed one on top of the other.

. As a

**Figure 1.** Visualization of 5 samples (stacked on top of each other) of the analyzed corrugated cardboard. **Figure 1.** Visualization of 5 samples (stacked on top of each other) of the analyzed corrugated cardboard.

Table 1 presents the geometrical parameters of both wavy layers (flutes). The second and third columns of Table 1 shows the wave period (pitch) and the wave amplitude (height), respectively. The take-up ratio, which defines the ratio of the length of the non-Table 1 presents the geometrical parameters of both wavy layers (flutes). The second and third columns of Table 1 shows the wave period (pitch) and the wave amplitude (height), respectively. The take-up ratio, which defines the ratio of the length of the non-fluted corrugated medium to the length of the fluted web, is specified in the last column of Table 1.

fluted corrugated medium to the length of the fluted web, is specified in the last column


**Table 1.** The geometrical features of both corrugated layers of EB-650. **Table 1.** The geometrical features of both corrugated layers of EB-650.

Paperboard, which is a main component of corrugated board, is made of cellulose fibers. The orientation of fibers is not random, but rather results from the production process, which causes that their vast majority is arranged along the web, called the machine direction (MD). The second direction, perpendicular to the MD, is called the cross direction (CD). Paperboard is both stronger and stiffer along the fiber direction. Paperboard, which is a main component of corrugated board, is made of cellulose fibers. The orientation of fibers is not random, but rather results from the production process, which causes that their vast majority is arranged along the web, called the machine direction (MD). The second direction, perpendicular to the MD, is called the cross direction (CD). Paperboard is both stronger and stiffer along the fiber direction.

In general, materials whose mechanical properties depend on fiber orientation are called orthotropic materials. As a component of corrugated cardboard is paper, it is also able to be considered as an orthotropic material. The orientation of the fibers, shown in Figure 2, makes the corrugated board stronger along the direction of the wave. Thus, the corrugated layers compensate (through the take-up factor) for the weaker mechanical properties of the board in CD. In general, materials whose mechanical properties depend on fiber orientation are called orthotropic materials. As a component of corrugated cardboard is paper, it is also able to be considered as an orthotropic material. The orientation of the fibers, shown in Figure 2, makes the corrugated board stronger along the direction of the wave. Thus, the corrugated layers compensate (through the take-up factor) for the weaker mechanical properties of the board in CD.

**Figure 2.** Material orientation in the corrugated board. **Figure 2.** Material orientation in the corrugated board.

Table 2 presents the material properties of the individual layers of the corrugated board. The compressive strength in CD, , is measured while using the short-span compression test according to DIN EN ISO 3037 [26]. The compressive strength of the combined corrugated board in CD, , specified by the producer–Aquila Września– is 7.6 kN/m (±10%), while the total thickness of the EB-650, is 4.3 mm (±0.2 mm). Table 2 presents the material properties of the individual layers of the corrugated board. The compressive strength in CD, *SCTCD*, is measured while using the short-span compression test according to DIN EN ISO 3037 [26]. The compressive strength of the combined corrugated board in CD, *ECTCD*, specified by the producer–Aquila Wrze´snia–is 7.6 kN/m (±10%), while the total thickness of the EB-650, *H* is 4.3 mm (±0.2 mm).


**Table 2.** Mechanical properties of individual layers of 5EB650C3. **Table 2.** Mechanical properties of individual layers of 5EB650C3.

#### *2.2. The Edge Crush Test 2.2. The Edge Crush Test*

The edge crush test (ECT) is a standard test to assess the compressive strength of corrugated board. The test is performed according to FEFCO DIN EN ISO 3037 [25,26], where a 100 mm long and 25 mm high specimen (see Figure 3a,b) is loaded between two rigid plates along its height (see Figure 4a). In order to preserve the parallelism of the cut edges of the sample, it should be cut on a special device, e.g., a FEMat CUT device [22] (see Figure 4b), where the samples are pneumatically cut with one-sided ground blades. All ECT tests were performed under controlled and standard air conditions, i.e., 23 °C and 50% relative humidity. The edge crush test (ECT) is a standard test to assess the compressive strength of corrugated board. The test is performed according to FEFCO DIN EN ISO 3037 [25,26], where a 100 mm long and 25 mm high specimen (see Figure 3a,b) is loaded between two rigid plates along its height (see Figure 4a). In order to preserve the parallelism of the cut edges of the sample, it should be cut on a special device, e.g., a FEMat CUT device [22] (see Figure 4b), where the samples are pneumatically cut with one-sided ground blades. All ECT tests were performed under controlled and standard air conditions, i.e., 23 ◦C and 50% relative humidity.

As already mentioned above, the typical ECT is only used to determine the compressive strength of the corrugated board in CD. Here, the new ECT test setup was also used to determine all of the elastic orthotropic properties of the in-plane tension/compression behavior of corrugated cardboard. For this purpose, beside the traditional method, we

*Materials* **2021**, *14*, x FOR PEER REVIEW 5 of 18

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measure displacements (deformations or strains).

measure displacements (deformations or strains).

also tested samples cut at an angle of 45° to the wave direction (see Figure 3c,d). Since the measurement in standard testing machines is considerably affected by the clearance and susceptibility on the crosshead, non-contact optical techniques are required to credibly

also tested samples cut at an angle of 45° to the wave direction (see Figure 3c,d). Since the measurement in standard testing machines is considerably affected by the clearance and susceptibility on the crosshead, non-contact optical techniques are required to credibly

**Figure 3.** The sample for the standard and new edge crush test: (**a**) standard sample view; (**b**) standard ECT sample–front, back and top view; (**c**) new ECT sample–front and top view; (**d**) new ECT sample–back and top view. **Figure 3.** The sample for the standard and new edge crush test: (**a**) standard sample view; (**b**) standard ECT sample–front, back and top view; (**c**) new ECT sample–front and top view; (**d**) new ECT sample–back and top view. **Figure 3.** The sample for the standard and new edge crush test: (**a**) standard sample view; (**b**) standard ECT sample–front, back and top view; (**c**) new ECT sample–front and top view; (**d**) new ECT sample–back and top view.

**Figure 4.** Edge crush test: (**a**) Universal Testing Machine (Instron 5569); (**b**) FEMAT lab device. **Figure 4.** Edge crush test: (**a**) Universal Testing Machine (Instron 5569); (**b**) FEMAT lab device.

Additionally, measurement without direct contact does not affect the measurement itself. In contact measurements (e.g., traditional extensometers), noise is introduced into the measurement, which may distort the actual measured values. *2.3. Optical Measurements of Sample Deformation*  In this study, as mentioned, the specimen was tested while using optical displacement and strain measurements, i.e., virtual extensometry and digital image correlation (DIC). Two cameras (the stereo DIC setup) were employed to track the deformation on Additionally, measurement without direct contact does not affect the measurement itself. In contact measurements (e.g., traditional extensometers), noise is introduced into the measurement, which may distort the actual measured values. *2.3. Optical Measurements of Sample Deformation*  In this study, as mentioned, the specimen was tested while using optical displacement and strain measurements, i.e., virtual extensometry and digital image correlation (DIC). Two cameras (the stereo DIC setup) were employed to track the deformation on As already mentioned above, the typical ECT is only used to determine the compressive strength of the corrugated board in CD. Here, the new ECT test setup was also used to determine all of the elastic orthotropic properties of the in-plane tension/compression behavior of corrugated cardboard. For this purpose, beside the traditional method, we also tested samples cut at an angle of 45◦ to the wave direction (see Figure 3c,d). Since the measurement in standard testing machines is considerably affected by the clearance and susceptibility on the crosshead, non-contact optical techniques are required to credibly measure displacements (deformations or strains).

the front faces to account for the out-of-plane bending produced by the non-symmetrical the front faces to account for the out-of-plane bending produced by the non-symmetrical Additionally, measurement without direct contact does not affect the measurement itself. In contact measurements (e.g., traditional extensometers), noise is introduced into the measurement, which may distort the actual measured values.

#### *2.3. Optical Measurements of Sample Deformation* brackets. Two 5 MPx cameras (Manta G504-b, Allied Vision, Stadtroda, Germany) were used to record greyscale images during the test, see Figure 5. The video extensometry was

*Materials* **2021**, *14*, x FOR PEER REVIEW 6 of 18

In this study, as mentioned, the specimen was tested while using optical displacement and strain measurements, i.e., virtual extensometry and digital image correlation (DIC). Two cameras (the stereo DIC setup) were employed to track the deformation on the front faces to account for the out-of-plane bending produced by the non-symmetrical section, and single a camera was employed on the back faces for standard optical extensometry, per the test setup shown in Figure 5a. Each of the two faces of the specimen were printed with the speckle pattern for both optical methods, i.e., DIC and video extensometry. Here, three models of deformation measurements were used, namely: performed using the MatchID DIC platform (v. 2020.2.0, MatchID, Ghent, Belgium). The cameras were calibrated while applying the MatchID calibration plate (MatchID, Ghent, Belgium) to acquire the pixel (px) to mm conversion rate of ~50 µm/px. The specimen was manually preloaded with a very small load (15 N) to ensure that both edges of the specimen were touching the loading plates. Then, the measured load cell and the displacement were zeroed, and the L-brackets supporting the sample were removed. The load and the crosshead displacement were synchronized with the cameras. The accuracy of the measurement was estimated using a set of 25 static images (without any movement); the stand-

The specimen was sandwiched between two platens and aligned using 3D printed L-

section, and single a camera was employed on the back faces for standard optical extensometry, per the test setup shown in Figure 5a. Each of the two faces of the specimen were printed with the speckle pattern for both optical methods, i.e., DIC and video extensome-

try. Here, three models of deformation measurements were used, namely:

Stereo (2.5D) DIC on the front (see Figure 5b) plus extensometry on the back.

• Crosshead from the machine. ard deviation of the measured elongation was evaluated to be 4 µm, which can be consid-

Crosshead from the machine.

Extensometry on the front and back.


**Figure 5.** Setup of the optical measurements: (**a**) configuration of cameras on the front and back face; (**b**) cameras recording the front face. **Figure 5.** Setup of the optical measurements: (**a**) configuration of cameras on the front and back face; (**b**) cameras recording the front face.

In total, 5 samples in CD and 5 samples using the 45° direction were tested. Unfortunately, data from one of the samples in the CD experiment were not recorded properly on the PC and were removed from the statistics. The loading rate was set to 5 mm/min (which is different from the standard rate 12.5 mm/min) because the samples failed too quickly for cameras to get enough data. The following stereo DIC procedures, with camera "Cam1" as the main, were utilized in this research: Perform DIC on the sample's face while using images from Cam1 and Cam0; region of interest (ROI) visible in Figure 6b. Align the data coordinate system with the specimen material direction, i.e., 11 = , 22 = , = vertical (see Figure 6a). Calculate strain from the displacements. The specimen was sandwiched between two platens and aligned using 3D printed L-brackets. Two 5 MPx cameras (Manta G504-b, Allied Vision, Stadtroda, Germany) were used to record greyscale images during the test, see Figure 5. The video extensometry was performed using the MatchID DIC platform (v. 2020.2.0, MatchID, Ghent, Belgium). The cameras were calibrated while applying the MatchID calibration plate (MatchID, Ghent, Belgium) to acquire the pixel (px) to mm conversion rate of ~50 µm/px. The specimen was manually preloaded with a very small load (15 N) to ensure that both edges of the specimen were touching the loading plates. Then, the measured load cell and the displacement were zeroed, and the L-brackets supporting the sample were removed. The load and the crosshead displacement were synchronized with the cameras. The accuracy of the measurement was estimated using a set of 25 static images (without any movement); the standard deviation of the measured elongation was evaluated to be 4 µm, which can be considered the level of uncertainty. The optical displacements were averaged for each face and compared against the crosshead displacement.

In total, 5 samples in CD and 5 samples using the 45◦ direction were tested. Unfortunately, data from one of the samples in the CD experiment were not recorded properly on the PC and were removed from the statistics. The loading rate was set to 5 mm/min (which is different from the standard rate 12.5 mm/min) because the samples failed too quickly for cameras to get enough data.

The following stereo DIC procedures, with camera "Cam1" as the main, were utilized in this research:

	- Align the data coordinate system with the specimen material direction, i.e., 11 = *MD*, 22 = *CD*, *yy* = vertical (see Figure 6a). Select a subregion and extract the data; all data in the subregion is averaged giving
	- Calculate strain from the displacements.
	- Select a subregion and extract the data; all data in the subregion is averaged giving one value of desired quantities per image, namely: *ε*11, *ε*22, *ε*12, *εyy*. Select a subregion and extract the data; all data in the subregion is averaged giving one value of desired quantities per image, namely: , , , . one value of desired quantities per image, namely: , , , . Shear strains reported as tensor shear strain component , need to be doubled for
	- Shear strains reported as tensor shear strain component *ε*12, need to be doubled for the engineering component. Shear strains reported as tensor shear strain component , need to be doubled for the engineering component. the engineering component.

**Figure 6.** Virtual optical gauges (**a**) sample in CD and in 45°; (**b**) ROI visualization. The \* denotes a material orientation in the sample cut in the 45°. **Figure 6.** Virtual optical gauges (**a**) sample in CD and in 45◦ ; (**b**) ROI visualization. The \* denotes a material orientation in the sample cut in the 45◦ . in the sample cut in the 45°. On the other hand, the video extensometry main procedures utilized in this study,

On the other hand, the video extensometry main procedures utilized in this study, were as follows: On the other hand, the video extensometry main procedures utilized in this study, were as follows: were as follows: Use a speckle pattern compatible with DIC (pen marks would work equally well, per


(**a**) (**b**) **Figure 7.** Virtual optical gauges (**a**) sample in CD; (**b**) sample in 45°. **Figure 7.** Virtual optical gauges (**a**) sample in CD; (**b**) sample in 45◦ .

**Figure 7.** Virtual optical gauges (**a**) sample in CD; (**b**) sample in 45°. Using the tests for CD, (in the CD direction) and (in the MD direction) were measured from each image either by averaging large region from the DIC (see Figure 6b) or by using virtual extensometers: 3 vertical plus 1 horizontal (see Figure 7a). The front and back data were averaged to remove artificial bending data. A similar methodology was used in case of the ECT in a 45° direction. All stiffnesses, e.g., vs. were cal-Using the tests for CD, (in the CD direction) and (in the MD direction) were measured from each image either by averaging large region from the DIC (see Figure 6b) or by using virtual extensometers: 3 vertical plus 1 horizontal (see Figure 7a). The front and back data were averaged to remove artificial bending data. A similar methodology was used in case of the ECT in a 45° direction. All stiffnesses, e.g., vs. were calculated from the linear portion of the graphs. Using the tests for CD, *εyy* (in the CD direction) and *εxx* (in the MD direction) were measured from each image either by averaging large region from the DIC (see Figure 6b) or by using virtual extensometers: 3 vertical plus 1 horizontal (see Figure 7a). The front and back data were averaged to remove artificial bending data. A similar methodology was used in case of the ECT in a 45◦ direction. All stiffnesses, e.g., *Fyy* vs. *εyy* were calculated from the linear portion of the graphs.

culated from the linear portion of the graphs.

#### *2.4. Proposed Method to Identify Matrix* **A**

The identification of matrix **A** is based here on two sets of tests, namely: (a) the standard ECT, in CD and (b) the new ECT in 45◦ direction. The well-known relation between cross-sectional forces and general strains has the form:

$$
\begin{bmatrix} \sigma\_{11} \\ \sigma\_{22} \\ \sigma\_{12} \end{bmatrix} = \begin{bmatrix} A\_{11} & A\_{12} & 0 \\ A\_{12} & A\_{22} & 0 \\ 0 & 0 & A\_{66} \end{bmatrix} \begin{bmatrix} \varepsilon\_{11} \\ \varepsilon\_{22} \\ \varepsilon\_{12} \end{bmatrix} \tag{1}
$$

where *σij* are the components of the sectional force vector, in [N/mm]; *Aij* are the stiffness components, in [N/mm]; and *εij* are the membrane (in-plane) strains.

From Equation (1) two sets of equations can be extracted, namely in the CD test:

$$\begin{aligned} A\_{12}\varepsilon\_{11} + A\_{22}\varepsilon\_{22} &= \sigma\_{22}, \\ A\_{11}\varepsilon\_{11} + A\_{12}\varepsilon\_{22} &= 0, \end{aligned} \tag{2}$$

and in the 45◦ direction test:

$$\begin{array}{l} A\_{11}\varepsilon\_{11} + A\_{12}\varepsilon\_{22} = \sigma\_{11}^{45} = 0.5\sigma^{45}, \\ A\_{12}\varepsilon\_{11} + A\_{22}\varepsilon\_{22} = \sigma\_{22}^{45} = 0.5\sigma^{45}. \end{array} \tag{3}$$

By building up a matrix of those equations from two experiments and solving it in the least square sense (se e.g., [40]) the components of matrix *A* = [*A*11, *A*12, *A*22] can be easily obtained. Component *A*<sup>66</sup> can be obtained independently, from the ECT in the 45◦ direction.

If one uses stresses instead of sectional forces, the following equations can be derived from the test in the CD:

$$
\begin{bmatrix}
\frac{E\_{11}}{1-\nu\_{12}\nu\_{21}} & \frac{E\_{22}\nu\_{12}}{1-\nu\_{12}\nu\_{21}}\\\
\frac{E\_{11}\nu\_{21}}{1-\nu\_{12}\nu\_{21}} & \frac{E\_{22}}{1-\nu\_{12}\nu\_{21}}
\end{bmatrix}
\begin{Bmatrix}
\mathcal{E}\_{11} \\
\mathcal{E}\_{22}
\end{Bmatrix} = \begin{Bmatrix}
0 \\
\sigma\_{22}
\end{Bmatrix},\tag{4}
$$

and from the test in the 45◦ direction:

$$
\begin{bmatrix}
\frac{E\_{11}}{1-\nu\_{12}\nu\_{21}} & \frac{E\_{22}\nu\_{12}}{1-\nu\_{12}\nu\_{21}}\\\
\frac{E\_{11}\nu\_{21}}{1-\nu\_{12}\nu\_{21}} & \frac{E\_{22}}{1-\nu\_{12}\nu\_{21}}
\end{bmatrix}
\begin{Bmatrix}
\varepsilon\_{11}^{\*}\\\
\varepsilon\_{22}^{\*}
\end{Bmatrix} = \frac{1}{2} \begin{Bmatrix}
\sigma\_{45}\\\
\sigma\_{45}
\end{Bmatrix}.\tag{5}
$$

From the test in the CD only, just two constitutive components can be computed, namely Poisson's ratio:

$$\nu\_{21} = -\frac{\varepsilon\_{11}}{\varepsilon\_{22}} \, ^\prime \tag{6}$$

and the elastic modulus:

$$E\_{22} = \frac{\sigma\_{22}}{\varepsilon\_{22}}.\tag{7}$$

On the other hand, from both the CD and 45◦ tests, all orthotropic stiffness coefficients can be obtained, namely elastic stiffness in MD:

$$E\_{11} = -\frac{\sigma\_{22}\sigma\_{45}}{\varepsilon\_{11}\sigma\_{45} - 2\varepsilon\_{11}^\*\sigma\_{22}},\tag{8}$$

elastic stiffness in CD:

$$E\_{22} = \frac{\sigma\_{22}}{\varepsilon\_{22}} \, ^\prime \tag{9}$$

Poisson's ratio *ν*<sup>12</sup> :

$$\nu\_{12} = \frac{\varepsilon\_{11}\sigma\_{45}}{\varepsilon\_{11}\sigma\_{45} - 2\varepsilon\_{11}^\*\sigma\_{22}} \, ^\prime \tag{10}$$

Poisson's ratio *ν*21:

$$\nu\_{21} = 1 - \frac{2\varepsilon\_{22}^\* \sigma\_{22}}{\varepsilon\_{22} \sigma\_{45}},\tag{11}$$

$$\sigma\_{21} = 1 - \frac{\omega \varepsilon\_{22} \sigma\_{22}}{\varepsilon\_{22} \sigma\_{45}},$$

 − 2

*Materials* **2021**, *14*, 5768

$$\frac{E\_{22}}{\nu\_{21} = \nu\_{12}} \frac{E\_{22}}{E\_{11}} \frac{}{\cdot}$$

<sup>∗</sup>

, (10)

or using the symmetry principals: The stiffness in the 45° direction can be computed directly from the test in 45° direc-

Poisson's ratio :

or using the symmetry principals:

$$\nu\_{21} = \nu\_{12} \frac{E\_{22}}{E\_1}. \tag{12}$$

$$\mathbf{E\_{45}} = \underset{\mathbf{E\_{45}}}{\text{const}} \tag{12}$$

The stiffness in the 45◦ direction can be computed directly from the test in 45◦ direction: 

=

$$\begin{array}{ccccc} \text{E}\_{45} = \frac{\sigma\_{45}}{\sqrt{2}\mathfrak{w}\_{12}} & & & & \text{1} \\ & \text{C} = \begin{pmatrix} \mathfrak{B}\mathfrak{w}\_{12} & \mathbf{1} & \mathbf{1} & \mathbf{1} & \mathbf{4} \\ \end{pmatrix} \end{array} \tag{13}$$

and is used to compute the last missing coefficient, namely the in-plane shear stiffness: = 2 − − + . (14)

*Materials* **2021**, *14*, x FOR PEER REVIEW 9 of 18

$$G\_{12} = \left(\frac{2\nu\_{12}}{E\_{11}} - \frac{1}{E\_{11}} - \frac{1}{E\_{22}} + \frac{4}{E\_{45}}\right). \tag{14}$$

#### **3. Results** *3.1. The ECT Enhanced with Optical Measurement Techniques*

**3. Results** 

tion:

#### *3.1. The ECT Enhanced with Optical Measurement Techniques* First, four tests of the CD are presented. Figure 8 shows the differences in the dis-

First, four tests of the CD are presented. Figure 8 shows the differences in the displacements measured by optical techniques (solid line) and taken from the machine crosshead (dashed line). placements measured by optical techniques (solid line) and taken from the machine crosshead (dashed line).

**Figure 8.** Force-displacement curves. Optical extensometry–solid lines; from machine crosshead– dashed lines. **Figure 8.** Force-displacement curves. Optical extensometry–solid lines; from machine crosshead– dashed lines.

Table 3 shows the elastic stiffness index, which was computed from the linear part of the curves shown in Figure 8. It should be pointed out that the cross-sectional force is normalized by the sample length ( = 100 mm) but not by the sample thickness. This approach complies with the specifications of the corrugated board manufacturers and al-Table 3 shows the elastic stiffness index, which was computed from the linear part of the curves shown in Figure 8. It should be pointed out that the cross-sectional force is normalized by the sample length (*L* = 100 mm) but not by the sample thickness. This approach complies with the specifications of the corrugated board manufacturers and allows the presentation of results regardless of the sample thickness.

**Table 3.** Elastic stiffness index in CD computed from the displacement measurement by the optical extensometry and from machine crosshead, as well as the edgewise compression strength in CD.

**Test ID** 

**E—Optical (N/mm)** 

lows the presentation of results regardless of the sample thickness.

**E—crosshead (N/mm)** 

**ECT (N/mm)** 


**Table 3.** Elastic stiffness index in CD computed from the displacement measurement by the optical extensometry and from machine crosshead, as well as the edgewise compression strength in CD. *Materials* **2021**, *14*, x FOR PEER REVIEW 10 of 18

#### *3.2. DIC vs. Extensometry 3.2. DIC vs. Extensometry*

Then the stereo DIC and the extensometry approach were compared. For this analysis, the selected test in the direction 45◦ was carefully analyzed. The DIC data in the zones occupied by extensometers were averaged and compared (see Figures 9 and 10). Then the stereo DIC and the extensometry approach were compared. For this analysis, the selected test in the direction 45° was carefully analyzed. The DIC data in the zones occupied by extensometers were averaged and compared (see Figures 9 and 10).

**Figure 9.** Location of each strain gauge on the sample in the test in the 45° direction. **Figure 9.** Location of each strain gauge on the sample in the test in the 45◦ direction.

The results presented in Figure 10 are comparable, but not identical in terms of elasticity, mainly due to a certain inhomogeneity in the deformation caused by the crushing of the edges, which obviously affected the extensometers. However, this can be reduced, e.g., by shortening the gauge length, which appears to be a key a priori choice. The question of how long the extensometers should be is discussed in the next subsection.

It is known that the error in strain measurements comes from error in the measured displacements (here it is constant at ~0.01 px) and the length of the gauge. Although it seems that the longer the gauge, the better, but the longer the gauge, the greater the risk of taking into account the edge effects of the sample, where (especially in the case of unwaxed samples) the largest local deformations (i.e., crushing and wrinkling) are usually concentrated.

(**a**) (**b**)

*Materials* **2021**, *14*, x FOR PEER REVIEW 10 of 18

*3.2. DIC vs. Extensometry* 

1 1447.45 441.82 −7.548 2 1380.25 536.82 −7.151 4 1531.96 450.66 −7.609 5 1615.12 611.39 −7.640 Mean (N/mm) 1493.70 510.17 −7.487 Std (N/mm) 102.01 79.93 0.227 Cov (%) 6.829 15.668 −3.038

Then the stereo DIC and the extensometry approach were compared. For this analysis, the selected test in the direction 45° was carefully analyzed. The DIC data in the zones

occupied by extensometers were averaged and compared (see Figures 9 and 10).

**Figure 9.** Location of each strain gauge on the sample in the test in the 45° direction.

**Figure 10.** DIC vs. virtual extensometry comparison: (**a**) region 1; (**b**) region 2; (**c**) region 3; (**d**) mean from 3 regions; (**e**) back-to-front average; (**f**) strains resulting from forces. **Figure 10.** DIC vs. virtual extensometry comparison: (**a**) region 1; (**b**) region 2; (**c**) region 3; (**d**) mean from 3 regions; (**e**) back-to-front average; (**f**) strains resulting from forces.

#### The results presented in Figure 10 are comparable, but not identical in terms of elas-*3.3. Length of Virtual Extensometry*

*3.3. Length of Virtual Extensometry* 

lated from = ( − )/20.

concentrated.

ticity, mainly due to a certain inhomogeneity in the deformation caused by the crushing of the edges, which obviously affected the extensometers. However, this can be reduced, e.g., by shortening the gauge length, which appears to be a key a priori choice. The question of how long the extensometers should be is discussed in the next subsection. It is known that the error in strain measurements comes from error in the measured A study on the length of the optical extensometry was performed on the test number 3 data in the CD–full-field data was extracted (i.e., strains and displacements). Virtual extensometers were generated with varied lengths at different horizontal positions and compared against the averaged vertical strains from the DIC. For example, two points were selected in the center of the sample: one at *Y*<sup>1</sup> = +10 mm with respect to the center of the

displacements (here it is constant at ~0.01 px) and the length of the gauge. Although it seems that the longer the gauge, the better, but the longer the gauge, the greater the risk of taking into account the edge effects of the sample, where (especially in the case of un-

A study on the length of the optical extensometry was performed on the test number 3 data in the CD–full-field data was extracted (i.e., strains and displacements). Virtual extensometers were generated with varied lengths at different horizontal positions and compared against the averaged vertical strains from the DIC. For example, two points were selected in the center of the sample: one at = +10 mm with respect to the center of the sample height, the other at = −10 mm and the extensometer strain was calcusample height, the other at *Y*<sup>2</sup> = −10 mm and the extensometer strain was calculated from *εyy* = (*v*<sup>1</sup> − *v*2)/20.

Three horizontal positions of the virtual strain gauges were considered: (1) left at 25% of the width; (2) mid at 50% and (3) right at 75% of the sample width. They were also averaged. Figure 11 shows the location of the optical strain gauges. The length of each gauge varies from 4 to 20 mm. Three horizontal positions of the virtual strain gauges were considered: (1) left at 25% of the width; (2) mid at 50% and (3) right at 75% of the sample width. They were also averaged. Figure 11 shows the location of the optical strain gauges. The length of each gauge varies from 4 to 20 mm. Three horizontal positions of the virtual strain gauges were considered: (1) left at 25% of the width; (2) mid at 50% and (3) right at 75% of the sample width. They were also averaged. Figure 11 shows the location of the optical strain gauges. The length of each gauge varies from 4 to 20 mm.

*Materials* **2021**, *14*, x FOR PEER REVIEW 12 of 18

*Materials* **2021**, *14*, x FOR PEER REVIEW 12 of 18

**Figure 11.** Location of the virtual strain gauges. **Figure 11.** Location of the virtual strain gauges. **Figure 11.** Location of the virtual strain gauges.

Figure 12 shows a comparison of strain calculated while using different lengths of virtual gauges with the DIC measurements. Figure 12 shows a comparison of strain calculated while using different lengths of virtual gauges with the DIC measurements. Figure 12 shows a comparison of strain calculated while using different lengths of virtual gauges with the DIC measurements.

**Figure 12.** Comparison of strains measured by different lengths of virtual gauges with DIC measurements. (**a**) left set; (**b**) mid set; (**c**) right set; (**d**) averaged. **Figure 12.** Comparison of strains measured by different lengths of virtual gauges with DIC measurements. (**a**) left set; (**b**) mid set; (**c**) right set; (**d**) averaged. **Figure 12.** Comparison of strains measured by different lengths of virtual gauges with DIC measurements. (**a**) left set; (**b**) mid set; (**c**) right set; (**d**) averaged.

The main observation was that for the test in the 45° direction, the extensometers should be arranged in a rectangular configuration (15 mm × 15 mm box, with longer gauges on the diagonal) or circular gauges (so as to keep the gauge length of 15 mm).

The main observation was that for the test in the 45° direction, the extensometers should be arranged in a rectangular configuration (15 mm × 15 mm box, with longer gauges on the diagonal) or circular gauges (so as to keep the gauge length of 15 mm).

The main observation was that for the test in the 45◦ direction, the extensometers should be arranged in a rectangular configuration (15 mm × 15 mm box, with longer gauges on the diagonal) or circular gauges (so as to keep the gauge length of 15 mm). *Materials* **2021**, *14*, x FOR PEER REVIEW 13 of 18 not the case for the 45° tests. For each recorded level of the force, the measured strain components averaged back-to-front are plotted (see Figure 13). It is visible that the tests can be split into two, more consistent groups (see Figure 14). Group 2 had a stiffer re-

The last issue was to check the data consistency of the new test in the 45° direction. For all the CD tests, the force-strain data was very consistent, but unfortunately this was

#### *3.4. Consistency of Tests in 45 Deg Direction 3.4. Consistency of Tests in 45 Deg Direction*  sponse in the 11 (MD) direction.

*3.4. Consistency of Tests in 45 Deg Direction* 

*Materials* **2021**, *14*, x FOR PEER REVIEW 13 of 18

The last issue was to check the data consistency of the new test in the 45◦ direction. For all the CD tests, the force-strain data was very consistent, but unfortunately this was not the case for the 45◦ tests. For each recorded level of the force, the measured strain components averaged back-to-front are plotted (see Figure 13). It is visible that the tests can be split into two, more consistent groups (see Figure 14). Group 2 had a stiffer response in the 11 (MD) direction. The last issue was to check the data consistency of the new test in the 45° direction. For all the CD tests, the force-strain data was very consistent, but unfortunately this was not the case for the 45° tests. For each recorded level of the force, the measured strain components averaged back-to-front are plotted (see Figure 13). It is visible that the tests can be split into two, more consistent groups (see Figure 14). Group 2 had a stiffer response in the 11 (MD) direction.

stead of full DIC, the trend stayed the same. Group 1 had (accidentally) a different orien-

**Figure 13.** The consistency of the data from tests 6–10. **Figure 13.** The consistency of the data from tests 6–10. tation of fluting with respect to the plate than group 2 (Figure 3c,d).

**Figure 14.** The consistency of data in tests 6–10: (**a**) group 1 (tests 6 and 8); (**b**) group 2 (tests 7, 9 and 10).

(**a**) (**b**)

193

The reasons for the difference are not fully clear. One of the observations was that group 1 (i.e., test 6 and 8) had a high flute oriented towards the stereo DIC setup (front face as depicted in Figure 3c). Local buckling on that face is more pronounced and that could have affected the measured strain. However, even when using extensometers instead of full DIC, the trend stayed the same. Group 1 had (accidentally) a different orientation of fluting with respect to the plate than group 2 (Figure 3c,d). *Materials* **2021**, *14*, x FOR PEER REVIEW 14 of 18 **Figure 14.** The consistency of data in tests 6–10: (**a**) group 1 (tests 6 and 8); (**b**) group 2 (tests 7, 9 and 10).

#### *3.5. Full Matrix A Identification 3.5. Full Matrix A Identification*

First, by combining tests 2 and 6 and using Equations (2) and (3) with the least square approximation, one can identify the full A matrix (see Table 4). First, by combining tests 2 and 6 and using Equations (2) and (3) with the least square approximation, one can identify the full A matrix (see Table 4).


<sup>1</sup> Results obtained directly from test 2 in the CD using Equation (7) or (9). (N/mm) 1078 1061.0 946.0 1 Results obtained directly from test 2 in the CD using Equation (7) or (9).

The Poisson's ratio computed directly from the CD test (see Equation (6)) turned out to be ~0.07, which is much closer to the value cited here: *A*12/*A*<sup>22</sup> = 0.09. In all cases, force was normalized by specimen width (100 mm). In the investigation, test number 1 was removed from the data pool due to an artefact point. The Poisson's ratio computed directly from the CD test (see Equation (6)) turned out to be ~0.07, which is much closer to the value cited here: / = 0.09. In all cases, force was normalized by specimen width (100 mm). In the investigation, test number 1 was

Finally, the same procedure as above was used, but with the two separate groups discussed in previous subsection shown in Figure 14. In total, 178 (group 1) and 204 (group 2) points were used here to calculate the in-plane stiffnesses (*A*11, *A*12, *A*22). This separation made it possible to study the effects of positioning unsymmetric samples on the ECT apparatus. removed from the data pool due to an artefact point. Finally, the same procedure as above was used, but with the two separate groups discussed in previous subsection shown in Figure 14. In total, 178 (group 1) and 204 (group 2) points were used here to calculate the in-plane stiffnesses (, , ). This separation made it possible to study the effects of positioning unsymmetric samples on

The reconstructed elastic forces from the identified parameters are shown in Figures 15 and 16—multiple lines represent multiple tests. These data show good model fitting. the ECT apparatus. The reconstructed elastic forces from the identified parameters are shown in Figures 15 and 16—multiple lines represent multiple tests. These data show good model fitting.

**Figure 15. Figure 15.** Curves reconstructed from the identified A matrix vs. measured force. Curves reconstructed from the identified A matrix vs. measured force.

**Figure 16.** Curves reconstructed from the identified A matrix vs. measured force: (**a**) using tests in group 1; (**b**) using tests in group 2. **Figure 16.** Curves reconstructed from the identified A matrix vs. measured force: (**a**) using tests in group 1; (**b**) using tests in group 2.

#### **4. Discussion 4. Discussion**

The previous section provides the outcomes of the research, presenting, among others, typical ECT results enriched with digital image correlation and/or optical, virtual extensometry techniques. The results summarized in Table 3 clearly show that the use of the displacements obtained from the machine crosshead introduces an error in the estimation of the stiffness index, underestimating this value almost 3 times. The same observation can also be found in the recent work of Garbowski et al. [15]. The compressive strength given in Table 3 (shown in column 4) is consistent with the value provided by the manufacturer of the corrugated board, namely 7.6 N/mm ±10%. The previous section provides the outcomes of the research, presenting, among others, typical ECT results enriched with digital image correlation and/or optical, virtual extensometry techniques. The results summarized in Table 3 clearly show that the use of the displacements obtained from the machine crosshead introduces an error in the estimation of the stiffness index, underestimating this value almost 3 times. The same observation can also be found in the recent work of Garbowski et al. [15]. The compressive strength given in Table 3 (shown in column 4) is consistent with the value provided by the manufacturer of the corrugated board, namely 7.6 N/mm ±10%.

The comparison of strains obtained from the DIC and while using virtual extensometers is presented in Figure 10. These results were comparable, but not identical. The best fit can be observed for the vertical strain . Based on the observations regarding the length of the optical extensometer and its influence on the accuracy of the results, 15 mm segments were used for further analyses. This can be observed in Figure 12, where the calculated strains were compared while using DIC and extensometers of different lengths. The main conclusion is that when applying longer gauges, the results are more stable. However, if the optical extensometer is too long (i.e., longer than 15 mm) or too short (i.e., shorter than 8 mm), the differences can be as high as 15%. The comparison of strains obtained from the DIC and while using virtual extensometers is presented in Figure 10. These results were comparable, but not identical. The best fit can be observed for the vertical strain *εyy*. Based on the observations regarding the length of the optical extensometer and its influence on the accuracy of the results, 15 mm segments were used for further analyses. This can be observed in Figure 12, where the calculated strains were compared while using DIC and extensometers of different lengths. The main conclusion is that when applying longer gauges, the results are more stable. However, if the optical extensometer is too long (i.e., longer than 15 mm) or too short (i.e., shorter than 8 mm), the differences can be as high as 15%.

The use of extensometers with a length of ~20 mm causes false results due to the proximity of the measuring tip to the crushed edge of the sample (which is 25 mm high). On the other hand, the use of short gauges of ~5 mm is affected by larger noise and causes the measurements to have an error due to buckling from the plane of the sample (see Figure 17b). The moment when the sample buckles is shown in Figure 12d–image number 38 (for a strain gauge 4 mm long). The influence of buckling (which manifests in the form of an out-of-plane deformation) on the measurement of in-plane deformations can be easily eliminated using the stereo DIC procedure. However, if optical extensometry is to be used, a fairly large area where the results obtained with the extensometer match those obtained with the DIC should be in the range of 8–16 mm. The use of extensometers with a length of ~20 mm causes false results due to the proximity of the measuring tip to the crushed edge of the sample (which is 25 mm high). On the other hand, the use of short gauges of ~5 mm is affected by larger noise and causes the measurements to have an error due to buckling from the plane of the sample (see Figure 17b). The moment when the sample buckles is shown in Figure 12d–image number 38 (for a strain gauge 4 mm long). The influence of buckling (which manifests in the form of an out-of-plane deformation) on the measurement of in-plane deformations can be easily eliminated using the stereo DIC procedure. However, if optical extensometry is to be used, a fairly large area where the results obtained with the extensometer match those obtained with the DIC should be in the range of 8–16 mm.

in group 2.

obtained with the DIC should be in the range of 8–16 mm.

*Materials* **2021**, *14*, x FOR PEER REVIEW 15 of 18

(**a**) (**b**) **Figure 16.** Curves reconstructed from the identified A matrix vs. measured force: (**a**) using tests in group 1; (**b**) using tests

facturer of the corrugated board, namely 7.6 N/mm ±10%.

shorter than 8 mm), the differences can be as high as 15%.

The previous section provides the outcomes of the research, presenting, among others, typical ECT results enriched with digital image correlation and/or optical, virtual extensometry techniques. The results summarized in Table 3 clearly show that the use of the displacements obtained from the machine crosshead introduces an error in the estimation of the stiffness index, underestimating this value almost 3 times. The same observation can also be found in the recent work of Garbowski et al. [15]. The compressive strength given in Table 3 (shown in column 4) is consistent with the value provided by the manu-

The comparison of strains obtained from the DIC and while using virtual extensometers is presented in Figure 10. These results were comparable, but not identical. The best fit can be observed for the vertical strain . Based on the observations regarding the length of the optical extensometer and its influence on the accuracy of the results, 15 mm segments were used for further analyses. This can be observed in Figure 12, where the calculated strains were compared while using DIC and extensometers of different lengths. The main conclusion is that when applying longer gauges, the results are more stable. However, if the optical extensometer is too long (i.e., longer than 15 mm) or too short (i.e.,

The use of extensometers with a length of ~20 mm causes false results due to the proximity of the measuring tip to the crushed edge of the sample (which is 25 mm high). On the other hand, the use of short gauges of ~5 mm is affected by larger noise and causes the measurements to have an error due to buckling from the plane of the sample (see Figure 17b). The moment when the sample buckles is shown in Figure 12d–image number 38 (for a strain gauge 4 mm long). The influence of buckling (which manifests in the form of an out-of-plane deformation) on the measurement of in-plane deformations can be easily eliminated using the stereo DIC procedure. However, if optical extensometry is to be used, a fairly large area where the results obtained with the extensometer match those

**4. Discussion** 

**Figure 17.** The ECT sample during the CD test: (**a**) sample during the CD test–no buckling; (**b**) sample during the CD test–buckling. buckling. Table 4 shows the identified components of matrix A. The second column shows the

Table 4 shows the identified components of matrix A. The second column shows the results obtained during tests 2 and 6, while columns 3 and 4 show the results obtained while using two different test groups. The groups included samples with a higher flute from the front (on the side of the DIC stereo set) and samples with a lower flute from the front. It is evident that the results for group 2, especially in the case of *A*<sup>11</sup> and *A*12, differed significantly from the results obtained in the first procedure, while considering group 1. This was due to the asymmetric cross-section of the sample and the different level of buckling on the sample side with the higher flute. Out-of-plane deformation related to buckling distorts measurement and therefore introduces noise that distorts the results. Other components of matrix A did not differ more than 10% when using different measurement techniques, which was very promising. results obtained during tests 2 and 6, while columns 3 and 4 show the results obtained while using two different test groups. The groups included samples with a higher flute from the front (on the side of the DIC stereo set) and samples with a lower flute from the front. It is evident that the results for group 2, especially in the case of and , differed significantly from the results obtained in the first procedure, while considering group 1. This was due to the asymmetric cross-section of the sample and the different level of buckling on the sample side with the higher flute. Out-of-plane deformation related to buckling distorts measurement and therefore introduces noise that distorts the results. Other components of matrix A did not differ more than 10% when using different measurement techniques, which was very promising. In order to validate the results presented in Table 4, the numerical homogenization

In order to validate the results presented in Table 4, the numerical homogenization procedure (for details see recent works by Garbowski and Gajewski [9] or Garbowski et al. [10]) of the cross-section of corrugated board BE-650 (see Figure 18) was used. The numerical homogenization technique used the geometrical and constitutive parameters presented in Tables 1 and 2. The following results were obtained while employing the homogenization technique: *A*<sup>11</sup> = 2620 N/mm, *A*<sup>12</sup> = 185 N/mm, *A*<sup>22</sup> = 1812 N/mm, *A*<sup>66</sup> = 906 N/mm. The results are in good agreement, which proves that the use of optical techniques in conjunction with the new setup of the ECT (samples cut at an angle of 45◦ with respect to the direction of corrugation) can be effective in determining the stiffness of corrugated cardboard. procedure (for details see recent works by Garbowski and Gajewski [9] or Garbowski et al. [10]) of the cross-section of corrugated board BE-650 (see Figure 18) was used. The numerical homogenization technique used the geometrical and constitutive parameters presented in Tables 1 and 2. The following results were obtained while employing the homogenization technique: = 2620 N/mm, = 185 N/mm, = 1812 N/mm, = 906 N/mm. The results are in good agreement, which proves that the use of optical techniques in conjunction with the new setup of the ECT (samples cut at an angle of 45° with respect to the direction of corrugation) can be effective in determining the stiffness of corrugated cardboard.

**Figure 18.** Visualization of the finite element model of corrugated board BE-650. **Figure 18.** Visualization of the finite element model of corrugated board BE-650.

#### **5. Conclusions 5. Conclusions**

results.

The main conclusion is that stereo DIC and/or optical extensometry techniques can be used to evaluate stiffness in a standard edge crush test. In order to determine all the stiffness coefficients, it is necessary to use an additional, new test specimen cut at an angle of 45° to the direction of the corrugation. By applying the results from the two samples The main conclusion is that stereo DIC and/or optical extensometry techniques can be used to evaluate stiffness in a standard edge crush test. In order to determine all the stiffness coefficients, it is necessary to use an additional, new test specimen cut at an angle of 45◦ to the direction of the corrugation. By applying the results from the

simultaneously and using a least squares minimization approach, all of the stiffness com-

tests. However, this is easily remedied by using a larger sample set and averaging the

two samples simultaneously and using a least squares minimization approach, all of the stiffness components can be easily identified. The only concern is proper surface selection in unsymmetrical corrugated cardboard samples for stereo DIC measurement, especially in the 45◦ tests. However, this is easily remedied by using a larger sample set and averaging the results.

**Author Contributions:** Conceptualization, T.G.; methodology, T.G.; software, T.G. and A.M.; validation, A.M., A.K.-P. and T.G.; formal analysis, A.M. and T.G.; investigation, A.M., A.K.-P. and T.G.; resources, A.M.; data curation, A.M.; writing—original draft preparation, A.K.-P. and T.G.; writing—review and editing, A.K.-P., T.G. and A.M.; visualization, A.M. and T.G.; supervision, T.G.; project administration, T.G.; funding acquisition, A.K.-P. and T.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** The APC was funded by the Ministry of Science and Higher Education, Poland, the statutory funding at Poznan University of Life Sciences, grant number 506.569.05.00 and the statutory funding at Poznan University of Technology, grant number 0411/SBAD/0004.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Acknowledgments:** Special thanks to the FEMat Sp. z o. o. company (Pozna ´n, Poland) (www.fematsystems. pl—accessed on 21 July 2021) for providing the laboratory equipment and commercial software.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

