Modeling of load-displacement relationships for corrugated cardboard containers

. This article discusses the nonlinear behavior of the corrugated packaging during compression. The model of this behavior is a load-displacement curve with an ascending branch, a peak point and a descending branch. In some cases, the descending branch shows signs of plastic transition and/or loss of stability of the container walls. The purpose of this work is to predict the transition point of a corrugated container into a plastic (or pseudo-plastic) state. This condition is unacceptable because there are residual deformations that reduce the quality of the container. The article proposes and implements an approach based on the joint application of the well-known equation of the dependence of the load on displacement and the differential fracture criterion. The simulation results are consistent with experiments known from the literature. The conducted research makes a certain contribution to the creation of new tools, the use of which expands the possibilities of analyzing the mechanical behavior of corrugated cardboard containers for the purpose of rational use of resources in accordance with the concept of sustainable development.


Introduction
Corrugated cardboard is one of the most widely used packaging materials for a number of reasons. It is inexpensive, has a high specific strength and is harmless to the environment. Corrugated cardboard is also one of the building materials in architecture [1]. Corrugated cardboard boxes are widely used for transporting various goods, from fresh food to industrial equipment [2,3]. During transportation, these boxes subjected to various loads, such as compression load, cyclic load, as well as the effects of moisture, temperature and other factors associated with changing environmental conditions [4]. In addition, ventilation holes design features, manufacturing technology and other factors significantly affect the reliability of containers [5,6].
To ensure the functionality of the corrugated packaging under the above loads, the corresponding strength and rigidity of these boxes, which are essentially structures, is necessary. Accordingly, strength refers to the ability of the structure to withstand external influences without destruction; stiffness refers to the ability of the structure to withstand the same external influences without experiencing excessive deformation. Therefore, in order to predict the functionality of corrugated cardboard boxes, the ratio of load and displacement must be known for them. This dependence is nonlinear [7,8], which complicates both numerical and analytical modeling of the objects under discussion [5]. Modern computer technologies make it possible to overcome these modeling difficulties [5,6,8], which, however, does not exclude the expediency of developing analytical models. The use of two types of models (numerical and analytical) makes it possible to analyze an object from different points of view, improving the understanding of its mechanical behavior for the purpose of rational use of resources in accordance with the concept of sustainable development [9].
However, at present, numerical approaches dominate in the analysis of corrugated cardboard structures [3,5,7,8], therefore, in order to use all the modeling possibilities, this article discusses the use of an analytical model to analyze the dependence of the load on the movement of a box and a system of corrugated cardboard boxes. The practical significance of the research using the analytical model is to expand the set of tools for analyzing the loaddisplacement ratio for both the box and the corrugated cardboard box system. In addition, the use of analytical models reduces the time and cost of calculations, since the solution of the problem reduced to the use of simple equations. However, analytical methods are not as universal as numerical ones.
This paper is devoted to the analysis of the insufficiently studied dependence of the load on the movement for containers. If the load is small, then this dependence is almost linear. As the load increases, the influence of plastic (or residual) deformations arising in the material of the structure increases. Residual deformations distort the shape and dimensions (geometry) of the corrugated cardboard container and create a potential danger to the contents of the container. Therefore, it is important to predict the occurrence of residual deformations of the container in real conditions.
Many studies carried out in this field of applied research and a number of important results obtained, which presented in the literature cited above. However, the question of the transition of the quasi-elastic stage of container deformation to the inelastic (quasi-plastic) stage of deformation remains relevant. Such a question formulated and solved at the empirical level in [10]. Since the appearance of this work (1996), new approaches and models have appeared [3,5,6,7,8]. However, the use of analytical models to model the relationship between the load and movement of corrugated cardboard containers and stacks of such containers still insufficiently investigated. The practical value of analytical models as an additional method of analysis lies in the fact that their use saves time and costs for predicting the mechanical state of the containers in question.
Considering the above, we formulate the goal of this work: at the stage of modeling to determine the transition point of the considered structure in an almost plastic state in order to predict the load at which there are large residual deformations of corrugated cardboard packaging.

Methodology and results
To analyze the load-displacement ratio, we will use the Blagojevich model [11,12], which we will write in terms of load-displacement [13]: Model (1) belongs to the class of the simplest, but rather universal models, an overview of which and application experience are reflected in [12]. To calculate the load (F) depending on the offset (u), the peak load (F peak ) and the corresponding offset (u peak ) must be known, as well as the model parameter n. Parameter n depends on the stiffness of the structure: for rigid structures and materials, the parameter values are higher than for soft ones. The values of this parameter can be determined using the test results at the pre-peak stage, by analogy with [14].
In this article, the peculiarities of application of model (1) and comparison with experiment considered on the example of load-displacement relation for corrugated cardboard boxes. To determine F peak and u peak , the results of tests of corrugated cardboard boxes presented in the form of graphs in [8] are used. The model parameter n can be determined, for example, using the method of least squares [12]. Another approach is to choose such a value of n, at which the criterion R 2 is equal, for example, to 0.99 (by analogy with determination of the tangential modulus of elasticity in [5]). Initial data for the example under consideration shown in Table 1, in which the values n 1 and n 2 are equal to the parameter n in equation (1) when modeling, respectively, the left and right branches of the curve total load-displacement (Figure 1).  . 1. Load-displacement curves from the uniaxial box compression Curves with markers are adapted from [8]. Curves without markers obtained by model (1). Figure 1 shows that the ascending branch and the initial fragment of the descending branch of the curve (1) for all samples correspond to the experimental data. However, if the displacement is greater than 3.5 mm, the discrepancy between the experimental and theoretical data increases rapidly, what can be explained by the influence of inelastic (plastic) deformations, i.e., the non-linearity of the material and structures. In addition, at a sufficiently large load, the container walls under longitudinal compression may lose stability, i.e., the flat state of the wall will be unstable and then the compressed wall will transition into a curved state, which will be stable (by analogy with the stability of rods under axial compression). The transition of the container wall under longitudinal compression accompanied by a loss of stiffness and an increase in displacement under almost constant load, which reflected in the graph by an almost horizontal post-peak branch. This pattern (lines with markers in Figure  1) perceived as the transition of the container from an almost brittle state to an almost plastic state, similar to the axial compression of dry wood. A similar test behavior of corrugated cardboard containers investigated, for example, in [6,10].
Similar patterns of mechanical behavior are recorded in experiments during uniaxial compression of wood along the fibers [15,16]: under small loads, the displacements are almost linear; if the load increases, the influence of elastic deformation decreases, but the contribution of plastic deformations to the total strain increases. The criterion for transition of wood to the plastic state under uniaxial compression considered in [16]. This criterion can be adapted to the load-displacement relation modeling for corrugated cardboard containers to determine the transition point to the near-plastic state, which will allow predicting the load at which large residual deformations of the container appear.
The application of the above-mentioned criterion for transition to the plastic state reduced to the following steps (see Figure 2). Using the load-displacement graph, we determine the tangential stiffness S of the container (by analogy with determining the tangential modulus of elasticity, as shown in [5]). Through the origin of coordinates, draw straight lines (dashed line in Figure 2) parallel to the tangents. Write the equation of the line y = 0.5 • S • u (solid lines in Figure 2). Determine the point of intersection of this line with the descending branch of the curve by equation (1) (for each of the four samples in Table 1 and Figure 1). The point thus found is the point of transition of the studied object into the plastic state. The rationale for this criterion can be find in [17]. The stage of plastic deformation shows Figure 2 with a solid line for each sample. The dotted lines correspond to the unstable states of the container; the unstable state not realized if there is no forced displacement control. Comparison of experimental data according to [8] simulation results by equation (1) shows Figure 3.   Fig. 3. Comparison of theoretical data by model (1) (solid lines without markers) with experimental data by [8] (semitransparent lines with markers).  Figure 3 shows that the plastic state appears only after passing the peak point in the loaddisplacement curve. This means that the peak point defines the upper limit for the safe functioning of the container. The permissible load can be equal to e.g., 0.85F peak . Figure 3 confirms the realistic results of modeling a corrugated cardboard container under compression. Analysis of previous studies of the mechanical behavior of corrugated cardboard [5,6,8] and containers [4,7,10] has shown that it is difficult to use the same equation (1) to model the pre-peak and post-peak conditions of a corrugated container ( Figure  1). An attempt to use equation (1) together with the differential brittle fracture energy criterion [17] was successful in simulating uniaxial compression of wood [16] and a corrugated container (Figure 3).

Discussion
Note that the amount of research using the approach discussed above is small, but good agreement with experiments is a reason to continue research. Difficulties in this direction can appear when determining the model parameters u peak , F peak and n (1). In such cases, these parameters can be determined using the results of prepeak tests [14]. It is important to note that the listed parameters of the model (1) indirectly (integrally) take into account the influence of ventilation holes in containers and all their other features.
The approach proposed and realized in this work, based on the application of model (1) and the differential criterion of brittle failure [17], contributes to the creation of new tools for analyzing the behavior of corrugated cardboard containers.
The practical significance of this study lies in the possibility of additional justification of safe loading of corrugated cardboard containers. Alternative analytical approaches and models are valid only in special cases. Attempts to generalize models have led to complex equations, which makes their practical application difficult; these aspects discussed in more detail in [5]. These difficulties are smoothed to some extent by the approach implemented in this paper, which uses a relatively small number of model parameters (1) combined with the failure criterion to model both individual corrugated cardboard containers and systems of such containers [17]. Further research needed to discuss the aspects raised in more detail.

Conclusion
This paper substantiates the relevance of research related to ensuring the strength and stiffness of corrugated cardboard containers.
An approach based on the joint application of model (1) and the differential brittle fracture criterion proposed and implemented [17]. A very small amount of initial data is an important feature of the developed approach.
Comparison with experiments known from the literature confirmed the adequacy of the approach and the consistency of the results of modeling and tests of a real corrugated cardboard container.
This study contributes to the creation of new tools, the use of which improves the ability to analyze the mechanical behavior of corrugated cardboard containers for the rational use of resources in accordance with the concept of sustainable development [9].
Given the small amount of research using the developed approach, it is necessary to continue research in this direction, despite the good agreement between the simulation results and the experimental data known in the literature.