# Advanced Numerical Analysis of Transport Packaging
## 2 Materials and Methods
### Mechanical Tests on the Cardboards  
A crucial aspect of this scientific work is the verification of the influence of corrugated cardboard material on the decrease in compressive strength of packaging subjected to prior transport loads. To achieve this, a series of material tests on corrugated cardboard were conducted, providing material data for computer simulations utilising the finite element method.  
In the initial phase, nine types of corrugated cardboard with varied properties were selected, distinguished by differences in the thickness of the constituent papers and the height of the flutes. In the second phase, four additional types of corrugated cardboard were tested to verify the relations observed in the first phase. The cardboard samples were chosen to achieve maximum diversity in mechanical characteristics. The study focused on 3-ply corrugated cardboards with grammages ranging from 350 to 550 g/m\({}^{2}\) and thicknesses from 1.5 mm to 4.0 mm.  
All cardboard samples were conditioned in a climatic chamber under standardised laboratory conditions [43; 44], specifically at a temperature of 23 \({}^{\circ}\)C and a relative humidity of 50%.  
Material testing was conducted using a multifunctional device, the Box Strength Estimation (BSE) System [45], which, among others functions, served in determining the mechanical properties of cardboard. The following tests on cardboard samples were performed on the testing machine: the edge crush test, four-point bending test in two directions (the machine direction--BNTMD; the cross-machine direction--BNTCD), torsional stiffness test in two directions (TSTMD and TSTCD), and shear stiffness test.  
For each type of cardboard, material testing was conducted on five sets of samples. One set of samples consisted of specimens for all tests conducted. All samples were cut using a Computer Numerical Control (CNC) laser device. Figure 2 presents images from the material testing procedures. The dimensions of the samples used in these tests are as follows:  
* ECT: 100 \(\times\) 25 mm;
* BNTMD and BNTCD: 250 \(\times\) 50 mm;
* TSTMD and TSTCD: 150 \(\times\) 30 mm;
* SST: 85 \(\times\) 85 mm.  
Figure 1: Schematic presentation of the workflow of this study.  
Representative and averaged results from the material testing of all types of cardboard are presented in Table 1.  
Based on data obtained from the experimental campaign (Table 1), it was possible to calculate the material constants for the adopted constitutive model. An orthotropic material model was applied within the elastic range, along with directional plasticity through the use of Hill's potential. This model has demonstrated effectiveness in numerous previous scientific studies [30,39].  
The material constants used in all numerical analyses of corrugated cardboard boxes considered in this work are presented in Table 2, where the following are shown:  
* \(E_{1}\), the elastic modulus in machine direction;
* \(E_{2}\), the elastic modulus in cross-machine direction;
* \(\
u_{12}\), the Poisson's ratio;  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{  
\(G_{12}\), \(G_{13}\), and \(G_{23}\), the shear moduli;
* \(\sigma_{0}\), the yield strength;
* \(R_{11}\), the yield stress factor accounted for in the Hill's potential.