Energies 2021 , 14 , 3203
5of 16
deformed shape of the sample (Figure 5c) with exemplary displacement fields (obtained via finite element method modelling and analytical approximation) are illustrated.
( a )
( b )
( c )
Figure 5. The cardboard crushing: ( a ) cardboard testing device; ( b ) loaded sample scheme; ( c) de- formed sample scheme. 2.2. Estimation Error by Coefficient of Determination To explore the relationship between a crushing and a decrease of the corrugated board stiffness values, the coefficient of determination was computed for each board quality, defined by the formula: R 2 = 1 − ∑ n i = 1 ( x i − ˆ y i ) 2 ( n − 1 ) · var ( x ) , (1) where: x i —the expected ratio of the measured value of the crushed sample to the initial value (CRS = 0% → x i = 1.0, CRS = 10% → x i = 0.9, etc.), ˆ y i —the values computed on the basis of the formula (2) describing the linear regression, var ( x ) — the variance of the expected ratio of the measured value of the crushed sample to the initial value: ˆ y i = a ( x i − x )+ y , (2) where: x —the mean value of the expected ratio of the measured value of the crushed sample to the initial value, y —the mean value of the measured quantities (SST-MD, TST-CD etc.), a —the slope of the linear regression:
∑ n i = 1 ( x i − x )( y i − y ) ∑ n i = 1 ( x i − x ) 2 .
(3)
a =
The higher the value of the coefficient of determination, R 2 , the better the fit of the regression line and the estimation error is considered to be smaller. 2.3. Numerical Approach for Modelling Crushing In this paper, apart from laboratory tests on crushed corrugated cardboards, also a simple numerical approach to consider the crushed properties of the corrugated board is proposed. The derived method does not require the modelling of the plasticization of the fluting, because its analytical equivalent (presented later) in FE model can be used. The aim of this part of the study is to validate the approach by using torsion test modelling [31,32]. The numerical study consists of several steps, illustrated by scheme in Figure 6: • Building initial geometry of the intact corrugated cardboard (stage a). • Performing FE analysis of corrugated cardboard crushing with plasticity included (stage b)—substituted with an analytical, simplified crushed flute shape approxima- tion. • Using the crushed geometry to build the classical material stiffness matrix of corru- gated cardboard (stage c,d). • Homogenizing the crushed corrugated cardboard by composite properties, based on Garbowski and Gajewski method [25] (stage d,e). • Computing torsion/bending response of crushed corrugated cardboard sample using composite properties (stage f).
Made with FlippingBook Online document maker