# Compressive strength

In-plane compressive strength is one of the most important properties of paperboard. In corrugated board, the compressive strengths of the liner and fluting control the compressive strength of the box. In a boxboard container, half of its compressive strength comes from the bending stiffness of the board and the other half from the edgewise compressive strength ^{1}.

Compressive strength is typically about one-third of tensile strength. The ratio between compression and tension strength is significant in severe bending. In bending, paper or board yields first on the compressed side. This is exploited in converting, because compression failure prevents the other side from breaking in tension. Thus, the breaking of the material into two in bending is prevented, and paperboard can be easily creased and bent to make packages.

## Definitions

Compressive strength is the largest compressive force that a test piece tolerates without failing. Compression strength can be characterised by the quantities defined in Table 1. With paper or board, compressive strength is usually defined as the force per unit width. This is not a material property because it depends on grammage.

Compressive failure stress would in principle be a material property, but the trouble is that the thickness of paper or board is not a well-defined quantity. For example, if board is calendered more, the compressive failure stress increases, although the force needed to break a specimen does not change (or it may even decrease). For these reasons, the compression index is often used as the material parameter characterising the compression strength of paper or board. It does not depend on grammage, if the furnish and network structure do not change.

*Table 1. Definition of compression strength properties. Symbols: F, the failure force, A, the cross-sectional area and W, the width of the test piece; b, grammage, ρ, density.*

Property | Symbol | Definition | Units |

Compressive failure stress | σ_{c} |
F/A |
N/m^{2} |

Compressive strength | s |
F/W |
N/m |

Compression index | P |
σ_{c}/ρ = s/b |
Nm^{2}/g |

Although seldom used in practice, compressive failure stress is needed when the behaviour of board is calculated using engineering mechanics (for example ^{2–5}). Also, in multi-layer products compressive strength is easy to approximate through the compressive failure stress *σ _{c}*. For example, in a symmetric three-layer structure

*σ*is approximately given by Equation 1, following the superposition principle of linear engineering mechanics

_{c}(1)

where *σ _{c,1}* and

*σ*are the compressive failure stresses of the middle layer and the outer layers, respectively,

_{c,2}*d*is the thickness of the middle layer and d the thickness of the board. The equation is valid provided that the layers have similar compressive breaking strain. If the failure strains differ considerably, the layer with the lowest strain will dominate. Consider, for example, the situation where the strain to failure of the middle ply is half the value of the other plies. When the middle ply reaches its ultimate load, it fails, and all stress moves to the not yet failed parts, which would then also fail because of the suddenly increased stress.

_{1}Measurement methods and equipment for compressive strength are discussed and described in Compression resistance tests.

## Significance of compressive strength

The behaviour of board boxes under a compressive load depends on the compressive strength and bending stiffness of the board. There is a large difference between corrugated board boxes and boxboard containers in the relative importance of the two terms.

The box crush strength of a rectangular boxboard container in compression is ^{1}

(2)

where *F* is the vertical force needed to crush the box, *S _{b}* bending stiffness in either MD (x) or CD (y),

*σ*the edgewise compressive failure stress of the board in the loading direction of the box,

_{c}*d*the thickness of the board and

*K*a parameter whose theoretical value is

*K = 2π*. In practice, the value of

*K*is lower and it has to be determined experimentally. Box compression strength depends weakly also on the box perimeter, approximately as perimeter to the power of 0.2–0.25.

According to Eq. (2) the box crush strength is determined by the geometric average of the MD and CD bending stiffnesses, , and by the compressive strength of the board in the loading direction. Therefore, the MD/CD compressive strength ratio of the board affects the box crush strength, but the MD/CD bending stiffness ratio does not, if *S _{b,geom}* is constant. The relative importance of bending stiffness and compressive strength is the same, and box strength is proportional to the square root of both. However, it is much easier to improve bending stiffness than compressive strength, so in boxboard the former is in practice more important than the latter.

The box crush strength of a *corrugated box* is given by the widely used McKee equation ^{6}, whose approximate form is

(3)

where *ECT* is the Edge Crush Test, and *z* is the perimeter of the box. Compressive strength enters through the ¾ power while the power of bending stiffness is ¼. Thus, in a corrugated box, the compressive strength of corrugated board is more important than the bending stiffness.

The compressive strength of the liner can also be improved by making it denser (and thus thinner) without reducing the bending stiffness of corrugated board, because the thickness of the board is maintained by the corrugating profile. The density of the fluting, however, cannot be too high, because then the stiffness of the corrugating profile becomes so low that the profile can no longer keep the liners apart. Also, too high a density of the fluting would disturb the wetting of the fluting on the corrugator.

Compressive strength is also important in some converting processes that employ strong bending, such as the folding of books and magazines, or creasing of paperboard. In bending, the convex side of the sheet is in tension and the concave side in compression (Figure 1). The stress-strain behaviour of the paper or board determines the precise stress distribution. If compressive strength were larger than tensile strength, the convex side would break first and the tensile failure could propagate through the entire sheet. However, since the compressive failure stress in paper and board is between a third and a half of the tensile failure stress, the compressed side fails first. This leads to an irreversible drop in bending resistance but, perhaps surprisingly, to no loss in tensile strength.

*Figure 1. Distribution of stress in the bending of a paper or board sheet, in the elastic region (a) and after failure (b) ^{8}.*

The compression strength of coatings may also be significant. In heat-set offset printing of LWC paper, the paper sometimes breaks at the fold. At that point, paper is very dry and thus brittle after drying. It has been postulated that the fold break may originate from the high compressive strength of the coating ^{7}. Because in an LWC paper the thickness of the coating is considerable compared to the overall thickness, the coating has sufficient strength in compression so that it forces the side in tension to extend so much that a tensile failure occurs. Coatings typically have a compressive strength larger than their tensile strength. The tensile failure on the concave side then leads to propagation of the failure through the thickness and the page breaks in two.

Traditionally, compressive strength has been used only to characterise liner and fluting. In liner only the CD strength has been considered important but in fluting both the MD and CD strengths have been relevant. Liners and flutings have quite similar values of the CD compression index, usually 10–12 kNm/kg when measured with the *RCT* method and 20–22 kNm/kg with the *SCT* method (Table 2). Even the difference between primary and secondary fibres is small.

*Table 2. Compression index values (in kNm/kg) of various boards. For the outer plies of folding boxboard, the values are approximations derived from the values of the whole board.*

Outer plies of folding boxboard | Middle ply of folding boxboard | Liners, flutings | |

SCT, MD | 20–25 | 9–11 | |

SCT, CD | 12–17 | 5–7 | 20–22 |

The compression index of boxboard is usually lower than that of liner or fluting (Table 2). Also in boxboards, differences between grades (folding boxboard / recycled fibre / solid board) are relatively small. Because the middle ply accounts for 70–85% of the thickness, depending on the grammage, its role is decisive in the formation of compressive strength (Eq. 1). There is no direct information on the compressive strength of pulps used in folding boxboard, but the compressive strength of the plies can be deduced from the values of the whole board. Typical *SCT* values are given in Table 2.

## Stress-strain curves in compression

The stress-strain curve of a paperboard in compression is compared with the tension curve in Figure 2. Some important features can be seen from the curves:

- The elastic modulus is the same in tension and compression.
- The non-linear region is very short in compression, both in MD and CD.
- The compressive strength is about one third of the tensile strength in MD and one half in CD.
- The compressive breaking strain is one quarter of the tensile breaking strain in MD and one fifth in CD.

*Figure 2. Stress-strain curve of a paperboard in tension and in compression ^{8}.*

Although the curves in Figure 2 represent just one example, the principal features are almost the same in all ordinary paper and board grades. Thus, the ratios of tensile and compressive strength and breaking strain can be used as rules of thumb. The largest deviations from them occur in breaking strain. For example, beating does not really change the strain to failure in compression, although it increases the strain to failure in tension ^{8,9}. On the other hand, drying shrinkage increases the strain to failure both in compression and in tension ^{8}.

The compressive stress-strain curve has the interesting feature that if the fibre network has low density or low inter-fibre bonding, then even the non-linear part of the compression curve is reversible, so the material is non-linearly elastic. In tension, the non-linear part of the stress-strain curve is almost exclusively irreversible.

This observation of the non-linear elasticity is consistent with both of the theories of the origin of compressive strength. At high densities the non-linear region corresponds to plastic irreversible compression ^{8}.