Elasticity

The elastic modulus is an important property because it controls the behaviour of paper in practice. For example, the elastic modulus determines how web tension depends on the speed difference in open draws in printing presses and other web-fed end-use processes. Through bending stiffness, it also controls the performance of paper and board in sheet form. The elastic modulus can also be a practical indicator of other paper properties, such as various strength properties and dimensional stability, because the elastic modulus can be measured quickly using ultrasonic methods.

The relationship between the elastic properties and structure of paper is much better understood than other mechanical properties. Strength properties depend on the “weakest link” in a specimen and often need a stochastic description in terms of their probability distributions and of how these depend on sample size.

We shall first define the elastic constants of paper and describe how they are measured. Then we outline the theoretical background of the elastic modulus of random fibre networks and apply it to experiments, as furnish properties and wet pressing vary. The anisotropic elastic modulus of machine-made paper is sensitive to the drying shrinkage and wet straining of paper and the fibre orientation. Baum describes extensively the elastic properties of paper and board in a review on subfracture mechanical properties ¹.

Elastic constants and their measurement

The ordinary elastic modulus or Young’s modulus, E [GPa], measures the force necessary for a small elongation, when these are in linear proportion. If σ is the applied stress, or force over the cross-sectional area of the specimen (Figure 1), and ε is the corresponding strain [%], then E is the following:

E = d σ ⁄ d ε a t ε \to 0

(1)

When the stress-strain curve or load-elongation curve is linear at small ε, one can also write E = σ/ε. At larger elongations, the slope, dσ/dε, is a tangent modulus, which is usually smaller than the true elastic modulus.

The elastic modulus of paper is different in the three principal directions: the machine direction (E_MD or E_x ), cross-machine direction (E_CD or E_y ), and thickness direction (E_ZD or E_z ). In packaging boards especially, all three moduli are often relevant. For the in-plane directions, measurement of the modulus is usually in the tensile mode by stretching the specimen. In the z-direction, the compression mode has more use than tension.

The elastic modulus is usually the same in tension and compression. This is the case for microscopically homogeneous materials such as perfect crystals. For paper, differences between tension and compression are possible because the structure consists of fibre segments that can bend or buckle differently depending on whether the network stretches or compresses.

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Figure 1. Measurement of elastic modulus, E, Poisson ration, ν, (a) and shear modulus, G, (b) for a specimen of length, L, width, w, and thickness, d.

Another important elastic property of paper is the Poisson ratio, ν, which gives the lateral strain for a given longitudinal strain, as indicated in Figure 1. For example, if a paper web stretches in MD, it contracts in CD. Too much contraction or variation in contraction may cause web wrinkling or problems with print registration. The ratio of CD contraction over MD stretch is the MD Poisson ratio denoted by ν_x, ν_MD or ν_xy. The significance of the MD and CD Poisson ratios on a biaxial stress state is as follows ²:

$σ_{x} = (E_{x} (ε_{x} + υ_{y} x ε_{y})) ⁄ ((1 - υ_{x} y υ_{y} x))$
$σ_{y} = (E_{y} (ε_{y} + υ_{x} y ε_{x})) ⁄ ((1 - υ_{x} y υ_{y} x))$	(2)

For example, under uniaxial stress such as the ordinary stress-strain test, σ_y = 0, and ε_y = -ν_xy ε_x. Under uniaxial strain, ε_y = 0, σ_y > 0, and σ_x = E_xε_x /(1-ν_xyν_yx) > E_xε_x.

In isotropic paper such as handsheets, the in-plane value is ν ≈ 0.3. Otherwise, ν = 0.1–0.5 depending on the anisotropy ³. Continuum mechanics implies ²

v_{M D} / v_{C D} = E_{M D} / E_{C D}

(3)

An elongation in MD therefore causes a larger lateral contraction than an elongation in CD. Equation 3 is the Maxwell relation. Its accuracy has not been verified for paper that is definitely not a continuum material. In the thickness direction, the Poisson ratios vary considerably, and they can even be negative. We shall return to the tri-axial deformations in a later section.

Shear modulus, G = τ / γ, gives the stress, τ, needed for a given skew deformation, γ, defined in Figure 1. Shear modulus is important in packaging board since shear stresses occur in box loading. In paper webs, shear stresses arise if the web does not run straight, possibly causing wrinkles in the web ⁴. For the same reason, the shear modulus is difficult to measure because wrinkling must be prevented in the measurement ⁵. In practice, G is often estimated from E_MD and E_CD. As a rough estimate, one can use one-third of the geometric mean of the MD and CD elastic moduli ⁶,

G \approx \sqrt{E_{M D} E_{C D}} / 3

. The in-plane shear modulus is independent of the loading direction.

The evaluation of elastic modulus from the load-elongation curve requires care. Good accuracy favours a linear regression fit to the curve over a long range but using an excessively long range will cause underestimation of the modulus. The linear behaviour often occurs only in a small elongation range. It may be completely absent, as shown in Figure 2. In that case, the pragmatic definition of E is the maximum slope of the curve. The nonlinearity of the curve is sometimes removed by mechanical conditioning, i.e., by small strain cycles before the load-elongation measurement.

Figure 2. Load-elongation curve measured at a very fine resolution, exhibiting no initial linearity ⁷.

In standard engineering mechanics, the elastic modulus has the units of stress ². The values in paper range from 2 to 20 GPa. A typical value would be E = 5 GPa. The quantity that is directly measured is force over specimen width, F/W, for a given strain and the modulus is obtained from E ≡ F/Wdε. Paper thickness therefore requires a separate measurement. This causes problems because then the elastic modulus depends on the method adopted for the thickness measurement. The specific modulus of elasticity or modulus divided by density, E/ρ ≡ F/Wbε, where b is grammage, is often more appropriate because no thickness measurement is necessary. The specific modulus is analogous to the ordinary tensile index.

The specific elastic modulus can be estimated from ultrasonic measurements ⁸. If c is the speed of longitudinal waves in paper, the specific elastic modulus is given by

E ⁄ ρ = c^{2} (1 - υ_{x} y υ_{y} x) \approx c^{2}

(4)

The last approximation assumes ν_xyν_yx << 1. The other elastic constants require additional measurements ⁸. Equation 4 applies strictly for orthotropic continuum materials. As illustrated by Figure 3, the “sonic” modulus of paper from Eq. 4 is larger than the mechanically measured modulus. The reason for this presumably lies in the non-linear creep that takes place during the measurement ⁹.

Figure 3. Elastic modulus, E, of bleached Kraft pulp handsheets of different grammages measured ultrasonically (diamonds) and mechanically (triangles). Ultrasonic value for G shown by squares (KCL, unpublished data).

Theory

Unlike tensile strength that depends on the “weak” points of the specimen, the elastic modulus is insensitive to formation-like and other fluctuations in the paper structure. The “self-averaging” nature follows from the relationship of modulus to average elastic energy, U = average energy per unit volume:

U = E ε^{2} / 2 = σ ε / 2 o r E = 2 U / ε^{2}

(5)

in the elastic region at small ε, in a uniaxial load-elongation test. This follows from elementary mechanics where force, F, is by definition equal to the derivative of total energy, UV, with respect to the distance travelled by the clamps, Lε, or F = d(UV)/dLε. Then σ = dU/dε from which Eq. 5 follows since σ ≡ Eε, and E is constant.

We see that the specific modulus of elasticity is equivalent to the total elastic energy at ε = 1 divided by the specimen mass, M, or E/ρ = 2UV/Mε² (N.B. units: GPa/(kg/m³) = MJ/g). If we assume, for example, that all the elastic energy in the network is contained in the axial elongations of the fibres, then

U = \frac{1}{2} ρ \frac{E_{f}}{ρ_{f}} ⟨ ε_{i}^{2} ⟩

(6)

where E_f is the elastic modulus of fibres,
ρ_f their density, and
the average of axial strain squared.

In the elastic region (small ε), the fibre strains ε_i relate linearly to the external strain, ε, or ε_i/ε = k_i. The “activation constant”, k_i, of a fibre depends on its orientation and on the surrounding random network structure. Thus

E = ρ \frac{E_{f}}{ρ_{f}} ⟨ k_{i}^{2} ⟩

(7)

If the axial strain, ε_i, of a fibre were equal to the macroscopic strain of the sheet in the same direction, then = 1/3 would apply for an isotropic fibre orientation distribution ¹⁰. In other words, the network modulus would be . This is equivalent to assuming that fibres are infinitely long.

In reality, is smaller than 1/3 primarily because fibre strain goes to zero at the fibre ends. The theoretical density dependence for a fibre of finite length is approximately ^11,12

E = \frac{1}{3} E^{*} (ρ - ρ_{0})

(8)

provided that all fibre properties are constant. Of the two factors, E* describes fibre properties and ρ₀ accounts for the inactive fibre ends and their effect on the surrounding network. At a high density, the inactive fibre ends are short (in proportion to pore sizes, and the network modulus is high. Deviations from Eq. 8 occur at low densities or short fibre lengths.

Equation 8 can also be derived as a high density or high bonding degree approximation to the two-dimensional shear-lag theory or Cox theory ¹³. The result is as follows:

ρ_{0} = \frac{w_{f}}{l_{f}} \frac{ρ}{R B A} {\sqrt{\frac{E_{f}}{2 G_{f}} a n d E * = E_{f}}}^{}

(9)

where w_f and l_f are the width and length of the fibres,
RBA the two-dimensional relative bonded area, and
G_f the shear modulus of the fibres.

Thus, ρ₀ is constant, since density is linearly related to RBA.

One must be cautious with the microscopic interpretation that Eq. 9 gives for the threshold density, ρ₀, and effective fibre modulus, E*, because the shear-lag model is only an approximation ¹². For example, according to Eq. 9, ρ is inversely proportional to fibre length. This holds in low-grammage handsheets ¹⁴, but in ordinary handsheets, ρ₀ is almost independent of l_f , as shown in Figure 4. These theories also assume that bonds are very effective in transferring stress from a fibre to a crossing one, and that the relative roles played by shear or bending and axial deformations can be neglected. This can obviously be wrong for very slender fibres, for bonds that have a very small shear modulus (as in wet sheets) and for short segments ¹⁵.

Figure 4. Elastic modulus vs. density of paper in wet pressing (a) and beating (b) for fibres cut to different lengths as given in the legend [mm] ^16,17.

An alternative explanation to the density dependence of the elastic modulus comes from the internal stresses. The internal stresses form in fibres during paper drying and influence their mechanical properties. A simple analysis of the distribution of internal stresses in the fibre network ¹⁸ can reproduce the observed density dependence of the elastic modulus. In other words, it is possible that the density dependence of the elastic modulus of paper does not arise from the shear lag mechanism.

Elasticity

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Elasticity

Elastic constants and their measurement

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