Mechanical characterisation of fibres for engineered wood products: a scanning force microscopy study

Abstract

Mechanical properties of individual wood fibres and the characterisation of the interaction between wood fibres and resins are of interest to the composite wood panel industry and others involved in the fabrication of engineered wood products. However, the size of such fibres, typically a few millimetres in length, makes characterisation of their mechanical properties difficult. Gripping fibres is problematic, not to mention the measurement of meaningful load displacement data. Using a novel three-point bend test technique, the Young’s moduli of single wood fibres were determined. Fibres were placed on a specially designed test rig, and a scanning probe microscope was used to apply a load and to measure the deflection at the centre of each fibre. A model of the fibre was produced in order to facilitate data analysis. The technique proved to be feasible, resulting in an average Young’s modulus value of 24.4 GPa for Pinus Sylvestris softwood fibres. This compares well with other values in the literature, but there is scope for improvement in the methodology to lead to more accurate measurements.

Appendix: Beam bending model

The beam bending model for a beam under symmetric three-point bending with unconstrained ends, as shown in Fig. 10, was considered to be the most applicable model to be used in the study.

Figure 10

Symmetric three-point bending

The bending moment, M, is $\frac{ - Fx}{2}$. This can be substituted into elastic beam bending equation given in Eq. 2.

$$ EI \frac{{{\text{d}}^{2} y}}{{{\text{d}}x^{2} }} = M $$

(2)

where E = Young’s modulus of the beam material, M = bending moment, and I = second moment of area of the beam cross section.

The resulting equation can be integrated as shown to produce an equation for the deflection of the beam, y, in terms of the distance from the end, x, the distance between the pivots, L, the Young’s modulus, E, and the second moment of area of the beam cross section, I (Eq. 3).

$$ EI \frac{{{\text{d}}^{2} y}}{{{\text{d}}x^{2} }} = \frac{ - Fx}{2} $$

$$ EI \frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{ - Fx^{2} }}{4} + c_{1} $$

At $x = \frac{L}{2}$, $\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0$. Therefore $c_{1} = \frac{{FL^{2} }}{16}$.

$$ EI \frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{ - Fx^{2} }}{4} + \frac{{FL^{2} }}{16} $$

$$ EIy = \frac{{ - Fx^{3} }}{12} + \frac{{FL^{2} x}}{16} + c_{2} $$

At x = 0, y = 0. Therefore c ₂ = 0.

$$ EIy = \frac{{ - Fx^{3} }}{12} + \frac{{FL^{2} x}}{16} $$

$$ y = \frac{Fx}{EI}\left( {\frac{{L^{2} }}{16} - \frac{{x^{2} }}{12}} \right) $$

$$ y = \frac{Fx}{48EI}\left( {3L^{2} - 4x^{2} } \right) $$

The deflection at the centre of the beam, δ, is the value of y when $x = \frac{L}{2}$:

$$ \delta = \frac{{FL^{3} }}{48EI} $$

(3)

where E = Young’s modulus of the beam material, L = distance between pivots, I = second moment of area of the beam cross section, F = applied force, and δ = total deflection of the beam.

The basic beam bending equation presented above assumes that the beam displays linear elastic behaviour during the application of the bending force and that the deflection is small (proportionally to the size of the specimen). Force–displacement graphs from the AFM display regions of significant adhesion forces. Wood fibre samples also appeared to show hysteresis and a degree of nonlinear and non-elastic behaviour. Therefore, the gradient of the best fit line from the linear region of each curve was measured instead of the ‘peak applied force’ and the ‘total deflection of the fibre’ under bending. This gradient, K, is the bending stiffness (Eq. 4). Factoring in this gradient leads a modified three-point bend equation specific to this study (Eq. 5).

$$ K = \frac{\Delta F}{\delta } $$

(4)

where K = bending stiffness, ΔF = change in applied force, and δ = deflection corresponding to ΔF.

$$ E = \frac{1}{48}\frac{{L^{3} }}{I}K $$

(5)

where E = Young’s modulus, L ₁ = distance between pivots, I = second moment of area of the cross section, and K = bending stiffness.

For simplicity, it was assumed that the cross section of an uncoated wood fibre was in the shape of an annulus, as shown in Fig. 11. It was also assumed that the fibre cross section was constant along the length of the fibre. This allowed the following equation to be used to calculate the second moment of area (Eq. 6):

$$ I = \frac{\pi }{4}\left( {r_{1}^{4} - r_{2}^{4} } \right) $$

(6)

where I = second moment of area, r ₁ = outer radius, and r ₂ = inner radius.

Figure 11

Illustration of an annulus

The equation was modified to be more specific to the study by writing r₁ and r₂, in terms of the average outer mid-section diameter, Avg D, and the wall thickness of a fibre (Eq. 7).

$$ I = \frac{\pi }{4}\left( {\left( {\frac{{{\text{Avg}} D}}{2}} \right)^{4} - \left( {\frac{{{\text{Avg}} D}}{2} - t} \right) ^{4} } \right) $$

(7)

where I = second moment of area of the fibre cross section, Avg D = average mid-section diameter of the fibre, and t = wall thickness of the fibre.