Microscopic yielding phenomena

When a paper sheet stretches, two phenomena occur at the microscopic level. Fibre segments elongate — in part irreversibly — and some bonds open gradually. These processes must consume energy because the area under the load elongation curve (TEA) represents work done on the system. Only part of that work is recoverable in elastic recoil upon release of the external stress.

Despite the irreversible work when paper stretches in the plastic region, the number of load-carrying elements in the fibre network does not change dramatically. Remember that the elastic modulus is a measure of the elastic energy content. Since the elastic modulus does not change significantly, as shown in Figure 4 in Macroscopic observations, the number of fibre segments that bear the load must remain almost constant during the irreversible elongation.

Ebeling ¹ has measured the heat flow into and from paper during load-elongation tests. In the elastic region, heat flows into paper when strained, as shown in Figure 11. In thermodynamics, this is the Kelvin effect. The internal energy of the system increases as the sum of the external work and heat flow. If the external load is removed in the elastic region, paper releases the same amount of heat that it absorbed during elastic elongation, and the internal energy returns to its original value. The elastic behaviour of paper is thermodynamically reversible.

Figure 1. Rate of mechanical energy input or output (top) and rate of heat flow into or out of rag paper in CD. The strain rate being constant, the rate of mechanical work is directly proportional to external stress indicated on the right ¹.

In the plastic region, Figure 1 shows that paper releases heat with decreasing and increasing elongation. Plastic elongation is thermodynamically irreversible. Since the elastic modulus does not change correspondingly, the irreversible heat generation must come from changes in the microscopic or molecular configuration within fibres and bonds. Complete rupture of fibres or bonds cannot explain the thermodynamic observations ¹.

Seth and Page have demonstrated that average behaviour of fibres can qualitatively explain the nonlinear load-elongation behaviour of paper when differences in elastic modulus are removed ². For example, the influence of drying shrinkage on the load-elongation curve of paper comes from an increase in plastic elongation, as indicated in Figure 2. Experiments with single fibres ³ show that fibres exhibit a similar increase in plastic elongation as paper does when dried under compressive stress, and likewise fibres dried under tensile stress have a more linear load-elongation curve such as paper does. Aside from the changes in elastic modulus, the actual shape of the load-elongation curve of paper changes little with drying shrinkage except for the location of the end point.

Figure 2. Stress-strain curves of handsheets with different drying restraints (a) and wet pressing levels (b). Stress values are normalised to a constant elastic modulus. The result is equivalent to the elastic strain ² ε_el = σ/E.

Changes in the bonding degree of the network have no systematic effect on the shape of the load-elongation curve. In order to demonstrate this, we divide stress values by the elastic modulus, which leads to the curve of elastic strain, ε_el = σ/E, vs. total strain. As shown by Figure 2, the shape of this curve is independent of changes in density induced by wet pressing. Density only affects the elastic modulus of paper and the location of macroscopic failure on the load-elongation or elastic strain-total strain curve. Changes in density alter the bonding degree, but since this has no effect on the shape of the load-elongation curve, neither can the opening of bonds explain the shape of the curve.

At the microscopic level, the effect of drying shrinkage or drying stress is largest on bonded fibre segments. Drying shrinkage induces “microcompressions” and other deformations into the fibre wall at bond sites. Drying stress and wet strain pull fibres straight and tight. The bonded fibre segments exhibit more plastic yielding in freely dried paper than in restraint-dried paper. Giertz and Roedland ⁴ demonstrated this directly by measuring segment elongations in paper sheets, see Figure 3. At small external strain, all fibre segments elongate fairly uniformly. At large strain, some bonded segments elongate more than the free segments between bonds. Permanent elongation occurred primarily in bonded segments. Free segments recovered elastically except in freely dried paper where free segments also underwent irreversible elongation.

Figure 3. Elongation of bonded and free fibre segments (top and bottom, respectively) vs. external strain of paper ⁴.

When external stress returns to zero after a loading cycle into the plastic region, bonded fibre segments are, on average, under compressive stress and free segments under tensile stress. This follows because free segments have not elongated plastically. Therefore, they must be under tensile stresses whenever the external strain is larger than zero. This is the case after loading the paper into the plastic region. As the total stress of the network is zero, we conclude that bonded segments are under compression.

Ebeling ¹ confirmed the generation of local compressions and tensions in his thermodynamic measurements, Figure 4. He found that the internal energy of paper increases in a loading cycle into the plastic region. The increase in the internal energy in such a cycle equals the mechanical work minus the net heat released from paper. Internal energy increases because there is elastic energy stored in the compressed and stretched segments even though the average stress is zero.

Figure 4. Tensile energy absorption (TEA, solid line), hysteresis work (dashed line), and change in internal energy (dash-dotted line) against the maximum strain turning point in a loading-deloading cycle of a rag paper in CD. Hysteresis work equals TEA minus the work recovered in deloading ¹.

Inter-fibre bonds open gradually when paper stretches. The opening of bonds is not the cause of the nonlinear curve, as noted earlier. Page, Tydeman and Hunt studied the opening of bonds under a light microscope⁵. Most bonds opened gradually and did not break completely even when paper was strained to failure. Complete bond failures were rare.

The gradual opening of bonds can be understood by the distribution of internal stress within the often irregular bond areas ^6,7. The reduction of bonding area with paper elongation is reflected in the optical properties of paper. For example, the light scattering coefficient increases. For a given paper, the increase in light scattering coefficient is linearly proportional to the plastic elongation, as shown in ⁸ Figure 5.

Figure 5. Light scattering coefficient vs. plastic strain of paper in a load-elongation test ⁸.

Since the load-elongation behaviour of paper must be to a great degree controlled by the corresponding fibre behaviour, one might try to derive a model that gives the link in a quantitative form. For example, in micromechanical models one considers a typical fibre embedded in a homogeneous background ^9,10. The predictive power of such models is limited because one cannot measure all the detailed properties of fibres and bonds that the models require. Moreover, these properties, as for example bond strength, have statistical distributions that should be included.

These complications are reflected in the stress distribution in fibre segments. It is stochastic because of the random network structure. Segments of high stress start yielding plastically at smaller external strain than segments of low stress. Theoretically, the segment stress distribution could determine the shape of the load-elongation curve of paper. A qualitatively reasonable load-elongation model for paper results if one combines a plausible network stress distribution with a fibre model where the stress-strain curve is composed of two linear parts ¹¹.