Figure 1 shows a typical load-elongation curve of paper. In MD, the curve is relatively short and linear. In CD, it extends to large strains and is markedly nonlinear. Qualitatively, the MD behaviour is near that of a linearly elastic material, and the CD behaviour is similar to that of an elastic — ideally plastic — material. Ideal plasticity means that above a yield threshold the load does not increase although elongation grows. The primary difference between the MD and CD curves is in the effect of drying stress. Drying stress is the most important single factor that affects the shape of the curve.
Figure 1. Cyclic stress-strain curve of a copy paper in MD and CD. Stress is divided by elastic modulus to obtain the elastic strain (KCL, unpublished data).
One could think that the nonlinearity in the load-elongation curve arises from a decreasing elastic modulus. In that case, elongation would be entirely reversible. In reality, most of the nonlinearity arises from irreversible or plastic elongation. If a paper specimen is stretched beyond the yield point and then released, it becomes permanently longer due to irreversible strain. A small part of the initial nonlinearity is reversible, as demonstrated by the shape of the deloading-loading cycles in Figure 1.
The elastic modulus of paper, E, usually changes relatively little during elongation. The reversible or elastic strain, εel, can therefore be calculated approximately by dividing stress, σ, with elastic modulus, εel = σ/E. The remainder, ε – σ/E, is equal to the plastic, irreversible strain component, εpl. Considering the elastic strain vs. total strain curve instead of the normal load-elongation curve is often useful. Figure 1 is a case in point.
In the load-elongation curve, an approximately linear section exists at small strains whose slope is the elastic modulus. The linear section ends at a yield point, and plastic elongation only becomes significant above that. Figure 2 demonstrates that the yield point has no unique definition because the deviation from a linear load- elongation curve begins to grow gradually when elongation increases. One common criterion for the yield point is a 0.2% strain deviation from the linear trend. Although the deviation increases gradually, a clearly non-zero value for the yield point is the result if one uses any finite strain offset as the criterion.
Figure 2. Onset of nonlinearity in the load-elongation curve of paper expressed by the plastic strain, εpl = ε – σ/E (on logarithmic scale) vs. total strain, ε. The inset shows the definition of plastic strain (KCL, unpublished data).
The evolution of the elastic modulus outside the linear region can be evaluated from load-elongation cycles. If one removes the external stress completely and then raises it again, the slope at the turning points gives two estimates, which are shown in Figure 3: the deloading and reloading elastic moduli, Ed and Er. One can also use high-frequency, small-amplitude strain cycles that give a dynamic measure for the elastic modulus, Ec, at any point along the load-elongation curve 1. The dynamic modulus exhibits behaviour that is similar to the deloading and reloading moduli.
Figure 3. Definition of the dynamic elastic modulus, Ec, deloading modulus, Ed, and reloading modulus, Er.
In the measurements with machine-made papers in Figure 4, the MD values of Ed and Er were almost equal and decreased only slightly at 10–15% over the entire load-elongation curve. In CD, Ed and Er vary more than in MD. At the macroscopic failure, Ed is typically 10% larger than Er. In some cases in Figure 4, Ed even increased. The increase in CD is presumably due to the straightening of micro-compressions and other structural deformations created in drying. In MD, the structure of machine-made paper has been rectified in papermaking by wet strain and drying stress. In some handsheet trials, Ed has increased with elongation by as much as a factor of two 2.
Figure 4. Reloading modulus Er (over initial elastic modulus, E0) vs. elastic strain (equal to stress/initial modulus) in some machine-made papers for MD and CD (a and b, respectively) (KCL, unpublished data).
The load-elongation curve ends when the specimen fails. The end point gives the breaking strain of paper, and the maximum stress along the curve defines the tensile strength. In standard terminology, one uses breaking strain rather than breaking elongation. The area under the load-elongation curve until the failure point is the work necessary to break the specimen or the tensile energy adsorption (TEA).
The shape of the load-elongation curve depends on the rate of elongation. At high strain rates, the curve is steeper than at low strain rates, as shown in 3 Figure 5. The time-independent component implies that the elongation of paper is not merely a visco-elastic phenomenon but contains a truly plastic permanent component. Paper is therefore a visco-elastic plastic material, together with the other time-dependent rheological properties of paper.
Figure 5. Stress-strain curve of an MG paper (a) and cellophane (b) at different strain rates 3.