Triaxial deformations

We have so far considered uniaxial stress-strain behaviour, i.e., the case where the stress and strain are measured in the same in-plane direction. In the triaxial case, one measures the full three-dimensional strain state for a given load. In such measurements, complications arise from the low thickness of paper and board. These make measurement of out-of-plane strains difficult. In addition, thin sheets buckle and wrinkle easily if not loaded in the principal directions.

Baumgartner and Göttshing ¹ have comprehensively studied the triaxial deformations of different paper grades under in-plane tensile stress ε_x. Figure 1 shows the effect of increased beating in handsheets, and Figure 2 compares machine-made papers. In the isotropic handsheets, the lateral in-plane contraction, -ε_y, increases almost linearly with elongation, ε_x. The ratio, ε_y/ε_x, is equal to the ordinary Poisson ratio at the limit ε_x → 0. In the machine-made papers, the corresponding curve turns upwards. It therefore seems that the ratio, ε_y/ε_x, generally increases at large stress levels. At first, this is surprising because the function of the lateral contraction, ε_y, is to make the lateral stress, σ_y, vanish, and σ_y should increase at the same rate as σ_x. A plausible explanation is that the fibres aligned in the lateral y-direction are under compressive stress and lose their load-carrying capacity earlier than the fibres in the longitudinal x-direction that are in tension.

In the thickness direction, the handsheets contract under in-plane elongation, although the contraction decreases at large elongations. In the machine-made papers, no contraction occurs, and paper thickness increases even faster when elongation increases. The initial contraction of the handsheets is the ordinary behaviour of any homogenous material. Elongation in one direction causes contraction in the transverse directions. The opposite phenomenon is transverse expansion. This may relate to the gradual opening of inter-fibre bonds. As bonds open, z-directional fibre undulations in the network relax and thickness increases.

Figure 1. Stress (top), in-plane lateral contraction (middle), and out-of-plane contraction (bottom) vs. external elongation of handsheets with low and high beating (open and solid symbols, respectively) ¹.

Figure 2. Stress (top), in-plane lateral contraction (middle), and out-of-plane contraction (bottom) vs. MD elongation of machine-made papers for uncoated printing paper (open circles), coated printing paper (open triangles), newsprint (open squares), sack paper (closed triangles), and folding boxboard (closed squares) ¹.