Web breaks and fracture toughness
Tensile strength gives the maximum tension-carrying capacity of paper but does not directly measure the “runnability” on high-speed printing presses, for example. The web tension in such applications is far below the ordinary tensile strength of paper. Because web failures are infrequent, large amounts of paper pass an open draw before a failure occurs. The size of the “specimen” is therefore huge, and its strength is much lower than the ordinary tensile strength determined from a few tests with exceptionally low values excluded on purpose.
The failure of a running web is usually due to a defect in the paper or a transient peak in the web tension that causes failure at an otherwise harmless defect. Fracture toughness measures the ability of paper to resist the growth of small incipient cracks. Web breaks are very infrequent. If one measures the runnability as the mean distance between breaks in papermaking or printing, a typical result is then 106 m or more.
Even at such a low rate, web breaks cause significant losses in production. The low frequency of breaks also makes it difficult to evaluate the runnability of a particular paper grade. Under normal circumstances, 1,000 to 10,000, rolls of paper would need to be run on a printing press to distinguish statistically between papers. Compiling the necessary data could take months during which time climatic conditions and paper properties could vary significantly 1.
A paper web fails if the local strength is too low somewhere in the web or the momentary load is too high. The frequency of breaks increases with increasing web tension, perhaps even exponentially, as shown in Figure 1. Significant numbers of breaks occur at web tensions that are an order of magnitude lower than the tensile strength of paper. They are caused by spatial fluctuations in the local strength of paper or temporal fluctuations in local load. The variance in local strength and load must therefore be sufficiently low. A low break frequency relies heavily on uniform operation of the paper machine or printing press.
Figure 1. Break frequency of newsprint webs in linear and logarithmic scale vs. tension on a winding device run at 400 m/min with curves corresponding to different papers 2.
Studies have indicated that web breaks can be caused by distinct flaws or defects in the paper web 1,2,3, while also opposite conclusions have been drawn, i.e., that defects do not explain web breaks 4. In papers made from a mechanical furnish, shives or rough fibres can sometimes create such defects 5. Disturbances in the wet end chemistry of the paper machine and creases, edge cuts and other physical defects are other possible causes of breaks. In coating and printing, paper is softened by water and heat. In the other extreme, a low moisture content makes paper brittle. The mechanisms of breaks are probably quite different in dry and wet paper, but little research has been published.
The local strength minima causing breaks in a paper web strongly depend on the nature and size of flaws or defects. Conducting a statistically sensible measurement of the local strength of paper in the presence of different flaws would be very difficult. Fracture toughness or fracture resistance is a material property that describes the ability of paper to resist a flaw from becoming a growing crack. To allow larger flaws in a paper web the fracture toughness of paper must be greater. Statistically correlating the break frequency with any paper property is difficult and controversial. Some evidence that fracture toughness relates to the frequency of web breaks does exist 1,5, while other analyses find the best correlation with low tensile strength 4.
To characterise the endurance of paper in the presence of flaws, one must consider what happens in the vicinity of an existing flaw. What are the paper properties affecting the extensibility of the flaw and the break propensity of the web? If an existing crack grows, elastic energy is released because stress vanishes at the crack faces. According to the basic concept of linear elastic fracture mechanics, LEFM, the Griffith criterion, a crack cannot grow unless this released energy is at least equal to the energy necessary to overcome the fracture resistance, or toughness, of the material 6. The Griffith argument needs to be augmented for materials such as a paper that are heterogeneous and exhibit non-linearities such as plastic yielding.
In a linearly elastic body, the condition for crack growth is:
- G = -dΠ/dA is the decrease in elastic energy per crack area increment (or the “crack-driving force”),
- β is a geometric factor,
- σ the remote stress far from the crack,
- a the crack length,
- E’ an elastic constant, and
- R = Gc the fracture energy of the material.
In isotropic paper under plane stress, E’ is equal to the ordinary elastic modulus. In anisotropic paper, E’ is given by a combination of the elastic constants 6.
A web with a crack releases elastic energy at a rate, G, if the crack grows. According to Eq. 1, the energy release rate, G, is proportional to web tension squared. If the web tension increases, at some point the energy release rate, G, reaches the fracture energy of the material, Gc = R. An infinitesimal increment of tension will then cause the crack to grow. If the crack keeps on growing even when tension is constant, the web will break.
Fracture toughness Kc is the following:
If the paper web contains a flaw of a given size, a, then the critical web tension or the apparent breaking stress – apparent tensile strength – of the web is directly proportional to Kc. More precisely, it follows from Eq. 1 that
- a is the flaw size, and
- β a geometric factor that depends on the location of the defect in the web.
In reality, Eq. 3 must be revised for two related reasons. First, if the plastic yielding inside the fracture process zone (FPZ) dominates, the flaw size, a, needs to be substituted by the sum of a and the intrinsic FPZ scale ξ:
This implies that the role of the plasticity is to cut off the expected divergence of the LEFM stress field in the vicinity of the crack tip. Equation 4 also accommodates for the limiting case where the flaw size, a, goes to zero. Even then, paper should naturally have a finite tensile strength, not infinite as Eq. 3 would predict.
Analogues of Eq. 4 abound in the fracture mechanics of quasi-brittle materials such as concrete, which share with paper the property of an inhomogeneous structure 7,8. If the fracture process and surrounding plastic deformations occur in a zone that is large compared to the size of the flaw or that of the object, one needs other approaches. Textbooks on basic fracture mechanics address this 9. In many paper web applications, the speed difference in open draws and not the web tension is the controlling parameter. The relative speed increase (in per cent) in an open draw directly gives the strain imposed on the web. For flaws of a fixed size, the critical strain level Kc/E follows from Eq. 3.
All in all, the critical web tension is thus a function of the fracture toughness, defect size and geometry. Figure 2 illustrates the application of Eq. 4 to large paper sheets. The geometric factor β is different for cuts in the centre and in the edge of the sheet. Small enough cuts have no effect on strength, presumably because strength is then determined by the formation-like non-uniformity and not by the small cuts, as we will discuss next.
Figure 2. Tensile strength vs. cut location (centre, edge) in large (1 m x 1.8 m) paper sheets. Tensile strength is given as the deviation from the strength of the sheet without a cut. The +24% value at zero cut size refers to the standard tensile strength measured from 15-mm-wide paper strips 10.