# Measurement of fracture energy

Compared with many other mechanical properties of paper, fracture energy is difficult to measure because the elastic energy release rate, G, easily shoots over the critical value, Gc. The latter is therefore difficult to determine. One must try to determine G right at the onset of crack propagation or measure G during stable sub-critical crack propagation where external work must be constantly supplied. In both cases, a complication is that the total fracture energy is independent of size and geometry only for very large objects. In elastic-plastic materials especially, it is difficult to confine the plastic yielding to a sufficiently small area and other methods are necessary to separate the actual fracture energy from plastic work.

Two notes on the terminology are necessary before proceeding with the fracture energy measurements. First, in accordance with the bulk of fracture mechanics literature, we use the term fracture toughness for Kc defined in Eq. 2 in Web breaks and fracture toughness. Often Kc is also the “critical stress intensity factor” and sometimes “tenacity” 1. In turn, the fracture energy (Gc or R) is sometimes also called fracture toughness. We prefer the notation where fracture toughness is calculated from Eq. 2 in Web breaks and fracture toughness and denoted Kc.

Second, one should make sure not to confuse fracture energy with the tensile energy absorption, TEA, that is the integral of the load-elongation curve:

 $TEA={\int }_{0}^{{\epsilon }_{br}}\sigma d\epsilon$ (1)

where εbr is the breaking strain. If stress, σ, is in MPa, Eq. 1 gives the energy per unit volume. The corresponding TEA index [J/g] results when dividing by density. Since TEA is the integral of the load-elongation curve, it grows with increasing tensile strength and breaking strain and above all depends on the shape of the load-elongation curve.

Fracture energy, in turn, gives the energy per unit length of the fracture line, and therefore has the units of J/m, or Jm/kg if indexed with grammage. In the J-integral method 2, one uses specimens with different initial notch lengths, a. For each length, one determines the energy adsorption, W(ε), as a function of external strain, ε. The energy release rate, or J-integral, for any given strain is then calculated numerically from the following

 $J\mathit{=}\mathit{–}\mathit{d}\mathit{W}\mathit{\left(}\mathit{\epsilon }\mathit{\right)}}{\mathit{d}\mathit{a}}$ (2)

The critical value of the J-integral, Jc, is determined by the strain, ε, at which the notches start to crack. The method is tedious because W(ε) is necessary for a few notch lengths, a.

To avoid the use of multiple specimens, methods are available for approximating Jc from specimens with a single notch length 2,3. In the method developed at STFI 4, the J-integral is numerically determined for a centre-notched specimen as a function of displacement. To do this, a material model first requires measurements with ordinary flaw-free tensile specimens. The critical displacement at maximum load is then measured for the notched specimen, and the corresponding critical value, Jc, is calculated from the numerical model for the J-integral.

In a completely different method, one determines the fracture energy or “essential work of fracture” (EWF)5 using specimens with a double-edge notch. One measures the total dissipated energy, Wtot. This energy is consumed by rupture at the crack and by plastic yielding in an elliptic zone outside the crack, as shown in Figure 1. The first energy contribution is therefore proportional to the ligament length, L, of the specimen of thickness t and the second to the size of the plastic zone or approximately L2. Linear extrapolation of Wtot/Lt to L → 0 gives we as the result. Proper specimen dimensions are necessary to keep the fracture stable. Some older publications give Wtot / Lt, (called fracture resistance by them), directly as an estimate of the fracture energy. Figure 1. Regions of energy dissipation in the measurement of essential work of fracture (EWF).

The fracture energy may also be measured using short specimens 6,7. With a short specimen, the fracture process is stable because the stored elastic energy shown by the hatched area in Figure 2 is not sufficient to break the specimen. After the maximum load and initiation of fracture, the external strain must continuously increase to break the specimen completely. In a long specimen, the process would be unstable since there is elastic energy available in abundance for the fracture. Figure 2. Stress vs. elongation for a short and long specimen. In both cases, the total energy consumption is the area under the curve, the stored elastic energy is given by the hatched area, and the fracture energy is the grey area 7.

The fracture energy is the grey area in Figure 2 — assuming that the fracture process occurs along a narrow zone that extends across the specimen. The width of this zone is uz = u – ur, where u is the total external elongation and the elongation of the undamaged specimen is

 ${u}_{r}={u}_{c}–\left({\sigma }_{c}–\sigma \right)L}{E}$ (3)

where

• uc and σc are the values of displacement and stress at maximum load, respectively,
• σ stress,
• L specimen length, and
• E elastic modulus.

After the maximum load ur decreases because stress decreases. The fracture energy, Gf is then given by

 ${\mathrm{G}}_{\mathrm{f}}={\int }_{0}^{\infty }{\mathrm{\sigma du}}_{\mathrm{z}}$ (4)

where the fracture zone width uz is by definition zero before the maximum stress occurs.

The various measures of fracture energy often give different values, due to fundamental differences or problems in the practical implementation. Only in the case of linearly elastic paper does one expect that all the fracture energies are equal, Gf = we = Jc = Gc.

Because of the easy measurement, in-plane tear test has been often used to estimate fracture energy. Usually the in-plane tear values correlated closely with the J-integral values 8, 9. The in-plane tear test gives the total energy per tear length consumed when a specimen of a given geometry undergoes tearing 10. Load is applied in the plane of paper, often at a 2–6° angle, as shown in Figure 3(a). In the out-of-plane version or Elmendorf tear test of Figure 3(b), load is in the out-of-plane direction. Figure 3. Principle of the in-plane and out-of-plane tear test, (a) and (b), respectively.

Since a tear test measures the total energy consumption, the result may include energy spent in plastic elongation outside the crack path, or elastic energy dissipated in the system. In both the in-plane and out-of-plane tear test, some energy may also go toward out-of-plane deformation and delamination of paper. In the past, out-of- plane tear was used to quantify the runnability of paper. However, extensive analysis 11 has later shown that there is no statistically valid connection between tearing resistance and web break frequency.