*2.3. Essential Work of Fracture (EWF) Approach*

In order to characterize the polymers fracture, the linear elastic fracture mechanics (LEFM) approach is generally adopted. However, the characterization of fracture toughness by LEFM theory is difficult when relatively ductile polymers are considered due to the formation of a large plastic zone prior to crack initiation that violates the validity of LEFM approach. The fracture event of a relatively ductile polymer, having a marked plastic deformation zone at the crack tip, can be investigated by the essential work of fracture approach (EWF) originally suggested by Broberg [59] and then developed by Cotterell, Mai and co-workers [48,60–62]. According to this methodology, the fracture process zone is divided into two regions: an inner region (where the fracture process occurs) and an outer region (where the plastic deformation occurs). The total work of fracture follows this regions division and it can be separated in two contributions: the first contribution is related to the work spent in the inner fracture zone (work of fracture) and the work spent in the plastic deformation zone (non-essential work of fracture) [48].

For the application of the EWF theory, two different kinds of specimen can be used: the single edge notched (SENT) specimen and the double edge notched (DENT) specimen. Generally, the EWF approach is applied to thin sheets having a thickness between 1 mm and 0.2 mm [63]; the thickness of the specimen used (1.5 mm), although slightly higher than the standard range, is very low and consequently an attempt of applying the EWF approach was maintained.

The SENT specimen configuration illustrated in Figure 1 was adopted.

The total work of fracture, *Wf*, is defined as:

$$\mathcal{W}\_f = \mathcal{W}\_\varepsilon + \mathcal{W}\_p \tag{9}$$

where *We* is the work spent for the formation of two new fracture surfaces and it is spent during the fracture process and corresponds to the resistance to crack initiation. For specimens having a given thickness, *We* is proportional to the ligament length, *l* (where *l* = *W* − *a*). In the Equation (9), *Wp* is a volume energy, it is proportional to *l* 2, and corresponds to the energy for activating the plastic deformation mechanism antagonists to the crack propagation. Consequently, Equation (9) can be rewritten in the following form (Equation (10)):

$$\mathcal{W}\_f = w\_\varepsilon t l + \beta w\_p t l^2 \tag{10}$$

The specific total work of fracture can be written in the following form:

$$w\_f = \left(\frac{W\_f}{tl}\right) = w\_\varepsilon + \beta w\_p l \tag{11}$$

where *we* and *wp* are the specific essential work of fracture and the specific non-essential work of fracture, respectively; *β* is the plastic zone shape factor while *t*, and *l* are related to the specimen geometry and correspond to the thickness and ligament length of the SENT specimen, respectively.

With the assumption that *we* is a material constant and *wp* and *β* are independent from the ligament length, it is possible to plot Equation (11) as a straight line in a graph *wf* vs. *l*. Consequently, *we* can be determined from the intercept of the *Y*-axis of the *wf* versus *l* plot, while *βwp* is the slope of the straight line. It must be pointed out that β depends on the specimen geometry and on the initial crack length so the straight line relationship can be obtained only if the geometric similarity is retained for all ligaments lengths.
