# Graphical Representation of Fracture Processes

The energetic equations allow graphical interpretation which, in many instances, supply. vivid pictures helpful for problem-solving and explaining. A loading process of a specimen or structure is sometimes best followed on a load-displacement plot. In the case of a single load P, the displacement to consider is the load-point displacement u. Let the plot of a quasi-static P-и curve for a given specimen be as shown in Fig. 2.1.4a. The work supplied to the specimen from the beginning of the loading, point O, up to point A, is the integral of Pdu, which is equal to the area OMAA’O shown in Fig. 2.1.4b. The area that complements this to rectangle О A" A A’ is the integral of udP, and is called the complementary work.

If all of the material remains linear elastic except for a zone of negligible volume along the crack path, the (elastic) strain energy U is represented by the area of the triangle OAA’ in Fig. 2.1.4c. The complementary energy U* is the area of the triangle OAA" which complements OAA’ to the rectangle О A’ /1/1", of area Pu.

The energy supplied for fracture is the difference between the work and the fracture energy, hence, the area О MAO in Fig. 2.1.4d (it is also the difference between the complementary energy and the complementary work). If the curve shown corresponds to an actual quasi-static fracture process, then the energy supplied for fracture must coincide with the energy consumed by fracture; hence, the area OMAO in Fig. 2.1.4d is also the energy consumed in fracture.

An equilibrium fracture process from point A, where the crack length was a, to a nearby point B, where the crack length has increased by Да, may be represented as shown in Fig. 2.1.5a. The energy release available for fracture is the area of triangle OAB. This area coincides, except for second-order small terms, with those corresponding to the virtual (nonequilibrium) processes represented in Figs. 2.1.5b, c, and d, respectively, corresponding to constant displacement, constant load, and arbitrary ДР/Д«. The energy release rate Q is the limit of the ratio of the area of any of the shaded triangles to the crack extension. This shows, again, that Q is path independent, hence, a state function. When Fig. 2.1.5b is used, Eq. (2.1.15) is obtained. When Fig. 2.1.5c is used, Eq. (2.1.21) is obtained. Both turn out to be identical to Eq.(2.1.32) as the reader may easily check. For example, taking the shaded triangle in Fig. 2.1.5c, we express the fracture energy as

g bAa = area(0/lB”) = X-P (Ш) = X-P{PC{a + Да) – PC(a)] = X-P1C'{a)Aa (2.1.40) from which Eq. (2.1.32) immediately follows.

As previously stated, only two of the four variables P, u, a, and Q can be taken its independent variables. Any pair of them may be used to define the entire fracture process. However, it is useful to take conjugate variable pairs, as is customary in thermodynamics, because then the areas in the graphical representation have direct energetic interpretations. One of such pair is the P-и representation just analyzed. The other is the Q-a representation. This representation has the advantage that Q is the “driving force’’ for crack growth which is directly related to the material property 71, the “resisting force”.

If one then imagines a plot of a loading process in a G-a plane, such as that in Fig. 2.1.6a, one finds that OM is a loading at constant crack length under increasing Q. At point M, the crack starts to increase under increasing Q up to point A. The total energy released is the integral of 0 da, equal, henceforth,   Figure 2.1.6 (a) Loading path in a Q – ■ a plot, (b) Area representing the total energy supply for fracture and the energy dissipated in fracture.

to the area О MAD’О in Fig. 2.1.6b. Moreover, if the process is an actual quasi-static (equilibrium) process, this coincides with the total I’raclure energy.

Furthermore, the instantaneous Q on the M A portion, where the crack is actually growing, must coincide with the instantaneous fracture energy which, in this example, is not constant. This is an example of the so-called Л-curve behavior, a short for resistance curve behavior, in which the crack growth resistance increases with the crack extension (Chapter 5). In Section 2.1.6 we argue that this is not a kind of behavior consistent with the hypotheses of LEFM, which really imply that the crack growth resistance must be a constant.