The General Energy Balance

Consider a plane structure of thickness b in which a preexisting straight crack of length a is present (Fig. 2.1.2). Assume that upon quasi-static loading, a certain load level is attained at which the crack advances an elemental length 6a in its own plane, sweeping an element of area 6A = b6a. The energy SWP required to do so is the increment of area times the crack growth resistance:

SWF = 1ib6a (2.1.1)

%

Alternative notations found in the literature for the crack growth resistance Ті are Gr when it is history – dependent, and Gc and Gjc (critical energy release rate) when it is a material property not dependent on the cracking history. In this book, we use Gf for the latter case because this is the most widespread notation in the field of concrete fracture. Gf is called the specific fracture energy, or fracture energy, for short. In Section 2.1.5 we will justify that the only case consistent with the hypotheses of elastic fracture, where the inelastic zone is negligibly small, is that of Ті = constant — G/.

The total energy supply to the structure is the external work, which, in the infinitesimal process under consideration, is denoted as 6Y>. From this total supply, a part is stored in the structure as elastic energy, 6U. The remainder is left to drive other processes and to generate kinetic energy SIC. When the only energy-consuming process is fracture, and the process is quasi-static (SIC – 0), this remainder is the
available energy for fracture, or elemental energy release 8WR:

6WR = 6W-SU (2.1.2)

Although Eq. (2.1.2) may be directly handled in many cases (as done by Griffith 1921), it is often more convenient to work with specific energies (energies per unit area of crack growth). The specific available energy, usually called the energy release rate, Q, is thus defined so that

Q b6a = 6WR = 8W -6U (2.1.3)

The essential advantage of using Q is that, as it will turn out in the next paragraph, Q is a state function. This means that Q depends on the instantaneous geometry and boundary conditions, but not on how they vary in the actual fracture process, or on how they have been attained.

The balance’ of energy requires that, in a quasi-static process,

G 8a — 715a for quasi-static growth (2.1.4)

and in a more general incipiently dynamic situation (initial kinetic energy 1C — 0, kinetic energy increase 81C > 0),

G5a = 718a + 8K./b (2.1.5)

Since 5K > 0 (because initially K, = 0 and always К > 0), the equations may be made to hold in any circumstance (as they should, the balance of energy being a universal law, the first law of thermodynamics)

if the following fracture criterion

is met:

if

Q <71

then

8a — 0

and

6K = 0

No crack growth (stable)

(2.1.6)

if

G = 71

then

8a > 0

and

SK. = 0

Quasi-static growth possible

(2.1.7)

if

G>71

then

8a > 0

and

V

О

Dynamic growth (unstable)

(2.1.8)

This system of conditions summarizes what seems obvious: If the energy available is less than that required, then the crack cannot grow (and the structure is stable). If the energy available equals the required energy, then the crack can grow statically, i. e., with negligible inertia forces (and the structure can be stable or unstable depending on the variation of G – 71 with displacements). If the energy available exceeds that required, then the structure is unstable and the crack will run dynamically (the excess energy being turned into kinetic energy).

The central problems of elastic fracture mechanics are to measure the crack growth resistance, Ті, for particular materials and situations, on one hand, and to calculate the energy release rate Q, on the other. This latter problem may be handled in various equivalent ways, the bases of which are explored next.