General Forms of the Expressions for Kj and Q

Since crack growth in LEFM is defined by the condition Q = Gj or К/ = К/с, we need to know the structure of the equations for Ki and Q if we want to investigate the influence of the size. And since size effect is one of the main topics of this book, it is also convenient to define the conventional forms of the equations we are going to use so that the size D is made explicit. Systematization of the presentation of the existing results also requires using general mathematical forms of the equations for Kj and Q, so that a single experimental or numerical result may be used for any similar specimen or structure.

To determine the general form, we consider a family of geometrically similar structures subjected to the same type of loading (for example, the center cracked panel in Fig. 2.1.1 or the DCB specimens in Fig. 2.1.3). Let D be a characteristic dimension (for example, the panel width or the arm depth in the DCB specimen), all the remaining dimensions being proportional (for example, the height-to-width ratio of the panel and the total length-to-depth ratio for the DCB), except for the crack-to-depth ratio a/D, which is free to vary. The purpose of the analysis is to obtain the general expression for Q and Кj showing explicitly the dependence on the variables P (or одг), D and a = a/D. Let us first elaborate the examples:

Example 2.3.1 For the center cracked panel with a short crack (a — a/D 1) the expression for the stress intensity factor (2.2.3) can be written in either of the two following forms:

К і = сгц/Ъ/тта = (2.3.9)

where we set <7дг — <j — P/bD, and b is the panel thickness. D

Example 2.3.2 For the DCB specimen of Fig. 2.1.3b, we take the approximate expression (2.1.29), substitute it into Irwin’s relationship (2.2.22) (assuming plane stress, E’ E), and then we get

Подпись:Ki — a jvfD2ois/b = —=2a/3


where we substituted D = ft, <j/v = P/bD. D

Подпись: K, General Forms of the Expressions for Kj and Q Подпись: (2.3.11)

The resemblance of the expressions for the these two simple cases is evident. They only differ in the factors depending on a (yAra for the panel and 2uJb for the DCB specimen). This result is general. Indeed, because the response is linear elastic, the stress intensity factor must be proportional to the force per unit thickness Pjb. Since the stress intensity factor must have dimensions of ForcexLength-3/2 and must depend on the relative crack depth a, the only possible forms of the expression based on P and о-дг are:

where k(a) and k(a) are dimensionless functions, a — a/D is the relative crack depth, and k(a) -■= k{ct)/cN. The convention of using ‘hatted’ к for expressions based on P and plain к for expressions based on <T/y will be retained throughout the book.

The general forms for the energy release rate Q may be obtained directly from the foregoing by using Irwin’s relationship (2.2.22). They are

Q ШШ’ 0Г G = 1?D 9^ (2.3.12)


g(a) — k2(a) and g(a) = —g(a) — k2(a) (2.3.13)


In what follows, we systematically use the forms (2.3.11) and (2.3.12). Other equivalent forms can be found in the literature, as discussed latter.

Another simple argument leading to (2.3.12) is_to note that the complementary energy of the structure must be expressible as U* — 2UVf (a), where U •- a2N/2E’ is a nominal energy density, V = bD2 is the volume of the structure and /(alpha) is a dimensionless function. Then Q = dll" fbda = (8U*/da)/bD — (a2N/E’)Df'(a). Setting g(a) = f(a), one gets Eq. (2.3.12).