# Elastic Potentials and Energy Release Rate

Consider the plane elastic specimen in Fig. 2.1.2, in which the crack length a can take any value. I. et P be the load and и the load-point displacement. By definition, the elementary work is

8W — PSu (2.1.9)

for any incremental process. For an equilibrium situation and given any crack length a, there is a unique relationship between the equilibrium force and the displacement (which can be calculated by solving the elastic problem). So we can write

P = P(u, a) (2.1.10)

where P(u, a) can be determined by clastic equilibrium analysis of the structure. Based on the corre­sponding elastic solution, the stored (elastic) strain energy can also be calculated for any и and a:

U — U{u, a)

Consider, then, that the elastic body with a crack of length a is subjected, in a static manner, to displacement 6u, with no crack growth. In this situation, ail the work is stored as strain energy, so that

SW – [6U}a = 0

where subscript a indicates that the crack length remains constant in this process.

Consider now a general process where both и and a are allowed to vary. Then, Eq. (2Л. З),’which is the definition of G, can be written as

(2.1.13)

Considering equilibrium variation at 6a = 0, one obtains the well-known second Casligliano’s theorem:

This basic result shows that Ihe energy release rate G is indeed a state function, because it depends only on the instantaneous boundary conditions and geometry (in this case uniquely defined by и and a).

Sometimes one may prefer to use the equilibrium load P rather than the equilibrium displacement и as an independent variable. In such case, it is preferable to introduce a dual elastic potential, the complementary energy U*, defined as

U’-Pu-U (2.1.16)

Substituting U from this equation in the expression for the available energy (2.1.2), together with the expression (2.1.9) for the elemental work, one gets for the elemental energy release

6WR r.-. 6W-6U = PSu – 6{Pu – U*) = 6W ~ и 6P (2.1.17)

Writing the complementary energy and the displacement as functions of the applied load and the crack length

и u(P, a) , U* =U*(P, a) (2.1.18)

and considering an equilibrium process in which both P and a are allowed to vary, Eq. (2.1.17) yields

(2.1.19)

Considering equilibrium variation at 6a — 0, one gets the well-known first Castigliano’s theorem

dU*(P, a

dP

The couple of equations (2.1.20) and (2.1.21) are strictly equivalent to the couple (2.1.14) and (2.1.15). Indeed, in this single-point-load problem there are 4 mechanical variables, namely, P, и, a, and G, but only two of them are independent variables (Elices 1987). The choice of the independent variables is arbitrary, and is usually done depending on the boundary conditions and the available data.

Remark: Under isothermal conditions (slow loading, slow crack growth), U represents the Helmholtz’s free energy of the structure and U" its Gibbs’ free energy. Under isentropic (or adiabatic) conditions (rapid loading,
rapid growth), U represents the internal (total) energy of the structure and W represents the enthalpy (see e. g., Bazant and Cedolin 1991, Sec. І0.1).

Other potentials can be used to perform the foregoing analysis. For example, the potential energy П of the structure-load system, defined as ГІ — U – 1 V„(u), where Wa{u) = Pa(u’) dv! is the work of the applied load P„(u), which is assumed to be defined independently of the structure. The energy release rate is easily expressed in terms of the potential energy as

Q = G(u, a) = – i 6

Same as for the strain energy, a dual potential can be defined for the potential energy, the complementary potential energy ІГ of the structure-load system, defined as ІГ —U’-: fV„*(P), where Wa{P) = JP u(P’) dP’ is the complementary work of the applied load (which is 0 for dead loads). The energy release rate is easily expressed in terms of the complementary potential energy as

In this book, the potential energy and complementary potential energy of the structure-load system will not be used. Д

The foregoing results may seem too particular because no distributed loads were considered. This limitation may be overcome in most practical cases by defining generalized forces and displacements. A generalized force Q and its associated generalized displacement q are defined in such a way that the external work 6W may be written as

5W = QSq (2.1.24)

With this definition, all the foregoing expressions hold as long as one interprets P as a generalized force and и as its generalized displacement.

There are many well known cases of generaliz. ed forces used in engineering. For example: the general­ized displacement associated with a torque is the angular rotation; the generalized displacement associated with a pressure acting inside a cavity is the volume variation of the cavity.

Example 2.1.1 To illustrate the application of the above equations, consider a long-arm double can­tilever beam (DCB) specimen subjected to constant moments M as depicted in Fig. 2.1.3a. Assume further that the material is linear elastic, and that the arms are slender enough for the classical theory of bending to apply. With these hypotheses, the elastic or complementary energy per unit length of the beam is known from the theory of strength of materials (e. g., Timoshenko 1956):

dU _ dW _ M2 dx dx 2 El

where x is the coordinate of a cross-section along the beam axis, E the elastic modulus, and I the inertia moment of the cross-section of the beam. We thus compute the clastic or complementary energy of the specimen as the energy of two pure bent cantilever beams of length a:

with M taking the place of P. D

Example 2.1.2 As another example, consider the double cantilever specimen in Fig. 2.1.3b. The bending moment distribution for the upper arm, M — Px, is also shown in this figure. Within the

(b) p|

Figure 2.1.3 Long double cantilever beam specimen subjected to (a) pure bending, and (b) opening end forces.

classical beam theory (neglecting shear), the corresponding complementary energy per unit length of one arm is given now, according to (2.1.25), by dU"/dx = P2x2 /2ЕЇ. The total complementary energy is obtained by integration along both arms of the specimen

With this, Equations (2.1.20) and (2.1.21) provide expressions for the relative displacement between the load points, и, and for the energy release rate Q

2 Pa? 8 Pa? P2a2 ПР2а2

~ 3El ~ ЕЫР ’ G’~ bEI " Eb2h2

in which we set I = bh?/2. Except for a factor 2, the first expression for и in the previous equation is very well known in the field of strength of materials. The factor 2 comes from the relative displacement of the forces (working displacement) being twice the deflection of one beam. D

2.1.1 The Linear Elastic Case and the Compliance Variation

The foregoing general results are greatly simplified in the particular, yet essential, case of linear elasticity, because of the linear relationship between и and P at constant a. This may be written as

и = C(a) P (2.1.30)

where C(a) is the (secant) compliance for a crack length a. After substituting и from Eq. (2.1.30) into Eq. (2.1.20), it immediately follows by integration that the complementary energy must be

Substitution of this expression into Eq. (2.1.21) gives the following result for the energy release rate:

%

4 P2dC{o) P* „ . G(P, a)_2b da ~ 2Ь° a ’

where, in the second expression, the first derivative of the compliance has been briefly denoted as C'{a).

In the foregoing derivation, (P, a) were taken as independent variables. But one can equally well use (u, a) as independent variables. Substituting P from Eq. (2.1.32) into Eq. (2.1.14), it follows by immediate integration that the elastic energy must be

Henceforth, from Eq. (2.1.21), the energy release rate is found to be

which, in view of (2.1.30), turns out to be identical to the previous Eq. (2.1.32), as it must.

At this point it is worth to recall the well-known fact that, in linear elasticity, the elastic energy and the complementary energy always take the same value (although they are conceptually different, as graphically shown in the next subsection). In the case of a single point load, it is sometimes useful to rewrite Eqs. (2.1.31) and (2.1.33) in the form

U = W = -Pu 2

Example 2.1.3 Consider again the pure bent DCB in Fig. 2.1.3a with the same hypotheses as stated in the previous section. Taking M as the generalized force, the relative rotation 6 of the arm ends is the corresponding generalized displacement. Since the rotation of each beam end is 6/2, and such rotation has an expression well known from the strength of materials: 6/2 = aM/EI. Therefore, the generalized compliance is

_ 6 2a

C-M = E1 (2’L36)

The use of (2.1.32) leads again to Eq. (2.1.27) for Q. D

Example 2.1.4 Consider again the long-arm DCB specimen of Fig. 2.1.3b subjected to loads P at the arm tips. In this case, the deflection of each arm is well known to be ki = Pa?/ЗЕІ. Thus the displacement over which the loads P work is ti — 2u = 2Po?/ЗЕІ, from which it follows that

2a3

3 El

Using (2.1.32) again yields the result (2.1.29).

Example 2.1.5 Consider the center cracked panel depicted in Fig. 2.1.1a. Let the dimensions of the panel —width, height, thickness— be, respectively, D, H, and b; and assume a central crack of total length 2a. (Note: it is customary to use 2a instead of a for the crack length for this kind of internal cracks; this requires special care when differentiating with respect to crack length, see below.) A detailed elastic analysis (Chapter 4) delivers the relative displacement of the upper and lower edges of the panel as a function of the crack size. For small cracks (2a D,2a H) and plane stress, this displacement turns out to be

all ( 27ra2 PH ( 2-а a?

= IT l + ~DH J = ~вШ Iі + ~DH

where we wrote that the resultant load is P — aBD. From the last expression we get C — u/P and using (2.1.32) with a replaced by 2a, we get

P2 (1C P2ira a2

9 " 2B d(2a) ~ B2D2E ~~ E™

This is one of the most celebrated Griffith’s results (although Griffith, obtained it in a different way).

 Figure 2.1.4 (a) Quasi-static load-displacement curve, (b) Area representing the total work supply, (c) Areas representing elastic strain energy and complementary energy, (d) Area representing energy supply for fracture and energy dissipated in fracture.