2.1. The GSM Method
This method is based on (i) a formal description of the
J integral for a circular notch from the definition, (ii) substitution of the relative strain energy density along the periphery of the notch by a third power of the cosine function of the polar angle, and (iii) allowing the radius of the circular notch to go to zero in the final expression for the
J integral. The idea of a formal description of the
J integral, for a circular notch followed by reducing the notch radius to zero in the final expression for obtaining the
J integral of a crack, is not new. There are some papers by Matvienko and Morozov [
13,
14] and by Matvienko [
15] which demonstrate such an approach.
As shown by Norio and Yasuhiro [
16], stress intensity factors can be determined from the limiting values of elastic stress concentration factors as the root radius
ρ of the notch approaches zero. In the derivation of the
J integral, the GSM method considers a symmetrically loaded notch with its tip embedded in a mode
I stress field. The maximum stress
σmax occurs directly ahead of the notch. Dimensional considerations of the crack-tip stress field for an isotropic elastic body lead to:
According to the definition, the
J integral for a cracked body is given by the expression
where
is the strain energy density
Γ is any contour encircling the tip of the crack in a counterclockwise direction
Ti are the components of the traction vector
ui are the displacement vector components
ds is a length increment along the contour Γ
Let us consider a body with external notches on both sides, loaded perpendicularly to the plane of the notches. A section of the notched body around the notch is shown in
Figure 1.
The shape of the notch root is semicircular, the radius being
ρ. By the concept of the invariance of the
J integral, the value of the
J integral does not depend on the path of integration. In its derivation the
J integral is formally written for the notched body, and the path of integration is chosen so as to coincide with the periphery of the semicircular notch root (see
Figure 1). Since the path of integration leads over a free surface, the second term on the right-hand side of Equation (2) becomes zero, so that after the transformation of Cartesian to polar co-ordinates of the points on the semicircular notch root (
Figure 2), the
J integral for the notched body takes the form:
The GSM method relates the strain energy density
w(θ) at point M on the periphery of the notch root, characterized by the polar angle
θ, to the maximum strain energy density
wmax =
w(θ=0). The relation between
w(θ) and
wmax depends not only on the polar angle
θ but also on the magnitude of the load. The results of finite element investigations into the strain energy density along a notch root in a double-edge notch panel [
1] showed that the relative strain energy density (
w(θ)/wmax) can be substituted with a certain approximation by the function cos
3θ. Considering this, Equation (3) can be rewritten as:
The infinitesimal strain energy density
dw is given in principal stresses and strains as:
In the notch root, characterized by
θ = 0, the stress
σ2 is always zero because of the free surface; the stress
σ3 is zero for the plane stress state; and the strain
ε3 is zero for the plane strain state. This means that the infinitesimal strain energy in the notch root (
θ = 0) is reduced to
so that the strain energy density becomes
or, with the notation used before:
The GSM method supposes that material obeys the Ramberg–Osgood dependence (9) and that the hypothesis of equivalent strain energy density at the notch tip [
17] can be applied:
According to the concept of this hypothesis, the following equation holds
where
wn is the energy density due to the net section stress
σn.
By combining (8) and (10), we arrive at:
By differentiating the Ramberg–Osgood relation (9), and considering that
ε0 =
σ0/E, it is possible to arrive at:
When applying this equation for
εn and
σn to Equation (11), the following expression for
w is obtained:
By substituting
wmax in Equation (4) with this expression, the
J integral for a notch obtains the form:
Recalling Equation (1), it is seen that the stress intensity factor for a crack can be expressed by:
From there it follows as:
By combining Equations (14) and (15) we obtain:
A multiple of four/three is applied to the first term in expression (17); this does not have a theoretical basis, but was incorporated to provide a well-known form of the elastic component of the J integral: Jel = K2/E′, where E′ = E for plane stress and for plane strain, ν being Poisson´s number.
Equation (17) then obtains the form:
Since the strain energy density
w(θ) in Equation (4) was substituted with
wmax cos
3θ regardless of whether plane stress or plane strain conditions were concerned, the resulting Formula (18) can be used as a basis for the
J integral assessment at conditions of plane stress and plane strain. As is known, the EPRI estimation scheme for the
J integral [
18] comes from stresses given by the HRR singularity and it arrives at the relationship simply expressed as:
P0 can be defined arbitrarily, e.g., as the limit load
PL. In order to make Equation (18) comply with this, a so-called limit load parameter
C, by which the uniaxial yield stress
σ0 in (18) is to be multiplied, is introduced into the GSM method. The
C parameter is given by Equation (20):
Equation (18) then obtains the form:
The limit load PL in Equation (20) can be determined as the product of the yield stress σ0 and a certain geometrical function, which is specific for each panel and depends on the crack length a and the width b of a cracked panel of unit thickness. It is seen that the difference between the J assessment in the plane stress condition and in the plane strain condition is given (besides Young´s modulus E′) by the level of the limit load parameter C.
2.2. The FC Method
As already mentioned, the FC method was proposed in Addendum A16 of the French nuclear code [
2] as the
Js method, and its further development was published by Marie et al. [
10]. The basis of this method was the R6 procedure [
19], which made it possible to arrive at the following formula for the
J calculation:
In Equation (22),
Je is the elastic component of the
J integral,
εref is the reference strain corresponding to the reference stress
σref defined by Equation (23),
εe =
σref/
E is the elastic strain, and
φ is the plastic zone size-correction factor given in [
10] by Equation (24):
It can be pointed out that, owing to Equation (20), the reference stress
σref can also be written as:
The first term in the brackets of Equation (22) reflects what experimentalists observed a long time ago, namely that at a certain load the
J integral is proportional to the ratio of the actual strain to its elastic component. The quantities used in expression (22) are illustrated in
Figure 3 for the Ramberg–Osgood approximation of the tensile curve of the material.
The Ramberg–Osgood dependence (9) can be rewritten by substituting
σ with
σref and
ε with
εref to obtain the form:
By denoting
and considering
Equation (26) obtains the form:
According to Equation (22), and considering Equations (24) and (28), the
J integral is then expressed by Equation (29):
This type of equation is also used in later editions of the RCC-MR code. The very last edition from 2018, denoting the RCC-MRx code, is not readily available from public sources. As follows from [
11], it has been designed primarily for the mechanical components of high-temperature structures of nuclear installations; however, it can also be used for mechanical components of other types of nuclear installations. Although not mentioned explicitly in [
11], it is likely that an equation of the type in (29) is also used in the RCC-MRx code, at least for some specific conditions like force-imposed mechanical loading, the modified limit load basis for the reference stress, and the Ramberg–Osgood description of the stress–strain curve.
Coming back to Equation (29), the fraction
in Equations (26) and (27) can be substituted, according to Equation (25), by
to obtain
where