A Novel Analysis of the Size Effect on Stress Intensity Factor for Center and Edge Cracks in Tension by the Net-Section Mechanics

The sizes of mechanical structures and components can vary over a wide range in practical applications. Reliable design and safety assurance of such structures require a knowledge of how the structural size affects the fracture behavior, especially the critical value of stress intensity factor (SIF) for unstable fracture from cracks. The important questions in this context are (i) whether the SIF is truly independent of structure or specimen size for standard crack configurations and (ii) whether the experimental fracture toughness value (K_c) of a material can be considered as size-independent. It is well known that there is a size effect on strengths of concrete structures [1,2], which is explained through the linear elastic fracture mechanics, but this is not the focus of this study. Rather, the focus here is on the effect of structure/specimen size or aspect-ratio on the stress intensity factor itself, in standard fracture mechanics crack configurations.

It should be noted that most of the standard stress intensity factor expressions in the Stress Analysis of Cracks Handbook [3] have been empirically constructed from numerical data for cracks in center- and edge-cracked tension plates. The SIF expression for the center-cracked tension (CCT) geometry is applicable only for specimens with L/W > 3 (L is the specimen length and W is the width) [3]. Isida [4] has shown the strong variation of stress intensity factor with L/W for 0.8 ≤ L/W ≤ 3.6. Additionally, Bowie et al. [5] have shown, by numerical analysis of single-edge-cracked specimens, that the SIF can vary significantly with L/W for 1 ≤ L/W ≤ 16. Bowie et al.'s data do not support the idea that SIF is independent of L/W for L/W > 3 as stated in the handbook [3]. Also, the size-independence of SIFs suggested in the handbook has not been experimentally demonstrated. The numerical data of Bowie et al. cover much larger range of L/W and they clearly show that there is a strong size or aspect-ratio effect on the stress intensity factors for edge cracks loaded in tension.

There is significant experimental data supporting the size-dependence of fracture and fatigue behavior. For instance, recent studies [6,7] have shown that the brittle fracture strength values depend on specimen length in edge-cracked and center-cracked tension configurations, which is possible only if the critical stress intensity factor for fracture (fracture toughness) is depended on the specimen size. In polymer films, the observations [8] of the dependence of fracture energy on the length of the tensile specimen, with a constant value of the edge-crack size in all specimens, can only be explained through the size-dependence of stress intensity factor or work of fracture. There is evidence that the size effect is present in other specimen/crack geometries as well. The effect of the width of compact tension (C(T)) specimen on J_max during crack growth was demonstrated by Kaiser and Hagedorn [9].

There are also experimental evidence indicating that fatigue crack growth (FCG) behavior is indeed affected by the dimensions of specimens. The presence of size effect in compact tension (C(T)) specimens was shown in the work of Garr and Hresko [10], where different sized C(T) specimens (Inconel-718 alloy) produced different FCG rates. In geometrically similar C(T) specimens with H/W ∼ 1, Brose and Dowling [11] found that an increase of width (W) from ∼25 mm to ∼400 mm caused a factor of five decrease in FCG rates in AISI 304 steel. The cause of this size effect was not fully explained although it was speculated that the variations in the extent of plasticity, in the net-sections of specimens of varying width, was responsible for the size effect. However, even in quasi-brittle materials (concrete) where plasticity is not an issue, a clear size effect on FCG in Paris' Law regime of crack growth was seen [12,13] in geometrically similar specimens. Therefore, the apparent size effects are beyond that which could be accounted by plasticity alone. The physical origin of the size effect on fatigue and fracture behavior, especially on the stress intensity factor, is yet to be clearly established in fracture mechanics.

The specimen size effect on crack tip stress intensity factor, as noted here, is a very difficult problem to solve analytically by mechanics, due to variations in boundary locations with respect to the crack tip. Here, simple analytical methods based on net-section mechanics can come handy, especially for solving the problem approximately. The net-section approach has been successful for SIF determinations in compact tension [14] and single-edge-crack bending configurations [15]. It has also been successful in explaining the size effect on fatigue crack growth behavior [16]. In this study, the net-section approach is used to analytically determine the size effect on stress intensity factor in center-cracked as well as edge-cracked tension plates with uniform stress or fixed displacement loading condition at the edges. The derived SIF expressions agree well with extensive numerical data from the literature validating the net-section approach.

It is important to emphasize that in deriving the strain energy release rate (G) for a crack in an infinite body, Griffith [17,18] had actually determined the increase in elastic strain energy stored in the body, due to the introduction of a crack in an infinite plate, by holding the stress constant at the boundary. This has later become to be called as energy released due to the crack, but this term actually applies to the decrease in strain energy stored in the body caused by the introduction of the crack, when the displacement at the boundary is fixed before and after the introduction of the crack. However, if we actually consider the work done at the boundary (see Ref. [19] for clarification), there is indeed a potential energy decrease at the loading point for the uniform stress boundary condition (load-control). Hence, no harm is done by calling the strain energy increase due to the crack, as determined by Griffith, as the energy released due to the crack although this can lead to some confusion. The increase in strain energy was then differentiated by Griffith with respect to crack length to find the notional strain energy release rate (which is actually the rate of strain energy increase in the body with crack length, to be precise) from which of course Irwin's concept of stress intensity factor in fracture mechanics emerged.

Consistent with Griffith's work on infinite plate, this study emphasizes the increase in elastic strain energy due to the introduction of crack in a finite-sized tension plate and its determination by simple net-section mechanics principles. Since the increase in elastic strain energy occurs over the net-section of the finite-sized specimen, it is termed here as the change in net-section strain energy. This is considered as the most important quantity, because by Clapeyron's theorem [20], this change in net-section strain energy is exactly equal to the work done at the plate boundary by the introduction of the crack. In the following, how the SIF is determined from this change in net-section strain energy, as a function of the plate size, is illustrated.

Figure 1(a) illustrates a plate in tension without a crack and loaded to P, producing a total displacement of “δ_o” over its length, L. The work done at the boundary, in the crack-free state, is equal to the area under the elastic P–δ line, up to the point (P, δ_o) (Fig. 1(c)). When an edge-crack of size “a” is introduced, the increase in compliance of the plate will lead to the increased boundary displacement δ₁ (Fig. 1(b)) at the same load. Note that the load in Fig. 1(b) is considered to act along the central axis of net-section, thereby not creating any bending moment, which is required to preserve the pure tension loading of the net-section for all crack lengths. The work done by the introduction of the crack is then equal to the area under the curve of the cracked plate, up to the point (P, δ₁). The change in work done (ΔW) is the difference between the amounts of work done for the two states, which is given by the green colored area in Fig. 1(c). By Clapeyron's theorem, this must be equal to the change in the strain energy (ΔC) of the plate, which is stored in the net-section, as shown in Fig. 1(d). Hence,

The ΔC parameter is therefore the change in net-section strain energy due to the crack under load control and it varies with a/W as in Eq. (5) for plane stress condition. For plane strain, E should be replaced with E/(1 − ν²) in the above equation.

The ΔC in the above equation is the change (decrease) in net-section strain energy due to the crack under clamped-end condition or under fixed displacement at the boundary of the plate for plane stress. For plane strain, E in the above equation should be replaced with E/(1 − ν²).

Hence, for the load-controlled (uniform stress at the boundary) condition, the net-section stress intensity factor from Eq. (5) is

To facilitate comparisons with numerical SIF data, which are to be made in terms of “L/W ” and “a/W,” the “W ” in the normalized parameters “L/W ” and “a/W ” is retained in the above equation—they automatically apply to specimens of unit width as well.

Here also, for comparison with numerical SIF data that will be shown later, the “W ” in the normalized parameters “L/W ” and “a/W ” are retained in the above equation—they automatically apply to specimens of unit width, when the data comparisons are made in terms of L/W and a/W.

Figures 3(a) and 3(b) illustrate the variation of stress intensity factor with specimen aspect-ratio for constant stress and constant displacement conditions at the specimen edge, at selected cracks lengths. It can be seen that the effect of specimen aspect-ratio on SIF is much stronger for the constant stress loading, relative to that is constant displacement loading. This is to be expected because constant stress is a load-controlled situation with an increase in net-section strain energy with increased crack length. However, the variation of SIF with specimen aspect-ratio is relatively less strong, due to the strain energy decrease with an increase in crack length in displacement-controlled situtation.

In addition to the good agreement with numerical SIF data, an interesting observation is that the spread in Isida's numerical SIF data is proportionate to the crack size, a/W, with the spread in SIF data being higher for relatively higher values of a/W. This spread is also reflected nearly exactly in the net-section-based stress intensity factor data in Fig. 4. That is, the effect of H/W on SIF is relatively small at low a/W values, but increases significantly with a/W, showing a strong effect of specimen size on SIF at relatively large crack lengths that are common in fracture toughness testing. But, even at relatively smaller a/W values, a significant size effect is present. For example, at a/W = 0.1, the SIF for unit stress increases from ∼0.21 to ∼0.45, when H/W increases from 0.4 to 1.8.

It is also interesting to note that for H/W = 2.3, the net-section stress intensity factors agree well with the well-known SIF equation, given in the Stress Analysis of Cracks Handbook by Tada et al. [3], which was obtained by modifying Koiter's formula. The difference between the two sets of data here is less than 5% over the range of 0.2 < (a/W) < 0.99. Isida's numerical data are limited to the range: 0.4 ≤ (H/W) ≤ 1.8. Also, the empirical equations in the handbook [3] do not show that the SIF is independent of plate length for H/W > 1.8, but it was implied there to assume that the SIF values do not change with H/W for H/W > 1.8, which remains to be tested. However, the net-section approach predicts a significant increase of SIF with H/W for H/W > 1.8. This suggests that additional numerical or finite-element method (FEM) modeling is needed to validate such a variation in SIF with specimen size at larger H/W values.

Equation (27) is used here to calculate the numerical stress intensity factors, as a function of a/W and H/W from Isida's tabulated F₅(a/W) data.

Figure 5 illustrates the comparison between Isida's numerical SIF values for center-cracked plates clamped at the end, with the K_NS data determined from the net-section approach. For the latter, the form of equation used here for plane stress, from Eq. (16a), is

In Fig. 5, the agreement between the two sets of data is quite good at relatively low H/W values. At H/W values >0.8, the agreement is good only in the range of 0.5 ≤ a/W ≤ 0.8. Unlike the SIF data for load-controlled (uniform stress) condition (Fig. 4), there appears to be some discrepancy in Isida's numerical data for the clamped-end condition. That is, at a/W < 0.5, the SIF values are not separated in proportion to the specimen size parameter, H/W. Such a separation is expected in light of the systematic separation of numerical data for a/W < 0.5 for the uniform stress loading condition (see Fig. 4). On the other hand, the separation of numerical SIF data in Fig. 5 is quite consistent with the net-section SIF data for a/W > 0.5. Therefore, the discrepancies between the numerical data and the net-section SIF data are likely to emerge from the numerical method itself. A potential reason for such discrepancies may be due to how strictly the enforced boundary conditions (the horizontal displacement must be strictly zero at all points in the boundary while also ensuring τ_xy = 0) conform to the required boundary displacement situation, in the numerical models at low a/W values. Nevertheless, it is not consistent to have a set of numerical data showing a lack of H/W effect at low a/W values, while showing a significant effect of H/W at higher a/W values.

The stress intensity factors of cracks in edge-cracked tension plates are also quite sensitive to the L/W ratio. John and Rigling [22] performed FEM analysis of edge-cracked plates under uniform stress loading and from these simulations, provided F(a/W) data applicable to 2 ≤ L/W ≤ 10, in table form.

Figure 6 illustrates the comparison between the finite-element SIF values obtained by John and Rigling [22] with the K_NS data determined from the net-section approach. When closely examined, the SIF values from John and Rigling were off by a factor of about √2, relative to Isida's numerical data for an equivalent central and edge-crack configuration (e.g., H/W = 1 for Isida and L/W = 2 for John and Rigling), the reason for which is not clear. Assuming Isida's data as reference, all the SIF values of John and Rigling were divided by √2 for comparison with the net-section approach here. The net-section SIF calculations used appropriate L/W values in Eq. (16a) for plane stress. After the modification, the agreement between the two sets of data is quite good at low L/W values, from 2 to 6, for most of the a/W values. At L/W values >6, however, the agreement is good only in the crack length range of 0.5 ≤ a/W ≤ 0.8. This discrepancy in the FEM data here, at low a/W values, is also quite similar to that found in Isida's numerical data for the clamped-end condition of center-cracked plate (see Fig. 5). That is, the SIF values are not separated in proportion to the specimen size parameter, L/W, in spite of the fact that the separation of FEM SIF data is quite consistent with net-section data for most values of a/W > 0.5. Again, it is not clear how consistently the boundary conditions were enforced in the finite-element K calculations at various a/W values, in the study of John and Rigling.

The comparison between the numerical SIF values obtained by Torvik [23] and Fett and Bahr [24] for the clamped-end plate with edge-crack, against the SIF values determined from the net-section approach (Eq. (16a)), is shown in Fig. 7. The agreement between the net-section-based SIF values calculated using Eq. (16a) and that from both numerical data sets is generally good at all L/W values, for all a/W values >0.3. The discrepancy in numerical data over 0 < a/W < 0.3 here is also similar to that found in Isida's numerical data for the clamped-end condition (Fig. 4), but the discrepancy here is limited to the range 0 < a/W < 0.3. This is also similar to the discrepancy found in the SIF data for load-controlled condition of edge-cracked specimen (Fig. 6). Since the application of boundary conditions in the numerical methods can differ, such discrepancies can arise due to varied enforcement of boundary conditions for K calculations in the range of a/W values in question.

A noteworthy aspect in Fig. 7 is that the agreement of the net-section SIF data is generally good against the SIF data obtained by two different numerical methods—Torvik used the variational technique, whereas Fett and Bahr used the weight function approach to determine the SIF values. There are, however, some differences between the two sets of the numerical data especially at L/W = 2, which could be due to the different levels of numerical errors and varied boundary conditions, which may be intrinsic in each method of numerical analysis.

It is remarkable that the net-section-based estimates of SIF for tension plates of various sizes generally agree very well with the numerical SIF data obtained by vastly different methods: (ii) complex potential approach by Isida [4], (ii) finite-element modeling by John and Rigling [22], (iii) variational principle with mixed-boundary conditions by Tarik [23], and (iv) weight function technique by Fett and Bahr [24]. This appears to suggest that the common phenomena controlling the stress intensity factor in center- and edge-cracked tension plates is the change in net-section strain energy, in the same sense as the change in strain energy in Griffith crack theory for infinite plate. Therefore, the present method can be broadly useful to estimate stress intensity factors of other geometries in complex loading conditions if the total change in net-section strain energy (due to tension and/or bending) can be determined as applicable to the specimen geometry. This was in fact already demonstrated for generalized compact tension specimen [14] as well as for the three-point bending specimen [15] elsewhere, where excellent agreement of the net-section-based SIF solutions with the traditional stress intensity factor solutions were obtained, validating the net-section approach. Also, the specimen size effect observed during fatigue crack growth in singe-edge-cracked specimens could be correlated very well, as shown elsewhere [16], following the analysis presented in this work.

The present results show that the size effect on SIF can be easily captured through the change in net-section strain energy due to the crack. It was shown previously by the author [25] that the effect of geometric correction factor, F(a/W), on K is to capture the net-section stress increase in finite-width fracture mechanics samples as the crack (mode-I) extends in the specimen. The physical meaning of F(a/W) in fracture mechanics is that it serves as the stress amplification factor accounting for the increase in the average net-section stress at various crack lengths. Therefore, it should not be surprising that the change in net-section strain energy, which is the origin of the net-section stress amplification, produces here relatively simple forms of equations describing accurately the variation of stress intensity factors with size in center- and edge-cracked tension plates.

The idea of using net-section strain energy to determine stress intensity factors seems to have not been considered before. Almost the entire focus in fracture mechanics has been to characterize K as that which is determined by singular stress field at the crack tip, without considering what happens in the rest of the net-section of the infinite plate. However, when the specimens are finite, the F(a/W) factors come to the rescue, providing accurate SIF values, by capturing the stress amplification in the net-section. The common fracture mechanics methodologies, however, have not allowed introduction of specimen length as a variable in stress intensity factor determinations, under the notion that the specimen size is not important. Hence, fracture mechanics calculations or experiments are mostly restricted to the limited sizes and geometries presented in the stress analysis of cracks handbook [3]. Hence, the present approach can be useful to obtain SIF solutions for crack problems that are not provided by the conventional fracture mechanics approaches. Additionally, mode-I crack problems with irregular specimen cross-sections can be easily handled by the net-section approach.

There is also significant experimental support validating the use of the change in net-section strain energy to characterize fatigue crack growth, thereby supporting its use here to determine the stress intensity factors as a function of specimen size. In recent works [26–28], the author has shown that the fatigue crack growth rates in center- and edge-cracked tension samples under various stress levels and stress ratios can be correlated extremely well using the change in net-section strain energies under cyclic loading. The excellent correlations of fatigue crack growth rates, based on the change in net-section cyclic strain energy, have validated well the net-section strain energy as a physically understandable and valid fracture mechanics parameter.

1. A net-section-based approach has been formulated to determine the stress intensity factor as a function of structure or specimen size for center-crack and edge-crack configurations of tension plates. This yielded simple analytical equations to calculate the size effect on stress intensity factors in standard fracture mechanics crack configurations. This can be quite useful because analytical equations capturing the size effect on stress intensity factor are not available.

2. The analytical equations capturing the size effect on stress intensity factor under load-controlled (uniform stress) and clamped-end (uniform displacement) conditions provide stress intensity factor values that are in good agreement with most of the numerical stress intensity factor data available in the literature. For the clamped-end condition, discrepancies in the numerical data, at relatively low crack lengths, have been pointed out in light of the net-section-based calculations.

3. It is also found that in the limit of an edge-crack reaching the farthest edge of the clamped-end plate, the net-section-based stress intensity factor recovers the Rice's J-integral-based stress intensity factor for a semi-infinite crack. This supports the validity of the present net-section approach.

4. The discrepancies within the numerical stress intensity factor data appear to depend on the crack configuration and the boundary condition used. The Isida's numerical data show discrepancies at low a/W values under clamped-end condition but such a discrepancy is absent in the numerical data for uniform stress boundary condition. In contrast, for the edge-crack configuration, the John–Rigling data for uniform stress condition showed similar discrepancy at low a/W values, but such discrepancy was somewhat absent in the clamped-end condition, as evident from the data of Torvik and Fett and Bahr. The present net-section approach highlights such discrepancies in numerical data, especially when the SIF data are compared over wide range of a/W and L/W values.

There are no conflicts of interest.

The authors attest that all data for this study are included in the paper.

a =

crack length

t =

thickness of the specimen

E =

elastic modulus

G =

strain energy release rate

H =

specimen half-length (L/2)

K =

stress intensity factor

L =

specimen full length (2H)

P =

applied load

W =

specimen width

v_o =

uniform displacement at the boundary

J_max =

maximum value of J-integral

K_c =

fracture toughness

$KI∞$ =

stress intensity factor for crack in an infinite plate

K_NS =

stress intensity factor from net-section approach

P₁ =

decreased load at the boundary in clamped-end condition

$uy*$ =

constant displacement at the loading boundary of the plate

C(T) =

compact tension specimen

F(a/W) =

geometric correction factor

F₁(a/W) =

geometric correction factor for uniform stress loading (Isida)

F₅(a/W) =

geometric correction factor for clamped-end condition (Isida)

F_Tor(a/W) =

geometric correction factor for clamped-end condition (Torvik)

F_FB(a/W) =

geometric correction factor for clamped-end condition (Fett–Bahr)

CCT =

center-cracked tension

FEM =

finite-element method

FCG =

fatigue crack growth

SIF =

stress intensity factor

δ_o =

elastic displacement of crack-free plate

δ₁ =

elastic displacement of cracked plate

ΔC =

change in net-section strain energy

ΔW =

work done corresponding to the change in net-section strain energy

ɛ_o =

average strain in crack-free plate

ɛ₁ =

average strain in the net-section of cracked plate

ν =

Poisson's ratio

σ =

applied stress

σ₁ =

hypothetical net-section stress if specimen ends are not clamped

σ_L =

average stress in net-section