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Fracture mechanics analysis of generalized compact tension specimen geometry using the mechanics of net-section
Engineering Fracture Mechanics 222 (2019) 106703
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Fracture mechanics analysis of generalized compact tension specimen geometry using the mechanics of net-section
T
K.S. Ravi Chandran Department of Materials Science and Engineering, 135 South 1460 East Rm. 412, The University of Utah, Salt Lake City, UT 84112, USA
A R T IC LE I N F O
ABS TRA CT
Keywords: Fracture mechanics Stress intensity factor Strain energy Net-section Geometric correction factor
Fracture mechanics analysis of a generalized compact tension specimen, on the basis of the mechanics of deformation of the net-section, is shown to provide a simple and a broadly useful expression for stress intensity factor calculations. A single analytical expression is found to be sufficient for the characterization of crack behavior in compact tension, extended compact tension and wedge splitting test specimens without any restriction on the specimen length (or height) and width. The analysis is enabled by the concept of the change in net-section energy, which is determined by summing the changes in strain energies of the net-section of the generalized specimen for tension and bending deformation modes, which result from the introduction of the crack. This is equivalent in concept to the increase in strain energy upon the introduction of the crack, as in the Griffith’s fracture theory. The square-root of the change in netsection strain energy parameter multiplied by the elastic modulus provides an expression that is equivalent to the conventional stress intensity factor expression. The application of the netsection based expression to the standard compact tension, the extended compact tension and the wedge-splitting test specimens shows very good agreements with the crack behaviors as expressed by the stress intensity factor expressions for the respective geometries. The proposed method enables easy analytical determination of stress intensity factors for any asymmetricallyloaded mode-I crack problem.
1. Stress intensity factor for compact tension specimens Stress intensity factors (K) in fracture mechanics are determined by a variety of methods, including analytical, numerical and finite element methods as well as experimental methods such as compliance measurements [1,2]. The standard expressions for the stress intensity factors are available in the well-known compendium by Tada Paris and Irwin [3]. In this reference, the stress intensity factor for the compact tension specimen is given as:
K=
(
a
)
2+W P a F⎛ ⎞ t W 1 − a 3/2 ⎝ W ⎠ W
(
)
where F(a/W) is the geometric correction factor (GCF) given by
a a a 2 a 3 a 4 F ⎛ ⎞ = 0.886 + 4.64 ⎛ ⎞ − 13.32 ⎛ ⎞ + 14.72 ⎛ ⎞ − 5.6 ⎛ ⎞ W W W W W ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
E-mail address: [email protected]. https://doi.org/10.1016/j.engfracmech.2019.106703 Received 26 July 2019; Received in revised form 12 September 2019; Accepted 26 September 2019 Available online 27 September 2019 0013-7944/ © 2019 Elsevier Ltd. All rights reserved.
(1)
Engineering Fracture Mechanics 222 (2019) 106703
K.S. Ravi Chandran
Nomenclature a a/W b b/Z c c/Z C(T) e E EC(T) F(a/W) G GCF I Io IN K M Mo M1 P L L/W L/Z s
t TPB W WST Z σ σL
crack length in C(T) specimen normalized crack length in C(T) specimen crack length in WST specimen normalized crack length in WST crack length in EC(T) or slot depth in WST specimens normalized crack length in EC(T) specimen compact tension eccentricity of load elastic modulus extended compact tension geometric correction factor Griffith strain energy release rate geometric correction factor moment of inertia of cross-section inertia of a crack-free beam inertia of a cracked beam stress intensity factor bending moment moment of a crack-free beam moment of a cracked beam applied load specimen length length-to-width ratio in C(T) specimen length-to-width ratio in EC(T) specimen distance of the loading point from left edge
δo δ1 δob δ1b ε1 εo θ θo θ1 ΔC ΔUb ΔUt ΔWb ΔWt
specimen thickness three-point bending specimen width wedge splitting test specimen total width of EC(T) specimen applied stress average stress in net-section after the introduction of crack load-point displacement in tension before the introduction of crack load-point displacement in tension after the introduction of crack load-point displacement in bending before the introduction of crack load-point displacement in bending after the introduction of crack average strain in net-section after the introduction of crack average strain in uncracked specimen rotation of a bending beam rotation of a crack-free beam rotation of a cracked beam the change in net-section strain energy the change in the strain energy of a bending beam the change in the strain energy in tension work done on net-section of a bending beam work done on net-section in tension
Note that the crack length a and width W, in the above expression, are defined from the loading line. This expression is claimed to be valid for 0.2 ≤ (a/W) ≤ 1 and was constructed in the basis of boundary collocation results of Srawley and Gross [4] and Wilson [5]. A specimen very close to the C(T) specimen is the extended compact tension specimen, EC(T). Fig. 1 illustrates the geometries of both specimens as well as the related wedge splitting test (WST) specimen. The SIF for the EC(T) specimen have been determined by boundary force method by Piasick et al. [6]. The expression for the K is given as
K=
P t Z
c ⎡ c c 3/2 c 1.4 + ⎛ ⎞⎤ ⎛1 − ⎞ G⎛ ⎞ Z ⎣ Z⎠ ⎝ Z ⎠⎦ ⎝ ⎝Z⎠
where
c c c 2 c 3 c 4 c 5 G⎛ ⎞ = 3.97 − 10.88⎛ ⎞ + 26.25⎛ ⎞ − 38.9⎛ ⎞ + 30.15⎛ ⎞ − 9.27⎛ ⎞ Z Z Z Z Z Z ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
(2)
where c is the crack length from the edge, Z is the full-width of the specimen, and W = Z−s, for the EC(T) specimen (Fig. 1a). There are important differences between the GCFs in Eqs. (1) and (2) in terms of how crack length and width are defined. The crack length and the width for the C(T) specimen are taken from the load line running through the centers of loading pin holes. On the other hand, for the EC(T) specimen they are measured from the leftmost edge of the sample (Fig. 1a&b). The latter sample also has 0.2 W as the distance of the pin-hole center from the left edge of the sample, whereas it is 0.25 W in the C(T) specimen. The standard length- or the height-to-width ratios are 1.2 (L/W) and 3 (L/Z) for the C(T) and EC(T) samples, respectively. The choices of the dimensions for the C(T) were made long time ago during the development of ASTM E399 test standard, but EC(T) is a recent modification of C(T) to enable easy experiments. The wedge splitting test (WST) is a technique that is used for determining the fracture energy during the slow stable cracking of brittle materials such as ceramics and concrete [7,8]. The WST specimen is illustrated in Fig. 1(c). Except for the loading arrangement, the WST specimen is the same as the conventional C(T) specimen, and often the GCF of the C(T) specimen is used for the WST specimen to calculate the stress intensity factor for the latter. In all the three geometries the GCFs are valid only for the specific specimen dimensions and shapes for which they were developed. This means that when any of the specimen size parameters need to be changed for whatever reason, one is required to do complex numerical analysis by collocation method or by finite-element-method to determine the GCF for the new geometry. The principal motivation of this work is to find a common analytical basis for the three specimens through the net-section fracture mechanics approach. There is a high degree of geometrical and loading-mode similarity between the three specimens. All the three specimens can be derived from a generalized asymmetric pin-loaded specimen geometry by varying s, W, Z and L, as evident from Fig. 1. The objective of the present work is to develop a generic analytical method and a common expression from which the stress 2
Engineering Fracture Mechanics 222 (2019) 106703
K.S. Ravi Chandran
Fig. 1. Schematics of C(T), EC(T) and WST specimens.
intensity factors for all the three geometries can be determined, while also allowing flexibility to vary any dimension of the generic specimen, in the same expression. 2. The net-section approach for fracture analysis The author has recently shown [9] that geometric correction factors (GCFs) in fracture mechanics may actually be interpreted as stress amplification factors in fracture mechanics specimens. Taking one step further, it has also been shown [10] that simple expressions, derived on the basis of the change in net-section strain energy, produce stress intensity factor values that are in excellent agreement with that provided by standard fracture mechanics expressions. These were the common fracture mechanics specimens such as the middle-crack tension, three-point bending and pin-loaded, single-edge-cracked specimens. Excellent correlations of fatigue crack growth behavior, in terms of the net-section fracture mechanics parameters, have also been demonstrated for these geometries [11–13]. In these works extended analytical expressions, to account for the asymmetry and the cyclic nature of fatigue loading, were developed on the basis of the effective work done such that the expressions are applicable to generalized cyclic loading situations. The present work builds on the success of the previous correlations. Specifically, the objective is to treat the compact tension specimen geometry in a generalized way in the net-section approach and see if the analysis provides results that are broadly applicable to the derivatives of the generalized geometry. The derivatives include the standard compact tension specimen, the extended compact tension specimen and the wedge splitting test specimen. The analysis yielded a simple and broadly applicable expression that provides stress intensity factors for any geometry derived from the generalized specimen geometry. The expression is validated by comparing it with the crack behaviors described by the conventional stress intensity factor expressions for the C(T), EC(T) and WST specimens, determined by others using numerical and FEM methods. A discussion, elaborating why the net-section approach is entirely consistent with Griffith’s theory of fracture, is also presented. 3. The change in net-section strain energy for the generalized specimen Fig. 2 illustrates the net-section approach for the generalized compact tension specimen, GC(T). The concept of the change in netsection strain energy, ΔC, as a fracture mechanics parameter was introduced in our previous works, which are referenced in the above section, in the context of crack propagation in fatigue. Here it is illustrated for the generalized compact tension specimen, as shown Fig. 2(a). For clarity, the deformations in the uncracked and the cracked specimen configurations are exaggerated. Note that in the 3
Engineering Fracture Mechanics 222 (2019) 106703
K.S. Ravi Chandran
Fig. 2. The generalized compact-tension specimen under asymmetric loading through pins. (a)–(c) illustrate the deformation of the specimen and how it is split into tension and the bending deformations, with and without the crack. (d) and (e) illustrates the work done and the change in netsection strain energy under pure tension. (f) shows the change in bending strain energy, which is given by the difference between the two highlighted areas, OM1θ1 and OMoθo. The figure is a modified form of an earlier version [13].
interest of wide applicability, a full specimen analysis is performed. In this, the width of the specimen is taken to be the entire width from the left to the right of the specimen, Z, with Z = s + W, where s is the portion of the specimen width to the left of the loading axis and W is the width in the conventional sense of the regular C(T) specimen, as in the ASTM E399 standard for fracture testing [14]. Additionally, the crack length, c, is taken from the left edge and this means c = s + a for c > s, where a is the crack length from the loading line, again, as defined in the regular C(T) specimen (Fig. 1(b)). It will be evident later that this basis is quite advantageous 4
Engineering Fracture Mechanics 222 (2019) 106703
K.S. Ravi Chandran
and clarifies some issues with the stress intensity factor expressions for the C(T) specimen. The deformations in the GC(T) sample consist of tension deformation in the net-section (Fig. 2(b)) as well as the bending deformation caused by the rotation of the net-section (Fig. 2(c)), due to asymmetric loading. When a crack size, c, is introduced at load, P, both deformations will lead to the accumulation of work at the loading point. The total work is captured by adding the work due to tension as well as the work due to bending on the net-section, which are determined as illustrated in the following. The total change in net-section strain energy for the GC(T) specimen, ΔC, is derived here following the Timoshenko [15] approach in mechanics of materials. First, the work done in tension, ΔWt, due to the formation of a crack, is given by (Fig. 2(d)):
ΔWt =
1 P (δ1 − δo) 2
(3)
The corresponding change in net-section strain energy, ΔUt, is given by the difference between the area “OσLε1O” and the area “OσεoO” in Fig. 2(e). Following the steps given elsewhere [11] this leads to
() c
ΔUt =
⎡ σ2 Z (LZt ) ⎢ 2E ⎢ 1− ⎣
(
c Z
)
⎤ ⎥ ⎥ ⎦
(4)
where σ is the nominal stress due to the tension component, E is the elastic modulus and t is the specimen thickness. All other specimen parameters are illustrated in Fig. 2. In terms of load, this is c
ΔUt =
P2 L ⎡ Z ⎛ ⎞ 2E ⎝ Zt ⎠ ⎢ 1 − ⎣
⎤ ⎥ ⎦
c Z
(5)
Next step is to determine the bending component of the change in net-section strain energy, caused by the introduction of the crack. This was previously determined [12] for pure bending as the difference between the bending energy of the beam made of the net-section defined by the crack and that of a crack-free beam. If the deflections due to bending, at load P, in the uncracked and cracked beams are δob and δ1b, respectively, the bending work done due to the crack introduction is given by
ΔWb =
1 P (δ1b − δob) 2
(6)
The corresponding change in strain energy can be written in terms of bending parameters in Fig. 2(c) as
ΔUb =
1 [M1 θ1 − Mo θo] 2
(7)
where M1 and Mo are the moments and θo and θ1 are the rotations of the crack-free and cracked beams, respectively, as shown in Fig. 2(c). Eq. (7) is applied here for the GC(T) specimen to determine the increase in strain energy as indicated by the highlighted area in Fig. 2(f). Using the relationship θ = ML/ EI ,
ΔUb =
M2L 1 ⎡ M12 L − o ⎤ ⎢ 2E ⎣ IN Io ⎥ ⎦
(8)
where L is the length of the beam, IN and Io are beam inertia terms given by: IN = t(Z − c) /12 and Io = tZ /12. As the crack length increases, the neutral axis in bending will move to the right relative to that in the uncracked specimen, creating an increasingly eccentric loading situation. For the uncracked specimen, the loading eccentricity is given by ((Z/2) − s). For the cracked specimen, the eccentricity is ((Z − 2 s + c)/2). Therefore Eq. (8) can be written explicitly as 3
P 1 ⎡ ⎢ ΔUb = 2E ⎢ ⎣
2 2 Z − 2s + c L 2
(
)
IN
P2 −
(
3
Z − 2s 2 L⎤ 2
)
Io
⎥ ⎥ ⎦
(9)
Simplifying Eq. (9),
(
{}
⎡ 1−2 s + Z 3P 2 L ⎢ ⎛ ⎞ ΔUb = c 3 2E ⎝ Zt ⎠ ⎢ 1− Z ⎢ ⎣
(
c 2 Z
)
)
⎤ s ⎞2⎥ − ⎛1 − 2 Z ⎠⎥ ⎝ ⎥ ⎦
{}
(10)
The total change in strain energy of the GC(T) specimen, ΔC, can be determined by adding the tension component (Eq. (5)) and bending component (Eq. (10)): c ⎧ P2 L ⎪ ⎡ Z ⎛ ⎞ ⎢ ΔC = 2E ⎝ Zt ⎠ ⎨ 1 − ⎪⎣ ⎩
(
{}
⎡ 1−2 s + Z ⎤ ⎢ 3 + c ⎥ c 3 ⎢ 1− Z Z ⎦ ⎢ ⎣
(
)
c 2 Z
)
⎤⎫ s ⎞2⎥ ⎪ − ⎛1 − 2 Z ⎠ ⎥⎬ ⎝ ⎪ ⎥ ⎦⎭
{}
(11)
In Griffith’s crack theory, the increase in elastic strain energy of an infinite plate due to the crack formation is determined. The 5
Engineering Fracture Mechanics 222 (2019) 106703
K.S. Ravi Chandran
relationship between the Griffith’s energy release rate, G, and the fracture mechanics stress intensity factor, K, is given by the Irwin’s relationship: K = G. E . Since ΔC also represents the increase in strain energy in a finite specimen, the parameter form that would be compatible with the stress intensity factor in fracture mechanics will be ΔC . E . Hence, Eq. (11) can be written as
ΔC . E =
P 2t
c
L Z ⎛ ⎞ ⎡ ⎢ ⎝Z⎠ ⎣1 −
(
{}
⎡ 1−2 s + Z ⎤ ⎢ 3 + c ⎥ c 3 ⎢ 1− Z Z ⎦ ⎢ ⎣
(
)
c 2 Z
)
⎤ s ⎞2⎥ − ⎛1 − 2 Z ⎠⎥ ⎝ ⎥ ⎦
{}
(12)
where L/Z is the length-to-width ratio of the generalized compact tension specimen. Eq. (12), in this form, directly applies to EC(T) specimen or C(T) specimen, where the normalized crack length is defined from the edge of the sample, c/Z, in both geometries. The same equation can be used for WST specimen, by replacing “c” with “b.” Both are the equivalent type of crack lengths from the edge: the former is for the EC(T) specimen and the latter is for the WST specimen (Fig. 1). The parameter ΔC . E can be given a letter or symbol as a single variable, once it is widely accepted. 4. Evaluation and discussion Fig. 3(a) illustrates the comparison of the net-section parameter ΔC . E from the present approach and the standard stress intensity factor data for EC(T) specimen (L/Z = 3) determined by Piascik et al. [6] by the body force method. In the calculations, however, L/Z = 3.2 was used as this corresponds to the actual loading length (distance) between the points of contacts of the loading pins with the holes in the specimen. The agreement between the Piascik et al.’s data and the present net-section-based calculation from Eq. (12) is surprisingly good over the entire range of normalized crack length, c/Z. The % difference between the two parameters, with respect to K, is plotted as a function of c/Z in Fig. 3(b)—the difference is less than 10% for 0.3 < (c/Z) < 0.95, which indicates that the present approach will be useful, especially for other L/Z ratios of EC(T) samples for which no GCF factors are available. The difference could be due to the errors inherent in each method. In the boundary element modeling of Piascik et al., the K expression for EC(T) was numerically determined and could have a certain amount of error with respect to the exact solution. The present method also will have some error, the magnitude of which could be on the order of the error in the bending formula of strength of materials. This is likely to be small, since the bending formula used here is considered to be exact in strength of materials. Both factors, however, can contribute to the observed difference between the two data and their relative error contribution can only be assessed when an exact solution is found. It is important to note the nature of the net-section parameter in Eq. (12) versus the stress intensity factor. For zero crack length, the change in net-section strain energy is zero, thus, the present method gives a correct result at zero crack length. However, such a
Fig. 3. Net-section fracture mechanics parameter ΔC . E (Eq. (12)) compared with (a) the stress intensity factor obtained by boundary force method for the EC(T) specimen and (b) the % difference between the two parameters, with respect to K. 6
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limiting value is not possible for K, since it is not defined at this limit. The K-behavior should be considered to be valid at small crack sizes, as small as possible (since LEFM definitions are increasingly accurate at vanishing crack lengths), but not zero. An exploration of why K cannot be defined at the limit but its values agree with the limiting trend near the zero crack length is beyond the scope of this work. In Fig. 4(a) the finite-element-modeling (FEM) results of wedge splitting test specimen (Fig. 1(c)) having L/Z = 2, generated for c/ Z = 0.25 by Guinea et al. [16], are used to compare the calculations from Eq. (12). Here, c represents the depth of the slot, as shown in Fig. 1(b). In this FEM model, the loading points were located at s = 0.05Z. The data corresponding to L/Z ratio of 2 with Z/c = 4 is chosen for the present comparison. The data, presented in terms of a/W, were converted to the full-specimen normalized crack length (b/Z) basis, by calculating as: b/Z = (c + (a/W)W)/Z. Additionally, the normalized K values, which were presented in terms of Kt√c/ P were converted into Kt√W/P, by using the fact that c = W/3 in Guinea et al.’s FEM model. Fig. 4(a) illustrates that the crack behavior as expressed by ΔC . E also agrees very well with the stress intensity factor data provided by Guinea et al., except for small values of b/Z. The proximity of the crack to the slot end, at b/Z = 0.25, in the FEM model of the WST specimen, may have influenced the deviation of the K from the main trend, around this b/Z value. The % difference between the two parameters, with respect to K, is plotted as a function of b/Z in Fig. 4(b)—this difference is also less than 10% for 0.3 < (b/Z) < 0.95, which indicates that the present approach will be useful, especially for other L/Z ratios of WST samples for which no GCF factors are available. For 0.3 < (b/ Z) < 0.95, the trend in the difference is approximately similar to that for the EC(T) specimen, as shown in Fig. 3(b). As stated before, this difference could be due to the inherent error in each method. The FEM modeling of Guinea et al, for WST specimen will produce a certain amount of error in the data, which is typically a few %. The present model also will have some error, the magnitude of which could be the same as that in the bending formula. Although the magnitude of these errors are not known, they are not zero with respect to the exact solution. Fig. 5 illustrates the comparison for the conventional C(T) specimen in fracture mechanics. Here, a value of L/W = 2 is used for calculating ΔC . E from Eq. (12). Note that the data needs to be plotted in terms of full crack length ratio, c/Z, as applicable to the C(T) specimen (Fig. 1(a)) for a better comparison. Since the crack length and specimen width for the C(T) specimen are defined from the loading line as a and W, then c = a + s, Z = W + s and, a/W = (c − s)/(Z − s), in terms of full-specimen parameters. Here, the GCFs provided by Tada [3] and the tabulated data from Srawley and Gross [4] for L/W = 2 with a load point distance of 0.3 (F/H in Srawley and Gross’s analysis), are used in the comparison. The abscissa for these data were chosen as c/Z for the comparison. The present parameter √(ΔC.E) from Eq. (12), for the C(T) specimen has about −22% difference at c/Z = 0.36, raising to about −38% at c/Z = 0.84, with respect to the numerical data of Srawley and Gross. This discrepancy is likely due to the differences in the loading point locations between the C(T) specimen in Srawley and Gross’s analysis and that in the present analysis. In the present analysis, the loading points are implicitly located at L/W = ± 1 from the crack plane and along the loading line, which correspond to the top and the bottom edges of the modeled specimen. However, in Srawley and Gross’s analysis, these locations are at distances ± 0.7 L/W form the crack plane or −0.3/+0.3 L/W from the top and bottom edges, respectively. A better comparison K-data set would be that from a
Fig. 4. Net-section fracture mechanics parameter ΔC . E (Eq. (12)) compared with (a) the stress intensity factor obtained by FEM method for the WST specimen and (b) the % difference between the two parameters, with respect to K. 7
Engineering Fracture Mechanics 222 (2019) 106703
K.S. Ravi Chandran
50
50
45
45
K-Tada (L/W=1.2) K-Srawley (L/W=2)
40
SQRT(ΔC.E); L/W=2 SQRT(ΔC.E); L/W=3
40
35
35
30
30
Sqrt (ΔC.E)
Sqrt (ΔC.E) and K for Unit Load
SQRT(ΔC.E); L/W=1
SQRT(ΔC.E); L/W=2
25 20
25 20
15
15
10
10
5
5
0
SQRT(ΔC.E); L/W=5
0 0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
c/Z
c/Z
(a)
(b)
0.8
1
Fig. 5. The calculations from the net-section fracture mechanics parameter ΔC . E (Eq. (12)) for the C(T) specimen geometry. (a) comparison with the K values from standard stress intensity factor data of fracture mechanics and (b) for various L/W ratios of the C(T) specimen.
boundary collocation calculation with the forces acting exactly at the specimen edges. An important point here is that the benefit of the full-specimen analysis can be recognized in Fig. 5(a). It shows that the extrapolation of Tada et al’s GCF for c/Z < 0.1 leads to negative values of K, which may be a problem. This GCF for the standard C(T) specimen was constructed by Tada et al. by defining the crack size, a/W, from the loading line. On the basis of this definition, the K values from Tada et al’s GCF can be determined for the “negative” crack length region, that is for −0.25 < a/W < 0. This region represents the “notch region” in the specimen. Although this region is excluded from Tada et al.’s GCF, it not clear how this notched region was handled in Tada’s GCF determination. Here, however, it leads to negative values for C/Z < 0.1 during the extrapolation, which is not correct, because at any notch length, there should be a positive K value for the C(T) geometry. These negative values may indicate the poorness of the GCF polynomial fit, which was made using the numerical results for 0.2 < a/W < 1, perhaps not considering the limiting value at the crack length tending to zero. This, in turn, may impact the accuracy of the GCF for the valid crack length region, 0.2 < a/W < 1, of the C(T) specimen. Therefore, it is evident that the correct crack behavior in C(T) can be assessed better when the full specimen crack length is used, as is the case for the EC(T) and WST samples discussed here. From a full specimen analysis perspective, the “notch” section of the C(T) specimen is included in the present analysis. In Fig. 5(a), the netsection fracture parameter correctly goes to zero as the full-specimen-crack-size (c/Z) reduces to zero in the C(T) specimen. Finally, a set of curves for the net-section fracture parameter were generated, for the conventional C(T) specimen, for varying L/W ratios, and are presented in Fig. 5(b). The data indicate an increase in stress intensity factor level with an increase in L/W. This trend is to be expected and is consistent with the increase in the stress intensity factor of the EC(T) specimen with L/Z = 3, relative to that of the C(T) specimen with L/W = 2. The Srawley and Gross’s analysis [4], however, indicates a weak dependence of K on the specimen length-to-width ratio, which is somewhat in contradiction to the behavior of fracture mechanics K values of EC(T) relative to that of C(T). A more detailed analysis appears to be necessary. The good agreement of the net-section fracture mechanics analysis results with that of the conventional fracture mechanics can be explained on the basis of Griffith’s crack theory [17,18]. The increase in strain energy of the net-section, as determined here, is equivalent to the increase in the strain energy of a solid due to crack formation in Griffith’s theory. The determination of this quantity is made easy and straight forward here through Timoshenko’s mechanics of materials approach. The present approach is also consistent for the determination of the stress intensity factor by numerical or FEM methods. The compliance-based numerical approach by Mowbray [19], the stiffness derivative method of Parks [20], FEM model of Dixon and Strannigan [21], and a recent lattice network model [22] implicitly consider the deformation of the entire body, in determining the energy release rate or the stress intensity factor. Finally, it is apparent from this work that the actual role of the GCF factors in conventional fracture mechanics is to capture empirically the increase in elastic strain energy of the specimen, upon introduction of the crack. The proposed approach simplifies the determination of fracture parameter for the characterization of fracture in various fracture mechanics specimens while providing a clear physical meaning of the fracture parameter.
8
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K.S. Ravi Chandran
5. Conclusions 1. The net-section-mechanics based analysis of the generalized compact tension specimen, provides a simple and a broadly useful single expression for stress intensity factor calculations of several derivative specimens, including compact tension, extended compact tension and wedge-splitting test specimens. The square-root of the change in net-section strain energy parameter multiplied by the elastic modulus provides an expression that is equivalent to the stress intensity factor expression for a generalized compact tension specimen of any shape. 2. Evaluations of the proposed analytical expression with the conventional stress intensity factor data for the EC(T) and WST specimens indicated very good agreement. The agreement was not that good for the C(T) specimen, but this may be explained given the difference between the loading point location in the present analysis and that in the conventional fracture mechanics analysis. 3. The expression also is applicable for any aspect ratio of the generalized and asymmetrically pin-loaded tension specimen, as long as the validity of the tension and bending formulae are met by the proper choice of the specimen dimensions. 4. The proposed net-section based fracture parameter represents the increase strain energy upon the introduction of the crack and it is in the same spirit of the Griffith’s fracture theory. Therefore, it may be concluded that the role of the geometric correction factors in fracture mechanics may be to actually capture the increase in strain energy in the net-section in a finite specimen. Declaration of Competing Interest The authors declared that there is no conflict of interest. Appendix A. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.engfracmech.2019.106703. References [1] Sanford RJ. Principles of fracture mechanics. NJ: Pearson College Division; 2003. p. 77–81. [2] Cartwright DJ, Rooke DP. Evaluation of stress intensity factors. A general introduction to fracture mechanics. Journal of Strain Analysis Monograph. London: Mechanical Engineering Publications Ltd.; 1978. p. 73–4. [3] Tada H, Paris PC, Irwin GR. The analysis of cracks handbook. NY: ASME Press; 2000. [4] Srawley JE, Gross B. Stress intensity factors for bend and compact specimens. Eng Fract Mech Mech 1972;4(3):587–9. [5] Wilson WK. Stress intensity factors for deep cracks in bending and compact tension specimens. Eng Fract Mech 1970;2(2):169–71. [6] Piascik RS, Newman Jr. JC, Underwood JH. The extended compact tension specimen. Fat Fract Eng Mater Struct 1997;20(4):559–63. [7] Sakai M, Bradt RC. Graphical methods for determining the nonlinear fracture parameters of silica and graphite refractory composites. Fracture mechanics of ceramics. Boston, MA: Springer; 1986. p. 127–42. [8] Brühwiler E, Wittmann FH. The wedge splitting test, a new method of performing stable fracture mechanics tests. Eng Fract Mech 1990;35(1–3):117–25. [9] Ravi Chandran KS. Insight on physical meaning of finite-width-correction factors in stress intensity factor (K) solutions of fracture mechanics. Eng Fract Mech 2017;186:399–409. [10] Ravi Chandran KS. The net-section approach to fracture mechanics and fatigue. Trans ASME J App Mech 2019. [Submitted for Publication, October 2019]. [11] Ravi Chandran KS. A new approach to the mechanics of fatigue crack growth in metals: correlation of mean stress (stress ratio) effects using the change in netsection strain energy. Acta Mater 2017;135:201–14. [12] Ravi Chandran KS. Fatigue crack growth in bending: Successful correlation of mean stress (stress ratio) effects using the change in net-section strain energy. Fat Fract Eng Mater Struct 2018;41(12):2566–76. [13] Ravi Chandran KS. Net-section based approach for fatigue crack growth characterization using compact tension specimen: physical correlation of mean-stress or stress-ratio effects. Int J Fat 2019;124:473–82. [14] ASTM E399-90. Standard test method for plane-strain fracture toughness of metallic materials. Annual book of ASTM Standards, Metals test method and analytical procedures, vol. 03.01; 2003, p. 451–82. [15] Timoshenko SP. Strength of materials: Part 1-elementary theory and problems. 3rd ed. Princeton (NJ): D Van Nostrand Company Inc.; 1955. [16] Guinea GV, Elices M, Planas J. Stress intensity factors for wedge-splitting geometry. Int J Fract 1996;81(2):113–24. [17] Griffith AA. The phenomena of rupture and flow in solids. Phil Trans Roy Soc Lond, Series A 1921;221(582–593):163–98. [18] Griffith AA. The theory of rupture. Proc First Int Cong App Mech, Delft, The Netherlands. 1924. p. 55–63. [19] Mowbray DF. A note on the finite element method in linear fracture mechanics. Eng Fract Mech 1970;2(2):173–6. [20] Parks DM. A stiffness derivative finite element technique for determination of crack tip stress intensity factors. Int J Fract 1974;10(4):487–502. [21] Dixon JR, Strannigan JS. Determination of energy release rates and stress-intensity factors by the finite-element method. J Strain Anal 1972;7(2):125–31. [22] Mohammadipour A, Kaspar W. Lattice approach in continuum and fracture mechanics. ASME J App Mech 2016;83(7):071003.
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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Fracture mechanics analysis of generalized compact tension specimen geometry using the mechanics of net-section
T
K.S. Ravi Chandran Department of Materials Science and Engineering, 135 South 1460 East Rm. 412, The University of Utah, Salt Lake City, UT 84112, USA
A R T IC LE I N F O
ABS TRA CT
Keywords: Fracture mechanics Stress intensity factor Strain energy Net-section Geometric correction factor
Fracture mechanics analysis of a generalized compact tension specimen, on the basis of the mechanics of deformation of the net-section, is shown to provide a simple and a broadly useful expression for stress intensity factor calculations. A single analytical expression is found to be sufficient for the characterization of crack behavior in compact tension, extended compact tension and wedge splitting test specimens without any restriction on the specimen length (or height) and width. The analysis is enabled by the concept of the change in net-section energy, which is determined by summing the changes in strain energies of the net-section of the generalized specimen for tension and bending deformation modes, which result from the introduction of the crack. This is equivalent in concept to the increase in strain energy upon the introduction of the crack, as in the Griffith’s fracture theory. The square-root of the change in netsection strain energy parameter multiplied by the elastic modulus provides an expression that is equivalent to the conventional stress intensity factor expression. The application of the netsection based expression to the standard compact tension, the extended compact tension and the wedge-splitting test specimens shows very good agreements with the crack behaviors as expressed by the stress intensity factor expressions for the respective geometries. The proposed method enables easy analytical determination of stress intensity factors for any asymmetricallyloaded mode-I crack problem.
1. Stress intensity factor for compact tension specimens Stress intensity factors (K) in fracture mechanics are determined by a variety of methods, including analytical, numerical and finite element methods as well as experimental methods such as compliance measurements [1,2]. The standard expressions for the stress intensity factors are available in the well-known compendium by Tada Paris and Irwin [3]. In this reference, the stress intensity factor for the compact tension specimen is given as:
K=
(
a
)
2+W P a F⎛ ⎞ t W 1 − a 3/2 ⎝ W ⎠ W
(
)
where F(a/W) is the geometric correction factor (GCF) given by
a a a 2 a 3 a 4 F ⎛ ⎞ = 0.886 + 4.64 ⎛ ⎞ − 13.32 ⎛ ⎞ + 14.72 ⎛ ⎞ − 5.6 ⎛ ⎞ W W W W W ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
E-mail address: [email protected]. https://doi.org/10.1016/j.engfracmech.2019.106703 Received 26 July 2019; Received in revised form 12 September 2019; Accepted 26 September 2019 Available online 27 September 2019 0013-7944/ © 2019 Elsevier Ltd. All rights reserved.
(1)
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Nomenclature a a/W b b/Z c c/Z C(T) e E EC(T) F(a/W) G GCF I Io IN K M Mo M1 P L L/W L/Z s
t TPB W WST Z σ σL
crack length in C(T) specimen normalized crack length in C(T) specimen crack length in WST specimen normalized crack length in WST crack length in EC(T) or slot depth in WST specimens normalized crack length in EC(T) specimen compact tension eccentricity of load elastic modulus extended compact tension geometric correction factor Griffith strain energy release rate geometric correction factor moment of inertia of cross-section inertia of a crack-free beam inertia of a cracked beam stress intensity factor bending moment moment of a crack-free beam moment of a cracked beam applied load specimen length length-to-width ratio in C(T) specimen length-to-width ratio in EC(T) specimen distance of the loading point from left edge
δo δ1 δob δ1b ε1 εo θ θo θ1 ΔC ΔUb ΔUt ΔWb ΔWt
specimen thickness three-point bending specimen width wedge splitting test specimen total width of EC(T) specimen applied stress average stress in net-section after the introduction of crack load-point displacement in tension before the introduction of crack load-point displacement in tension after the introduction of crack load-point displacement in bending before the introduction of crack load-point displacement in bending after the introduction of crack average strain in net-section after the introduction of crack average strain in uncracked specimen rotation of a bending beam rotation of a crack-free beam rotation of a cracked beam the change in net-section strain energy the change in the strain energy of a bending beam the change in the strain energy in tension work done on net-section of a bending beam work done on net-section in tension
Note that the crack length a and width W, in the above expression, are defined from the loading line. This expression is claimed to be valid for 0.2 ≤ (a/W) ≤ 1 and was constructed in the basis of boundary collocation results of Srawley and Gross [4] and Wilson [5]. A specimen very close to the C(T) specimen is the extended compact tension specimen, EC(T). Fig. 1 illustrates the geometries of both specimens as well as the related wedge splitting test (WST) specimen. The SIF for the EC(T) specimen have been determined by boundary force method by Piasick et al. [6]. The expression for the K is given as
K=
P t Z
c ⎡ c c 3/2 c 1.4 + ⎛ ⎞⎤ ⎛1 − ⎞ G⎛ ⎞ Z ⎣ Z⎠ ⎝ Z ⎠⎦ ⎝ ⎝Z⎠
where
c c c 2 c 3 c 4 c 5 G⎛ ⎞ = 3.97 − 10.88⎛ ⎞ + 26.25⎛ ⎞ − 38.9⎛ ⎞ + 30.15⎛ ⎞ − 9.27⎛ ⎞ Z Z Z Z Z Z ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
(2)
where c is the crack length from the edge, Z is the full-width of the specimen, and W = Z−s, for the EC(T) specimen (Fig. 1a). There are important differences between the GCFs in Eqs. (1) and (2) in terms of how crack length and width are defined. The crack length and the width for the C(T) specimen are taken from the load line running through the centers of loading pin holes. On the other hand, for the EC(T) specimen they are measured from the leftmost edge of the sample (Fig. 1a&b). The latter sample also has 0.2 W as the distance of the pin-hole center from the left edge of the sample, whereas it is 0.25 W in the C(T) specimen. The standard length- or the height-to-width ratios are 1.2 (L/W) and 3 (L/Z) for the C(T) and EC(T) samples, respectively. The choices of the dimensions for the C(T) were made long time ago during the development of ASTM E399 test standard, but EC(T) is a recent modification of C(T) to enable easy experiments. The wedge splitting test (WST) is a technique that is used for determining the fracture energy during the slow stable cracking of brittle materials such as ceramics and concrete [7,8]. The WST specimen is illustrated in Fig. 1(c). Except for the loading arrangement, the WST specimen is the same as the conventional C(T) specimen, and often the GCF of the C(T) specimen is used for the WST specimen to calculate the stress intensity factor for the latter. In all the three geometries the GCFs are valid only for the specific specimen dimensions and shapes for which they were developed. This means that when any of the specimen size parameters need to be changed for whatever reason, one is required to do complex numerical analysis by collocation method or by finite-element-method to determine the GCF for the new geometry. The principal motivation of this work is to find a common analytical basis for the three specimens through the net-section fracture mechanics approach. There is a high degree of geometrical and loading-mode similarity between the three specimens. All the three specimens can be derived from a generalized asymmetric pin-loaded specimen geometry by varying s, W, Z and L, as evident from Fig. 1. The objective of the present work is to develop a generic analytical method and a common expression from which the stress 2
Engineering Fracture Mechanics 222 (2019) 106703
K.S. Ravi Chandran
Fig. 1. Schematics of C(T), EC(T) and WST specimens.
intensity factors for all the three geometries can be determined, while also allowing flexibility to vary any dimension of the generic specimen, in the same expression. 2. The net-section approach for fracture analysis The author has recently shown [9] that geometric correction factors (GCFs) in fracture mechanics may actually be interpreted as stress amplification factors in fracture mechanics specimens. Taking one step further, it has also been shown [10] that simple expressions, derived on the basis of the change in net-section strain energy, produce stress intensity factor values that are in excellent agreement with that provided by standard fracture mechanics expressions. These were the common fracture mechanics specimens such as the middle-crack tension, three-point bending and pin-loaded, single-edge-cracked specimens. Excellent correlations of fatigue crack growth behavior, in terms of the net-section fracture mechanics parameters, have also been demonstrated for these geometries [11–13]. In these works extended analytical expressions, to account for the asymmetry and the cyclic nature of fatigue loading, were developed on the basis of the effective work done such that the expressions are applicable to generalized cyclic loading situations. The present work builds on the success of the previous correlations. Specifically, the objective is to treat the compact tension specimen geometry in a generalized way in the net-section approach and see if the analysis provides results that are broadly applicable to the derivatives of the generalized geometry. The derivatives include the standard compact tension specimen, the extended compact tension specimen and the wedge splitting test specimen. The analysis yielded a simple and broadly applicable expression that provides stress intensity factors for any geometry derived from the generalized specimen geometry. The expression is validated by comparing it with the crack behaviors described by the conventional stress intensity factor expressions for the C(T), EC(T) and WST specimens, determined by others using numerical and FEM methods. A discussion, elaborating why the net-section approach is entirely consistent with Griffith’s theory of fracture, is also presented. 3. The change in net-section strain energy for the generalized specimen Fig. 2 illustrates the net-section approach for the generalized compact tension specimen, GC(T). The concept of the change in netsection strain energy, ΔC, as a fracture mechanics parameter was introduced in our previous works, which are referenced in the above section, in the context of crack propagation in fatigue. Here it is illustrated for the generalized compact tension specimen, as shown Fig. 2(a). For clarity, the deformations in the uncracked and the cracked specimen configurations are exaggerated. Note that in the 3
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K.S. Ravi Chandran
Fig. 2. The generalized compact-tension specimen under asymmetric loading through pins. (a)–(c) illustrate the deformation of the specimen and how it is split into tension and the bending deformations, with and without the crack. (d) and (e) illustrates the work done and the change in netsection strain energy under pure tension. (f) shows the change in bending strain energy, which is given by the difference between the two highlighted areas, OM1θ1 and OMoθo. The figure is a modified form of an earlier version [13].
interest of wide applicability, a full specimen analysis is performed. In this, the width of the specimen is taken to be the entire width from the left to the right of the specimen, Z, with Z = s + W, where s is the portion of the specimen width to the left of the loading axis and W is the width in the conventional sense of the regular C(T) specimen, as in the ASTM E399 standard for fracture testing [14]. Additionally, the crack length, c, is taken from the left edge and this means c = s + a for c > s, where a is the crack length from the loading line, again, as defined in the regular C(T) specimen (Fig. 1(b)). It will be evident later that this basis is quite advantageous 4
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K.S. Ravi Chandran
and clarifies some issues with the stress intensity factor expressions for the C(T) specimen. The deformations in the GC(T) sample consist of tension deformation in the net-section (Fig. 2(b)) as well as the bending deformation caused by the rotation of the net-section (Fig. 2(c)), due to asymmetric loading. When a crack size, c, is introduced at load, P, both deformations will lead to the accumulation of work at the loading point. The total work is captured by adding the work due to tension as well as the work due to bending on the net-section, which are determined as illustrated in the following. The total change in net-section strain energy for the GC(T) specimen, ΔC, is derived here following the Timoshenko [15] approach in mechanics of materials. First, the work done in tension, ΔWt, due to the formation of a crack, is given by (Fig. 2(d)):
ΔWt =
1 P (δ1 − δo) 2
(3)
The corresponding change in net-section strain energy, ΔUt, is given by the difference between the area “OσLε1O” and the area “OσεoO” in Fig. 2(e). Following the steps given elsewhere [11] this leads to
() c
ΔUt =
⎡ σ2 Z (LZt ) ⎢ 2E ⎢ 1− ⎣
(
c Z
)
⎤ ⎥ ⎥ ⎦
(4)
where σ is the nominal stress due to the tension component, E is the elastic modulus and t is the specimen thickness. All other specimen parameters are illustrated in Fig. 2. In terms of load, this is c
ΔUt =
P2 L ⎡ Z ⎛ ⎞ 2E ⎝ Zt ⎠ ⎢ 1 − ⎣
⎤ ⎥ ⎦
c Z
(5)
Next step is to determine the bending component of the change in net-section strain energy, caused by the introduction of the crack. This was previously determined [12] for pure bending as the difference between the bending energy of the beam made of the net-section defined by the crack and that of a crack-free beam. If the deflections due to bending, at load P, in the uncracked and cracked beams are δob and δ1b, respectively, the bending work done due to the crack introduction is given by
ΔWb =
1 P (δ1b − δob) 2
(6)
The corresponding change in strain energy can be written in terms of bending parameters in Fig. 2(c) as
ΔUb =
1 [M1 θ1 − Mo θo] 2
(7)
where M1 and Mo are the moments and θo and θ1 are the rotations of the crack-free and cracked beams, respectively, as shown in Fig. 2(c). Eq. (7) is applied here for the GC(T) specimen to determine the increase in strain energy as indicated by the highlighted area in Fig. 2(f). Using the relationship θ = ML/ EI ,
ΔUb =
M2L 1 ⎡ M12 L − o ⎤ ⎢ 2E ⎣ IN Io ⎥ ⎦
(8)
where L is the length of the beam, IN and Io are beam inertia terms given by: IN = t(Z − c) /12 and Io = tZ /12. As the crack length increases, the neutral axis in bending will move to the right relative to that in the uncracked specimen, creating an increasingly eccentric loading situation. For the uncracked specimen, the loading eccentricity is given by ((Z/2) − s). For the cracked specimen, the eccentricity is ((Z − 2 s + c)/2). Therefore Eq. (8) can be written explicitly as 3
P 1 ⎡ ⎢ ΔUb = 2E ⎢ ⎣
2 2 Z − 2s + c L 2
(
)
IN
P2 −
(
3
Z − 2s 2 L⎤ 2
)
Io
⎥ ⎥ ⎦
(9)
Simplifying Eq. (9),
(
{}
⎡ 1−2 s + Z 3P 2 L ⎢ ⎛ ⎞ ΔUb = c 3 2E ⎝ Zt ⎠ ⎢ 1− Z ⎢ ⎣
(
c 2 Z
)
)
⎤ s ⎞2⎥ − ⎛1 − 2 Z ⎠⎥ ⎝ ⎥ ⎦
{}
(10)
The total change in strain energy of the GC(T) specimen, ΔC, can be determined by adding the tension component (Eq. (5)) and bending component (Eq. (10)): c ⎧ P2 L ⎪ ⎡ Z ⎛ ⎞ ⎢ ΔC = 2E ⎝ Zt ⎠ ⎨ 1 − ⎪⎣ ⎩
(
{}
⎡ 1−2 s + Z ⎤ ⎢ 3 + c ⎥ c 3 ⎢ 1− Z Z ⎦ ⎢ ⎣
(
)
c 2 Z
)
⎤⎫ s ⎞2⎥ ⎪ − ⎛1 − 2 Z ⎠ ⎥⎬ ⎝ ⎪ ⎥ ⎦⎭
{}
(11)
In Griffith’s crack theory, the increase in elastic strain energy of an infinite plate due to the crack formation is determined. The 5
Engineering Fracture Mechanics 222 (2019) 106703
K.S. Ravi Chandran
relationship between the Griffith’s energy release rate, G, and the fracture mechanics stress intensity factor, K, is given by the Irwin’s relationship: K = G. E . Since ΔC also represents the increase in strain energy in a finite specimen, the parameter form that would be compatible with the stress intensity factor in fracture mechanics will be ΔC . E . Hence, Eq. (11) can be written as
ΔC . E =
P 2t
c
L Z ⎛ ⎞ ⎡ ⎢ ⎝Z⎠ ⎣1 −
(
{}
⎡ 1−2 s + Z ⎤ ⎢ 3 + c ⎥ c 3 ⎢ 1− Z Z ⎦ ⎢ ⎣
(
)
c 2 Z
)
⎤ s ⎞2⎥ − ⎛1 − 2 Z ⎠⎥ ⎝ ⎥ ⎦
{}
(12)
where L/Z is the length-to-width ratio of the generalized compact tension specimen. Eq. (12), in this form, directly applies to EC(T) specimen or C(T) specimen, where the normalized crack length is defined from the edge of the sample, c/Z, in both geometries. The same equation can be used for WST specimen, by replacing “c” with “b.” Both are the equivalent type of crack lengths from the edge: the former is for the EC(T) specimen and the latter is for the WST specimen (Fig. 1). The parameter ΔC . E can be given a letter or symbol as a single variable, once it is widely accepted. 4. Evaluation and discussion Fig. 3(a) illustrates the comparison of the net-section parameter ΔC . E from the present approach and the standard stress intensity factor data for EC(T) specimen (L/Z = 3) determined by Piascik et al. [6] by the body force method. In the calculations, however, L/Z = 3.2 was used as this corresponds to the actual loading length (distance) between the points of contacts of the loading pins with the holes in the specimen. The agreement between the Piascik et al.’s data and the present net-section-based calculation from Eq. (12) is surprisingly good over the entire range of normalized crack length, c/Z. The % difference between the two parameters, with respect to K, is plotted as a function of c/Z in Fig. 3(b)—the difference is less than 10% for 0.3 < (c/Z) < 0.95, which indicates that the present approach will be useful, especially for other L/Z ratios of EC(T) samples for which no GCF factors are available. The difference could be due to the errors inherent in each method. In the boundary element modeling of Piascik et al., the K expression for EC(T) was numerically determined and could have a certain amount of error with respect to the exact solution. The present method also will have some error, the magnitude of which could be on the order of the error in the bending formula of strength of materials. This is likely to be small, since the bending formula used here is considered to be exact in strength of materials. Both factors, however, can contribute to the observed difference between the two data and their relative error contribution can only be assessed when an exact solution is found. It is important to note the nature of the net-section parameter in Eq. (12) versus the stress intensity factor. For zero crack length, the change in net-section strain energy is zero, thus, the present method gives a correct result at zero crack length. However, such a
Fig. 3. Net-section fracture mechanics parameter ΔC . E (Eq. (12)) compared with (a) the stress intensity factor obtained by boundary force method for the EC(T) specimen and (b) the % difference between the two parameters, with respect to K. 6
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limiting value is not possible for K, since it is not defined at this limit. The K-behavior should be considered to be valid at small crack sizes, as small as possible (since LEFM definitions are increasingly accurate at vanishing crack lengths), but not zero. An exploration of why K cannot be defined at the limit but its values agree with the limiting trend near the zero crack length is beyond the scope of this work. In Fig. 4(a) the finite-element-modeling (FEM) results of wedge splitting test specimen (Fig. 1(c)) having L/Z = 2, generated for c/ Z = 0.25 by Guinea et al. [16], are used to compare the calculations from Eq. (12). Here, c represents the depth of the slot, as shown in Fig. 1(b). In this FEM model, the loading points were located at s = 0.05Z. The data corresponding to L/Z ratio of 2 with Z/c = 4 is chosen for the present comparison. The data, presented in terms of a/W, were converted to the full-specimen normalized crack length (b/Z) basis, by calculating as: b/Z = (c + (a/W)W)/Z. Additionally, the normalized K values, which were presented in terms of Kt√c/ P were converted into Kt√W/P, by using the fact that c = W/3 in Guinea et al.’s FEM model. Fig. 4(a) illustrates that the crack behavior as expressed by ΔC . E also agrees very well with the stress intensity factor data provided by Guinea et al., except for small values of b/Z. The proximity of the crack to the slot end, at b/Z = 0.25, in the FEM model of the WST specimen, may have influenced the deviation of the K from the main trend, around this b/Z value. The % difference between the two parameters, with respect to K, is plotted as a function of b/Z in Fig. 4(b)—this difference is also less than 10% for 0.3 < (b/Z) < 0.95, which indicates that the present approach will be useful, especially for other L/Z ratios of WST samples for which no GCF factors are available. For 0.3 < (b/ Z) < 0.95, the trend in the difference is approximately similar to that for the EC(T) specimen, as shown in Fig. 3(b). As stated before, this difference could be due to the inherent error in each method. The FEM modeling of Guinea et al, for WST specimen will produce a certain amount of error in the data, which is typically a few %. The present model also will have some error, the magnitude of which could be the same as that in the bending formula. Although the magnitude of these errors are not known, they are not zero with respect to the exact solution. Fig. 5 illustrates the comparison for the conventional C(T) specimen in fracture mechanics. Here, a value of L/W = 2 is used for calculating ΔC . E from Eq. (12). Note that the data needs to be plotted in terms of full crack length ratio, c/Z, as applicable to the C(T) specimen (Fig. 1(a)) for a better comparison. Since the crack length and specimen width for the C(T) specimen are defined from the loading line as a and W, then c = a + s, Z = W + s and, a/W = (c − s)/(Z − s), in terms of full-specimen parameters. Here, the GCFs provided by Tada [3] and the tabulated data from Srawley and Gross [4] for L/W = 2 with a load point distance of 0.3 (F/H in Srawley and Gross’s analysis), are used in the comparison. The abscissa for these data were chosen as c/Z for the comparison. The present parameter √(ΔC.E) from Eq. (12), for the C(T) specimen has about −22% difference at c/Z = 0.36, raising to about −38% at c/Z = 0.84, with respect to the numerical data of Srawley and Gross. This discrepancy is likely due to the differences in the loading point locations between the C(T) specimen in Srawley and Gross’s analysis and that in the present analysis. In the present analysis, the loading points are implicitly located at L/W = ± 1 from the crack plane and along the loading line, which correspond to the top and the bottom edges of the modeled specimen. However, in Srawley and Gross’s analysis, these locations are at distances ± 0.7 L/W form the crack plane or −0.3/+0.3 L/W from the top and bottom edges, respectively. A better comparison K-data set would be that from a
Fig. 4. Net-section fracture mechanics parameter ΔC . E (Eq. (12)) compared with (a) the stress intensity factor obtained by FEM method for the WST specimen and (b) the % difference between the two parameters, with respect to K. 7
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K.S. Ravi Chandran
50
50
45
45
K-Tada (L/W=1.2) K-Srawley (L/W=2)
40
SQRT(ΔC.E); L/W=2 SQRT(ΔC.E); L/W=3
40
35
35
30
30
Sqrt (ΔC.E)
Sqrt (ΔC.E) and K for Unit Load
SQRT(ΔC.E); L/W=1
SQRT(ΔC.E); L/W=2
25 20
25 20
15
15
10
10
5
5
0
SQRT(ΔC.E); L/W=5
0 0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
c/Z
c/Z
(a)
(b)
0.8
1
Fig. 5. The calculations from the net-section fracture mechanics parameter ΔC . E (Eq. (12)) for the C(T) specimen geometry. (a) comparison with the K values from standard stress intensity factor data of fracture mechanics and (b) for various L/W ratios of the C(T) specimen.
boundary collocation calculation with the forces acting exactly at the specimen edges. An important point here is that the benefit of the full-specimen analysis can be recognized in Fig. 5(a). It shows that the extrapolation of Tada et al’s GCF for c/Z < 0.1 leads to negative values of K, which may be a problem. This GCF for the standard C(T) specimen was constructed by Tada et al. by defining the crack size, a/W, from the loading line. On the basis of this definition, the K values from Tada et al’s GCF can be determined for the “negative” crack length region, that is for −0.25 < a/W < 0. This region represents the “notch region” in the specimen. Although this region is excluded from Tada et al.’s GCF, it not clear how this notched region was handled in Tada’s GCF determination. Here, however, it leads to negative values for C/Z < 0.1 during the extrapolation, which is not correct, because at any notch length, there should be a positive K value for the C(T) geometry. These negative values may indicate the poorness of the GCF polynomial fit, which was made using the numerical results for 0.2 < a/W < 1, perhaps not considering the limiting value at the crack length tending to zero. This, in turn, may impact the accuracy of the GCF for the valid crack length region, 0.2 < a/W < 1, of the C(T) specimen. Therefore, it is evident that the correct crack behavior in C(T) can be assessed better when the full specimen crack length is used, as is the case for the EC(T) and WST samples discussed here. From a full specimen analysis perspective, the “notch” section of the C(T) specimen is included in the present analysis. In Fig. 5(a), the netsection fracture parameter correctly goes to zero as the full-specimen-crack-size (c/Z) reduces to zero in the C(T) specimen. Finally, a set of curves for the net-section fracture parameter were generated, for the conventional C(T) specimen, for varying L/W ratios, and are presented in Fig. 5(b). The data indicate an increase in stress intensity factor level with an increase in L/W. This trend is to be expected and is consistent with the increase in the stress intensity factor of the EC(T) specimen with L/Z = 3, relative to that of the C(T) specimen with L/W = 2. The Srawley and Gross’s analysis [4], however, indicates a weak dependence of K on the specimen length-to-width ratio, which is somewhat in contradiction to the behavior of fracture mechanics K values of EC(T) relative to that of C(T). A more detailed analysis appears to be necessary. The good agreement of the net-section fracture mechanics analysis results with that of the conventional fracture mechanics can be explained on the basis of Griffith’s crack theory [17,18]. The increase in strain energy of the net-section, as determined here, is equivalent to the increase in the strain energy of a solid due to crack formation in Griffith’s theory. The determination of this quantity is made easy and straight forward here through Timoshenko’s mechanics of materials approach. The present approach is also consistent for the determination of the stress intensity factor by numerical or FEM methods. The compliance-based numerical approach by Mowbray [19], the stiffness derivative method of Parks [20], FEM model of Dixon and Strannigan [21], and a recent lattice network model [22] implicitly consider the deformation of the entire body, in determining the energy release rate or the stress intensity factor. Finally, it is apparent from this work that the actual role of the GCF factors in conventional fracture mechanics is to capture empirically the increase in elastic strain energy of the specimen, upon introduction of the crack. The proposed approach simplifies the determination of fracture parameter for the characterization of fracture in various fracture mechanics specimens while providing a clear physical meaning of the fracture parameter.
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Engineering Fracture Mechanics 222 (2019) 106703
K.S. Ravi Chandran
5. Conclusions 1. The net-section-mechanics based analysis of the generalized compact tension specimen, provides a simple and a broadly useful single expression for stress intensity factor calculations of several derivative specimens, including compact tension, extended compact tension and wedge-splitting test specimens. The square-root of the change in net-section strain energy parameter multiplied by the elastic modulus provides an expression that is equivalent to the stress intensity factor expression for a generalized compact tension specimen of any shape. 2. Evaluations of the proposed analytical expression with the conventional stress intensity factor data for the EC(T) and WST specimens indicated very good agreement. The agreement was not that good for the C(T) specimen, but this may be explained given the difference between the loading point location in the present analysis and that in the conventional fracture mechanics analysis. 3. The expression also is applicable for any aspect ratio of the generalized and asymmetrically pin-loaded tension specimen, as long as the validity of the tension and bending formulae are met by the proper choice of the specimen dimensions. 4. The proposed net-section based fracture parameter represents the increase strain energy upon the introduction of the crack and it is in the same spirit of the Griffith’s fracture theory. Therefore, it may be concluded that the role of the geometric correction factors in fracture mechanics may be to actually capture the increase in strain energy in the net-section in a finite specimen. Declaration of Competing Interest The authors declared that there is no conflict of interest. Appendix A. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.engfracmech.2019.106703. References [1] Sanford RJ. Principles of fracture mechanics. NJ: Pearson College Division; 2003. p. 77–81. [2] Cartwright DJ, Rooke DP. Evaluation of stress intensity factors. A general introduction to fracture mechanics. Journal of Strain Analysis Monograph. London: Mechanical Engineering Publications Ltd.; 1978. p. 73–4. [3] Tada H, Paris PC, Irwin GR. The analysis of cracks handbook. NY: ASME Press; 2000. [4] Srawley JE, Gross B. Stress intensity factors for bend and compact specimens. Eng Fract Mech Mech 1972;4(3):587–9. [5] Wilson WK. Stress intensity factors for deep cracks in bending and compact tension specimens. Eng Fract Mech 1970;2(2):169–71. [6] Piascik RS, Newman Jr. JC, Underwood JH. The extended compact tension specimen. Fat Fract Eng Mater Struct 1997;20(4):559–63. [7] Sakai M, Bradt RC. Graphical methods for determining the nonlinear fracture parameters of silica and graphite refractory composites. Fracture mechanics of ceramics. Boston, MA: Springer; 1986. p. 127–42. [8] Brühwiler E, Wittmann FH. The wedge splitting test, a new method of performing stable fracture mechanics tests. Eng Fract Mech 1990;35(1–3):117–25. [9] Ravi Chandran KS. Insight on physical meaning of finite-width-correction factors in stress intensity factor (K) solutions of fracture mechanics. Eng Fract Mech 2017;186:399–409. [10] Ravi Chandran KS. The net-section approach to fracture mechanics and fatigue. Trans ASME J App Mech 2019. [Submitted for Publication, October 2019]. [11] Ravi Chandran KS. A new approach to the mechanics of fatigue crack growth in metals: correlation of mean stress (stress ratio) effects using the change in netsection strain energy. Acta Mater 2017;135:201–14. [12] Ravi Chandran KS. Fatigue crack growth in bending: Successful correlation of mean stress (stress ratio) effects using the change in net-section strain energy. Fat Fract Eng Mater Struct 2018;41(12):2566–76. [13] Ravi Chandran KS. Net-section based approach for fatigue crack growth characterization using compact tension specimen: physical correlation of mean-stress or stress-ratio effects. Int J Fat 2019;124:473–82. [14] ASTM E399-90. Standard test method for plane-strain fracture toughness of metallic materials. Annual book of ASTM Standards, Metals test method and analytical procedures, vol. 03.01; 2003, p. 451–82. [15] Timoshenko SP. Strength of materials: Part 1-elementary theory and problems. 3rd ed. Princeton (NJ): D Van Nostrand Company Inc.; 1955. [16] Guinea GV, Elices M, Planas J. Stress intensity factors for wedge-splitting geometry. Int J Fract 1996;81(2):113–24. [17] Griffith AA. The phenomena of rupture and flow in solids. Phil Trans Roy Soc Lond, Series A 1921;221(582–593):163–98. [18] Griffith AA. The theory of rupture. Proc First Int Cong App Mech, Delft, The Netherlands. 1924. p. 55–63. [19] Mowbray DF. A note on the finite element method in linear fracture mechanics. Eng Fract Mech 1970;2(2):173–6. [20] Parks DM. A stiffness derivative finite element technique for determination of crack tip stress intensity factors. Int J Fract 1974;10(4):487–502. [21] Dixon JR, Strannigan JS. Determination of energy release rates and stress-intensity factors by the finite-element method. J Strain Anal 1972;7(2):125–31. [22] Mohammadipour A, Kaspar W. Lattice approach in continuum and fracture mechanics. ASME J App Mech 2016;83(7):071003.
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