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International Journal of Fracture 127: 283–302, 2004. © 2004 Kluwer
Academic Publishers. Printed in the Netherlands.

Characterization of crack-tip field and constraint for bending specimens under large-scale yielding

Y.J. CHAO1,∗, X.K. ZHU1, Y. KIM1, P. S. LAR2, M.J. PECHERSKY2 and M.J. MORGAN2

1Department of Mechanical Engineering, University of South Carolina 2Savannah River Technology Center, Savannah River Company ∗Author for correspondence (E-mail: [email protected])

Received 16 November 2003; accepted in revised form 1 April 2004

Abstract. Elastic-plastic crack-tip fields and constraint levels in bending specimens under large-scale yielding (LSY) are examined. The J − A2 three-term solution is modified by introducing an additional term caused by the global bending. Three different methods, i.e. two-point matching, constant A2 and elastic stress estimation method, are proposed to determine the fourth term. It is shown that the elastic stress estimation method is the simplest, yet effective, in that the fourth term can be derived from the strength theory of materials and the concept of plastic hinge, and effectively quantifies the contribution of the global bending moment on the crack-tip field. Consequently, the modified J − A2 solution, with the inclusion of the correction for global bending, does not introduce any new parameter. The two parameters remain as the loading (J and M) and the constraint level (A2). To validate the present solution, detailed finite element analyses (FEA) were conducted for a Three Point Bend (TPB) specimen with a/W = 0.59 in A285 steel, and Single Edge Notched Bend (SENB) specimen subjected to pure bending with a/W = 0.5 in A533B steel at different deformation levels ranging from small-scale yielding (SSY) to LSY. Results show that the modified J − A2 solution matches fairly well with the FEA results for both TPB and SENB specimens at all deformation levels considered. In addition, the fourth stress term is (a) propor- tional to the global bending moment and inversely proportional to the ligament length; (b) negligibly small under SSY; and (c) significantly large under LSY or fully plastic deformation. Accordingly, the present model effectively characterizes the crack-tip constraint for bending dominated specimens with or without the large influence from the global bending stress on the crack-tip field.

Key words: Crack-tip field, fracture constraint, TPB, SENB, large-scale yielding.

1. Introduction

Fracture constraint refers to the effect of geometric and loading configurations of a flawed specimen or structure on the mechanics behavior at the crack-tip. It has been shown that the crack-tip constraint has significant effect on the crack-tip field, fracture toughness and crack extension resistance to ductile tearing, and therefore is an important parameter in structural design and integrity assessment using fracture mechanics methodology.

It is well known that for elastic-plastic materials, the HRR singularity field (Hutchinson, 1968; Rice and Rosengren, 1968) provides an effective characterization of the crack-tip stress field using the single parameter J -integral for high constraint specimens. The two-parameter approaches including the J − T approach (Betegon and Hancock, 1991), J − Q theory (O’Dowd and Shih, 1991; 1992) and J − A2 three-term solution (Yang et al., 1993; Chao et al., 1994; Chao and Zhu, 1998) are valuable in describing the crack-tip stress fields for low constraint specimens beyond the single parameter characterization. However, extensive work (Shih and German, 1987; O’Dowd and Shih, 1992; Parks, 1992; Wang and Parks, 1995; Chao

284 Y.J. Chao et al.

and Zhu, 1998; Lam et al., 2003) have shown that for bending dominated fracture specimens, such as the Three Point Bend (TPB), Single Edge Notched Bend (SENB) under pure bending, Single Edge Notched Tension (SENT) and Compact Tension (CT) specimens, both the single and two-parameter crack-tip solutions are limited to small scale yielding (SSY) conditions, because the crack-tip field is significantly affected by the global bending moment under the conditions of large-scale yielding (LSY) or fully plastic deformation. As a result, none of the available crack-tip solutions can accurately characterize the crack-tip field for such bending specimens within the crack-tip region prone to ductile fracture. Accordingly, all existing the- ories fail to quantify the crack-tip constraint in the bending specimens under LSY conditions, as outlined in the following reviews.

For shallow cracked bending specimens, the global bending has limited influence on the crack-tip field, and the two-parameter solutions can well characterize the crack-tip field even under LSY conditions (Al-Ani and Hancock, 1991; Zarzour et al., 1993; Chao and Zhu, 1998). For deeply cracked bending specimens, the crack-tip field exhibits high constraint behavior and is fairly well described by the single parameter J -integral through the HRR asymptotic solution under relatively low load or SSY conditions. As the load increases beyond the SSY level, however, the J gradually losses its dominance since the global bending progressively builds up at and impinges on the crack-tip. Chao and Zhu (1998) showed that under contained yielding conditions, the J −A2 solution can still approximately describe the crack-tip field of SENB specimens at a deformation level between SSY and LSY. However, under LSY or fully plastic deformation when bσ0/J ≤ 30, where b is the ligament length ahead of the crack-tip and σ0 is the yield stress of the material, the global bending strongly affects the crack-tip stress field of SENB specimens with deep cracks for low hardening materials (e.g. strain hardening exponent n = 10). The size of the J − A2 dominance zone r shrinks to a small value, e.g. r/(J/σ0) < 2, which is much smaller than the region at the crack-tip prone to ductile fracture, i.e. 1 < r(J/σ0) (Ritchie et al., 1973).

Similar phenomena are also observed when using the J − Q theory. O’Dowd and Shih (1992) showed that in a deeply cracked bend bar, the crack-opening stress decreases rapidly with distance away from the crack-tip for the fully yielding condition (i.e. J/σ0 > 0.05b or bσ0/J < 20), and the stress gradient across the ligament is very large. The opening stress is compressive near the free surface and becomes tensile as the crack tip is approached. The global bending stress distribution prevails near the crack-tip when J/σ0 > 0.06b. Wang and Parks (1995) discussed certain limits of the two parameter J − T and J − Q characterization of elastic-plastic crack-tip fields, and concluded that the opening stress is dominated by the global bending stress for a TPB specimen with a/W = 0.4 and J/aσ0 > 0.096. Wei and Wang (1995) and Karstensen et al. (1997) pointed out that at LSY or fully plastic deformation state, the difference of the full field numerical solution and the SSY crack-tip field for the SENB specimen, which is used to determine the Q-stress, becomes strongly distance de- pendent as the global bending stress is significant. This influence arises because the SENB specimen is subjected to a global bending moment and the ligament far from the crack-tip is in compression. In summary, for deep bend specimens under LSY or fully plastic deformation, all available two-parameter solutions for quantifying the constraint including the J −T , J −Q

and J − A2 break down since the global bending significantly impinges on the crack-tip and alters the stress fields.

To solve this problem, Wei and Wang (1995) modified the J −Q two-parameter solution to include a third parameter k2, and proposed the J −Q − k2 three-parameter solution, where Q

and k2 are determined by matching with the FEA results. However, the physical meaning of the

Characterization of crack-tip field and constraint for bending specimens 285

third parameter is ambiguous, i.e. whether this third parameter (k2) is a constraint parameter, a loading parameter, or simply a numerical fit is unclear. Likewise, Karstensen et al. (1997) also modified the J − Q scheme by decomposing the parameter Q into two parts. One is a distance independent term QT , which is formally related to the T stress under the SSY; the other is a distance dependent term Qp, which is related to the global bending stress field and is regarded as the difference between the total loss of constraint given by Q and the loss of constraint given by a negative T . They then suggested a crack-tip constraint estimation scheme for SENB specimens using the three parameters, J , QT , and Qp.

The present paper revisits the problem stated above and extends the J − A2 three-term solution to include an additional term caused by the global bending to characterize the crack- tip stress field of deeply cracked TPB and SENB specimens under LSY conditions. Three different methods, i.e. two-point matching, constant A2 and elastic stress estimation method, are presented to determine the additional term. It is shown that the simplest one is the elastic stress estimation method which uses the strength theory of materials and the concept of plastic hinges. In the J − A2 four-term solution, J as usual represents the intensity of applied loads, A2 describes the crack-tip constraint level, and the additional or the fourth term is induced by the global bending moment directly related to the applied load. Therefore, the fourth term does not introduce a third parameter since it is related to the applied load or specifically it can be evaluated from the global bending moment determined by the simple strength of materials approach. The results from the proposed method are then compared with the FEA full field solutions. Comparisons show that the modified J − A2 solution agrees well with the full field solutions. Consequently, the parameter A2 can effectively quantify the constraint level of bending specimens in both LSY and SSY deformation states.

2. The J − A2 three-term solution and its modification

Under LSY and deep bend specimens, Chao and Zhu (1998) have shown that the J − A2

three-term solution is valid to characterize the crack-tip field only within a small fraction of the ligament. For the SENB specimen, the influence of the global bending stress on the crack-tip field is very small within the J − A2 valid region, but becomes large gradually beyond the region. The J − A2 solution is briefly reviewed in this section and then modified to characterize more accurately the crack-tip stress field and constraint of bending specimens under LSY conditions

2.1. THE J − A2 THREE-TERM SOLUTION

Yang et al. (1993) and Chao et al. (1994) developed the J − A2 three-term solution with the J -integral as the applied loading intensity and A2 the constraint parameter. Under plane strain conditions, the three-term asymptotic stress field can be expressed as

σij

( r

L

)s2

2

( r

L

)s3

] , (1)

where the stress angular functions σ (k) ij (θ) (k = 1, 2, 3) and the stress power exponents sk

(s1 < s2 < s3) only depend on the hardening exponent n, and is independent of the other material constants (i.e. hardening parameter α, yield strain ε0, and yield stress σ0) and the applied loads. L is a characteristic length parameter and L = 1 mm is taken in this work. The parameters A1 and s1 have the same values as given in the HRR field:

286 Y.J. Chao et al.

Figure 1. Comparison of the opening stresses along the ligament for CCP and SENB specimens.

A1 = (

J

αε0σ0InL

)−s1

n + 1 (2)

When n ≥ 3, the third stress power exponent depends on the first and second stress exponents, i.e. s3 = 2s2−s1. The angular values of σ

(k) ij (θ) and sk are reported by Chao and Zhang (1997).

The parameter A2 can be regarded as a measure of the crack-tip constraint. Using the point matching method, one can determine the A2 value by matching the opening stress from the three-term solution with the FEA result at r/(σ0/J ) = 1 ∼ 2 (Chao and Zhu, 2000).

2.2. A MODIFICATION OF THE J − A2 THREE-TERM SOLUTION

Before modifying the J − A2 solution, a simple comparison of the opening stress ahead of a crack-tip for bending and tension specimens is demonstrated first. The opening stresses for a tension specimen, such as the Central Cracked Panel (CCP) or the Double Edge Cracked Plate (DECP), are tensile along the entire ligament, and approach to uniform distributions away from the crack-tip (Zhu and Chao, 2000). Consequently, the stresses in such tension dominant specimens can be well described by the J−A2 three-term solution, as shown in Chao and Zhu (1998). For bending dominant specimens like TPB and SENB, the opening stress is tensile along the ligament near the crack-tip, but becomes compressive near the specimen surface on the other side of the crack-tip. Figure 1 illustrates the difference of the opening stress distribution along the ligament for the CCP and SENB specimens under LSY conditions within the small strain framework. It is apparent that the compressive stress near the free surface of the bending specimen is caused by the global bending.

For elastic-plastic crack problems, the applied bending stress is independent of the crack- opening stress, but the opening stress depends on the applied bending stress. Since both the opening and bending stresses on the ligament are nonlinear under LSY conditions, it is extremely difficult to accurately account for the global bending stress in the asymptotic crack- tip field. As an approximation, however, it is assumed here that the opening stress, referred

Characterization of crack-tip field and constraint for bending specimens 287

to as σθθ at θ = 0 hereafter, for bending specimens is a simple superposition of the J − A2

solution, σ JA2 θθ and the global bending induced stress, σM

θθ :

θθ (3)

Extensive finite element analyses (FEA), as reported by O’Dowd and Shih (1992), Wang and Parks (1995), Wei and Wang (1995), Karstensen et al. (1997), Chao and Zhu (1998) and the present work, indicate that the global bending stress dominates the crack-tip stress field of bending specimens under LSY conditions, and the opening stress linearly distributes on the ligament away from the crack-tip even within the region of interest 1 < r/(J/σ0) < 5. Furthermore, the present calculation further reveals that at a specific location (i.e. a fixed r) on the ligament ahead of the crack-tip, the opening stress is also linearly related to the applied load or moment, as shown in Figures 6 and 10 later. Therefore, it is reasonable to assume that the global bending induced stress σM

θθ in Equation (3) is a linear function of the distance from the crack-tip, r, and the global bending moment, M, on the ligament plane of the specimen as

σM θθ (r, θ = 0) = C

Mr

b3 , (4)

where C is an undetermined load-independent constant, and M is the moment per unit length, e.g. the thickness of the specimen. The ligament length b is arbitrarily inserted in Equation (4) so that the C is a non-dimensional constant. The next section will confirm that Equation (4) is a reasonable form for the bending induced stress.

Substituting Equations (1) and (4) into Equation (3), we obtain the general form of the modified J − A2 solution for the opening stress ahead of the crack-tip, i.e. θ = 0, as follows

σθθ(r, 0)

σ0 = A1

2.3. DETERMINATION OF THE CONSTANT C

To determine magnitude of the constant C in Equation (5) and understand its physical mean- ing, three different methods, i.e. two-point matching, constant A2 and elastic stress estimation methods are discussed in this section.

(a) Two-point matching method. For a given applied load, it is assumed that the opening stress ahead of a crack-tip within the interest region 1 < r(J/σ0) < 5 has been determined in a FEA calculation. Letting σFEA1

θθ (0) and σFEA2 θθ (0) are the FEA results of the opening

stress at two specific points at r = r1 and r = r2, respectively, then the constraint parameter A2 and the constant C in Equation (5) can be determined by solving the following equations simultaneously.

σFEA1 θθ (0)

(r1

L

)s2

2

(r1

L

)s3

(r2

L

)s2

2

(r2

L

)s3

(6)

(b) Constant A2 method. Yang et al. (2003) and Chao and Zhu (2000) have shown that the constraint parameter A2 is invariable under LSY and fully plastic deformation. If so, determination of the constant C is straightforward. For a given bending specimen, FEA

288 Y.J. Chao et al.

calculations can be carried out first under different loading levels from SSY to LSY, and then the corresponding A2 values are determined by matching FEA results with the conventional J − A2 three-term solution in Equation (1). The A2 becomes a constant or approximate con- stant when the applied load increases from SSY to LSY, this constant can be thus adopted as the value of the constraint parameter A2 for the bending specimen under LSY conditions. It should be noted that as the applied load increases from SSY to LSY, the A2 may approach a constant first, and then deviates from the constant later for deeply cracked bending specimens when only using Equation (1).

With the invariant value of A2, the constant C in Equation (5) can be easily determined using a FEA result of the opening stress σFEA

θθ (0) under LSY near the crack-tip, for example, at r = 2J/σ0 by

C= σ 2 0 b3

2JM

{ σFEA

( 2J

Lσ0

)s2

2

( 2J

Lσ0

)s3

]} (7)

(c) Elastic stress estimation method. Both methods proposed above, as shown in Equa- tions (6) and (7), require the FEA calculations first, and does not reveal the physical meaning of the constant C. Accordingly, a simple and intuitive method to estimate the bending induced stress in Equation (3) or determine the constant C in Equation (5) is introduced here.

To determine the crack-tip opening displacement (CTOD) for a TPB specimen, the concept of plastic hinge was introduced in fracture mechanics analysis (see Broek, 1973 for detailed discussion) as the crack surfaces rotate when loaded. Veerman and Muller (1972) found that the center of this rotation is located at about 0.47b from the crack-tip. For a plain un-cracked beam with a height equal to the ligament length b, the neutral axis of the rectangular cross section is at the centerline of the rectangle, i.e. 0.5b, which is close to 0.47b. It is therefore assumed that the neutral axis of an un-cracked beam is approximately the axis of plastic rotation (or plastic hinge) of the bending specimens. The global bending induced stress in Equation (3) can thus be estimated by that for the plain un-cracked beam.

In the theory of plastic bending for beams (Johnson and Mellor, 1983; Dowling, 1993), it is assumed that the originally plane cross section remains plane even under plastic deformation. It results in a linear variation of the normal strain with distance from the neutral axis. Hence, if yielding occurs, the nonlinear stress-strain behavior of the beam material causes the stress distribution to be nonlinear and the distribution should have the same shape as the correspond- ing stress-strain curve up to the maximum strain at the edges of the beam. For a beam having rectangular cross-section of b in height and B in thickness, the bending stress under elastic condition is given as

σM(r) = Mx∗

Iy

, (8)

where M is the applied bending moment, x∗ is the distance from the neutral axis or x∗ = b 2 −r,

and Iy = Bb3

12 is the second moment of inertia of the area of the cross section about the neutral axis. It should be noted that the elastic bending stress σM in Equation (8) is different from the bending induced stress σM

θθ assumed in Equation (3). As illustrated in Figure 2, the global bending stress on the ligament MN separates the

ligament plane of a bending specimen into a tensile and a compressive zone. Both zones experience partly plastic yielding under LSY conditions. And, the plastic hinge is approxim- ately located at the neutral axis, i.e. point O. Let us focus our attention on the tensile zone,

Characterization of crack-tip field and constraint for bending specimens 289

Figure 2. Illustration of the global bending stress estimation.

i.e. region MO in Figure 2, and the local singularity stress at the crack-tip is not considered for the moment. For elastic-plastic materials under LSY, the global bending stress is linear in the region at and close to the neutral axis or point O, but nonlinear near the crack-tip or point M. While the curved triangle MDHO denotes the schematic distribution of the elastic- plastic tensile stress, the right triangle MDO stands for the elastic tensile stress. Near point M, the curve DH can be approximated by line DI which is the bisector line of triangle ODF with FG = GO or EI = IJ. Recall that the J − A2 three-term solution (as well as the HRR solution) is developed or well-suited for a crack under remote uniform tension, such as the uniform tensile stress indicated by the rectangle MDFO. For bending specimen, however, the applied tensile bending stress on the ligament is non-uniformly distributed, e.g. MDHO. As a result, extra tensile stress generated from DEFOH is enforced in the crack-tip J − A2

solution. This contribution must be deducted from the crack-tip opening stress for bending specimens. From an arbitrary point K to the crack-tip M, the global bending stress denoted by the quadrilateral MDHK can be approximated by the sum of the uniform tensile stress MDEK and linear compressive stress denoted by the triangle DIE. Since EI = EJ/2=(EK-JK)/2, the additional stress induced by the global bending in Equation (3) can then be estimated using the elastic bending stress as

σM θθ (r) = −1

2 [σM |r=0 − σM(r)] for r ≥ 0 (9)

Combining Equations (1), (3), (8) and (9), one obtains a modification of the J −A2 solution for the opening stress in bending specimens:

290 Y.J. Chao et al.

Figure 3. Geometry of TPB and SENB specimens.

σθθ(0)

( r

L

)s2

2

( r

L

)s3

σ0b 3 , (10)

where the plane strain condition is assumed, and the thickness is taken as unity, i.e. B = 1 mm. Comparing Equation (10) with Equation (5), we have C = −6, which indicates that the constant C is purely a coefficient without special physical meaning. Moreover, Equation (10) shows that the regular J − A2 three-term solution does not diminish when the global bending stress dominates at the crack-tip, but has the same order of magnitude as the bending induced stress.

From Equation (10), the constraint parameter A2 can be determined at r/(J/σ0) = 1 ∼ 2 by solving the following equation

A2 2

σ0A1 = 0, (11)

where both J (and thus A1) and σFEA yy (0) can be obtained from the FEA calculation, and M is

the global bending moment applied at the cross section of the bending specimen. In summary, the modified J − A2 four-term solution as shown in Equation (10) have three

‘parameters’: A1, A2 and M. However, both A1 (or J , see Equation (2)) and M are related to the applied load. Consequently, the modified J − A2 solution does not introduce any new parameter after incorporating the additional stress term induced by the global bending. The two parameters remain as the loading (J and M) and the constraint level (A2).

3. Finite element modeling

To demonstrate the global bending effect on the crack-tip stress field in bending specimens under LSY and validate the present model, detailed plane strain FEA calculations are per- formed for two bending specimens. Specimen geometry, material response and FEA models are described as below, and full-field FEA results are reported in the next section.

3.1. SPECIMEN GEOMETRY

Two commonly used bending specimens, i.e. TPB under three point loading and SENB under pure bending, are considered in this work and illustrated in Figure 3. Note that these two bending specimens have similar geometry, but different loading configuration. For the TPB specimen, the specimen geometry and the material were taken from the experimental results of a related project (Lam et al., 2003). The specimen length is L = 142.88 mm, width W = 31.75 mm, span S = 127 mm, crack depth a = 18.73 mm, and thickness B = 15.875 mm. Hence, the ratio of crack depth to specimen width is a/W = 0.59 and the ligament is b = W − a = 13.02 mm. In addition, a 10% side groove is cut on each side of the specimen to

Characterization of crack-tip field and constraint for bending specimens 291

Figure 4. FEA mesh for TPB and SENB specimens.

help maintain plane strain conditions at the crack-tip. For the SENB specimen, W = 50 mm, a = 37.5 mm, b = 12.5 mm, a/W = 0.75 and S = 4 W. Both the TPB and SENB belong to deeply cracked, high constraint specimens.

3.2. MATERIAL RESPONSE

Two materials considered in this work are A285 steel for the TPB specimen, and A533B steel for the SENB specimen. These two steels are commonly used in nuclear waste storage tanks and nuclear reactor vessels, respectively, and their material properties are representatives of a high and a low hardening material. Plastic deformation behavior of the material is modeled by the deformation theory of plasticity within the framework of the small strain theory, since this plasticity theory is simple and effective for elastic-plastic crack problems under LSY to fully plastic deformation. The constitutive response of the material obeys the power-law hardening stress-strain relation, which can be expressed as:

εij

σ0 (12)

where σ0 is the initial yield stress; ε0 is the yield strain and σ0 = Eε0 with E as the Young’s modulus; ν is the Poisson ratio; n is the strain-hardening exponent; α is a material hardening constant; sij = σij − σkkδij /3 are the deviatoric stress components; and σe = (2sij sij /3)1/2 is the Mises effective stress. The index i, j and k range from 1 to 3.

For the A285 steel, the yield stress is σ0 = 251 MPa; the ultimate tensile strength is 415 MPa; Young’s modulus is E = 207 GPa; Poisson’s ratio is ν = 0.3; the strain hardening exponent n = 5; and the hardening coefficient α = 3.2. The experimental stress-strain curve is given in Lam et al. (2003). For the A533B steel, the material constants are σ0 = 450 MPa, ε0 = 0.002, ν = 0.3, α = 3/7 and n = 10. These two steels represent a moderate and moderately low hardening material, respectively.

292 Y.J. Chao et al.

Figure 5. Radial distribution of the numerical opening stress for TPB specimen under the six loadings. Plotted versus (a) real distance and (b) normalized distance from the crack-tip.

3.3. FINITE ELEMENT MODEL

Due to symmetry, only one half of the specimens was modeled. The FEA mesh for the TPB and SENB specimens is illustrated in Figure 4. Totally 945 eight-node elements with reduced integration and 2,976 nodes are used in this FEA model. The smallest element size at the crack-tip is 3.45 × 10−3 mm. The specimen is loaded by a concentration force on the top of the TPB specimen in the symmetric plane, and by a bending moment at the end of the SENB specimen. The support and symmetric boundary conditions are used in this model. All FEA calculations are conducted using the commercial FEA code ABAQUS (HKS, 2002). Detailed FEA results are discussed next.

Characterization of crack-tip field and constraint for bending specimens 293

4. Numerical results and analyses

4.1. TPB SPECIMEN WITH a/W = 0.59

Six different loading levels, i.e. bσ0/J = 200, 120, 60, 30, 15, and 10.7, are selected in the FEA calculations for the TPB specimen with a/W = 0.59 to cover a wide range of deformation levels to demonstrate the global bending influence on the crack-tip field. It is noted that bσ0/J = 10.7 corresponds to the load level at the fracture toughness of A285 steel, i.e. JIC ≈ 305 kJ m2, which is in the LSY regime.

4.1.1. Full-field numerical results The radial distributions of the opening stress along the ligament near the crack-tip are depicted in Figure 5a for the six loading levels of bσ0/J = 200, 120, 60, 30, 15, 10.7. This figure indicates that for the six loadings, (a) all opening stresses increase as the loading increases; (b) the distributions of the stress are non-linear within r < 1 mm, and nearly linear as r ≥ 1 mm; (c) the opening stresses become zero for all loading cases at r ≈ 5.2 mm. Accordingly, the global bending stress dominates the crack-tip stress field in the region r ≥ 1 mm, and the location r ≈ 5.2 mm is equivalent to the neutral axis of a beam under simple bending.

Figure 5b shows the distributions of the normalized opening stress, σθθ/σ0, as a function of the normalized radial distance, r/(J/σ0) for the same loadings as in Figure 5a. It is apparent that the crack-tip opening stress decreases as the applied load increases within the region of interest, 1 ≤ σ0/J ≤ 5. Moreover, at LSY loading levels bσ0/J ≤ 60 (Shih and German, 1981) the opening stress gradually deviates from the results of the moderate yielding state bσ0/J = 200, and shows a strong linear distribution. At the fully plastic deformation level bσ0/J = 10.7, the opening stress equals zero at rσ0/J ≈ 4.2, and then become negative beyond this point. This indicates that the global bending stress has significantly impinged the crack-tip stress field within this region of interest.

Since the critical region at the crack-tip prone to ductile fracture is generally recognized as 1 < rσ0/J < 5 in normalized form, the corresponding actual critical region for the six loadings, i.e. bσ0/J = 200, 120, 60, 30, 15 and 10.7, is shown in Table 1. From Figure 3a, the region strongly affected by the global bending stress appears in r > 1 mm. It reveals that the region of interest 1 ≤ σ0/J ≤ 5 for the load of bσ0/J = 10.7 is beyond r = 1 mm, therefore, the opening stress is strongly affected by the global bending. Similarly, the opening stress at r/(J/σ0) = 5 is strongly affected by the global bending as loading bσ0/J ≤ 60. These quantitative results are consistent with the observation from Figure 3b.

Figure 6 shows the variation of the opening stress with the applied bending moment at three different locations from the crack-tip, i.e. r = 0.538 mm, 1.063 mm, and 2.903 mm, which corresponds to r = b/24.213, b/12.246, and b/4.478. The three straight lines in Figure 6 reveal that the opening stress from the near tip to the middle of the ligament is a linear function of the bending moment. It strongly suggests that the global bending stress dominates the crack- tip field of the TPB specimen for this load range of P ≈ 450 ∼ 830 N. This observation provides an evidence of the assumption previously made for Equation (5) in this work.

4.1.2. Determination of the constraint parameter A2. Using the FEA results of the opening stress in the ligament at r/(J/σ0)1 ∼ 2, the A2 are determined using the elastic stress estimation method as developed in Equation (11). For the six loading cases, i.e. bσ0/J = 200, 120, 60, 30, 15 and 10.7, the magnitudes of the applied

294 Y.J. Chao et al.

Table 1. The critical region near the crack-tip for the six loading levels of the TPB specimen.

σ0/J range of r (mm) range of r (in term of b)

200 0.065 < r < 0.325 b/200 < r < b/40

120 0.108 < r < 0.542 b/120 < r < b/24

60 0.217 < r < 1.085 b/60 < r < b/12

30 0.434 < r < 2.169 b/30 < r < b/6

15 0.878 < r < 4.339 b/15 < r < b/3

10.7 1.217 < r < 6.083 b/10.7 < r < b/2.14

Figure 6. Variation of the opening stress with applied moment for TPB specimen at r = 0.538 mm, 1.063 mm, and 2.903 mm.

moment M on the remaining ligament section of the TPB specimen are 14,732, 16,510, 19,050, 21,749, 24,695 and 26,226 , respectively. The corresponding J -integrals are 16.36, 27.13, 54.39, 108.8, 217.7 and 305.3 KJ/m2. Using these values and Equation (11), the corres- ponding A2 are −0.3099, −0.3038, −0.2945, −0.2895, −0.2905 and −0.2916, respectively. If the two-point matching method is adopted, similar values of A2 and the constant C ≈ −6 are obtained by solving Equation (6). If the constant A2 method is used, the A2 determined by Equation (1) approaches to −0.29 under LYS, and then from Equation (7) one has C ≈ −6. These results indicate that the proposed three methods to determine the bending stress term in the modified J − A2 are practically equivalent to each other.

4.1.3. Stress field under contained yielding Figures 7 show the radial distributions of the opening stress of the TPB specimen from the FEA, the J − A2 solution (denoted by JA2 in the figure) and the modified J − A2 solution (denoted by JA2-M in the figure) along the un-cracked ligament for the three loading cases, bσ0/J = 200, 120 and 60. These three loading levels are relatively low to median, i.e. the first is a SSY case, the second is moderate yielding and the third is close to LSY. It is seen

Characterization of crack-tip field and constraint for bending specimens 295

Figure 7. Radial distributions of the opening stress from FEA results and the asymptotic solutions for TPB specimen. (a) bσ0/J = 200, (b) bσ0/J = 120, (c) bσ0/J = 60.

296 Y.J. Chao et al.

from Figures 7a and 7b that the J − A2 solution is almost identical to the modified J − A2

solution, and both asymptotic solutions match well with the FEA numerical results within the region of interest, 1 ≤ σ0/J ≤ 5. It indicates that the J − A2 solution can well describe the crack-tip stress field for the deeply cracked bending specimens under contained yielding (SSY) or moderate yielding conditions.

Figure 7c shows that the J −A2 solution is also very close to the modified J −A2 solution, and reasonably matches with the FEA numerical results within the region of interest, 1 ≤ rσ0/J ≤ 5. However, the J −A2 solution begins to deviate from the modified J −A2 solution and the FEA numerical results as rσ0/J > 4.5. Therefore, for the loading level of bσ0/J = 60 (or r < b/12) which is close to LSY deformation, the J − A2 solution can still reasonably represent the crack-tip fields of the deeply cracked bending specimens, and the effect of the global bending is negligible.

4.1.4. Stress field under large-scale yielding Figures 8a to 8c show the radial distributions of the opening stress of the TPB specimen from the FEA results, the J − A2 solution and the modified J − A2 solution along the ligament for the three large load cases, i.e. 1 ≤ rσ0/J = 30, 15, 10.7. The results show that (a) the modified J − A2 solution can match well with the FEA numerical results within the region of interest 1 ≤ σ0/J ≤ 5 for all three cases; (b) the valid zone of the J − A2 solution shrinks gradually from rσ0/J ≈ 4 as bσ0/J = 30 to rσ0/J ≈ 1.5 as bσ0/J = 10.7; (c) the maximum valid zone of the J − A2 solution is r ≈ b/8. Therefore, the effect of the global bending must be considered in the asymptotic solution for the LSY loading or at least for the deformation level at bσ0/J ≤ 15.

4.2. SENB SPECIMEN WITH a/W = 0.75

Likewise, four different loading levels, i.e. bσ0/J = 600, 200, 60, and 30 are selected in the FEA calculations for the SENB specimen with a/W = 0.75 to cover from SSY to LSY deformation levels. The objective is to further demonstrate the influence of the global bending on the crack-tip field in the A533B steel, and validate the modified J − A2 four-term solution for describing such problems by comparing to FEA results.

4.2.1. Full-field numerical results Figure 9a plots the radial distribution of the opening stress along the ligament near the crack- tip for the four deformation levels of bσ0/J = 600, 200, 60, and 30. Similar trends of the global bending stress as in the TPB specimen made of the A285 steel are observed from this figure for the SENB specimen made of the A533B steel. Particularly, it is evident from this figure that the linear portion of the opening tensile stress on the ligament increases as the loading increases.

Figure 9b shows the distributions of the normalized opening stress, σθθ/σ0, as a function of the normalized radial distance, r/(J/σ0) for the same four loads. It is clear that the crack- tip opening stress decreases with increasing applied load using this normalized distance. Moreover, the opening stresses are almost identical to each other for the two SSY loads of bσ0/J = 600 and 200. Again at the LSY loading levels, i.e. bσ0/J = 60 and 30, the opening stress gradually deviates from the SSY results at bσ0/J = 200, and shows a strong linear distribution. This result further demonstrates that the global bending stress significantly impinges the crack-tip stress field within the region of interest.

Characterization of crack-tip field and constraint for bending specimens 297

Figure 8. Radial distributions of the opening stress from FEA results and the asymptotic solutions for TPB specimen. (a) bσ0/J = 30, (b) bσ0/J = 15, (c) bσ0/J = 10.7.

298 Y.J. Chao et al.

Figure 9. Radial distributions of numerical opening stress for SENB specimen under the six loadings (a) real sizes; (b) normalized sizes.

Figure 10 shows the variation of the opening stress with the applied bending moment at three different locations from the crack-tip, i.e. r = 0.483 mm, 1.160 mm, and 2.762 mm, which corresponds to r = b/25.859, b/10.776, and b/2.762. Once again this figure reveals that the opening stress from the near tip to the middle point of ligament is a linear function of the bending moment at load levels ranging from SSY to LSY, which further confirms our assumption previously made for Equation (5).

4.2.2. Determination of the constraint parameter A2

Again, using the FEA results of the opening stress on the ligament at r/(J/σ0) = 1 ∼ 2, the values of A2 are determined using the elastic stress estimation method, Equation (11). For the four loading cases, i.e. bσ0/J = 600, 200, 60 and 30, the applied moment M are given for the SENB specimen as 16,328, 24,971, 31,560 and 34,431 N mm. The corresponding J -integrals are obtained from FEA as 9.27, 27.81, 92.71 and 185.40 KJ/m2, and then the A2 values are

Characterization of crack-tip field and constraint for bending specimens 299

Figure 10. Variation of the opening stress with applied moment for SENB specimen at r = 0.483 mm, 1.160 mm, and 2.762 mm.

determined as −0.2178, −0.1649, −0.1647, and −0.1664, respectively. Likewise, the two- point matching method and constant A2 method previously discussed give similar values of A2 and the constant C ≈ −6.

4.2.3. Stress field under large-scale yielding Since the J − A2 solution can characterize the crack-tip stress field very well for deeply cracked bending specimens under SSY conditions as shown in Figure 9b, the stress fields for the SENB specimen are discussed here only for the two LSY loadings. Figures 11a and 11b show the radial distributions of the opening stress for the SENB specimen from the FEA, the J −A2 solution and the modified J −A2 solution along the ligament for the two LSY loading cases, bσ0/J = 60 and 30. It is evident again from these figures that the J − A2 solution deviates gradually from the full-field FEA results, but the modified J − A2 solution match well with the FEA numerical results within the region of interest, 1 ≤ σ0/J ≤ 5.

5. Conclusions

The detailed plane strain FEA calculations were performed for the TPB specimens with a/W = 0.59 and SENB with a/W = 0.75 under loading levels ranging from SSY to LSY to demonstrate the effect of the global bending on the crack-tip stress fields. Two different strain hardening materials, A285 and A533B steels, are considered in the FEA calculations. A simple modification of the J − A2 solution has been proposed in this work to consider this effect in the asymptotic crack-tip solution. The primary results are summarized as follows:

(1) The J − A2 three-term solution is only a good approximation of the crack-tip fields up to the contained yielding conditions or bσ0/J > 60 for bending specimens.

(2) The influence of the global bending stress on the crack-tip fields of bending speci- mens gradually increases with the increasing load. The effect of this global bending becomes significant under LSY conditions or bσ0/J ≤ 30.

300 Y.J. Chao et al.

Figure 11. Radial distributions of the opening stress from FEA results and the asymptotic solutions for SENB specimen. (a) bσ0/J = 60, (b) bσ0/J = 30.

(3) The crack-opening stress is a linear function of distance and applied moment under LSY conditions.

(4) Three methods, i.e. the two-point matching, constant A2 and elastic stress estimation methods are presented for determining the additional bending stress term in the modified J − A2 solution. Numerical results indicate they give practically the same results. The elastic stress estimation method is the simplest, yet effective, and is therefore recommended in practice.

(5) The modified J − A2 four-term solution given in Equation (10) contains only the two usual parameters, i.e. J (the load level) and A2 (the constraint level), without introducing any new parameter. Comparison with FEA results show that it can effectively characterize the crack-tip field for bending specimens beyond the validity range of the J − A2 solution under LSY conditions.

(6) Having the last term included in Equation (10) for reflecting the global bending effect on the crack-tip field, the parameter A2 can be determined by Equation (11) and it preserves its

Characterization of crack-tip field and constraint for bending specimens 301

ability to accurately quantify the constraint level at the crack-tip of bending specimens under LSY conditions.

Acknowledgements

This work was sponsored by the United States Department of Energy under Contract Number DE-AC09-96SR18500 and through the South Carolina Universities Research and Education Foundation.

References

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Betegon, C. and Hancock, J.W. (1991). Two-parameter characterization of elastic-plastic crack-tip fields. Journal of Applied Mechanics 58, 104–110.

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Mechanics and Physics of Solids 16, 13–31. Hibbitt, Karlsson and Sorensen, Inc., 2002. ABAQUS Users Manual, Version 6.2, Providence, R. Johnson, W. and Mellor P.B. (1983). Engineering Plasticity. Ellis Horwood Limited Publishers, New York. Karstensen, A.D., Nekkal, A. and Hancock, J.W. (1997). Constraint estimation for edge cracked bending bars,

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fracture applications. Journal of the Mechanics and Physics of Solids 40, 939–963. Parks, D.M. (1992). Advances in characterization of elastic-plastic crack-tip fields, Fracture and Fatigue (edited

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Characterization of crack-tip field and constraint for bending specimens under large-scale yielding

Y.J. CHAO1,∗, X.K. ZHU1, Y. KIM1, P. S. LAR2, M.J. PECHERSKY2 and M.J. MORGAN2

1Department of Mechanical Engineering, University of South Carolina 2Savannah River Technology Center, Savannah River Company ∗Author for correspondence (E-mail: [email protected])

Received 16 November 2003; accepted in revised form 1 April 2004

Abstract. Elastic-plastic crack-tip fields and constraint levels in bending specimens under large-scale yielding (LSY) are examined. The J − A2 three-term solution is modified by introducing an additional term caused by the global bending. Three different methods, i.e. two-point matching, constant A2 and elastic stress estimation method, are proposed to determine the fourth term. It is shown that the elastic stress estimation method is the simplest, yet effective, in that the fourth term can be derived from the strength theory of materials and the concept of plastic hinge, and effectively quantifies the contribution of the global bending moment on the crack-tip field. Consequently, the modified J − A2 solution, with the inclusion of the correction for global bending, does not introduce any new parameter. The two parameters remain as the loading (J and M) and the constraint level (A2). To validate the present solution, detailed finite element analyses (FEA) were conducted for a Three Point Bend (TPB) specimen with a/W = 0.59 in A285 steel, and Single Edge Notched Bend (SENB) specimen subjected to pure bending with a/W = 0.5 in A533B steel at different deformation levels ranging from small-scale yielding (SSY) to LSY. Results show that the modified J − A2 solution matches fairly well with the FEA results for both TPB and SENB specimens at all deformation levels considered. In addition, the fourth stress term is (a) propor- tional to the global bending moment and inversely proportional to the ligament length; (b) negligibly small under SSY; and (c) significantly large under LSY or fully plastic deformation. Accordingly, the present model effectively characterizes the crack-tip constraint for bending dominated specimens with or without the large influence from the global bending stress on the crack-tip field.

Key words: Crack-tip field, fracture constraint, TPB, SENB, large-scale yielding.

1. Introduction

Fracture constraint refers to the effect of geometric and loading configurations of a flawed specimen or structure on the mechanics behavior at the crack-tip. It has been shown that the crack-tip constraint has significant effect on the crack-tip field, fracture toughness and crack extension resistance to ductile tearing, and therefore is an important parameter in structural design and integrity assessment using fracture mechanics methodology.

It is well known that for elastic-plastic materials, the HRR singularity field (Hutchinson, 1968; Rice and Rosengren, 1968) provides an effective characterization of the crack-tip stress field using the single parameter J -integral for high constraint specimens. The two-parameter approaches including the J − T approach (Betegon and Hancock, 1991), J − Q theory (O’Dowd and Shih, 1991; 1992) and J − A2 three-term solution (Yang et al., 1993; Chao et al., 1994; Chao and Zhu, 1998) are valuable in describing the crack-tip stress fields for low constraint specimens beyond the single parameter characterization. However, extensive work (Shih and German, 1987; O’Dowd and Shih, 1992; Parks, 1992; Wang and Parks, 1995; Chao

284 Y.J. Chao et al.

and Zhu, 1998; Lam et al., 2003) have shown that for bending dominated fracture specimens, such as the Three Point Bend (TPB), Single Edge Notched Bend (SENB) under pure bending, Single Edge Notched Tension (SENT) and Compact Tension (CT) specimens, both the single and two-parameter crack-tip solutions are limited to small scale yielding (SSY) conditions, because the crack-tip field is significantly affected by the global bending moment under the conditions of large-scale yielding (LSY) or fully plastic deformation. As a result, none of the available crack-tip solutions can accurately characterize the crack-tip field for such bending specimens within the crack-tip region prone to ductile fracture. Accordingly, all existing the- ories fail to quantify the crack-tip constraint in the bending specimens under LSY conditions, as outlined in the following reviews.

For shallow cracked bending specimens, the global bending has limited influence on the crack-tip field, and the two-parameter solutions can well characterize the crack-tip field even under LSY conditions (Al-Ani and Hancock, 1991; Zarzour et al., 1993; Chao and Zhu, 1998). For deeply cracked bending specimens, the crack-tip field exhibits high constraint behavior and is fairly well described by the single parameter J -integral through the HRR asymptotic solution under relatively low load or SSY conditions. As the load increases beyond the SSY level, however, the J gradually losses its dominance since the global bending progressively builds up at and impinges on the crack-tip. Chao and Zhu (1998) showed that under contained yielding conditions, the J −A2 solution can still approximately describe the crack-tip field of SENB specimens at a deformation level between SSY and LSY. However, under LSY or fully plastic deformation when bσ0/J ≤ 30, where b is the ligament length ahead of the crack-tip and σ0 is the yield stress of the material, the global bending strongly affects the crack-tip stress field of SENB specimens with deep cracks for low hardening materials (e.g. strain hardening exponent n = 10). The size of the J − A2 dominance zone r shrinks to a small value, e.g. r/(J/σ0) < 2, which is much smaller than the region at the crack-tip prone to ductile fracture, i.e. 1 < r(J/σ0) (Ritchie et al., 1973).

Similar phenomena are also observed when using the J − Q theory. O’Dowd and Shih (1992) showed that in a deeply cracked bend bar, the crack-opening stress decreases rapidly with distance away from the crack-tip for the fully yielding condition (i.e. J/σ0 > 0.05b or bσ0/J < 20), and the stress gradient across the ligament is very large. The opening stress is compressive near the free surface and becomes tensile as the crack tip is approached. The global bending stress distribution prevails near the crack-tip when J/σ0 > 0.06b. Wang and Parks (1995) discussed certain limits of the two parameter J − T and J − Q characterization of elastic-plastic crack-tip fields, and concluded that the opening stress is dominated by the global bending stress for a TPB specimen with a/W = 0.4 and J/aσ0 > 0.096. Wei and Wang (1995) and Karstensen et al. (1997) pointed out that at LSY or fully plastic deformation state, the difference of the full field numerical solution and the SSY crack-tip field for the SENB specimen, which is used to determine the Q-stress, becomes strongly distance de- pendent as the global bending stress is significant. This influence arises because the SENB specimen is subjected to a global bending moment and the ligament far from the crack-tip is in compression. In summary, for deep bend specimens under LSY or fully plastic deformation, all available two-parameter solutions for quantifying the constraint including the J −T , J −Q

and J − A2 break down since the global bending significantly impinges on the crack-tip and alters the stress fields.

To solve this problem, Wei and Wang (1995) modified the J −Q two-parameter solution to include a third parameter k2, and proposed the J −Q − k2 three-parameter solution, where Q

and k2 are determined by matching with the FEA results. However, the physical meaning of the

Characterization of crack-tip field and constraint for bending specimens 285

third parameter is ambiguous, i.e. whether this third parameter (k2) is a constraint parameter, a loading parameter, or simply a numerical fit is unclear. Likewise, Karstensen et al. (1997) also modified the J − Q scheme by decomposing the parameter Q into two parts. One is a distance independent term QT , which is formally related to the T stress under the SSY; the other is a distance dependent term Qp, which is related to the global bending stress field and is regarded as the difference between the total loss of constraint given by Q and the loss of constraint given by a negative T . They then suggested a crack-tip constraint estimation scheme for SENB specimens using the three parameters, J , QT , and Qp.

The present paper revisits the problem stated above and extends the J − A2 three-term solution to include an additional term caused by the global bending to characterize the crack- tip stress field of deeply cracked TPB and SENB specimens under LSY conditions. Three different methods, i.e. two-point matching, constant A2 and elastic stress estimation method, are presented to determine the additional term. It is shown that the simplest one is the elastic stress estimation method which uses the strength theory of materials and the concept of plastic hinges. In the J − A2 four-term solution, J as usual represents the intensity of applied loads, A2 describes the crack-tip constraint level, and the additional or the fourth term is induced by the global bending moment directly related to the applied load. Therefore, the fourth term does not introduce a third parameter since it is related to the applied load or specifically it can be evaluated from the global bending moment determined by the simple strength of materials approach. The results from the proposed method are then compared with the FEA full field solutions. Comparisons show that the modified J − A2 solution agrees well with the full field solutions. Consequently, the parameter A2 can effectively quantify the constraint level of bending specimens in both LSY and SSY deformation states.

2. The J − A2 three-term solution and its modification

Under LSY and deep bend specimens, Chao and Zhu (1998) have shown that the J − A2

three-term solution is valid to characterize the crack-tip field only within a small fraction of the ligament. For the SENB specimen, the influence of the global bending stress on the crack-tip field is very small within the J − A2 valid region, but becomes large gradually beyond the region. The J − A2 solution is briefly reviewed in this section and then modified to characterize more accurately the crack-tip stress field and constraint of bending specimens under LSY conditions

2.1. THE J − A2 THREE-TERM SOLUTION

Yang et al. (1993) and Chao et al. (1994) developed the J − A2 three-term solution with the J -integral as the applied loading intensity and A2 the constraint parameter. Under plane strain conditions, the three-term asymptotic stress field can be expressed as

σij

( r

L

)s2

2

( r

L

)s3

] , (1)

where the stress angular functions σ (k) ij (θ) (k = 1, 2, 3) and the stress power exponents sk

(s1 < s2 < s3) only depend on the hardening exponent n, and is independent of the other material constants (i.e. hardening parameter α, yield strain ε0, and yield stress σ0) and the applied loads. L is a characteristic length parameter and L = 1 mm is taken in this work. The parameters A1 and s1 have the same values as given in the HRR field:

286 Y.J. Chao et al.

Figure 1. Comparison of the opening stresses along the ligament for CCP and SENB specimens.

A1 = (

J

αε0σ0InL

)−s1

n + 1 (2)

When n ≥ 3, the third stress power exponent depends on the first and second stress exponents, i.e. s3 = 2s2−s1. The angular values of σ

(k) ij (θ) and sk are reported by Chao and Zhang (1997).

The parameter A2 can be regarded as a measure of the crack-tip constraint. Using the point matching method, one can determine the A2 value by matching the opening stress from the three-term solution with the FEA result at r/(σ0/J ) = 1 ∼ 2 (Chao and Zhu, 2000).

2.2. A MODIFICATION OF THE J − A2 THREE-TERM SOLUTION

Before modifying the J − A2 solution, a simple comparison of the opening stress ahead of a crack-tip for bending and tension specimens is demonstrated first. The opening stresses for a tension specimen, such as the Central Cracked Panel (CCP) or the Double Edge Cracked Plate (DECP), are tensile along the entire ligament, and approach to uniform distributions away from the crack-tip (Zhu and Chao, 2000). Consequently, the stresses in such tension dominant specimens can be well described by the J−A2 three-term solution, as shown in Chao and Zhu (1998). For bending dominant specimens like TPB and SENB, the opening stress is tensile along the ligament near the crack-tip, but becomes compressive near the specimen surface on the other side of the crack-tip. Figure 1 illustrates the difference of the opening stress distribution along the ligament for the CCP and SENB specimens under LSY conditions within the small strain framework. It is apparent that the compressive stress near the free surface of the bending specimen is caused by the global bending.

For elastic-plastic crack problems, the applied bending stress is independent of the crack- opening stress, but the opening stress depends on the applied bending stress. Since both the opening and bending stresses on the ligament are nonlinear under LSY conditions, it is extremely difficult to accurately account for the global bending stress in the asymptotic crack- tip field. As an approximation, however, it is assumed here that the opening stress, referred

Characterization of crack-tip field and constraint for bending specimens 287

to as σθθ at θ = 0 hereafter, for bending specimens is a simple superposition of the J − A2

solution, σ JA2 θθ and the global bending induced stress, σM

θθ :

θθ (3)

Extensive finite element analyses (FEA), as reported by O’Dowd and Shih (1992), Wang and Parks (1995), Wei and Wang (1995), Karstensen et al. (1997), Chao and Zhu (1998) and the present work, indicate that the global bending stress dominates the crack-tip stress field of bending specimens under LSY conditions, and the opening stress linearly distributes on the ligament away from the crack-tip even within the region of interest 1 < r/(J/σ0) < 5. Furthermore, the present calculation further reveals that at a specific location (i.e. a fixed r) on the ligament ahead of the crack-tip, the opening stress is also linearly related to the applied load or moment, as shown in Figures 6 and 10 later. Therefore, it is reasonable to assume that the global bending induced stress σM

θθ in Equation (3) is a linear function of the distance from the crack-tip, r, and the global bending moment, M, on the ligament plane of the specimen as

σM θθ (r, θ = 0) = C

Mr

b3 , (4)

where C is an undetermined load-independent constant, and M is the moment per unit length, e.g. the thickness of the specimen. The ligament length b is arbitrarily inserted in Equation (4) so that the C is a non-dimensional constant. The next section will confirm that Equation (4) is a reasonable form for the bending induced stress.

Substituting Equations (1) and (4) into Equation (3), we obtain the general form of the modified J − A2 solution for the opening stress ahead of the crack-tip, i.e. θ = 0, as follows

σθθ(r, 0)

σ0 = A1

2.3. DETERMINATION OF THE CONSTANT C

To determine magnitude of the constant C in Equation (5) and understand its physical mean- ing, three different methods, i.e. two-point matching, constant A2 and elastic stress estimation methods are discussed in this section.

(a) Two-point matching method. For a given applied load, it is assumed that the opening stress ahead of a crack-tip within the interest region 1 < r(J/σ0) < 5 has been determined in a FEA calculation. Letting σFEA1

θθ (0) and σFEA2 θθ (0) are the FEA results of the opening

stress at two specific points at r = r1 and r = r2, respectively, then the constraint parameter A2 and the constant C in Equation (5) can be determined by solving the following equations simultaneously.

σFEA1 θθ (0)

(r1

L

)s2

2

(r1

L

)s3

(r2

L

)s2

2

(r2

L

)s3

(6)

(b) Constant A2 method. Yang et al. (2003) and Chao and Zhu (2000) have shown that the constraint parameter A2 is invariable under LSY and fully plastic deformation. If so, determination of the constant C is straightforward. For a given bending specimen, FEA

288 Y.J. Chao et al.

calculations can be carried out first under different loading levels from SSY to LSY, and then the corresponding A2 values are determined by matching FEA results with the conventional J − A2 three-term solution in Equation (1). The A2 becomes a constant or approximate con- stant when the applied load increases from SSY to LSY, this constant can be thus adopted as the value of the constraint parameter A2 for the bending specimen under LSY conditions. It should be noted that as the applied load increases from SSY to LSY, the A2 may approach a constant first, and then deviates from the constant later for deeply cracked bending specimens when only using Equation (1).

With the invariant value of A2, the constant C in Equation (5) can be easily determined using a FEA result of the opening stress σFEA

θθ (0) under LSY near the crack-tip, for example, at r = 2J/σ0 by

C= σ 2 0 b3

2JM

{ σFEA

( 2J

Lσ0

)s2

2

( 2J

Lσ0

)s3

]} (7)

(c) Elastic stress estimation method. Both methods proposed above, as shown in Equa- tions (6) and (7), require the FEA calculations first, and does not reveal the physical meaning of the constant C. Accordingly, a simple and intuitive method to estimate the bending induced stress in Equation (3) or determine the constant C in Equation (5) is introduced here.

To determine the crack-tip opening displacement (CTOD) for a TPB specimen, the concept of plastic hinge was introduced in fracture mechanics analysis (see Broek, 1973 for detailed discussion) as the crack surfaces rotate when loaded. Veerman and Muller (1972) found that the center of this rotation is located at about 0.47b from the crack-tip. For a plain un-cracked beam with a height equal to the ligament length b, the neutral axis of the rectangular cross section is at the centerline of the rectangle, i.e. 0.5b, which is close to 0.47b. It is therefore assumed that the neutral axis of an un-cracked beam is approximately the axis of plastic rotation (or plastic hinge) of the bending specimens. The global bending induced stress in Equation (3) can thus be estimated by that for the plain un-cracked beam.

In the theory of plastic bending for beams (Johnson and Mellor, 1983; Dowling, 1993), it is assumed that the originally plane cross section remains plane even under plastic deformation. It results in a linear variation of the normal strain with distance from the neutral axis. Hence, if yielding occurs, the nonlinear stress-strain behavior of the beam material causes the stress distribution to be nonlinear and the distribution should have the same shape as the correspond- ing stress-strain curve up to the maximum strain at the edges of the beam. For a beam having rectangular cross-section of b in height and B in thickness, the bending stress under elastic condition is given as

σM(r) = Mx∗

Iy

, (8)

where M is the applied bending moment, x∗ is the distance from the neutral axis or x∗ = b 2 −r,

and Iy = Bb3

12 is the second moment of inertia of the area of the cross section about the neutral axis. It should be noted that the elastic bending stress σM in Equation (8) is different from the bending induced stress σM

θθ assumed in Equation (3). As illustrated in Figure 2, the global bending stress on the ligament MN separates the

ligament plane of a bending specimen into a tensile and a compressive zone. Both zones experience partly plastic yielding under LSY conditions. And, the plastic hinge is approxim- ately located at the neutral axis, i.e. point O. Let us focus our attention on the tensile zone,

Characterization of crack-tip field and constraint for bending specimens 289

Figure 2. Illustration of the global bending stress estimation.

i.e. region MO in Figure 2, and the local singularity stress at the crack-tip is not considered for the moment. For elastic-plastic materials under LSY, the global bending stress is linear in the region at and close to the neutral axis or point O, but nonlinear near the crack-tip or point M. While the curved triangle MDHO denotes the schematic distribution of the elastic- plastic tensile stress, the right triangle MDO stands for the elastic tensile stress. Near point M, the curve DH can be approximated by line DI which is the bisector line of triangle ODF with FG = GO or EI = IJ. Recall that the J − A2 three-term solution (as well as the HRR solution) is developed or well-suited for a crack under remote uniform tension, such as the uniform tensile stress indicated by the rectangle MDFO. For bending specimen, however, the applied tensile bending stress on the ligament is non-uniformly distributed, e.g. MDHO. As a result, extra tensile stress generated from DEFOH is enforced in the crack-tip J − A2

solution. This contribution must be deducted from the crack-tip opening stress for bending specimens. From an arbitrary point K to the crack-tip M, the global bending stress denoted by the quadrilateral MDHK can be approximated by the sum of the uniform tensile stress MDEK and linear compressive stress denoted by the triangle DIE. Since EI = EJ/2=(EK-JK)/2, the additional stress induced by the global bending in Equation (3) can then be estimated using the elastic bending stress as

σM θθ (r) = −1

2 [σM |r=0 − σM(r)] for r ≥ 0 (9)

Combining Equations (1), (3), (8) and (9), one obtains a modification of the J −A2 solution for the opening stress in bending specimens:

290 Y.J. Chao et al.

Figure 3. Geometry of TPB and SENB specimens.

σθθ(0)

( r

L

)s2

2

( r

L

)s3

σ0b 3 , (10)

where the plane strain condition is assumed, and the thickness is taken as unity, i.e. B = 1 mm. Comparing Equation (10) with Equation (5), we have C = −6, which indicates that the constant C is purely a coefficient without special physical meaning. Moreover, Equation (10) shows that the regular J − A2 three-term solution does not diminish when the global bending stress dominates at the crack-tip, but has the same order of magnitude as the bending induced stress.

From Equation (10), the constraint parameter A2 can be determined at r/(J/σ0) = 1 ∼ 2 by solving the following equation

A2 2

σ0A1 = 0, (11)

where both J (and thus A1) and σFEA yy (0) can be obtained from the FEA calculation, and M is

the global bending moment applied at the cross section of the bending specimen. In summary, the modified J − A2 four-term solution as shown in Equation (10) have three

‘parameters’: A1, A2 and M. However, both A1 (or J , see Equation (2)) and M are related to the applied load. Consequently, the modified J − A2 solution does not introduce any new parameter after incorporating the additional stress term induced by the global bending. The two parameters remain as the loading (J and M) and the constraint level (A2).

3. Finite element modeling

To demonstrate the global bending effect on the crack-tip stress field in bending specimens under LSY and validate the present model, detailed plane strain FEA calculations are per- formed for two bending specimens. Specimen geometry, material response and FEA models are described as below, and full-field FEA results are reported in the next section.

3.1. SPECIMEN GEOMETRY

Two commonly used bending specimens, i.e. TPB under three point loading and SENB under pure bending, are considered in this work and illustrated in Figure 3. Note that these two bending specimens have similar geometry, but different loading configuration. For the TPB specimen, the specimen geometry and the material were taken from the experimental results of a related project (Lam et al., 2003). The specimen length is L = 142.88 mm, width W = 31.75 mm, span S = 127 mm, crack depth a = 18.73 mm, and thickness B = 15.875 mm. Hence, the ratio of crack depth to specimen width is a/W = 0.59 and the ligament is b = W − a = 13.02 mm. In addition, a 10% side groove is cut on each side of the specimen to

Characterization of crack-tip field and constraint for bending specimens 291

Figure 4. FEA mesh for TPB and SENB specimens.

help maintain plane strain conditions at the crack-tip. For the SENB specimen, W = 50 mm, a = 37.5 mm, b = 12.5 mm, a/W = 0.75 and S = 4 W. Both the TPB and SENB belong to deeply cracked, high constraint specimens.

3.2. MATERIAL RESPONSE

Two materials considered in this work are A285 steel for the TPB specimen, and A533B steel for the SENB specimen. These two steels are commonly used in nuclear waste storage tanks and nuclear reactor vessels, respectively, and their material properties are representatives of a high and a low hardening material. Plastic deformation behavior of the material is modeled by the deformation theory of plasticity within the framework of the small strain theory, since this plasticity theory is simple and effective for elastic-plastic crack problems under LSY to fully plastic deformation. The constitutive response of the material obeys the power-law hardening stress-strain relation, which can be expressed as:

εij

σ0 (12)

where σ0 is the initial yield stress; ε0 is the yield strain and σ0 = Eε0 with E as the Young’s modulus; ν is the Poisson ratio; n is the strain-hardening exponent; α is a material hardening constant; sij = σij − σkkδij /3 are the deviatoric stress components; and σe = (2sij sij /3)1/2 is the Mises effective stress. The index i, j and k range from 1 to 3.

For the A285 steel, the yield stress is σ0 = 251 MPa; the ultimate tensile strength is 415 MPa; Young’s modulus is E = 207 GPa; Poisson’s ratio is ν = 0.3; the strain hardening exponent n = 5; and the hardening coefficient α = 3.2. The experimental stress-strain curve is given in Lam et al. (2003). For the A533B steel, the material constants are σ0 = 450 MPa, ε0 = 0.002, ν = 0.3, α = 3/7 and n = 10. These two steels represent a moderate and moderately low hardening material, respectively.

292 Y.J. Chao et al.

Figure 5. Radial distribution of the numerical opening stress for TPB specimen under the six loadings. Plotted versus (a) real distance and (b) normalized distance from the crack-tip.

3.3. FINITE ELEMENT MODEL

Due to symmetry, only one half of the specimens was modeled. The FEA mesh for the TPB and SENB specimens is illustrated in Figure 4. Totally 945 eight-node elements with reduced integration and 2,976 nodes are used in this FEA model. The smallest element size at the crack-tip is 3.45 × 10−3 mm. The specimen is loaded by a concentration force on the top of the TPB specimen in the symmetric plane, and by a bending moment at the end of the SENB specimen. The support and symmetric boundary conditions are used in this model. All FEA calculations are conducted using the commercial FEA code ABAQUS (HKS, 2002). Detailed FEA results are discussed next.

Characterization of crack-tip field and constraint for bending specimens 293

4. Numerical results and analyses

4.1. TPB SPECIMEN WITH a/W = 0.59

Six different loading levels, i.e. bσ0/J = 200, 120, 60, 30, 15, and 10.7, are selected in the FEA calculations for the TPB specimen with a/W = 0.59 to cover a wide range of deformation levels to demonstrate the global bending influence on the crack-tip field. It is noted that bσ0/J = 10.7 corresponds to the load level at the fracture toughness of A285 steel, i.e. JIC ≈ 305 kJ m2, which is in the LSY regime.

4.1.1. Full-field numerical results The radial distributions of the opening stress along the ligament near the crack-tip are depicted in Figure 5a for the six loading levels of bσ0/J = 200, 120, 60, 30, 15, 10.7. This figure indicates that for the six loadings, (a) all opening stresses increase as the loading increases; (b) the distributions of the stress are non-linear within r < 1 mm, and nearly linear as r ≥ 1 mm; (c) the opening stresses become zero for all loading cases at r ≈ 5.2 mm. Accordingly, the global bending stress dominates the crack-tip stress field in the region r ≥ 1 mm, and the location r ≈ 5.2 mm is equivalent to the neutral axis of a beam under simple bending.

Figure 5b shows the distributions of the normalized opening stress, σθθ/σ0, as a function of the normalized radial distance, r/(J/σ0) for the same loadings as in Figure 5a. It is apparent that the crack-tip opening stress decreases as the applied load increases within the region of interest, 1 ≤ σ0/J ≤ 5. Moreover, at LSY loading levels bσ0/J ≤ 60 (Shih and German, 1981) the opening stress gradually deviates from the results of the moderate yielding state bσ0/J = 200, and shows a strong linear distribution. At the fully plastic deformation level bσ0/J = 10.7, the opening stress equals zero at rσ0/J ≈ 4.2, and then become negative beyond this point. This indicates that the global bending stress has significantly impinged the crack-tip stress field within this region of interest.

Since the critical region at the crack-tip prone to ductile fracture is generally recognized as 1 < rσ0/J < 5 in normalized form, the corresponding actual critical region for the six loadings, i.e. bσ0/J = 200, 120, 60, 30, 15 and 10.7, is shown in Table 1. From Figure 3a, the region strongly affected by the global bending stress appears in r > 1 mm. It reveals that the region of interest 1 ≤ σ0/J ≤ 5 for the load of bσ0/J = 10.7 is beyond r = 1 mm, therefore, the opening stress is strongly affected by the global bending. Similarly, the opening stress at r/(J/σ0) = 5 is strongly affected by the global bending as loading bσ0/J ≤ 60. These quantitative results are consistent with the observation from Figure 3b.

Figure 6 shows the variation of the opening stress with the applied bending moment at three different locations from the crack-tip, i.e. r = 0.538 mm, 1.063 mm, and 2.903 mm, which corresponds to r = b/24.213, b/12.246, and b/4.478. The three straight lines in Figure 6 reveal that the opening stress from the near tip to the middle of the ligament is a linear function of the bending moment. It strongly suggests that the global bending stress dominates the crack- tip field of the TPB specimen for this load range of P ≈ 450 ∼ 830 N. This observation provides an evidence of the assumption previously made for Equation (5) in this work.

4.1.2. Determination of the constraint parameter A2. Using the FEA results of the opening stress in the ligament at r/(J/σ0)1 ∼ 2, the A2 are determined using the elastic stress estimation method as developed in Equation (11). For the six loading cases, i.e. bσ0/J = 200, 120, 60, 30, 15 and 10.7, the magnitudes of the applied

294 Y.J. Chao et al.

Table 1. The critical region near the crack-tip for the six loading levels of the TPB specimen.

σ0/J range of r (mm) range of r (in term of b)

200 0.065 < r < 0.325 b/200 < r < b/40

120 0.108 < r < 0.542 b/120 < r < b/24

60 0.217 < r < 1.085 b/60 < r < b/12

30 0.434 < r < 2.169 b/30 < r < b/6

15 0.878 < r < 4.339 b/15 < r < b/3

10.7 1.217 < r < 6.083 b/10.7 < r < b/2.14

Figure 6. Variation of the opening stress with applied moment for TPB specimen at r = 0.538 mm, 1.063 mm, and 2.903 mm.

moment M on the remaining ligament section of the TPB specimen are 14,732, 16,510, 19,050, 21,749, 24,695 and 26,226 , respectively. The corresponding J -integrals are 16.36, 27.13, 54.39, 108.8, 217.7 and 305.3 KJ/m2. Using these values and Equation (11), the corres- ponding A2 are −0.3099, −0.3038, −0.2945, −0.2895, −0.2905 and −0.2916, respectively. If the two-point matching method is adopted, similar values of A2 and the constant C ≈ −6 are obtained by solving Equation (6). If the constant A2 method is used, the A2 determined by Equation (1) approaches to −0.29 under LYS, and then from Equation (7) one has C ≈ −6. These results indicate that the proposed three methods to determine the bending stress term in the modified J − A2 are practically equivalent to each other.

4.1.3. Stress field under contained yielding Figures 7 show the radial distributions of the opening stress of the TPB specimen from the FEA, the J − A2 solution (denoted by JA2 in the figure) and the modified J − A2 solution (denoted by JA2-M in the figure) along the un-cracked ligament for the three loading cases, bσ0/J = 200, 120 and 60. These three loading levels are relatively low to median, i.e. the first is a SSY case, the second is moderate yielding and the third is close to LSY. It is seen

Characterization of crack-tip field and constraint for bending specimens 295

Figure 7. Radial distributions of the opening stress from FEA results and the asymptotic solutions for TPB specimen. (a) bσ0/J = 200, (b) bσ0/J = 120, (c) bσ0/J = 60.

296 Y.J. Chao et al.

from Figures 7a and 7b that the J − A2 solution is almost identical to the modified J − A2

solution, and both asymptotic solutions match well with the FEA numerical results within the region of interest, 1 ≤ σ0/J ≤ 5. It indicates that the J − A2 solution can well describe the crack-tip stress field for the deeply cracked bending specimens under contained yielding (SSY) or moderate yielding conditions.

Figure 7c shows that the J −A2 solution is also very close to the modified J −A2 solution, and reasonably matches with the FEA numerical results within the region of interest, 1 ≤ rσ0/J ≤ 5. However, the J −A2 solution begins to deviate from the modified J −A2 solution and the FEA numerical results as rσ0/J > 4.5. Therefore, for the loading level of bσ0/J = 60 (or r < b/12) which is close to LSY deformation, the J − A2 solution can still reasonably represent the crack-tip fields of the deeply cracked bending specimens, and the effect of the global bending is negligible.

4.1.4. Stress field under large-scale yielding Figures 8a to 8c show the radial distributions of the opening stress of the TPB specimen from the FEA results, the J − A2 solution and the modified J − A2 solution along the ligament for the three large load cases, i.e. 1 ≤ rσ0/J = 30, 15, 10.7. The results show that (a) the modified J − A2 solution can match well with the FEA numerical results within the region of interest 1 ≤ σ0/J ≤ 5 for all three cases; (b) the valid zone of the J − A2 solution shrinks gradually from rσ0/J ≈ 4 as bσ0/J = 30 to rσ0/J ≈ 1.5 as bσ0/J = 10.7; (c) the maximum valid zone of the J − A2 solution is r ≈ b/8. Therefore, the effect of the global bending must be considered in the asymptotic solution for the LSY loading or at least for the deformation level at bσ0/J ≤ 15.

4.2. SENB SPECIMEN WITH a/W = 0.75

Likewise, four different loading levels, i.e. bσ0/J = 600, 200, 60, and 30 are selected in the FEA calculations for the SENB specimen with a/W = 0.75 to cover from SSY to LSY deformation levels. The objective is to further demonstrate the influence of the global bending on the crack-tip field in the A533B steel, and validate the modified J − A2 four-term solution for describing such problems by comparing to FEA results.

4.2.1. Full-field numerical results Figure 9a plots the radial distribution of the opening stress along the ligament near the crack- tip for the four deformation levels of bσ0/J = 600, 200, 60, and 30. Similar trends of the global bending stress as in the TPB specimen made of the A285 steel are observed from this figure for the SENB specimen made of the A533B steel. Particularly, it is evident from this figure that the linear portion of the opening tensile stress on the ligament increases as the loading increases.

Figure 9b shows the distributions of the normalized opening stress, σθθ/σ0, as a function of the normalized radial distance, r/(J/σ0) for the same four loads. It is clear that the crack- tip opening stress decreases with increasing applied load using this normalized distance. Moreover, the opening stresses are almost identical to each other for the two SSY loads of bσ0/J = 600 and 200. Again at the LSY loading levels, i.e. bσ0/J = 60 and 30, the opening stress gradually deviates from the SSY results at bσ0/J = 200, and shows a strong linear distribution. This result further demonstrates that the global bending stress significantly impinges the crack-tip stress field within the region of interest.

Characterization of crack-tip field and constraint for bending specimens 297

Figure 8. Radial distributions of the opening stress from FEA results and the asymptotic solutions for TPB specimen. (a) bσ0/J = 30, (b) bσ0/J = 15, (c) bσ0/J = 10.7.

298 Y.J. Chao et al.

Figure 9. Radial distributions of numerical opening stress for SENB specimen under the six loadings (a) real sizes; (b) normalized sizes.

Figure 10 shows the variation of the opening stress with the applied bending moment at three different locations from the crack-tip, i.e. r = 0.483 mm, 1.160 mm, and 2.762 mm, which corresponds to r = b/25.859, b/10.776, and b/2.762. Once again this figure reveals that the opening stress from the near tip to the middle point of ligament is a linear function of the bending moment at load levels ranging from SSY to LSY, which further confirms our assumption previously made for Equation (5).

4.2.2. Determination of the constraint parameter A2

Again, using the FEA results of the opening stress on the ligament at r/(J/σ0) = 1 ∼ 2, the values of A2 are determined using the elastic stress estimation method, Equation (11). For the four loading cases, i.e. bσ0/J = 600, 200, 60 and 30, the applied moment M are given for the SENB specimen as 16,328, 24,971, 31,560 and 34,431 N mm. The corresponding J -integrals are obtained from FEA as 9.27, 27.81, 92.71 and 185.40 KJ/m2, and then the A2 values are

Characterization of crack-tip field and constraint for bending specimens 299

Figure 10. Variation of the opening stress with applied moment for SENB specimen at r = 0.483 mm, 1.160 mm, and 2.762 mm.

determined as −0.2178, −0.1649, −0.1647, and −0.1664, respectively. Likewise, the two- point matching method and constant A2 method previously discussed give similar values of A2 and the constant C ≈ −6.

4.2.3. Stress field under large-scale yielding Since the J − A2 solution can characterize the crack-tip stress field very well for deeply cracked bending specimens under SSY conditions as shown in Figure 9b, the stress fields for the SENB specimen are discussed here only for the two LSY loadings. Figures 11a and 11b show the radial distributions of the opening stress for the SENB specimen from the FEA, the J −A2 solution and the modified J −A2 solution along the ligament for the two LSY loading cases, bσ0/J = 60 and 30. It is evident again from these figures that the J − A2 solution deviates gradually from the full-field FEA results, but the modified J − A2 solution match well with the FEA numerical results within the region of interest, 1 ≤ σ0/J ≤ 5.

5. Conclusions

The detailed plane strain FEA calculations were performed for the TPB specimens with a/W = 0.59 and SENB with a/W = 0.75 under loading levels ranging from SSY to LSY to demonstrate the effect of the global bending on the crack-tip stress fields. Two different strain hardening materials, A285 and A533B steels, are considered in the FEA calculations. A simple modification of the J − A2 solution has been proposed in this work to consider this effect in the asymptotic crack-tip solution. The primary results are summarized as follows:

(1) The J − A2 three-term solution is only a good approximation of the crack-tip fields up to the contained yielding conditions or bσ0/J > 60 for bending specimens.

(2) The influence of the global bending stress on the crack-tip fields of bending speci- mens gradually increases with the increasing load. The effect of this global bending becomes significant under LSY conditions or bσ0/J ≤ 30.

300 Y.J. Chao et al.

Figure 11. Radial distributions of the opening stress from FEA results and the asymptotic solutions for SENB specimen. (a) bσ0/J = 60, (b) bσ0/J = 30.

(3) The crack-opening stress is a linear function of distance and applied moment under LSY conditions.

(4) Three methods, i.e. the two-point matching, constant A2 and elastic stress estimation methods are presented for determining the additional bending stress term in the modified J − A2 solution. Numerical results indicate they give practically the same results. The elastic stress estimation method is the simplest, yet effective, and is therefore recommended in practice.

(5) The modified J − A2 four-term solution given in Equation (10) contains only the two usual parameters, i.e. J (the load level) and A2 (the constraint level), without introducing any new parameter. Comparison with FEA results show that it can effectively characterize the crack-tip field for bending specimens beyond the validity range of the J − A2 solution under LSY conditions.

(6) Having the last term included in Equation (10) for reflecting the global bending effect on the crack-tip field, the parameter A2 can be determined by Equation (11) and it preserves its

Characterization of crack-tip field and constraint for bending specimens 301

ability to accurately quantify the constraint level at the crack-tip of bending specimens under LSY conditions.

Acknowledgements

This work was sponsored by the United States Department of Energy under Contract Number DE-AC09-96SR18500 and through the South Carolina Universities Research and Education Foundation.

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