D Stress intensity factor and T-stresses ( T 11 and T 33 ) formulations for a Compact Tension specimen

The paper describes test specimen thickness effect on stress intensity factor (KI), and T-stresses stresses (T11 and T33) for a Compact Tension specimen. Formulations to estimate 3D KI, T11 and T33 stresses are proposed based on extensive 3D Finite element analyses. These formulations help to estimate magnitudes of 3D KI and T11 and T33 which are helpful to quantify in-plane and out-ofplane constraint effect of the crack tip. The proposed formulations are validated with the similar results available in literature and found to be within acceptable error.


INTRODUCTION
tress tri-axiality at the crack-tip can alter crack-tip constraint and fracture toughness values of a material.This is the reason transferability of fracture toughness data estimated using laboratory test specimens to a full-scale cracked structure is an important issue in structural integrity assessment of engineering materials.In LEFM non-zero nonsingular terms in the series expansion of three-dimensional stress field [1] referred as T-stresses (T 11 and T 33 ) can alter the crack-tip stress tri-axiality and are considered as constraint parameters [2].T 11 (the second term of William's extension acting parallel to the crack plane) plays an important role on the in-plane constraint effect.The thickness at the crack tip contributes to the out-of-plane constraint, T 33 (the second term of William's extension acting along the thickness).To transfer fracture toughness data under different constraints, both in-plane and out-of-plane constraint effect should be considered for the specimens.

S
Several researchers [3][4][5][6][7][8][9][10][11][12][13] have shown that the specimen thickness has major effect on the magnitude of stress intensity factor and T stresses.Kwon and Sun [7] have presented 3D FE analyses, to investigate the stress fields near the crack-tip and suggested a simple technique to determine 3D K I at the mid-plane by knowing 2D K I and Poisson's ratio () of the material using the Eq. ( 1): For quantifications of in-plane and out-of-plane constraint issues, magnitudes of T11 and T33 stresses are to be computed.
But simple formulations such as Eq. ( 1) to estimate 3D K I , are not available to compute constraint parameters, T 11 and T 33 stresses.Usually they are obtained by complex 3D numerical methods.The aim of this investigation is to study the variation of KI, T11 and T33 along the crack-front considering a CT specimen geometry having varied thickness, B, crack length to width ratio (a/W) and applied stress, , using 3D elastic FE analysis.
Based on the present finite element results an effort is made to formulate approximate analytical equations to estimate the magnitudes of maximum 3D K I , T 11 and T 33 for a CT specimen.The proposed analytical formulations can be used to estimate the maximum 3D K I , T 11 and T 33 for the CT specimen, which are helpful in quantifying in-plane and out-ofplane constraint issues in fracture.By means of numerical analyses, it is shown that specimen thickness and crack length play an important role on the constraint effects.

FINITE ELEMENT ANALYSIS
ommercial FE software ABAQUS 6.5 [14] is used for the 3D FEA.The dimensions of CT specimen have been chosen according to ASTM standard E1820 [15] and the specimen geometry is shown in the Fig. 1.One-half of the specimen geometry is modeled due to specimen symmetry.Twenty noded quadratic brick elements available in the ABAQUS are used to discretize the analysis domain.This kind of elements was used in the earlier works [8,9] available in the literature.Initially, three-dimensional FE analyses on CT specimens were made by varying number of layers in thickness direction (each layer is of element thickness).It is observed that the variation in results of KI is insignificant for 8-14 layers.Consequently, in the present analyses 11 layers along the thickness direction were chosen as it gives ten numbers of elements along the thickness direction to extract the KI and T-stress.Due to half symmetry, the symmetrical boundary conditions have been imposed (ux2=0) along the ligament of the model.Load is applied on approximately 1/3 rd portion nodes of the loading-hole circle perpendicular to the ligament.To keep the loading in perfectly Mode-I condition corresponding nodes are arrested except x2-direction.A typical mesh used in the analyses along with boundary conditions is shown in Fig. 2. The magnitudes of KI and T-stresses have been extracted by using ABAQUS post processor.The details of extraction of stress 3D stress intensity factor (K I ) and T-stresses are discussed elsewhere [11,12,16,17].The variation of KI and T-stresses along the crack-front has been studied for different specimen thickness (B/W=0.1-1.0) and crack length to width ratio (a/W=0.45-0.70).In this work the magnitude of applied stress, , for the CT specimen is computed using the relation [18]: where, P is applied load, W is width of the specimen, a is crack length.In these finite element calculations, the material behaviour has been considered to be linear elastic pertaining to an interstitial free (IF) steel possessing yield stress y=155MPa, elastic modulus of 197 GPa, Poisson's ratio=0.30.

Stress Intensity Factor (KI)
typical variation of normalized stress intensity factor (K I /(a) 1/2 ) and the distance along the crack front (specimen thickness direction, x 3 ) for specimens having various a/W is shown in Fig. 3.The nature of variation of normalized stress intensity factor shown in Fig. 3 is in good agreement with the similar results [7,19].The A magnitudes of 3D maximum stress intensity factors (at the center of the specimen) are compared with the analytical 2D value computed by Eq. ( 3).
Typically, such comparison for a specimen with a/W=0.65 is shown in Fig. 3.It is estimated that magnitude of 2D KI is about 10% lower than 3D KI.
It is well known that variation of stress intensity factor against     and a/W (Fig. 4) is nonlinear, the data is fit with a suitable polynomial.In such exercise, it is found that a polynomial equation of third order fits the 3D FEA results with least error (Regression co-efficient=0.998).This polynomial fit (Eq.( 4)) is superimposed on the 3D FEA results plotted in Fig. 4 and shows an excellent agreement.From this third order polynomial fit, the relation between K I-max , a/W and  can be expressed as:  The Eq. ( 4) reduces to: Eq. ( 6) is similar to Eq. ( 3) and the constant C1 shown in Eq. ( 5) is similar to the geometric factor, Y as used in Eq. (3).Hence, C 1 in this work is referred as 3D geometric factor.Eq. ( 6) can be used to estimate magnitudes of K I-max for the CT specimen.To validate the proposed formulation given in Eq. ( 6), the computed values of K I-max using Eq.( 6) for various a/W are compared with the present 3D FEA results and the results computed by Eq.( 1) proposed by Kwon and Sun [7] in Fig. 4. The figure shows that the results computed by Eq. ( 6) are in good agreement with the results obtained by present 3D FEA for a/W 0.45 to 0.70 (entire range of study).The values of KI-max estimated from the Eq.( 1) [7] are found to in good agreement with the present results for specimens with a/W 0.45 to 0.65, but for a specimen having a/W>0.65 the Eq.( 1) [7] is showing higher error.The error analysis between the results obtained by present 3D FEA and the Eq.( 6) is conducted.The maximum percentage of error estimated in use of Eq.( 6) for B=2 to 20mm and a/W=0.45 to 0.70 is < 2.95 %.

T11-stress
The magnitudes of T 11 are extracted from 3D FE analysis for varied , B and a/W.A typical variation of T 11 along the crack-front for various B, a/W=0.50 and  =86.8 MPa is shown in Fig. 5.This figure shows that the variation of T 11 along the crack-front (x 3 ) depends on the specimen thickness.It is observed that the magnitude of T 11 is maximum at the centre of the specimen than on the surface.However, for the specimen thickness, B<6mm the maximum value of T11 is found to be just below the surface.It is seen from Fig. 5 that for specimen with higher thickness, the magnitude of T 11 decreases for a short distance from the specimen surface and again increases at the centre of the specimen thickness.Similar kind of variation was observed in the work of Pavel et al. [20].To study the effect of a/W, a typical variation of T11/ for various a/W, B=10mm and  =86.8 MPa is plotted in Fig. 6.The figure illustrates that T11 depends on the specimen a/W, and it is observed to be maximum for higher a/W.The results in Fig. 5-6 indicate that the in-plane constraint parameter, T11, is not a unique value (as obtained in 2D analysis) for a specimen thickness, but it varies along the thickness and it is maximum at the centre.The variation of T11 at the center (T11-max) vs. normalized KI (KI-max/(πB) 1/2 ) for various specimen thicknesses is studied.
A typical plot of T 11-max against K I-max /(πB) 1/2 for various a/W and B=10 mm is shown in Fig. 7.It is seen from the Fig. 7 that the variation of T11-max vs. KI-max/(πB) 1/2 is linear and is independent of a/W.The slopes of T11-max vs. KI-max/(πB) 1/2 are computed for various a/W by fitting a straight-line equation to all the results shown in Fig. 7.The estimated slopes T11max /(K I-max /(πB) 1/2 ) are plotted against normalized thickness (B/W) for various a/W in Fig. 8.This plot indicates that, the nature of variation of slopes T 11max / (K I-max /(πB) 1/2 ) against B/W is nonlinear and is almost independent of a/W.The result plotted in Fig. 8 is used to obtain a relationship between T11-max and KI-max.As there is a small difference is observed in the results presented in Fig. 8 for various a/W, the average slope of all a/W for particular B/W is used.The average results are fit by a polynomial equation of third order (Regression Co-efficient=0.994),which give a best fit to all the data.From this third order polynomial fit, the relation between T11-max and KI-max is obtained and can be expressed as:    Let, The Eq. ( 7) reduces to: Substituting for K I-max from Eq.( 6) The above Eq.(11) can be used to estimate T 11-max for the CT specimen for a given specimen dimensions and applied load, .A typical plot of T11-max computed from Eq.( 11) is superimposed (Red curve) on 3D FEA results shown in Fig. 8.This plot shows a good agreement with the results estimated by Eq.( 11) and 3D FEA.An error analysis is carried out between the estimated values from Eq.( 11) and the present 3D FEA results.The maximum percentage of error in use of Eq.( 11) for various B, a/W and  used in this study is found to be < 5.1%.The proposed analytical Eq.( 11) is a simple method to compute T 11-max for the CT specimen geometry.T 33 -stress The magnitudes of T 33 are estimated using Eq.( 12) by substituting the extracted  33 (strain) from ABAQUS post processor and T11 for varied , B and a/W.
The variation of T33-max/ against B/W for various a/W is shown in Fig. 9.This figure indicates that, T33 strongly depend on B/W.It is observed that the magnitude of T 33 is negative for all cases that were considered in this analysis, and approached to zero as B/W increased to 1.For B/W=0.5, ASTM requirement for K IC test specimen [15], it is seen that T 33 / is negative indicating loss of out-of-plane constraint.T 33 also showed dependence on a/W, for B/W<0.7,T 33 found to be maximum for thinner specimens (B/W=0.1)with higher a/W =0.7.As T 33 strongly depends on the specimen thickness, it is not possible to get a simple relation between T33 and specimen geometry as obtained in case of KI and T11.
To obtain expressions between T 33 , specimen geometry and the applied load, the results in Fig. 9 are given a polynomial fit to suit the 3D FEA results.In this exercise, it is found that the 5 th order polynomial fits the data with least error.A typical equation for estimation of T33-max is given by Eq. ( 13).The equations for the 3D geometric factors (C3) to compute T33-max for various a/W obtained by fitting 5 th order polynomial are tabulated in Tab.3.
The computed values of C3 for various a/W are given in Tab.4.The maximum percentage of error in the use of equations given in Tab.3 for various B, a/W and  is found to be < 7.8%.
To evaluate the worthiness of the analytical expressions to estimate KI-max, T11-max and T33-max proposed in this analysis, the estimated magnitudes KI-max, T11-max and T33-max obtained by Eq.( 6), Eq.( 11) and Tab.3 are compared with the similar results on the CT specimen presented by Toshiyuki and Tomohiro [12].The authors [12] have given the variation of T 11- max and T33-max for the toughness value of the material (0.55% Carbon Steel) KI-max= 66 MPam 1/2 .Fig. 10 shows a typical variation of estimated KI-max, T11-max and T33-max at the specimen thickness center against B/W (0.1 -1.0) obtained by proposed analytical expressions (Eq.( 6), (11) and Tab.3) for K I-max = 66 MPam 1/2 and results of Toshiyuki and Tomohiro [12].The authors [12] have computed various parameters for B/W=0.25,0.4 and 0.5 only.According to Fig. 10, the magnitudes of KI-max were not affected by B/W as expected and are in excellent agreement with Toshiyuki and Tomohiro [12].T 11 showed visible dependence on B/W, though the variation was less than 20% and it is in good agreement with the earlier reported results [12].In summary, the in-plane parameters at the specimen thickness center showed small dependence on B, and are in excellent agreement with the earlier results [12] providing validation for the proposed Eq.( 6) and (11).On the other hand, out-of-plane constraint factor, T 33 showed strong dependence on B/W.T 33 was found negative and approached zero as B/W increased from 0.1 to 1. Fig. 10 shows that the nature of variation of present results of T33 (obtained by Tab.3 expressions) is in similar manner to the one presented by Toshiyuki and Tomohiro [12].However, some difference in the magnitude of T 33 between both the results for B/W=0.25-0.5 is observed.This difference is attributed to the effect of side grooves in the CT specimen used in the work of Toshiyuki and Tomohiro [12].The side grooves in the specimen affects the strain distribution in the out-of-plane direction and restricts the value of T 33 (Ref.Eq.( 12)).To study this effect, we have conducted 3D FEA on CT specimen used by Toshiyuki and Tomohiro [12] without side groove and for the same material properties.These computed results are superimposed in Fig. 10 by red colored lines for comparison.This plot show that the results (KI-max, T11-max and T33-max) for the specimen without side groove match with the one computed by the analytical expressions (Eq.( 6), (11) and Tab.3) proposed in this work.This exercise infers that the use of side grooves in a specimen controls 33 and improves the out-of-plane constraint (Ref.Fig.10: for a/W=0.5,T 33 is improved from -150 MPa to -84 MPa (44%) by using 10% side groove [12]).The Fig. 10 provides validation to the proposed analytical formulations to estimate K I-max , T 11-max and T 33-max .9)) is highly negative for B/W=0.1 (thin specimens), and almost zero for B/W=1 (thick specimens), indicating that the thick specimens have higher out-of-plane constraint.For B/W=0.5, ASTM requirement for KIC test [15], it is observed I that T33 / is negative indicating some loss of out-of-plane constraint.For the specimens with similar a/W and B/W the loss of out-of-plane constraint (T 33 ) is much significant than the in plane constraint (T 11 ) (Ref.Fig. 10).This infers that the major constraint loss in a CT specimen is due to the out-of-plane effects.The out-of-plane constraint loss can be corrected to some extent by providing side grooves to a CT specimen.Using the present 3D FEA results, approximate analytical formulations are proposed to evaluate the K I-max, T 11-max and T 33-max by knowing only applied stress and specimen dimensions.These formulations can be helpful in the analysis of in-plane and out-of-plane constraint issues.

Figure 1 :
Figure 1: The geometry of the CT specimen used in the analysis (W=20 mm).
a/W are plotted in Fig 4. As the relation between KI-max/   a 1/2

Figure 5 :
Figure 5: A typical variation of T11/ along the crack-front for various B and a/W=0.50.

Figure 6 :
Figure 6: A typical variation of T 11 / along the crack-front for various a/W and B=10mm.

Figure 9 :
Figure 9: Variation of T 33 / against B/W for various a/W.

Figure 10 :
Figure 10: A typical variation of KI-max, T11-max and T33-max against B/W obtained by proposed analytical expressions and results of Toshiyuki and Tomohiro [12].

Table 4 :
Values of C 3 computed from formulations given in Tab.3.