Three-dimensional finite element analyses of T-stress for different experimental specimens

Accepted Manuscript Three-dimensional finite element analyses of T-stress for different experimental specimens Xin-Ting Miao, Chang-Yu Zhou, Fei Lv, Xiao-Hua He PII: DOI: Reference:

S0167-8442(17)30107-6 http://dx.doi.org/10.1016/j.tafmec.2017.04.018 TAFMEC 1850

To appear in:

Theoretical and Applied Fracture Mechanics

Received Date: Revised Date: Accepted Date:

2 March 2017 1 April 2017 26 April 2017

Please cite this article as: X-T. Miao, C-Y. Zhou, F. Lv, X-H. He, Three-dimensional finite element analyses of Tstress for different experimental specimens, Theoretical and Applied Fracture Mechanics (2017), doi: http:// dx.doi.org/10.1016/j.tafmec.2017.04.018

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Three-dimensional finite element analyses of T-stress for different experimental specimens Xin-Ting Miao, Chang-Yu Zhou*, Fei Lv, Xiao-Hua He School of mechanical and power engineering, Nanjing Tech University, Nanjing 211816, China *Corresponding author. Tel: +86-25-58139951; fax: +86-25-58139951. E-mail address: [email protected]

Abstract Due to the non-negligible role of T-stress on fracture parameter, T-stresses of three typical specimens, including central cracked plate (CCP), compact tension specimen (CTS) and four point bending specimen (FPB), are calculated using 3D finite element method. From the results, the tensile dominant specimen like CCP is in a lower-constraint state when the component of mode I is significant, and in a higher-constraint state when the component of mode II is significant, compared with bending dominant specimen like CTS and FPB specimens. Further, the crack initiation angles and plastic zones are assessed for the three specimens based on modified maximum tangential stress (MMTS) and modified R-criterion (MR). From the comparison, the effect of T-stresses or constraints induced by specimen geometries and loading modes play a significant role on crack initiation angle and plastic zone around the crack tip, which in turn can affect the fracture of specimen. Keywords: T-stress, mixed mode crack, experimental specimens, constraint effect Nomenclature a W a/W T E B Me v r, θ β rc σy θ0 σrr σθθ τ KI KII KIC

crack size width of specimens crack size ratio elastic T-stress Young’s modulus biaxility ratio mode mixity Poisson's ratio coordinates in conventional polar systems loading angle for I-II mixed crack radius from crack tip dependent on the material yielding stress or ultimate stress dependent on the material crack initiation angle radial stress in polar systems circumferential stress in polar systems tangential stress in polar systems mode I stress intensity factors mode II stress intensity factors fracture toughness for pure mode I crack 1

CCP CTS specimen FPB specimen MTS criterion SED criterion G criterion MMTS criterion

center cracked plate compact tension-shear specimen four point bending specimen maximum tangential stress criterion minimum energy density criterion maximum energy release rate criterion modified maximum tangential stress criterion

MR criterion

modified R-criterion

1 Introduction In 1939, Westergaard [1] developed a description of a biaxial stress field for internal cracks in pressurized cylinders. The equations originally developed for biaxial stress fields were subsequently applied for uniaxial stress fields. So for a purely uniaxial load an extra transverse component at the boundary edges along the crack direction occurred, which was due to Westergaard’s solution developed for a biaxial loading. To eliminate this transverse component, Irwin [2] has suggested the use of the transverse stress. Later, Williams [3] showed the presence of T-stress at the edge of a crack through the eigen series expansion of the stress field, as described in Eq. (1), where the high order terms are neglected.  rr 

1 1      3 cos   K I 1  sin 2   K II  sin   2 tan    T cos 2  2   2 2  2 r 2

  

1 1   3  cos   K I cos 2  K II sin    T sin 2  2 2 2 2 r  



1 1 cos   K I sin   K II  3cos   1  T sin  cos  2 2 2 r

(1a) (1b) (1c)

where KI and KII are the mode I and mode II stress intensity factors (SIF) and r, θ are co-ordinates in conventional polar systems with the crack tip at the origin. The term T is the stress acting parallel to the crack. The earlier experiments conducted by Williams and Ewing [4] using polymethylmethacrylate (PMMA) showed that there was a discrepancy between mixed mode fracture based on maximum tangential stress (MTS) criterion and their experiments. One reason for the differences was the presence of T-stress. Later work by Ueda et al [5] illustrated similar effects. Their work demonstrated that non-singular terms (T-stress) had a significant influence on brittle fracture in linear elastic materials subjected to mixed mode loading. Since then there have been considerable studies on T-stress through experimental, analytical and finite element methods which relate T-stress with phenomena in fracture mechanics. Larsson and Carlsson [6] showed that in small scale yielding T-stress had a significant effect on size and shape of plastic zones around the crack tip for mode I crack. Nazarali et al [7] and Smith et al [8] have studied the effect of T-stress on crack-tip plastic zones under mixed-mode loading conditions. They found that T-stress played a 2

very important role in plastic zone variations. The work of Ayatollahi [9-12] showed that modifications of MTS criterion with the non-singular term T-stress and other higher order terms improved the correlation with the experimental results, through combined experimental, numerical and analytical work. And the studies [13-14] demonstrated that the nonsingular T-stress term is needed to be considered for more accurate fracture resistances through experiments. Recent work by Ayatollahi also demonstrated that T-stress played an important role on fracture strength and crack trajectory [15-16]. The plastic strains in the vicinity of crack-tip are restrained which will result in a triaxial stress state at the crack tip, whose effect is called crack tip constraint. The constraint effect will affect the fracture toughness of a material and crack paths of specimens. For fracture under conditions of contained yielding, T-stress has been used as a parameter of constraint due to the effect on the plastic zone size and shape. And specimens with a negative T-stress have lower constraint compared with those having a positive T-stress. For I-II mixed mode cracks, crack growth path is dependent on both loading mode and specimen geometry. There have been studies on the crack path and methods arresting the crack or extending the fatigue life [17-18]. For linear elastic materials, several criteria have been developed to describe brittle failure, MTS criterion [19] developed by Erdogan and Sih, minimum strain energy density (SED) criterion [20], and maximum energy release rate (G-criterion) criterion [21] and so on. Several studies have been found that the crack initiation angles and the fracture toughness based on these criteria will be different with various T-stresses, and the results from modified criteria with considering T-stress will be closer to experiments. In the literature [22], it was concluded that SED criterion was less be affected by T-stress as compared with MTS criterion. And the effect of T-stress in M-criterion (maximum triaxial stress criterion) and G-criterion was lower than that in MTS criterion. So based on different criteria, the influence of T-stress is somewhat different. Khan [23] proposed a new criterion (R-criterion) for I-II mixed mode crack initiation angles based on the characteristics of plastic core region surrounding the crack tip. The proposed criterion stated that the crack extended in the direction of the local or global minimum of plastic core region boundary depending on the resultant stress state at the crack tip. Later in the paper [8], the authors associating the R-criterion with T-stress. In the present paper, the role of T-stress on fracture parameter is reviewed. And then three widely used experimental specimens including central cracked plate (CCP), compact tension specimen (CTS) and four point bending specimen (FPB), which are capable of creating mixed mode crack, are modeled by 3D finite element method to calculate T-stress. The specimens are with different mode mixities and crack sizes, which will induce different values of T-stress. Further, the crack initiation angles and plastic zones of CCP, CTS and FPB are assessed using modified MTS criterion (MMTS) and modified R-criterion (MR). 3

2 The role of T-stress 2.1 Crack initiation angle and fracture toughness MTS criterion is one of the widely used criteria for I-II mixed mode crack, in which Authors have taken T-stress into consideration [24]. MTS criterion includes two aspects: The fracture occurs at the crack tip where the circumferential stress σθθ is the maximum; and the circumferential stress σθθ at the position rc along the direction θ0 exceeds σy. Among which rc and σy are two material parameters. In this paper MMTS is called for MTS criterion considering T-stress, and the simplified progress is listed. According to the MTS requirement, θ0 can be obtained by Eq. (2), which is expressed by Eq. (1b) and the first order derivation of Eq. (1b).

 |r  r,    c

0

(2a)

y

  |r rc , 0  0 

(2b)

Finally, Eqs. (3) (4) (5) of the crack propagation angles and fracture toughness can be obtained by Eq. (2), where   K I / K II , KI and KII are stress intensities of pure mode I and mode II crack respectively. And here in Eqs. (4) and (5), KI/KIC and KII/KIC both represent the fracture toughness for pure mode I, mode II and I-II mixed mode crack, KIC is the constant representing fracture toughness of pure mode I crack,. 

 T sin 2  0  2 sin  0sin 2 0   y        16 T T sin 0 cos  0 cos3 0  sin  0 1  sin 2  0     3 y 2 2 y  

 3cos 0  1 1 

T

y

T

sin 2  0 y K II  K Ic  cos3  0  3 sin  cos  0 0 2 2 2 1

(3)

KI K  II KI c K Ic

(4)

(5)

The ratio of mode I and mode II stress intensity factors is often presented by a dimensionless parameter called mode mixity Me which is written as Me 

2 tan 1  . The value of Me is unity for 

pure mode I and zero for pure mode II. From Eq. (3), the relationship of θ0 versus Me can be obtained, and the relationship of KI/KIC and KII/KIC can be received from Eqs. (4) and (5), which are plotted in Fig. 1.

4

(a) Crack initiation angle

(b) Stress intensity factors Fig. 1 The effect of T-stress on the crack propagation

Fig. 1 presents the effects of T-stress on the crack propagation and fracture toughness for 5

mode I, mode II and I-II mixed mode cracks with various normalized T-stresses (T/σy). It can be found that T-stress has a significant role on crack path and fracture toughness for mode I, mode II and I-II mixed mode crack. For mode I fracture, T /  y  0.375 is a critical point. When T /  y  0.375 , the crack initiation angle is 0 and K I / K Ic is 1, T-stress has no effect on it. When T /  y  0.375 , T-stress has some effects, the intersection point is K I / K Ic  1 , and the crack initiation angle (-θ0) is greater than zero. For I-II mixed mode crack and pure mode II crack, the effect of T-stress cannot be ignored in Fig. 1. When T /  y  0 , the crack initiation angles (-θ0) are greater than those without considering T-stress, and the fracture toughness are smaller than traditional results. When T /  y  0 , the crack initiation angles (-θ0) are smaller and the fracture toughness are greater than traditional results. That is to say negative T-stresses can improve the fracture resistance.

2.2 Plastic zone The effect of T-stress on plastic zone for I-II mixed mode crack is plotted in Fig. 2, and a simplified description has been presented. The plastic zone radius rp is according to the function in [7], where the crack-tip stress field is incorporated into Von Mises yield criteria that determines the crack-tip plastic zone, which is validated reasonable by Broek [25]. The normalized form of rp is as Eq. (6) 2  11  T T r      *  b  b2  ac*   2  2  c  y y 1 K         y 

rp   

2

(6)

where, K  K I 2  K II 2 V  1  2 

2

1  T c    3  V   2    y *

2

   2 

  tan 1  9 3    a  sin 2   sin 2   V (cos   1)   cos  sin   2sin (3cos   V)  cos2  6  sin 2   V (cos   1)  2 2            b  sin  cos (3(cos 2  cos 2 )  V )   cos  sin (3(cos 2  cos 2 )  V )  2 2 2    2  6

Note that ϕ=90° (Me=1) is for pure mode I crack, ϕ=0° (Me=1) is for pure mode II crack, and 0°<ϕ<90° (0
T/σy = -0.3 T/σy = 0 T/σy = 0.3

T/σy = 0.6

(a) Me=1

(b) Me=0.667

(c) Me=0.333

(d) Me= 0

Fig. 2 Plastic zones for I-II mixed mode cracks 7

3 T-stress analyses for different experimental specimens According to analyses in Section 2, T-stress has an effect on both crack initiation angle and fracture toughness. Also the non-singular crack tip parameter (T-stress) is influenced considerably by the specimen geometries and loading conditions [26]. In this section, three widely used specimens are included to investigate the T-stress. The CCP experiment was first conducted by Williams and Ewing [4] using PMMA. CTS specimen with I-II loading device designed by Richard [27] is one of the most popular used specimens, which can provide complete mode mixities from pure mode I to pure mode II [11]. Four-point loading on a single-edge-notched specimen [22, 28] is also a common setup for I-II mixed mode test, which can also provide complete I-II mixed-mode range from pure mode I to pure mode II.

3.1 Finite element analyses The geometries of the specimens (CCP, CTS, and FPB) and their loading fixtures are shown schematically in Fig. 3. The crack lengths are defined as a and the specimen widths are W in a uniform way. In this paper, three crack sizes are considered including a/W= 0.3, 0.5, 0.7. For CCPs, I-II mixed mode loading conditions with different mode-mixity ratios Me are obtained by changing the angle β in Fig. 3(a). β = 90° means mode I crack (Me=1), 0<β<90° means I-II mixed mode crack (0
β

W

β

8

(a) CCP

(b) CTS specimen P 2S

a

w

S

S0

2S

S

(c)Four-point bend specimen Fig. 3 Geometries of three different specimens Three-dimensional finite element models are simulated and analyzed using finite element code ABAQUS (version 6.13). The region around the crack tip is meshed using a cylinder with 6 rings of elements, each consists 24 quadrilateral elements in the circumferential direction from -180° to 180°. And the number of elements through the thickness is considered to be 12. The minimum size of the finite element close to the crack tip along the radial direction is 0.1 mm, which is 0.004 times of the width of the specimen width, and the minimum size has been validated accurately in our previous paper [26]. The thickness number of element has also been validated in [26], which affects little to finite element results, and the number of 12 is larger enough for the specimens in this paper. Fig. 4 shows the typical finite element meshes for CCP, CTS specimen and four-point bend specimen, and the details of crack tips are in the enlarged figures. For CCP in Fig. 4(a), the pressures are applied on the upper and lower surfaces. As shown in Fig. 4(b), the CTS specimen is fixed to the loading fixture through 6 pin-holes. In order to prevent plastic deformation around the pin-holes, the region around these holes in the CTS specimen is designed thicker than the central part. And the fixture is much stiffer than CTS specimen, so it is modeled as a three-dimensional rigid body. To transmit the loads from the fixture to CTS specimen, contact conditions are considered between the rigid fixture and the specimen described. In addition, to simulate closer to the experimental conditions, a pin is modeled described to apply loadings, whose material is set much stiffer than the specimen’s material, similarly the contact conditions are set between the pin and the loading fixture. The loading angle β can be changed by choosing appropriate loading holes on the loading fixture in order to provide different combinations of mode I and mode II loading. For FPB described in Fig. 4(c), a displacement-controlled loading applies to the top of a bar, which simulates the load applications in the experimental procedures. The experimental setup utilizes two rollers to transfer the forces from the loading bar to the fracture specimen. The numerical model shown in Fig. 4(c) simplifies the loading bar and the rollers as a whole, the applied load is transferred from the ends of the whole loading bar to the 9

fracture specimen. And the material of the loading bar is defined much stiffer than that of the specimen. The support rollers are taken as two rigid bodies at the bottom.

(a) CCP pin

rigid body surface-surface contact specimen

crack tip

(b) CTS specimen

10

(c)Four- point bend specimen Fig. 4 Typical meshes of finite element models for different specimens

3.2 T-stress results for different specimens The values of T-stresses are extracted directly from ABAQUS which makes use of an interaction integral method to compute. To normalize the T-stress related with the stress intensity factors KI and KII, a dimensionless parameter called the biaxility ratio B for mixed mode loading is used. It was proposed by Leevers and Radon [29], written as B

T a

(7)

K I2  K II2

The equation shows that the contribution of T-stress increases with B, relative to the singular terms (KI and KII). Smith et al [24] have shown that mixed mode brittle fracture was significantly affected by T-stress for specimens with a significant value of B. Fig. 5(a) shows the values of biaxility ratio B versus the mode mixities Me for CCP specimen with different crack sizes a/W. The values of B are all decreasing with the increase of mode mixities Me for CCP specimens. The value of B is positive when Me is smaller than 0.6, and negative when Me is larger than 0.6. For different crack sizes, the values of B increase as the crack size a/W increases when Me is smaller than 0.6, and decrease when Me is larger than 0.6. However, the differences for different crack sizes are small, especially for larger Me.

(a) CCP 11

(b) CTS specimen

(c) Four-point bend specimen Fig. 5 Relationships of B and mode mixities Me for three different specimens

Fig. 5(b) presents the functions between values of B and mode mixities Me for CTS 12

specimens with different crack sizes a/W. Also the results in the previous paper [12] are presented to compare, from the comparison it can be found that the results in this paper almost agree with those in the paper [12], so it validates that finite element method in this paper is accurate and believable. From Fig. 5(b), there exist three different trends of B for three different crack sizes. When a/W is 0.5, the values of B almost keep unchanged with Me, and the values of B are close to zero. When a/W is 0.3, B decreases as Me increases, and the values of B are almost negative for different mode mixities. When a/W is 0.7, B increase as Me increase, and the values of B are almost positive for different mode mixities. For CTS specimens, T-stresses are close to zero and vary little in mode II state as the crack size changes. As the mode mixities Me become far away from zero, that is the component of mode I is becoming significant, the differences of T-stresses from different crack sizes are increasing. Fig. 5 (c) presents the functions between the values of B and mode mixities Me for four-point bend specimen with different crack sizes. When a/W=0.3, B almost keep stable with Me changing, and the values of B are close to zero. For a/W=0.5 and 0.7, B increases as Me increases. For the same mode mixities Me, the values of B are increasing with the crack size increasing. And similar with the CTS specimen, T-stresses vary little in mode II state as the crack size changes. As the component of mode I becomes significant, the differences of T-stresses from different crack sizes are increasing, and when the mode mixity is close to 1, the difference reaches the maximum. T-stress has been used as a measure of constraint, specimens with higher T-stresses are in a higher constraint state compared with those with lower T-stresses. For mode I crack and mode I dominant crack (Me>0.6) in Fig. 5, the values of B of CCP are negative and lower than those of CTS specimen and FPB specimen. For mode II dominant crack, the values of B of CCP are positive and greater than those of CTS specimen and FPB specimen. So, for mode I dominant crack, tensile dominant specimens (CCP) are in low-constraint states, and bending dominant specimens (CTS specimen and FPB specimen) are in high-constraint states. For elastic-plastic material, Zhu thought that the bending dominant specimens (like CTS specimen and FPB in this paper) were in a high constraint state [30], which would obtain a conservative fracture toughness. Here from the results in Fig. 5, it can be concluded that for elastic materials or the small scale yielding, the bending dominant specimens are in high constraint states when the component of mode I is dominant. While for mode II dominant crack, the conclusions are just opposite, tensile dominant specimens like CCP are in higher-constraint states compared with bending dominant specimens. Also from the conclusion that, if a conservative result is wanted, bending dominant specimen is a priority when the component of mode I is significant, and tensile dominant specimen is a priority when the component of mode II is significant.

13

3.3 Discussion Firstly, the modified R-criterion (MR criterion) is defined in this section for convenience, which is the criterion associating R-criterion with T-stress and has been reported in [8]. The following will discuss the effect of T-stress on crack initiation angle and plastic zone for specimens (CCP, CTS and FPB) based on MMTS criterion and MR criterion. Fig. 6 shows the effect of T-stress on crack initiation angle for the three specimens in this paper, and three crack lengths are considered. The caption of a/W=0.3 based on MMTS in Fig. 6 means that the crack initiation angles for specimens with crack length a/W=0.3 are obtained based on MMTS, and so on for other captions. MTS criterion in Fig. 6 is the criterion without considering T-stress. Fig. 7 presents the effect of T-stress on plastic zone for the three specimens with mode I dominant cracks and mode II dominant cracks, also three crack lengths are considered.

(a) CCP

14

(b) CTS

(c) FPB Fig. 6 Crack initiation angles for different specimens Fig. 6(a) presents the crack initiation angles versus Me for CCP, in which MTS criterion is included. From the figure, crack initiation angles do not agree with those based on MTS criterion. 15

As Me are larger, crack initiation angles are above the curve of MTS criterion. And as Me decrease, crack initiation angles (-θ) are gradually becoming greater compared with those of MTS criterion. In addition, initiation angles (-θ) according to MMTS for CCPs with different crack lengths are smaller than those according to MR criterion. The differences between the two criterions become smaller as the values of Me decrease. Fig. 6(b) presents the crack initiation angles versus Me for CTS. From the figure, initiation angles according to MMTS criterion for CTS agree with MTS criterion when the component of mode II is significant. And as the component of mode I increases, the differences become significant. The initiation angles of CTS with a/W=0.5 agree well with MTS criterion, initiation angles of CTS with a/W=0.3 are above it, and initiation angles of CTS with a/W=0.7 are below it. Oppositely, initiation angles based on MR criterion agree better with MTS when Me is higher, and as the component of mode II becomes significant, crack initiation angles based on MR criterion are away from those based on MMTS criterion. Likely with CTS, initiation angles according to MTS criterion for FPB in Fig. 6(c), agree with MTS criterion when the component of mode II is significant. As the component of Mode I increases, the differences become significant. The initiation angles of CTS with a/W=0.3 agree well with MTS without T-stress, and initiation angles CTS with a/W=0.5 and 0.7 are below it. Oppositely, initiation angles based on MR criterion agree with MTS when Me is larger. And as the component of mode II becomes significant, crack initiation angles based on MR criterion are away from those based on MTS criterion.

a/W=0.3 a/W=0.5 a/W=0.7

CCP, β =15°

CTS, β =15°

(a) Mode I dominant crack

16

FPB, S=20

a/W=0.3 a/W=0.5 a/W=0.7

CCP, β =75°

CTS, β =75°

FPB, S=2

(b) Mode II dominant crack Fig. 7 Plastic zones for different specimens Fig. 7(a) presents plastic zones for CCP, CTS and FPB with mode I dominant cracks. From the figure, plastic zones for CCP with different crack lengths vary little, while the effects of crack sizes on plastic zones for CTS and FPB are much obvious. In addition, plastic zones are the largest for CCP and CTS with the minimum crack sizes a/W=0.3. While for FPB, the opposite conclusion is obtained, specimens with the largest crack sizes a/W=0.7 are with the largest plastic zone. For mode II dominant cracks in Fig. 7(b), plastic zones for FPB with different crack lengths vary little, while the effects of crack sizes on plastic zones for CCP and CTS are much obvious. Likely with mode I dominant crack, plastic zones with the minimum crack sizes a/W=0.3 are the largest for CCP and CTS, and with the largest crack sizes a/W=0.7 are with the largest plastic zone for FPB. So it can be seen that, the variation of crack initiation angles and plastic zones are dependent on the T-stress induced by different specimen geometries and loading modes.

4 Conclusions 1) The biaxility ratios (B) are calculated by finite element method for three widely used test specimens. For CCP, the values of B vary little as the crack size changes. For CTS specimens and FPB specimens, there exist some differences of B for different crack sizes. T-stresses do not vary a lot in a mode II state as the crack size changes. When the component of mode I is becoming larger, the differences of T-stress from crack sizes are increasing. 2) For mode I dominant crack, CCP specimens are in a low-constraint state, and CTS specimen and four-point bending specimen are in a high-constraint state. For mode II dominant crack, the CCP is in a higher-constraint state compared with CTS specimen and four-point bending specimen. 3) Specimen with high constraint (a higher T-stress) is a priority if a conservative result is wanted. So, bending dominant specimen is a priority when the component of mode I is significant, and tensile dominant specimen is a priority when the component of mode II is significant. 17

4) For tensile dominant specimen (CCP), the trends of initiation angles based on MR criterion and MMTS criterion are similar. For bending dominant specimens, there exist some differences between initiation angles based on MMTS and MR criterion. Acknowledgements The authors gratefully acknowledge the financial supports of the National Natural Science Foundation of China (51475223, 51675260).

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PVP2016-63662.

Highlights CCP is in a low-constraint state for mode I dominant crack, and in a higher-constraint state for mode II dominant crack compared with CTS specimen and FPB specimen. Bending dominant specimen is a priority for conservative results when the component of mode I is significant, and tensile dominant specimen is a priority when the component of mode II is significant. The effect of T-stresses or constraints produced by specimen geometries and loading modes play a significant role on crack initiation angle and plastic zone around the crack tip, which in turn can affect the fracture of specimen.

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