Journal of Mechanical Science and Technology 26 (11) (2012) 3549~3554 www.springerlink.com/content/1738-494x
DOI 10.1007/s12206-012-0870-0
Energy release rates for various defects under different loading conditions† Y. Eugene Pak1,*, Dhaneshwar Mishra2 and Seung-Hyun Yoo2 1
Advanced Institutes of Convergence Technology, Seoul National University, Suwon, Korea 2 Department of Mechanical Engineering, Ajou University, Suwon, Korea
(Manuscript Received December 7, 2011; Revised April 6, 2012; Accepted June 17, 2012) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Abstract It is well known that the energy release rate associated with translation, rotation and self-similar expansion of defects in solids are expressed by the path-independent integrals J, L, and M, respectively. It is shown that these integrals for a crack or a circular hole may be obtained by first considering an elliptical cavity and then performing a limiting process. This obviates dealing with singularities at the crack tip. The energy release rates for these defects under various mechanical, thermal and electromechanical loading conditions are calculated. Keywords: Energy release rate; Path-independent J, M and L integrals; Defects; Piezoelectric material ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1. Introduction The strength of most brittle solids is determined by defects such as cracks. Thus, finding the stress distribution on the basis of theory of elasticity is of interest. The linear theory leads to a so-called singular boundary value problem. The plane stress distribution near a sharp crack in an infinite body subjected to biaxial tension was first treated by Westergaard [1] whose result was shown to give singular (infinite) stresses at the crack tip. Still, the stress intensity factors and the energy release rates related to material toughness can be calculated. A path-independent integral called J-integral was introduced by Rice [2] and provides alternate means for evaluating energy release rates and stress intensity factors. Other pathindependent integrals, later called L and M, were discovered by Günther [3] and by Knowles and Sternberg [4]. All these path-independent integrals are useful in fracture mechanics in that they are related to energy release rates. Freund [5] demonstrated the use of the M-integral in finding stress intensity factors for various fracture mechanics problems. Later, other authors [6-10] applied M- and L-integrals in studying interface and microcrack interaction problems. King and Hermann [11] demonstrated an alternative non-destructive experimental measurement technique for evaluating the J- and M-integrals. The purpose of this contribution is to show that the pathindependent integrals leading to energy release rates for a crack can be evaluated by first considering the corresponding *
Corresponding author. Tel.: +82 31 888 9056, Fax.: +82 31 888 9040 E-mail address:
[email protected] † Recommended by Associate Editor Seong Beom Lee © KSME & Springer 2012
expression for an elliptical cavity and then passing to the limit of a vanishingly small semi-axis of the ellipse, i.e., a crack. This approach completely avoids dealing with singular stress fields. Another purpose is to provide closed-form expressions of M- and L-integrals, which are the measures of energy release rate for elliptical and circular cavities as well as for a slit crack embedded in an infinite body under various loading conditions. This will help the readers to have a better understanding on energy release rates for various defects under different types of loading conditions. Some of the results presented in this paper have been published elsewhere by Pak [12-16]. Here, we have added the closed-form of J, M and L expressions for elliptical and circular cavities under far-field antiplane shear, τ0. The same procedure is carried out for a piezoelectric material under antiplane shear, τ0, and inplane electric field, E0. For the sake of completeness, previously found results are tabulated along with the new results in Table 1.
2. Path-independent integrals In a two-dimensional field in Cartesian coordinates x and y, the J-integral [2] is defined as Jp =
∫ (Wdy − T u ) ds i i ,x
(1)
CJ
where CJ is a contour surrounding a crack tip (Fig. 1), W is the strain energy density, ui is the displacement vector and Ti is the traction vector defined as σijnj , where nj is the outward unit
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Table 1. Closed form expressions of path independent integrals under various far field loading conditions. Jw
M
L
Case 1. Uniaxial Tension at an Arbitrary Angle β [Fig. 3(a)]
π a 2T 2 (1 − cos 2β ) / E
π a 2T 2 sin 2 β / E
Crack
0
Circular cavity
0
3π a 2T 2 / E
0
Elliptical cavity
0
π a + b + ab − (a 2 − b 2 ) cos 2β T 2 / E
π T 2 ( a 2 − b 2 ) sin 2β / E
Crack
0
2π a 2T 2 / E
2
2
Case 2. Biaxial Tension [Fig. 3(b)] 0
Circular cavity
0
4π a T / E
0
Elliptical cavity
0
π ( 2a 2 + b 2 ) T 2 / E
0
Crack
0
2π a 2 S 2 / E
0
0
8π a S / E
0
0
2π a ( a + b ) S 2 / E
0
2
2
Case 3. In-plane Shear [Fig. 3(c)] Circular cavity Elliptical cavity
2
2
Case 4. Antiplane Shear at an Arbitrary Angle β [Fig. 3(d)] Crack
0
π a 2τ 02 (1 − cos 2β ) / 2G
Circular cavity
0
2π a 2τ 0 2 / G
Elliptical cavity
0
π ( a + b )τ ( a + b ) − ( a − b ) cos 2β / 2G
Crack
0
π Eα 2 a 4τ 2 sin 2 β / 8
0
π Eα a τ / 2
Elliptical cavity
0
π Eα ( a + b ) b − ( a − b ) sin β τ / 8
Crack
0
π a 2 τ 02 − kE02 (1 − cos 2 β ) / 2G
π a 2 τ 02 − kE02 sin 2β / 2G
Circular cavity
0
2π a 2 τ 02 − kE02 / G
0
Elliptical cavity
0
π ( a + b ) τ 02 − kE02 ( a + b ) − ( a − b ) cos 2 β / 2G
π ( a 2 − b 2 ) τ 02 − kE02 sin 2 β / 2G
π a 2τ 02 sin 2β / 2G 0
πτ
2 0
2 0
(a
2
)
− b sin 2β / 2G 2
Case 5. Uniform Heat Flow at an Arbitrary Angle β [Fig. 3(e)]
Circular cavity
2
2
2
4
0
2
0
2
2
2
(
)
Eα ab b − a 2 τ 2 sin 2β / 8
2
2
2
Case 6. In-plane Electrical and Antiplane Mechanical Loading [Fig. 3(f)]
(
Where k = e15M + C44M ε11M 2
)
intensity factors for Mode I and Mode II fracture, respectively, which are defined as K I = σ yy∞ π a K II = σ xy∞ π a .
Fig. 1. Paths for J, M and L-Integrals.
normal vector to the contour. In the presence of far-field homogeneous stresses σ xx∞ , σ xy∞ and σ yy∞ , J is related to the stress intensity factors in plane stress condition as J = (K + K 2 I
2 II
)/ E .
.
(3)
In the case of a thermal loading, the stress intensity factors are functions of thermal loading parameters, while for electrical loading, they contain electrical displacement or electric field loading parameters. The J-integral is useful in fracture mechanics because it can be shown to be equal to the crack extension force G, obeying Irwin’s relationship, i.e, J = G = ( K I2 + K II2 ) / E .
(4)
(2)
Here, E is Young’s modulus, and KI and KII are the stress
It has also been related to the potential energy release rate associated with unit crack tip advancement, namely
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J =−
lim ∆U el + ∆U lm 2 ∆a ∆a → 0
(5)
where ∆Uel and ∆Ulm are the increase in the elastic strain energy of the body and the potential energy of the loading mechanism, respectively, due to the elongation of the crack. In the classical mechanics sense of a generalized force as a potential energy release rate, J can be called the “material force” or the “crack extension force” which acts on the defect enclosed by the contour CJ. For piezoelectric materials, the J-integral can be generalized by taking the electric enthalpy density, H, to be the Lagrangian density, and simply differentiating it with respect to the spatial coordinate, xi. This operation leads to the J-integral for piezoelectric materials, JP, as Jp =
∫ ( Hn
− Ti ui , x + Di ni Ex ) ds
x
∫ (Wx n −T u x ) ds. i
i
k
(7)
k ,i i
CM
For the particular case of a crack undergoing self-similar expansion such that there is a relative scale change ∆a/a, the M-integral represents a measure of energy release rate with respect to this change. For the case of a crack, a simple relationship can be established between M and J [5], i.e., M = 2aJ
(8)
which is useful when it is more convenient to evaluate M using a closed contour surrounding the whole crack rather than J using a contour around a crack tip (e.g., if there is loading on the crack faces which make the J path-dependent). The generalized M-integral given by Pak [12] for piezoelectric materials can be written as
∫ ( Hx n k
k
− Tk uk ,i xi + Dk nk Ei xi ) ds.
(9)
CM
The last path-independent integral to be discussed is the Lintegral, first defined by Günther [3] as
(
)
L = ∫ ezij Wx j n i +Tiu j − Tk uk ,i x j ds CL
2π a 2 ∞ ∞ σ xy σ yy + σ xx∞ E
L=
2 K II K I + σ xx∞ π a . E
(
(10)
)
(11)
)
(12)
or
(
They have also shown L to be the rotational energy release rate that can be interpreted as the measure of energy release rate of a crack per unit rotation. It can be regarded as the “material moment” acting on the defect. The relationship has the form L=−
where Di and Ei are the electric displacements and the electric fields, respectively. Pak [13] has shown that this expression is physically meaningful in that it can be related to the electroelastic energy release rate for defects in piezoelectric materials. Another path-independent integral, M, has been shown to be the energy release rate for a self-similarly expanding defect that is completely enclosed by a contour CM, as shown in Fig. 1 [4, 17]. The M-integral expression can be written as
MP =
L=
(6)
CJ
M=
where ezij is the alternating tensor. By evaluating this integral around the whole crack subjected to far-field loads σxx∞, σxy∞ and σyy∞, Herrmann and Herrmann [18] obtained the result
∂U ∂φ
(13)
where U is the total potential energy (Uel + Ulm), and φ is a small angle through which the crack rotates with respect to the applied field. The value of the L-integral, with its relationship to the stress intensity factor given by Eq. (12), provides an added independent relationship to J so that the stress intensity factors KI and KII can be obtained separately for a crack subjected to a mixed-mode load. We have generalized the L-integral expression for piezoelectric materials introduced by Eishen and Hermann [20] by employing the electrical enthalpy density in place of the strain energy density:
(
)
LP = ∫ ezij Hx j ni + T i u j − Tk uk ,i x j + Di ni E j x j ds.
(14)
CL
This relationship has been used to evaluate the rotational energy release rate for cavities embedded in an infinite piezoelectric matrix.
3. Validity of the limiting procedure In this section, the method of evaluating path-independent integrals for various defects, e.g., a circular cavity or a crack, is demonstrated (Fig. 2) by first considering an elliptical cavity and then performing a limiting process. To show the validity of this limiting procedure, Jw- (a J-integral enclosing the whole defect), M- and L-integrals have been calculated for the elliptical cavity embedded in an infinite body subjected to farfield antiplane shear, τ0 (Fig. 3(d)). In addition, a case of a piezoelectric material subjected to combined antiplane shear, τ0, and inplane electric field, E0, is also considered as shown in Fig. 3(f). The M- and L-integrals for the circular cavity embedded in an infinite body can be obtained as a special case of the elliptical solution by letting a → b, while for the slit crack they can be obtained by letting b → 0.
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Fig. 2. (a) circular cavity; (b) elliptical cavity; (c) crack in an infinite medium.
(a)
(b)
(c)
loading was zero. This is due to the result of the path chosen. Physically, this means that the net material force, Jw, acting on the whole cavity in an infinite medium is zero. Since the far field loads are symmetric with respect to the midsection (plane x = 0) of the defect, the material force acting on the right half of the defect in the x-direction is equal and opposite to the one acting on the left half. Because the chosen contour surrounds the whole defect, the net material force, Jw, in the x-direction is zero. In other words, there is no change in the energy of the system due to an infinitesimal translation of the defect in an infinite body under any arbitrary loading. There is a difference between J surrounding only one crack tip (CJ) and Jw surrounding the whole defect (CJw). When we choose the path that surrounds one crack tip (CJ) only, the Jintegral results validate relation (8), i.e. M = 2aJ. For example, the Jp-integral for a crack in an infinite piezoelectric medium obtained by Pak [13], Jp =
πa E
(τ
2 0
− kE02 )
where
(
k = e15M + C44M ε11M (d)
(e)
2
)
(15)
(f)
Fig. 3. Elliptical cavity in an infinite medium subjected to: (a) far-field uniaxial tension, T, at an arbitrary angle β; (b) biaxial tension, T; (c) inplane shear, S; (d) antiplane shear τ0; (e) uniform heat flow, τ, at an arbitrary angle β; (f) antiplane shear, τ0, and inplane electric field E0 at an arbitrary angle β.
The J-integral expression for a crack can then be obtained by making use of the relation (8). These relations deduced from the elliptical solutions are compared with the expressions explicitly calculated for the circular hole and the crack by Pak [13, 14] to validate the correctness of the method. Similar work done previously are shown in Figs. 3(a), (b), (c), and (e) and have also been tabulated in Table 1 [12, 15, 16] for the cases of: (a) remote uniaxial tension, T, at an arbitrary angle β, (b) biaxial tension, T, (c) inplane shear, S, and (e) uniform heat flow, τ. The paths taken have been along the boundary of the cavity, employing already existing plane stress elasticity solutions from Timoshenko and Goodier [19] and piezoelectric solution by Pak [13]. Since we have evaluated these integrals along the traction-free boundary, the terms containing traction, Ti, in the integrand of Eqs. (1), (6), (7), (9), (10) and (14) do not contribute to the final value of these integrals. We only need to integrate the strain energy density term along the rim. For the circular and elliptical cavity, only the hoop stress contributes to the strain energy density.
4. Discussion It is shown that Jw for the elliptical cavity in all cases of
is in complete agreement with the result we found in this work when we used the limiting procedure (b → 0) on the piezoelectric elliptical cavity solution along with relation (8). Also, for the piezoelectric case, the Jp-integral when evaluated on a path that surrounds the whole defect is shown to be zero as explained above. However, the self-similar expansion force M is not zero because when a defect expands in a self-similar manner, the energy of the system changes. The results for the M-integral expressions for a circular cavity (b → a) and a slit crack (b → 0) deduced from the embedded elliptical cavity considered here are again in complete agreement with previously calculated results for all loading cases. For example, the expression of the M-integral for a circular cavity, M=
2π a 2 2 (τ 0 − kE02 ) E
(16)
deduced from the elliptical cavity solution for a coupled electromechanical load, agrees with the same solution developed by Pak [14]. All loading cases tabulated in Table 1 show the same trend in that the M-integral is maximum when loading is perpendicular to the major-axis of the ellipse and minimum when it is parallel. The loading angle does not have any effect for the circular cavity. This is due to an obvious reason; as a circular cavity expands in a self-similar manner, the energy change or energy release rate is independent of the loading direction. The L-integrals are zero when defects are in equilibrium with respect to the orientation of the external load in all cases.
Y. E. Pak et al. / Journal of Mechanical Science and Technology 26 (11) (2012) 3549~3554
There is no first-order change in the total energy with respect to the cavity rotation about the current cavity orientation. We can say that the net ‘material moment’ acting on the defect is zero. As a result, the propagating crack will extend in its own plane under such circumstances. The energy release rate per infinitesimal rotation of the cavity, i.e., the material moment, is maximum when β = ± π/4 in all loading cases (uniaxial tension, antiplane mechanical shear, uniform heat flow and antiplane mechanical shear and inplane electric field). An equal and opposite moment is experienced by the elliptical cavity if β is shifted by π/2. Here again, the direction of the rotational material moment is such that it tries to orient the defect so that maximum M, or a state of maximum energy (electric enthalpy for piezoelectric material) release rate would occur. In case of the circular cavity (b → a), L comes out to be zero for every case. There is no preferential direction for the circular cavity with respect to far-field loading. For a crack (b → 0) under a uniform thermal load, L is zero, regardless of the heat flow angle β. Physically, this means that there is no resultant ‘material moment’ acting on the crack, and it can be predicted that the crack will not undergo an angled propagation, but rather propagate in its own plane when the critical crack extension force is reached at the crack tip [15]. However, for a piezoelectric material under electromechanical loading, the energy release rate behavior is similar to the purely elastic case. In all but thermal loading cases, the crack will undergo a skewed propagation to position itself to a more energetically favorable orientation, i.e., an orientation at which L would be zero that coincides with M being maximum. Thus, if L is zero, then there would be no crack skewing, i.e., a state of single-mode fracture. It is interesting to note that for a blunt crack (b ≠ 0) in all cases, L is also zero for the cases β = 0, π/2, and crack skewing can occur at all other loading angles.
5. Conclusions It is shown that the energy release rates for defects such as cracks and circular cavities under various loading conditions can be obtained by first evaluating the path-independent integrals for an elliptical cavity and then performing a limiting process. This limiting process with certain special properties of the M-integral makes it possible to find the J-integral or the crack extension force for a crack. This work provides all the closed-form expressions for various loading conditions in a tabulated form so that one can compare the results and understand how different types of loading can affect the energy release rates. This work also discusses the variation in the energy release rate due to the far-field loading angle, hence providing a good understanding of the energetics of various defects under different loading conditions.
Acknowledgment This work was supported by the grant 2011-P2-21 from
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Seoul National University’s Advanced Institutes of Convergence Technology. Authors DM and SHY acknowledge the support provided by the Government of Korea under its BK21 program.
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661-673. [16] Y. E. Pak, Elliptical inclusion problem in antiplane piezoelectricity: implication for fracture mechanics, International Journal of Engineering Science, 48 (2010) 209-222. [17] B. Budiansky and J. R. Rice, Conservation laws and energy release rates, ASME Journal of Applied Mechanics, 40 (1973) 201. [18] G. A. Herrmann and G. Herrmann, On the energy release rate for a plane crack, ASME Journal of Applied Mechanics, 48 (1981) 525-528. [19] S. P. Timoshenko and J. N. Goodier, Theory of elasticity, Mc-Graw-Hill (1970). [20] J. W. Eishen and G. Hermann, Energy release rates and related balance laws in linear elastic defect mechanics, Journal of Applied Mechanics, 109 (1987) 388-392.
Y. Eugene Pak is currently serving as Director of the Convergence Research Division at the Advanced Institutes of Convergence Technology in Seoul National University. He received his B.S. degree in Mechanical Engineering from the State University of New York at Buffalo in 1980 and M.S. and Ph.D degrees in Mechanical Engineering from Stanford University in 1982 and 1985, respectively. After obtaining his Ph.D. degree, he worked as a Senior Research Scientist at Northrup-Grumman Corporate Research Center. He worked with the Samsung Advanced Institute of Technology (SAIT) from 1995 to 2009 where he conducted research in MEMS, nano and biotechnology.
Dhaneshwar Mishra received his B.E. Degree from MNREC (presently known as MNNIT), India in 1998 and his M.E. Degree at Anna University, Chennai, India in 2003. He completed his Ph.D Degree at Ajou University, Suwon, Korea in 2011 August, and is currently working as a Post Doctoral researcher at the Research Center for Automotive Parts Technology, Ajou University. His areas of interest are computational mechanics of materials, especially defect and fracture mechanics of bio and piezoelectric materials, Design and CAE. Seung-Hyun Yoo is a professor of the Department of Mechanical Engineering and the Director of the AUCHRD (Ajou University Center for Human Resource Development), Ajou University. He received his B.S. and M.S. degree from Seoul National University in 1977 and 1981 respectively after fulfilling military service as an artillery officer in the Korean army. He then received his Ph.D degree from Stanford University in 1987. Thereafter he worked as a Research Fellow at the University of Michigan and as a Senior Researcher at KIMM (Korea Institute of Machinery and Materials) till he moved to Ajou University in 1990. His research interests cover all aspects of mechanics of thin structures with discontinuities, biomechanics and CAI (computer aided innovation), and SI (systematic innovation) for creative engineering design.
