Appl. Math. Mech. -Engl. Ed.   2016, Vol. 37 Issue (6): 695-706     PDF       
http://dx.doi.org/10.1007/s10483-016-2083-6
Shanghai University
0

Article Information

Dianhui HOU, Xiao CHONG, Guixiang HAO, Yao DAI.
Higher-order crack tip fields for functionally graded material plate with transverse shear deformation
Appl. Math. Mech. -Engl. Ed., 2016, 37(6): 695-706
http://dx.doi.org/10.1007/s10483-016-2083-6

Article History

Received Sept. 2, 2015;
Revised Oct. 19, 2015
Citation
Dianhui HOU, Xiao CHONG, Guixiang HAO, Yao DAI. Higher-order crack tip fields for functionally graded material plate with transverse shear deformation[J]. Appl. Math. Mech. -Engl. Ed., 2016, 37(6): 695-706.
Higher-order crack tip fields for functionally graded material plate with transverse shear deformation
Dianhui HOU1, Xiao CHONG2, Guixiang HAO3, Yao DAI4        
1. Institute of Command and Information Systems, PLA University of Science and Technology, Nanjing 210007, China;
2. Changping School of Noncommissioned Officer, PLA Academy of Equipment, Beijing 102200, China;
3. Army 73111, Xiamen 361025, China;
4. Academy of Armored Force Engineering, Beijing 100072, China
Received Sept. 2, 2015; Revised Oct. 19, 2015
Fund program: Project supported by the National Natural Science Foundation of China (Nos. 90305023 and 11172332)
Corresponding author: Yao DAI, E-mail: dai yao@sina.com
ABSTRACT: The crack tip fields are investigated for a cracked functionally graded material (FGM) plate by Reissner's linear plate theory with the consideration of the transverse shear deformation generated by bending. The elastic modulus and Poisson's ratio of the functionally graded plates are assumed to vary continuously through the coordinate y, according to a linear law and a constant, respectively. The governing equations, i.e., the 6th-order partial differential equations with variable coefficients, are derived in the polar coordinate system based on Reissner's plate theory. Furthermore, the generalized displacements are treated in a separation-of-variable form, and the higher-order crack tip fields of the cracked FGM plate are obtained by the eigen-expansion method. It is found that the analytic solutions degenerate to the corresponding fields of the isotropic homogeneous plate with Reissner's effect when the in-homogeneity parameter approaches zero.
Keywords: crack tip field     Reissner's effect     plate     functionally graded material (FGM)     eigen-expansion method    
1 Introduction

Although the plate is the pervasive structure in engineering, its research keeps expanding along with the development of materials,e.g.,homogeneous materials,non-homogeneous materials, functionally graded materials (FGMs),and piezoelectric materials. It is well-known that the early studies of plates were for homogeneous materials based on Kirchhoff's theory[1]. However, the fracture analysis on the classical plates with Kirchhoff's theory led to an unreasonable conclusion that the transverse shear resultants had a singularity of the order r-3/2,which led to the unbounded strain energy[2]. To solve such problems,the first-order plate theory,i.e.,Reissner's theory[3] considering transverse shear deformation,is developed and used. Up to now,a number of valuable results have been obtained. Knowles and Wang[4] and Hartranft and Sih[5] first presented all of the stress components with the inverse square-root singularity of r-1/2. Murthy et al.[6] studied the symmetrical I-mode crack in homogeneous plates,and obtained the general term of the expansion equation of the crack tip strain field. Liu[7] and Liu and Jiang[8] studied the crack tip field in homogeneous plates by means of the eigen-expansion method,and gave the first several items of the eigen-function solutions. Qian and Long[9] analyzed the stress and strain fields at the tip of a notch,derived the eigen-functions of the problem,and calculated the eigen-values of different notches with different angles by the Muller iteration method. Xu[10] extended the expansion form of the stresses and deformation near a crack tip to a general case,and showed that the eigen-problem of Reissner's plate was equivalent to a combination of a plane problem and an anti-plane problem.

In the 1980s,FGMs appeared,and widely applied on aviation, aerospace,nuclear industry and so on due to their superior performance compared with that of traditional materials. From then on,FGM plates have been increasingly emphasized. First,the higher-order plate theory was developed,e.g.,the third-order theory[11, 12, 13]. Next,the inverse square-root singularity of r-1/2 was proved to be still valid to crack problems of Reissner's plates only if the material function ahead of the crack was continuous. Furthermore,various crack problems were studied by different methods,among which the integral transform method was often adopted to obtain the stress intensity factors (SIFs)[14, 15, 16, 17] and the effects of various factors on the SIFs.

It is well-known that,the properties of FGMs are functions of the spatial coordinates generally,and the governing equations are the system of partial differential equations with the variable coefficients,whose analytical solutions are difficult to be obtained. It can be found that only the crack tip singular field is involved in the researches mentioned above. This is because that the eigen-functions or the higher-order crack tip fields of the FGM plates with Reissner's effect as Williams' solutions of homogeneous materials have not been available up to now. The full-fields of homogeneous material plates have sometimes been considered in the investigations for FGMs. However,this usually results in errors, and even makes the analysis meaningless. Butcher et al.[18] pointed out: "a full-field asymptotic description for cracked FGMs would be of great value for improving the fracture parameter estimation and a subsequent development of a fracture criterion and verification." Therefore,the main effort of this paper is to find out the higher-order crack tip fields of the cracked FGM plate with Reissner's effect by the eigen-expansion method.

2 Governing equations

Assume that the analysis is limited to the small elastic deformation. Then,the geometric relation of the deformation of a plate in the polar coordinate system can be written as follows[3]:

where φr and φθ are the rotation angles,and εr,εθ,and γrθ are the strains at the point (r, θ, z) (see Fig. 1). Hooke's law is

where E is Young's modulus,and μ is Poisson's ratio.

Fig. 1 Crack problem of FGM plate
Figure options

Substituting Eq. (1) into Eq. (2) and integrating the derived expression with respect to z from -h/2 to h/2,where h is the thickness of the plate,yield the following equations of moment components Mr,Mθ,and Mrθ:

where

In addition,the transverse shear components of Reissner's plate are

where w is the deflection of the neutral plane,and

The equilibrium equations can be expressed by these internal force components as follows:

Figure 1 shows the studied FGM plate with a through crack. Young's modulus of the plate is assumed to be

where E0 is Young's modulus at the crack tip,and γ≥0 is the in-homogeneity parameter. The effect of Poisson's ratio on the stress intensity factor (SIF) is far less than that of the elastic modules[19]. Therefore,Poisson's ratio is assumed to be a constant μ. The transverse loading of the plate is assumed to be zero in this paper.

Substituting Eqs. (3) and (4) into Eq. (5) yields the governing equations as follows:

Since the crack surface is free,the boundary conditions are

where θ=± π.

3 Higher-order crack tip field

The crack tip stress field can be equipped with the same square root singularity as that of homogeneous materials when the properties of different composite materials at the interfaces are continuous[8, 20]. Therefore,the generalized displacements can be expressed as follows[21, 22]:

where fn(θ),gn(θ),and jn(θ) are eigen-functions.

Substituting Eq. (9) into Eqs. (7) and (8),we can obtain that the coefficients of r-n/2 are linearly independent,and each coefficient term must be zero. Then,the systems of equations to determine the different order terms of the eigen-functions can be obtained as follows (only 6 terms are shown by the limit of length):

where

Equation (8) can be treated similarly. Then,the systems of equations with the corresponding boundary conditions can be solved, and the first six items of the eigen-functions fn(θ),gn(θ),and jn(θ) can be given as follows:

where Bij(i=1, 2, …, n; j=1, 2, 3) are undetermined coefficients.

Substituting Eqs. (16)--(20) into Eq. (9),we can obtain the generalized displacement field and then the stress field based on the relationship between the generalized displacements and stresses. The results show that the first two items of the higher-order crack tip fields of FGMs have the same mathematical form as that of the corresponding terms of homogeneous plates. The effect of the in-homogeneity on the fields is mathematically explicit only in the higher-order items. In order to display the effect qualitatively,as a special example,it is assumed that only the in-homogeneity parameter γ varies and the undetermined coefficients are arbitrary constants. Let μ= 0.3,r=0.05a(a is a half of the crack length),and the symmetric functions for Mode Ⅰ and the asymmetric functions for Mode Ⅱ be separated in Eqs. (9) and (18)-(21). The variations of ør with the in-homogeneity parameter γ of the six-order solution in this paper are shown in Figs. 2 and 3 for Mode Ⅰ and Mode Ⅱ,respectively.

Fig. 2 Variations of ør with γ of 6th-order solution for Mode Ⅰ
Figure options
Fig. 3 Variations of ør with γ of 6th-order solution for Mode Ⅱ
Figure options

From the curves in Figs. 2 and Fig. 3,it can be seen that the in-homogeneity parameter γ has an obvious effect on the higher-order crack tip fields. The larger the parameter γ is,the more significant the effect is. Furthermore,the crack tip fields of FGM plates will degenerate to the crack tip fields of homogeneous plates if γ =0[8].

4 Conclusions

The crack tip higher-order fields of FGM plates based on the first-order plate theory,i.e.,Reissner's theory,are obtained in the polar coordinate system by use of the eigen-expansion method, which extends the famous Williams' solutions to homogeneous materials to the FGM plates.

When the in-homogeneity parameter approaches zero,the analytic solutions given herein will degenerate to the corresponding fields of the isotropic homogeneous plate with Reissner's effect.

The effect of the in-homogeneity parameter on the crack tip higher-order fields increases when γ becomes larger. In order to explicitly display the effect,at least,the first three terms of the higher-order fields must be considered apparently according to the theoretical results.

The crack tip higher-order fields are equivalent to the eigen-functions mathematically. Therefore,they provide the theoretical basis for the numerical simulations,engineering applications (e.g.,determinations of SIFs),experimental analysis of cracked FGM plates,etc.

References
[1] Timoshenko, S. and Woinowsky-Kriege, S. Theory of Plates and Shells, McGraw-Hill Book Com-pany, New York, 1-81 (1959)
[2] Williams, M. L. The bending stress distribution at the base of a stationary crack. Journal of Applied Mechanics, 28, 78-82 (1976)
[3] Reissner, E. On bending of elastic plates. Quarterly of Applied Mathematics, 5, 55-68 (1947)
[4] Knowles, J. K. and Wang, N. M. On the bending of an elastic plate containing a crack. Journal of Mathematics and Physics, 39, 223-236 (1960)
[5] Hartranft, R. J. and Sih, G. C. Effect of plate thickness on the bending stress distribution around through cracks. Journal of Mathematical Physics, 47, 276-291 (1968)
[6] Murthy, M. V. V., Raju, K. N., and Viswanath, S. On the bending stress distribution at the tip of a stationary crack from Reissner's theory. International Journal of Fracture, 17, 537-552 (1981)
[7] Liu, C. T. Stresses and deformations near the crack tip for bending plate. Acta Mechanica Solid Sinica, 3, 441-448 (1983)
[8] Liu, C. T. and Jiang, C. P. Fracture Mechanics for Plates and Shells, Defense Industry Press, Beijing, 139-163 (2000)
[9] Qian, J. and Long, Y. Q. The expression of stress and strain at the tip of notch in Reiss-ner's plate. Applied Mathematics and Mechanics (English Edition), 13(4), 297-306 (1992) DOI 10.1007/BF02451417
[10] Xu, Y. J. Eigen-problem in fracture mechanics for a Reissner's plate. Acta Mechanica Solid Sinica, 25, 225-228 (2004)
[11] Reddy, J. N. A simple higher order theory for laminated composite plates. Journal of Applied Mechanics, 51, 745-752 (1984)
[12] Reddy, J. N. Analysis of functionally graded plates. International Journal for Numerical Methods in Engineering, 47, 663-684 (2000)
[13] Reddy, J. N. and Khdeir, A. A. Buckling and vibration of laminated composite plates using various plate theories. AIAA Journal, 27, 1808-1817 (1989)
[14] Erdogan, F. and Wu, B. H. The surface crack problem for a plate with functionally graded properties. Journal of Applied Mechanics, 64, 449-456 (1997)
[15] Li, Y. D., Jia, B., Zhang, N., Dai, Y., and Tang, L. Q. Anti-plane fracture analysis of a functionally gradient materials infinite strip with finite width. Applied Mathematics and Mechanics (English Edition), 27(6), 683-689 (2006) DOI 10.1007/s10483-006-0608-z
[16] Huang, G. Y., Wang, Y. S., and Yu, S.W. A new multi-layered model for in-plane fracture analysis of functionally graded materials. Acta Mechanica Sinica, 37, 1-8 (2005)
[17] Cheng, Z. Q. and Zhong, Z. Fracture Analysis of a functionally graded strip. Chinese Quarterly of Mechanics, 19, 114-121 (2006)
[18] Butcher, R. J., Rousseau, C. E., and Tippur, H. V. A functionally graded particulate composite: preparation, measurements and failure analysis. Acta Materialia, 47, 259-268 (1999)
[19] Delale, F. and Erdogan, F. The crack problem for a nonhomogeneous plane. Journal of Applied Mechanics, 50, 609-614 (1983)
[20] Jin, Z. H. and Noda, N. Crack-tip singular fields in nonhomogeneous materials. Journal of Applied Mechanics, 61, 738-740 (1994)
[21] Dai, Y., Zhang, L., Zhang, P., Li, S. M., Liu, J. F., and Chong, X. The eigen-functions of anti-plane crack problems in non-homogeneous materials. Science China: Physica, Mechanica & Astronomica, 8, 852-860 (2012)
[22] Liu, C. T. and Li, Y. Z. Stress strainfields at crack tip and stress intensity factors in Reissner's plate. Acta Mechanica Sinica, 16, 351-362 (1984)