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Engineering, 2011, 3, 517-524 doi:10.4236/eng.2011.35060 Published Online May 2011 (http://www.SciRP.org/journal/eng) Copyright © 2011 SciRes. ENG Analytical Solution for Bending Stress Intensity Factor from Reissner’s Plate Theory Lalitha Chattopadhyay National Aerospace Laboratories, Bangalore, India E-mail: lalitha@nal.res.in Received December 30, 2010; revised January 27, 2011; accepted April 7, 2011 Abstract Plate-type structural members are commonly used in engineering applications like aircraft, ships nuclear re- actors etc. These structural members often have cracks arising from manufacture or from material defects or stress concentrations. Designing a structure against fracture in service involves consideration of strength of the structure as a function of crack size, dimension and the applied load based on principles of fracture me- chanics. In most of the engineering structures the plate thickness is generally small and in these cases though the classical plate theory has provided solutions, the neglect of transverse shear deformation leads to the limitation that only two conditions can be satisfied on any boundary whereas we have three physical bound- ary conditions on an edge of a plate. In this paper this incompatibility is eliminated by using Reissner plate theory where the transverse shear deformation is included and three physically natural boundary conditions of vanishing bending moment, twisting moment and transverse shear stress are satisfied at a free boundary. The problem of estimating the bending stress distribution in the neighbourhood of a crack located on a single line in an elastic plate of varying thickness subjected to out-of-plane moment applied along the edges of the plate is examined. Using Reissner’s plate theory and integral transform technique, the general formulae for the bending moment and twisting moment in an elastic plate containing cracks located on a single line are derived. The thickness depended solution is obtained in a closed form for the case in which there is a single crack in an infinite plate and the results are compared with those obtained from the literature. Keywords: Reissner Plate Theory, Integral Transform, Stress Intensity Factor, Singular Integral Equation. 1. Introduction In the classical theory of bending of thin plates, it is pos- sible to satisfy stress-free conditions at an edge only in an approximate way, since only two boundary conditions may be enforced in connection with the bi-harmonic dif- ferential equation. It is the purpose of this paper to ex- amine the crack problem by using the theory of bending of elastic plates developed by Reissner[1] in which the three physically natural boundary conditions of vanish- ing bending moment, twisting moment and transverse shear stress must be satisfied at a free boundary. The present problem is concerned with the problem of an infinite plate under uniform uniaxial bending far from the crack (Figure 1). In the present work the complete solution is obtained for bending stresses in the vicinity of a crack tip in a plate taking transverse shear deformation into account through the use of Reissner’s plate theory. Using Reissner’s theory and integral transform tech- niques, the general formulae for the bending mo- ment,twisting moment and bending stress distribution in an elastic plate containing cracks located on a single line are derived. The procedure employed here is to formulate the problem in terms of Reissner’s equations. The stress in- tensity factor is obtained for the case in which there is a single crack in an infinite plate and the results are com- pared with those given in the literature. The mechanical behavior near the crack tip is modeled using Reissner’s plate theory in the case of an elastic plate in [2,3].Effect of plate thickness on the bending stress distribution is included by Hartranft and Sih [2]. The general solution for bending stress in the vicinity of a crack tip in a plate taking shear deformation into ac- count through the use of a sixth order plate bending the- ory of Reissner’s theory is developed by Viswanath [3]. L. CHATTOPADHYAY 518 Figure 1. Plate containing a single crack and subjected to symmetric bending load. The solution of the thin plate-bending problem was pio- neered by Williams [4], who made use of the eigen func- tion expansion technique and determined the stress dis- tribution in the neighborhood of a crack. Sih et al. [5] applied a complex variable method for evaluating the strength of stress singularities at crack tips in plate ex- tension and bending problems. The general solution for finite number of cracks using anisotropic elasticity is presented by Krenk[6]. Alwar and Ramachandran [7] showed that the stress intensity factor is nearly linear through the thickness for thin plates, in the absence of crack closure. Using finite element method, Mark et al. [8], Alberto Zucchini et al. [9] computed stress intensity factors for thin cracked plates. Using complex variable method Zehnder et al. [10] calculated stress intensity factor for a finite crack in an infinite isotropic plate. The present method uses an integral transform technique and does not assume any symmetry about the co-ordinate axes. Also the constants appearing in the solution of the governing differential equations are obtained from the displacement boundary conditions by defining the de- rivative of the displacement discontinuities on the crack surfaces apart from the moment boundary conditions and continuity conditions. In the present study, the general formulae for the stress distribution in an infinite elastic plate containing cracks are derived and the stress inten- sity factor is determined in a closed form in the case of a single crack when the plate is subjected to out-of-plane moments and the results are compared with those from the literature. 2. Formulation of the Problem Let us consider the cases of bending or twisting actions of an infinite plate by moments that are uniformly dis- tributed along the edges of the plate containing collinear cracks. We take xy-plane to coincide with the middle plane of the plate before deformation. The z-axis is as- sumed to be perpendicular to the middle plane. We de- note the bending moment per unit length about x-axis by Myy and about y-axis by Mxx and the twisting moment per unit length by Mxy. Let Qxand Qy be the shear forces components. The thickness of the plate is h and we con- sider it to be small in comparison with other dimensions. Let us assume that during bending, the plate undergoes the displacement w perpendicular to xy-plane. In the present analytical method, we consider the problem in which an infinite elastic plate, contains cracks located on a single line is acted upon by applied moments. Let the co-ordinate system be so chosen that the x-axis coincides with the line on which the cracks are located. Let L de- note the union of intervals occupied by the cracks on the x-axis and M is the interval not occupied by the cracks. Suppose that a thin plate containing a crack is subjected to uniform bending or twisting moments at infinity. Since the crack surface is traction-free the boundary conditions along the crack surface permitting all of the free edge conditions for the present problem is given by the following equations: The free boundary conditions on the crack surface are given by, ,0 0, xy M xxL (1) ,0 0, yy M xxL (2) ,0 0, y Qx xL (3) The solution to this problem may be obtained by su- perposing the simple solution of an uncracked plate un- der uniform bending moment or twisting moment to that of a cracked plate with bending or twisting moment ap- plied to the crack surfaces. That is, the solution may be obtained by using standard superposition technique and thus for the purpose of evaluating the crack tip singular stresses it is sufficient to consider the problem in which self-equilibrating crack surface loads are the only exter- nal loads. Thus, it suffices to solve the problem of speci- fying uniform bending and twisting moment on the crack segment of the plate. Let the desired system be com- posed of two parts, one the uniform moment field and the other a perturbation field due to the crack which dies out at infinity. While the boundary conditions along the free edges of the crack require traction free conditions, it is possible to formulate the problem as one of finding solu- tions for the perturbation solutions satisfying the field equations and the boundary conditions * ,0 , 2 xy Gx M xxL (4) Copyright © 2011 SciRes. ENG L. CHATTOPADHYAY519 * ,0 , 2 yy Hx M xxL (5) ,0 0 y Qx The equilibrium equations are given by, 0 xy y y MM Q xy (6) 0 yx x x MMQ yx (7) 0 y xQ Q xy (8) Also, the stress components are the linear combination of the variable z. 33 12 12 12 σ;σ;σ xy yy 3 x x xy yyxx Mz Mz M z hh h (9) The strain compatibility equation is given by, 22 2 22 yy xy xx x y yx (10) If we define the moment resultants in terms of the bi- harmonic function ),( yx as given by 22 2 22 ;; xyxy MMM x y yx (11) then the governing Equations (6)-(8) are satisfied. From the compatibility conditions (10), the present problem reduces to that of solving the bi-harmonic equation in (, ) x y 40 (12) where, 44 4 422 2 4 4 x xy y (13) Let be the Fourier transform of 1(, )G y , x y for . Then 0y 11 ,,ed, ix Gy xyxy 0 (14) Taking Fourier transformation of the bi-harmonic equation w. r. t the variable x, we get the ordinary differ- ential equation in as given by ,Gy 2 2 2 2 d, dGy y 0 (15) whose solutions are given by 1 11 ,e y GyP yQy 0 (16) 2 22 ,e y GyP yQy where is the Fourier transformation of 2 G , x y for 0y , and 12 1 2 ,,,PPQQ are the unknown functions to be determined. From the moment boundary conditions we have the following equations, 12 ,0,0 0, yy yy M xMx x (18) 12 ,0,0 0, xy xy M xMx x (19) The bending and twisting moments in terms of for are given by 1,G y0y 1 2 1 2 , 1 ,e 2π ix x Gy Mxy y y d,0 (20) 11 2 1 ,,e 2π ix y MxyGyy d,0 (21) 1 1, ,e 2π ix xy Gy i Mxy y y d,0 (22) Similarly we get the bending and twisting moments for 0y in terms of 2,Gy The bending and twisting moments in terms of for 2,G y0y are given by 2 2 2 2 , 1 ,e 2π ix x Gy Mxy y y d,0 (23) 22 2 1 ,,e 2π ix y MxyG yy d,0 (24) 2 2, ,e 2π ix xy Gy i Mxy y y d,0 (25) The displacement components are given by the follow- ing expressions: x w uz x (26) y w uz y (27) The displacement boundary conditions are given by 0, A xxM (28) 0,Bxx M (29) where the displacement discontinuities are defined by the functions A(x), B(x) 12 ,0,0 , xx A xuxuxx x L (30) 0 (17) 11 ,0,0 , yy Bxuxuxx L x (31) Copyright © 2011 SciRes. ENG L. CHATTOPADHYAY Copyright © 2011 SciRes. ENG 520 and the superscripts (1) and (2) denote the components in the upper half plane and lower half plane 0y0y respectively. From the continuity conditions and the moment boundary conditions we get the four simultaneous equa- tions for solving 12 1 2 ,,,PPQQ 2 1 PP (32a) 112 QPQ P 2 (32b) 2 2 112 2 12PPQQA D1 (32c) 2 112 2 13iPPiQQBD1 (32d) Solving the above equations we obtain the un- knowns as given by, 12 1 2 ,,,PPQQ 2 , y , M yG y (34a) 2 2 , xx G My y (34b) 2 12 2sgn 1 4 i PP BD (33a) , xy G Myi y (34c) 2 1 1sgn 1 4 QA iBD (33b) Substituting the values of 11 ,PQ into (21), we get the bending moment resultants in the upper half plane , in the transformed co-ordinates as given by the following equation, 0y 2 2 1sgn 1 4 QA iBD (33c) 2 11 ,sgn1e 8π ix y y D MxyAy Biyy d,0 (35a) Performing the inner integral in terms of A(s) and B(s) we get the bending moment resultants in the upper half-plane , in terms of the unknown displacement functions A(s) and B(s) as given by 0y 22 22 2 1 2 2222 3 1 ,d 4π y yxsxs y D MxyAsys Bsxssy xs yxs y d0 (35b) Substituting the values of 22 ,PQ into (24), we get the bending moment resultants in the lower half plane 0y , in the transformed coordinates as given by the following equation 2 21 ,sgn1e 8π ix y y D MxyAy Biyy d,0 (35c) Performing the inner integral in terms of A(s) and B(s) we get the bending moment resultants in the lower half-plane 0y , in terms of the unknown displacement functions A(s) and B(s) as given by 22 22 2 2 2 2222 3 1 ,d 4π y yxsxs y D MxyAsys Bsxssy xs yxs y d0, (35d) Combining the Equations (35b) and (35d) we get the expression for bending moment resultant, 22 22 2 2 2222 3 1 ,d 4π y yxsxs y D MxyAsys Bsxssy xs yxs y d0, (36) L. CHATTOPADHYAY Copyright © 2011 SciRes. ENG 521 Similarly the expression for the twisting moment , xy M xy as given by 22 22 2 22 22 22 3 1 ,d 4π xy xs yxs y D MxyAsxss Bsysy xs yxs y d0, (37) where () A sand are the unknown functions to be determined from the given boundary conditions. The limiting values as y 0 and y 0 of the bending and twisting moments along the crack line are given by, ()Bs 2 1 ,0 d 4π y DBs M xs x s (38a) 2 1 ,0 d 4π xy D A s M xs x s (38b) By using the conditions (4)-(5) in the above expres- sions, the interval of integration reduces to L. From the boundary conditions (4) and the above relations we get the singular integral equations 1 d L L As , s GxxL xs (39) 1 d L L Bs , s HxxL xs (40) where 2 1 2π D , for the determination of un- known functions A and B on the interval L. Once the functions A(s) and B(s), are known, the bending and twisting moments for the crack problem are deter- mined. Lx 3. Single Crack Problem In order to illustrate the present procedure, we give the details in the case of a single crack opened by the action of symmetric bending load applied at the edges of the plate. In this section, we consider the problem of deter- mining the distribution of bending stress in the vicinity of a Griffith crack of length 2c, occupying the interval (- c, c) on the x-axis in an infinite isotropic elastic plate. The bending moment resultants and the transverse shear force components are given by, 222 22 5 x x Q wwh MD x xy (41) 222 22 5 y y Q wwh MD y yx (42) 22 22 1 1 x ww QD x y xy (43) 22 22 1 1 y ww QD yx xy (44) Taking Fourier transform of the above equations w. r. t. x, we get the displacement component in terms of the bending moment components in the transformed co-ordinate system as given by, 2 2 21 x y MM w wD y (45) The transverse shear force y Q are given by, 1 xy y MM Q yi (46) where is calculated from the equation, 2 10 0 yh (47) Taking Fourier Transform of the above Equation (47) we get 2 22 20 d ydy (48) 22 22 22 22 12 ee yy cc (49) where 10 h Since and as we have 0 x Q0 y Qy 20c 22 22 1ey c (50) The constant 0is determined from ,0 0, 1 y xy y Qxxc MM Qi y (51) Substituting , x M y , , y M y and ,y in (46) and using the crack surface boundary condition, ,0 0, y Qxx c L. CHATTOPADHYAY 522 the constant is given by, 2 1 1 2π D C B (52) From (50), (52) the function , x y is given by, 22 1 ,e 2π c ixs y c Cdd x yBs e s (53) 22 2 22 1 1 ,d 8π c xsy c Dy x yBsKxs ys (54) 22 2 2 22 1 2 22 0 0 ,1 d 58π5 c c y y Kxsy xy D hh Bs s xyx xs y (55) 2 2 22 2 2 0 ,1 d 58π5 c c y xy DBs hh s Kx xyx s (56) 2 2 2 2 2 22 0 ,1 12 d 00 58π52 c c y xy DBsLt Lt hh s hh xyx sx (57) 2 2 2 0 ,1 d 058π c c y x yD BsLt h s h x yx s (58) The bending moment ,0 y M x along the crack line 0y , as is given by 0h 22 11 ,0 dd 4π8π cc y cc DBs DBs M xs xs xs s (59) Making use of the boundary condition (4) we get the singular integral equation for the determination of B(s) as given by, 2 13d 8π12 c c DBsHx s xc xs (60) Solving for B(s), we get the following expression for B(s), 222 22 114 31π d c c Bs Das Hxctx xs (61) Substituting the value of B(s) in (38a), the bending moment for a single crack problem in the limiting case as is given by, 0h 22 22 * 0 sgn 1 ,0 d 3 2π 2 c y c Hxc t x M xt tx xc Hx M (62) The bending moment resultant along the crack line from the above equation is given by, 0 22 sgn 1 ,0 3 y xx M xM xc (63) The bending stress ,0 y x along the crack line 0y , as is given by 0h 0 322 12 sgn1 ,0 , 3 yy Mz xx x xc hxc (64) The bending stress intensity factor KI due to bending moment at z = h/2 is given by 0 2 1 6 2,03 Iyy Lim Mc Kxcx xc h (65) Copyright © 2011 SciRes. ENG L. CHATTOPADHYAY Copyright © 2011 SciRes. ENG 523 For small h a, the bending moment on the crack line y = 0 is calculated as follows: 2 2 2 2 0 ,1 d 58π c c y x yD Bs h K xs x yx where 222 110 ln;; 0.5772 2 x Kx h x s (66) Hence the bending moment along the crack line y=0 is given by, 22 22 111 ,0d lnd 4π8π2 c y c DBs DBs x M xs s x sx x s (67) Substituting the value of B(s) from (61) into the above equation and performing the inner integral we get the bending moment along the crack line y = 0 as given by, 2 0 22 22 1 sgn 11 ,0 2ln, 3413 y xxM M xx x xc xc (68) The bending stress ,0 y x along the crack line 0y , is given by 2 0 22 322 1 12sgn 11 ,02ln , 3413 y xxzM x x x hxc xc (69) Stress intensity factor KI at z = h/2 ,due to bending mo- ment is given by 2 0 22 1 1 61 lim[2],02ln 3413 Iyy xc Mc Kxcx c hc 2 (70) The graph of non-dimensional stress intensity factor vs. thickness for 0.3 is plotted in Figure 2 and the stress intensity factor is in good agreement with the results in [2] and [3]. The stress variation near the crack tip, calcu- lated from Equation (69) is plotted in Figure 3. For ex- ample, a value of c/h = 5.0 is assumed for calculation. Figure 2. Variation of non-dimensional stress intensity with thickness of the plate. 4. Results and Discussion The variation of non-dimensional stress intensity stress intensity factor with thickness of the plate is shown in Figure 2. The small differences between the present re- sults and in the references [2] and [3] may be due to two Figure 3. Stress ,0 yx distribution near the crack tip for x c, c/h = 5.0. L. CHATTOPADHYAY 524 different approaches being used in [2] and [3]. Hartranft and Sih [2] used more rigorous method using eigenfunc- tion expansions for plate bending problem introducing the effect of plate thickness on crack-tip stress distribu- tion. The approximate method based partly on finite element analysis and partly on a continuum analysis us- ing Irwin’s [11] solution for an infinite plate is used in [3]. Figure 3 shows the exponential variation of normal stress component near the crack tip for x c, z = h/2. It decreases away from the crack tip as expected. Future work in this direction is planned to solve composite plate problems with delamination. 5. Conclusions A simple method for determining the analytical expres- sion for the bending stress distribution, in the vicinity of a crack in an infinite elastic plate using Reissner plate theory is explained. The general formulae for the bend- ing moment and twisting moment in an elastic plate con- taining cracks located on a single line are derived. The solution is obtained in a closed form for the case in which there is a single crack in an infinite plate and the stress intensity factor is calculated as a function of plate thickness, when the plate is subjected to symmetric bending loads. The stress intensity factor is compared with that obtained in the literature. 10. References [1] E. Reissner, “The Effect of Transverse Shear Deformation on the Bending of Elastic Plates,” ASME Journal of Ap- plied Mechanics, Vol. 12, 1945, pp. A68-A77. [2] R. J. Hartranft and G. C. Sih, “Effect of Plate Thickness on the Bending Stress Distribution around Through Cracks,” Journal of Mathematics and Physics, Vol. 47, 276-291,1968 [3] S. Viswanath, “On the Bending of Plates with through Cracks from Higher Order Plate Theories,” Ph. D Thesis, Indian Institute of Science, 1985. [4] M. L. Williams, “The Bending Stress Distribution at the Base of a Stationary Crack,” ASME Journal of Applied Mechanics, Vol. 28, 1961, pp. 78-82. [5] G. C. Sih, P. C. Paris and F. Erdogan, “Crack-Tip, Stress-Intensity Factors for Plate Extension and Plate Bending Problems,” ASME Journal of Applied Mechanics, Vol. 9, 1962, pp. 306-312. [6] S. Krenk, “The Stress Distribution in an Infinite Anisot- ropic Plate with Collinear Cracks,” International Journal of Solids and Structures, Vol. 11, No. 4, 1975, pp. 449-460. doi:10.1016/0020-7683(75)90080-3 [7] R. S. Alwar and K. N. Ramachandran, “Influence of Crack Closure on the Stress Intensity Factor for Plates Subjected to Bending—A 3-D Finite Element Analysis,” Engineering Fracture Mechanics, Vol. 17, No. 4, 1983, pp. 323-333. doi:10.1016/0013-7944(83)90083-8 [8] M. J. Viz, D. O. Potyondy, A. T. Zehnder, C. C. Rankin and E. Riks, “Computation of Membrane and Bending Stress Intensity Factors for Thin, Cracked Plates,” Inter- national Journal of Fracture, Vol. 72, No. 1, 1995, pp. 21-38. doi:10.1007/BF00036927 [9] A. Zucchini, C.-Y. Hui and A. T. Zehnder, “Crack Tip Stress Fields for Thin Plates in Bending, Shear and Twisting: A Three Dimensional Finite Element Study,” International Journal of Fracture, Vol. 104, No. 4, 2000, pp. 387-407. [10] A. T. Zehnder and C.-Y. Hui, “Stress Intensity Factors for Plate Bending and Shearing Problems,” Journal of Ap- plied Mechanics, Vol. 61, No. 3, 1994, pp. 719-722. doi:10.1115/1.2901522 [11] G. R. Irwin, “Analysis of Stresses and Strains near the End of a crack Traversing a Plate,” Journal of applied Mechanics, 1957, Vol. 24, pp. 361-364. Copyright © 2011 SciRes. ENG |