Growth simulation for 3D surface and through-thickness cracks using SGBEM-FEM alternating method

Transcription

1 Journal of Mechanical Science and Technology 5 (9) (011) 335~344 DOI /s Growth simulation for 3D surface and through-thickness cracks using SG-FEM alternating method Jai Hak Park 1,* and Gennadiy P. Nikishkov 1 Department of Safety Engineering, Chungbuk National University, Chungbuk , Korea Department of Computer Science and Engineering, The University of Aizu, Fukushima , Japan (Manuscript Received August 3, 010; Revised April 6, 011; Accepted April 9, 011) Abstract An SG-FEM alternating method had been proposed by Nikishkov, Park and Atluri for the analysis of three-dimensional planar and non-planar cracks and their growth. The proposed method is an ective method for fatigue or stress corrosion crack growth simulation. During crack growth simulation, however, an oscillation phenomenon is observed in crack advance or stress intensity factor distribution. If oscillating amplitude in SIF or crack advance does not decrease during next increment steps, the crack growth simulation fails. In this paper several methods are examined to remove the oscillation phenomenon. As a result, it is found that smoothing in stress intensity factor distribution or in crack front geometry can remove or weaken the oscillation phenomenon. Using the smoothing techniques, stress corrosion crack growth simulation is performed for a semi-elliptical surface crack and a through-thickness crack embedded in a plate. Crack front shape and stress intensity factor distribution are obtained after each increment during the crack growth. And the depth and length of a crack are obtained as a function of time. It is noted that the SG-FEM alternating method is a very ective method for SCC growth simulation for a surface crack and a through-thickness crack. Keywords: Symmetric Galerkin boundary element method; Finite element method; Alternating method; Stress corrosion cracking (SCC); Threedimensional crack Introduction This paper was recommended for publication in revised form by Associate Editor Chongdu Cho * Corresponding author. Tel.: , Fax.: address: jhpark@chungbuk.ac.kr KSME & Springer 011 For several decades, the Shwartz-Neumann alternating technique has been developed for three-dimensional cracks [1-4]. Nikishkov, Park and Atluri [5] proposed an SG-FEM alternating method to analyze planar or non-planar threedimensional cracks in a finite body. They used the symmetric Galerkin boundary element method (SG) [6, 7] for modeling a crack embedded in an infinite body. To perform fatigue or stress corrosion crack growth simulation, fracture parameters such as stress intensity factor (SIF) should be obtained along the crack front during crack growth. The well-established finite element method can be used for the simulation, but it is difficult to model growing cracks due to complications related to generation and modifications of finite element mesh during crack growth. The proposed SG- FEM alternating method is a convenient method to perform crack growth simulation. Since the boundary element mesh (crack mesh) is independent of the finite element mesh, crack growth can be simulated by just changing the boundary element mesh. During crack growth simulation, however, an oscillation phenomenon is observed in crack advance and stress intensity factor distribution. The phenomenon occurs for the following reason. If a crack front point advances less than adjacent crack front points due to calculation error or local geometry, the SIF of the point becomes larger than the values of adjacent crack front points. So in the next increment, the crack front point advances more than other adjacent points. Then after the increment, the SIF of the point becomes less than other points. If oscillating amplitude in SIF or crack advance does not decrease during next increment steps, the crack growth simulation fails. Fig. 1 shows typical oscillation phenomenon in SIF distribution of a growing surface crack. We propose a methodology that removes the oscillation phenomenon in crack growth simulation. As a result, it is found that geometrical smoothing of crack front and smoothing in SIF distribution can remove or weaken the oscillation phenomenon. Starting from a small initial surface crack growth simulation is made until the crack becomes a large surface crack using the smoothing method. If a growing surface crack satisfies the crack growth instability criterion, unstable crack growth occurs and the crack becomes a through-thickness crack. Since a through-thickness crack also grows due to stress corrosion, it will be convenient

2 336 J. H. Park and G. P. Nikishkov / Journal of Mechanical Science and Technology 5 (9) (011) 335~344 Fig. 1. Typical oscillation phenomenon in SIF distribution. A crack growth procedure is based on the ective stress intensity factor. It is supposed that the crack grows in the direction of the J-integral and the crack growth rate is determined by the value of the ective stress intensity factor K. The value of K is calculated through the J-integral value. The crack front advancement is performed by adding a new boundary element layer to the existing crack model. Crack growth rate in the case of SCC is determined with an empirical model [1, 13] using the ective stress intensity factor. The developed crack growth procedure for threedimensional mixed mode cracks is implemented in the Java SG-FEM code. Examples of non-planar growth of semielliptical cracks under SCC conditions are presented. The SG equilibrium equation system for a crack in an infinite medium is written in the following matrix form: [ K ]{ U } = { T}. (1) Fig.. Typical crack mesh. to include SCC growth of a through-thickness crack for a complete crack growth simulation. Park, Kim and Nikishkov [8] examined the applicability of the SG-FEM alternating method to a through-thickness and found that accurate SIF can be obtained using the method. Starting from a small surface crack, SCC growth simulation is performed including the later stage of the growth of a through-thickness crack. From the growth simulation, crack shape and stress intensity factor distribution are obtained after each increment during the crack growth. The crack depth and crack length are obtained as a function of time.. Analysis method.1 SG-FEM alternating method In this paper, the modeling of SCC crack growth is performed using the combination of the finite element method (FEM) and the symmetric Galerkin boundary element method (SG) [6, 9, 10]. The uncracked structural component is discretized with finite elements. The crack is modeled with the SG. The alternating method [5, 11] is used to combine both methods using the superposition principle. The equilibrium state for the system of the structural component (FEM) and the crack (SG) is reached as a result of iterations that alternate between two methods. The crack is represented by a set of quadratic eight-node boundary elements. Singular boundary elements are used at the crack front. Fig. shows typical crack mesh for a semi-elliptical surface crack with the crack depth a and the crack length C. The lower mesh with the length L ext is located outside the body. The stress intensity factors K I, K II and K III at the crack front are calculated through displacements of nodes near the crack front. Here [ K ] is the global SG matrix, { U } is the global displacement discontinuity vector and { T } is the nodal equivalent of the crack surface forces. Eight-node quadratic boundary elements are employed for crack modeling. Elements can have curved edges and consequently, a curved surface. This allows representing arbitrary three-dimensional cracks with non-planar surface. Solution of the boundary value problem for a structural component with a crack is sought as superposition of the finite element solution (uncracked finite body) and the boundary element solution (crack in an infinite medium). For a correct superposition, fictitious forces on the boundary of the finite element model should be found in order to compensate for the stresses caused by the presence of a crack in an infinite body. These fictitious forces can be iciently found with the alternating procedure: (0) (0) (0) { UFEM } = {0}, { Ψ } = { P}, { U } = {0} do iterations 1 ( i 1) { Δ UFEM } = [ KFEM ] { Ψ } ( i 1) { UFEM } = { UFEM } + { ΔUFEM } { Δ σ FEM } = [ E][ B]{ ΔUFEM } Δ = Δσ FEM S 1 { Δ U } = [ K ] { ΔT } ( i) ( i 1) ( i) { U } = { U } + { ΔU } { Δ σ } = { Δσ({ ΔU })} { Ψ } = [ FEM ][ ]{ Δσ } S { T } [ N ][ n]{ } ds until Ψ / P < ε. N n ds Equilibrium superposition is determined as a result of iterations. Superscripts in parentheses denote iteration numbers. During iterations global matrices [ K FEM ] and [ K ] do not change. In the above relations, { Ψ } is the residual force vec- ()

3 J. H. Park and G. P. Nikishkov / Journal of Mechanical Science and Technology 5 (9) (011) 335~ J x x 1 α J J x 3 1 Fig. 4. Components J 1 and J of the J-integral at the crack front. It is supposed that the crack grows in the direction of the J-integral. Fig. 3. The global coordinate system X 1, X, X 3 and the crack front coordinate system x 1, x and x 3. tor, { σ } is the stress vector, [ B ] is the finite element displacement differentiation matrix, and [ E ] is the elasticity matrix. The iterative loop ends when the relative norm of the finite element residual becomes less than the specified error tolerance ε. After termination of the iterative procedure, correct tractions at the crack surface are determined thus making possible to estimate values of the stress intensity factors at the crack front.. Calculation of fracture mechanics parameters Once the displacement discontinuities are obtained at nodes, the stress intensity factors K I, K II and K III can be easily determined using the following relations [14]: K K K I II III = = E u 3 (1 ν ) 4 E π π r u (1 ν ) 4 r E π u1 = (1 + ν ) 4 r where E is the elasticity modulus; ν is Poisson s ratio; r is the distance from the point to the crack front; and u 1, u and u 3 are components of the displacement discontinuities at points on the crack surface in a local crack front coordinate system, x 1, x and x 3, which are illustrated in Fig. 3. The axis x 1 of the crack front coordinate system is parallel to the crack front, and the axis x 3 is normal to the crack surface. The following procedure for the stress intensity factor calculation is used in the current work: G (1) Obtain the displacement discontinuities ui in the global coordinate system for the quarter-point node and for the corner node of a singular crack front element. () Extrapolate u G i r to the crack front, using values at the quarter-point node (L/4) and at the corner node (L). Here r is the distance along the line normal to the crack front and (3) G ui components of displacement discontinuities in the global coordinate system. (3) Transform the extrapolated displacement discontinuities from the global coordinate system to the crack front coordinate system, ui = αijui where α ij G are direction cosines of the transformation. (4) Calculate the stress intensity factors using Eq. (3). 3. Modeling crack growth 3.1 Modeling of non-planar crack growth The SG-FEM alternating method is quite suitable for crack growth simulation. Since the crack is modeled separately, the finite element model need not be modified during crack growth. Only the boundary element model (crack model) should be changed during crack growth. For crack growth simulation of a non-planar crack, it is necessary to know the direction of crack growth and the amount of crack growth. The J-integral is used to determine the crack growth direction and the amount of crack growth as follows [8]: (1) Crack grows in the direction of J-integral vector as shown in Fig. 4; () Crack growth rate is determined by the ective stress intensity factor K based on the J-integral. In an elastic three-dimensional case, the J-integral components are evaluated using the stress intensity factors as: 1 ν 1+ ν ( I II ) III J1 = K + K + K E E 1 ν J = KIKII E J = J + J. 1 The crack growth angle α, which is the angle between the axis x 1 and the crack growth direction, is determined by the direction of J-integral vector: tan. J1 (4) J α = (5)

4 338 J. H. Park and G. P. Nikishkov / Journal of Mechanical Science and Technology 5 (9) (011) 335~344 It is worth noting that the J-integral vector is normal to the crack front. Hence, a point at the crack front moves in the plane normal to the crack front at the angle α, from the plane which is tangential to the crack surface. A typical crack growth model, suitable for fatigue or SCC crack growth simulation can be expressed using the ective stress intensity factor K as follows: K max Δa max K Δa da f( K ) dt = (6) where da/dt is the crack growth rate and K is related to the J- integral as: K = JE 1 ν. (7) Fig. 5. Advancement of the crack front. Points at the crack fronts are moved in the directions of the J-vector. Maximum crack advance Δ a. max x x Crack growth algorithm The following algorithm is used to model mixed mode SCC crack growth for a surface crack [8]: (1) Solve the boundary value problem for the current crack configuration using the SG-FEM alternating method. () Obtain the stress intensity factors K I, K II and K III for the element corner nodes located at the crack front and calculate the ective stress intensity factor K according to Eq. (7) max and select the maximum value K. (3) Estimate increment of the crack life by the following integration and accumulate the crack life t = t+δ t : a+δa da Δ t =. (8) f ( K ( a)) a (4) If no value of crack advance Δa max is left in the input data, then stop. (5) For each corner node, determine the crack front coordinate system by averaging the coordinate axis vectors determined at the corner point of two neighboring boundary elements. Also determine x 1, x and x 3 local coordinate system illustrated in Fig. 4. (6) For each corner node, calculate the crack growth angle α according to Eq. (5). (7) Determine crack advance Δa for the corner nodes at the crack front using the following equation (see Fig. 5): f( K ) Δ a =Δ amax (9) f K max ( ) (8) Move the corner nodes along the J-integral vector according to computed Δa values. (9) Transform the coordinates of the neighbor nodes into the x 1, x and x 3 coordinate system. Perform curve fitting for x 1 and x coordinates with respect to x 3 coordinate using a polynomial expression. Fig. 6. Geometrical crack front smoothing. (10) Using the fitted polynomial coicients, obtain the x 1 and x coordinates of the advance point. For the crack front points located outside the body, we use the same Δa and α as the values calculated at the nearest crack front point on the body boundary. (11) Find the locations of crack front midside nodes, using linear or cubic spline interpolation. (1) Shift the quarter-point nodes of the previous crack front elements to midside position. Put quarter-point nodes on element sides nearly normal to the crack front. (13) Generate one layer of boundary elements between old and new crack fronts. (14) Go to step (1). 3.3 Geometrical smoothing of crack front When crack growth simulation is performed using the alternating method, the oscillation phenomenon is observed in crack advance and SIF distributions as illustrated in Fig. 1. To eliminate the oscillation phenomenon, geometrical smoothing of crack front can be used. Fig. 6 illustrates the procedure of geometrical smoothing. Consider a procedure to find a new advancing crack front point A corresponding to the current crack front point A. First obtain the crack advance points corresponding to the current crack front points. The open circles in Fig. 6 denote the crack advance points. Next, transform the coordinates of the crack advance points into the x 1, x and x 3 coordinate system in Fig. 4, which is the local coordinate system at A. Note that the local coordinate system changes according to the current crack front point. Curve fitting is performed for the x 1 and x coordinates with respect to x 3 coordinate using a polynomial expression. Let the number of points used in the curve fitting be n fit. An even n fit value can remove the oscillating phenome-

5 J. H. Park and G. P. Nikishkov / Journal of Mechanical Science and Technology 5 (9) (011) 335~ non more ectively than an odd n fit value. In this study, the second order polynomial is used in the curve fitting and n fit = 6 is used. Using the fitted polynomials, the x 1 and x coordinates of A can be calculated easily. This procedure can be applied to planar and non-planar crack growth. Normally, the SIF value on the body boundary is less than the value inside the body. If we use the SIF values on the boundary in crack growth simulation without modification, it may be the source of the oscillation phenomenon. Instead of the obtained SIF value, an extrapolated SIF value using inner or 3 points can be used. It is also possible to perform crack front geometrical smoothing excluding the front points on the body boundary. In this case, the obtained SIF value on the body boundary is not used. (a) 3.4 Smoothing of stress intensity factor distribution Smoothing of the SIF distribution can be used instead of geometrical smoothing technique. After calculating SIF values for crack front corner nodes, we can modify the values through smoothing technique. To obtain a modified SIF value, curve fitting is performed for modes I, II and III SIF values. Consider a crack front point A, at which a modified SIF value is desired. First, choose n fit points including the point A and the nearest adjacent crack front corner points. Transform the coordinates of the chosen points into the x 1, x and x 3 local coordinate system of the point A. Perform curve fitting for the SIF values of the points with respect to the local x 3 coordinate using a polynomial expression. Using the obtained polynomials, the mixed mode SIF values at point A can be obtained easily. As in the geometrical smoothing, an even n fit value can remove the oscillating phenomena more ectively than an odd n fit value. (b) Fig. 7. Crack front shifting technique. (a) 3.5 Crack front shifting In the current technique, one crack front element layer is added to simulate the increase in crack size. The maximum increment can be controlled by using the input data. But if the SIF values for a part of crack front are much less than the maximum SIF value, a crack front element with short edge is generated and the simulation fails because of the poor element quality. To prevent generating an element with poor quality, crack shifting can be used as shown in Fig. 7. In the method, the crack front nodes are moved to the new position without generating a new crack front element layer. In Fig. 7(a), the open circles denote the new positions of crack front points. This technique also curbs the increase in number of boundary elements and eventually reduces the simulation time. When a crack front element with short edge is generated, the element quality can be improved by adjusting the element heights of two current crack front element layers as shown in Fig. 8. After generating a new crack front element layer, the coordinates of the previous crack front points are modified so that the two current element layers have nearly the same width. If this technique is used in conjunction with crack shifting, a small increment becomes possible in the crack growth simulation. 3.6 Crack growth material model As an example, consider crack growth under SCC conditions. Several material models for determining the SCC crack growth rate in the stainless steel-water systems have been developed [1, 13]. Currently for testing the developed crack growth procedure, we use the mechanochemical model proposed by Saito and Kuniya [1]. The model is represented by the following equation: ( ( ISCC )) /( 1) m da n+ = A 0 C1exp C C3 C4( K K ). dt (b) Fig. 8. Adjustment of the heights of two current crack front elements. (10)

6 340 J. H. Park and G. P. Nikishkov / Journal of Mechanical Science and Technology 5 (9) (011) 335~344 σ c β a h t W ϕ (a) (b) Fig. 9. Inclined semi-elliptical surface crack in a plate subjected to a uniform tensile stress. Location of points on the crack front is characterized by elliptical angle ϕ. Here, K is the ective stress intensity factor calculated through the J-integral value, A 0, C 1, C, C 3 and C 4 are material constants, K ISCC is the threshold stress intensity factor, n is the Ramberg-Osgood type strain hardening coicient, m is the parameter representing the ect of environment and material chemistry. Fig. 10. Example of finite element mesh and initial boundary element mesh (crack mesh) used in the study. Only half of the finite element mesh is plotted. 3.7 Through-thickness crack The initiated crack due to SCC is very small, but it can grow to a large surface crack with time. If a growing crack satisfies an instability criterion it becomes a through-thickness crack. For a long through-thickness crack two-dimensional SIF solutions can be used to simulate crack growth. But if the crack length is short or the crack front is not straight or normal to the surface, a three-dimensional SIF solution could be used. For this purpose, the SG-FEM alternating method is adequate. The accuracy of SG-FEM alternating method is fully examined for a surface crack [5, 15] and for a throughthickness crack [8] in other references. Park, Kim and Nikishkov [8] verified that accurate SIF values can be obtained and crack growth simulation can be performed for a throughthickness crack using SG-FEM alternating method. The crack growth algorithm for a through-thickness crack is also given in the reference. 4. SCC growth simulation for a semi-elliptical surface crack SCC growth simulation is performed for a plate with an initial inclined semi-elliptical surface crack. A plate has thickness t, width W and height h as illustrated in Fig. 9. It is subjected to tension with surface intensity. Initial semielliptical crack with aspect ratio a/c is located at the center of the specimen surface and oriented under angle β to the horizontal plane. Plate and loading parameters have the following values: t = 0. m, W = 0.64 m, h = 0.6 m, σ = 00 MPa. And crack parameters are: a = 0.01 m, a/c = 0.5, β = 0, 15, Fig. 11. Initial boundary element mesh (crack mesh). 45. The body is assumed as an elastic material with elastic modulus E = 10 GPa and Poisson s ratio ν =0.3. The SCC Saito-Kuniya model Eq. (10) is used for prediction of SCC crack growth with the following parameters: A 0 = , C 1 = , C =1.9199, C 3 =3.0, C 4 =0.15, K ISCC =9.0 MPa m 1/ and n=5. These values are those for type 304 stainless steel in water at 88 C and the unit of da/dt is m/s in Eq. (10). Fig. 10 shows an example of finite element mesh and boundary element mesh (crack mesh). Only a half of the finite element mesh is plotted. In the finite element mesh, 4,907 nodes and node three-dimensional solid elements are used. Fig. 11 shows boundary element mesh for an initial crack. The boundary element mesh consists of 57 nodes and 7 8-node boundary elements. To represent stress singularity at the crack front, the midside nodes are moved to the quarter positions in crack front elements. In Fig. 11 the lower element layer with the height of L ext is the fictitious portion of the boundary element mesh and located outside the body. The fictitious portion improves the accuracy of SIF solution. The run time is 73 sec for one increment on an Intel 3 MHz personal computer for the typical model in Fig. 10. The run time is strongly dependent on the size of SG model. The specified da max value has an ect on the accuracy and stability of the simulation. If an inaccurate SIF value is obtained, it is necessary to adjust the da max value, or to use the crack front shifting technique.

7 J. H. Park and G. P. Nikishkov / Journal of Mechanical Science and Technology 5 (9) (011) 335~ Fig. 14. Boundary element mesh after 15 increments for a crack with β = 0. Fig. 1. Normalized SIF distributions during the crack growth for an initial crack with β =0. The geometrical smoothing technique is used. Fig. 13. Normalized increments of crack depth and half crack length as a function of time. The geometrical smoothing is used. Fig. 15. Normalized SIF distributions during the crack growth for an initial crack with β =0. The SIF smoothing technique is used. 4.1 Planar crack growth First planar crack growth is considered. The problem is the case when β = 0 in Fig. 9. Fifteen crack advances are performed with specified da max values, whose values are m, m, m 10 and m 3. Fig. 1 shows the Mode I SIF distributions during the crack growth when the geometrical smoothing is used. The SIF is normalized by K 0, where K 0 is [14]: π a K0 = σ (11) Q 1.65 a Q = c (1) After each increment the normalized SIF is plotted as a function of angle θ. Angle θ is defined by θ = arctan( y/ x). For the initial crack, the SIF values at the deep points are greater than the values at the points near surface. As the crack grows, the SIF values become nearly constant excluding the values on the surface. During the crack growth, Mode II and Mode III SIF maintain very small value. The maximum Mode II or Mode III SIF is less than 1% of Mode I SIF. Fig. 13 shows the variation of Δa/a and Δc/a as a function of time. Δa and Δc are the total increments in the crack depth and the half crack length respectively. In the initial stage the growth rate of Δa/a is larger than that of Δc/a. As time goes on, the two growth rates become equal. Fig. 14 shows the final Fig. 16. Normalized ective SIF distributions during the crack growth for an initial crack with β =15. The geometrical smoothing technique is used. boundary element mesh after 15 increments. Fig. 15 shows the Mode I SIF distributions during the crack growth when SIF smoothing is used. Each curve represents SIF distribution before smoothing after each increment. So each SIF distribution becomes smoother after the smoothing procedure. On the boundary, the SIF values obtained by linear extrapolation using two internal points are used in crack growth simulation. 4. Non-planar crack growth Next consider SCC growth simulation for an inclined initial surface crack with β = 15 in Fig. 9. Fifteen crack advances are performed with specified da max values, whose values are m, m, m 8, m and m 3. Fig. 16 shows normalized ective SIF, K /K 0 distributions

8 34 J. H. Park and G. P. Nikishkov / Journal of Mechanical Science and Technology 5 (9) (011) 335~344 Fig. 17. Normalized Mode II and III SIF distributions during the crack growth for an initial crack with β =15. The geometrical smoothing technique is used. Fig. 19. Normalized ective SIF distributions during the crack growth for an initial crack with β =45. The geometrical smoothing technique is used. Fig. 18. Boundary element mesh after 15 increments for a crack with β = 15. during the crack growth. Here, K is calculated using Eq. (7) and K 0 is defined by Eq. (11). In the initial stage of growth the SIF at deep points are larger than those of the points near the surface. But as the crack grows, the SIF values become equal at all points except the points on or just near the surface. Normalized Mode II and Mode III SIF distributions during the first three increments are given in Fig. 17. Before growth, the maximum normalized Mode II SIF is During the next three increments, the values become -0.09, and It can be noted that the magnitude of Mode II SIF decreases as the crack grows. Mode III SIF also decreases as the crack grows, but the decreasing rate is small. Fig. 18 shows the final boundary element mesh after 15 increments. Initially inclined crack surface changes to almost a flat surface which is normal to the loading direction as the crack grows. Figs. 19 and 0 show normalized ective and Mode I SIF distributions during the crack growth for an initial inclined surface crack with β = 45. The specified da max values are the same as in the case when β = 15. Fig. 1 shows normalized Mode II and Mode III SIF distributions during the first three increments. The magnitude of Mode II SIF decreases rapidly as the crack grows, but Mode III SIF maintains nearly the same value during the first three increments. According to the results, the Mode III SIF decreases to a small value very slowly as the crack grows. Fig. shows the final boundary element mesh after 15 increments. Fig. 0. Normalized Mode I SIF distributions during the crack growth for an initial crack with β =45. The geometrical smoothing technique is used. Fig. 1. Normalized Mode II and III SIF distributions during the crack growth for an initial crack with β =45. The geometrical smoothing technique is used. Fig.. Boundary element mesh after 15 increments for a crack with β = 45.

9 J. H. Park and G. P. Nikishkov / Journal of Mechanical Science and Technology 5 (9) (011) 335~ Fig. 3. SCC growth model used in this study. Fig. 5. Final boundary element mesh for a growing surface crack. Fig. 6. Final boundary element mesh for a growing through-thickness crack. Fig. 4. FEM mesh and boundary element mesh (crack mesh) for a through-thickness crack. Only a half of the finite element mesh is plotted. 5. SCC growth simulation from a surface crack to a through-thickness crack If a growing surface crack satisfies an instability criterion, unstable crack growth occurs and it becomes a throughthickness crack. For complete SCC growth simulation the growth of a through-thickness crack should be included. Starting from a small semi-elliptical surface crack SCC growth simulation is performed until the crack becomes a throughthickness crack with moderate length. Net section yielding or tearing modulus can be used as an instability criterion [14]. But in this study, it is assumed that if the crack depth reaches 80% of the thickness it becomes a through-thickness crack. Fig. 3 shows an initial shape of a through-thickness with unequal surface lengths a 1 and a. For a conservative analysis, we can use the assumption, a = a 1 = c f for an initial through-thickness. Here c f is the half crack length of the surface crack when the instability criterion is satisfied. In this study we assume that a 1 = c f, a = 0.5 a 1. A plate has thickness t, width W and height h. It is subjected to tension with surface intensity σ. Plate and loading parameters have the following values: t = 0.05 m, W = 0.64 m, h = 0.6 m, σ = 00 MPa. And the initial crack parameters are: a = 0.01 m, a/c = 0.5. The finite element mesh and boundary element mesh for a surface crack are the same as in Figs. 10 and 11. Fig. 4 shows finite element mesh and boundary element mesh (crack mesh) for a through-thickness crack. Fig. 7. Variation of normalized crack size as a function of time. Only a half of the finite element mesh is plotted. In the finite element model, 4,907 nodes and node threedimensional solid elements are used and in the crack mesh, 789 nodes and 40 8-node boundary elements are used. Fig. 5 shows the final boundary element mesh when the crack depth of the surface crack reaches 80% of the thickness. The final half crack length c f is m. Fig. 6 shows the final boundary element mesh for the through-thickness crack. Only the mesh inside of the body is plotted. The shaded area represents the boundary element mesh for the initial throughthickness crack. Fig. 7 shows variation of normalized crack size as a function of time. For the crack size parameter, crack depth is chosen for a surface crack and half crack length at the middle of the plate is chosen for a through-thickness crack. The crack size is normalized by the thickness t. The discontinuity in the curve is because the crack is changed from a surface crack to a through-thickness crack. It is noted that the crack growth rate gradually increases with time. 6. Conclusion Crack growth simulation is performed using the SG- FEM alternating method. It is found that the oscillation phe-

10 344 J. H. Park and G. P. Nikishkov / Journal of Mechanical Science and Technology 5 (9) (011) 335~344 nomenon can be removed using geometrical smoothing of crack front or smoothing of stress intensity factor distribution. For complete SCC growth simulation, the growth of a through-thickness crack is also considered. Starting from a small initial surface crack, crack growth simulation is made in a stainless steel plate until the crack becomes a large surface crack or a through-thickness crack. Crack shape and stress intensity factor distribution are obtained after each increment during the crack growth. The crack depth and crack length are obtained as a function of time. It is found that the SG- FEM alternating method can be used as an ective method to analyze SCC growth simulation for a surface crack and a short through-thickness crack. References [1] S. N. Atluri, Structural integrity and durability, Tech Science Press, Forsyth, GA, [] T. Nishioka and S. N. Atluri, Analytical solution for embedded elliptical cracks and finite element alternating method for elliptical surface cracks, subjected to arbitrary loadings, Eng. Fract. Mech., 17 (1983) [3] K. Vijaykumar and S. N. Atluri, An embedded elliptical crack, in an infinite solid, subject to arbitrary crack-face tractions, J. Appl. Mech., 103 (1) (1981) [4] Z. D. Han and S. N. Atluri, SG (for cracked local subdomain)-fem (for uncracked global structure) alternating method for analyzing 3D surface cracks and their fatiguegrowth, Computer Modeling in Engineering & Sciences, 3 (6) (00) [5] G. P. Nikishkov, J. H. Park and S. N. Atluri, SG-FEM alternating method for analyzing 3D non-planar cracks and their growth in structural components, Comp. Modeling in Engng & Sci., (3) (001) [6] M. Bonnet, G. Maier and C. Polizzotto, Symmetric Galerkin boundary element methods, Appl. Mech. Rev., 51 (1998) [7] S. Li and M.E. Mear, Singularity-reduced integral equations for displacement discontinuities in three-dimensional linear elastic media, Int. J. Fract., 93 (1998) [8] J. H. Park, M. W. Kim and G. P. Nikishkov, SG-FEM alternating method for simulating 3D through-thickness crack growth, Computer Modeling in Engineering & Sciences, 68 (010) [9] S. Li, M. E. Mear and L. Xiao, Symmetric weak-form integral equation method for three-dimensional fracture analysis, Comput. Meth. Appl. Mech. Engng, 151 (1998) [10] A. Frangi, G. Novati, R. Springhetti and M. Rovizzi, Fracture mechanics in 3D by the symmetric Galerkin boundary element method, VIII Conf. on Numerical Methods in Continuum Mechanics, 19-4 Sept. 000, Liptovsky Jan, Slovak Republic, 000. [11] G. P. Nikishkov and S. N. Atluri, Combining SG and FEM for modelling 3D cracks. In: Engineering computational technology (Ed. B. H. V. Topping and Z. Bittnar), Saxe-Coburg (00) [1] K. Saito and J. Kuniya, Mechanochemical model to predict stress corrosion crack growth of stainless steel in high temperature water. Corrosion Science, 43 (001) [13] D. O. Harris and D. Dedhia, WinPRAISE 98: PRAISE code in windows, ETM TR , [14] Q. J. Peng, J. Kwon and T. Shoji, Development of a fundamental crack tip strain rate equation and its application to quantitative prediction of stress corrosion cracking of stainless steels in high temperature oxygenated water, Journal of Nuclear Materials, 34 (004) [15] T. L. Anderson, Fracture mechanics, 3rd ed., CRC Press, 005. [16] J. H. Park and G. P. Nikishkov, Examination and improvement of accuracy of three-dimensional elastic solutions obtained using finite element alternating method, Transactions of the KSME A, 34 (010, (in Korean). Jai Hak Park received his M.S. and Ph.D. in Mechanical Engineering from KAIST. He is currently a professor at Chungbuk National University. His research interests are in the area of fracture mechanics, computational mechanics and probabilistic assessment of structure. Gennadiy Nikishkov received his Ph.D. and D.Sc. in Computational Mechanics from the Moscow Engineering Physics Institute. He held a Professor position at the Moscow Engineering Physics Institute. Dr. Nikishkov is currently a Professor at the University of Aizu, Japan. His research interests include computational modeling, high performance computing, visualization and computer graphics.