WAVELET-NUMERICAL METHOD IN CRACK ANALYSIS *

Transcription

1 Applied Mamatics and Mechanics (English Edition, Vol 21, No 10, Oct 2000) Published by Shanghai University, Shanghai, China Article ID: (2000) WAVELET-NUMERICAL METHOD IN CRACK ANALYSIS * SHEN Yuan-tong (~:~)1, YI Xu-ming (~_~flj~)~" ( 1. Department Mamatics and Physics, China University Geosciences, Wuhan , P R China; 2. School Mamatics Sciences, Wuhan University, Wuhan , P R China) (Communicated Abstract by ZHOU Huan-wen) The Abstract: one-dimensional Properties problem wavelet good motion localization a rigid were flying used to plate approximate under explosive displacement attack has an analytic fields solution near only crack when tip. Wavelet-numerical polytropic index algorithm detonation and simulation products singularity equals problem to three. In general, a numerical crack tip were analysis established. is required. As an In example, this paper, stress however, intensity factors by utilizing were obtained. "weak" The shock behavior numerical reflection results show shock that in this algorithm explosive htz~ products, good precision. and applying small parameter purterbation method, an analytic, first-order approximate solution is obtained for problem flying plate driven Key by words: various wavelet-numerical high explosives with algorithm; polytropic scaling indices function; or than stress but intensity nearly equal factor; three. Final velocities flying singularity plate obtained agree very well with numerical results by computers. Thus an analytic CLC formula numbers with : 0242 two ; parameters 0343 Document high explosive code: (i.e. A detonation velocity and polytropic index) for estimation velocity flying plate is established. Introduction Fracture mechanics has been widely used in various engineering fields. In fracture Explosive driven flying-plate technique ffmds its important use in study behavior mechanics, fields stresses and displacements near crack tip and stress intensity factors are materials under intense impulsive loading, shock synsis diamonds, and explosive welding and cladding important metals. contents The method research [1] estimation. For infinite flyor velocity plate with and cracks way we raising can it seek are questions analytical solutions common through interest. methods complex analysis and integral transform. However, for finite plate Under with cracks assumptions we have to select one-dimensional numerical methods plane detonation properly. and Owing rigid to flying existing plate, singularity normal approach near crack solving tip, as far problem as dealing motion with singularity flyor is is to concerned, solve following traditional system numerical equations methods governing are certain flow trouble. field At detonation present, products boundary behind collocation flyor method, (Fig. I): analytical-variational method and finite element method [2'31 are mainly used. In recent years, wavelet ory is quickly developing ap +u_~_xp + au as a relatively new mamatical tool. It has been widely used in signal process, image data compression, pattern identification and solution difference equation and so on. Since y wavelet =0, analysis is good localization in both time area and frequency area, it is superior to Fourier analysis and can focus on any details as as with increment wavelet space. a--t Hence, se properties are important signification in analysis singularity [4' 5]. In this paper, we make use properties wavelet. For cracked plate, we use scaling function to approximate displacement fields plane crack and where establish p, p, wavelet-numerical S, u are pressure, algorithm. density, specific Meanwhile, entropy numerical and particle analysis velocity is completed detonation for products finite respectively, with trajectory R reflected shock detonation wave D as a boundary and trajectory F flyor as anor boundary. Both are unknown; position R and state parameters * on Received it are governed date: : by flow Revised field I date: central rarefaction wave behind detonation wave D and by Foundation initial stage item: motion National flyor Natural also; Science position Foundation F and China state ( ) parameters ; Innvotion products and Backbone Foundation Wuhan University Biography: SHEN Yuan-tong ( 1963 ~ ), Associate Pressor 1139

2 1140 SHEN Yuan-tong and YI Xu-ming plains with central crack and symmetric edge cracks. Numerical results show that this algorithm is reliable and is good precision. 1 Basic Concept Wavelet Let N,~ (x) be m-order B-spline function defined recursively by Eq. ( 1 ), namely, N,, (x) is convolution N,~_92 (x) and N1 (x), where N1 (x) is characteristic function on interval [ 0,1 ] Express Na(x) = X[o.,](x), Nm(x ) = (N._, * N1)(x) = [~N._1(x - t)dt (m = 2,3,4--'). (l) 30 Abstract The one-dimensional problem 9(x) = N=(x), motion 9j,k(X) a rigid = 9(2ix- flying plate k). under explosive attack has (2) an analytic solution only when polytropic index detonation products equals to three. In general, For a anyj numerical E Z, analysis define Vj is = required. span{~j,k(x) In this I paper, k E Z}, however, and n by utilizing { Vj }/6zhave "weak" properties shock behavior as follows : reflection shock in explosive products, and applying small parameter purterbation method, an analytic, first-order approximate solution is obtained for problem flying i) V~. C V/+I, Vj E Z; ie~('l Vj = {0}; i?zv/ = LZ(R); plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final velocities 2) f(x) ff flying Vj r f(2x) plate obtained E Vj+I, agree Vj very E Z; well with numerical results by computers. Thus an analytic 3) {~9(x formula - k) with I k two ~ Z} parameters form a non-conditional high explosive Riesz (i.e. basis detonation V0 velocity ; and polytropic index) for estimation velocity flying plate is established. If Wj is defined to be orthogonal complement space Vj in Vj+~, n have 4) Vj Wj; justjgzw j -- L"(R). From above, following orems can be obtained [6] : Explosive Theorem driven 1 9 flying-plate (x) is scaling technique function ffmds its L important 2 (R), and use in study behavior materials i ) under suppp(x) intense = impulsive [0,m]; loading, shock synsis diamonds, and explosive welding and cladding metals. The method ll) 9(x) = NaO-r~+' (7) estimation flyor velocity and way raising it are questions common interest. z..a- 9(2x - k) ; k=0 Under assumptions one-dimensional plane detonation and rigid flying plate, normal iii ) 9*(x) = gr~_l(x) - g~,_,(x - approach solving problem motion 1). flyor is to solve following system equations governing Theorem flow 2 field Let N,, detonation (x) be m-order products B-spline behind function. flyor (Fig. Define I): a sequence [(- i_), ~--~.(m]n,,~(k + 1- l) (k = O,"',3m - 2), q, = { 2 m-~ ~.~z/ ap - +u_~_xp + 0, orwise. Then we have next results: as as 3m-2 a--t i ) (a(x) = ~ q~c?(2x - k) is a wavelet function; k=o [l ) supp~b(x) : [0,2m - 1]. where Since p, p, S, u sequence are pressure, { Vj density, }j6 z construct specific one-dimensional entropy and particle mulfiresolution velocity analysis detonation (MRA), products and respectively, n V} = with ~ 1/i. construct trajectory two-dimension R reflected MRA shock L detonation 2 (R z). Moreover, wave D as scaling a boundary function and trajectory F flyor as anor boundary. Both are unknown; position R and state parameters space on V~ it is are 9 ( governed x, y) = by ~ ( x ) ~ flow (y). field Let I W~. be central orthogonal rarefaction complement wave behind space detonation V} in wave V~+I, D namely and by V~+ initial 1 = stage V~ (~ motion W~, where flyor W} also; = ( Vj position Wj ) 9 F ( Wj and Vj state ) 9 parameters ( Wj Wj ), products and its corresponding wavelet functions are 9 ( x ) ~b ( y ), ~b ( x ) 9 (Y) and ~b ( x ) ~b ( y ). au

3 Wavelet-Numerical Method in Crack Analysis Wavelet-Numerical Algorithm on Crack Problem Consider following plane crack problems on boundary B1 and B: imposed next boundary conditions : E(u)a = ~ onboundary B 1, (3) u = ~ on boundary B~. (4) From minimum potential energy principle, general potential energy system [71 is as follows 1~ ~ f~ Wdu - ffvf'~ud'lj - f,tudb1, (5) v Abstract B 1 where The W one-dimensional is strain energy problem density, f and p motion are body a force rigid and flying surface plate force under respectively. explosive attack has an analytic Displacements solution only u = when (u, v )T are polytropic decomposed index space detonation V~., n products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock behavior reflection {/ shock = ~-a in ~"~ai'"'q~j"(x)qgj'l(y) explosive products, and = "~(x'y)a' applying small parameter pur- (6) terbation method, an analytic, first-order approximate solution is obtained for problem flying plate driven by various high explosives ~ ~jbi,k,tgi,k(x)gi,,(y) with polytropic indices or ~(x,y)b. than but nearly equal to three. Final velocities flying plate obtained agree very well with numerical results by computers. Thus an (6) analytic can be formula expressed with as two follows parameters high explosive (i.e. detonation velocity and polytropic index) for estimation velocity flying plate is established. {~ {;} where c = b are undetermined coefficients. u = = ~(x,y) = ~c, (7) 7) Explosive driven flying-plate technique ffmds its important use in study behavior Consider relationships between strains and displacements, we have materials under intense impulsive loading, shock synsis diamonds, and explosive welding and cladding metals. The method estimation flyor velocity and way raising it are questions common interest. Under assumptions one-dimensional plane detonation and rigid flying plate, normal approach v~ solving t3u/gy problem + 9v/3x motion J a3~/ay flyor is + to b~ril/~x solve ~ following L3o)/3y system 3r equations j governing flow field detonation products behind flyor (Fig. I): Moreover (8) can be rewritten in next form ap +u_~_xp + au {s}= [B]{c}. (9) Take next relation stresses and strains into account as {a}= as [DI{6}. (10) a--t (5) can be rewritten by combining (7), (9) and (10) as follows (here body force is not considered) : where p, H p, S, = fl u are ~ 1 pressure, -~'TDe'dv density, - f fit specific udb1 entropy = ~f -5- and l ctb particle TDBcdvvelocity f ~T~cdB detonation 1 products = respectively, with tt ~-~ trajectory R B 1 reflected shock tl ~ detonation wave D B as t a boundary and trajectory F flyor as anor boundary. Both are unknown; position R and state parameters on it are -5-c l governed r fy BTDBdvc by flow - y,'r field I ~db central 1 c. rarefaction wave behind detonation wave (11) D and by initial stage motion flyor also; position F and state parameters products By Lagrange' s method multipliers, inlxoducing boundary conditions into (11), n (8)

4 1142 SHEN Yuan-tong and YI Xu-ming Let and )~ : =,BTDBdvc -,gtodb~c + (Oc - ~)XdB,. B B z arl( c,,~ ) _ o a.~( c,;t ) o. 8c ' 9,l - Then we can obtain following equations associated with undetermined coefficients c EK][c] + [R][;t] = EF], [O][c] = [ii], (12) where [ K ], [ R ] and [ F ] are constant coefficient matrix, and have following expressions respectively [K-]--'ffvBTOnd'l), [R].=f Abstract I~)de 2, IF]-- I,T (~db1. B,. B The By solving one-dimensional linear problem algebraic equations motion (12), a we rigid can flying obtain plate under solutions explosive c, and attack reby, has an analytic solution only when polytropic index detonation products equals to three. In displacement fields near crack tip can be obtained. general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock behavior 3 Analysis reflection Numerical shock in Results explosive products, and applying small parameter purterbation method, an analytic, first-order approximate solution is obtained for problem flying plate driven To verify by various effectiveness high explosives wavelet-numerical with polytropic indices algorithm or that than we but establish, nearly equal we to choose three. Final finite velocities plates which flying is imposed plate obtained unidirectional agree very force well conditions with numerical on results boundary by computers. and which Thus have an analytic formula with two parameters high explosive (i.e. detonation velocity and polytropic central crack (see Fig. 1) and symmetric edge crack (see Fig.2). index) for estimation velocity flying plate is established. For symmetry problem we can only calculate 1/4 part it. For plate with central crack and symmetric edge ~ack, coordinates axes are established as follows and displacement boundary conditions are determinated respectively (see Fig. 3 or Fig.4). Explosive driven flying-plate technique ffmds its important use in study behavior materials under intense impulsive loading, shock synsis diamonds, and explosive welding and cladding metals. The method estimation flyor velocity and way raising it are questions common interest. Under assumptions one-dimensional plane detonation and rigid flying plate, normal L approach solving problem motion flyor is to solve following system equations governing flow field detonation products behind flyor (Fig. I): Fig. 1 ap +u_~_xp + au Finite plates with symmetau au Fig. 2 1 ric central crack Finite plates with symmetric edge cracks y y as as a--t t where p, p, S, u are pressure, density, specific entropy and particle IE> velocity detonation products b respectively, with trajectory R reflected shock detonation wave D as a boundary and tt~ A A\ trajectory F flyor as anor boundary. Both are unknown; position creak creak R and state parameters on it are governed by flow field I central rarefaction wave behind detonation wave Fig.3 1/4 central cracked plate Fig. 4 1/4 symmetric edge cracked plate D and by initial stage motion flyor also; position F and state parameters products and boundary conditions and boundary conditions

5 Wavelet-Numerical Method in Crack Analysis 1143 As is well-known that near crack tip displacement y-direction is KI ~/~(k + 1) (r << a) (13) v- 2G where r is distance from crack tip to a point considered along with crack edge. Therefore stress intensity factor K1 can be written 2Gv ~ (14) KI = (k + 1)~/ r " As stress intensity factors near crack tip can be extrapolated to crack tip, t_he numerical solutions stress intensity factor are gotten. For plates with central crack and symmetric edge crack, if h/b = 1, we get stress intensity factors from different ratio values a/b (see Abstract Table 1). For plates with central crack, if a/b = 0.4, we can also get stress intensity factors from different ratio values h/b (see Table 2). The For one-dimensional convenience problem comparison, motion following a rigid data flying stress plate intensity under explosive factors are attack all non- has an analytic solution only when polytropic index detonation products equals to three. In dimensional stress intensity factors Z I = K I/P'/-Ja. general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock behavior reflection shock in explosive products, and applying small parameter pur- Table 1 Stress intensity factors Z I from different ratio values a/b ( h/b = 1) terbation method, an analytic, first-order approximate solution is obtained for problem flying plate driven by various high I 0.1 explosives 0.2 with polytropic 0.3 indices 0.4 or than 0.5 but nearly 0.6 equal to 0.7 three. Final velocities flying plate obtained agree very well with numerical results by computers. Thus an analytic central crack formula! with two parameters high i.994 explosive (i.e. detonation velocity and polytropic index) I symmetric for estimation edge crack] velocity flying plate is established Table 2 Stress intensity factors Z I 1. from Introduction different ratio values h/b ( a/b = 0.4) Explosive driven flying-plate technique ffmds 0.9 its important use 1.5 in study behavior >3 materials under intense impulsive loading, shock synsis diamonds, and explosive welding and , cladding metals. The method estimation flyor velocity and way raising it are questions common interest. Under Known from assumptions Table 2, when one-dimensional h/b > 3 its plane non-dimensional detonation and stress rigid flying intensity plate, factor normal Z I is approach But solving from expression problem central motion crack plate flyor stress is to intensity solve factor following Z I : system equations governing flow field detonation products behind flyor (Fig. I): K I r 1 2 b 1/,_ Z I - _ [~-~atan ~a] (15) result non-dimensional stress intensity ap +u_~_xp factor + au Z I is 1.906, relative error is 1.4%. We also compare numerical results au with au that obtained 1 by or methods, results are listed in Table 3. All results indicate that it is good for wavelet to be used to solve stress intensity factors. Wavelet-numerical algorithm as has good as precision to approximate displacement fields near crack tip. a--t Table 3 Stress intensir r factor results different methods where p, p, S, u are pressure, types density, specific entropy and particle velocity detonation products central crack symmetric edge crack respectively, with trajectory R reflected shock detonation wave D as a boundary and trajectory wavelet-numerical F flyor as anor method boundary. Both are unknown; position R and state parameters on it are governed by flow field I central rarefaction wave behind detonation wave analysis- variational method D and by initial stage motion flyor also; position F and state parameters products boundary couocation method

6 1144 SHEN Yuan-tong and YI Xu-ming References : [I] DING Shui-dong. Fracture Mechanics[M]. Beijing:Publishing House Mechanic Industry,1997. (in Chinese) [2] ZHANG Hang. Method Stress Intensity Factor in Fracture Mechanics [ M]. Beijing: Publi.~hing House National Defence Industry, (in Chinese) [ 3] Barsoum R S. On use isoparamelric finite elements in linear fracture mechanics[ J]. Internal J Numer Methods Eng, 1976,10( 1 ) : 25 ~ 37. [4] Mallat S, Wen L H. Singularity Detect and processing with wavelets[ J~. IEEE Transaction on Information Theory, 1992,38(2) :617 ~ 634. [5] SHF_N Yuan-tong, &re Bi-quan, YI Xu-ming. Wavelet-collocation method for solving a sort sin- gular differential equation [ J ]. Mamatics Abstract Magazine, 1997,17 (4) : 517 ~ 521. ( in Chinese) [6] Chui C K. A Introduction to Wavelet[M]. Boston: Academic Press Inc,1992. [7] The H-U one-dimensional Hal-chang. Variation problem Principle motion Elasticity Mechanics a rigid flying and plate Applicability under explosive [ M]. Beijing:Science attack has an analytic Press, solution (in only Chinese) when polytropic index detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock behavior reflection shock in explosive products, and applying small parameter purterbation method, an analytic, first-order approximate solution is obtained for problem flying plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final velocities flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two parameters high explosive (i.e. detonation velocity and polytropic index) for estimation velocity flying plate is established. Explosive driven flying-plate technique ffmds its important use in study behavior materials under intense impulsive loading, shock synsis diamonds, and explosive welding and cladding metals. The method estimation flyor velocity and way raising it are questions common interest. Under assumptions one-dimensional plane detonation and rigid flying plate, normal approach solving problem motion flyor is to solve following system equations governing flow field detonation products behind flyor (Fig. I): ap +u_~_xp + au as a--t as where p, p, S, u are pressure, density, specific entropy and particle velocity detonation products respectively, with trajectory R reflected shock detonation wave D as a boundary and trajectory F flyor as anor boundary. Both are unknown; position R and state parameters on it are governed by flow field I central rarefaction wave behind detonation wave D and by initial stage motion flyor also; position F and state parameters products