A boundary element method with analytical integration for deformation of inhomogeneous elastic materials

Transcription

1 Journal of Physcs: Conference Seres PAPER OPEN ACCESS A boundary element method wth analytcal ntegraton for deformaton of nhomogeneous elastc materals To cte ths artcle: Moh. Ivan Azs et al 2018 J. Phys.: Conf. Ser Vew the artcle onlne for updates and enhancements. Ths content was downloaded from IP address on 22/01/2019 at 06:28

2 A boundary element method wth analytcal ntegraton for deformaton of nhomogeneous elastc materals Moh. Ivan Azs, Syamsuddn Toaha, Maward Bahr and Nrwan Ilyas Department of Mathematcs, Hasanuddn Unversty, Maassar, INDONESIA E-mal: Abstract. Ths paper s concerned wth obtanng solutons to the equaton governng statc deformatons of nhomogeneous elastc materals. The materal parameters are assumed to vary contnuously wth the spatal varables. A boundary element method (BEM) wth analytcal ntegraton s used to fnd the solutons. The results show that the BEM s feasble to be used to fnd the solutons of the problems and wth analytcal ntegraton the BEM gves very accurate solutons n a shorter computaton tme. 1. Introducton A varous classes of problems have been solved numercally usng BEM. The classes of problems nclude heat conducton problems, nfltraton problems, deformaton problems of elastc materals, pollutant transport problems and many others. The use of BEM has also been appled to homogeneous as well as to nhomogeneous materals, and not only to sotropc but also to ansotropc materals. For example, 1 used the BEM for solvng nfltraton problems of sotropc homogeneous meda, 2 derved a fundamental soluton, 3 consdered dffusonconvecton problems for ansotropc homogeneous materals, 4 solved elastcty problems for sotropc nhomogeneous materals and 5 studed elastcty problems, 6 wored on ellptc boundary value problems, 7 solved heat conducton problems, 8 wored on elastcty problems and 9 consdered transent heat conducton problems for ansotropc nhomogeneous materals. Apparently n the mplementaton of the BEM the partal dfferental equaton whch governs the system has to be converted to a boundary ntegral equaton. And eventually after a dscretsaton of the boundary nto a number of segments, ntegraton over each segment has to be calculated ether numercally or analytcally f possble. To some extent analytcal ntegraton has of course advantages over numercal ntegraton n the aspect of accuracy and computaton tme. In recent years some progress has been made toward solvng numercally the deformaton problem of nhomogeneous elastc materals usng the BEM. The wor done by 4, for example, deals wth the case of sotropc materals, whereas n 5, 8 consdered the case for ansotropc materals. However, n all of the papers the authors used numercal ntegraton for fndng the BEM solutons. The current study s concerned wth fndng the solutons of the problems usng the BEM wth analytcal ntegraton. Content from ths wor may be used under the terms of the Creatve Commons Attrbuton 3.0 lcence. Any further dstrbuton of ths wor must mantan attrbuton to the author(s) and the ttle of the wor, journal ctaton and DOI. Publshed under lcence by Ltd 1

3 2. The boundary value problem Referred to a Cartesan frame Ox 1 x 2 x 3 the equlbrum equatons governng small deformatons of an nhomogeneous ansotropc elastc materal occupyng a regon Ω n R 3 wth boundary Ω whch conssts of a fnte number of pecewse smooth closed curves may be wrtten n the form x j c (x) u = 0 (1) where, j,, l = 1, 2, 3, x = (x 1, x 2, x 3 ), u denotes the dsplacement, c (x) the elastc parameters and the repeated summaton conventon (summng from 1 to 3) s used for repeated Latn suffces. Troughout the paper t s assumed that u s ndependent of x 3 and taes the form u = A f(x 1 + τx 2 ) where A, τ are constants and f s a twce dfferentable functon. relatons are gven by σ j = c u and the tracton vector P on the boundary Ω s defned as The stress dsplacement P σ j n j = c u n j (2) where n = (n 1, n 2 ) denotes the outward pontng normal to the boundary Ω. For all ponts n Ω the coeffcents c (x) are requred to satsfy the usual symmetry condtons and also suffcent condtons for the stran energy densty to be postve. Ths requrement ensures that the system of partal dfferental equatons s ellptc throughout Ω. A soluton to (1) s sought whch s vald n the regon Ω and satsfes the boundary condtons that on Ω 1 the dsplacement u s specfed and on Ω 2 the tracton P s specfed, where Ω = Ω 1 Ω Reducton to a lnear constant coeffcents equaton The coeffcents n (1) are now requred to tae the form where the Consder the transformaton c (x) = g(x) (3) are constants. Equaton (1) thus may be wrtten g(x) u = 0 (4) Use of (5) n (4) provdes the equaton u = g 1/2 ψ (5) Thus f g1/2 2 ψ ψ 2 g 1/2 = 0 (6) 2 ψ = 0 (7) 2 g 1/2 = 0 (8) 2

4 then (6) wll be satsfed. Thus when g satsfes the system (8) the transformaton gven by (5) transforms the lnear system wth varable coeffcents (4) to the lnear system wth constant coeffcents (7). Equaton (7) comprses a system of three constant coeffcents lnear ellptc partal dfferental equatons n the three dependent varables u for = 1, 2, 3. A soluton to ths system may be readly obtaned n terms of arbtrary analytc functons. As a result of the symmetry property c = c lj equaton (8) conssts of a system of sx constant coeffcents partal dfferental equatons n the one dependent varable g 1/2. In general the soluton to ths system conssts of a lnear functon of the three ndependent varables x 1, x 2, x 3. Thus n the general case g may be wrtten n the form g(x) = (α x ) 2 (9) where the α are constants whch may be used to ft the elastc parameters c = g(x) to gven numercal data. Although n the general case t s necessary for g to be restrcted to the form gven by (9) t s approprate to note at ths stage that for partcular classes of deformatons the tranformaton from (4) to (7) may be acheved for a more general class of functons g(x). Some of these cases are now consdered. Now substtuton of (3) and (5) nto (2) yelds where P = P g ψ + P ψ g 1/2 (10) P g = g 1/2 n j (11) P ψ = ψ n j (12) The boundary ntegral equaton for the soluton of (7) may be wrtten n the form Use of (5) and (10) n (13) yelds ηψ m (x 0 ) = Φ m P ψ Γ m ψ ds (13) Ω ηg 1/2 u m (x 0 ) = (g 1/2 Φ m )P (g 1/2 Γ m P g Φ m)u ds Ω Ths equaton provdes a boundary ntegral equaton for determnng u m and P m at all ponts of Ω. Also Φ m and Γ m are gven by Φ m = 1 { 3 } 2π R A α N α log(z α c α ) d m Γ m = 1 { 3 } 2π R L jα N α (z α c α ) 1 n j d m α=1 α=1 (14) where R denotes the real part of a complex number, z α = x 1 + τ α x 2, and c α = a + τ α b, where τ α are the three roots wth postve magnary part of the polynomal n τ 11 + c(0) 21 τ + c(0) 12 τ + c(0) 22 τ 2 = 0 3

5 The A α occurng n (14) are the solutons of the system Also the N α, L jα and d m are defned by ( 11 + c(0) 21 τ α + 12 τ α + ) 22 τ α 2 A α = 0 δ = α A α N α L jα = ( j1 + τ α j2 )A α δ m = α=1 {L 2α N α L 2α N α } d m where bar denotes the complex conjugate. 4. A further perturbaton method In ths secton a procedure s obtaned when the coeffcents c (x) s perturbed about ts expresson presented n (3) whle retanng the condton (8). Specfcally the coeffcent c (x) s supposed to have the form where the c (x) = g(x) + ɛc(1) (x) (15) are constants, ɛ s a small parameter, c(1) satsfes (8). Substtuton of (15) nto (1) and use of the transformaton (5) gve g (g 1/2 ψ ) = ɛ c (1) (g 1/2 ψ ) s a dfferentable functon and g(x) Use the analyss used to derve (7) from (4) for the left hand sde and a smplfcaton for the rght hand sde of (16) yelds where 2 ψ = ɛ Aj x j ψ + (A j + B ) ψ 2 ψ + B x j A j (x) = c (1) g 1/2, B (x) = c (1) x g 1/2 l A soluton to equaton (17) s sought n the form ψ (x) = r=0 (16) (17) ɛ r ψ (r) (x) (18) Substtuton of (18) nto (17) and equatng coeffcents of powers of ɛ yelds for r = 1, 2, ψ (0) = 0 2 ψ (r) = Aj x j ψ (r 1) + (A j + B ) ψ(r 1) x j 2 ψ (r 1) + B (19) (20) 4

6 The ntegral equatons for (19) and (20) are respectvely ηψ m (0) (x 0 ) = where P ψ(0) = ( ψ(0) /)n j and ηψ m (r) (x 0 ) = Ω Ω Φ m P ψ(r) Φ m P ψ(0) Γ m ψ (0) ds (21) Γ m ψ (r) ds + h (r) Φ m ds (22) Ω where P ψ(r) = ( ψ(r) /)n j and h (r) s the rght hand sde of (20) for r = 1, 2,.... The correspondng value of P may be wrtten as where P g P = P g ψ(0) + P ψ(0) g 1/2 + ɛ r P g ψ(r) + P ψ(r) g 1/2 + G (r) r=1 ψ and P are gven by (11) and (12), and G (r) = c (1) (g 1/2 ψ (r 1) ) n j (23) To satsfy the boundary condtons n Secton 2 t s requred that ψ (0) = g 1/2 u where u taes on ts specfed value on Ω 1. Also t s requred that on Ω 2 P ψ(0) = g 1/2 (P + P g ψ(0) ) where P taes on ts specfed value on Ω 2. It then follows from (12) and (18) that ψ (r) = 0 on Ω 1 and P ψ(r) = g 1/2 (P g ψ(r) G (r) ) on Ω 2 for r = 1, 2,.... The ntegral equatons (21) and (22) may thus be wrtten n the form ηψ m (0) (x 0 ) = Φ m P ψ(0) (Γ m g 1/2 )u ds m (x 0 ) = + ηψ (r) Ω 1 Ω 2 (g 1/2 Φ m )P (Γ m g 1/2 P g Φ m)ψ (0) Φ m P ψ(r) Ω 1 Ω 2 Ω ds (g 1/2 Φ m )G (r) (Γ m g 1/2 P g Φ m)ψ (r) ds (24) ds h (r) Φ m ds for r = 1, 2,... (25) The boundary ntegral equaton (24) serves to determne the unnown ψ (0) (x) for x Ω 2 and the unnown P ψ(0) (x) for x Ω 1. Once these boundary values are nown equaton (24) serves to determne ψ (0) and ts dervatves for all ponts n the doman Ω. Smlarly the ntegral equaton (25) serves to determne the unnown ψ (r) (x) for x Ω 2 and the unnown P ψ(r) (x) for x Ω 1 for r = 1, 2,.... Once these boundary values are nown equaton (25) serves to determne ψ (r) and ts dervatves for all ponts n the doman Ω. At each stage n usng (25) to determne ψ (r) and the G (r) (23) whch may be evaluated from the prevously determned dervatves of ψ (r 1) occurng n (25) may be obtaned from. Havng determned the values of ψ (r) for all r equatons (18) and (5) then provde successvely, the values of ψ (x) and u (x). 5

7 x 2 (0, 1) D C A B (1, 0) x 1 Fgure 1. The geometry for the numercal examples 5. Numercal results To show the usefulness of the formulas and the valdty of the technques derved n the prevous sectons some partcular plane stran problems, for whch the dsplacements u 1 = u 1 (x 1, x 2 ), u 2 = u 2 (x 1, x 2 ) and u 3 = 0, of nhomogeneous ansotropc materals wll be consdered. The constant BEM s used to fnd numercal solutons to the problems. The doman Ω s taen to be a unt square (see Fgure 1). Each sde of the boundary Ω s dvded nto N segments of equal length and nodal ponts are taen to be the md ponts of each segment. To save the computaton tme and to ncrease the accuracy, the evaluaton of the lne ntegrals of Φ m, Γ m and ts dervatves along a segment q 1, q jonng the pont q 1 to the pont q and excludng the sngular ponts of Φ m and Γ m s done analytcally by utlsng the followng results, q log(z α c α )ds(x) = R 1 + log(a α + B α ) + A α log(1 + B α ), q 1 B α A α where q (z α c α ) 1 ds(x) = R log(1 + B α ) q 1 B α A α a log(z α c α ) ds(x) = R log(1 + B α ) A α q q 1 q q 1 q q 1 q q 1 B α b log(z α c α ) ds(x) = R τ α log(1 + B α ) B α A α (z α c α ) 1 ds(x) = R A α (A α + B α ) 1 a b (z α c α ) 1 ds(x) = R τ α A α (A α + B α ) 1 q = (x, y ), x = x 1, y = x 2 R = q q 1 A α = x 1 a + τ α (y 1 b) B α = x x 1 + τ α (y y 1 ) The last four formulas are used for the calculaton of the stress dstrbuton n the doman Ω. Along a segment q 1, q whch ncludes a sngular pont (e. the nodal or md pont of a 6

8 segment) the evaluaton of the ntegral of Φ m s done by splttng the segment nto two parts on ether sde of the sngular pont and summng the values of ntegrals along both sub segments. Ths then gves the followng formula q q 1 log(z α c α )ds(x) = R ( 1 + log log B α) Meanwhle the ntegral of Γ m s zero due to the fact that n ths case the vectors x x 0 and n are perpendcular and therefore the normal dervatve of Φ m (and so Γ m ) s zero. For the methods requrng doman ntegrals the doman Ω s dscrtsed nto N 2 equal subsquares and the followng analytcal results are used for the evaluaton of area ntegrals over a partcular sub square {(x 1, x 2 ) : s x 1 t, v x 2 w} t w s t w s v v log x 1 a + τ α (x 2 b) dx 2 dx 1 = q 1 (v b)(t a) 1 (t a) 2 1 2τ α 2 τ α(v b) 2 + q 2 (v b)(s a) + 1 (s a) τ α 2 τ α(v b) 2 + q 3 (w b)(s a) 1 (s a) 2 1 2τ α 2 τ α(w b) 2 + q 4 (w b)(t a) + 1 (t a) τ α 2 τ α(w b) 2 3 (t s)(w v) 2 x 1 log x 1 a + τ α (x 2 b) dx 2 dx 1 = 1 q 1 {2q5 3 6aq (a 2 t 2 )q 5 2a 3 4t 3 + 6at 2 }+ 12τ α q 2 { 2q aq 2 5 6(a 2 s 2 )q 5 + 2a 3 + 4s 3 6as 2 }+ q 3 {2q 3 6 6aq (a 2 s 2 )q 6 2a 3 4s 3 + 6as 2 }+ q 4 { 2q aq 2 6 6(a 2 t 2 )q 6 + 2a 3 + 4t 3 6at 2 }+ τ α q 7 { 4b + 2(v + w)} + q 7 { 4a 7(s + t)} where t w s v x 2 log x 1 a + τ α (x 2 b) dx 2 dx 1 = 1 12τ 2 α q 1 {2τ 3 αq 8 + 6τ 2 α(v 2 b 2 )(t a) + 6τ α b(t a) 2 2(t a) 3 }+ q 2 { 2τ 3 αq 8 6τ 2 α(v 2 b 2 )(s a) 6τ α b(s a) 2 + 2(s a) 3 }+ q 3 {2τ 3 αq 9 + 6τ 2 α(w 2 b 2 )(s a) + 6τ α b(s a) 2 2(s a) 3 }+ q 4 { 2τ 3 αq 9 6τ 2 α(w 2 b 2 )(t a) 6τ α b(t a) 2 + 2(t a) 3 }+ τ α q 7 {4b + 7(v + w)} + q 7 {4a 2(s + t)} q 1 = log t a + τ α (v b) q 2 = log s a + τ α (v b) q 3 = log s a + τ α (w b) q 4 = log t a + τ α (w b) q 5 = τ α (v b) q 6 = τ α (w b) q 7 = τ α (t s)(w v) q 8 = b 3 + 2v 3 3v 2 b q 9 = b 3 + 2w 3 3w 2 b 7

9 Table 1. Numercal and analytcal results Poston BEM 12 segments BEM 40 segments Analytcal (x 1, x 2 ) u 1 /u u 2 /u u 1 /u u 2 /u u 1 /u u 2 /u (0.1,0.5) (0.3,0.5) (0.5,0.5) (0.7,0.5) (0.9,0.5) A test problem for Secton 4 Consder the sotropc case for whch the elastc modul tae the form (15) wth A/c = λ (0) +2µ (0), B/c = λ (0), D/c = λ (0) + 2µ (0) and F/c = µ (0), and the boundary condtons ɛ = 0.1 g(x) = ( x 1 /l) 2 λ (0) = µ (0) = 0.5 λ (1) = ( x 1 /l) 2 µ (1) = 0 u 1 /u = 5 5/( x 1 /l) u 2 /u = 5 4.9/( x 1 /l) on AB u 1 /u = u 2 /u = on BC u 1 /u = 5 5/( x 1 /l) u 2 /u = 5 4.9/( x 1 /l) on CD u 1 /u = 0 u 2 /u = 0.1 on AD Ths problem admts the analytcal soluton u 1 /u = 5 5/( x 1 /l) and u 2 /u = 5 4.9/( x 1 /l). An approxmate soluton s taen to be the sum of the frst two terms of the seres soluton (18). The so called regular method s used to avod a sngular pont by locatng the source ponts outsde the doman at the poston of a dstance as long as the length of the boundary segment measured from the md pont of each boundary segment. It should be noted that η = 0 f x 0 / Ω. The area ntegral over a doman cell Ω l s approxmated wth h (r) Ω l Φ m ds h (r) (x) Φ m ds Ω l where x s the centre pont of Ω l. Table 1 compares the analytcal and BEM results for some nteror ponts. It s observed that the accuracy mproves as the number of segments ncreases. It also mproves as we move further from the boundary ponts to the centre pont of the doman. 6. Summary Some BEMs for the soluton of certan classes of boundary value problems of elastcty for ansotropc nhomogeneous meda has been derved. The methods are generally easy to mplement to obtan numercal values for partcular problems. They can be appled to a wde class of mportant practcal problems for nhomogeneous ansotropc materals. The numercal results obtaned usng the methods ndcate that they can provde accurate numercal solutons. Acnowledgments Ths wor was fnancally supported by the LP2M (Lembaga Peneltan dan Pengabdan pada Masyaraat) of Hasanuddn Unversty Maassar and The Hgher Educaton Mnstry of Indonesa. 8

10 References 1 Azs M I, Clements D L and Lobo M 2003 A boundary element method for steady nfltraton from perodc channels ANZIAM J. 44(E) C61 2 Azs M I 2017 Fundamental solutons to two types of 2D boundary value problems of ansotropc materals Far East Journal of Mathematcal Scences Haddade A, Salam N, Khaeruddn and Azs M I 2017 A Boundary Element Method for 2D Dffuson- Convecton Problems n Ansotropc Meda Far East Journal of Mathematcal Scences Clements D L and Azs M I 2000 A Note on a Boundary Element Method for the Numercal Soluton of Boundary Value Problems n Isotropc Inhomogeneous Elastcty Journal of the Chnese Insttute of Engneers Azs M I and Clements D L 2001 A Boundary Element Method for Ansotropc Inhomogeneous Elastcty Internatonal Journal of Solds and Structures Azs M I, Clements D L and Budh W S 2003 A boundary element method for the numercal soluton of a class of ellptc boundary value problems for ansotropc nhomogeneous meda ANZIAM J. 44(E) C79 7 Azs M I and Clements D L 2008 Nonlnear transent heat conducton problems for a class of nhomogeneous ansotropc materals by BEM Engneerng Analyss wth Boundary Elements Azs M I and Clements D L 2014 On some problems concernng deformatons of functonally graded ansotropc elastc materals Far East Journal of Mathematcal Scences Azs M I and Clements D L 2014 A Boundary Element Method for Transent Heat Conducton Problem of Nonhomogeneous Ansotropc Materals Far East Journal of Mathematcal Scences