Approximation of complex potentials as a uniform approach for solving classical and non-classical boundary value problems of plane elasticity

Transcription

1 Boudary Elemets XXVII 255 Approximatio of complex potetials as a uiform approach for solvig classical ad o-classical boudary value problems of plae elasticity A. N. Galybi School of Civil ad Resource Egieerig, The Uiversity of Wester Australia, Australia Abstract This article aims to develop a uiform method for solvig plae elastic boudary value problems, PEBVPs. Stress or displacemet vectors are assumed to be give o the boudary i the classical formulatio of PEBVP, while o classical formulatios iclude boudary coditios i terms of orietatios of stresses, forces or displacemets. It is show that the approximatio of the complex potetials by the liear combiatios of holomorphic fuctios ca be used to obtai solutios i these cases ad that the other well ow umerical methods ca be represeted as particular cases of this approach. Two examples are preseted. Keywords: plae elasticity, boudary value problems, complex potetials, stress traectories. Itroductio Geeral solutio of plae elastic boudary value problems, PEBVP is give i terms of two holomorphic fuctios (complex potetials, see Mushelishvili []) that are to be determied from boudary coditios posed i displacemets ad stresses that are foud via the Kolosov-Mushelishvili formulae. This presets a uiform method for solvig PEBVPs. It covers both classical ad o-classical formulatios. Stress (or displacemet) vector is give o the boudary of a domai i classical formulatios of PEBVP, while o classical formulatios iclude boudary coditios i terms of orietatios of stresses, forces or ISSN X (o-lie)

2 256 Boudary Elemets XXVII displacemets. As show i precedig authors papers, o-classical PEBVPs may ot have uique solutios or may be usolvable. It is proposed to see represetatios for complex potetials as liear combiatios of idepedet holomorphic fuctios, followed by the determiatio of uow (complex) coefficiets from the boudary coditios by the collocatio method. I geeral, the umber of collocatio poits is assumed to be greater tha the umber of the sought coefficiets, which results i a overdetermied system of liear algebraic equatios. A approximate solutio of the system is obtaied by iversio of the matrix by usig the sigular value decompositio method (SVD). Two umerical examples are preseted to illustrate the approach. The first oe deals with the classical PEBVP for a ifiite plae weaeed by a elliptic hole with its cotour subected to uiform pressure. I this case the obtaied solutios (with differet sets of approximatio fuctios) are verified agaist the aalytical solutio. The secod problem is of o-classical type i which a admissible solutio is foud by employig the orietatios of pricipal stresses o the boudary of the domai. Calculatios are performed for the elastic model of Atarctica with the real data available from the World Stress Map Proect. 2 Differet umerical methods ad the problem of approximatio of complex potetials by holomorphic fuctios For plae isotropic elastic domai, the Kolosov-Mushelishvili solutio give i terms of complex potetials is valid (o body forces) σxx + σ yy = P( z, z) = Φ( z) + Φ( z) 2 σ yy σxx + iσxy = D( z, z) = zφ ( z) + Ψ( z) () 2 2G( u + iv) = W ( z, z) = κϕ( z) zφ( z) ψ( z), Φ( z) = ϕ ( z), Ψ( z) = ψ ( z) Here the followig otatios are used: harmoic fuctio P ad complex-valued fuctio D represet fuctios of stress compoets σ xx, σ yy ad σ xy ; W is a complex-valued fuctio proportioal to the displacemet vector (u,v); G is the shear modulus, κ=3-4ν for plae strai ad κ=(3-ν)/(+ν) for plai stress, ν is Poisso s ratio. Arbitrary holomorphic fuctios ϕ ad ψ (or Φ ad Ψ) are foud from boudary coditios. Two boudary value problems are cosidered further: (A) PEBVP i terms of stresses ad (B) i terms of stress orietatios. Utilizig otatios for the stress fuctio oe ca preset the boudary coditios i problem A i the followig form [] d P + D = N( ) + it ( ), (2) d ISSN X (o-lie)

3 Boudary Elemets XXVII 257 Here N ad T are ormal ad shear compoets of tractios o the boudary of the cosidered domai Ω (iterior or exterior); P ad D are boudary values of the stress fuctios defied i (). Note that hereafter the same otatios are used for the stress fuctios ad their boudary values, although the latter are fuctios of cotour variable, i order to distiguish betwee them, o (sigle) argumets are show for boudary values. Boudary coditios i problem B are expressed as follows (Galybi ad Muhamediev [2]) arg D arg D = α( ), = α ( ), (3) Here is the outward uit ormal to the cotour; α ad α' are give fuctios o the cotour. The first coditio i (3) prescribes the directio of pricipal stresses o, while the secod oe determies the rate of chage of the stress traectory icliatio agle o i the directio of the outward ormal. I both these problems it is ecessary to determie holomorphic fuctios Φ(z) ad Ψ(z). The use of differet represetatios for holomorphic fuctios evetually leads to differet computatioal methods. Thus, a fiite elemet formulatio is observed if oe uses piecewise liear approximatios for ϕ(z)=a z+b ad ψ(z)=c z+d, the it follows from hat displacemets are also liear, while stresses are costats withi the elemets. The differece with the classical FEM is that the approximatio of displacemets withi the elemets is doe here by usig sigle (complex) variable i cotrast to two idepedet real variables used i the covetioal formulatio. Although the umber of real uows i each elemet apparetly icreases by two, this does ot affect the determiatio of stresses expressed via three real uows (Re(a ), Re(c ) ad Im(c )), i.e. for the first PEBVP other uows ca be eglected. For the secod PEBVP (i terms of displacemets) the umber of real uows remais six if the followig two complex parameters are itroduced istead of a, c ad d e = κa a, f = κb d (4) Expasio of complex potetial ito Fourier series (Mushelishvili []) is aother well-ow approach for solvig PEBVPs. It is modified here by assumig that complex potetials, Φ(z) ad Ψ(z), ca be approximated by liear combiatio of idepedet holomorphic fuctios H (z) as follows Φ ( z) = A H ( z) Ψ( z) = A H ( z) =, (5) where 2 complex coefficiets A are uow (for simplicity the umber of terms i both sums are the same). They are foud from a liear system of algebraic equatios followed from specified boudary coditios. Two examples of usig approximatios (5) are preseted i the ext sectio. Below it is show that the boudary elemet formulatios ca also be preseted i the form (5). Let the holomorphic fuctios be preseted through the Cauchy itegrals = + ISSN X (o-lie)

4 258 Boudary Elemets XXVII ( z) ϕ = 2πi ( t) g dt, t z ψ ( z) = 2πi ( t) h dt t z where the uow complex valued fuctios g=g(t) ad h=h(t) satisfy the Hölder coditio. These fuctios are ot idepedet from each other due to the fact that two idepedet real-valued fuctios are eeded to determie both holomorphic fuctios. This arbitrariess is used further i order to obtai sigular itegral equatios for the cosidered problems (A ad B). Taig ito accout that for the ()-derivative of the holomorphic fuctios (e.g., Mushelishvili []) ( ) ( ( ) ) ( ) ϕ z = g t 2πi t z dt (7) oe ca derive the followig represetatios for the stress fuctios g ( ) zg ( ) + h dt P z, z = Re dt, D z, z = (8) πi t z 2πi t z These represetatios are valid for ay poit of iterior as well as exterior domais of the etire complex plae separated by for the cotour. Boudary values of holomorphic fuctios are foud by the Sohotsi- Plemel formulae 2 ϕ ± = ± g + I( g) (e.g., []). The for the boudary values of the stress fuctio oe obtais ± 2P = ± g ± g + I( g ) + I( g ) (9) ± 2D = ±g ± h + I g + I h ( ) ( ) where superscripts ± deote boudary value for iterior ad exterior domais respectively ad I(..) is sigular operators as follows ( ) g t ( g) ( ) (6) I = πi t dt (0) Substitutio of (9) ito (2) results i d ± g + g + ( g + h ) + 2 d () d I( g ) + I( g ) + ( I( g ) + I( h )) = N + it 2 d It ca be oted that the expressio i the first bracets i (9) is the derivative d ( h + g + g ), therefore usig the arbitrariess i the choice of the d fuctios g ad h oe ca exclude the o-itegral term i (9) by ISSN X (o-lie)

5 Boudary Elemets XXVII 259 puttig h = g g. This agrees with the approach used by Liov [3] ad evetually leads to the sigular itegral equatio, SIE, for the first PEBVP, which solutio should be foud umerically. To do this oe ca approximate the fuctio g by usig some systems of basis fuctios. If, for istace, power series are used g = = 0 = 0 c t, oe obtais the followig represetatios for the holomorphic fuctios (iterior domai) ctt ϕ( z) = c z, ψ( z) = ϕ( z) dt (2) 2πi t z = If orthogoal polyomials are used, this results i differet fuctios H (z), however the geeral form of (5) remais. Approximatios of g by cubic splies also lead to the represetatio (5) because the fuctios eterig ito the Cauchy itegrals satisfy the Hölder coditio; ad thus the itegrals represet holomorphic fuctios. It is evidet that (2) is a particular case of (5). Similar coclusio ca be draw for the secod PEBVP formulated i displacemets ad for o-classical PEBVP specified by equatio (3). The latter however may ot have a uique solutio. It has bee show [2] that the umber of idepedet solutios i this problem is defied by the idex, 2Κ, of the correspodig system of SIEs that is calculated through the icremet of α() after the complete traverse of the boudary 2K = Id D = 2π dα (3) Thus, for a arbitrary, simply coected domai, bouded by a smooth closed cotour ad for ay o-egative idex, 2K, the solutio for stresses cotais ot less tha 8K+5 arbitrary real costats. For ay egative idex 2K<- o bouded solutios exist. This aalysis has to be acowledged whe oe assumes form (5). Therefore, the system for the determiatio of uow coefficiets from boudary coditios (3) should have less ra the the umber of uows (provided that 2K 0, i.e. several idepedet solutios exist), which meas that 8K+5 real parameters eterig i complex coefficiets a caot be determied. 3 Numerical examples 3. Plae with elliptic hole Let us cosider a elastic isotropic plae with elliptic hole havig semi-axes a ad b. Let the boudary of the hole be subected to ormal pressure p (Fig a). A aalytical solutio ca be derived for plae strai (ad plae stress) coditios i the form (Mushelishvili, []) ISSN X (o-lie)

6 260 Boudary Elemets XXVII pm p pm + mξ a b ϕ ( z) =, ψ( z) =, m =, b a (4) 2 ξ ξ ξ ξ m a + b This solutio has bee obtaied by coformal mappig of the exterior of ellipse oto the exterior of uite circle, which is performed by the followig fuctio m ω( ξ) = ξ + (5) ξ Firstly a approximate solutio decayig at ifiity is sought i the form (5) where H (z)=z - (= ). Substitutio of (5) ito (2) followed by itroductio of N equidistat collocatio poits,, results i the followig system of 4 liear algebraic equatios MC = B (6) where C is 4-vector of uows with the compoets C =Re(A ), = 2, C =Im(A ), =2+ 4; M is Nx4 matrix of the system; B is 2N vector of applied load with the compoets B m =p, m= N, B m =0, m=n+ 2N. The compoets of the matrix M are as follows M, = F ( ) (7) where F ( ) = i 2 2iα( ) + + e 2iα( ) e + + 2iα( ) i e 3 2iα( ) ( 2) e 2 i 2 < 2 2 < 3 3 < 4 (8) If the umber of collocatios ca be chose greater tha the umber of the sought coefficiets, the system (6) is overdetermied. Its iverse ca be foud by the sigular value decompositio, SVD (see e.g. Golub ad va Loa [4]). This method allows to cotrol the coditio umber of the matrix Cod(M) ad if ecessary to perform regularisatio if Cod(M) is large. The solutio of (6) is preseted as follows T C = MregB, Mreg = V D U (9) Here U (2Mx2N) ad V (4x4) are orthogoal matrices i the SVD of the matrix M, M=UDV T, D is (4x4) diagoal matrix formed from the sigular values, d, of the matrix M placed i descedig order, d d 2 d, D is the diagoal matrix of the ra as follows D =diag{d -,d - 2, d -,0 0}. If o regularisatio has bee made the D =D -. Calculatios have bee performed for differet m ad N. It has bee foud that satisfactory results ca be obtaied for ellipsis with 0 m<0.5, for arrow ISSN X (o-lie)

7 Boudary Elemets XXVII 26 ellipsis it is required to icrease the order of approximatio ad the hece the umber of collocatio poits, which leads to large matrices ad as a cosequece to the icrease i Cod(M). Startig from certai of values the regularisatio does ot provide sufficiet accuracy. Therefore, the represetatio for the basis fuctios has bee revised i order to cover all shapes as follows. Boudary coditios (2) are rewritte i the form (Mushelishvili []) iθ Φ + Φ() t [ ω Φ + ω Ψ() t ] = N + it, t = e (20) 2 t ω the the fuctios i (8) become F = t i 2 it + t t t 2 it i ω( t) t ω 2 ω ω 2 ( 2) t ω ω ω ω < 2 2 < 3 3 < 4 (2) The compariso of results calculated by usig (8) with the aalytical solutio shows that satisfactory results ca be obtaied for ellipses with m varyig i a wide rage 0 m<0.95. Fig illustrates the boudary stresses for m=0.7, =48, N=3=44, p=. It is evidet that results of calculatios are very close to the aalytical solutio. The solutio for σ ρ calculated (Fig b) is accurate withi %. Peas i Fig d that shows the ratio of σ θ calculated/ σ θ aalytical are associated with the small values of stresses (Fig c) (a) σ ρ (b) y b=0.3 p 0 m=0.7 a=.7 x θ θ θ Figure : Compariso with aalytical solutio for the case of uit pressure, p=: (a) scheme, (b) calculated ormal stress, (c) calculated circumferetial stress, (d) ratio of calculated ad exact circumferetial stresses. σ θ (c) (d) ISSN X (o-lie)

8 262 Boudary Elemets XXVII 3.2 A model of elastic stress field i Atarctica This is a example of recoverig stress traectories i Atarctica. The data used for the recostructio are available through the World Stress Map proect [5]. The mai feature of these data is that the iformatio o stress orietatios is maily available i a relatively arrow regio located ear the boudary of the Atarctic plate (show by shaded areas i Fig. 2). The aalysis of data has show that the idex of the problem, 2K, determied by (3) ca be posed equal to zero. Therefore the solutio obtaied may cotai some additioal parameters that caot be uidetified from the data. O the other had o reliable iformatio o the secod boudary coditio i formulae (3) ca be derived. For that reaso the approach employed i [6] has bee used to cosider a model of the stress field i Atarctica. It does ot require the owledge of the ormal derivative o the boudary: istead all data o stress orietatios ear the plate boudary are used. This actually ca be cosidered as a approximatio of the secod boudary coditio i (3). Therefore the oly coditio used i modellig ca be writte as Im iα( z, z ) [ e D( z, z) ] = 0, z Ω (22) where Ω is the domai where the data is available. It is show i Fig 2 by light colour. Substitutio of (5) ito (22) leads to the followig form for the stress fuctio D 2 ( z, z) c F ( z, z), F ( z, z) = = 0 zh = H ( z), ( z), < 2 (23) Equatio (23) beig substituted ito (22) results i the followig fuctioal equatio Im 2 = 0 c ( z, z ) F ( z, z) = 0 z Ω iα e, (24) Equatio (24) is further reduced to a system of liear algebraic equatios with respect to uow costats C. Sice the fuctio α is ow at the poits z oe obtais by discretizatio of (25) the followig system of N complex equatios for the determiatio of 2+ complex costats Im 2 = 0 e ( z, z ) C = 0, = N i α F (25) ISSN X (o-lie)

9 Boudary Elemets XXVII 263 Equatio (25) is a homogeeous oe, thus at least oe extra coditio is required i order to fid its o-trivial solutio. As it is evidet that the multiplicatio of (22) by ay real costat does ot violate this system, which meas that ay solutio satisfyig (22) ca be ormalised by a real costat. This costat ca be chose from the coditio that the average modulus of D over the domai is uity. Sice D =e -iα D, the extra equatio assumes the form N 2 = = 0 e ( z z ) c N iα F =, (26) It is coveiet to rewrite system (25)-(26) i a matrix form by presetig the complex costats as A =C +ic 2++ (ad hece itroducig a vector C of 4+3 real uows) ad separatig the real ad imagiary parts of complex equatios, which evetually leads to system (6) i which 4 is replaced by 4+3. Thus, M is (N+) (4+3) matrix of the system with the coefficiets M, defied below; C is (4+3) vector of real uows C ; ad B is (N+) vector all compoets of which are zero except of the last oe that is equal to N due to (27) M M, N +, Im = Im N e = = N i ie = [ ( )] iα e F z, z, iα F ( z, z ) [ ie 2 ], iα α F F ( z, z ), 2 ( z, z ), 0 2, 2 < < 4 + = N (27) Solutio of (27) is give by (9). It should be oted that stress traectories are defied uiquely; they are schematically show i Fig 2 by dotted lies together with the data o stress orietatios (blac segmets) ad plate boudaries. Polyomials of the secod order were used for approximatio. The data cotai 357 poits of differet quality (raged from A to C, see [5]). 4 Coclusios It is show that the approximatio of the stress potetial by liear combiatios of holomorphic fuctios ca be used as the uiform approach for solvig the PEBVP ad that other ow umerical methods ca be reduced to the this problem. Two cosidered examples idicate that the proposed approach is capable to solve both classical ad o-classical PEBVPs. ISSN X (o-lie)

10 264 Boudary Elemets XXVII Ido-Australia plate Atarctic plate Atarctica Greewich Data mostly related to plate boudaries Scotia plate Plate boudaries Figure 2: Stress field i Atarctica: data (blac segmets) ad stress traectories (dotted lies). Acowledgmet This wor is supported by the MNRF ACcESS proect. Refereces [] Mushelishvili, N.I. Some basic problems of the mathematical theory of elasticity, P. Noordhoff, Groige: the Netherlads, 963. [2] Galybi, A.N., Muhamediev, Sh.A. Plae elastic boudary value problem posed o orietatio of pricipal stresses. Joural of the Mechaics ad Physics of Solids. 47 (), pp , 999. [3] Liov, A.M. Complex boudary itegral equatio method for elasticity. Naua, St Petersburg, 999. [4] Golub, G.H. ad va Loa C.F. Matrix computatios (2 d editio), The Johs Hopis Baltimore / Lodo: Uiversity Press, 989. [5] Reiecer, J., Heidbach, O., Tigay, M., Coolly, P. ad Müller, B., The 2004 release of the World Stress Map ( [6] Galybi, A.N. ad Muhamediev, Sh.A. Determiatio of elastic stresses from discrete data o stress orietatios. It. Joural of Solids ad Structures. 4 (8-9), pp , ISSN X (o-lie)