UNIVERSITY OF CALIFORNIA BERKELEY Structural Engineering,Department of Civil Engineering Mechanics and MaterialsFall 2013 Professor: S. GovindjeeMichell’s Solution to the Bi-HarmonicFor the case where stress in not a function of θ then the solution to∇4Φ =∂∂r2+1r∂∂r+1r2∂∂θ2Φ,rr+1rΦ,r+1r2Φ,θθ= 0 (1)is given byΦ = a0oθ + aoln(r) + bor2+ cor2ln(r) (2)If the coordinate system origin is in the body (is a material point) thena0o= co= ao= 0. If the body is multiply connected, then to ensure singlevaluedness the following integrability conditions on each internal cavity mustbe satisfied:ZCiddn(∇2Φ) ds = 0 (3)ZCiydds(∇2Φ) + xddn(∇2Φ)ds = −11 − νBx(4)ZCiyddn(∇2Φ) − xdds(∇2Φ)ds = −11 − νBy(5)where d/ds and d/dn respectively denote tangential and normal derivativesto the cavity boundary and Bxand Byare the body force components. Thefirst of these conditions requires ao= 0 if the origin is surrounded by aninternal boundary.The Michell solution to ∇4Φ = 0 is given by consider a separable solutionin the spirit of Levy as fn(r) exp[±inθ]. The result of much algebra and thecareful application of “variation of parameters” to treat multiple repeated1roots is:Φ = aoln(r) + bor2+ cor2ln(r) + dor2θ + a0oθ (6)+ a1rθ sin(θ) +b1r3+ a01/r + b01r ln(r)cos(θ) (7)+ c1rθ cos(θ) +d1r3+ c01/r + d01r ln(r)sin(θ) (8)+∞Xn=2anrn+ bnrn+2+ a0n/rn+ b0n/rn−2cos(nθ) (9)+∞Xn=2cnrn+ dnrn+2+ c0n/rn+ d0n/rn−2sin(nθ) (10)where aietc. are constants.Remarks:1. The term cor2ln(r) implies that uθis proportional to θ. Therefore,co= 0 if the origin is surrounded by any body boundary.2. The term dor2θ implies that uris proportional to θ. Therefore, do= 0if the origin is surrounded by any body boundary.3. The terms a1rθ sin(θ) and b01r ln(r) cos(θ) imply a multiple valued resultif the origin is surrounded by any body boundary. It can be shownthat this requires b01= −a1(1 − 2ν)/(2(1 − ν)); further, if the origin isa material point then a1= b01= 0.4. The terms c1rθ cos(θ) and d01r ln(r) sin(θ) imply a multiple valued resultif the origin is surrounded by any body boundary. It can be shown thatthis requires d01= c1(1−2ν)/(2(1−ν)); further, if the origin is a materialpoint then c1= d01= 0.5. The solution above comes from a paper by J.H. Michell, Proc. Lon-don Math Soc., vol. 31, p.100, 1899. A disscusion on the use of thissolution can be found in Theory of Elasticity by S.P. Timoshenko andJ.N. Goodier, Art. 43-46. A more comprehensive discussion of theproperties of the solution to the biharmonic equation in the context ofelasticity may be found in Mathematical Theory of Elasticity by I.S.Sokolnikoff, Art. 69-70. The essense of the art of using the generalsolution is to understand through the computation of examples whatthe individual terms do. Then to consider their linear combinations insuch a way as to solve practical problems of interest.26. Note that at times the general solution can be a bit too general. Forinstance, while expressions of the form θ ln(r) and r2θ ln(r) are formallypart of the solution through the approriate setting of inter-relationsbetween the coefficients, it can require the computation of many termsfor reasonable convergence. Thus it can be advantageous to approachthe solution to the bi-harmonic equation through separable expansionsof the form θf(r).7. The wikipedia page http://en.wikipedia.org/wiki/Michell solution con-veniently lists the resulting expressions for the stress components anddisplacement components for each term in the above expansion. Notethat each term by itself satisfies the bi-harmonic equation. Thus bylooking through the listing on the website you can find the terms youneed by mixing an matching to the boundary conditions you have inyour