95%(44)42 out of 44 people found this document helpful
This preview shows page 167 - 170 out of 247 pages.
rrθ2xy=1xx=Figure P7-27ˆzgρ= −feLzFig. P7.30
166SOLUTIONS MANUALSolution:Using the cylindrical coordinate system shown in the figure above, the bound-ary conditions of the problem can be stated as:Atz= 0, L:σzz= 0,σzr= 0,σzθ= 0,Atr=a:ur= 0,uθ= 0,uz= 0.(1)The body force component isfz=-ρg.Neglecting the end effects, we can assume the following form of solution for the antiplanestrain problem:ur= 0,uθ= 0,uz=U(r).(2)The nonzero strains and stresses are given by2εrz=dUdr,σzr=μdUdr.(3)Substitution into the third equilibrium equation yields (the other two equilibrium equa-tions are identically satisfied)dσzrdr+1rσzr-ρg= 0→1rddrrdUdr=ρgμ.(4)The solution of the equation is given byrdUdr=ρgμr22+A,ordUdr=ρgμr2+Ar,U(r) =ρgμr24+Alogr+B,whereAandBare constants of integration that must be determined using the boundaryconditions. The boundary conditions associated with the antiplane strain problem arethat (a)U(r) is finite atr= 0, and (b)U(a) = 0.The first condition givesA= 0and the second one leads toB=-(ρg/4μ)a2.The first boundary condition can bereplaced by the requirement thatσrz= 0 atr= 0. This will lead to the conclusionthatdU/dr= 0 atr= 0, from which we arrive at the same result (that is,A= 0). Thesolution becomesuz(r) =U(r) =-ρga24μ1-r2a2.(5)The stress field becomesσθz= 0,σzr=ρg2r.(6)Note that the boundary conditions (1) of the 3D problem are not satisfied atz= 0, L.Hence, it is only an approximate solution.7.31An external hydrostatic pressure of magnitudepis applied to the surface of a sphericalbody of radiusbwith a concentricrigidspherical inclusion of radiusa, as shown inFig. P7.31. Determine the displacement and stress fields in the spherical body. Usingthe stress field obtained, determine the stresses at the surface of a rigid inclusion in aninfinite elastic medium.abpRigid spherical coreSpherical shell ( ),μ λFigure P7-32Fig. P7.31
CHAPTER 7: LINEARIZED ELASTICITY167Solution:We use the semi-inverse method to solve the problem.Assume thatuφ=uθ= 0 anduR=U(R). The boundary conditions are:Atr=b:σRR=-p;AtR=a:uR=U(a) = 0(1)The solution of the Navier equations give the displacement [see Eq. (7.3.13)]U(R) =AR+1R2B,σRR(R) = (2μ+ 3λ)A-4μR3B,(2)whereμandλare the Lam´e constants.Using the boundary conditions, we obtain(2μ+ 3λ= 3K)U(a) = 0→B=-a3A;σRR(R) =3K+4a3μR3A,(3)σRR(b) =-p→A=-p3K+4a3μb3.(4)Hence the displacementuRand stress field are given byuR(R) =-b3pR3Kb3+ 4μa31-a3R3(5)σRR(R) =-1 + 2α(a3/R3)1 +βp,α=2μ3K,β= 2αa3b3(6)σθθ(R) =σφφ=-1-α(a3/R3)1 +βp.(7)To obtain the stresses at the surface of a rigid inclusion in an infinite elastic medium,we letb→ ∞and obtainσRR=-1 +4μa33KR3p,σθθ=σφφ=-1-2μa33KR3p .(8)At the interface of the rigid inclusion and the elastic medium (R=a), the stresses areσRR=-1 +4μ3Kp,σθθ(R) =σφφ=-1-2μ3Kp .(9)7.32Consider the concentric spheres shown in Fig. P7.32. Suppose that the core is elastic