弹性力学 作业答案 2

Homework21.Solution:(a)tan(2α)=2τxyσx-σyAccordingtotheproblem,wecaninferthat,α=14arctan()231414πarctan()andarctan()+23232So,theprincipaldirectionsare(b)σ=σxcosα+σysinα+2τxysinαcosαSubstitutingα=1414πarctan()andα=arctan()+respectivelyintheformulaabove,2323222wecaninferthat,⎧σ1=110MPa⎨⎩σ2=10MPa(c)Iftheprincipaldirectionsaretakenasthexandyaxes,⎧σ=σxcos2α+σysin2α⎪(1)⎨1⎪τ=(σy-σx)sin2α⎩2⎧⎪σy=σ1=110MPaAnd⎨⎪⎩σx=σ2=10MPaWecanfigureoutthatwhenα=τmax=50MPa.π4,thein-planeshearingstressachievesmaximum:AccordingtoEqs.(1),wecanfigureoutthestateofstressontheelementcontainingthe⎧σ=60MPamaximumin-planeshearingstressis⎨⎩τ=50MPa⎧σ1=110MPa⎪(d)⎨σ2=10MPa⎪σ=0MPa⎩3τabs=σ1-σ32=55MPa2.Solution:ThecircleinthepictureaboveistheMohrCircleforthestrainrosette,sowecanfigureoutthat⎧ε1=2015⨯10-6⎪-6⎨ε2=935⨯10⎪φ=6.5︒⎩3.Solution:ψθ=⎛∂v∂u⎫∂u22cosθ+-sinθcosθ-sinθ⎪∂x∂y⎝∂y∂x⎭∂vSo,ψθ'=-2⎛∂v⎝∂x+⎛∂v∂u⎫∂u⎫sin2θ+2-⎪⎪cos2θ∂y⎭∂y∂x⎝⎭Ifψθ'=0,wecanobtainthat,⎛∂v∂u⎫⎛∂v∂u⎫-2+sin2θ+2-⎪⎪cos2θ=0⎝∂x∂y⎭⎝∂y∂x⎭∂vOr,tan2θ=∂y∂v∂x-+∂u∂x.∂u∂yTherefore,wecanmakeaconclusionthatthelineelementsatthepointx,ythathavethemaximumandminimumrotationarethoseinthetwoperpendiculardirectionsθ∂v∂y∂v∂x-+∂u∂x.∂u∂ydeterminedbytan2θ=4.Solution:Inproblemsofplanestress,wehavethestress-strainrelations:⎧Eσ=ε+vεy)⎪x2(x1-v⎪E⎪σ=ε+vεx)⎨y2(y1-v⎪E⎪τ=γxy⎪xy2(1+v)⎩Eqs.(a):⎧∂u⎪εx=∂x⎪⎪∂v⎨εy=∂y⎪⎪∂u∂v+⎪γxy=∂y∂x⎩SubstitutingEqs.(a)inthestress-strainrelations,⎧E⎛∂u∂v⎫σ=+v⎪x⎪21-v∂x∂y⎝⎭⎪⎪E⎛∂v∂u⎫⎪σ=+v⎨y⎪21-v∂y∂x⎝⎭⎪⎪⎛∂u∂v⎫E⎪τxy=+⎪21+v∂y∂x⎪()⎝⎭⎩Eqs.(18):⎧∂σx∂τxy++X=0⎪∂x∂y⎪⎨⎪∂σy+∂τxy+Y=0⎪∂y∂x⎩Accordingtotheproblem,wecanseethatX=0,Y=0.So,⎧∂σx∂τxy+=0⎪∂y⎪∂x⎨⎪∂σy+∂τxy=0⎪∂y∂x⎩⎧E⎛∂u∂v⎫σ=+v⎪x⎪21-v∂x∂y⎧∂σx∂τxy⎝⎭⎪+=0⎪⎪∂yE⎛∂v∂u⎫⎪⎪∂x+vSubstituting⎨σy=in,wecanobtain,⎨⎪21-v⎝∂y∂x⎭⎪⎪∂σy+∂τxy=0⎪⎪∂y∂x⎛∂u∂v⎫E⎩⎪τxy=+⎪2(1+v)⎝∂y∂x⎭⎪⎩⎧∂2u∂2u1+v+⎪2+2∂x∂y1-v⎪⎨22⎪∂v∂v1+v++⎪∂x2∂y21-v⎩∂⎛∂u∂v⎫+⎪=0∂x⎝∂x∂y⎭∂⎛∂v∂u⎫+⎪=0∂y⎝∂y∂x⎭5.Solution:TakethesmalltriangularprismABCasshowninthepictureabove.DivideitintotwosmallrighttriangularprismABDandBCD,anduseboundaryconditionsrespectively.ForABD,⎧⎪lσx+mτxy=0⎨⎪⎩mσy+lτxy=0ForBCD,⎧⎪lσx'-mτxy=0⎨mσ-lτ=0⎪yxy⎩Put⎨⎧⎪lσx+mτxy=0⎪⎩mσy+lτxy=0and⎨⎧⎪lσx'-mτxy=0⎪⎩mσy-lτxy=0together,wecanobtain,⎧lσx+mτxy⎪⎪mσy+lτxy⎨⎪mσy-lτxy⎪lσ'-mτxy⎩x=0⎧σx=0⎪=0⎪σx'=0⇒⎨σy=0=0⎪⎪τ=0=0⎩xySincestressesvarycontinuouslythroughouttheregionandtheprismABCcanbemadesmallerandsmaller,therefore,wecanmakeaconclusionthatthereisnostressatallattheapexofthetooth.6.Solution:εx=γxy=∂u∂x，εy=+∂v∂x∂v∂y，εz=∂u∂z∂w∂z+∂w∂x，γyz=∂v∂z+∂w∂y∂u∂y，γxz=Accordingtotheproblem,u=a+ω1z-ω2yv=b+ω2x-ω3zw=c+ω3y-ω1xTherefore,wecanfind