8 CHAPTER 1 Basic Elasticity
Fig.1.
Stresses on the faces of an element at a point in an elastic body.
whichsimplifiesto
τxyδyδzδx+
∂τxy
∂x
δyδz
(δx)^2
2
−τyxδxδzδy−
∂τyx
∂y
δxδz
(δy)^2
2
= 0
Dividingbyδxδyδzandtakingthelimitasδxandδyapproachzero.
Similarly,
τxy=τyx
τxz=τzx
τyz=τzy
⎫
⎬
⎭
(1.4)
Wesee,therefore,thatashearstressactingonagivenplane(τxy,τxz,τyz)isalwaysaccompaniedby
anequalcomplementary shearstress(τyx,τzx,τzy)actingonaplaneperpendiculartothegivenplane
andintheoppositesense.
Nowconsideringtheequilibriumoftheelementinthexdirection
(
σx+
∂σx
∂x
δx
)
δyδz−σxδyδz+
(
τyx+
∂τyx
∂y
δy
)
δxδz
−τyxδxδz+
(
τzx+
∂τzx
∂z
δz
)
δxδy
−τzxδxδy+Xδxδyδz= 0