东南大学：《弹性力学》课程教学课件（英文讲稿）04 Stress and Equilibrium

Stress and Equilibrium

Stress and Equilibrium

OutlineBodyand SurfaceForcesTraction/Stress Vector Stress TensorTraction on Oblique PlanesPrincipal Stresses and DirectionsMohr's Circles of Stresses Octahedral StressesSpherical and Deviatoric StressesConservation of Linear Momentum Conservation of Angular MomentumEquilibrium Equations: Equilibrium Equations in Curvilinear Coordinates2

Outline • Body and Surface Forces • Traction/Stress Vector • Stress Tensor • Traction on Oblique Planes • Principal Stresses and Directions • Mohr’s Circles of Stresses • Octahedral Stresses • Spherical and Deviatoric Stresses • Conservation of Linear Momentum • Conservation of Angular Momentum • Equilibrium Equations • Equilibrium Equations in Curvilinear Coordinates 2

Body and Surface ForcesExternal loads include body and surface forcesP:Body Forces: F(x)++1Surfaceforces. Forces are vectors (unit: N)F= Fe, +Fze2 + Fe, = Fe,: Often interpreted in terms of density: body force densityand surface force densityF, = (l,T"(x)dSF, = JF(x)dV3

Body and Surface Forces • External loads include body and surface forces. Surface forces • Forces are vectors (unit: N) F e e e e     F F F F 1 1 2 2 3 3 i i • Often interpreted in terms of density: body force density and surface force density 3  d b V  V  F F x  d S S  S  n F T x

Traction/Stress VectorAF: Given △F as the force transmitted across △A, a stresstraction vector can be defined asAFT" (x) = limM-0 △AUnits: Pa (N/m2), 1 MPa = 106 Pa, 1 GPa = 109 PaDecomposition of the traction vectorT"(x)=on+tt =on+t't'+t"t"4

Tn σ  Units: Pa (N/m2 ), 1 MPa = 106 Pa, 1 GPa = 109 Pa. • Given ΔF as the force transmitted across ΔA, a stress traction vector can be defined as F A n 4 Traction/Stress Vector   0 lim  A A    n F T x                 n T x n t n t t • Decomposition of the traction vector

Stress TensorXoer+to.eTry+treOeOTT.L1VZ2Z福yxJZ2Xz0.TTZX1ZTy0xvon KarmanNotation5

5 Stress Tensor         x x xy y xz z n T x e e e   yx x y y yz z       n T x e e e   zx x zy y z z       n T x e e e x xy xz ij yx y yz zx zy z                         von Karman Notation