东南大学：《固体力学基础》课程教学课件（英文讲稿）03 Stress Measures

Stress Measuresmi@se.ed.cn

Stress Measures

Outline·Bodyand surfaceforces（体力与面力）Traction/stressvector（应力矢量）Cauchystresstensor（柯西应力张量）Tractiononobliqueplanes（斜面上的应力）Differentstressmeasures（应力度量）Stressmeasuresfor infinitesimaldeformation（小变形应力度量）Principal stressesanddirections（主应力与主方向），Octahedral stresses（八面体应力）· Hydrostatic, deviatoric and von Mises effective stresses (平均应力、偏应力、米泽斯等效应力）Conservationoflinearmomentum（线动量守恒）Conservationofangularmomentum（角动量守恒）·Rateof workdoneby stresses（应力做功的速率）2

Outline • Body and surface forces（体力与面力） • Traction/stress vector（应力矢量） • Cauchy stress tensor （柯西应力张量） • Traction on oblique planes（斜面上的应力） • Different stress measures（应力度量） • Stress measures for infinitesimal deformation（小变形应力 度量） • Principal stresses and directions（主应力与主方向） • Octahedral stresses（八面体应力） • Hydrostatic, deviatoric and von Mises effective stresses（平 均应力、偏应力、米泽斯等效应力） • Conservation of linear momentum（线动量守恒） • Conservation of angular momentum（角动量守恒） • Rate of work done by stresses（应力做功的速率） 2

Body and SurfaceForces External loads include body and surface forcesP3Body Forces: F(x)PSurfaceforcesForces are vectors (unit: N)F= Fe, +Fe2 +Fe, = Fe Often interpreted in terms of density: body force densityand surface force densityT"(x)dSFRF(x)dVS3

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

Traction/Stress VectornAFnLARDeformedOriginalconfigurationconfigurationGiven AF as the force transmitted across △A, a stresstraction vector can be defined asAFT() (x,n)= limM->0 AUnits: Pa (N/m2), 1 MPa = 106 Pa, 1 GPa = 109 PaDecomposition of the traction vectorT(") (x,n) =o n+tt=o n+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    ,                n T x n n t n t t • Decomposition of the traction vector

Cauchy Stress Tensorr(n) (T(x,n=e)=oe+twe,+txe.x.n=+te.+=Te+o.e.OOxzxyVZZXV2A福0zy7Txy0xvonKarmanNotation5

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

Sign Convention: Normal stress: tension positive / compression negative: Shear stress: product of the surface normal (the firstsubscript) and the stress direction (the second subscript): All stress components shown in the figure are positiveQytyxtyzTxyxzOz6

6 Sign Convention • Normal stress: tension positive / compression negative • Shear stress: product of the surface normal (the first subscript) and the stress direction (the second subscript) • All stress components shown in the figure are positive

Traction on an Oblique Plane - 2D[0=ZF-x1[0=ZF2-福T()A= C,AcosO+txAsin 0(n)A = tAAcosO+o,AAsin0-(no.n.+t.nyx1yxXT(n)T.n.+on1x1XxVineoo2D Cauchy's relationnn.o7

                    0 0 cos sin cos sin x y x x yx y xy y x x x yx y y xy x y y x xy x y x y yx y F F T A A A T A A A T n n T n n T T n n T n                                                                n n n n n n n n T = n σ n t  x  y yx  xy  n T 7 Traction on an Oblique Plane - 2D 2D Cauchy’s relation

Traction on an Obliue Plane - 3D. The state of stress at a point is defined bynsOx, Ty, x- Tyx,Oy,TyeTexTy,O.oZ: Consider the tetrahedron with unit normal nTxyn·etz(acos(n,e.)n;ane;T(e)T(ntr[0=ZF21K0ZF,0=toz?yT(")A=o,AAn,+twAn,+tAn,T(")A=TAAn, +,AAn, +T,AnT()A= TxAn, +T,AAn, +o,An,(nn.03D Cauchy's relationn.o8

  T x n   T y n   T z n • The state of stress at a point is defined by: , , , , , , , ,          x xy xz yx y yz zx zy z • Consider the tetrahedron with unit normal n             cos , 0 0 i i i i x y x x x yx y zx z y xy x y y zy z z xz z yz y z z i j ji n F F T A An An An T A An An An T A An An An T n                                                    n n n n n n e n e n e T = n σ 3D Cauchy’s relation 8 Traction on an Oblique Plane - 3D

Different Stress Measures: Cauchy stress o; (the actual force per unit area acting on an actual,deformed solid) is the most physical measure of internal force: Other definitions of stress often appear in constitutive equations: Other stress measures regard forces as acting on the undeformedsolid.dp(n)0dp(n)dAdAu(x)xOriginalDeformed-configurationconfiguration9

Different Stress Measures • Cauchy stress σij (the actual force per unit area acting on an actual, deformed solid) is the most physical measure of internal force. • Other definitions of stress often appear in constitutive equations. • Other stress measures regard forces as acting on the undeformed solid. 9

Different Stress Measures. Cauchy stress Cj: Kirchhoff stress has no obvious physical significanceT= JoT,=Jo,Nominal (1st Piola-Kirchhoff) stress / PK1 stress: S? Material (2nd Piola-Kirchhoff) stress / PK2 stress: Zfjdp(n)dp(a) = dAnkOkidp(nAdp(a) = dAon S udAu(x)DeformeddP(no) = dAgnZkiurationconfiguration10

Different Stress Measures • Cauchy stress σij • Kirchhoff stress has no obvious physical significance. ij ij τ   J J σ   • Nominal (1st Piola-Kirchhoff) stress / PK1 stress: Sij • Material (2nd Piola-Kirchhoff) stress / PK2 stress: Σij       0 0 0 0 0 i k ki i k ki i k ki dP dAn dP dA n S dP dA n      n n n 10