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

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