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1 Stress and Equilibrium

2 Outline Bod and Surface Forces( 体力与面力 ) raction/stress Vector( 应力矢量 ) Stress ensor ( 应力张量 ) raction on Oblique Planes( 斜面上的应力 ) Principal Stresses and Directions( 主应力与主方向 ) Mohr s Circles of Stresses( 莫尔应力圆 ) Octahedral Stresses( 八面体应力 ) Spherical and Deviatoric Stresses( 球应力与偏应力 ) Conservation of Linear Momentum( 线动量守恒 ) Conservation of Angular Momentum( 角动量守恒 ) Equilibrium Equations( 平衡方程 ) Equilibrium Equations in Curvilinear Coordinates( 柱坐标与球坐标系下的平衡方程 )

3 Bod and Surface Forces Eternal loads include bod and surface forces. Surface forces Forces are vectors (unit: N) F Fe F e F e Fe i i Often interpreted in terms of densit: bod force densit and surface force densit 3

4 raction/stress Vector A n F n σ Given ΔF as the force transmitted across ΔA, a stress traction vector can be defined as n, n lim A0 F A Units: Pa (N/m ), 1 MPa = 10 6 Pa, 1 GPa = 10 9 Pa. Decomposition of the traction vector n, n n t n t t 4

5 Stress ensor n n e e e e, z z n n e e e e, z z n n e e e e, z z z z z z ij z z z z von Karman Notation 5

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. 6

7 raction on an Oblique Plane - D t n A n 0 F 0 F n A Acos Asin n A Acos Asin n n n n n n n n n n n n n = n σ D Cauch s relation 7

8 raction on an Oblique Plane - 3D n z n n 3D Cauch s relation he state of stress at a point is defined b:,, z,,, z, z, z, z Consider the tetrahedron with unit normal n nei ni cos ne, i n e 0 F 0 F n A An An An n A An An An n A An An An n n i z z z z z z z z z z n i j ji = n σ 8

9 Stress ransformation - D Q Q σ QσQ cos sin cos sin σ sin cos sin cos cos sin cos sin sin cos ave ave ; Mohr s stress circle Stress smmetr will be proved shortl. 9

10 Stress ransformation - 3D ake (the simplest) spherical transformation as an eample e cos sin 0 ez cos sin e z sin cos 0 r sin cos 0 0 cos sin e e e sin cos e e e QQ ; σqσq ij ik jl kl Q Q Q Q Q Q z z z z z Q z Q Q z Q z Q Q Q Q Q Q Q Q z z z No Mohr s sphere eists. Not meaningful in terms of finding principal stresses. An arbitrar transformation can be realized b three successive rotations, i.e. the Euler s angles. 10

11 Principal Stresses and Directions Seeking the solution through an Eigen-equation n = n σ σ n = nn det ij nij 0 I I I 3 n 1 n n 3 hree invariants of the stress tensor 1 I 1 kk, I ii jj ij ji, I3 det ij ij I, I I 1 1 3,

raction Vector Decomposition n N S n n n N n n n n n n n i i ji j i k k ik i jk j 1 1 3 3 n n n n n 1 1 3 3 1 1 3 3 1 S N N

12 raction Vector Decomposition n N S n n n N n n n n n n n i i ji j i k k ik i jk j n n n n n S N N n n n n N n1 n n3 1 S N N n3 he principal space is taken as the reference S N 3 N 1 1 n1 n 3n3 N S n

13 Mohr s Circles of Stress Admissible N and S values lie in the shaded area. For 1 3 S ( N )( N 3) 0 S ( N 3)( N 1) 0 S ( N 1)( N ) 0 1 Sma 1 3 he principal space is taken as the reference. 13

Octahedral Stress n e e e 1 3 1n1 n 3n3 N n n n N S 1 1 3 3 1 1 N I 3 3 3 8 1 3 1 Octahedral shear stress is the equivalent stress of the

14 Octahedral Stress n e e e 1 3 1n1 n 3n3 N n n n N S N I Octahedral shear stress is the equivalent stress of the maimum distortion energ criterion. Etremel significant for plastic deformation 1 S N I 6 1 I

15 Sample Problem For the following state of stress, determine the principal stresses and directions and find the traction vector on a plane with the given unit normal. Also, determine the normal and shear stresses on this plane σ 1 0, n Solution: I 3, I 6, I n n n 4, 1, n1 n n n1 4n n n1 n 4n3 0 n e e e n e1 e e3 3 3 n e e = n σ e e e 1 0 n n n N N n e e e e e S

16 Spherical and Deviatoric Stress Decomposition of the stress tensor ˆ ij ij ij Spherical (mean) stress tensor: volume change + isotropic 1 1 ij mij kkij z ij 3 3 Deviatoric (octahedral) stress tensor: shape change ˆij ij ij elationships among principal stresses and directions ˆ nˆ ˆ nˆ nˆ ˆ nˆ nˆ ˆ nˆ ij j n i ij m ij j n i ij j n m i n n ij j n i nˆ i ni ˆ n n m 16

17 Conservation of Linear Momentum d d d d d d n ji, j Fi dv F ds FdV n ds FdV i i i ji j i S V S V V 3-D z F z 0 z ji, j Fi 0 F z z z z Fz z 0 0 -D 0 F d d d d d d FXd d FX 0 FY 0 17

18 Conservation of Angular Momentum d d d d d d 3-D -D n ds S V rf r rfdv S n ds F dv V ijk j lk l V ijk j k F dv ijk j lk, l ijk j k V V ijk jl lk ijk j lk, l ijk j k ijk jk dv z jk kj z z z F dv M d d d d d 1 1 d d d d d 18

19 Equilibrium Equations σ F σ F 0 ji, j Fi ij, j Fi 0 z F 0 z z F 0 z z z z Fz 0 z z F 0 z F 0 z z z F z 0 z z z z z z z z ij σ e e e e e e e F F F ij i j k i j k ij, j i k k ji σ e e e e e e e k σ σ k ji j i k j i ji, j i k 19

20 Clindrical Equilibrium Equations σ contraction on the first and third inde of σ r 1 r zr r σ er r r z r r 1 z r r e r r z r rz 1 z z rz ez r r z r σ F 0 r 1 r rz r Fr r r z r r 1 z r F 0, r r z r rz 1 z z rz Fz 0. r r z r 0, σ rz r r rz r z z z r e e e r r r rz z θ e e e r r z z z e e e rz r z z z 0

21 Spherical Equilibrium Equations σ contraction on the first and third inde of σ 1 1 cot σ sin 1 1 cot sin σ F sin cot e 1 1 cot F 0, sin cot F 0, sin cot F 0. sin e e σ e e e φ e e e θ e e e 1

22 Outline Bod and Surface Forces( 体力与面力 ) raction/stress Vector( 应力矢量 ) Stress ensor ( 应力张量 ) raction on Oblique Planes( 斜面上的应力 ) Principal Stresses and Directions( 主应力与主方向 ) Mohr s Circles of Stresses( 莫尔应力圆 ) Octahedral Stresses( 八面体应力 ) Spherical and Deviatoric Stresses( 球应力与偏应力 ) Conservation of Linear Momentum( 线动量守恒 ) Conservation of Angular Momentum( 角动量守恒 ) Equilibrium Equations( 平衡方程 ) Equilibrium Equations in Curvilinear Coordinates( 柱坐标与球坐标系下的平衡方程 )

Untitled-3

Untitled-3 SEC.. Separable Equations In each of problems 1 through 8 solve the given differential equation : ü 1. y ' x y x y, y 0 fl y - x 0 fl y - x 0 fl y - x3 3 c, y 0 ü. y ' x ^ y 1 + x 3 x y 1 + x 3, y 0 fl