GIÁO TRÌNH KHAI THÁC PHẦN mềm TRONG GIA CÔNG KHUÔN mẫu chapter IV stress

1 Chapter IV: Stress 1 Content • Stress Components • State of Stress (2D & 3D) • Transformation of Stress Components • Principal Stresses • Stress Invariants • Local Equilibrium Equations • Hydrostatic Stress Components • Deviatoric Stress Components 01.058 01.10.2001 Chapter IV: Stress 2 Stress Components on a Plane P ∆ A plane body : normal of plane : tangent to plane : simplest static equivalent force on n t F A ∆ ∆ r r ur Normal stress component at point P on given plane: Shear stress component at point P on given plane: 0 lim n n A F d F A dA σ ∆ → ∆ = = ∆ uur uur 0 lim t t A F d F A dA τ ∆ → ∆ = = ∆ uur uur 2 Chapter IV: Stress 3 Stress StressStress Stress General notion of Stress General notion of Stress General notion of Stress General notion of Stress (Kh¸i niÖm chung) (Kh¸i niÖm chung)(Kh¸i niÖm chung) (Kh¸i niÖm chung) Internal forces and stresses in the body Definition of Stress at the point: Normal and tangential Stresses Stresses in 3 directions: On the one plane there are 3 terms of stresses: - 1 normal stress - 2 tangential stresses P 1 P 2 P 3 P 4 P 5 P n Chapter IV: Stress 4 Engineering versus True Stress l 0 F F dl A 0 A Assuming that the axial stress is ‘uniform’ over the cross-secction we can define: Engineering Stress: True Stress: 0 0 F A σ = F A σ = We use only true stresses in plasticity! 3 Chapter IV: Stress 5 Stresses on the co-ordination planes (ø øø øng suÊt trªn c¸c mÆt to¹ ®é) ng suÊt trªn c¸c mÆt to¹ ®é)ng suÊt trªn c¸c mÆt to¹ ®é) ng suÊt trªn c¸c mÆt to¹ ®é) σ σ σ σ σ σ σ σ σ xx xy xz yx yy yz zx zy zz           τ τ τ τ τ τ xy yx xz zx zy yz = = = ; ; σ τ τ σ τ σ x x y x z y yz z • • •           Co-ordination sysytem of Decac Chapter IV: Stress 6 State of Stress at a Point 01.061 01.10.2001 xx xy xz ij yx yy yz zx zy zz σ τ τ σ τ σ τ τ τ σ     =       The state of stress at point P: Because of moment equilibrium: = , , xy yx yz zy zx xz τ τ τ τ τ τ = = Note: All components shown are positive! 4 Chapter IV: Stress 7 Special Stress States Plane Stress State ( Biaxial Stress State): 0 0 0 zz zx zy σ τ τ  =  = ⇒   =  0 0 0 0 0 xx xy ij yx yy σ τ σ τ σ     =       Uniaxial Stress State: Only 0 xx σ ≠ ⇒ 0 0 0 0 0 0 0 0 xx ij σ σ     =       Chapter IV: Stress 8 Stress Vector Let’s define the “true (Cauchy or Euler) stress tensor” in “vector notation”: { } x y z xy i yz zx yx zy xz T T σ σ σ τ τ τ τ τ τ               = =                 [ ] x xy xz ij yx y yz zx zy z σ τ τ σ σ τ σ τ τ τ σ     = =       5 Chapter IV: Stress 9 Stress on the sloping plane (ø øø øng suÊt trªn mÆt ph¼ng nghiªng) ng suÊt trªn mÆt ph¼ng nghiªng)ng suÊt trªn mÆt ph¼ng nghiªng) ng suÊt trªn mÆt ph¼ng nghiªng) It can be demonstrated, that stress state of on point is completely determined if all term of stress tensor drawing through this point are known. It means, from the term of known stresses, we can calculate the normal and tangential stresses of any sloping plane drawing throung the point. Call N is a normal line of sloping plane. The position of N can be determined by cosin directions: cosα = cos(N,x) = a x cosβ = cos(N,y) = a y cosγ = cos(N,z) = a z A B C Chapter IV: Stress 10 Call ∆F is an area of sloping plane ABC, area of rest planes of tetrahedron (khối tứ diện) coresponding to their position will be ∆F x , ∆F y and ∆F z . (OAB is projection of ABC on plane Oxy) Supposing (giả thiết) that the total stress on sloping plane is S, in which normal stress is σ n and tangential stress τ. The terms of total stress S following the directions of co-ordinations in succession (lần lượt) are s x , s y vµ s z . The tetrahedron OABC lies on the equilibrium state, if it satisfies (thỏa mãn) the following conditions: X S F F F F x x x xy y xz z ∑ = − − − =∆ ∆ ∆ ∆σ τ τ 0 Y S F F F F y yx x y y yz z ∑ = − − − =∆ ∆ ∆ ∆τ σ τ 0 Z S F F F F z zx x zy y z z ∑ = − − − =∆ ∆ ∆ ∆τ τ σ 0 S a a a S a a a S a a a x x x xy y x z z y yx x y y yz z x zx x zy y z z = + + = + + = + +      σ τ τ τ σ τ τ τ σ σ σ σ σ τ τ τ n x x y y z z xy x y yz y z zx z x a a a a a a a a a= + + + + + 2 2 2 2 2 2 τ σ 2 2 2 = −S n 6 Chapter IV: Stress 11 Main normal Stresses (principal stresses) Through on point in the stress state, it can be always found 3 planes perpendiculary (vuông góc) to each other, on which there are only normal stresses, the tangential stresses are 0. 3 normal stresses are called main normal stresses. σ σ σ 1 2 3 ; ; T ijσ σ σ σ σ = = • • •           11 22 33 0 0 0 T ij xx xy xz yy yz zz σ σ σ σ σ σ σ σ = = • • •           This is a linear homogeneous (thuần nhất) equation system. This system has the roots (nghiệm số) if: ( ) ( ) ( ) σ σ τ τ τ σ σ τ τ τ σ σ x y z − − − = xy xz yx yz zx zy 0 ( ) ( ) ( ) σ σ σ σ σ σ σ σ σ σ σ σ τ τ τ σ σ σ τ τ τ σ τ σ τ σ τ 3 2 2 2 2 2 2 2 2 0 − + + − − − − + + + − − + − − − = x y z x y y z z x xy yz zx x y z xy yz zx x yz y zx z xy Stress Invariants Chapter IV: Stress 12 - Invariant I 1 : first grade I const x y z1 = + + = σ σ σ ( ) I const x y y z z x xy yz zx2 2 2 2 = − + + + + + =σ σ σ σ σ σ τ τ τ I const x y z xy yz zx x yz y zx z xy3 2 2 2 2= + − − − =σ σ σ τ τ τ σ τ σ τ σ τ σ σ σ 3 1 2 2 3 0− − − =I I I Stress Tensor has also 3 independent stress invariants: - Invariant I 2 : second grade - Invariant I 3 : third grade Therefore, we get an equation: Solve this equation, we get 3 principal normal stresses σ σ σ 1 2 3 ; ; 7 Chapter IV: Stress 13 Stress Invariants Three independent stress invariants are: 1 2 2 2 2 2 2 2 3 , , 2 xx yy zz xy yz zx xx yy yy zz zz xx xx yy zz xy yz zx xx yz yy zx zz xy I I I σ σ σ τ τ τ σ σ σ σ σ σ σ σ σ τ τ τ σ τ σ τ σ τ = + + = + + − − − = + − − − In terms of principal stresses: ( ) 1 1 2 3 2 1 2 2 3 3 1 3 1 2 3 , , I I I σ σ σ σ σ σ σ σ σ σ σ σ = + + = − + + = The principal stresses can be also found as the root of the equation: 3 2 1 2 3 0 I I I σ σ σ − − − = Chapter IV: Stress 14 Main tangential Stresses (principal tangential stresses) Determination: on which planes the tangential stresses reach maximum ??? τ σ σ σ σ σ σ 2 1 2 1 2 2 2 2 2 3 2 3 2 1 1 2 2 2 2 3 3 2 2 = + + − + +a a a a a a( ) a a a 1 2 2 2 3 2 1+ + = ( ) ( ) [ ] τ σ σ σ σ σ σ 2 1 2 1 2 2 2 2 2 3 2 1 2 2 2 1 1 2 2 2 2 3 1 2 2 2 2 1 1= + + − − − + + − −a a a a a a a a To determine the extrema, we derivative to a 1 ,a 2 ,a 3 and assign them to 0. 8 Chapter IV: Stress 15 τ σ σ τ σ σ τ σ σ 12 1 2 1 2 3 23 2 3 1 2 3 31 3 1 1 1 2 3 2 1 2 1 2 0 2 0 1 2 1 2 2 1 2 0 1 2 = ± − = ± = ± =       = ± − = = ± = ±       = ± − = ± = = ±                  ; ; ; ; ; ; ; ; ; a a a a a a a a a Value of principal tangential stresses: Main tangential Stresses (principal tangential stresses) Chapter IV: Stress 16 Some Remarks on Stress Components Theorem 3.1: The limits of force divided by area for diminishing area exist. Theorem 3.2: The complete internal force state at a point can be represented fully by the stress components on three mutually orthogonal planes passing through point P. 9 Chapter IV: Stress 17 Transformation of Stress Components σ τ (cw) τ (ccw) σ σ σ mean =( + )/2 xx yy σ max σ min ( , ) σ τ yy yx ( , ) σ τ xx xy 2 θ τ max τ xy σ yy σ xx τ yx x y σ max σ min θ Chapter IV: Stress 18 Three Dimensional Mohr’s Circle σ 3 σ 2 σ 1 σ τ locus of all possible stress components 1 2 3 0 0 0 0 0 0 ij σ σ σ σ     =       Principal Stress State: 10 Chapter IV: Stress 19 Example (1) Example 3.1: Consider the stress state at a point given by its components: a) Determine the principal stresses and their orientation. b) Determine the largest shear stress. 100 MPa, 50 MPa, 0, 30 MPa, 0 xx yy zz xy xz yz σ σ σ τ τ τ = = = = = = Chapter IV: Stress 20 Local Equilibrium Equations 0, 0, 0. x y z F F F = = = ∑ ∑ ∑ Assumption: No body forces and body moments [...]...    Chapter IV: Stress 26 13 Remarks on Hydrostatic Stress It is observed that a material exposed to a hydrostatic stress state does not deform plastically However, the formability of metals increase with increasing compressive hydrostatic stress Chapter IV: Stress 27 Deviatoric Stress Tensor The true stress tensor can be splitted additively as: h ′ σ ij = σ ij + σ ij where, the deviatoric stress. .. 1′ ) , ′ ′ J 3 = σ 1′σ 2σ 3 The principal stresses can be also found as the root of the equation: σ ′3 − J 2σ ′ − J 3 = 0 Chapter IV: Stress 29 Example (2) Example 3.2: Consider the given uniaxial stress state as it is present in a uniaxial specimen:  240 MPa 0 0  σ ij =  0 0 0    0 0 0   Find the hydrostatic and deviatoric stress tensors Chapter IV: Stress 30 15 ... 3 stress stress ∂τ   + τ zx + zx dz  ⋅ dx dy − σ xx ⋅ dy dz − τ zx ⋅ dy dx − τ yx ⋅ dz dx = 0 ∂z   Chapter IV: Stress 21 Equilibrium in x-Axis Direction (2) ∑F x ∂τ yx   ∂σ xx   =  σ xx + dx  ⋅ dy dz +  τ yx + dy  ⋅ dx dz + { { ∂x ∂y   area  1442443 4 4 144 244  area 3 stress stress ∂τ   +  τ zx + zx dz  ⋅ dx dy − σ xx ⋅ dy dz − τ zx ⋅ dy dx − τ yx ⋅ dz dx = 0 ∂z   Chapter IV: ... ∂y ∂z Chapter IV: Stress Assumption: No body forces and body moments 24 12 Complete Equilibrium Equations in Two Dimensions ∑F = 0: ∑F = 0: x y or: ∂σ xx ∂τ yx + =0 ∂x ∂y ∂τ xy ∂σ yy + =0 ∂x ∂y Assumption: No body forces and body moments 2 ∂ 2σ xx ∂ σ yy = ∂x 2 ∂y 2 Chapter IV: Stress 25 Hydrostatic Stress By definition hydrostatic stress is computed as: σh = Note that: The hydrostatic stress state... is the deviatoric stress state which deforms material plastically! Chapter IV: Stress 28 14 Deviatoric Stress Invariants Three independent stress invariants are: ′ ′ ′ J 1 = σ xx + σ yy + σ zz = 0, 2 2 2 ′ yy ′ ′ ′ ′ J 2 = τ xy + τ yz + τ zx − σ xxσ ′ − σ yyσ zz − σ zzσ xx , ′ ′ ′ ′ 2 ′ 2 ′ 2 J 3 = σ xxσ yyσ zz + 2τ xyτ yzτ zx − σ xxτ yz − σ yyτ zx − σ zzτ xy In terms of principal stresses: ′ ′ J 1... Stress 22 11 Equilibrium in x-Axis Direction (3) ∑F x  ∂σ =  xx  ∂x  ∂τ  ⋅ dx dy dz +  yx    ∂y   ∂τ zx   ⋅ dx dy dz +   ⋅ dx dy dz = 0  ∂z   ∂σ xx ∂τ yx ∂τ zx + + =0 ∂x ∂y ∂z Chapter IV: Stress 23 Complete Equilibrium Equations in Three Dimensions ∑ Fx = 0 : ∑F = 0: ∑F = 0: y z ∂σ xx ∂τ yx ∂τ zx + + =0 ∂x ∂y ∂z ∂τ xy ∂σ yy ∂τ zy + + =0 ∂x ∂y ∂z ∂τ xz ∂τ yz ∂σ zz + + =0 ∂x ∂y ∂z Chapter . dA τ ∆ → ∆ = = ∆ uur uur 2 Chapter IV: Stress 3 Stress StressStress Stress General notion of Stress General notion of Stress General notion of Stress General notion of Stress (Kh¸i niÖm chung) (Kh¸i. increasing compressive hydrostatic stress. Chapter IV: Stress 28 Deviatoric Stress Tensor The true stress tensor can be splitted additively as: h ij ij ij σ σ σ ′ = + where, the deviatoric stress tensor. principal tangential stresses: Main tangential Stresses (principal tangential stresses) Chapter IV: Stress 16 Some Remarks on Stress Components Theorem 3.1: The limits of force divided by area for