slide cơ học vật chất rắn chapter 3 new stress equilibrium

th an co ng c om h"p://incos.tdt.edu.vn   cu u du o ng Chapter 3: Stress & Equilibrium HCM University of Science 2015 CuuDuongThanCong.com https://fb.com/tailieudientucntt .c om h"p://incos.tdt.edu.vn   3.1 Body and Surface Forces   co ng 3.2 Traction Vector and Stress Tensor an 3.3 Stress Transformation th 3.4 Principal Stresses & Directions du o ng 3.5 Spherical, Deviatoric, Octahedral and Von Mises Stresses 3.6 Equilibrium Equations cu u 3.7 Relations in Cylindrical and Spherical Coordinates   CuuDuongThanCong.com https://fb.com/tailieudientucntt .c om h"p://incos.tdt.edu.vn   3.1 Body and Surface Forces   co ng 3.2 Traction Vector and Stress Tensor th 3.4 Principal Stresses & Directions an 3.3 Stress Transformation du o ng 3.5 Spherical, Deviatoric, Octahedral and Von Mises Stresses 3.6 Equilibrium Equations cu u 3.7 Relations in Cylindrical and Spherical Coordinates   CuuDuongThanCong.com https://fb.com/tailieudientucntt ng - Body forces are proportional to the body’s c om h"p://incos.tdt.edu.vn   Body  Forces:  F(x)   co mass and are reacted with an agent outside of the th an body Example: gravitational-weight forces, ng magnetic forces, inertial forces du o - By using continuum mechanics principles, a body force density (force per unit volume) F(x) (a)  Can*lever  Beam  Under  Self-­‐Weight  Loading   cu u can be defined such that the total resultant body force of an entire solid can be written as a volume integral over the body FR = ∫∫∫ F ( x ) dV V   CuuDuongThanCong.com https://fb.com/tailieudientucntt co result from physical contact with another body ng - Surface forces always act on a surface and c om h"p://incos.tdt.edu.vn   an - The resultant surface force over the entire th surface S can be expressed as the integral of a ng Surface  Forces:  T(x)   du o surface force density function Tn(x) cu S u FS = ∫∫ T n ( x ) dS S - The surface force density is normally referred to as the traction vector (b)  Sec*oned  Axially  Loaded  Beam                     CuuDuongThanCong.com https://fb.com/tailieudientucntt .c om h"p://incos.tdt.edu.vn   3.1 Body and Surface Forces   co ng 3.2 Traction Vector and Stress Tensor th 3.4 Principal Stresses & Directions an 3.3 Stress Transformation du o ng 3.5 Spherical, Deviatoric, Octahedral and Von Mises Stresses 3.6 Equilibrium Equations cu u 3.7 Relations in Cylindrical and Spherical Coordinates   CuuDuongThanCong.com https://fb.com/tailieudientucntt - The stress or traction vector is defined by P3   ng ΔF T (x, n) = lim Δ A→ Δ A c om h"p://incos.tdt.edu.vn   P2   n   co n ΔF   an th - Notice that the stress vector depends on ΔA   ng both the spatial location and the unit (Sec4oned  Body)    p   (Externally  Loaded  Body)   P1   - In order to define the stress tensor, we consider special Tn (x, n = e1 ) = σ xe1 + τ xye + τ xz e3 cases in which unit normal vectors of ΔA point along the Tn (x, n = e ) = τ yxe1 + σ ye + τ yz e3 positive coordinate axes For these cases, the traction Tn (x, n = e3 ) = τ zxe1 + τ zye + σ z e3 cu u du o normal vector to the surface under study vectors on each face are   CuuDuongThanCong.com https://fb.com/tailieudientucntt   c om h"p://incos.tdt.edu.vn   ng σ is called the stress tensor (2rd order) σy ⎡σ x τ xy τ xz ⎤ ⎢ ⎥ σ = ⎢τ yx σ y τ yz ⎥ ⎢τ zx τ zy σ z ⎥ ⎣ ⎦ These components y co τyx τyz τzy σx τzx τxz are called the stress an σz ng th components x z y   du o σx is normal stress, τxy is shearing stress where x shows τxy u plane of action and y shows direction of stress Tn   cu n   x   z     CuuDuongThanCong.com https://fb.com/tailieudientucntt   c om h"p://incos.tdt.edu.vn   σy ng Traction on an Oblique Plane - Consider the traction vector on an oblique plane with y co τyx τyz τzy σx τzx τxz arbitrary orientation The unit normal to the surface is th an σz x ng - Using the force balance between tractions on the z y   du o oblique and coordinate faces gives τxy Tn = nx Tn ( n = e1 ) + n y Tn ( n = e ) + nz Tn (n = e3 ) Tn   u n   cu Tn = (σ x nx + τ yx n y + τ zx nz ) e1 + (τ xy nx + σ y n y + τ zy nz ) e + (τ xz nx + τ yz n y + σ z nz ) e3 Ti n = σ ji n j x   z     CuuDuongThanCong.com https://fb.com/tailieudientucntt .c om h"p://incos.tdt.edu.vn   3.1 Body and Surface Forces   co th an 3.3 Stress Transformation 3.4 Principal Stresses & Directions ng 3.2 Traction Vector and Stress Tensor du o ng 3.5 Spherical, Deviatoric, Octahedral and Von Mises Stresses 3.6 Equilibrium Equations cu u 3.7 Relations in Cylindrical and Spherical Coordinates 10   CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   c om det ⎡⎣σ ij − σδ ij ⎤⎦ = −σ + I1σ − I 2σ + I = σ1 , σ , σ ng I1 = σ + σ + σ ; I = σ  σ + σ 2σ + σ 3σ ; I = σ 1σ 2σ co σy τyz τzy du o τzx τxz σz σx ng y τxy σ1 th τyx σ2 an σ3 u x z cu   (General Coordinate System) (Principal Coordinate System) 14   CuuDuongThanCong.com https://fb.com/tailieudientucntt Traction Vector Components N = Tn ⋅ n =Ti n ni = σ ji n jni N   N = Tn ⋅ n ΔA   T  n   = σ 1n12 + σ n22 + σ 3n32 ng S = (| Tn |2 − N ) n   c om h"p://incos.tdt.edu.vn   co 1/2 = σ 12 n12 + σ 22 n22 + σ 32 n32 du o N = σ 1n12 + σ n22 + σ 3n32 ng th an S   | Tn |2 = Tn ⋅ Tn = Ti nTi n = σ ji n jσ ki nk S + N =σ n +σ n +σ n 2 1 = n12 + n22 + n32 2 2 2 3 u cu 2 S + ( N − σ )( N − σ ) n1 = (σ − σ ) (σ − σ ) S + ( N − σ )( N − σ1 ) n = (σ − σ ) (σ − σ ) 2 S + ( N − σ )( N − σ ) n = (σ − σ )(σ − σ ) 15   CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn     c om S   S + ( N − σ )( N − σ ) = - Without loss in generality, we can co σ3 And applying the conditions the σ3   an positivity of square of unit normal N   σ2   σ1   S + ( N − σ )( N − σ ) ≥ S + ( N − σ1 )( N − σ ) = S + ( N − σ )( N − σ1 ) = cu u S + ( N − σ )( N − σ ) ≤ du o ng th vectors , we get Mohr’s Circles of Stress ng rank the principal stresses as σ1 > σ2 > S + ( N − σ )( N − σ ) ≥ Admissible N and S values lie in the shaded area 16   CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   For the equality, the above equations c om   S   S + ( N − σ )( N − σ ) = ng represent three circles in an S-N σ3   N   σ2   σ1   th Three above inequalities imply that all an circles of stress co coordinate system which is called Mohr’s ng admissible values of N and S lie in the du o shaded regions bounded by three circles u Note that for the ranked principal S + ( N − σ1 )( N − σ ) = S + ( N − σ )( N − σ1 ) = cu stresses, the largest shear is easily determined as 17   CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Example  3-­‐1  Stress  Transforma4on   1⎤ 2⎥ ⎥ 0⎥⎦ ng co ⎡3 σ ij = ⎢1 ⎢ ⎢⎣1 c om For the given state of stress below, determine the principal stresses and directions and find the traction vector on a plane with unit normal n = (0,1,1)/√2 an The principal stress problem is started by calculating the three invariants, giving the result I1 = 3, I2 = -6, I3 = -8 This yields the following characteristic equation th − σ + 3σ + 6σ − = du o ng The roots of this equation are found to be σ = 4, 1, -2 Back-substituting the first root into the fundamental system (1.6.1) gives − n1(1) + n2(1) + n3(1) = n1(1) − 4n2(1) + 2n3(1) = cu u n1(1) + 2n2(1) − 4n3(1) = Solving this system, the normalized principal direction is found to be n(1) = (2, 1, 1)/√6 In similar fashion the other two principal directions are n(2) = (-1, 1, 1)/√3, n(3) = (0, -1, 1)/√2 The traction vector on the specified plane is calculated using the relation Ti n CuuDuongThanCong.com ⎡3 = ⎢⎢1 ⎢⎣1 ⎤ ⎡ ⎤ ⎡2 / ⎢ 2⎥⎥ ⎢⎢1 / ⎥⎥ = ⎢2 / 0⎥⎦ ⎢⎣1 / ⎥⎦ ⎢⎣2 / 2⎤ ⎥ 2⎥ ⎥⎦ 18   https://fb.com/tailieudientucntt .c om h"p://incos.tdt.edu.vn   3.1 Body and Surface Forces   co ng 3.2 Traction Vector and Stress Tensor th 3.4 Principal Stresses & Directions an 3.3 Stress Transformation du o ng 3.5 Spherical, Deviatoric, Octahedral and Von Mises Stresses 3.6 Equilibrium Equations cu u 3.7 Relations in Cylindrical and Spherical Coordinates 19   CuuDuongThanCong.com https://fb.com/tailieudientucntt .c om h"p://incos.tdt.edu.vn   co ng 1 σ! ij = σ kkδ ij ; σˆ ij = σ ij − σ kkδ ij ; σ ij = σ! ij + σˆ ij 3 an Consider the normal and shear stresses (tractions) that act on a special plane whose normal th makes equal angles with three principal axes This plane is referred to as the octohedral 1 (σ + σ + σ ) = σ kk = I1 3 1⎡ 2 2 S = τ oct = (σ − σ ) + (σ − σ ) + (σ − σ ) ⎤ ⎦ 3⎣ 1 = ( I1 − I ) du o N = σ oct = u (1,1,1) cu ni = ± ng plane The unit normal vector to the octohedral plane is 20   CuuDuongThanCong.com https://fb.com/tailieudientucntt ng c om h"p://incos.tdt.edu.vn   th an The effective or von Mises stress is given by co The octahedral shear stress τoct is directly related to the distortional strain energy σ e = σ vonMises 2 ⎡ = σ − σ + σ − σ + σ − σ + (τ xy2 + τ xz2 + τ yz2 )⎤ ( ) ( ) ( ) x y y z x z ⎥⎦ ⎢⎣ cu u du o ng σ e = σ vonMises 1 ⎡ 2 2 = σ − σ ) + (σ − σ ) + (σ − σ ) ⎤ ( ⎦ 2⎣ 21   CuuDuongThanCong.com https://fb.com/tailieudientucntt .c om h"p://incos.tdt.edu.vn   3.1 Body and Surface Forces   co ng 3.2 Traction Vector and Stress Tensor th 3.4 Principal Stresses & Directions an 3.3 Stress Transformation du o ng 3.5 Spherical, Deviatoric, Octahedral and Von Mises Stresses 3.6 Equilibrium Equations cu u 3.7 Relations in Cylindrical and Spherical Coordinates 22   CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   S Ti dS + ∫∫∫ Fi dV = V σ ji n j dS + ∫∫∫ Fi dV = V ng ∫∫ S c om ∑ F = ⇒ ∫∫ T  n   n Applying the divergence theorem (1.8.7), then ji , j + Fi ) dV = co V F   V   an ∫∫∫ (σ S   th Because the region V is arbitrary, and the integrand is continuous, then by the zero-value σ ji , j + Fi = cu u du o ng theorem (1.8.12), the integrand must vanish ∂σ x ∂τ yx ∂τ zx + + + Fx = ∂x ∂y ∂z ∂τ xy ∂σ y ∂τ zy + + + Fy = ∂x ∂y ∂z ∂τ xz ∂τ yz ∂σ z + + + Fz = ∂x ∂y ∂z 23   CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   ijk S S ijk V x jσ lk nl dS + ∫∫∫ ε ijk x j Fk dV = V co Applying the divergence theorem (1.8.7), then V V du o Using equilibrium equations (3.6.4) σ jk − ε ijk x j Fk + ε ijk x j Fk ⎤⎦ dV = ∫∫∫ ε ijkσ jk dV = u ijk cu ∫∫∫ ⎡⎣ε V   x j ,lσ lk + ε ijk x jσ lk ,l + ε ijk x j Fk ⎤⎦ dV = ∫∫∫ ⎡⎣ε ijkδ jlσ lk + ε ijk x jσ lk ,l + ε ijk x j Fk ⎤⎦ dV = V F   th ijk ng ∫∫∫ ⎡⎣ε S   an ⎡(ε x σ ) + ε x F ⎤ dV = ijk j k ∫∫∫ ⎣ ijk j lk ,l ⎦ V T  n   ng ∫∫ ε x jTkn dS + ∫∫∫ ε ijk x j Fk dV = c om ∑ r × F = = ∫∫ ε V τ xy = τ yx ∑ r × F = ⇒ ε ijkσ jk = ⇒ σ ij = σ ji ⇒ τ yz = τ zy ⇒ σ ij , j + Fi = τ zx = τ xz CuuDuongThanCong.com https://fb.com/tailieudientucntt 24   .c om h"p://incos.tdt.edu.vn   3.1 Body and Surface Forces   co ng 3.2 Traction Vector and Stress Tensor th 3.4 Principal Stresses & Directions an 3.3 Stress Transformation du o ng 3.5 Spherical, Deviatoric, Octahedral and Von Mises Stresses 3.6 Equilibrium Equations cu u 3.7 Relations in Cylindrical and Spherical Coordinates 25   CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   z   σz   σθ   x1   cu u th dr   τrθ   σr   x2   ⎡σ r σ = ⎢τ rθ ⎢ ⎢⎣τ rz Tr = σ r e r τ rθ τ rz ⎤ σ θ τ θ z ⎥⎥ τ θ z σ z ⎥⎦ + τ rθ eθ + τ rz e z Tθ = τ rθ e r + σ θ eθ + τ θ z e z Tz = τ rz e r + τ θ z eθ + σ z e z ng du o Equilibrium Equations θ   dθ   an r   τrz   co τθz   ng Cylindrical Coordinates c om x3   ∂σ r ∂τ rθ ∂τ rz + + + [σ r − σ θ ] + Fr = ∂r r ∂θ ∂z r ∂τ rθ ∂σ θ ∂τ θ z + + + τ rθ + Fθ = ∂r r ∂θ ∂z r ∂τ rz ∂τ θ z ∂σ z + + + τ rz + Fz = ∂r r ∂θ ∂z r 26   CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   x3   Spherical Coordinates c om τRθ   σR   τRΦ   R   x2   Tϕ = τ Rϕ e R + σ ϕ eϕ + τ ϕθ eθ ng Tθ = τ Rθ e R + τ ϕθ eϕ + σ θ eθ du o Equilibrium Equations th x1   an τΦθ     θ   σΦ   co Φ   ng σθ   ⎡ σ R τ Rϕ τ Rθ ⎤ ⎢ ⎥ σ = ⎢τ Rϕ σ ϕ τ ϕθ ⎥ ⎢⎣τ Rθ τ ϕθ σ θ ⎥⎦ TR = σ Re R + τ Rϕ eϕ + τ Rθ eθ cu u ∂σ R ∂τ Rϕ ∂τ Rθ + + + ( 2σ R − σ ϕ − σ θ + τ Rϕ cotϕ ) + FR = ∂R R ∂ϕ Rsinϕ ∂θ R ∂τ rϕ ∂σ ϕ ∂τ ϕθ + + + ⎡⎣(σ ϕ − σ θ ) cotϕ + 3τ Rϕ ⎤⎦ + Fϕ = ∂R R ∂ϕ Rsinϕ ∂θ R ∂τ rθ ∂τ ϕθ ∂σ θ + + + ( 2τ ϕθ cotϕ + 3τ Rθ ) + Fθ = ∂R R ∂ϕ Rsinϕ ∂θ R 27   CuuDuongThanCong.com https://fb.com/tailieudientucntt .c om ng co an th ng du o u cu CuuDuongThanCong.com https://fb.com/tailieudientucntt ... m2 m3 + σ z n2 n3 + τ xy (l2 m3 + m2l3 ) + τ yz (m2 n3 + n2 m3 ) + τ zx (n2l3 + l2n3 ) cu u τ ′zx = σ xl3l1 + σ y m3m1 + σ z n3n1 + τ xy (l3m1 + m3l1 ) + τ yz (m3n1 + n3m1 ) + τ zx (n3l1 + l3n1... co ng 3. 2 Traction Vector and Stress Tensor th 3. 4 Principal Stresses & Directions an 3. 3 Stress Transformation du o ng 3. 5 Spherical, Deviatoric, Octahedral and Von Mises Stresses 3. 6 Equilibrium. .. xl32 + σ y m32 + σ z n32 + 2(τ xy l3m3 + τ yz m3n3 + τ zx n3l3 ) du o τ ′xy = σ xl1l2 + σ y m1m2 + σ z n1n2 + τ xy (l1m2 + m1l2 ) + τ yz (m1n2 + n1m2 ) + τ zx (n1l2 + l1n2 ) τ ′yz = σ xl2l3 +