GIÁO TRÌNH KHAI THÁC PHẦN mềm TRONG GIA CÔNG KHUÔN mẫu chapter v yield criteria

1 Chapter V: Yield Criteria 1 Content • Yielding under a Uniaxial Stress States • Yielding under Multiaxial Stress States • Tresca Yield Criterion • v. Mises Yield Criterion • Plastic Work • Definition of the Equivalent Plastic Strain • Plastic Power • Heat Dissipation Chapter V: Yield Criteria 2 Yielding under a Uniaxial Stress State ε xx σ xx Y 1 Y 0 λ λλ λ F F 1 2 3 4 5 (ε ) xx pl 0 0 0 0 0 0 0 0 xx ij σ σ     =       1: Elastic State 2: Plastic State (Initial yield Y 0 ) 3: Unloading 4: Reloading Elastically 5: Plastic State (Initial yield Y 1 ) Initial transition to plastic state at end of path 1 if: 0 xx Y σ = Initial transition to plastic state at end of path 4 if: 1 xx Y σ = 2 Chapter V: Yield Criteria 3 Strain Hardening/Softening 1 0 Y Y > 1 0 Y Y < 1 0 . Y Y const = = If: Material is said to be strain hardening. Material is said to be strain softening. Material is said to be non-hardening. In the most general case the yield stress depends on various factors such as: Y = f (prior plastic strain, current strain-rate, current temperature, current microstructure, current stress state, prior stress-path, current loading direction) Chapter V: Yield Criteria 4 Multidimensional Yield Hypothesis The beginning of yielding can be described in terms of the current stress state components and a constant. Hypothesis: Hence, if: ( ) ij f c σ = yielding starts. where: c= f (prior plastic strain, current strain-rate, current temperature, current microstructure) 3 Chapter V: Yield Criteria 5 Tresca Yield Criterion (1) By Henri Edouard Tresca (1864) yielding starts if: max c τ = To find the constants c consider the uniaxial tension test: max , 2 2 xx Y σ τ = = 2 Y c ∴ = On the other hand, the maximum shear stress can be expressed in terms of principal stresses: max min max , 2 2 Y σ σ τ − = = max min Y σ σ ∴ − = Chapter V: Yield Criteria 6 Tresca Yield Criterion (2) Similarly the constant c could be found from the pure shear tests, for which yielding occurs if: k τ = where k is the yield stress in shear. Hence: max k τ τ = = c k = max min 2 k σ σ − = or: The yield criterion reads then: By the way, for the simple tension test max min , 0 Y σ σ = = ⇒ 2 Y k = 4 Chapter V: Yield Criteria 7 The Tresca Yield Criterion in Plane Stress I σ III σ Y -Y -Y Y Plane stress: 0! II σ = Region I 0 II 0 III 0 IV 0 V 0 VI 0 I III I III I III III I III I III I I I III III I III I III Y Y Y Y Y Y σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ > > = > > = > > = − > > = − > > = − > > = − σ I σ III I σ Ι σ III VI σ Ι σ III II σ I σ III III σ I σ ΙΙΙ V σ Ι σ ΙΙΙ IV plastic Chapter V: Yield Criteria 8 v. Mises Yield Criterion (1) ( ) ( ) ( ) 2 2 2 1 2 3 max max max 1 3 c τ τ τ   = + +     By R. von Mises (1913) material yields if 2 max τ 1 max τ 3 max τ where the constant c is found from standard experiments. 5 Chapter V: Yield Criteria 9 Richard von Mises • Born 1883 in Lemberg (Austria) • Studied mathematics, physics & engineering at the University of Vienna • Doctorate 1907 University of Vienna • Died 1953 in Boston, MA/USA • An international authority on the Austrian poet Rainer Maria Rilke • Interest in philosophy • Professor of applied mathematics at University of Strasburg from 1909 until 1918 • 1919 chair of hydrodynamics and aerodynamics at the Technische Hochschule in Dresden • 1920 Institute of Applied Mathematics at the University of Berlin • 1933 Chair at University of Istanbul • 1939 professor of Aerodynamics and Applied Mathematics at Harvard University Chapter V: Yield Criteria 10 v. Mises Yield Criterion (2) ( ) ( ) ( ) 2 2 2 1 2 2 3 3 1 1 2 Y σ σ σ σ σ σ   = − + − + −   The v. Mises yield criterion can be written also as: and more generally: ( ) ( ) ( ) ( ) . . 2 2 2 2 2 2 1 6 2 v M xx yy yy zz zz xx xy yz zx Y σ σ σ σ σ σ σ τ τ τ   = − + − + − + + +     14444444444444244444444444443 where the v. Mises equivalent stress is defined by: ( ) ( ) ( ) ( ) 2 2 2 2 2 2 . . 1 6 2 v M xx yy yy zz zz xx xy yz zx σ σ σ σ σ σ σ τ τ τ   = − + − + − + + +     6 Chapter V: Yield Criteria 11 v. Mises Yield Criterion (3) If the v. Mises yield criterion is applied to the pure shear test, where: xy k τ = we obtain: { { { { { { { 2 2 2 2 2 2 2 0 0 0 0 0 0 0 1 6 2 xx yy yy zz zz xx xy yz zx k Y σ σ σ σ σ σ τ τ τ                   = − + − + − + + +                       14243 or: 3 Y k = Chapter V: Yield Criteria 12 v. Mises Yield Criterion in Plane Stress II 2 Plane stress: 0! σ σ = = Yielding starts if: 45 o -Y -Y Y Y s I s III majoraxis minoraxis 2 2 . . v M I I III III Y σ σ σ σ σ = = − + Mises Yield mod.cdr 7 Chapter V: Yield Criteria 13 Comparing Tresca with v. Mises (1) compare yield criteria mod.cdr -Y -Y Y Y s I s III 1.155Y 1.155Y v.Mises Tesca Maximum deviation between both criteria 15.5%! Chapter V: Yield Criteria 14 Comparing Tresca with v. Mises (2) Common properties of both criteria: 1) Both are independent of the hydrostatic stress 2) They are symmetric regarding the sense of stress (no Bauschinger effect!) 3) Both can be expressed in terms of invariants. v. Mises: Tresca: 2 2 k J = 3 2 2 2 4 6 2 3 2 2 4 27 36 96 64 0 J J J k J k k − − + − = 8 Chapter V: Yield Criteria 15 Johann Bauschinger German Born: 11 June 1834 Died: 25 November 1893 Professor at the Technical University of Munich and Director of the Materials Testing Laboratory Work on Bauschinger Effect is published 1886 Chapter V: Yield Criteria 16 -Y -Y Y Y s I s III 1.155Y 1.155Y v.Mises Tesca Examples (1) Example 4.1: Evaluate the yield criteria by Tresca and by v. Mises for the following forming tests and processes: a) Tensile test, b) Bulging test, c) Torsion test, d) Frictionless upsetting test, 9 Chapter V: Yield Criteria 17 Plastic Work in Uniaxial Tension λ λλ λ F F d λ λλ λ A Consider the plastic extension of a rod: The infinitesimal plastic work done by F is: dW F d = ⋅ l The specific infinitesimal plastic work (per unit volume) is dW F d F d dw A A A ⋅     = = = ⋅ ⇒     ⋅ ⋅     l l l l l dw d σ ε = ⋅ Chapter V: Yield Criteria 18 Plastic Work in 3-D The uniaxial relation for plastic work can be generalized for three-dimensional loadings as: 3 3 ij ij i j dw d σ ε = ⋅ ∑∑ +2 2 2 xx xx yy yy zz zz xy xy yz yz zx zx dw d d d d d d σ ε σ ε σ ε σ ε σ ε σ ε = + + + + + Or: 10 Chapter V: Yield Criteria 19 Definition of Equivalent Plastic Strain The equivalent plastic strain is derived now from the relationship: 3 3 ij ij i j dw d d σ ε σ ε = ⋅ = ⋅ ∑∑ Now, depending on the yield criterion used, the equivalent stress will differ and hence the equivalent strain increment: For v. Mises: ( ) ( ) ( ) 2 2 2 1 2 2 3 3 1 1 2 σ σ σ σ σ σ σ   = − + − + − ⇒   2 2 2 1 2 3 2 3 d d d d ε ε ε ε = + + For Tresca: max max min 2 σ τ σ σ = = − ⇒ max i d d ε ε = Chapter V: Yield Criteria 20 Examples (2) Example 4.2: For the given processes below, compute the infinitesimal specific plastic work using, a) the components of the stress and infinitesimal strain tensor, b) the equivalent stress and the infinitesimal equivalent strain: i) Uniaxial tension ii) Plane strain compression with 2 1 3 2 1 3 1 , 0, 0, 2 d d d σ σ σ ε ε ε = = = = − [...]... ⋅ d ε V V ε   dV  Total heat energy: Wheat = ∫ ρ ⋅ c ⋅ ∆T ⋅ dV V Wheat = ( 0.85 to 0.95) ⋅ W plastic Assuming homogeneous deformation and utilizing the mean value theorem: ( 0.85 to 0.95) ⋅ σ mean ⋅ ε ∆T = ρ ⋅c Chapter V: Yield Criteria ρ: density c: spec heat capacity 22 11 Examples (3) Example 4.3: Compute the increase in temperature for the upsetting process of the various materials given in...Plastic Power The specific power p can be derived from the specific work expression as: dε dw 3 3 dε = ∑∑σ ij ⋅ ij = σ ⋅ dt dt dt i j 3 3 i j & & p = ∑∑ σ ij ⋅ ε ij = σ ⋅ ε Hence: Total specific plastic work is given by: w = ∫ p ⋅ dt = ∫ σ ⋅ d ε ε t Chapter V: Yield Criteria 21 Heat Dissipation 85 to 95 % of the plastic work is irreversible and is therefore dissipated as heat Assuming adiabatic... process of the various materials given in the table below at an heigth reduction corresponding to an equivalent strain of 1 ρ Steels ca 7.8 c J/(kg K) ca 500 Al-Alloys ca 2.7 ca 1000 100-300 Cu-Alloys ca 8.7 ca 400 200-400 Ti-Alloys ca 4.5 ca 600 750-1500 kg/dm3 σ mean N/mm2 400-1200 Chapter V: Yield Criteria 23 12 . − + Mises Yield mod.cdr 7 Chapter V: Yield Criteria 13 Comparing Tresca with v. Mises (1) compare yield criteria mod.cdr -Y -Y Y Y s I s III 1.155Y 1.155Y v. Mises Tesca Maximum deviation between both. > = − > > = − σ I σ III I σ Ι σ III VI σ Ι σ III II σ I σ III III σ I σ ΙΙΙ V σ Ι σ ΙΙΙ IV plastic Chapter V: Yield Criteria 8 v. Mises Yield Criterion (1) ( ) ( ) ( ) 2 2 2 1 2 3 max. Berlin • 1933 Chair at University of Istanbul • 1939 professor of Aerodynamics and Applied Mathematics at Harvard University Chapter V: Yield Criteria 10 v. Mises Yield Criterion (2) ( ) (