1 Chapter VI: Flow Rule 1 Content • Terminology • Elastic Stress-Strain Relationship • Uniaxial Observations • Levy-Mises Flow Rule • Derivation of Equivalent Plastic Strain Rate • Principle of Normality • Work-Hardening Case • Hoogenboom’s Experiment Chapter VI: Flow Rule 2 Terminology • Solid Mechanics: Stress-Strain Relationship (especially in elasticity) • Continuum Mechanics: Constitutive Equations, Material Law • Plasticity: Flow Rule The material dependent relationship between the stress and the strain is described in literature by various terms: 2 Chapter VI: Flow Rule 3 Elastic Stress-Strain Relationships For small strain elastic behaviour, the stress-strain relation is given by: ( ) ( ) ( ) 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 2 2 1 1 2 2 1 1 2 2 xx xx yy zz yy yy xx zz zz zz xx yy xy xy xy yz yz yz zx zx zx e E e E e E e G e G e G σ ν σ σ σ ν σ σ σ ν σ σ γ τ γ τ γ τ   = − +     = − +     = − +   = = = = = = ⇒ 0 1 2 ij ij e G σ ′ ′ = ( ) 0 where, is the engineering strain tensor the deviatoric engineering strain tenso r the nominal stress tensor the Young's modulus the Poisson's ratio the shear modulus and 2 1 ij ij ij e e E E G G σ ν ν ′ = + Chapter VI: Flow Rule 4 Uniaxial Observations Consider the given simple tension test: 1 1 2 3 0 0 3 h σ σ σ σ σ ≠  ⇒ = ⇒  = =  λ λλ λ F F d λ λλ λ 1 1 2 3 1 1 2 3 2 1 , 3 3 2 2 σ σ σ σ σ σ σ σ ′ ′ ′ = = = − ′ ′ ′ = − = − Stress State: Strain Increment State: 1 2 3 2 2 d d d ε ε ε = − = − Hence: 1 2 3 1 2 3 d d d d ε ε ε λ σ σ σ = = = ′ ′ ′ 3 Chapter VI: Flow Rule 5 Levy-Mises Flow Rule (1) These observations lead to the general Levy-Mises flow rule: ij ij d d ε λ σ ′ = ⋅ where: is the plastic true strain increment the deviatoric true stress a non-negative real number ij ij d d ε σ λ ′ Remark: The Levy-Mises flow rule assumes rigid-plastic deformation, i.e. ignores elastic deformations. Hence the first invariant of the strain increment tensor is zero. Chapter VI: Flow Rule 6 Levy-Mises Flow Rule (2) ij ij ε λ σ ′ = ⋅ & & By dividing both sides of the Levy-Mises equation by dt we obtain an alternative form in terms of strain rates: To determine the non-negative proportionality factor in the above equation consider following development: 3 3 ij 1 1 3 3 1 1 3 with we obtain: 2 1 3 2 ij ij ij i j ij ij i j ε σ σ σ σ λ σ ε ε λ = = = =   ′ ′ ′ = =       =     ∑∑ ∑∑ & & & & & 3 3 1 1 3 2 ij ij i j ε ε λ σ = =       = ∑∑ & & & or: 4 Chapter VI: Flow Rule 7 Levy-Mises Flow Rule (3) Notice that 3 3 2 1 1 1 2nd invariant of 2 ij ij ij ij i j I ε ε ε ε = =   ≡ ≡     ∑∑ & & & & Hence: 2 3 ij I ε λ σ ⋅ = & & Furthermore, note that the material must be always in the plastic state for above equation to hold, so that: f σ σ → and hence: 2 3 ij f I ε λ σ ⋅ = & & where σ f is the so-called flow stress. Chapter VI: Flow Rule 8 Levy-Mises Flow Rule (4) So, the Levy-Mises flow rule can be rewritten as: 2 3 ij ij ij f I ε ε σ σ ⋅ ′ = & & This equation can be further simplified by recalling the equivalent plastic strain rate using the identity: 3 33 3 ij i ij i j j ji i j p σ ε σ ε εσ = ⋅ ⋅= ′ = ⋅ ∑∑ ∑∑ & & & 5 Chapter VI: Flow Rule 9 Derivation of the Equivalent Strain Rate (1) Replacing in 3 3 ij ij i j σ ε σ ε ′ ⋅ = ⋅ ∑∑ & & 2 3 ij ij ij f I ε ε σ σ ⋅ ′ = & & the strain rate tensor by: we obtain: 3 3 2 3 ij ij ij i j f I ε σ ε σ σ σ   ⋅   ′ ′ ⋅ = ⋅     ∑∑ & & Chapter VI: Flow Rule 10 Derivation of the Equivalent Strain Rate (2) 2 3 3 2 2 3 ij ij ij i j f J I ε σ ε σ σ σ ⋅   ⋅   ′ ′ ⋅ = ⋅     ∑∑ & & 1442443 Or: But recall that: 3 3 2 3 = = 3 2 f ij ij i j J σ σ σ σ   ′ ′ = ⋅ ⋅     ∑∑ 2 2 3 2 3 ij f f f I ε σ ε σ σ   ⋅     ⋅ =         & & 6 Chapter VI: Flow Rule 11 Derivation of the Equivalent Strain Rate (3) 3 3 2 1 1 4 2 3 3 ij ij ij i j I ε ε ε ε = =   = ⋅ =     ∑∑ & & & & Hence: So, the Levy-Mises flow rule can be rewritten finally as: 3 2 ij ij f ε ε σ σ   ′ =       & & 3 2 ij ij f d d ε ε σ σ   ′ =       or, equivalently, as Chapter VI: Flow Rule 12 Comparison of Hooke’s Law and Levy- Mises Flow Rule The Levy-Mises equation can be written also in terms of the total stresses: ( ) 1 1 2 3 1 etc. 2 f d d ε ε σ σ σ σ   = − +     Comparing this form with the Hooke’s law: ( ) 1 2 3 0 0 0 1 1 e E σ ν σ σ   = − +   Note: 1. The Poisson’s ratio is replaced by the factor 0.5 in the Levy-Mises law 2. The constant Young’s modulus is now a variable depending on the flow stress and the equivalent plastic strain increment in the plastic case 3. Levy-Mises equations are relating stresses to strain increments or rates 7 Chapter VI: Flow Rule 13 Normality Rule (1) In fact the Levy-Mises flow rule is one typical application of the plastic potential theory, which establishes the flow rule by: ij ij f d d ε λ σ ∂ = ⋅ ∂ where f is the so-called plastic potential. In case of associative plasticity this potential is taken identical to the yield function. Above equation indicates also that the strain increment “vector” is normal to the yield surface in the stress space. Chapter VI: Flow Rule 14 Normality Rule (2) For the two-dimensional plane stress case the normality rule can be illustrated by: One important fact about flow surfaces is that they must be convex always for stable material flow. 8 Chapter VI: Flow Rule 15 Work-Hardening Case 3 2 ij ij f d d ε ε σ σ   ′ =       For non-hardening (perfect plastic) case we found: For an hardening material according to the flow curve ( ) f σ ε we can define the local slope H as: f f d d H d d H σ σ ε ε = ⇒ = Inserting the latter into the first equation above gives: 3 2 f ij ij f d d H σ ε σ σ   ′ =       Chapter VI: Flow Rule 16 Work-Hardening Case: Ludwik Eqn. n f C σ ε = For a flow curve of Ludwik’s type: (C and n are material constants) we can find the slope as: 1 n n f f d H C n d C σ σ ε −   = = ⋅ ⋅     9 Chapter VI: Flow Rule 17 Remark on Levy-Mises Flow Rule 3 2 f ij ij f d d H σ ε σ σ   ′ =       From the flow rule it is apparent that: has the same "direction" as and NOT ! ij ij ij d ε σ σ ′ In case of Hooke’s law: has the same "direction" as ! ij ij d ε σ ′ ′ Chapter VI: Flow Rule 18 Hoogenboom’s Experiment (1) Stan Hoogenboom (Ankara, 2002) Experimental procedure: 1. Load the wire with weights until it elongates plastically. 2. Keeping this axial load, start to apply a torque to the wire by rotating it. 10 Chapter VI: Flow Rule 19 Hoogenboom’s Experiment (2) z θ τ ⇒ where m is the mass of the weights, g the gravitational accelaration and A the current cross- sectional arae of the wire. z m g A σ ⋅ = due to the torque T Chapter VI: Flow Rule 20 Hoogenboom’s Experiment (3) The strain increment due to axial load is simply: 3 3 1 2 2 3 f f f zz zz zz zz zz f f f d d d d d H H H λ σ σ σ ε σ σ σ σ σ σ σ         ′ = ⋅ = ⋅ − = ⋅                       14243 Now, after the axial load is kept constant, the torque T is applied: ( ) after torque 1 2 f xx σ σ = yy σ − ( ) 2 yy σ + ( ) 2 zz zz xx σ σ σ − + − ( ) 2 2 6 xy τ + 2 2 yz zx τ τ + + ( )       Therefore: ( ) ( ) ( ) after torque after torque before torque 0 f f f d σ σ σ ≈ − > Hence: 0 f zz zz f d d H σ ε σ σ   = ⋅ >       during torque application [...]... strain state Chapter VI: Flow Rule 21 Examples (2) Example 5.2: Consider the uniform rod with current diameter D= 25 mm and current length of λ = 120 mm, which is plastically deformed with a velocity of vtool=3 mm/s at each end If the material has a flow stress of 250 MPa at the current strain, strain rate and temperature, determine all the external forces acting on the specimen Chapter VI: Flow Rule 22 . ′ 3 Chapter VI: Flow Rule 5 Levy-Mises Flow Rule (1) These observations lead to the general Levy-Mises flow rule: ij ij d d ε λ σ ′ = ⋅ where: is the plastic true strain increment the deviatoric. ij f I ε λ σ ⋅ = & & where σ f is the so-called flow stress. Chapter VI: Flow Rule 8 Levy-Mises Flow Rule (4) So, the Levy-Mises flow rule can be rewritten as: 2 3 ij ij ij f I ε ε σ σ ⋅ ′ = & & This. Levy-Mises flow rule assumes rigid-plastic deformation, i.e. ignores elastic deformations. Hence the first invariant of the strain increment tensor is zero. Chapter VI: Flow Rule 6 Levy-Mises Flow Rule