Non-Associated Plastic Flow and Effects on Macroscopic Failure Mechanisms by Vikranth Racherla Mechanical Engineering IIT Kharagpur
Overview Background and motivation • Basis for non-associated plastic flow • Multi-scale nature of plastic deformation • Generalized yield criteria for single crystals undergoing multiple slip Constitutive models for non-associated flow • Uniqueness and stability of solutions to incremental BVPs • Implication of second-order work (SOW) • Classical rate-independent non-associated flow theory and negative SOW • Rate-dependent theory and effect of strain-rate sensitivity on SOW Effects on macroscopic deformation mechanisms • Cavitation instabilities • Localized necking • Strain bursts – local instabilities • Crack tip fields for associated and non-associated flow
BCC Metals and Intermetallic Compounds in High-Performance Applications • High strength and fracture toughness • Creep resistance at elevated temperatures Nickel-Aluminum super-alloy in turbine blade applications High temperature furnace with Tungsten carbide drill bit nuclear technology molybdenum hot zone (fusion reactors)
Multiscale Analysis of Non- Component Response Associated Plastic Flow • Atomistic studies of defect structures are the basis of models at progressively higher length-scales which ultimately are used to study macroscopic Macroscopic Simulations response and, in particular, failure mechanisms. • The strategy here is to pass only the most essential information on to higher length scales. Polycrystals Single Crystal Dislocation Core Homogenization Crystal Plasticity Atomistics effective macroscopic behavior multi-slip models identification of slip planes & non-glide stress components
Single Crystal Yield Surfaces with Non-Glide Stresses Single crystal yield function τ ∗ α ≡ τ α τ α τ α τ α + a + a + a 1 1 2 2 3 3 τ ∗ α τ = = cr o τ α α ⋅ ⋅ α m σ n = (Schmid stress) τ α τ α α ⋅ ⋅ α σ n = = m 1 1 ( ) α α α α α τ τ × ⋅ ⋅ σ n = = n m ⊥ 2 ( ) α α α α α τ τ × ⋅ ⋅ σ n = = n m ⊥ 3 1 1 1 non-glide stress coefficients obtained for each slip system ( N =24 for BCC) from fits to atomistic simulations specify 3 vectors, m , n , n 1 ( a 1 , a 2 , a 3 ) = (0.24, 0, 0.35) Ref: Qin and Bassani (1992)
Single Crystal Plasticity with Non-Glide Stresses α α α α α α α τ = τ + τ ≡ σ = σ = τ ∑ yield criteria: * * * * a : d d η η ij ij cr η α α α α ∑ ∑ p = γ = γ ∂ τ ∂ σ D d flow equation: α α Crystallographic tensors σ Maximum Principle – Let σ denote the actual stress and an allowable stress: ( ) ( ) ( ) α α α α α α α α τ − τ γ = τ − τ γ = − γ ≥ σ σ d * * * * * : 0 cr Summing over all systems gives the convexity inequality: ( ) ∑ ∑ σ ( ) ( ) α α α α α τ − τ γ = − γ ≡ − ≥ σ d σ σ ε * * * * : : 0 cr α α ( ) ∑ ∑ α α α α α α α α α τ − τ γ γ ≥ γ = σ = τ ≤ τ p * * ; * * * subj to: 0 , d D , d QPP – minimize: cr ij ij ij cr ij α α Ref: Yin and Bassani (2006)
Macroscopic Constitutive Theory: Non-Associated Flow = = + e p e e-1 e p p-1 e-1 Kinematics – F F F L F F F F F F , σ Strain-rate elastic part plastic part ( ) = = + e p ε D sym L D D ij ij ij ij Velocity gradient yield Constitutive Equation for Plastic Strain-rate surface ( ) F σ ij flow potential yield function ∂ ∂ G 1 G p = λ σ = D F ij ∂ ∂ σ p E ij ij t flow Plastic tangent potential ( ) G σ modulus ij F = G for classical associated flow behavior Isotropic yield and flow surfaces predicted using a Taylor model of a random BCC polycrystal Ref: Yin and Bassani (2006)
Yield and Flow Functions For Random Polycrystals Yield function: isotropic – in terms of stress invariants 1/3 3 ( ) 3/ 2 = + F J b J ( ) 2 3 σ 1/3 + 2 3 3 2 b non-associated σ 1 flow parameter 1 = ( ) = ( ) = − ( ) ′ ′ ′ ′ ′ ′ 1 ′ ⋅ 1 ⋅ ⋅ σ σ σ J 2 tr σ σ 3 tr σ σ σ 3 tr δ J 2 3 b = -0.7, SD = 0.2 A single parameter b characterizes the isotropic yield surface Best fit to yield surface ( ) σ − σ 2 = t c ≈ − SD 0.26 b ( ) σ + σ t c Strength σ differential 2 Yield stress Yield stress in in tension compression σ 1 b is expressed in terms of strength-differential of a material Flow Potential ~ ~ ~ ~ b 0, SD 0 = 3 G J 2 Ref: Racherla and Bassani (2007) Best fit to flow surface
Effects of Non-Associated Flow on Cavitation Instabilities Constitutive Equations P cr ∂ σ 1 G σ = = p e , where and D F E o ij t ∂ σ p ε p E ij t e σ ε ε ε ≤ ε p p if N is the strain o e o e o hardening exponent σ = = N F ε p e σ ε ≥ ε p e if o e o ε o P p cr cr ( ) P cr A void in an infinite matrix grows unbounded as the mean ( ) σ stress approaches the cavitation limit 3 P cr kk
Local Sufficient Condition for Uniqueness and Stability in Incremental Boundary Value Problems Governing Equations ∂ P = ρ Ω ij u in Equilibrium equations ∂ j t X i ∂ ( ) ( ) v Rate-independent = = = ∂ m A A A P L with k X , L and L ij ijkl kl ijkl ijkl p q mn mn constitutive equations x n = Γ σ N P t on i ij i t Nominal traction-rates and velocity boundary conditions = Γ u x x on j j t > Sufficient Condition for Uniqueness and Stability: P L 0 ij ji ∇ ≅ σ > P L D 0 infinitesimal deformations: ij ij ji ij Second-order work (SOW) At equilibrium, the sufficient condition ensures an increase in potential energy for any admissible perturbation in the displacement field
Non-Associated Flow and Negative SOW Classical Constitutive Equation: ∂ G ∂ F ∂ ∂ ∂ σ G 1 G σ ∂ σ ij = λ σ = F p D F 2 ij ij ∂ ∂ σ p E ij ij t ∇ σ G ij Plastic Part of Second Order Work ∂ ∇ ∇ 1 G 1 σ = σ = < p σ D F F G 0 ij ij ij ∂ σ 1 p p E E ij Stress rates in hashed region t t result in negative plastic part of second-order work and, therefore, can lead to negative SOW In classical rate-independent non-associated flow second-order work can be negative even with positive hardening modulus so uniqueness and stability can be lost even at small strains
Rate-Dependent Theory and Positive SOW Hardening Relation ( ) ( ) ∂ G m σ = ε ε p p m is the strain-rate g ∂ σ σ eff e e sensitivity parameter ε ij F p 2 π ij θ < 2 σ G ij Constitutive Equation p = + G D qN ij ij ∂ 2 δ ∇ ∇ q t G σ + σ σ G F N N m σ kl kl ij e σ ∂ σ ∂ σ k l 1 m e ij kl For a moderately large m, the angle between plastic strain rate and stress π / 2 rate is less than A corner-like term always makes a positive contribution to SOW and for a moderately large strain-rate sensitivity parameter m the SOW is always positive
Effect of Non-Associated Flow on Sheet Necking Bifurcations If n i represents the normal to a localized band = and A ijkl the incremental modulus ( ) A P F ijkl ijkl kl the bifurcation condition is given by ( ) = A det n n 0 i l ijkl A sheet is deformed under affine BCs. At bifurcation the IBVP admits two solutions: a uniform field and a Critical strains at bifurcation for field with localized deformation in a band various loading strain ratios The bifurcation strains are obtained using a corner theory
Effect of Non-Associated Flow on Sheet Necking Bifurcations Corner theory: Elastic-plastic transition function α ∂ ∂ α ∂ 2 ∇ ∇ G F G p = σ + σ D c F kl kl ij o ∂ σ ∂ σ ∂ σ ∂ σ p p E E ij kl ij kl t t Corner coefficient Effect of corner coefficient on critical bifurcation strains Localized band orientations
Strain Localization: Growth of Inhomogenieties Configuration Critical necking strains for various strain ratios Condition for sheet necking h → m 0 h b To obtain the forming limit diagram a uniform sheet with a thickness inhomogeneity in the form of groove (or band) is deformed at a fixed strain ratio until the sheet necks Effect of non-associated flow on strain localization
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