Damage Fracture using Continuum Damage Mechanics Prakash M. Dixit Department of Mechanical Engineering Indian Institute of Technology Kanpur Acknowledgements Dr. Sankar Dhar, ME Dept., Jadavpur University Dr. Sachin S. Gautam, ME Dept., IIT Guwahati Mr. Manoj Kumar, ME Dept., IIT Kanpur 1
Plan of Presentation Introduction to Ductile Fracture Continuum Damage Mechanics (CDM) Damage Growth Law Crack Initiation Criterion (Critical Damage) Ductile Fracture in Taylor Rod and Tube Impact Problems Damage Growth Law (Revisited) Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA
Introduction to Ductile Fracture Micro-Structural Observation: Ductile fracture in metals involves the following three stages. Nucleation of microvoids : due to fracture of secondary particles or their debonding from ductile matrix. Growth of voids induced by plastic straining where the triaxiality may also increase. Void coalescence, i.e., due to necking of inter-void matrix, leading to micro-crack initiation. Figure 1 : Three stages of ductile fracture Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA
Introduction to Ductile Fracture... Two commonly used continuum models for the prediction of ductile fracture based on micro-structural observations of void nucleation, growth and coalescence : 1. Porous Plasticity Model (Gurson, 1977; Tvergaard and Needleman 1984) 2. Continuum Damage Mechanics (CDM) Model (Chaboche, 1981; Lemaitre, 1985; Rousselier, 1987) Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA
Porous Plasticity Model Material with voids idealized as a porous material . Gurson (1977) proposed the following expression for plastic potential : 2 σ σ ( ) 3 q Φ = + − + = eq 2 2 m 2 q f cosh 1 q f 0 σ σ 1 3 2 y y f = void volume fraction (porosity), σ = mean stress of the porous aggrgrate, m σ = equivalent stress of the porous aggrgrate, eq σ = yield stress of t he mat rix mate ri al, y = = Gurson as sume d q q q =1. 1 2 3 ( ) = = 2 Tve rgaard 1981 proposed 1.5, = 1 , q q q q 1 2 3 1 for better agreement of the model with numerical predictio ns Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA
Porous Plasticity Model … Evolution Law for Void Volume Fraction (Porosity): = + f f f , nucleation growth = ε p f A , rate of change of porosity due to void nucleation nucleation eq = − ε p f (1 f ) , rate of change of porosity due to void gr ow h t g rowth kk = Constant, A ε = p Mean plastic strain rate, kk ε = p Equivalent plastic strain rate eq The critical value of porosity is obtained by either a void coalescence condition or experimentally. Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA
Continuum Damage Mechanics (CDM) In CDM, the effect of void growth on material behaviour is incorporated by introducing a continuum variable, called damage , in the formulation Damage: Assumed isotropic. It quantifies the void density at a point. It is defined as ∆ Av lim = ≤ ≤ D ; 0 D 1 ∆ → ∆ A 0 A Δ A v = Area of void (and crack) traces contained in Δ A Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA
Continuum Damage Mechanics (CDM) … In a damaged material, the specific Helmholtz free energy ( ψ ) or thermodynamic potential is given by: 1 ψ = ε ε − − e e (1 ) C D Ts ρ ijkl ij kl 2 ρ = ε = e Density, Elastic part of strain tensor ij = C Fourth order elasticity tensor ijkl = T = Absolute temperature, s Specific entropy Elastic constitutive relation is obtained as ∂ ψ σ = ρ = ε − σ = e C (1 D ), Cauchy stress tensor ∂ ε ij ijkl kl ij e ij Conservative part of the thermodynamic force Y corresponding to damage is given by : ∂ ψ 1 = ρ = − ε ε e e Y C ∂ ijkl ij kl D 2 These are called State Laws Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA
Continuum Damage Mechanics (CDM) … Using Isotropic stress-strain relations, the conservative part of the thermodynamic force Y corresponding to damage can be expressed in terms of stress: σ 2 σ = − eq m Y f ( ) σ 2 2 E 1- D eq 2 σ σ 2 1 ( ) ( ) = + ν + ν m m f 3 1- 2 σ σ 3 eq eq ν = E, Elastic constants, σ = σ (1/ 3) ; Mean stress, m ii 1/2 ′ ′ σ = σ σ (3 / 2) ; Equivalent stress, eq ij ij ′ = σ Deviatoric part of the stress tensor, ij σ m The ratio is called Triaxiality σ eq Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA
Continuum Damage Mechanics (CDM) … Φ = ≥ i TS 0 Dissipation Potential (per unit volume) : i S = Entropy production rate per unit volume ( i for irreversible) Φ For isothermal process, depends on the following state variables ( ) Φ = Φ ε ε ε ε p p p p and their rates: , , , , , D D ij eq ij eq ε = ε = p p Plastic part of strain tensor; Plastic part of strain rate tensor; ij ij t 2 ∫ ε = ε ε = ε ε p p p p p dt ; ; (Equivalent plastic strain and strain rate) eq eq eq ij ij 3 0 ∂Φ ∂Φ ∂Φ σ = − = − = Complementary Laws : ; R ; Y ∂ ε ∂ ε ∂ ij p p D ij eq - and - R Y : Dissipative parts of the thermodynamic forces ε p corresponding to and respectively. D eq − is the negative of the corresponding conservative Y Y Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA
Continuum Damage Mechanics (CDM) … Dissipation Potential and Complementary Laws: Legendre-Fenchel transformation: ( ) ( ) Φ ε ε ε ε ⇒ Φ ∗ σ − − Φ p p p p , , D , , , D , R , Y , Dual of ij eq ij eq ij The complementary laws can be expressed as the evolution laws of the rates of state variables: ∂Φ ∗ ε = ∂ p σ ij ij ∗ ∂Φ ε = ∂ − p ( ) eq R ∂Φ ∗ = ∂ − D ( ) Y Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA
Continuum Damage Mechanics (CDM) ... Dissipation Potential and Evolution Laws: Since Φ and Φ * are governed by an equality, evolution laws not useful for computations The dissipation potential Φ * is maximized subject to the constraint = σ − − that the yield function/plastic potential F F ( , R , Y ) ij ∗ δ Φ − λ = should be zero on the yield surface: ( F ) 0 λ Here, δ is the variation and is the Lagrange multiplier ∂ ∂ ∂ F F F ε = λ ε = λ = λ p p ; ; D ( ) ∂ σ ∂ − ∂ − ij eq R ( Y ) ij The above laws called the plastic flow rules. Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA
Continuum Damage Mechanics (CDM) ... Plastic Potential Assume: Plastic potential can be decomposed as (Lemaitre, 1985) ( ) ( ) = σ ε + − p F F , , D F Y 1 ij eq D Where F 1 and F D are the plastic potentials associated with yielding/hardening and damage respectively. The variables ( -R ) ε p and (- Y) in F 1 replaced by and D respectively. eq Plastic Potential due to Yielding/Hardening: von Mises yield function for damaged material: σ = − σ eq F , − 1 Y 1 D The variable yield stress usually modelled by a power law σ ≡ ε = σ + ε p p n H ( ) ( ) K ( ) Y eq Y 0 eq σ = ( ) Initial yield stress; K, n = Hardening parameters Y 0 H = Hardening function Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA
Continuum Damage Mechanics (CDM) ... Incremental Elasto-Plastic Constitutive Relation First Plastic Flow Rule in Incremental Form: ′ ∂ σ σ ∂ λ 3 F d ε = λ σ = = = eq ij p 1 d d a ; a ∂ − ∂ σ σ ij ij ij 1 D 2 ij ij eq The scalar d λ found from the consistency condition: ∂ ∂ F F ≡ σ + ε = p 1 1 dF d d 0 ∂ σ ∂ ε 1 ij eq p ij eq Incremental Elasto-Plastic Constitutive Relation: ( ) σ = ε − ε = ε p EP d C d d C d ij ijkl ij ij ijkl ij Substitution of the plastic flow rule and expression for d λ in the above equation leads to C a a C σ = ε = − ijmn mn pq pqkl EP EP d C d , C C ′ + ij ijkl ij ijkl ijkl a C a H rs rsuv uv ( ) ( ) ( ) n = ε = σ + ε ' p p Derivative of the hardenin g function : H H K eq Y eq 0 In the above relation, d σ ij has to be objective. One common choice is the product of the Jaumann stress rate and incremental time. Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA
Recommend
More recommend
