Configurational Equilibrium of Cracks Affected by Surface Stress and - PowerPoint PPT Presentation

Configurational Equilibrium of Cracks Affected by Surface Stress and Crack-Tip Point Load Chien H. Wu University of Illinois at Chicago cwu cwu cwu cwu UIC 062999-01

Configurational Equilibrium b g Potential Energy ∂ 0 b g = Configuration ∂ Effect of Surface Stress and Crack-Tip Load cwu UIC cwu cwu cwu 062999-02

Surface Stress Stretch Then Split 1+ ε 1+ ε 1 Split Then Stretch 1+ ε 1 1 Stretch unit cube along x, then split W( ) 2 Γ ( ) Work A ε + ε = Split unit cube, then stretch along x b g 2 ( ) 0 W ε ( ) 2 Work B Γ + + Σε = Work A = Work B / Σ = ∂Γ ∂ε cwu UIC cwu cwu cwu 062999-03

Referential and Spatial Forms 1+ ε 1 a f : Surface Energy per Unit Referential Area Γ ε b g : Surface Energy per Unit Spatial Area γ ε a f a f a f a f a f 1 = ∂Γ + ∂γ 1 , γ ε = Γ ε = − ε Γ ε Σ ∂ε = γ ε 1 + ε ∂ε a f a f Linear Density: Γ , ( ) ε = Γ + Σ ε γ ε = Γ + Σ − Γ ε o o o o o Γ o : Surface Tension Coefficient Σ o : Surface Stress Coefficient cwu cwu UIC cwu cwu 062999-04

Surface Stress Coefficient Behaves like a pre-stressed membrane that is perfectly fitted on the bounding surface of a bulk material body. Interacts with the deformation of the bulk material through the curvature of the surface. Becomes increasingly important at reduced scales. Effect on linear and curvilinear cracks. cwu cwu cwu cwu UIC 062999-05

Configurational Energy Griffith Crack (1921 & 1924) Configurational Energy: σ 22 = Π Π 0 + ∆Π Stress Intensity Factor: K = σ a π 1 22 σ 11 For Configurational Equilibrium: 1 κ + 2 2 4 a a ∆Π = − µ π σ + Γ 22 0 8 16 ∂∆Π µΓ 2 2 0 K K C 0 = ⇒ = ≡ R → ∞ 1 1 2 a 1 ∂ κ + cwu UIC cwu cwu cwu 062999-06

Configurational Energy Σ 0 Griffith Crack with Surface Stress σ 22 Configurational Energy: = Π Π 0 + ∆Π σ 11 Stress Intensity Factor: K a (same) = σ π R → ∞ 1 22 For Configurational Equilibrium: a a f 1 1 κ + κ + 2 2 4 4 a a ∆Π = − µ π σ + Γ − Σ σ − σ 22 0 0 22 11 8 8 µ F I 2 G J ∂∆Π Σ σ 2 2 0 K 0 H 1 11 K K K C 0 = ⇒ + − − = 1 1 1 2 a a ∂ π σ 22 cwu cwu UIC cwu cwu 062999-07

Configurational Energy Griffith Crack with Surface Stress and Crack-Tip Point Load P Σ 0 σ 22 Configurational Energy: = Π Π 0 + ∆Π P P σ 11 Stress Intensity Factor: R → ∞ K a (still the same) = σ π 1 22 For Configurational Equilibrium: P a f 1 1 κ + κ + 2 2 ( 4 2 P ) ( 4 2 ) a a a ∆Π = − µ π σ + Γ − − Σ − σ − σ 22 0 0 22 11 8 8 µ F I F I 2 P P G J G J ∂∆Π Σ − σ 2 2 0 K 0 H 1 11 K K H 1 K K C 0 = ⇒ + − − − = 1 1 1 2 2 a a ∂ π σ Γ 22 0 cwu cwu UIC cwu cwu 062999-08

Configurational Energy Surface Stress, Internal Pressure p,and Crack-Tip Load P Configurational Energy: R → ∞ P P = Π Π 0 + ∆Π Stress Intensity Factor: Pressure p K = p π a 1 For Configurational Equilibrium: 1 1 κ + κ − 2 2 p ( 4 2 P ) ( 2 P p ) a a a ∆Π = − µ π + Γ − + Σ − 0 0 8 2 µ a f F I 2 1 ( 2 P ) P G J κ − ∂∆Π Σ − 2 2 0 K 0 K H 1 K K C 0 = ⇒ + − − = 1 1 1 2 ( 1 ) 2 a a ∂ π κ + Γ 0 cwu cwu UIC cwu cwu 062999-09

Configurational Energy Circular-Arc Cracks, Remote Tension and Surface Stress σ R → ∞ ρ σ 2 φ cwu cwu cwu cwu UIC 062999-10

Configurational Energy Circular-Arc Cracks, Remote Tension and Surface Stress Remote Tension: F I 2 φ 2 2 2 ρ H K (o) ( ) o ( ) o 2 2 K K K sin / 1 sin ≡ + = σ πρ φ + I II 2 2 φ Remote Tension and Surface Stress: F I F I 2 2 G J Σ φ φ H K 2 2 2 2 H K K K K 0 sin / 1 sin ≡ + = σ − πρ φ + I II 2 Remote πρ Tension For Configurational Equilibrium: σ a fa f F I 2 1 1 cos κ + − φ Σ φ H K a f 4 0 ∆Π = ρφΓ − πσρ − 0 2 3 cos µ − φ π − L O ∂∆Π = φ M P K 2 2 2 0 4 Σ sin / sin K K IC 0 N Q ⇒ πρ φ − = 0 2 2 ρ∂φ cwu UIC cwu cwu cwu 062999-11

Elasticity Solution Remote Loading, Surface Stress and Crack-Tip Load σ 22 R → ∞ The Bulk-Surface σ 11 System P P Free-Body Of P P The “Surface” 2 Σ − P 0 cwu UIC cwu cwu cwu 062999-12

Elasticity Solution Circular-Arc Crack,Remote Loading and Surface Stress σ 2 R → ∞ Σ 0 ρ Σ 0 Σ 0 σ 2 φ ρ ρ 2 Σ 0 Free-Body Of The Bulk-Surface System The “Surface” cwu cwu UIC cwu cwu 062999-13

Elasticity Solution - Griffith Crack Remote Loading, Surface Stress and Crack-Tip Load c h d i d i a f a f a f a f a f a f o o o s s s u , , u , , u , , ε τ = ε τ + ε τ α αβ αβ α αβ αβ α αβ αβ Total Load Surface Stress Strain Energies: z z a f a f a f a f c h R 2 π U W dA u R , u R , d ≡ ε = σ θ θ + σ θ θ θ e rr r r αβ θ θ 2 0 a f a f a f 1 P 2 u 0 u 0 a, a, + − Σ − − 0 1 1 2 z z a f a f a f a f d i R a f a f a f a f 2 π o o o o U W dA u R , u R , d ≡ ε = σ θ θ + σ θ θ θ e rr r r αβ θ θ 2 0 z a f a f a f d i 1 a f a f a f a f s s s s U W dA P 2 u 0 u 0 a, a, ≡ ε αβ = − Σ − − e 0 1 1 2 cwu cwu cwu UIC cwu 062999-14

Total Elastic Energy π a f a f z a f a f R a f a f 2 s s U u R , u R , d = σ θ θ + σ θ θ θ e rr r r θ θ 2 0 a f a f a f 1 a f a f o o P 2 u 0 u 0 a, a, + − Σ − − 0 1 1 2 a f a f o s U U + + e e a f a f a f a f a f a f a f o o o s P 2 u 0 u 0 U U a, a, = − Σ − − + + 0 1 1 e e Total Elastic Strain Energy: a f a f a f o s os U U U U = + + e e e e Interaction Energy: a f a f a f a f a f a f os o o U P 2 u a, 0 u a, 0 = − Σ − − e 0 1 1 cwu UIC cwu cwu cwu 062999-15

Energy of the Bulk-Surface System a f a f a f o s os Elastic Energy of the Bulk: U U U U = + + e e e e Surface Energy: z a f a f a f +a U 2 dx 4 2 u 0 u 0 a a, a, s = Γ ε = Γ + Σ − − 1 0 0 1 1 a − Potential of External Loads: π a f a f z a f a f 2 V R u R , u R , d = σ θ θ + σ θ θ θ rr r r θ θ 0 a f a f P 2 a + u a, 0 u a, 0 + − − 1 1 Total Potential Energy: a f a f a f a f o s os U U V U U U 4 2 P a Π = + − = − − − + Γ − s e e e e 0 cwu cwu UIC cwu cwu 062999-16

Final Results For σ , , P, and R σ Σ → ∞ 11 22 0 a fa f a f 2 R 1 a f π κ + 2 2 o 2 2 U 1 2 a = κ − σ + σ + σ − σ + µ π σ + ⋅⋅⋅ e 11 22 11 22 22 16 8 µ a f σ 22 2 1 Σ ln z a a f κ + − s 0 U z a = − → e 2 z a πµ + P P σ 11 a f a fa f a f P 2 1 a − Σ κ + os 0 U = σ − σ e 11 22 4 µ P a f 1 1 κ + κ + 2 2 ( 4 2 P ) ( 4 2 ) a a a ∆Π = − µ π σ + Γ − − Σ − σ − σ 22 0 0 22 11 8 8 µ F I F I 2 P P G J G J ∂∆Π Σ − σ 2 2 0 K 0 H 1 11 K K H 1 K K C 0 = ⇒ + − − − = 1 1 1 2 2 a a ∂ π σ Γ 22 0 cwu cwu UIC cwu cwu 062999-17

The Role of P σ 22 P P σ 11 Let = 0 σ = σ 11 22 Then ( 4 2 P a ) and ∆Π = Γ − 0 ∂∆Π 0 P 2 = ⇒ = Γ 0 2 a ∂ cwu cwu UIC cwu cwu 062999-18

Splitting A Double-Cantilever Beam a α P / 2 ∆ F b g P b P b g a a f a A Smooth Wedge of Angle 2 α L O 2 2 2 EI α ∂Π M P , 0 P = 6 a α Π = + Γ = ⇒ Γ N Q 0 0 α 3 a a ∂ a f b A Smooth Edge of Thickness 2 ∆ L O 2 2 3 EI ∆ ∂Π M P , 0 P = 2 a ∆ Π = + Γ = ⇒ Γ N Q 0 0 ∆ 3 2 a a ∂ cwu cwu cwu UIC cwu 062999-19