L10: Plastic yielding and ductile fracture

Yield Criteria

Tresca Criterion

• Definition: Material yields when the maximum shear stress reaches the yield strength in shear (k).

• Equation: τmax=σ1−σ32=k

• Limitation: Conservative but less accurate for most metals.

von Mises Criterion

• Definition: Material yields when the effective stress (σe) reaches the yield strength in uniaxial loading.

• Equation:

  σe=(σ1−σ2)2+(σ2−σ3)2+(σ3−σ1)22=σy​

• Advantage: Better captures yield behaviour for most metals (e.g., copper, aluminium, steel).

When von Mises ≈ Tresca

• Conditions:

  • Uniaxial tension/compression (σ2=σ3=0).

  • Pure shear (σ1=−σ3​, σ2=0).

• Note: In other multiaxial cases, von Mises predicts yielding at higher stresses than Tresca.

Yield Condition

• Yield Function (f):

  • f=0: Material is at the yield point.

  • f<0: Elastic deformation (no yielding).

  • f>0: Not physically meaningful (yield surface cannot be exceeded).

2. Ductile Failure & Necking

Necking

• Definition: Localized deformation in ductile materials prior to fracture.

• True Stress-Strain: Power law σ=Kϵ^n, where K = strength coefficient, n = strain-hardening exponent.

Instability Criterion

• Uniaxial Case: Instability occurs when σ=Kn^n.

• Multiaxial Case (von Mises): Effective stress must satisfy σe=σy.

  

3. Elasto-Plastic Fracture Mechanics (EPFM)

Limitations of LEFM

• Not valid for materials with significant plastic deformation.

J-Integral Approach

• Definition: Energy-based method for nonlinear (elastic-plastic) materials.

• Path Independence: J can be calculated far from the crack tip.

  

• Physical Meaning: Energy release rate for nonlinear elastic materials.

• Equation:

  J=−dΠda​

  where Π = potential energy, a = crack length.

Experimental Determination

1. Measure load-displacement curves for varying crack lengths.

2. Plot J vs. displacement to find critical Jc at fracture initiation.

4. Crack Tip Opening Displacement (CTOD)

Definition

• Measures crack blunting due to plasticity.

Relation to Stress Intensity (K)

• Plane Stress:

  CTOD=K2σyE

  where σy​ = yield strength, E = Young’s modulus.

• Wells' Postulate: CTOD is a valid fracture parameter when LEFM is invalid.

  

  
  

5. Crack Growth Resistance (R-Curve)

Initiation Toughness (JIc​)

• Onset of crack extension.

R-Curve Behaviour

• Steep Gradient: Stable crack growth (safe).

• Shallow Gradient: Unstable growth (dangerous).

  

Validity of Parameters

• K: Small-scale yielding (plastic zone ≪ crack size).

• J-Integral: Large-scale yielding (plastic zone ≈ crack size).

  

6. Plastic Zone Behaviour Near Crack Tip

Effective Stress in Plastic Zone

• Within the plastic zone: σe=σy (equal to yield strength).

• Outside the plastic zone: σe<σy​ (elastic region).

Implications for EPFM

• The plastic zone is where stresses redistribute to maintain σe=σy​ (perfect plasticity assumed).

Takeaways

• Yield Criteria: von Mises is generally more accurate except in uniaxial/pure shear cases.

• Ductile Failure: Governed by necking and multiaxial instability.

• EPFM Tools:

  • J-Integral (energy-based) for large plasticity.

  • CTOD (displacement-based) for crack tip blunting.

• Plastic Zone: σe=σy within the zone, critical for fracture analysis.

Useful for analysing fracture in ductile materials (e.g., metals) beyond LEFM limits.