Uniaxial Monotonic Plasticity Lecture Notes

Introduction to Plasticity

• Plasticity: Branch of mechanics dealing with inelastic behavior of materials & structures.
• Different sub-branches: rate-dependent, time-dependent recovery, etc.
• Focus of the class: Metal plasticity (initially rate-independent loading).
• Situations may include work hardening or softening, cyclic loading.

Elastic vs. Plastic Behavior

Elasticity

• Definition: Deformation is recovered upon unloading.
  • Example: Steel bar stretches under load and returns to original length when unloaded.
• Alternative Definition: Complete energy recovery during a loading/unloading cycle.
• Stress-Strain Curve:
  • Hysteresis loop indicates permanent deformation.
• Elastic behavior is an assumption, not guaranteed at atomic level.
• Permanent changes can occur even below yield stress.

Plasticity

• Definition: Permanent deformation occurs when loaded.
• Observable after unloading (final length > original length).
  • Not easily distinguishable during loading.
• "Plastic" in this context refers to permanent deformation, not polymer materials.

Stress-Strain Curves

• Tension Tests: Can be done under load control or displacement control.
  • Load (P) vs. Displacement (Delta) measures.
• Displacement control is common in electromechanical testing machines.
• Engineering Stress:
  • Stress (σ_E) = Force / Original Cross-sectional Area (A₀).
• Engineering Strain:
  • Strain (ε_E) = Extensometer Displacement / Original Gauge Length.

Key Features of Stress-Strain Curves

• Ultimate Stress: Maximum stress point.
• Fracture Stress: Stress at failure.
• Proportional Limit Stress (σ_PL): Point where stress-strain relationship deviates from linearity.
• Yield Stress: Mark where permanent deformation begins, often estimated using a 0.2% offset method.

0.2% Offset Yield Stress Method

• The 0.2% refers to strain; involves drawing a line parallel to the initial slope (modulus of elasticity) at 0.002 strain to find yield stress.

Data Collection and Analysis

• Students will analyze data from 6061 T6 aluminum tension tests.
• Preload affects initial strain measurements; must account for this in analysis.
• Necking: Occurs after ultimate stress; leads to localized deformation.
• Engineering and true stress/strain definitions:
  • True Stress: Force / Instantaneous Area.
  • True Strain: ln(Instantaneous Length / Original Length).
  • True Stress (σ) = Engineering Stress × (1 + Engineering Strain).
  • True Strain (ε) = ln(1 + Engineering Strain).

Homework Assignment

• Construct engineering stress-strain curve.
• Adjust strain for preload.
• Find 0.2% offset yield stress.
• Construct true stress-strain curve.
• Plot both curves together.

Note: All definitions and methods are applicable until the necking point.