3: Flow Stress

Mechanical Properties of Materials

Introduction

• Overview of mechanical properties, focusing on dislocations and dislocation motion.

• Review of tensile test curves:

  • Elastic region (linear stress vs. strain).

  • Transition to plastic deformation (nonlinear, permanent deformation).

Yield Point and Yield Strength

• Definition of Yield Point: Transition from elastic (reversible) to plastic (permanent) deformation.

• Yield Stress: Determined using a 2% offset on the stress-strain curve.

• Yield Point characteristics:

  • Below yield point: Material returns to original state after unloading.

  • Beyond yield point: Material has permanent deformation when unloaded.

• Dislocation movement as a source of plastic deformation.

Transition from Uniaxial to Three-Dimensional Stress States

• Review of uniaxial tensile test:

  • Only one direction of stress (c3_11 = c3; c3_22 = c3_33 = 0).

• Importance of understanding three-dimensional stress states:

  • Real-world applications (e.g., bridge trusses, car parts) involve complex stress states.

  • Yielding criteria in complex stress conditions will be discussed.

Yielding Criteria

Tresca Theory

• Definition: Maximum shear stress approach to yielding.

• Shear Stress from Uniaxial Test: Maximum shear stress from the uniaxial test is c3_y / 2.

• Yielding Condition:

  • If c4_max (three-dimensional stress state) e0 c4_max (uniaxial test), yielding occurs.

• Example Calculation:

  • In principal coordinates: c4_max = (c3_11 - c3_33) / 2.

Von Mises Theory

• Definition: Energy per unit volume approach to yielding.

• Yielding Condition:

  • Energy comparison method; yielding when elastic strain energy is exceeded.

• Yield Stress Equation derived from stress tensor:

  • c2_y = 1/c3(2(c3_xx - c3_yy)^2 + 2(c3_yy - c3_zz)^2 + 2(c3_zz - c3_xx)^2 + 6(c3_xy^2 + c3_yz^2 + c3_xz^2))^1/2

• Importance of understanding energy absorption during yielding.

Plastic Region Behavior

• Understanding the shape of the plastic region of the stress-strain curve.

• Characteristic behaviors include:

  • Flow Stress: Related to dislocation motion.

  • Dislocation generation and interactions influence plastic deformation shape.

• Effects of plastic deformation on yield strength:

  • After initial yield, reloading requires additional stress beyond initial yield strength for further plastic deformation.

Strain and Strain Rate Hardening

Strain Hardening

• Definition: Increase in yield strength due to plastic deformation.

• Concept of strain hardening exponent:

  • c3 = K * b5^n; where n is the strain hardening exponent and K is a constant.

• Mechanisms contributing to strain hardening include increased dislocations that can block each other, resulting in a higher resistance to motion.

Strain Rate Hardening

• Definition: Dependence of yield strength on the rate of deformation.

• Higher strain rates can increase resistance due to insufficient time for dislocation motion.

• Flow stress behavior is defined as:

  • c3_flow = K * b5^n * b5_dot^m; where m is the strain rate hardening exponent.

Temperature Dependence

• Impacts of temperature on dislocation motion; kinetic limitations affect both strain hardening and strain rate behavior.

• Importance of conducting experiments at controlled temperatures for accurate measurements of strain and strain rate exponents.