2: Inner Atomic Energy Potential

Inner Atomic Energy Potential

• Discussed graph of inner atomic energy potential (U vs R).

• Identified regions exhibiting elastic behavior, encompassing both linear and nonlinear elasticity.

• Elastic behavior characterized by restoring forces that oppose applied force, returning atoms to original position.

Plastic Behavior

• Defined as permanent deformation in materials, resulting from bond breakage due to applied stress or force.

• Transition from elastic (temporary) to plastic (permanent) behavior occurs when sufficient energy breaks molecular bonds.

Theoretical Break Force

• Explored the concept of using inner atomic potential to calculate the theoretical force required to separate two atoms completely.

• Maximum bond energy when atoms are infinitely far apart implies bond energy approaches zero.

• At the inflection point of the energy curve, the maximum force a bond can sustain is reached.

Understanding Force Responses and Inflection Points

• Slope Analysis:

  • Slope increases positively as more force is applied up to the inflection point.

  • Initially zero slope (no response), transitions to positive slope (increasing force response), then curvature changes at the inflection point (max force).

  • Concavity Changes:

    • Identifies transition from positive to negative slope (response starts to decrease).

Identifying the Inflection Point

• The theoretical break point of a bond can be defined mathematically as the point where the second derivative (d²U/dr²) equals zero.

• Mathematical representation for finding slope and inflection point:

  • First derivative: dU/dr = - (6a/r^5) - (12b/r^7)

  • Second derivative: d²U/dr² = - (42a/r^8) + (157b/r^{14})

• Setting the second derivative to zero provides the value of R at which maximum force exists.

Elastic vs. Plastic Transitions in Experiments

• Real-world tensile tests analyze stress versus strain to determine yield stress—transition from elastic to plastic deformation.

• Yield Stress: Defined at the point of permanent deformation, often lower than theoretical force predictions.

Influence of Defects on Material Properties

• Identified discrepancies between theoretical maximum stress and actual stress values found in experiments.

• Atoms may arrange imperfectly, leading to defects responsible for yielding at lower energies.

• Defects include bond misalignments and broken bonds, impacting the material's functional properties.

Summary

• The study of atomic forces and deformations shows a gap between theoretical predictions and experimental observations in tensile testing.

• The presence of material defects plays a critical role in the mechanical properties of materials, explaining why deformation can occur under lower stress than expected.