# Bounds on Precipitate Hardening of Line and Surface Defects in Solids

## TL;DR

This paper analyzes how precipitates affect the movement of different crystal defects, revealing that line defects are more strongly hindered than surface defects due to differences in critical forces and scaling with precipitate size.

## Contribution

It provides a mathematical explanation for the differential impact of precipitates on line versus surface defects in crystalline solids.

## Key findings

- Critical force for surface defects scales with precipitate radius squared.
- Critical force for line defects scales linearly with precipitate radius.
- Surface defects are less affected by precipitates than line defects.

## Abstract

The yield behavior of crystalline solids is determined by the motion of defects like dislocations, twin boundaries and coherent phase boundaries. These solids are hardened by introducing precipitates -- small particles of a second phase. It is generally observed that the motion of line defects like dislocations are strongly inhibited or pinned by precipitates while the motion of surface defects like twin and phase boundaries are minimally affected. In this article, we provide insight why line defects are more susceptible to the effect of precipitates than surface defects. Based on mathematical models that describe both types of motion, we show that for small concentrations of a nearly periodic arrangement of precipitates, the critical force that is required for a surface defect to overcome a precipitate is smaller than that required for a line defect. In particular, the critical forces for surface and line defects scale with the radius of precipitates to the second and first power, respectively.