Critical resolved shear stress

Introduction:

When a material is subjected to an external load, stresses develop inside it. In crystalline materials (such as metals), plastic deformation primarily occurs through slip—the motion of dislocations along specific crystallographic planes and directions.

Slip plane: The plane with the highest atomic density.
Slip direction: The direction within that plane with the highest linear atomic density.

The combination of slip plane and slip direction forms a slip system.

However, not all the applied stress is effective in driving slip. Only the component of stress resolved along the slip direction on the slip plane contributes to dislocation motion. This component is called the resolved shear stress (RSS).

The Critical Resolved Shear Stress (CRSS) is the minimum RSS required to initiate slip. Once the RSS on a particular slip system reaches this critical value, the crystal will start to deform plastically.

Schmid’s Law

The relationship between applied stress and resolved shear stress is given by Schmid’s Law:τ=σ⋅cos⁡ϕ⋅cos⁡λτ=σ⋅cosϕ⋅cosλ

Where:

ττ = resolved shear stress
σσ = applied normal stress
ϕϕ = angle between the applied stress axis and the slip plane normal
λλ = angle between the applied stress axis and the slip direction

According to Schmid’s Law, plastic deformation begins when:τ≥τcτ≥τc

where τcτc is the CRSS for that material.

Typical Values of CRSS

For pure metals, CRSS is typically very low (in the range of 0.5–5 MPa), because dislocations move relatively easily.
For alloys or materials with strong obstacles to dislocation motion (solid solution strengthening, precipitation hardening, work hardening), CRSS can be much higher.

This explains why hardened steels, nickel superalloys, or titanium alloys resist plastic deformation under high stresses, while pure aluminum or copper deform more readily.

Why is CRSS Important?

For Students:

Fundamental to plasticity theory – CRSS explains why materials begin to yield at certain stress levels.
Connects microstructure to mechanics – It highlights how crystal structure influences mechanical properties.
Foundation for advanced concepts – Understanding CRSS sets the stage for topics like strengthening mechanisms, creep, and fatigue.

For Engineers:

Material selection – Knowing CRSS helps in choosing alloys that can resist or allow plastic deformation, depending on the application.
Design for strength – In mechanical design, predicting yield strength from CRSS (and vice versa) aids in reliable load-bearing design.
Manufacturing processes – Forging, rolling, and extrusion rely on dislocation motion. Understanding CRSS helps optimize processing conditions.
Failure analysis – In cases of unexpected plastic deformation or fracture, CRSS-based analysis can identify if slip systems were activated.

A Practical Example

Consider a single crystal of copper subjected to a tensile stress of 50 MPa50MPa. The most favorable slip system in FCC crystals is the {111}⟨110⟩{111}⟨110⟩ system.

If the angles ϕ=45∘ϕ=45∘ and λ=45∘λ=45∘, then:τ=50⋅cos⁡45∘⋅cos⁡45∘=50⋅(0.707⋅0.707)≈25 MPaτ=50⋅cos45∘⋅cos45∘=50⋅(0.707⋅0.707)≈25MPa

If copper’s CRSS is around 0.5 MPa, the resolved shear stress is much greater than CRSS, meaning slip will definitely occur.

Key Takeaways

CRSS is the threshold shear stress required for slip in a crystal.
It explains why materials yield and how crystal orientation affects strength.
Schmid’s Law links applied stress to resolved shear stress.
For engineers, CRSS is central to material selection, mechanical design, and manufacturing process optimization.

👉 In summary, Critical Resolved Shear Stress bridges the gap between atomic-scale mechanisms and macroscopic mechanical behavior. For students, it’s a cornerstone of materials science. For engineers, it’s a tool for smarter, stronger, and safer designs.

Elegant Engineer