The Universal Tension Machine (UTM) vs. Reality
In a lab, we pull a specimen until it yields at a stress $S_{yt}$. The stress state is simple: $\sigma_1 = S_{yt}, \sigma_2 = 0, \sigma_3 = 0$.
But imagine a transmission shaft. It experiences bending ($\sigma$) and torsion ($\tau$) simultaneously. This creates a complex state of Principal Stresses ($\sigma_1, \sigma_2, \sigma_3$). Failure theories give us an mathematical "envelope". If our stress state lies inside the envelope, the design is safe.
The General Approach
Calculate Principal Stresses $\to$ Apply Failure Theory $\to$ Compare to $S_{yt}$ or $S_{ut}$.
Maximum Principal Stress Theory (Rankine)
The Rule: Failure occurs when the maximum principal stress ($\sigma_1$ or $\sigma_2$) reaches the ultimate strength of the material ($S_{ut}$ or $S_{uc}$).
- Forms a Square failure envelope.
- Ignores the effect of other principal stresses.
- Best for: Brittle materials (e.g., Cast Iron, Chalk, Glass).
- Highly unsafe for ductile materials subjected to shear.
Maximum Shear Stress Theory (Tresca)
The Rule: Yielding begins when the maximum shear stress in a complex stress state equals the maximum shear stress at the yield point in a simple tensile test.
- Forms a Hexagon failure envelope.
- Predicts yield in pure shear at exactly $0.5 \times S_{yt}$.
- Best for: Ductile materials. It is highly conservative (safe).
Maximum Distortion Energy (von Mises)
The Rule: Yielding occurs when the distortion strain energy per unit volume reaches the distortion energy at yield in simple tension. It mathematically removes "hydrostatic" stresses (which don't cause yielding).
- Forms an Ellipse failure envelope.
- Predicts yield in pure shear at $0.577 \times S_{yt}$.
- Best for: Ductile materials. The most accurate theory compared to experimental data.
The Holy Trinity of Failure Envelopes
Comparing Rankine, Tresca, and von Mises on a single 2D Principal Stress plane.
Rankine (Square)
Extends furthest into the quadrants. If a ductile material operates near the corner of quadrant 2 or 4 (pure shear), Rankine dangerously predicts it's safe.
von Mises (Ellipse)
The gold standard for metals. Fits perfectly with actual experimental test data points. Circumscribes the Tresca hexagon.
Tresca (Hexagon)
Inscribed inside the von Mises ellipse. Because it is smaller, it is more conservative. If it passes Tresca, it definitively passes von Mises.
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