6.3 Theories of Failure for Ductile and Brittle Materials

Failure Theories

Maximum Stress and Strain Theories

These theories each propose a different answer to the same question: what condition inside the material actually triggers failure? Choosing the right one depends on whether your material is ductile or brittle and what kind of loading it sees.

Maximum Normal Stress Theory predicts failure when the largest principal stress reaches the material's ultimate tensile strength ($S_{ut}$). This theory works best for brittle materials under static loading, since brittle materials tend to fracture on planes of maximum normal stress rather than shearing first.

Maximum Shear Stress Theory (Tresca Criterion) predicts yielding when the maximum shear stress reaches half the yield strength:

$\tau_{max} = \frac{\sigma_1 - \sigma_3}{2} = \frac{S_y}{2}$

• Yielding begins when shear stress on any plane hits this critical value
• Conservative and straightforward to apply for ductile materials (mild steel, aluminum)
• Always predicts failure at or before the von Mises criterion does, so it's the more conservative of the two ductile theories

Distortion Energy Theory (von Mises Criterion) predicts yielding when the distortion energy per unit volume equals the distortion energy at yield in a simple tension test. The von Mises stress is:

$\sigma' = \sqrt{\frac{(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2}{2}}$

Failure occurs when $\sigma' \geq S_y$.

• Accounts for the combined effect of all three principal stresses, not just the extreme ones
• More accurate than Tresca for most ductile materials (structural steel, copper) because it matches experimental data more closely
• Slightly less conservative than Tresca, typically by about 15%

Coulomb-Mohr Theories

These theories address materials that have unequal tensile and compressive strengths, which is common in brittle and quasi-brittle materials.

Coulomb-Mohr Theory predicts failure based on a combination of normal and shear stresses. For a biaxial stress state with principal stresses $\sigma_1$ and $\sigma_3$ (where $\sigma_1 > \sigma_3$), the failure criterion is:

$\frac{\sigma_1}{S_t} - \frac{\sigma_3}{S_c} \geq 1$

where $S_t$ is the tensile strength and $S_c$ is the compressive strength (entered as a positive value).

• Suitable for materials like cast iron and concrete, where compressive strength can be 3 to 10 times the tensile strength
• Graphically, the failure envelope connects the tensile and compressive Mohr's circles with straight tangent lines

Modified Mohr Theory adjusts the Coulomb-Mohr envelope to better fit experimental data, particularly in the fourth quadrant of the $\sigma_1$-$\sigma_3$ plane (tension-compression region).

• Provides more accurate predictions for brittle materials than the standard Coulomb-Mohr theory
• The modification essentially uses the maximum normal stress line in the first quadrant and a modified Coulomb-Mohr line in the fourth quadrant
• Applicable to cast iron, some composite materials, and rock mechanics

Fracture Mechanics

Fracture Mechanics Concepts

Fracture mechanics shifts the question from "will the material yield?" to "will an existing crack grow to cause sudden failure?" This matters because real components almost always contain small flaws from manufacturing, fatigue, or corrosion.

Stress Intensity Factor ($K$) quantifies the severity of the stress field near a crack tip. For a through-crack of half-length $a$ in an infinite plate under remote stress $\sigma$:

$K = \sigma \sqrt{\pi a} \cdot Y$

where $Y$ is a dimensionless geometry correction factor (equals 1 for the infinite plate case, but varies for real geometries like edge cracks or holes).

• $K$ depends on three things: applied stress, crack size, and component geometry
• Fracture toughness ($K_{Ic}$) is the critical value of $K$ at which a crack propagates unstably. It's a material property measured under plane strain conditions.
• Failure occurs when $K \geq K_{Ic}$

Typical $K_{Ic}$ values range widely: around 1 $MPa\sqrt{m}$ for glass, 25–50 $MPa\sqrt{m}$ for aluminum alloys, and 50–150+ $MPa\sqrt{m}$ for structural steels.

Brittle-Ductile Transition

Many materials, especially BCC metals like low-carbon steel, shift from ductile to brittle behavior as temperature drops. This transition is characterized by a temperature range, not a single point.

• Below the transition temperature, the material fractures with little plastic deformation and low energy absorption (cleavage fracture)
• Above the transition temperature, the material yields and deforms significantly before fracture (ductile tearing)
• The transition temperature shifts upward (toward more brittle behavior) with higher strain rates, the presence of notches or cracks, and increased material thickness
• The Charpy V-notch impact test is the standard method for measuring this transition

This is not just academic: the brittle fracture of Liberty ships in WWII and the Titanic's hull plates are classic examples of failures caused by operating below the brittle-ductile transition temperature.

Stress States

Plane Stress and Plane Strain

These are simplifying assumptions about the stress or strain state in a component. Which one applies depends primarily on the component's thickness relative to its other dimensions and the crack or load geometry.

Plane Stress occurs when one principal stress is zero (typically $\sigma_3 = 0$). This applies to:

• Thin plates and shells loaded in-plane (aircraft skins, thin-walled pressure vessels)
• The free surface of any component

Because the material is free to contract through the thickness, there's less constraint, and the material behaves in a more ductile manner. Stress analysis simplifies to two dimensions, and failure theories are applied using only $\sigma_1$ and $\sigma_2$.

Plane Strain occurs when one principal strain is zero (typically $\epsilon_3 = 0$). This applies to:

• Thick components where through-thickness deformation is constrained (thick-walled cylinders, dams, deep underground structures)
• The interior of a thick component near a crack front

The constraint generates a nonzero third principal stress ($\sigma_3 = \nu(\sigma_1 + \sigma_2)$ for elastic behavior), which raises the hydrostatic stress and reduces ductility. This is why fracture toughness $K_{Ic}$ is measured under plane strain conditions: it represents the worst-case (lowest) toughness value.