Failure mechanism theories

Failure mechanisms are essential for understanding how materials break down under various conditions.

These mechanisms help predict the failure of structures and materials, allowing for safer and more reliable designs.

The main failure mechanism theories include:

1. Maximum Stress Theory (Rankine Criterion)

This theory suggests that failure occurs when the maximum principal stress in a material reaches a critical value, which is the material’s ultimate tensile or compressive strength. The criterion is defined as: $\sigma_{\text{max}} = \sigma_1$
Where $\sigma_1$ is the maximum principal stress. The material is assumed to fail if:
$\sigma_1 \geq \sigma_{\text{ult}}$
where $\sigma_{\text{ult}}$ is the ultimate tensile stress of the material.

Applications and Limitations:

This theory is best suited for brittle materials where failure occurs by cracking rather than yielding.
It does not account for shear stress and is less accurate for ductile materials.

2. Maximum Strain Theory (St. Venant’s Theory)

This theory suggests that failure occurs when the maximum normal strain in the material exceeds a critical value. The strain in each principal direction is considered, and failure is predicted if:
$\epsilon_{\text{max}} \geq \epsilon_{\text{ult}}$
where $\epsilon_{\text{ult}}$ is the material’s ultimate strain.

Applications and Limitations:

Useful in cases where materials exhibit significant strain before failure, such as elastomers.
It doesn’t work well for materials that fail under low strain but high stress.

3. Maximum Shear Stress Theory (Tresca Criterion)

This theory asserts that failure occurs when the maximum shear stress in the material reaches a critical value. According to Tresca, the criterion is given by:
$\tau_{\text{max}} = \frac{1}{2}(\sigma_1 - \sigma_3)$
where $\sigma_1$ and $\sigma_3$ are the maximum and minimum principal stresses, respectively. Failure is predicted if:
$\tau_{\text{max}} \geq \tau_{\text{ult}}$
where $\tau_{\text{ult}}$ is the material’s ultimate shear stress.

Applications and Limitations:

This theory is more accurate for ductile materials, where shear deformation is significant before failure.
It is overly conservative for brittle materials since it ignores tensile stresses.

4. Maximum Distortion Energy Theory (Von Mises Criterion)

The Von Mises criterion, also known as the maximum distortion energy theory, predicts that failure occurs when the distortion energy in a material reaches a critical value. The theory is represented by:
$\sigma_{\text{v}} = \sqrt{\frac{1}{2}[(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2]}$
where $\sigma_1$ , $\sigma_2$ , and $\sigma_3$ are the principal stresses. Failure is predicted if:
$\sigma_{\text{v}} \geq \sigma_{\text{yield}}$
where $\sigma_{\text{yield}}$ is the yield stress of the material.

Applications and Limitations:

This theory is widely used for ductile materials and provides a good approximation of yielding under complex loading conditions.
Less accurate for predicting failure in brittle materials, which fail primarily under tensile stresses.

5. Coulomb-Mohr Theory (Mohr-Coulomb Criterion)

The Coulomb-Mohr theory combines the effects of normal and shear stresses on the failure plane. It assumes that failure occurs when a combination of shear and normal stress on a plane reaches a critical value. The failure condition is given by:
$\sigma_1 = \sigma_{\text{tension}} \quad \text{or} \quad \sigma_3 = \sigma_{\text{compression}}$
where the shear stress $\tau$ on the failure plane depends on the normal stress $\sigma_n$ according to:
$\tau = \tau_0 + \mu \sigma_n$
where $\tau_0$ is the material’s cohesive strength, and $\mu$ is the coefficient of internal friction.

Applications and Limitations:

This criterion is widely used for materials like concrete, soil, and rock, where shear and compressive stresses dominate.
It is more complex than other theories but provides a better approximation for materials with different tensile and compressive strengths.

6. Griffith’s Theory of Brittle Fracture

Griffith’s theory is specifically for brittle materials and predicts that failure occurs due to the propagation of pre-existing flaws or cracks in the material. According to Griffith, the condition for crack growth is:
$\sigma = \sqrt{\frac{2E \gamma_s}{\pi a}}$
where:

$\sigma$ is the applied stress.
$E$ is the Young’s modulus of the material.
$\gamma_s$ is the surface energy per unit area.
$a$ is the crack length.

Applications and Limitations:

This theory explains the fracture behaviour of brittle materials like glass and ceramics, which fail due to the growth of microscopic cracks.
It does not apply well to ductile materials where plastic deformation precedes fracture.

7. Drucker-Prager Criterion

The Drucker-Prager criterion is an extension of the Von Mises criterion for materials with pressure-dependent yield criteria, such as soils, concrete, and rock. The failure criterion is expressed as:
$\sqrt{J_2} = A + B I_1$
where:

$J_2$ is the second invariant of the deviatoric stress tensor.
$I_1$ is the first invariant of the stress tensor.
$A$ and $B$ are material constants.

Applications and Limitations:

This criterion is particularly suited for geomaterials and materials with different behaviours under compressive and tensile loads.
It is more complex and requires calibration of material parameters.

Summary

These failure theories provide a framework for understanding how different materials respond to stress and strain under various conditions. The appropriate theory is selected based on the material properties, the nature of the loads, and the specific application. The choice of theory can significantly affect the prediction of failure and the design of safe, reliable structures and components.

1. Maximum Stress Theory (Rankine Criterion)

2. Maximum Strain Theory (St. Venant’s Theory)

3. Maximum Shear Stress Theory (Tresca Criterion)

4. Maximum Distortion Energy Theory (Von Mises Criterion)

5. Coulomb-Mohr Theory (Mohr-Coulomb Criterion)

6. Griffith’s Theory of Brittle Fracture

7. Drucker-Prager Criterion

Summary

Share this:

Like this:

Leave a ReplyCancel reply

Discover more from Product Development Engineers Ltd