8.2 Yield criteria and failure theories

Yield criteria and failure theories are crucial tools for predicting when materials will break or deform. They help engineers determine safe stress levels for various materials under different loading conditions, ensuring structures and components don't fail unexpectedly.

These concepts are essential in understanding how materials behave under stress. By applying yield criteria and failure theories, we can design safer, more efficient structures and products, balancing strength requirements with material properties and loading conditions.

Yield Criteria for Material Failure

Concept and Role of Yield Criteria

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• Yield criteria are mathematical expressions that define the stress state at which a material begins to yield or plastically deform
• Used to predict the onset of material failure under various loading conditions (uniaxial, biaxial, or triaxial stress states)
• The yield strength of a material is the stress level at which plastic deformation begins, and it is a critical material property used in yield criteria
• Consider the principal stresses acting on a material and compare them to the yield strength to determine if yielding occurs
• The choice of an appropriate yield criterion depends on the material properties and the specific loading conditions

Common Yield Criteria

• Tresca yield criterion (maximum shear stress criterion)
• von Mises yield criterion (maximum distortion energy criterion)
• Drucker-Prager yield criterion
• Each criterion has its own mathematical expression and assumptions based on material behavior and loading conditions
• The selection of the appropriate yield criterion is crucial for accurate prediction of material failure

Tresca vs von Mises Yield Criteria

Tresca Yield Criterion

• Also known as the maximum shear stress criterion
• States that yielding occurs when the maximum shear stress reaches a critical value equal to half the yield strength in uniaxial tension
• Expressed as: $max(|σ1 - σ2|, |σ2 - σ3|, |σ3 - σ1|) = σy$, where $σ1$, $σ2$, and $σ3$ are the principal stresses, and $σy$ is the yield strength
• Suitable for materials with similar yield strengths in tension and compression (cast iron, some ceramics)

von Mises Yield Criterion

• Also known as the maximum distortion energy criterion
• States that yielding occurs when the distortion energy reaches a critical value
• Expressed as: $(σ1 - σ2)^2 + (σ2 - σ3)^2 + (σ3 - σ1)^2 = 2σy^2$, where $σ1$, $σ2$, and $σ3$ are the principal stresses, and $σy$ is the yield strength
• Widely used for ductile materials (most metals) due to its ability to account for the combined effect of all principal stresses
• Particularly suitable for materials that exhibit isotropic behavior and have similar yield strengths in tension and compression

Applying Tresca and von Mises Criteria

• To apply the Tresca or von Mises yield criteria, the principal stresses acting on the material must be determined from the given stress state
• The calculated principal stresses are then substituted into the respective yield criterion equation to check if the condition for yielding is met
• If the yield criterion is satisfied, the material is expected to undergo plastic deformation, indicating the onset of yielding
• Example: For a given stress state, if the von Mises stress exceeds the yield strength, the material will yield according to the von Mises criterion

Principal Stresses and Failure Theories

Principal Stresses

• Principal stresses are the normal stresses acting on mutually perpendicular planes where the shear stresses are zero
• The three principal stresses ($σ1$, $σ2$, and $σ3$) are ordered such that $σ1 ≥ σ2 ≥ σ3$, with $σ1$ being the maximum principal stress and $σ3$ being the minimum principal stress
• Represent the most critical stress states acting on a material
• The orientation of the principal stress planes can be determined using Mohr's circle or by solving the eigenvalue problem for the stress tensor

Relationship to Failure Theories

• Principal stresses are crucial in failure theories because they represent the most critical stress states acting on a material
• The difference between the maximum and minimum principal stresses ($σ1 - σ3$) is called the principal stress difference and is used in some failure theories (Tresca yield criterion)
• The hydrostatic stress, defined as the average of the three principal stresses, $(σ1 + σ2 + σ3) / 3$, does not contribute to material yielding or failure in most metals
• Failure theories, such as the maximum normal stress criterion (Rankine criterion) and the Mohr-Coulomb criterion, rely on principal stresses to predict material failure

Selecting Failure Theories for Materials

Factors Influencing Failure Theory Selection

• Material properties (ductile, brittle, isotropic, anisotropic)
• Loading conditions (uniaxial, multiaxial, cyclic, fatigue)
• Desired level of accuracy
• Limitations and assumptions of each failure theory
• Experimental validation of the chosen theory

Failure Theories for Different Materials and Loading Conditions

• Ductile materials (most metals): von Mises yield criterion
  • Accounts for the combined effect of all principal stresses
  • Suitable for materials with isotropic behavior and similar yield strengths in tension and compression
• Brittle materials (ceramics, some polymers): Maximum normal stress criterion (Rankine criterion) or Mohr-Coulomb criterion
  • Maximum normal stress criterion states that failure occurs when the maximum principal stress reaches the ultimate strength of the material
  • Mohr-Coulomb criterion considers the effect of shear stress and normal stress on failure, suitable for materials with different strengths in tension and compression
• Cyclic or fatigue loading: Specific fatigue failure theories (Goodman criterion, Soderberg criterion)
  • Account for the effect of mean stress and alternating stress on fatigue life
• Anisotropic materials or materials with different yield strengths in different directions: Advanced failure theories (Hill criterion, Tsai-Wu criterion)
  • Consider the anisotropic behavior of the material

Importance of Validation

• It is important to consider the limitations and assumptions of each failure theory
• Validate the chosen theory with experimental data when possible
• Ensure that the selected failure theory accurately predicts the material behavior under the given loading conditions