Glossary

8.2 Yield criteria and failure theories

4 min read•july 30, 2024

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 of a material is the stress level at which 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

  • (maximum shear stress criterion)
  • (maximum distortion energy 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∣)=σymax(|σ1 - σ2|, |σ2 - σ3|, |σ3 - σ1|) = σy, where σ1σ1, σ2σ2, and σ3σ3 are the principal stresses, and σyσy 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=2σy2(σ1 - σ2)^2 + (σ2 - σ3)^2 + (σ3 - σ1)^2 = 2σy^2, where σ1σ1, σ2σ2, and σ3σ3 are the principal stresses, and σyσy is the yield strength
  • Widely used for (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 (σ1σ1, σ2σ2, and σ3σ3) are ordered such that σ1≥σ2≥σ3σ1 ≥ σ2 ≥ σ3, with σ1σ1 being the maximum principal stress and σ3σ3 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 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−σ3σ1 - 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(σ1 + σ2 + σ3) / 3, does not contribute to material yielding or failure in most metals
  • Failure theories, such as the (Rankine criterion) and the , 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
  • (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 (, )
    • 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 (, )
    • 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

Key Terms to Review (24)

ASTM Standards: ASTM standards are guidelines and specifications developed by ASTM International, an organization that sets consensus standards for materials, products, systems, and services. These standards are vital in ensuring quality and safety across various industries, including construction, manufacturing, and materials science. They provide a basis for evaluating the performance and reliability of materials and components, which is essential for understanding yield criteria and failure theories.
Brittle materials: Brittle materials are substances that fracture or break without significant deformation when subjected to stress. They tend to have a limited ability to absorb energy before failure, meaning they do not undergo plastic deformation and typically fail suddenly under tensile or compressive forces. This behavior is crucial in understanding the yield criteria and failure theories, as it helps in predicting how such materials will respond under load, especially when assessing safety and structural integrity.
Compressive Loading: Compressive loading refers to the application of a force that tends to compress or shorten a material or structure, resulting in axial stress. This type of loading is crucial for understanding how materials behave under pressure and is directly related to yield criteria and failure theories, which help predict when materials will fail or deform irreversibly under such loads. Analyzing compressive loading is essential for ensuring the safety and integrity of structures designed to bear loads, as it informs engineers about the material limits and potential failure points.
Design Margin: Design margin is the additional safety factor included in the design of structures or materials to ensure they can withstand unexpected loads or conditions beyond the expected operating limits. This margin accounts for uncertainties in material properties, loading conditions, and potential defects, thus enhancing reliability and safety. By incorporating a design margin, engineers aim to prevent failure, ensuring that structures perform adequately under various scenarios.
Drucker-Prager Yield Criterion: The Drucker-Prager yield criterion is a mathematical model used to predict the yield behavior of materials under stress, particularly for materials like soils and rocks. This criterion generalizes the von Mises yield criterion for isotropic materials by accounting for the effects of pressure, making it particularly useful in geotechnical engineering and plasticity theory.
Ductile Materials: Ductile materials are substances that can undergo significant plastic deformation before rupture, allowing them to be stretched into wires or molded into shapes without breaking. This property is essential in engineering, as it enables materials to absorb energy during stress, providing a warning before failure occurs. Ductility plays a critical role in understanding how materials behave under load, especially regarding yield criteria and failure theories, as well as normal and shear strain.
Elastic Limit: The elastic limit is the maximum stress that a material can withstand while still being able to return to its original shape upon unloading. Beyond this point, the material will undergo permanent deformation and will not return to its initial dimensions. Understanding the elastic limit is crucial for predicting material behavior under load, especially when applying yield criteria and determining how structures will react to forces while maintaining their integrity.
Factor of Safety: The factor of safety (FoS) is a design principle that provides a safety margin in engineering by comparing the maximum load a structure can withstand to the actual load it is expected to carry. This concept is crucial as it helps prevent structural failure by ensuring that the materials used can handle more stress than they will encounter during normal use. Understanding the factor of safety is essential in evaluating material behavior under different loading conditions, ensuring reliability and durability in various applications.
Goodman Criterion: The Goodman Criterion is a method used to predict the failure of materials under fluctuating loads, particularly focusing on fatigue failure. This criterion provides a graphical representation that helps engineers assess the safe limits of alternating and mean stresses, establishing a relationship that allows for the determination of permissible stress levels in materials subjected to cyclic loading. It's particularly relevant in evaluating components that experience repeated loading conditions, ensuring safety and longevity.
Hill Criterion: The Hill Criterion is a yield criterion specifically developed for materials that exhibit anisotropic behavior, meaning their strength varies with direction. It is particularly useful for predicting the onset of plastic deformation in metals, especially those with different mechanical properties in different directions due to processes like rolling or forging. This criterion helps engineers understand when materials will yield under multi-axial loading conditions, offering a more accurate representation of real-world material behavior compared to isotropic models.
ISO Guidelines: ISO Guidelines refer to the standards set by the International Organization for Standardization (ISO) that provide a framework for the evaluation and classification of material properties, particularly in relation to yield criteria and failure theories. These guidelines help ensure consistency and reliability in assessing the performance and safety of materials under different loading conditions, which is crucial in engineering design and analysis.
Maximum normal stress criterion: The maximum normal stress criterion is a failure theory that states a material will fail when the maximum normal stress in the material exceeds a certain allowable limit. This concept is crucial for assessing material strength and is often used to predict failure under various loading conditions, linking directly to how materials behave under stress and the conditions leading to their ultimate failure.
Maximum Stress Theory: Maximum stress theory, also known as the Rankine failure criterion, is a principle used to predict the failure of materials under different loading conditions. It states that failure occurs when the maximum normal stress in a material exceeds the material's yield strength. This theory is particularly useful in understanding how materials behave under axial loads and is connected to yield criteria and failure theories in materials science.
Mohr-Coulomb Criterion: The Mohr-Coulomb criterion is a mathematical model used to describe the conditions under which materials fail due to shear stress and normal stress. It defines the failure envelope as a linear relationship between shear stress and normal stress, emphasizing the role of cohesion and internal friction angle of the material. This criterion is crucial for understanding material behavior under different loading scenarios and helps in predicting failure in structural components.
Mohr's Circle: Mohr's Circle is a graphical representation used to determine the state of stress at a point in a material. It provides a visual way to analyze the relationships between normal and shear stresses acting on different planes, making it easier to understand concepts like principal stresses, maximum shear stress, and failure criteria. By using Mohr's Circle, engineers can efficiently assess how materials will respond under various loading conditions, which is crucial for ensuring structural integrity.
Plastic Deformation: Plastic deformation refers to the permanent change in shape or size of a material when subjected to stress beyond its yield strength. This process occurs after the elastic limit has been exceeded, and the material cannot return to its original shape once the load is removed. Understanding plastic deformation is crucial for analyzing how materials behave under different loads and conditions, as it connects to yield criteria, stress-strain relationships, material behavior, and how structures respond to bending and other forces.
Soderberg Criterion: The Soderberg Criterion is a failure theory that assesses the safety of materials under fluctuating loads by relating the mean and alternating stresses to the material's yield strength and ultimate tensile strength. It provides a graphical representation to determine the limits of safe loading conditions, particularly for materials subjected to repeated or cyclic loading, making it a crucial tool in the evaluation of fatigue failure.
Stress-strain curve: A stress-strain curve is a graphical representation that illustrates the relationship between the stress applied to a material and the resulting strain it experiences. This curve provides crucial insights into how materials behave under various loads, showing important stages such as elastic deformation, yield point, and ultimate tensile strength, which are essential in understanding material failure and design considerations.
Tensile Loading: Tensile loading refers to the application of a force that stretches a material, resulting in an increase in length and a decrease in cross-sectional area. This type of loading is critical for understanding how materials respond to forces, particularly in the context of yield criteria and failure theories, which help predict when a material will fail or yield under stress.
Tresca Yield Criterion: The Tresca yield criterion is a theory used to predict the yield point of ductile materials under shear stress. This criterion states that yielding begins when the maximum shear stress in the material reaches a critical value, which is determined by the yield strength of the material in simple tension or compression. It is particularly useful for analyzing metals and helps engineers understand when materials will deform plastically under complex loading conditions.
Tsai-Wu Criterion: The Tsai-Wu criterion is a failure theory used to predict the failure of composite materials under multi-axial loading conditions. This criterion combines normal and shear stress components to provide a comprehensive assessment of material behavior, making it particularly useful for understanding the complex interactions in composite structures. By applying this criterion, engineers can determine the limits of strength in various loading scenarios and optimize design to prevent failure.
Ultimate Tensile Strength: Ultimate tensile strength (UTS) is the maximum stress a material can withstand while being stretched or pulled before breaking. It serves as a critical parameter in understanding the strength of materials and connects to their behavior under loading, allowing for assessments of yield criteria, stress-strain relationships, and the transition from elastic to plastic deformation.
Von Mises yield criterion: The von Mises yield criterion is a mathematical model used to predict the yielding of materials under complex loading conditions. It states that yielding occurs when the second deviatoric stress invariant reaches a critical value, specifically when the equivalent stress exceeds the yield strength of the material. This criterion is essential for understanding how materials behave under combined stresses and forms a foundational concept in analyzing safety and performance in structural engineering.
Yield Strength: Yield strength is the stress at which a material begins to deform plastically, meaning it will not return to its original shape after the load is removed. This concept is crucial as it helps determine the limits of material performance under various loading conditions, affecting design and safety in engineering applications.
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