Short Answer:

The yield criterion is a condition or rule used to determine when a material begins to yield or undergo plastic deformation under a complex state of stress. It helps in predicting the onset of yielding in materials when subjected to different combinations of stresses.

In simple terms, the yield criterion defines the limit between elastic behavior (where the material returns to its original shape) and plastic behavior (where permanent deformation starts). Common yield criteria include Tresca’s (maximum shear stress) criterion and Von Mises (distortion energy) criterion, which are mainly used for ductile materials.

Detailed Explanation :

Yield Criterion

The yield criterion is an essential concept in mechanics of materials and plasticity theory, which describes the condition under which a material begins to yield or plastically deform. It provides a relationship between the different stress components acting on a material at the point of yielding.

When a material is loaded, it initially deforms elastically, meaning it can return to its original shape once the load is removed. However, beyond a certain stress limit called the yield point, permanent or plastic deformation begins. The yield criterion defines this limit in terms of the stresses acting on the material.

It is important because, in most engineering applications, components are designed to remain within the elastic range to avoid permanent damage. Thus, knowing when yielding begins helps ensure safety and reliability.

Definition

The yield criterion can be defined as:

“A yield criterion is a mathematical condition that defines the combination of stresses at which a material begins to yield or deform plastically.”

It expresses a relationship between principal stresses or strain energy at the onset of yielding. Depending on the type of material (ductile or brittle), different yield criteria are used.

For example:

• Ductile materials (like steel or copper) usually fail by shear yielding.
• Brittle materials (like cast iron or glass) fail by fracture under normal stress.

Hence, separate yield criteria are developed for each material type.

Mathematical Representation

A general form of the yield criterion is expressed as:

where,

•  are the principal stresses, and
•  is a function that represents the material’s yield behavior.

This equation defines a yield surface in the stress space. Points inside the surface correspond to the elastic region, while points on the surface represent the onset of yielding. Points outside the surface indicate that plastic deformation has already occurred.

Types of Yield Criteria

Different materials follow different yielding conditions depending on their mechanical behavior. The most common yield criteria are explained below:

1. Maximum Shear Stress Theory (Tresca Criterion)
   • Proposed by Henri Tresca, it states that yielding occurs when the maximum shear stress in the material reaches the shear stress at yield in a simple tensile test.
   • Mathematically,

where  is the yield stress.

1. • This criterion is mainly used for ductile materials.
2. Distortion Energy Theory (Von Mises Criterion)
   • Proposed by Richard von Mises, it states that yielding begins when the distortion energy per unit volume in a complex stress condition equals that at yield in a simple tension test.
   • Mathematically,

1. • It provides more accurate results than Tresca’s theory for ductile materials.
2. Maximum Principal Stress Theory (Rankine’s Criterion)
   • This theory is used for brittle materials and states that failure occurs when the maximum principal stress equals the ultimate tensile strength of the material.
   • Mathematically,

1. Maximum Principal Strain Theory (Saint-Venant’s Criterion)
   • According to this theory, failure occurs when the maximum principal strain equals the strain at yield in a simple tension test.
   • This theory is less accurate and not widely used.
2. Maximum Strain Energy Theory (Haigh’s Criterion)
   • This theory assumes that yielding occurs when the total strain energy per unit volume reaches the same value as in a uniaxial test.
   • However, it is less accurate compared to Von Mises and Tresca theories.

Yield Surface Concept

In three-dimensional stress space, the yield criterion can be visualized as a yield surface.

• Inside the surface → Elastic region (no permanent deformation).
• On the surface → Onset of yielding.
• Outside the surface → Plastic deformation region.

For example, in the Von Mises theory, the yield surface is an ellipsoid, while in the Tresca theory, it is a hexagon in the principal stress plane. This graphical approach helps in visualizing how yielding depends on the combination of different stresses.

Importance of Yield Criterion

1. Design Safety:
   Helps engineers determine safe working stress limits for materials.
2. Prediction of Failure:
   Accurately predicts when and how a material will start to deform plastically.
3. Material Selection:
   Assists in choosing appropriate materials for specific applications based on stress behavior.
4. Structural Analysis:
   Used in finite element analysis (FEA) and plasticity studies to simulate material deformation.
5. Component Reliability:
   Ensures that parts like beams, shafts, and pressure vessels operate below the yield limit.

Yield Criterion for Ductile and Brittle Materials

• For ductile materials (e.g., steel, copper, aluminum):
  The Von Mises and Tresca criteria are most accurate because ductile materials fail by shear yielding.
• For brittle materials (e.g., glass, concrete, cast iron):
  The Rankine criterion is most suitable because brittle materials fail by fracture due to normal stress.

Practical Applications

• Design of pressure vessels and shafts under combined stresses.
• Finite element analysis (FEA) for predicting yield zones in machine parts.
• Aerospace and automotive industries for ensuring structural integrity.
• Metal forming and plastic deformation studies.

Conclusion

The yield criterion defines the condition at which a material transitions from elastic behavior to plastic deformation. It establishes the relationship between stresses at the onset of yielding, allowing engineers to design safe and reliable components. For ductile materials, the Von Mises (distortion energy) and Tresca (maximum shear stress) criteria are most accurate, while for brittle materials, the Rankine (maximum principal stress) criterion is preferred. In modern engineering, yield criteria play a vital role in analyzing and preventing material failure under complex loading conditions.