Short Answer:

The Maximum Strain Theory, also known as the Saint-Venant Theory, states that failure of a material occurs when the maximum principal strain in a complex stress system reaches the same value as the strain at the yield point in a simple tensile test. In simple words, this theory suggests that material failure is caused by excessive strain rather than stress.

It assumes that when the largest principal strain in a material under combined stresses equals the strain at yield during a uniaxial tension test, the material will start to fail. This theory is mainly applied to ductile materials but is less accurate than other advanced theories.

Detailed Explanation :

Maximum Strain Theory (Saint-Venant Theory)

The Maximum Strain Theory was proposed by Saint-Venant and is one of the earliest theories used to explain material failure under combined stresses. The theory is based on the assumption that the failure of a material depends on the strain produced within it rather than the stress.

It states that when a material is subjected to complex stresses in different directions, it will fail when the maximum principal strain in the material reaches the strain at yield obtained from a simple uniaxial tensile test. In other words, the strain responsible for failure in a complex stress system is equal to that which causes yielding in a simple tension test.

This theory is useful for understanding deformation behavior but not very accurate for predicting actual failure because it ignores the effects of shear and distortion energy, which are significant in ductile materials.

Definition

The Maximum Strain Theory can be defined as:

“Failure occurs when the maximum principal strain in a material reaches the limiting strain at the yield point in a simple tensile test.”

This means the material begins to yield or fail when its maximum principal strain equals the strain at which it starts to yield during a simple tension test.

Mathematical Expression

Let the principal stresses in the three perpendicular directions be .
The corresponding principal strains in the three directions can be expressed using Hooke’s Law as:

 

where,

•  = principal strains,
•  = Young’s modulus,
•  = Poisson’s ratio.

According to the Maximum Strain Theory, yielding or failure occurs when:

where,

•  is the strain at the yield point from a simple tensile test,
•  = yield stress in tension.

Substituting the expression for , we get:

Simplifying, the failure condition becomes:

This is the mathematical expression of the Maximum Strain Theory.

For Two-Dimensional Stress Condition

In most engineering applications, stress acts in two perpendicular directions (plane stress). In such a case, .
Hence, the failure condition becomes:

This equation defines the limit at which the material will start to yield or fail according to the maximum strain theory.

Graphical Representation

In the principal stress diagram (σ₁ vs σ₂), the yield condition represented by this theory forms an ellipse whose shape depends on the Poisson’s ratio .

• Points inside the ellipse represent safe stress conditions.
• Points on the ellipse indicate yielding.
• Points outside the ellipse indicate failure.

This ellipse lies between the shapes predicted by Rankine’s (square) and Von Mises’ (circular) criteria, indicating its intermediate nature.

Physical Meaning

This theory emphasizes that failure is due to excessive strain rather than stress. It assumes that a material fails when one of the principal strains (usually the maximum one) reaches the critical limit measured in a simple tension test.

However, it neglects the role of shear strain, which is responsible for distortion in ductile materials. Hence, it gives results that are sometimes inaccurate for materials that fail due to shear yielding, such as steel or aluminum.

Applicability

• The Maximum Strain Theory is suitable mainly for ductile materials under simple stress conditions.
• It provides a reasonable estimate for materials under uniform tension or compression.
• However, it is not suitable for brittle materials because they fail suddenly without large strains.
• In modern engineering, it is rarely used because energy-based theories like Von Mises provide more accurate results.

Advantages

1. Simple Concept:
   It is easy to understand and apply since it is based on the direct relationship between stress and strain.
2. Useful for Preliminary Design:
   Can be used in early design stages where detailed failure analysis is not required.
3. Applies to Elastic Range:
   Works well for materials that remain elastic up to the point of failure.

Limitations

1. Neglects Shear Effects:
   The theory does not consider shear strain, which plays a major role in ductile material failure.
2. Inaccurate for Ductile Materials:
   Ductile materials usually fail by yielding due to shear, not by normal strain.
3. Not Experimentally Supported:
   Experimental results often deviate from predictions made by this theory.
4. Limited Use in Modern Design:
   Replaced by more accurate theories like the Maximum Shear Stress Theory (Tresca) and Maximum Shear Strain Energy Theory (Von Mises).

Comparison with Other Theories

• The Maximum Stress Theory (Rankine) considers maximum stress, while this theory considers maximum strain.
• The Maximum Shear Stress Theory (Tresca) and Von Mises Theory are more suitable for ductile materials.
• The Maximum Strain Theory lies between stress-based and energy-based theories in accuracy.

Practical Example

If a steel plate is subjected to two principal stresses, say  and , and if Poisson’s ratio , and yield stress , the condition for failure according to Saint-Venant’s theory is:

Substituting values:

Since , the material is safe under these stresses.

Conclusion

The Maximum Strain Theory (Saint-Venant Theory) states that failure occurs when the maximum principal strain in a material equals the strain at yield in a simple tensile test. Although simple and useful for basic analysis, it does not accurately predict the failure of ductile materials, which fail due to shear. This theory is of more historical importance and has been replaced in practical applications by more reliable and energy-based theories such as the Von Mises theory.