# What You Should Know About the Four Strength Theory?

Since the failure of materials can be divided into brittle fracture and yield according to their physical essence, the strength theory can be divided into two categories accordingly.

Here are the four commonly used strength theories.

## 1. Maximum tensile stress theory (the first strength theory is the maximum principal stress):

This theory is also called the first strength theory.

This theory believes that the main cause of failure is the maximum tensile stress.

No matter the complex and simple stress state, as long as the first principal stress reaches the strength limit of uniaxial tension, it is fracture.

Failure form: fracture;

Damage conditions: σ1 =σb;

Strength condition: σ1≤[σ];

Experiments show that the strength theory can better explain the fracture phenomenon of brittle materials such as stone and cast iron along the section where the maximum tensile stress is located;

It is not suitable for cases without tensile stress, such as one-way compression or three-way compression.

Disadvantages: the other two principal stresses are not considered.

Scope of application: suitable for tension of brittle materials. Such as cast iron stretching and twisting.

## 2. Maximum elongation linear strain theory (the second strength theory is the maximum principal strain)

This theory is also called the second strength theory.

This theory holds that the main cause of failure is the maximum linear strain of elongation.

No matter the complex and simple stress state, as long as the first principal strain reaches the limit value of uniaxial tension, it is fracture.

Failure assumption: the maximum elongation strain reaches the limit of simple tension (assuming that Hooke’s law can still be used to calculate until fracture occurs).

Failure form: fracture.

Brittle fracture conditions:ε1= εu=σb/E;

ε1=1/E[σ1−μ(σ2+σ3)];

Damage conditions: σ1−μ(σ2+σ3) = σb;

strength condition: σ1−μ(σ2+σ3)≤[σ].

Experiments show that the strength theory can better explain the phenomenon of fracture along the cross section of brittle materials such as stone and concrete under axial tension.

However, its experimental results are only consistent with few materials, so it has been rarely used.

Disadvantages: the general law of brittle fracture failure cannot be widely explained.

Scope of application: suitable for axial compression of stone and concrete.

## 3. Maximum shear stress theory (the third strength theory, namely Tresca strength):

This theory is also called the third strength theory.

This theory believes that the main cause of failure is the maximum shear stress max τ。

No matter the complex and simple stress state, as long as the maximum shear stress reaches the ultimate shear stress value under uniaxial tension, it is yield.

Failure assumption: the maximum shear stress of the danger sign of complex stress state reaches the shear stress limit of the material during simple tension and compression.

Failure form: yield.

Failure factor: maximum shear stress.

τmax=τu=σs/2

Yield failure condition: τmax=1/2(σ1−σ3 )

Damage conditions: σ1−σ3 = σs;

Strength condition: σ1−σ3≤[σ]

Experiments show that this theory can better explain the phenomenon of plastic deformation of plastic materials.

However, since the influence of 2σ is not considered, the components designed according to this theory tend to be safe.

Scope of application: suitable for general conditions of plastic materials.

The form is simple, the concept is clear, and the machinery is widely used.

But the theoretical result is safer than the actual one.

## 4. Specific energy theory of shape change (fourth strength theory, namely von mises strength)

This theory is also called the fourth strength theory.

This theory holds that no matter what stress state the material is in, the reason for the material to yield is that the specific energy (DU) of the shape change has reached a certain limit value.

Thus, the following can be established:

Damage conditions: 1/2(σ1−σ2)2+2(σ2−σ3)2+(σ3−σ1)2=σs;

Strength condition:σr4= 1/2(σ1−σ2)2+ (σ2−σ3)2 + (σ3−σ1)2≤[σ].

According to the thin tube test data of several materials (steel, copper, aluminum), it is shown that the shape change specific energy theory is more in line with the experimental results than the third strength theory.

Unified form of four strength theories: let the equivalent stress σrn have unified expression of strength condition σrn≤[σ].

Expression of equivalent stress:

σr1=σ 1≤[σ]

σr2=σ1−μ(σ2+σ3)≤[σ]

σr 3=σ1−σ3≤ [σ]

σr4= 1/2(σ1−σ2)2+(σ2−σ3)2+(σ3−σ1)2≤ [σ]

## 5. Mohr strength theory

Mohr’s strength theory does not simply assume that the failure of materials is caused by a certain factor (such as stress, strain or specific energy) reaching its limit value.

It is a strength theory based on the failure test results of materials under various stress states, considering the differences between tensile and compressive strengths of materials, recognizing that the maximum shear stress is the main cause of yield shear, and considering the influence of normal stress on the shear plane.

Mohr’s strength theory takes into account the different tensile and compressive capacities of materials, which is consistent with brittle materials (such as rock concrete, etc.), but without considering the influence of intermediate principal stress 2a is its deficiency.

## 6. Scope of application of strength theory

It depends not only on the nature of the material, but also on the stress state at the dangerous point.

Generally, the strength theory of brittle fracture and Mohr’s strength theory are used for brittle materials, and the strength theory of yield is used for plastic materials.

However, the failure mode of materials is also related to the stress state.

For example, whether plastic or brittle materials will fail in the form of fracture under the condition of three-dimensional tensile stress, the maximum tensile stress theory should be adopted.

In the case of three-dimensional compressive stress, plastic deformation is caused, and the third or fourth strength theory should be adopted.

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