An overview of the four strength theories

Since the failure of materials is divided into two types of brittle fracture and yielding according to their physical nature, the strength theories are correspondingly divided into two categories. The following four commonly used strength theories will be introduced.

 

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

 

This theory is also known as the first strength theory. According to this theory, the main cause of failure is the maximum tensile stress. Regardless of the complex or simple stress state, as long as the first principal stress reaches the strength limit of uniaxial tension, it breaks.

 

Form of failure: fracture.

 

Destruction condition: σ1 =σb

 

Strength condition: σ1≤[σ]

 

Experiments have shown that this strength theory can better explain the fracture of brittle materials such as stone and cast iron along the section where the maximum tensile stress is located; however, it is not suitable for uniaxial compression or triaxial compression without tensile stress.

 

Disadvantage: The other two principal stresses are not considered.

 

Scope of use: suitable for brittle materials under tension. Such as cast iron stretching, twisting.

 

2. The maximum elongation line strain theory (the second strength theory is the maximum principal strain)

 

This theory is also known as the second strength theory. According to this theory, the main cause of failure is the maximum elongation line strain. Regardless of the complex or simple stress state, as long as the first principal strain reaches the limit value during uniaxial stretching, it is fractured. Failure assumption: The maximum elongation strain reaches the limit of simple stretching (assume Hooke's law can be used until fracture occurs).

 

Form of failure: fracture.

 

Brittle fracture condition: ε1= εu=σb/E

 

ε1=1/E[σ1−μ(σ2+σ3)]

 

Destruction condition: 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 when they are axially stretched. However, its experimental results are only consistent with very few materials, so it has been rarely used.

 

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

 

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

 

3. The maximum shear stress theory (the third strength theory is Tresca strength):

 

This theory is also known as the third strength theory. This theory holds that the main cause of failure is the maximum shear stress

 

maxτ. Regardless of the complex or simple stress state, as long as the maximum shear stress reaches the ultimate shear stress value in uniaxial tension, it yields. Failure assumption: Dangerous sign of complex stress state, the maximum shear stress reaches the limit of shear stress when the material is simply pulled and compressed.

 

Destruction form: yield.

 

Failure factor: maximum shear stress.

 

τmax=τu=σs/2

 

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

 

Destruction condition: σ1−σ3 = σs

 

Strength condition: σ1−σ3≤[σ]

 

Experiments show that this theory can better explain the phenomenon of plastic deformation in plastic materials. However, since the influence of 2σ is not considered, the components designed according to this theory are on the safe side.

 

Disadvantage: No 2σ effect.

 

Scope of use: suitable for general conditions of plastic materials. The form is simple, the concept is clear, and the machinery is widely used. However, the theoretical results are safer than the actual ones.

 

4. The specific energy theory of shape change (the fourth strength theory is the von Mises strength)

 

This theory is also known as the fourth strength theory. This theory holds that: no matter what stress state the material is in, the reason for the material force student to yield is that the specific energy (du) of the shape change reaches a certain limit value. This can be established as follows

 

Destruction condition: 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 consistent with the experimental results than the third strength theory.

 

The unified form of the four strength theories: let the equivalent stress σrn, there is a unified expression for strength conditions

 

σrn≤[σ].

 

Equivalent stress expression:

 

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. Moiré Intensity Theory

 

Moiré 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 based on the failure test results of materials under various stress states. , considering the difference in tensile and compressive strengths of materials, acknowledging that the maximum shear stress is the main cause of yield shearing, and considering the effect of normal stress on the shear plane and establishing a strength theory.

 

Moiré strength theory takes into account the unequal tensile and compressive capabilities of materials, which is consistent with brittle materials

 

(such as rock concrete, etc.) failure characteristics, but the influence of the intermediate principal stress 2σ is not considered, which is its insufficiency.

 

6. Scope of application of strength theory

 

Not only depends on the properties of the material, but also on the stress state at the point of danger. In general, the strength theory and Mohr strength theory for brittle fracture are selected for brittle materials, and the strength theory for yield is selected for plastic materials. However, the failure mode of the material is also related to the stress state. For example, whether plastic or brittle materials will fail in the form of fracture under three-dimensional tensile stress, the maximum tensile stress theory should be used. In the case of three-dimensional compressive stress, plastic deformation is caused, and the third or fourth strength theory should be used.

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