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An overview of the four strength theories
Release time: 2023-05-08
Since material failure is divided into brittle fracture and yield according to its physical nature, the strength theory is divided into two categories accordingly. The following are four commonly used strength theories.
1. 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 complex or simple stress states, as long as the first principal stress reaches the strength limit of unidirectional tensile, that is, fracture.
Failure mode: fracture.
Failure condition: σ1 =σb
Strength condition: σ1≤[σ]
The experimental results show that the strength theory can well explain the fracture phenomenon of brittle materials such as stone and cast iron along the section where the maximum tensile stress occurs. It is not suitable for unidirectional compression or three-way compression without tensile stress.
Disadvantages: The other two principal stresses are not considered.
Scope of application: Applicable to brittle material tension. Such as cast iron stretching, torsion.
2. Strain theory of maximum elongation line (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 complex or simple stress states, as long as the first principal strain reaches the limit value of unidirectional tensile, that is, fracture. Failure assumption: Maximum elongation strain reaches the limit of simple stretching (assuming Hooke's law can be calculated until fracture occurs).
Failure mode: fracture.
Brittle failure condition: ε1= εu=σb/E
Epsilon 1 = 1 / E [1 - mu sigma (sigma 2 + 3 sigma)]
Failure condition: σ1−μ(σ2+σ3) = σb
Strength conditions: σ1−μ(σ2+σ3)≤[σ]
Experimental results show that the strength theory can well explain the phenomenon of fracture along the cross section of brittle materials such as stone and concrete under axial tension. However, its experimental results only agree with a few materials, so it has been rarely used.
Disadvantages: The general law of brittle failure cannot be broadly explained.
Application scope: Suitable for stone, concrete axial compression.
3. Maximum shear stress theory (the third strength theory is Tresca strength)
This theory is also known as the third strength theory. According to this theory, the main cause of failure is the maximum shear stress
maxτ. Regardless of complex or simple stress states, as long as the maximum shear stress reaches the ultimate shear stress value of unidirectional tensile, that is, yield. Failure hypothesis: the maximum shear stress of the danger mark of complex stress state reaches the shear stress limit of the material during simple tension and compression.
Form of destruction: submission.
Failure factor: maximum shear stress.
Tau Max = tau u = sigma s / 2
Yield failure condition: τmax=1/2(σ1−σ3)
Failure condition: σ1−σ3 = σs
Strength condition: σ1−σ3≤[σ]
Experimental results show that this theory can explain the phenomenon of plastic deformation of plastic materials well. However, since the effects of two sigma are not taken into account, components designed according to this theory tend to be safer.
Disadvantages: There is no influence of two Sigma.
Application: Suitable for plastic materials in general. The form is simple, the concept is clear, the machine is widely used. But the theoretical results are safer than the actual ones.
4. Specific energy theory of shape change (the fourth intensity theory, namely von mises intensity)
This theory is also known as the fourth strength theory. According to this theory, no matter what stress state the material is in, the reason why the material will yield is that the specific energy of shape change (du) has reached a certain limit. This can be established as follows
Failure condition: 1/2(σ1−σ2)2+2(σ2−σ3)2+(σ3−σ1)2=σs
Strength conditions: σR4 = 1/2(σ1−σ2)2+ (σ2−σ3)2 + (σ3−σ1)2≤[σ]
According to the experimental data of several materials (steel, copper, aluminum), it is shown that the specific energy theory of shape change is more consistent with the experimental results than the third strength theory.
The unified form of the four strength theories: let equivalent stress σrn, there is a unified expression of strength conditions
σrn≤[σ].
Expression of equivalent stress:
σr1= σ1 ≤[σ]
Sigma r2 = 1 - mu sigma (sigma 2 + sigma 3) or less (sigma)
σr 3=σ1−σ3≤ [σ]
σR4 = 1/2(σ1−σ2)2+(σ2−σ3)2+(σ3−σ1)2≤ [σ]
5. Mohr's 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 based on the failure test results of materials under various stress states and takes into account the difference in tensile and compressive strength of materials. The maximum shear stress is the main cause of yield shear and the strength theory is established considering the influence of normal stress on shear plane.
Mohr's theory of strength takes into account the unequal tensile and compressive capacities of materials, which is consistent with brittle materials
Failure characteristics (such as rock concrete, etc.), but does not take into account the effect of intermediate principal stress 2σ is a shortcoming.
6. Applicable scope of strength theory
Not only depends on the nature of the material, but also depends on the stress state at the danger point. In general, brittle materials choose the strength theory of brittle fracture and Mohr's strength theory, plastic materials choose the strength theory of yield. But the failure mode of the material is also related to the stress state. For example, no matter plastic or brittle materials, they will fail in the form of fracture under the condition of three-way tensile stress, so the maximum tensile stress theory should be adopted. In the case of three compressive stresses, plastic deformation is caused, so the third or fourth strength theory should be adopted.
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