As the destruction of materials according to its physical nature is divided into brittle and yield two types of forms, so the strength theory is also divided into two categories accordingly, the following is to introduce the four commonly used strength theory.

1, the maximum tensile stress theory (the first strength theory that the maximum principal stress)

This theory is also known as the first strength theory. This theory believes that the main cause of damage is the maximum tensile stress. Regardless of the complex, simple stress state, as long as the first principal stress reaches the one-way tensile strength limit, that is, fracture.

Form of damage: Fracture.

Damage condition: σ1 = σb

Strength condition: σ1 ≤ [σ]

Experiments have proved that the strength theory better explains the phenomenon of fracture of brittle materials such as stone, cast iron, etc. along the cross section where the maximum tensile stress occurs; and it is not suitable for the cases without tensile stress such as unidirectional compression or three-way compression.

Disadvantage: the other two principal stresses are not considered.

Scope of use: applicable to brittle materials subjected to tension. Such as cast iron tensile, torsion.

2, the maximum elongation line strain theory (the second strength theory that the maximum principal strain)

This theory is also known as the second strength theory. This theory that the main cause of damage is the maximum elongation strain. Regardless of the complex, simple stress state, as long as the first principal strain reaches the limit value of unidirectional tension, that is, fracture. Damage assumption: the maximum elongation strain reaches the limit of simple stretching (assuming that until the occurrence of rupture can still be calculated by Hooke's law).

Form of damage: Fracture.

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

ε1=1/E[σ1-μ(σ2+σ3)

Damage condition: σ1-μ(σ2+σ3) = σb

Strength condition: σ1-μ(σ2+σ3) ≤ [σ]

Experiments have proved that the strength theory better explains the phenomenon of fracture along the cross-section when brittle materials such as stone and concrete are subjected to axial tension. However, its experimental results agree with only a few materials, so it has been rarely used.

Disadvantages: Cannot widely explain the general law of brittle fracture damage.

Scope of use: suitable for stone, concrete axial pressure.

3, the maximum shear stress theory (the third strength theory that Tresca strength)

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

Regardless of the complex, simple stress state, as long as the maximum tangential stress reaches the value of the ultimate tangential stress in unidirectional tension, that is, yielding. Damage assumptions: complex stress state danger sign maximum shear stress reaches the material simple tensile, compressive shear stress limit.

Destruction form: yielding.

Damage factor: the maximum tangential stress.

τmax=τu=σs/2

Yield damage conditions: τmax = 1/2 (σ1-σ3 )

Damage condition: σ1-σ3 = σs

Strength condition: σ1-σ3 ≤ [σ]

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

Disadvantage: No 2σ influence.

Scope of use: Suitable for the general case of plastic materials. The form is simple, the concept is clear, and the machinery is widely used. However, the theoretical results are more safe than the actual.

4, shape change specific energy theory (the fourth strength theory that von mises strength)

This theory is also known as the fourth strength theory. This theory that: no matter what the material is in the stress state, the material hair material mechanics yielding is due to the shape of the change than the energy (du) reached a certain limit value. This can be established as follows

Destruction 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]

Based on the test data of thin tubes of several materials (steel, copper, and 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: so that the equivalent stress σrn, there is a unified expression for the strength condition

σrn≤[σ].

Expression for 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、 Moore's strength theory

Moore strength theory is not simply assume that the destruction of the material is caused by a factor (such as stress, strain or specific energy) reached its limit value, it is based on a variety of stress state of the material damage test results, taking into account the material tensile and compressive strength of the different, recognizing the maximum shear stress is the main cause of yield shear and take into account the impact of the positive stress on the shear surface and the establishment of the strength theory. The theory of strength was established by recognizing that the maximum shear stress is the main cause of yield shear and considering the effect of positive stress on the shear surface.

Moore's strength theory takes into account the unequal tensile and compressive capabilities of materials, which is consistent with the brittle material

This is in line with the damage characteristics of brittle materials (e.g., rock concrete, etc.), but the failure to consider the effect of the intermediate principal stress 2σ is its shortcoming. 6.

6. Scope of application of strength theory

It depends not only on the nature of the material, but also on the state of stress at the point of danger. In general, brittle materials choose the theory of strength on brittle fracture and Mohr's theory of strength, plastic materials choose the theory of strength on yielding. However, the form of failure of a material is also related to the state of stress. For example, whether plastic or brittle materials, in the case of three-way tensile stress will fail in the form of fracture, it is appropriate to use the maximum tensile stress theory. In the case of three-way compressive stresses, all of which cause plastic deformation, the third or fourth strength theory is appropriate.