Since the damage of materials is divided into two forms of brittle fracture and yielding according to their physical nature, the strength theories are divided into two categories accordingly, and the following are the four strength theories commonly used at present.
1, the maximum tensile stress theory (the first strength theory that is the maximum principal stress)
This theory is also known as the first strength theory. This theory that the main cause of damage is the maximum tensile stress. Regardless of the complex, simple stress state, as long as the first main stress reaches the strength limit of the one-way stretch, that is, fracture.
Damage form: fracture.
Damage condition: σ1 = σb
Strength condition: σ1 ≤ [σ]
Experiments have proved that this strength theory better explains the phenomenon of fracture of brittle materials such as stone and cast iron along the cross section where the maximum tensile stress is located; it is not suitable for cases without tensile stresses such as one-way compression or three-way compression.
Disadvantage: The other two main stresses are not considered.
Use range: Applicable to brittle materials under tension. Such as cast iron tensile, torsion.
2、Maximum elongation line strain theory (second strength theory i.e. maximum principal strain)
This theory is also called the second strength theory. This theory believes that the main cause of damage is the maximum elongation line strain. Regardless of the complex, simple stress state, as long as the first main strain reaches the limit value of one-way stretching, that is, fracture. Damage assumption: The maximum elongation strain reaches the limit in simple tension (it is assumed that until fracture occurs it can still be calculated using Hooke's law).
Damage form: fracture.
Brittle fracture damage condition: 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]
It is proved that this strength theory better explains the phenomenon of fracture along the cross section of brittle materials such as stone and concrete when they are subjected to axial tension. However, its experimental results only agree with few materials, so it has been rarely used.
Disadvantage: It cannot widely explain the general law of brittle fracture damage.
Scope of use: Suitable for stone and concrete axially compressed.
3, 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
Regardless of the complex, simple stress state, as long as the maximum shear stress reaches the ultimate shear stress value in one-way stretching, that is, yielding. Damage assumption: complex stress state danger sign maximum shear stress reaches the limit of the material simple tensile, compressive shear stress.
Damage form: yielding.
Damage factor: maximum shear stress.
τmax = τu = σs / 2
Yield damage conditions: τmax=1/2(σ1-σ3 )
Damage condition: σ1-σ3 = σs
Strength condition: σ1-σ3 ≤ [σ]
Experimentally, it is proved that this theory can better explain the phenomenon of plastic deformation in plastic materials. However, the members designed according to this theory are on the safe side because the influence of 2σ is not considered.
Disadvantage: No 2σ effect.
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 result is safer than the actual one.
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 stress state the material is in, the material mechanics of the material yielded because the shape change ratio (du) reached a certain limit value. This can be established as follows
Damage 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]
Based on test data for thin tubes 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: so that the equivalent stress σrn, has the 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]
