Questions regarding material strength theory

Under complex stress conditions, it is not possible to determine the strength of materials through experiments; therefore, hypotheses must be formulated regarding the mechanical factors that lead to strength failure (failure) of the materials. This allows it to be possible to infer the strength of the same material under various complex stress conditions, based on its strength under simple stress conditions (tension, compression) or a few complex stress conditions. This hypothesis and the failure criteria derived from it are known as the strength theory of materials or mechanical strength theory; the latter is used to emphasize that such theories are based on macroscopic mechanical factors, distinguishing them from physical strength theories developed by studying the microscopic structure of materials. The failure of materials in terms of strength occurs in two forms: brittle fracture and plastic flow. Some basic strength theories are only applicable to a certain form of strength failure. 　　The maximum tensile stress theory (first strength theory) holds that brittle fracture of a material under any stress condition occurs because the largest tensile principal stress σ1 among the three principal stresses σ1, σ2, σ3 reaches the material’s limit value ； The corresponding strength condition is σ1≤〔σ〕, where〔σ〕 is the allowable stress of the material. 　　The maximum elongation strain theory (second strength theory) uses the maximum of the three principal strains, namely the elongation strain ε1, reaching its limit value for the material, as a criterion to determine brittle fracture ； The corresponding strength condition is 【σ1 - v(σ2 + σ3)】 ≤ 【σ】, where v is the Poisson’s ratio. Although this theory appears more refined than the first strength theory in form, it does not match experimental results well and is now used less frequently. 　　The maximum shear stress theory (third strength theory) holds that plastic flow in a material (including shearing) occurs because the maximum shear stress τ = (σ1–σ3)/2 reaches the material’s limit value ； The corresponding strength condition is (σ1-σ3)≤〔σ〕. The error resulting from this theory’s neglect of the effect of the intermediate principal stress σ2 is, for steel, at most 10% based on strength test data under biaxial stress conditions, and this value is on the safe side. This theory cannot be applied to materials with unequal flow limits under tension and compression. 　　The theory of specific energy for shape change (commonly known as the fourth strength theory) takes the limit value of the strain energy per unit volume of the material that corresponds to shape change (specific energy for shape change) as the criterion for plastic flow in the material ； The corresponding strength condition is as follows. This strength condition can also be derived from the perspective that the shear stress on the octahedron reaches its limit value, or from the perspective that the statistically average shear stress reaches its limit value. This theory can also only be applied to materials with equal flow limits under tension and compression. 　　Moor’s strength theory can be applied to materials for which the tensile and compressive strengths are different. It is a theory that is based directly on the results of material failure tests, and it has a certain element of empiricism. According to this theory, first, the limit stress circles for each stress state should be drawn based on the maximum principal stress σ1 and the minimum principal stress σ3 at which the material fails under that stress state (ignoring the intermediate principal stress σ2). Then, the envelope of a set of limit stress circles is formed (Figure 1). The criterion for strength failure of a material under any stress state is that the stress circle representing that stress state (based on σ1 and σ3) is tangent to the limit envelope. As an approximation, the limit envelope can be replaced by a straight line tangent to the limit stress circles for uniaxial tension and uniaxial compression, thereby simplifying the limit envelope. In this way, the aforementioned failure criterion can be expressed using geometric relationships (Figure 2) through a simple formula, and the corresponding strength condition is given by, where [σ1] and [σa] represent the allowable tensile and compressive stresses of the material under uniaxial tension and compression, respectively. This theory also has certain errors due to ignoring the effect of σ2.