YIELDING AND RUPTURE CRITERIA (limit hypothesis)

1. YIELDING AND RUPTURE CRITERIA (limit hypothesis)

2. 1 Rm RH 1 arctanE The knowledge of stress and strain states and displacements in each point of a structure allows for design of its members. The dimensions of these members should assure functional and safe exploitation of a structure. In the simplest case of uniaxial tension (compression) it can be easily accomplished as stress matrix is represented by one component 1 only, and displacement along bar axis can be easily measured to determine axial strain 1 Measurements taken during the tensile test allow also for determination of material characteristics: elastic and plastic limits as well as ultimate strength. With these data one can easily design tensile member of a structure to assure its safety. ? expl Ultimate strength expl<<Rm Elastic limit expl<RH s-1 expl = 1 =RH /s Safety coefficient

3. z tzx txz sx sx sx txz x tzx 2 z s2 a2 s1 x s1 a1 1 s2 2 s1 s2 1 |p| In the more complex states of stress (for example in combined bending and shear) the evaluation of safe dimensioning (related to elastic limit) becomes ambiguous. Do we need to satisfy two independent conditions x< RH x< RH where RHt i RHs denote elastic limits in tension and shear, respectively? Transformation to the principal axis of stress matrix does not help either, as we do not know whether the modulus of combined stresses is smaller then RH … Thus, we need to formulate a hypothesis defining which stress components should be taken as basis for safe structure design.

4. 0 In general case of 3D state of stress we introduce a function in 9-dimensional space of all stress components (or 3-dimensional in the case of principal axes) which are called the exertion function: In uniaxial sate of stress: We postulate that exertion function will take the same value in given 3D state of stress as that in uniaxial case. The solution of this equation with respect to 0: is called substitute stress according to the adopted hypothesis defining function F and thus – function ,as well.

5. stress vector in main principal axis Let the exertion measure be: The ratio: gives „the distance” from unsafe state. This distance can be dealt with as the exertion in a given point. SUCH A HYPOTHESIS DOES NOT EXIST ! A very similar one, which does exist is called Gallieo-Clebsh-Rankine hypothesis Associated  function appears to bo not-analytical one (derivatives on edges are indefinable)

6. GALIEO-CLEBSH-RANKINE hypothesis (GCR) It is seen, that materials which obey this hypothesis are isotopic with respect to their strength. They are also isonomic, as their strength properties are identical for tension and compression. For plane stress state it reduces to a square.

7. Exertion ≤ 100% Exertion ≤ 80% Exertion ≤ 60% Exertion ≤ 40% Exertion 0%

8. a when a>0 0when a<0 GALILEO hypothesis Material isonomic and isotropic Material insensitive to compression. (classical Galileo hypothesis) Material isotropic but not isonomic where

9. COULOMB-TRESCA-GUEST hypothesis CTG For torsion: Many materials are sensitive to torsion This hexagon represents Coulomb-Tresca hypothesis (for plane stress state); the measure of exertion is maximum shear stress: Uniaxial tension In uniaxial state of stress:

10. HUBER-MISES-HENCKY hypothesis HMH Small but important improvement has been made by M.T. Huber followed by von Mises and Hencky: It is distortion energy only which decides on the material exertion: For elastic materials (Hooke law obeys): In uniaxial state of stress: In 3D space of principal stresses (Haigh space) this hypothesis is represented by a cylinder with open ends. In 2Dplane stress state for is an ellipse shown above.

11. Hypothesis GCR CTG HMH Maximum normal stress Maximum shear stress Exertion measure Deformation energy Hexagonal prism with uniformly inclined axis Circular cylinder with uniformly inclined axis Cube with sides equal to 2R 3D image 2D image Substitute stress Substitute stress for beams

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