1/141/14 M.Chrzanowski: Strength of Materials SM2-10: Yielding & rupture criteria YIELDING AND RUPTURE CRITERIA (limit hypothesis)

2/142/14 M.Chrzanowski: Strength of Materials SM2-10: Yielding & rupture criteria 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. 11 11 arctanE RHRH RmRm  expl <<R m  expl <R H  expl =  1 =R H /s  expl s -1 Safety coefficient ? Ultimate strength Elastic limit

3/143/14 M.Chrzanowski: Strength of Materials SM2-10: Yielding & rupture criteria 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.  zx  xz xx xx xx z x 11 22 22 11 1 2 z x 22 11 Do we need to satisfy two independent conditions  x < R H  x < R H where R Ht i R Hs 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 R H … 11 22 p|p| 1 2 Thus, we need to formulate a hypothesis defining which stress components should be taken as basis for safe structure design.

4/144/14 M.Chrzanowski: Strength of Materials SM2-10: Yielding & rupture criteria 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. 00

5/145/14 M.Chrzanowski: Strength of Materials SM2-10: Yielding & rupture criteria Let the exertion measure be: 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) The ratio: gives „the distance” from unsafe state. stress vector in main principal axis This distance can be dealt with as the exertion in a given point.

6/146/14 M.Chrzanowski: Strength of Materials SM2-10: Yielding & rupture criteria It is seen, that materials which obey this hypothesis are isotopic with respect to their strength. GALIEO-CLEBSH-RANKINE hypothesis (GCR) They are also isonomic, as their strength properties are identical for tension and compression. For plane stress state it reduces to a square.

7/147/14 M.Chrzanowski: Strength of Materials SM2-10: Yielding & rupture criteria Exertion ≤ 100% Exertion ≤ 80% Exertion ≤ 60% Exertion ≤ 40% Exertion 0%

8/148/14 M.Chrzanowski: Strength of Materials SM2-10: Yielding & rupture criteria Material isonomic and isotropic Material isotropic but not isonomic Material insensitive to compression. (classical Galileo hypothesis) where a when a>0 0 when a<0 GALILEO hypothesis

9/149/14 M.Chrzanowski: Strength of Materials SM2-10: Yielding & rupture criteria COULOMB-TRESCA-GUEST hypothesis CTG For torsion: Uniaxial tension Many materials are sensitive to torsion This hexagon represents Coulomb- Tresca hypothesis (for plane stress state); the measure of exertion is maximum shear stress: In uniaxial state of stress:

10 /14 M.Chrzanowski: Strength of Materials SM2-10: Yielding & rupture criteria In uniaxial state of stress: 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 3D space of principal stresses (Haigh space) this hypothesis is represented by a cylinder with open ends. In 2D plane stress state for is an ellipse shown above. HUBER-MISES-HENCKY hypothesis HMH

11 /14 M.Chrzanowski: Strength of Materials SM2-10: Yielding & rupture criteria Hypothesis Exertion measure 3D image 2D image GCRCTGHMH Maximum normal stress Maximum shear stress Deformation energy Cube with sides equal to 2R Hexagonal prism with uniformly inclined axis Circular cylinder with uniformly inclined axis Substitute stress Substitute stress for beams

12 /14 M.Chrzanowski: Strength of Materials SM2-10: Yielding & rupture criteria  stop