yielding and rupture criteria (limit hypothesis)

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

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YIELDING AND RUPTURE CRITERIA (limit hypothesis).  1. R m. R H.  1. arctanE. - PowerPoint PPT Presentation

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Page 1: YIELDING AND RUPTURE CRITERIA (limit hypothesis)

1/14M.Chrzanowski: Strength of Materials

SM2-10: Yielding & rupture criteria

YIELDING AND RUPTURE CRITERIA

(limit hypothesis)

Page 2: YIELDING AND RUPTURE CRITERIA (limit hypothesis)

2/14M.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

1arctanE

RH

Rm

expl<<Rm

expl<RH

expl = 1 =RH /s

expl

s-1

Safety coefficient

?Ultimate strength

Elastic limit

Page 3: YIELDING AND RUPTURE CRITERIA (limit hypothesis)

3/14M.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

zx

xz

xz

xxx

z

x

1

2

2

1

1

2z

x2

1

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 …

12

p |

1

2

Thus, we need to formulate a hypothesis defining which stress components should be taken as basis for safe structure design.

Page 4: YIELDING AND RUPTURE CRITERIA (limit hypothesis)

4/14M.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:

),,()( 321 fFW ij

limW

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. )()( 0 FFW ij

The solution of this equation with respect to 0: )(0 ij is called substitute stress according to the adopted hypothesis defining function F and thus – function , as well.

)( 00 FWW

limWW

0

Page 5: YIELDING AND RUPTURE CRITERIA (limit hypothesis)

5/14M.Chrzanowski: Strength of Materials

SM2-10: Yielding & rupture criteria

23

22

21

2 pW

221 ,, f200 W

NR 023

22

21

10 0

N

N

R

R

Let the exertion measure be:

SUCH A HYPOTHESIS DOES NOT EXIST !A very similar one, which does exist

321 ,,max Wmis called Gallieo-Clebsh-Rankine hypothesis

Associated function appears to bo not-analytical one (derivatives on edges are indefinable)

NR

0WW

pmW

1

2

3

NRNR The ratio:

gives „the distance” from unsafe state.

stress vector in main principal axis

p

This distance can be dealt with as the exertion in a given point.

Page 6: YIELDING AND RUPTURE CRITERIA (limit hypothesis)

6/14M.Chrzanowski: Strength of Materials

SM2-10: Yielding & rupture criteria

NR1

321 ,,max Wm

NR2

NR3

NN RR 1

NN RR 2

NN RR 3

NRNR

NRNR

NR

NR

1

2

3

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.

Page 7: YIELDING AND RUPTURE CRITERIA (limit hypothesis)

7/14M.Chrzanowski: Strength of Materials

SM2-10: Yielding & rupture criteria

1

1 1

1

NR1

NR2

Exertion ≤ 100%

Exertion ≤ 80%

Exertion ≤ 60%

Exertion ≤ 40%

Exertion 0%

1

1 1 NR1

NR21

Page 8: YIELDING AND RUPTURE CRITERIA (limit hypothesis)

8/14M.Chrzanowski: Strength of Materials

SM2-10: Yielding & rupture criteria

scNR

1

2rNR

rNRsc

NR

Material isonomic and isotropic

Material isotropic but not isonomic

ncomrpessioN

tesionN RR

ncompressioN

tensionN RR

Material insensitive to compression. (classical Galileo hypothesis)

321 ,,max Wm

where

aa when a>0

0 when a<0

GALILEO hypothesis

Page 9: YIELDING AND RUPTURE CRITERIA (limit hypothesis)

9/14M.Chrzanowski: Strength of Materials

SM2-10: Yielding & rupture criteria

COULOMB-TRESCA-GUEST hypothesis CTG

scNR

1

2rNR

rNRsc

NR

rNR1

rNR2

12

0

0TFor 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:

1332210 ,,max

321 ,,max Wm

In uniaxial state of stress:

2

max0 oWm

2

,2

,2

max 133221 Wm

Page 10: YIELDING AND RUPTURE CRITERIA (limit hypothesis)

10/14M.Chrzanowski: Strength of Materials

SM2-10: Yielding & rupture criteria

NR

1

2

NR

NRNR DDm fW 2

1

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:

AADDvf 2

1

2

1

For elastic materials (Hooke law obeys):

2132

322

212

6

13

4

1 GG mijijf

20

0 26

1 Gf 213

232

2210

2

1

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. 03

HUBER-MISES-HENCKY hypothesis HMH

Page 11: YIELDING AND RUPTURE CRITERIA (limit hypothesis)

11/14M.Chrzanowski: Strength of Materials

SM2-10: Yielding & rupture criteria

NR

1

2

NR

NRNR

NR

1

2

NR

NRNR

NR

1

2

NR

NRNR

Hypothesis

Exertion measure

3D image

2D image

GCR CTG HMHMaximum

normal stressMaximum

shear stress Deformation energy

Cube with sides equal to 2R

Hexagonal prism with uniformly inclined axis

Circular cylinder with uniformly inclined axis

Substitute stress

),max( 210 210 2122

210

Substitute stress for beams

220 42

1xyxx 22

0 4 xyx 220 3 xyx

Page 12: YIELDING AND RUPTURE CRITERIA (limit hypothesis)

12/14M.Chrzanowski: Strength of Materials

SM2-10: Yielding & rupture criteria

stop