4.3 Yield Criteria - University of Florida

[Pages:16]4.3 Yield Criteria

? Cannot perform tests for all combinations of 3D loading so we need yield criterion to generalize from small number of tests.

? What are the ideal properties of a 3D yield

criterion?

( ) f ij, Y < 0 Elastic Behavior

( ) f ij, Y = 0 Onset of Inelastic Behavior

? We usually visualize yield criterion by a surface in principal stress space. Why?

? We also calculate "effective" stress to compare with yield stress.

Maximum principal stress criterion William Rankine (1820-1872)

? Applicable to brittle materials (mostly in tension)

f = max ( 1 , 2 , 3 ) = Y

Maximum principal strain criterion

Adh?mar Jean Claude Barr? de Saint-Venant 1797 - 1886

? Has the advantage that strains are often easier to measure than stresses

? Assume that epsilon1 is the largest principal strain

1

=

1 E

(1

- 2

- 3

)

f1 = 1 - 2 - 3 - Y

1 - 2 -3 = ? Y

e

=

max

i jk

i

-

j

- k

f

=e -Y

Anti-optimization for selecting test conditions

? What test will give us maximum difference between maximum principal stress and maximum principal strain criteria?

? Obviously 1 = 2 = 3 = ? With max principal stress

e =

? With max principal strain e = (1- 2 ) = 0.4

? Alternatively 1 = - 2 = -3 =

? Max principal strain e = (1+ 2 ) = 1.6

? What is bad about these test conditions?

In plane stress

? Figure 4.8

Strain energy density criterion (Eugenio Beltrami 1835-1900)

? Strain energy density

U0

=

1 2E

12

+

2 2

+

2 3

-

2

(1 2

+ 1 3

+ 2 3

)

Uniaxial test : 1 = Y 2 = 3 = 0

U0

=

1 2E

12

=

Y2 2E

? Criterion

12 + 22 + 32 - 2 (12 + 13 + 23 ) - Y2 = 0

? Effective stress f = (e )2 - Y2

( ) e = 12 + 22 + 32 - 2 12 + 13 + 23

? Extreme 1 = - 2 = -3 =

e = 3 + 2 = 1.90

Plane stress

? Depends on Poisson's ratio

? What is common to strain and energy criteria?

4.4. Yielding of ductile metals

? Maximum shear-stress criterion (Henri Eduard Tresca,

1814-1885) ? Uniaxial loading

1

=

Y; 2

=

0; 3

=

0

max

=

Y- 0 2

=

Y 2

? Criterion

f

=

e -

Y 2

? Effective stress

e = max = max (1,2,3 )

1

=

2

-3

2

2

=

3

-1

2

3

=

1

-2

2

? Conservative for metals in shear

? Extreme case

1 = - 2 = - 3 =

e =

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download