R. Rizza 3/22/2007

Equations for Static Failure Theory.

by

Dr. Robert Rizza Associate Professor

Department of Mechanical Engineering Milwaukee School of Engineering

1025 N. Broadway Milwaukee, WI 53202

(414) 277-7377 Fax:(414) 277-2222

Email: rizza@msoe.edu http://people.msoe.edu/~rizza

The equations are believed to be correct. But, if you find any errors please let me know by writing to rizza@msoe.edu.

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R. Rizza 3/22/2007

Nomenclature Symbol Meaning

a Half the length of a 2-D crack β Stress intensity modification factor

ΚΙ, ΚΙΙ, ΚΙΙ Stress intensity factors KIc Fracture toughness (mode I) n Safety factor

σmax, σmin Maximum and minimum principal stresses σ1, σ2, σ3 Principal stresses

σeff Effective stress (Von Mises) σ% Effective stress (Coulomb-Mohr) σys Yield stress

Sut, Suc Ultimate tensile strength in tension and compression

Sys Strength in shear Ssu Ultimate shear strength

Ductile

1. Tresca (Maximum Shear Stress Theory) The failure criterion is:

max

max minmin

max min max min

if σ and have the same sign

if σ and have opposite sign

ys

ys

ys

σ σσ

σ σ

σ σ σ σ

⎫= ⎪⎬

= ⎪⎭− =

Safety factor (n):

1 3

ysnσ

σ σ=

−

Strength in pure shear:

2ys

ysSσ

=

2

R. Rizza 3/22/2007

2. Von Mises (Distortion Energy Theorem) The failure criterion is:

( ) ( ) ( )2 2 2 2

1 2 2 3 3 12 2 21 1 2 2

2 (3-D)

(plane stress)

ys

ys

σ σ σ σ σ σ σ

σ σ σ σ σ

− + − + − =

− + =

Safety factor (n):

ys

effn

σσ

=

where

( ) ( ) ( ) ( )

2 2 21 2 3 1 2 2 3 1 3

2 2 2 2 2 2

2 2 2

6

2

3 (plane stress)

eff

x y y z z x xy yz x

eff x y x y xy

σ σ σ σ σ σ σ σ σ σ

σ σ σ σ σ σ τ τ τ

σ σ σ σ σ τ

= + + − − −

− + − + − + + +=

= + − +

z

Strength in pure shear:

su2, S

33ys ut

shearSS

σ= =

Brittle Let σut and σuc be the strength in tension and compression, respectively. 1. Rankine (Maximum Normal Stress Theory) The failure criterion is:

1

2

ut

ut

S

S

σ

σ

=

=

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R. Rizza 3/22/2007

2. Coulomb-Mohr (Modified Mohr) An approach by Dowling that does not require the drawing of the failure envelope (Figure 1, Figure 5-13 in Norton or Figure 6-22 Shigley) is based on the need to find an equivalent stress σ% . The safety factor is then

1 2 3 1 2 3Max(C , , , , , )0 if Max < 0

utSn

C Cσ

σ σ σ σσ

=

=

=

%

%

%

where

( )

( )

( )

1 1 2 1 2

2 2 3 2

3 3 1 3 1

212

212

212

ut uc

uc

ut uc

uc

ut uc

uc

C

C

C

σ σσ σ σ σ

σ

σ σσ σ σ σ

σ

σ σσ σ σ σ

σ

⎡ ⎤−= − + +⎢ ⎥

−⎢ ⎥⎣ ⎦⎡ ⎤−

= − + +⎢ ⎥−⎢ ⎥⎣ ⎦

⎡ ⎤−= − + +⎢ ⎥

−⎢ ⎥⎣ ⎦

3

Figure 1 (based on Norton Figure 5-13, page 273 or page 272 Shigley).

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R. Rizza 3/22/2007

Columb-Mohr for Plane Stress (σA and σB are principal stresses. Not good for fourth quadrant)

0

1 0

0

= ≥

− = ≥ ≥

= − ≥ ≥

utA A

A BA B

ut uc

ucB A

Sn

S S nSn

σ σ

σ σ

≥B

B

σ

σ σ

σ σ σ

Modified II-Mohr for Plane Stress (Good for fourth quadrant) ( ) 1 0 1

0

uc ut A B BA B

uc ut uc A

ucB A

S Sand

S S S nSn

σ σ σσ σσ

σ σ

−− = ≥ ≥ >

= − ≥ ≥ Bσ

Brittle Fracture Failure Modes

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Crack Geometry and Nomenclature

Stress Field 1. Mode I

( )

3cos 1 sin sin2 2 2 2

3cos 1 sin sin , 02 2 2 2

3sin sin cos2 2 2 2

0 plane stress

plane strain

Ix

Iy xz yz

Ixy

zx y

Kr

Kr

Kr

θ θ θσ σπ

θ θ θσ σ τ τπ

θ θ θτ σπ

σν σ σ

⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎧⎪= ⎨ +⎪⎩

= =

2. Mode II

( )

3sin 2 cos cos2 22

3sin cos cos2 2 22

31 sin sin2 22

0 plane stress

plane strain

0

IIx

IIy

IIxy

zx y

xz yz

Kr

Kr

Kr

2θ θ θσ

πθ θ θσ

πθ θτ

π

σν σ σ

τ τ

⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠⎛ ⎞ ⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞⎛ ⎞ ⎛ ⎞= − ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎧⎪= ⎨ +⎪⎩

= =

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3. Mode III

sin22

cos22

0

IIIxz

IIIyz

z x y xy

Kr

Kr

θτπ

θτπ

σ σ σ τ

⎛ ⎞= − ⎜ ⎟⎝ ⎠

⎛ ⎞= ⎜ ⎟⎝ ⎠

= = = =

Safety Factor

IC

I

KnK

=

Fracture Toughness of some Materials

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Theoretical Stress Intensity Factors (From Machine Design February 1, 1968)

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