9-distortion Energy Theory-derivation_0.ppt

UNIVERSITY OF NAIROBI DEPARTMENT OF MECHANICAL AND MANUFACTURING ENGINEERING ENGINEERING DESIGN II FME 461 PART 9 GO NYANGASI November 2008 1

DISTORTION ENERGY THEORY A THEORY OF FAILURE APPLICABLE TO DUCTILE MATERIALS 2

STATEMENT OF THE THEORY 1) When Yielding occurs in any material, 2) The distortion strain energy per unit volume 3) At the point of failure 4) Equals or exceeds 5) The distortion strain energy per unit volume 6) When yielding occurs in the tension test specimen. 3

DISTORTION ENERGY THEORY • Is based on yielding • Applies to ductile materials

4

STRAIN ENERGY AT A LOCATION OF THE ELEMENT • SEGREGATED INTO THREE CATEGORIES: 1) Total strain energy per unit volume of the stressed element, arising from the three principal stresses 2) Strain energy per unit volume arising from the hydrostatic stress that causes change of volume only, and which is uniform in all three directions 3) Strain energy per unit volume arising from stresses causing distortion of the element, and this can be expressed as the difference between category (1) and (2). 5

THREE DIMENSIONAL STRESS • General Case

y y

 yz

 yx

 zy  zx  xz

 xy

x x

z z

6

TRI-AXIAL STRESS SITUATION

1

3 2

7

ELASTIC STRESS-STRAIN RELATIONS • UNI-AXIAL STRESS • This is the case of a single principal stress • Principal strains are then given in terms of principal stresses by the expressions in next slide

8

ELASTIC STRESS-STRAIN RELATIONS: Uni-Axial stress The variables are: 1) Principal strain in the direction of the principal stress 2) Poisson’s ratio for the material 3) Modulus of elasticity for the material 4) Principal stress 1 

1 E

,

 2  

1 E

,

 3  

1 E 9

Uni-Axial Stress One dimensional (Normal/Shear)

1

1

10

ELASTIC STRESS-STRAIN RELATIONS: Bi-Axial stress • In this case the stress situation consists of two principal stresses, • The strains[1] are given by in terms of the two principal stresses as shown in next slide [1] Mechanical Engineering Design; Shigley, Joseph, pg 124, McGraw Hill, Seventh Edition, 2004 11

STRAINS IN BI-AXIAL STRESS • Stress situation consists of two principal stresses and strains are given by the expressions

1  1   1   2 , E 1  2   2   1 , E 1  3     1   2  E 12

Bi-Axial Stress Two Dimensional (Plane) 2 1

1

2

13

ELASTIC STRESS-STRAIN RELATIONS • • • •

Tri-Axial Stress This is the case of three principal stresses The most general case Three strains in the directions of the principal stresses • Given by in terms of the three principal stresses as shown in next slide 14

STRAINS IN TRI-AXIAL STRESS • Strains are given by the expressions

1 1   1    2   3  E 1  2   2    1   3  E 1  3   3    1   2  E

(1) (2) (3) 15

Tri-Axial Stress Three Dimensional stress 2

1

3

16

ENERGY PER UNIT VOLUME TriAxial Stress • Total strain energy • The total strain energy is the strain energy caused by the three principal stresses. It is given by the expressions •

1 1 1 U   1 1   2 2   3 3 2 2 2 17

TOTAL STRAIN ENERGY • Substituting for elastic strains 1  1   1    2   3  E 1  2   2    1   3  E 1  3   3    1   2  E Yields the Total strain energy



(1) ( 2) (3) in terms of stresses



1 U  12   2 2   3 2  2  1 2   2 3   1 3  2E

18

Tri-Axial Stress Three Dimensional stress 2

1

3

19

STRAIN ENERGY DUE HYDROSTATIC STRESS • Hydrostatic stress is the stress that causes change of volume only • Hydrostatic stress may be considered as the average of the three principal stresses and derived and expressed as

 av 

1   2   3 3 20

HYDROSTATIC STRAIN ENERGY • Using the equation for total strain energy yields an expression for hydrostatic strain energy:





1 U  12   2 2   3 2  2  1 2   2 3   1 3  2E 1 2 2 Uv  3 av  2 3 av 2E 2 3 av 31  2  2 1  2   Uv   av 2E 2E







21

HYDROSTATIC STRAIN ENERGY • Simplifying for hydrostatic strain energy Substituting  av 

1   2   3 3

int o previous exp ression

31  2    1   2   3  31  2   1   2   3  Uv     2E  3 9 * 2E   1  2  2  1   2   3  Uv  6E  1  2  2 Uv   1   2 2   3 2  2 1 2   2 3   1 3  6E 2





22

DISTORTION STRAIN ENERGY • This is the difference between total strain energy and the hydrostatic strain energy Ud  U Uv But Total Strain Energy is 1 U  1 2   2 2   3 2  2  1 2   2 3   1 3  2E and Hydrostatic strain energy 1  2 2 Uv   1   2 2   3 2  2 1 2   2 3   1 3  6E







 23

Tri-Axial Stress Three Dimensional stress 2

1

3

24

DISTORTION STRAIN ENERGY • Distortion strain energy





1 U  1 2   2 2   3 2  2  1 2   2 3   1 3  2E 1  2 2 Uv   1   2 2   3 2  2 1 2   2 3   1 3  6E Substituting for U and U v in U d  U  U v yields



Ud

 1  



2 1

2

2

3E  1 2 2 2       Ud  1   2   2   3  1   3 6E







  2   3   1 2   2 3   1 3 

 25

CASE OF SIMPLE TENSION • When yielding occurs in simple tension test Pr incipal stresses are  1  S y ,  2  0, and  3  0 Substituting int o exp ression for distortion energy  1  S y  02  0  02  S y  02 Ud  6E  1  2 Ud  2S y 6E

  



26

Uni-Axial Stress One dimensional (Normal/Shear)

1

1

27

DISTORTION ENERGY THEORY For the general three dimensional stress situation • When Yielding occurs in any material, • The distortion strain energy per unit volume • At the point of failure, Equals or exceeds • The distortion strain energy per unit volume • When yielding occurs in the tension test specimen. 28

THREE DIMENSIONAL STRESS WHEN YIELDING OCCURS • Comparing three dimensional case with simple tension When failure by yielding occurs in material

Ud

 1   

2

  2    2   3    1   3 



 1  2S specimen 

2

2

1

6E Equals or exceeds

U d in the tension test

2

6E

y



When yielding occurs 29

Tri-Axial Stress Three Dimensional stress 2

1

3

30

THREE DIMENSIONAL STRESS WHEN YIELDING OCCURS • Equating the two conditions

 1   2    2   3    1   3  2

2

2

 2S y

2

or   1   2 2   2   3 2   1   3 2     Sy 2  

31

EQUIVALENT STRESS • Left hand side of equation referred to as the Equivalent, or Von-Mises stress   1   2 2   2   3 2   1   3 2  e    2   Equivalent (Von Mises) Stress is then the significan t stress which is compared to the Design or Allowable Stress as below   1   2 2   2   3 2   1   3 2  S y e    2   f .s 32

APPLICATION OF DESIGN EQUATION Principal stresses are • Determined by stress analysis. • Stress analysis describes the principal stresses as a function of 1) Load carried, 2) Geometry and dimensions of the machine or structural element.

33

APPLICATION OF DESIGN EQUATION Left hand side of design equation 1) Equivalent stress in terms of Loads and Dimensions of machine or structural element, Right hand side of design equation 1) Indicator of strength expressed as Working, (design, allowable) stress a function of strength of the material, and a factor of safety. 34