stress state · 2016-11-11 · •plane stress - state of stress in which two faces of the cubic...

Stress State

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• The Stress State of a Point（点的应力状态）

• Simple, General & Principal Stress State（简单、一般和主应力状态）

• Ordering of Principal Stresses（主应力排序）

• On the Damage Mechanisms of Materials（材料失效机制）

• Plane Stress States（平面应力）

• 2-D Cauchy’s Relation（平面柯西关系）

• Plane Stress Transformation（平面应力转换）

• Mohr’s Circle（莫尔圆）

• Maximum Shearing Stress（最大切应力）

• Construction of Mohr’s Circle （莫尔圆的构建）

Contents

2

3

• 3-D Cauchy’s Relation（三维柯西关系）

• Stress Transformation of General 3-D Stresses（空间应力转换）

• Application of Mohr’s Circle in 3-D（莫尔圆在三维应力状态中的应用）

• Generalized Hooke’s Law（广义胡克定律）

• The Relation among E, ν and G（杨氏模量、泊松系数及剪切模量间的关系）

• Fiber-reinforced Composites（纤维增强复合材料）

• Strain Energy Density under Generalized Stress States（一般应力状态下的应变能）

• Volume Strain, Mean Stress and Bulk Modulus（体应变、平均应力和体积模量）

• Volumetric & Distortion Energy Density（体积应变能密度和畸变能密度）

Contents

• The stress state at a point

- uniformly distributed stress on each of the six surfaces

The Stress State of a Point

- is defined as the collection of the stress distributions

on all planes passing through the point.

- can be analyzed via the stress distributions acting on a

differential cube, arbitrarily oriented with reference

coordinates

• Method of differential cubes with:

- infinitesimal side length

- equal and opposite stress on parallel sides

4

F

FF A

F

• The stress distribution at

a point depends on the

plane orientation along

which the point is

examined.

2cos

sin 22

Stresses on Oblique Planes

5

FF A

MeMe

A

FA

B

C

A

A

A

B

C

Examples of the Stress State of a Point

6

Stress Under General Loadings

• A member subjected to a general

combination of loads is cut into

two segments by a plane passing

through a point of interest Q

• For equilibrium, an equal and

opposite internal force and stress

distribution must be exerted on

the other segment of the member.

A

V

A

V

A

F

xz

Axz

xy

Axy

x

Ax

limlim

lim

00

0

• The distribution of internal stress

components may be defined as,

7

• Stress components are defined for the planes

cut parallel to the x, y and z axes. For

equilibrium, equal and opposite stresses are

exerted on the hidden planes.

• It follows that only 6 components of stress are

required to define the complete state of stress

• The combination of forces generated by the

stresses must satisfy the conditions for

equilibrium:

0

0

zyx

zyx

MMM

FFF

yxxy

yxxyz aAaAM

0

zyyzzyyz andsimilarly,

• Consider the moments about the z axis:

General Stress State of a Point

8

• The most general state of stress at a

point may be represented by 6

components,

),, :(Note

stresses shearing,,

stresses normal,,

xzzxzyyzyxxy

zxyzxy

zyx

• Same state of stress is represented by a

different set of components if axes are

rotated.

• There must exist a unique orientation

(Principal Directions) of the

differential cube, having only normal

stresses (Principal Stresses) acting on

each of its six faces.

Principal Stress State of a Point

9

• Stress states classified based on the number of zero principal

stresses:

A2

1

3

Principal axes & stresses

max inter min

1 2 3

Ordering of Principal Stress State

- Tri-axial stress state: none zero principal stresses;

- Biaxial stress state: one zero principal stresses;

- Uniaxial stress state: two zero principal stresses;

10

• Damage (fracture) of brittle materials is governed by the maximum

tensile stress

On the Damage Mechanism of Materials

• Damage (yielding) of ductile materials is governed by the maximum

shearing stress

• The goal of stress state analysis is to determine the

surface/orientation on which normal/shearing stress achieves the

maximum values.

• Strength analysis can be subsequently performed in terms of the

maximum normal/shearing stresses.

11

• Plane Stress - state of stress in which two

faces of the cubic element are free of stress.

For the illustrated example, the state of

stress is defined by

.0,, and xy zyzxzyx

• State of plane stress also occurs on the

free surface of a structural element or

machine component, i.e., at any point of

the surface not subjected to an external

force.

• State of plane stress occurs in a thin plate

subjected to forces acting in the mid-plane

of the plate.

Plane Stress States

12

cos sin0

cos sin0

x xx yxx

y xy yyy

x xx x yx y

y xy x yy y

t A A AF

t A A AF

t n n

t n n

t n n

t = n σ = σ n

• Cauchy’s relation:

• Principal stresses & principal directions

1

2

I

0

0

x xy

xy y

n

n

t = σ n n

σ n 0

n

t

A

xx

yy yx

xy

t

2-D Cauchy’s Relation

13

• Solution

1

2

2

1 2

50 40 0

40 10 0

50 400 50 10 1600 0

40 10

40 2100 0 70, 30

n

n

• For the state of plane stress shown, determine the principal stresses

and principal directions.

1 1 1

1 2

2 2 2

1 1 1

2 1

2 2 2

250 70 40 0 20 40 0 5

240 10 70 0 40 80 0 1

5

50 30 40 0 80 40 02

40 10 30 0 40 20 0

n n nn n

n n n

n n nn n

n n n

15

25

2. Principal directions

14

Sample Problem

1. Principal stresses

sinsincossin

coscossincos0

cossinsinsin

sincoscoscos0

AA

AAAF

AA

AAAF

xyy

xyxyxy

xyy

xyxxx

• Consider the conditions for equilibrium of a

prismatic element with axis perpendicular to

the x, y, and x’.

• The equations may be rewritten to yield

cos2 sin 22 2

sin 2 cos2

2

cos2 sin 2

2

2 2

x y x y

x

x y x

xy

x y

x y x

y

y xy

y

2

2

2 2

1 cos2cos

21 cos2

sin2

sin 2 2sin cos

cos2 cos sin

15

Plane Stress Transformation

cos sin cos sin

sin cos sin cos

cos 2 sin 2 sin 2 cos 22 2 2

sin 2 cos 2 cos 2 sin 22 2 2

T

x x y x xy

x y y xy y

x y x y x y

xy xy

x y x y x y

xy xy

16

Plane Stress Transformation – Alternative Way

• The previous equations are combined to

yield parametric equations for a circle,

2

2 2 2 2

cos 2 sin 22 2

sin 2 cos 22

, ;2 2

x y x y

x xy

x y

x y xy

x y x y

x ave x y ave xyR R

• Principal stresses

2

2

max,min

o

0 sin 2 cos 22

2 2

tan 22

Note: defines two angles separated by 90

x y

x y xy

x y x y

x ave xy

xy

p

x y

R

17

Mohr’s Circle

• It is hard to predict which of the two angles results

in the maximum/minimum principal stress.

Maximum shearing stress occurs for

2

2

max

2

o

2 2

2

2

2tan 2

Note: defines two ang

cos 2 sin 22

les separated by 90 an

2

sin 2 cos 22

tan 2 tan 2 1

x ave x y

x y

x y x y

x xy

x y

x y xy

x y

x ave x

xy

x y

s

xy

y

s p

R

R

o

d

offset from by 45p

18

Maximum Shearing Stress

• The second way is typically chosen, such

that the direction of rotation of Ox to Oa is

the same as CX to CA.19

Construction of Mohr’s Circle

2 2 2

max,m

2

2

in

cos 2 sin 22 2

sin 2 cos 22

,

, tan 22

;2 2

x y x y

ave x

x y x y

x xy

x y

x y xy

x ave x y

xy

ave p

x y

yRR

R

0 ,

2 ,

,

or

,

, , ,

,

x xy

y xy

x xy y

x x y

x x y

xy

x xy y xy

X Y

X Y

• Two options for plotting the diameter XY

,x xyX

,y xyY

2 p

• With Mohr’s circle uniquely defined,

the state of stress at other axes

orientations may be depicted.

• For the state of stress at an angle θ

with respect to the xy axes, construct a

new diameter X’Y’ at an angle 2θ

with respect to XY.

• Normal and shearing stresses are

obtained from the coordinates

X’Y’.


20

• Mohr’s circle for centric axial loading:

0, xyyxA

P

A

Pxyyx

2

• Mohr’s circle for torsional loading:

0x y xy

P

T

W 0x y xy

P

T

W

21


For the state of plane stress

shown, (a) construct Mohr’s

circle, determine (b) the

principal planes, (c) the

principal stresses, (d) the

maximum shearing stress and

the corresponding normal stress.

Solution:

• Construction of Mohr’s circle

MPa504030

MPa40MPa302050

MPa202

1050

2

22

CXR

FXCF

yxave

22

Sample Problem

• Principal planes and stresses

5020max CAOCOA

MPa70max

5020max BCOCOB

MPa30max

40tan 2

302 53.1 , 233.1

p

p

FX

CF

26.6 , 116.6p

23

• Maximum shearing stress

45ps

6.161 ,6.71s

Rmax

MPa 50max

ave

MPa 20

24

For the state of stress shown,

determine (a) the principal

planes and the principal

stresses, (b) the stress

components exerted on the

element obtained by rotating

the given element counter-

clockwise through 30 degrees.

Solution:

• Construct Mohr’s circle

MPa524820

MPa802

60100

2

2222

FXCFR

yxave

25

Sample Problem

• Principal planes and stresses

4.672

4.220

482tan

p

pCF

XF

clockwise7.33 p

5280

max

CAOCOA

5280

max

BCOCOA

MPa132max MPa28min

26

6.52sin52

6.52cos5280

6.52cos5280

6.524.6760180

XK

CLOCOL

KCOCOK

yx

y

x

• Stress components after rotation

by 30o

Points X’ and Y’ on Mohr’s circle

that correspond to stress

components on the rotated

element are obtained by rotating

XY counterclockwise through 60o MPa3.41

MPa6.111

MPa4.48

yx

y

x

27

• State of stress at Q defined by:

zxyzxyzyx ,,,,,

• Consider the general 3D state of stress at a

point and the transformation of stress from

element rotation

• Consider tetrahedron with face

perpendicular to the line QN with direction

cosines: cos ,x

n xn x

n x

0

0

x xx x yx y zx z

x

y xy x yy y zy z

y

z xz z yz y zz z

i ji j ij j

t A A A AF

t A A A AF

t A A A A

t n n

t = n σ = σ n

• Cauchy’s relation:

28


xy zx

yz

yz

3 2

2 2 2

( )

( ) 0

( )

( )

( )

( 2

x

xy y

zx z

x y z

x y y z z x xy yz zx

x y z xy y

t = σ n n

2 2 2

3 2

1 2 3

)

0

0

z zx x yz y zx z xy

I I I

I1, I2, and I3 are known as stress invariants as they are independent

of coordinate choices.29


• Determine the normal and

shearing stresses on the

element in the x, y, and z

direction of a given reference;

• Calculate the three stress

invariants;

• Solve the cubic equation for

its three roots (principal

stresses);• Substitute the roots back into

original characteristic

equation to solve for the

principal directions.

• Solution

20 10 0

10 20 0

0 0 20

• For the state of 3-D stress shown, determine the principal stresses

and principal directions.

2. Principal directions

x

y

z

20MPa

10MPa

20MPa

20MPa

1

2 2 2

2

2 2 2

3

3 2 2

1 1 1

20 20 20 60

400 400 400 100 1100

2 8000 2000 6000

60 1100 6000 20 40 300 0

30 20 10 0

30; 20; 1

x y z

x y y z z x xy yz zx

x y z xy yz zx x yz y zx z xy

I

I

I

0

1

2

3

20 10 0 0

10 20 0 0

0 0 20 0

n

n

n

30

Sample Problem

1. Principal stresses

• The requirement leads to 0nF

2 2 2 2 2 2n x x y y z z xy x y yz y z zx z x

• Consider the conditions for

equilibrium of the tetrahedron

generated from an oblique plane

with the elemental volume.

31

Stress Transformation of General 3-D Stresses

nx ny nz

nx ny nz

ox oy oz

ox oy oz

px py pz

px py pz

nx ny nz x xy xz nx ox pxT

ox oy oz xy y yz ny oy py

px py pz xz yz z nz oz pz

x y z

na

o

p

a a

• Too complicated to find the principal stress states

through stress transformation…

32

Stress Transformation of General 3-D Stresses

• The three circles represent the

normal and shearing stresses

for rotation around each

principal axis.

• Points A, B, and C represent the

principal stresses on the principal

planes (shearing stress is zero)

minmaxmax2

1

• Radius of the largest circle yields the maximum shearing stress.

1

2

3

1

2

3

33

Application of Mohr’s Circle in 3-D

• For the state of 3-D stress shown below, determine the principal

stresses and maximum shearing stress.

x

y

z

40MPa

20MPa

20MPa

30MPa

• Solution

2

2 46 MPa40 - 20 40 2020

-26 MPa2 2

i

j

1 2 346 MPa, 26 MPa, 30 MPa

1 3max

46 ( 30)38 MPa

2 2

40 20 0

20 20 0

0 0 30

2. Find the two principal stresses in x-y plane.

3. Rearrange the three principal stresses.

4. Find the maximum shearing stress.

34

Sample Problem

1. Summarize the 3-D stress state.

E • Axial strain:

• For a bar under uniaxial tension:

E • Transverse strain:

,xx

xy z x

E

E

x

y

z

x

Uniaxial stress along x: Along y:

,y

y

y

z x y

E

E

x

y

z

y

Along z:

,zz

zx y z

E

E

x

y

zz

Generalized Hooke’s Law

35

x

y

z

1

1

1

yx zx x x y z

y xzy y x y z

yxzz z x y z

E E E E E

E E E E E

E E E E E

xy xyG

x

y

z

Shearing stress around z: Around y:

zx zxG

x

y

z

Around x:

yz yzG

x

y

z

Generalized Hooke’s Law

Shearing Stresses

22 2 2

2

2

2 1 2 1 2

12 1 2

12 1 2

bdL h h

hE

hE

2 1E G

2 1bdL h

2 2 2 2 2

2 2

2 cos 2 2 1 sin

2 1 2 1

bdL h h h h

h h G

Only two of the three elastic

constants are independent.

The Relation among E, ν and G

37

A circle of diameter d = 9 in. is

scribed on an unstressed aluminum

plate of thickness t = 3/4 in.

Forces acting in the plane of the

plate later cause normal stresses σx

= 12 ksi and σz = 20 ksi.

For E = 10x106 psi and ν = 1/3,

determine the change in:

a) the length of diameter AB,

b) the length of diameter CD,

c) the thickness of the plate, and

d)the volume of the plate.

Sample Problem

38

SOLUTION:

• Apply the generalized Hooke’s Law to

find the three components of normal

strain.

in./in.10600.1

in./in.10067.1

in./in.10533.0

ksi203

10ksi12

psi1010

1

3

3

3

6

EEE

EEE

EEE

zyxz

zyxy

zyxx

• Evaluate the deformation components.

in.9in./in.10533.0 3 dxAB

in.9in./in.10600.1 3 dzDC

in.75.0in./in.10067.1 3 tyt

in.108.4 3AB

in.104.14 3DC

in.10800.0 3t

• Find the change in volume

33

333

in75.0151510067.1

/inin10067.1

eVV

e zyx

3in187.0V

39

• For the bar shown, find the change in length of line segment BC

and in right angle ABC. Assume the Young’s modulus and

Poisson’s ratio as E and ν respectively.

FF

60o30o

A

B

C

b

h

x

30o

30o

30o

120o

• Solution

2. Stress state in transformed coordinates:

x F bh

All other stress components vanish.

40

Sample Problem

1. Stress state in reference

coordinate system

0

0

0

3

430

3

430

1

4120

cos 2 sin 22 2

sin 2 cos 22

cos 2 sin 22 2

x y x y

x xy x

x y

x y xy x

x y x y

y xy x

4. The change in length of BC

0 030 30

32

2BC BC

Fl l h

Eb

5. The change in right angle ABC

03

30 4

2(1 )

3(1 ) 3 (1 )

2 2x

E x

F

G E bhE

3. Longitudinal strain along BC

0 0 030 30 120

31 3

4 4x

F

E E Ebh

41

• Fiber-reinforced composite materials are formed

from lamina of fibers of graphite, glass, or

polymers embedded in a resin matrix.

z

zz

y

yy

x

xx EEE

• Normal stresses and strains are related by Hooke’s

Law but with directionally dependent moduli of

elasticity,

x

zxz

x

yxy

• Transverse contractions are related by directionally

dependent values of Poisson’s ratio, e.g.,

• Materials with directionally dependent mechanical

properties are anisotropic.

Fiber-reinforced Composites (Anisotropy)

42

du d

• Strain energy density of non-

linearly elastic material under

uniaxial stress state

• Strain energy density of

linearly elastic material

under uniaxial stress state

1

2u

43

Strain Energy under Generalized Stress States

x

y

zdx

dz

dy

ij ij

x x y y z z

xy xy yz yz zx zx

du d

d d d

d d d

• Strain energy density of non-


generalized 3-D stress states

• Strain energy density of


generalized 3-D stress states

2 2 2

2 2 2

xy yz zx

1

2

1

2 2

1

2G

x x y y z z

xy xy yz yz zx zx

x y z

x y y z z x

U

E

44

Strain Energy under Generalized Stress States

0

1

2

1 0

0

(1 ) (1 ) (1 )

1

1- 2

x y z

x y z

V x y z x y z

V dxdydz

V dx dy dz

dxdydz O

V V

V E

x

y

zdx

dz

dy

• For a given stress state, if no volume change happen, the first

invariant of the stress tensor must be zero.

0x y z

45

Volume Strain, Mean Stress and Bulk Modulus

• Bulk modulus can be defined as the ratio between mean stress and

volume strain

3 0 1 23 1 2

m V x y z V

EK

• Volume strain

(a) Mean stress

tensor

(b) Deviatoric stress

tensor

= +x

y

zdx

dz

dy

y

x

z x

y

zdx

dz

dy

m

m

m x

y

zdx

dz

dy

y m

x m

z m

V du u u

2 2 2 2 2 2

xy yz zx

1 1

6 2Gd x y y z z xu

E

22

3 1 2 1 21

2 2 6V m V m x y zu

E E

• Volumetric energy density:

• Distortion energy density:

46

Volumetric & Distortion Energy Density

• The Stress State of a Point（点的应力状态）

• Simple, General & Principal Stress State（简单、一般和主应力状态）

• Ordering of Principal Stresses（主应力排序）

• On the Damage Mechanisms of Materials（材料失效机制）

• Plane Stress States（平面应力）

• 2-D Cauchy’s Relation（平面柯西关系）

• Plane Stress Transformation（平面应力转换）

• Mohr’s Circle（莫尔圆）

• Maximum Shearing Stress（最大切应力）

• Construction of Mohr’s Circle （莫尔圆的构建）

Contents

47

48

• 3-D Cauchy’s Relation（三维柯西关系）

• Stress Transformation of General 3-D Stresses（空间应力转换）

• Application of Mohr’s Circle in 3-D（莫尔圆在三维应力状态中的应用）

• Generalized Hooke’s Law（广义胡克定律）

• The Relation among E, ν and G（杨氏模量、泊松系数及剪切模量间的关系）

• Fiber-reinforced Composites（纤维增强复合材料）

• Strain Energy Density under Generalized Stress States（一般应力状态下的应变能）

• Volume Strain, Mean Stress and Bulk Modulus（体应变、平均应力和体积模量）

• Volumetric & Distortion Energy Density（体积应变能密度和畸变能密度）

Contents

stress state · 2016-11-11 · •plane stress - state of stress in which two faces of the cubic...

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