EMSD Ch 6

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Chapter VI

Theory of isotropic linear elasticity

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VI-2


Elements oftensors calculus

ijk q lmn q ijk lmnv a

lmn q ijk v

F G dv n F G da

G F dv

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VI-3


Statics of

deformable solids

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VI-4


Balance equations

globally:

locally:

0

0

i iV A

ijk j k ijk j k V A

F dV T dA

e x F dV e x T dA

0j ji i

ij ji

j ji i

F

n T

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VI-5


Kinematics of deformable solids

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VI-6



compatibility of displacements

u e

eu

1

2ij j i i j D u D ue

0kk ij ij kk jk ik ik jk D D D De e e e

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VI-7



equations of compatibility

2 2 2

11 22 12

2 2

1 22 1

2 22

33 2322

2 2 2 33 2

2 22

33 1311

2 2

1 33 1

X XX X

X XX X

X XX X

e e

e e

e e

2

23 1311 12

2 3 1 1 2 3

2

13 2322 12

1 3 2 2 1 3

2

33 13 2312

1 2 3 3 2 1

2

2

2

X X X X X X

X X X X X X

X X X X X X

e

e

e

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VI-8


Virtual work principle

1

2

E i i i i v a

I ij ij v

ij j i i j

E I

W F u dv T u da

W dv

u u

W W

e

e

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VI-9


Constitutive law

elasticity: general case

ij ijkl kl

ij ijkl kl

C

D

e

e

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VI-10


Constitutive law

isotropic elasticity:

1 21 1 2

11

ij ij ll ij

ij ij ll ij

E

E

e e

e

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VI-11


Constitutive law


11 11

22 22

33 33

12 12

13 13

23 23

1 0 0 01 0 0 0

1 0 0 0

1 20 0 0 0 0

21 1 21 2

0 0 0 0 02

1 20 0 0 0 0

2

E

e

e

e

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VI-12


Constitutive law


11 11

22 22

33 33

12 12

13 13

23 23

1 0 0 0

1 0 0 0

1 0 0 01

0 0 0 2 1 0 00 0 0 0 2 1 0

0 0 0 0 0 2 1

E

e

e

e

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VI-13


Plane strain state (EPe):

11 11

22 22

12 12

1 0

1 01 1 2

1 20 0

2

E e

e

13 23 330 e

13 23 33 11 220

11 11

22 22

12 12

1 01

1 0

0 0 2E

e

e

2 2 2

11 22 12

2 2

1 22 1 X XX X

e e

Compatibility:

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VI-14


Plane stress state (EP):

11 11

22 222

12 12

1 0

1 0

1 10 0

2

E e

e

13 23 330

13 23

33 11 22 11 22

0

1E

e e e

11 11

22 22

12 12

1 0

1 1 0

0 0 2 1E

e

e

2 2 2

11 22 12

2 2

1 22 1X XX X

e e

Compatibility:

2 2 2

33 33 33

2 2 2

1 2 3

0X X X

e e e

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VI-15


3. St-Venants equations of compatibility:

plane strain state (EPe):

0 and 0z xz yz z

e

2 11 01

z x y

yxx y

FF

x y

2 2

2 2

1

1

yxx y

FF

x y x y

exact

2 22

2 2

y xyx

y x x y

e e

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VI-16


4. Equation of Beltrami-Mitchell:

3D case:

plane state

2

2

2

11 1 01

3 :

11 2 1 0

1

kk ij ij kk i j j i k k ij

yz x zz x y z

F F F

i j

FF F F

z z x y z

2

0

11 0

1

yxz

z

FF

x y

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VI-17

end

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VI-18


Compatibility equations in terms of stresses:

EPe:

EP:

From one state to the other:

EPEPe:

EPeEP:

2 11 221

div

1

F

2 11 22 1 div F

identicalifF= cst

2

' ';

1 ' 1 '

EE

2

' 1 2 ' ';

1 '1 '

EE

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VI-19


1. In-plane problems:

a) Plane stress state

2

1 01

1 0 0

0 0 2 1 0

01 0

1 0 01

1 0

0 0 2

z x yx x

y y xz

xy xy yz

zx x

y y xz

xy xy yz

E

E

E

e

e

e

e

e

0z xz yz

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VI-20


1. In-plane problems:

b) Plane strain state

1 01 0 0

1 1 21 2 0

0 0

2

01 01

1 0 0

0 0 2 0

z x yx x

y y xz

xy xy yz

zx x

y y xz

xy xy yz

E

E

e

e

ee

e

0z xz yz e

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VI-21


From one state to the other:

Problem in plane stress state:

Let be the solution

this solution depends on Eand

One gets the solution for the plane strain state by thesubstitutions:

NB: - this does not change G

- if= 0, both solutions coincide

x y xy

x y xy

e e

2;

1 1EE

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VI-22


2. Basic equations in plane state:a) Balance equations

b) Reciprocity

0

0

0

xyxx

x xy x

xy y xy y

j ij i j

y

i

y

ij

Fl m Tx y

l m TF

x

F T

y

xij i y yxj

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VI-23


2. Basic equations in plane state:c) Stresses on an oblique facet

2 2

2

*

2

2

cos sin 2 cos sin

sin2 cos2

2

x y xy

x y xy

y x

ij i j

i

xy

j i j

l m lm

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VI-24


2. Basic equations in plane state:d) Dilatation and shearing

*

*

2 2

Dilatation along

Shearing of angle

cos sin 2 cos sin2

sin2 cos2

2 2

ij i j

i

xy

x y

x

j

y x y

i j

e e e

e e

e e

e

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VI-25


2. Basic equations in plane state:e) Displacements-strains relationship

2

2

1ij i j j

x

y

xy xy

i

u

x

v

y

u v

y x

u u

e

e

e

e

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VI-26


3. St-Venants equations of compatibility:

3D case:

22 22

2 2

22 2 2

2 2

2 22 2

2 2

0

2

2

2

kk ij ij kk ik jk jk ik

yz xy x xzy xyx

y yz xy x z xz xz

y yzz z

y z x x y z y x x y

z x x z x z y x y z

z y y z x y

e e e e

e e e

e e e

e e e

yz xy xz

z x y z

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VI-27


3. St-Venants equations of compatibility:plane stress state (EP):

particular case: Fxand Fy= cst z= 0 : ok

general case: doing following substitutions in EPe case

(Best approximation consistent with the assumptions z=xz=yz=0)

NB: if= 0 or if volumetric forces Fiare constant,both plane states lead to

' 1and 1 '

1 ' 1

2 2

2 21

yxx y

FF

x y x y

approximation

2 22

2 2 0x y x yx y

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VI-28


3. St-Venants equations of compatibility:plane stress state (EP):

moreover:

in general the 3 equations above are not satisfied(they require ez = ax + by + c = linear function)

plane stress state is an approximation

0 ; 0 ; 0

xz yz z x yz e e e

2 22

2 2

y xyx

y x x y

e e

2 2 2

2 2

0 ; 0 ; 0z z z

x y x y

e e e

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VI-29


Engineering notation:

cartesian

coordinates x,y,z

cylindrical

coordinates r,q,z

spherical

coordinates r,j,q

r r rz

r z

zr z z

q

q q q

q

x xy xz

yx y yz

zx zy z

r r r

r

r

j q

j j jq

q qj q

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VI-30


Engineering notation:

cartesian

coordinates x,y,z

cylindrical

coordinates r,q,z

spherical

coordinates r,j,q

1 1

2 2

1 1

2 2

1 1

2 2

x xy xz

yx y yz

zx zy z

e

e e

e

1 1

2 2

1 1

2 2

1 12 2

r r rz

r z

zr z z

q

q q q

q

e

e e

e

1 1

2 2

1 1

2 2

1 12 2

r r r

r

r

j q

j j jq

q qj q

e

e e

e

Documents

EMSD Ch 6