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1/12

M.Chrzanowski: Strength of Materials

SM1-09: Continuum Mechanics: State of

deformation

CONTINUUM MECHANICS

(STATE OF DEFORMATION)

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M.Chrzanowski: Strength of Materials

SM1-09: Continuum Mechanics: State of

deformation



Under action of external loadings the body (structure) changes its

shape. The original positions of material points are shifted to a

new position – this isthe state of deformation.



This change in material point position influences interaction

between body material points resulting in raising internal forces.



If

the whole structure is in equilibrium then any part of it is in

equilibrium, too. This gives rise to the introduction of Navier

equation as shown in preceding chapters

.



This equation cannot be solved without further consideration of

the state of the body being deformed.

Deformation versus internal

forces

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M.Chrzanowski: Strength of Materials

SM1-09: Continuum Mechanics: State of

deformation

A

A’

'

r

u

r











'

'B

A

AB 

Displacement
vector

A

A’

'

r

u

r

B

B’

r

r

u









 '





0

'

'

,



B

A

AB



 

i

u

u



A

u



B

u







3

2

1

,

,

x

x

x

u

u

i

i



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within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education

4/12

M.Chrzanowski: Strength of Materials

SM1-09: Continuum Mechanics: State of

deformation

'

'B

A

AB 

j

i

ij

x

x

e

AB

B

A











~

2

...

'

'

2

2

 

i

x

A





i

i

x

x

B









































j

k

i

k

i

j

j

i

ij

x

u

x

u

x

u

x

u

e

~

2

denotes derivative
in an intermediate
point

A’

A

B

2

B

3



23



12



31

B

1

1

x



3

x



2

x



B’

1

B’

2

B’

3

x

1

x

3

x

2

 





j

i

i

x

u

x

A



'









j

j

i

i

i

x

x

u

x

x

B











'

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within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education

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M.Chrzanowski: Strength of Materials

SM1-09: Continuum Mechanics: State of

deformation

j

i

ij

x

x

e

AB

B

A











~

2

...

'

'

2

2





ik

k

i

k

x

x

B











































j

k

i

k

i

j

j

i

ij

x

u

x

u

x

u

x

u

e

~

2

'

A

k

B'

3

,

2

,

1



i





3

2

1

1

1

,

,

x

x

x

x

B









3

2

2

1

2

,

,

x

x

x

x

B









3

3

2

1

3

,

,

x

x

x

x

B





11

1

1

~

2

1

'

'

e

x

B

A







kk

k

k

e

x

B

A

~

2

1

'

'







22

2

2

~

2

1

'

'

e

x

B

A







 

i

x

A

………………………..

j

i

j

i

ij

B

A

B

A

B

A

B

A

'

'

'

'

'

'

'

'

cos









jj

ii

ij

ij

e

e

e

~

2

1

~

2

1

~

2

cos









when i

j

 no

summation

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within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education

6/12

M.Chrzanowski: Strength of Materials

SM1-09: Continuum Mechanics: State of

deformation

kk

k

k

e

x

B

A

~

2

1

'

'







k

k

x

AB





1

2

1

~

2

1

lim

'

'

lim

0

0





























kk

k

k

kk

k

x

k

k

k

x

e

x

x

e

x

AB

AB

B

A

k

k

kk

kk

e









1

2

1

Normal (linear)
strain in point A

A

B

k

A’

B’

k

when i=j=k

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7/12

M.Chrzanowski: Strength of Materials

SM1-09: Continuum Mechanics: State of

deformation





ij

jj

ii

ij

ij

e

e

e

















2

sin

~

2

1

~

2

1

~

2

cos

when i



j

jj

ii

ij

jj

ii

ij

A

B

A

B

ij

A

B

A

B

e

e

e

e

e

e

j

i

j

i

2

1

2

1

2

arcsin

2

1

~

2

1

~

2

1

~

2

arcsin

2

1

lim

2

2

1

lim







































 













A’

B’

i

B’

j

A

B

i

B

j



/2



ij

ij

jj

ii

ij

e

e

e









2

1

2

1

2

arcsin

2

1

Shear (angular)
strain

when
i

j



/2-



ij

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M.Chrzanowski: Strength of Materials

SM1-09: Continuum Mechanics: State of

deformation

when

i



j

when

i=j=k



ij



1

2

1





kk

e

jj

ii

ij

e

e

e

2

1

2

1

2

arcsin

2

1





1

2





























i

j

j

i

ij

x

u

x

u

e

1







j

i

x

u





































j

k

i

k

i

j

j

i

ij

x

u

x

u

x

u

x

u

e

2

ij

ij

e

e

2

2

sin



1

2

1







kk

kk

e



1

2

2

1

2









kk

kk

kk

e









2

1

2

1







kk

kk

e



kk

kk

e





ij

ij

ij

e

e 

 2

2

1



for

i



j

for

i=j=k



























i

j

j

i

ij

x

u

x

u

2

1



Cauchy
equation

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within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education

9/12

M.Chrzanowski: Strength of Materials

SM1-09: Continuum Mechanics: State of

deformation



























i

j

j

i

ij

x

u

x

u

2

1



Small
strains

Shear strains
when

i



j

Normal
strains when

i=j

32

2

3

3

2

23

2

1































x

u

x

u

2

2

22

x

u









3

3

33

x

u









1

1

1

1

1

1

11

2

1

x

u

x

u

x

u

































21

1

2

2

1

12

2

1































x

u

x

u

31

1

3

3

1

13

2

1































x

u

x

u





















33

32

31

23

22

21

13

12

11





















T

Strain matrix

– symmetrical by

definition of angular

strains

1







j

i

x

u

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M.Chrzanowski: Strength of Materials

SM1-09: Continuum Mechanics: State of

deformation

x

1

x

2

x

3

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11/12

M.Chrzanowski: Strength of Materials

SM1-09: Continuum Mechanics: State of

deformation





















33

32

31

23

22

21

13

12

11





















T

Eigenvalues of strain matrix are normal strains
on the planes where there are no shear strains.

Principal strains can be found by solving the
secular equation:





















3

2

1

0

0

0

0

0

0









T

0

3

2

2

1

3





















I

I

I

where

I

1

, I

2

, I

3

are invariants of strain

matrix

When transfer of strain matrix is made to the
new co-ordinate system then matrix
transformation rule holds:

kl

jl

ik

ij











,

In the co-ordinates
defined by principal
directions of strain
matrix it takes the
diagonal form.

3

2

1











- principal

strains

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12/12

M.Chrzanowski: Strength of Materials

SM1-09: Continuum Mechanics: State of

deformation



























i

j

j

i

ij

x

u

x

u

2

1



Cauchy equation can be viewed as a the set of 6 linear differential
equations for 3 unknown displacement functions:





3

2

1

,

,

x

x

x

u

i

To solve this set, appropriate kinematic boundary conditions (KBC) for
these functions and/or their derivatives given on the body surface

S

u

have to be formulated

...



u

S

i

u

...







u

S

j

i

x

u

To satisfy the compatibility of deformations only 3 of these 6 six
equations are independent, so the 3 have to be eliminated
If displacement functions are given (or known in advance) then the
Cauchy equation becomes a recipe for determination of all 6
components of the strain matrix.

and/o
r

Comment

s

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within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education

13/12

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SM1-09: Continuum Mechanics: State of

deformation



stop