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1/16

M.Chrzanowski: Strength of Materials

SM1-10: Continuum Mechanics: Constitutive

equations

CONTINUUM MECHANICS

(CONSTITUTIVE EQUATIONS -

- HOOKE LAW)

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2/16

M.Chrzanowski: Strength of Materials

SM1-10: Continuum Mechanics: Constitutive

equations

0























j

ij

i

x

P



j

ij

i

q











3

internal equilibrium

equations (Navier eq.)

6

unknown functions (stress

matrix components)

Boundary

conditions (statics)



























i

j

j

i

ij

x

u

x

u

2

1



6

kinematics equations (Cauchy

eq.),

9

unknown functions (6 strain

matrix components, 3
displacements)

Boundary

conditions

(kinematics)



u

S

i

u

9

equations

15

unknown functions (6

stresses, 6 strains, 3
displacements)

From the formal point of view
(mathematics) we are lacking 6
equations

From the point of view of physics –
there are no material properties
involved

Summary of stress and strain state

equations

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3/16

M.Chrzanowski: Strength of Materials

SM1-10: Continuum Mechanics: Constitutive

equations

Strain versus stress

An obvious solution is to exploit

already noticed interrelation

between strains and stresses

Material

deformability

properties

General property of majority of

solids is elasticity (instantaneous

shape memory)

P

u

Linear
elasticity

Deformation versus internal

forces

CEIIINOSSSTTUV

UT TENSIO SIC VIS

u

k

P







which reads:

„as much the extension as the

force is”

where

k

is a constant

dependent on a material

and

body shape

This observation was made

already in 1676 by Robert

Hooke:

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4/16

M.Chrzanowski: Strength of Materials

SM1-10: Continuum Mechanics: Constitutive

equations

Physical

quantities

(measurable)

Mathematical

quantities (non-

measurable)

Dynamics

Kinematic
s

P



u

ij



 

ij

ij

f







Hooke, 1678

Navier, 1822

ij



f

- linear function of all strain matrix

components defining all stress component

matrix

To make Hooke’s law independent of a body shape one has to use state

variables characterizing internal forces and deformations in a material point

i.e. stress and strain.

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5/16

M.Chrzanowski: Strength of Materials

SM1-10: Continuum Mechanics: Constitutive

equations

The coefficients of this equation do depend only on the material
considered,

but not on the body shape

.

As Navier equation is a set of 9 linear algebraic equations then the
number of coefficient in this set is 81 and can be represented as a
matrix of 3

4

=81 components:

kl

ijkl

ij

C







Summation

over

kl

indices reflects linear character of this

constitutive equation.

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6/16

M.Chrzanowski: Strength of Materials

SM1-10: Continuum Mechanics: Constitutive

equations

Nevertheless, the number of assumptions allows for the reduction of
the coefficients number: two of them are already inscribed in the
formula:

Universality of linear elasticity follows observation, that for loading
below a certain limit (elasticity limit) most of materials exhibit this
property.

kl

ijkl

ij

C







1. For zero valued deformations all stresses vanish: the body in a

natural state is free of initial stresses.

2. Coefficients

C

ijkl

do not depend on position in a body – material

properties are uniform (homogeneous).

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7/16

M.Chrzanowski: Strength of Materials

SM1-10: Continuum Mechanics: Constitutive

equations

3. Assumption of the existence of elastic potential yields symmetry of
group of indices

ij - kl

thus reducing the number of independent

coefficients to
36 [=(81-9)/2].

4. Symmetry of material inner structure allows for further reductions.
In a general case of lacking any symmetry (

anisotropy

) the number

of independent coefficients is 21 [=(36-6)/2+6].

5. In the simplest and the most frequent case of structural
materials (

except the composite materials

) – the number of

coefficients is 2 (

isotropy

):

ij

kk

ij

ij

G











2

or in an inverse
form:









ij

kk

ij

ij

v

E















1

1

The pairs of coefficients

G,



i

E, ν

are interdependent so there

are really only two material independent constants.

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8/16

M.Chrzanowski: Strength of Materials

SM1-10: Continuum Mechanics: Constitutive

equations

9

9

6

6

81 components of

C

ijkl

General anisotropy: 15+6=21
constants

Isotropy: 2
constants

0 0 0
0 0 0
0 0 0

0 0 0
0 0 0
0 0 0

0 0

0 0

0

0

Symmetrical
components

Identical components
Components dependent on
other components

ji

ij







ji

ij







Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union

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9/16

M.Chrzanowski: Strength of Materials

SM1-10: Continuum Mechanics: Constitutive

equations

ij

kk

ij

ij

G











2



,

G

Summation obeys !

Kronecker’s delta

Lamé
constants [Pa]

)

(

2

33

22

11

11

11



















 G

)

(

2

33

22

11

22

22



















 G

)

(

2

33

22

11

33

33



















 G

12

12

2





G



23

23

2





G



31

31

2





G



Normal stress and normal
strain dependences

Shear stress and shear
strain dependences

This equation consists of two

groups:

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10/16

M.Chrzanowski: Strength of Materials

SM1-10: Continuum Mechanics: Constitutive

equations

E

Summation

obeys !

Kronecker’s

delta

Poisson modulus [0]









)

(

1

1

33

22

11

11

11

























E





E

12

12

1



















ij

kk

ij

ij

v

E















1

1



Young modulus [Pa]





E

)

(

33

11

22

22





















E

)

(

33

22

11

11





















E

)

(

22

11

33

33





















E

23

23

1















E

31

31

1



















ij

kk

ij

ij

v

E















1

1

31

31

2





G













1

2

E

G

G

2

31

31







Normal stress and normal
strain dependences

Shear stress and shear
strain dependences

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11/16

M.Chrzanowski: Strength of Materials

SM1-10: Continuum Mechanics: Constitutive

equations

ij

kk

ij

ij

G











2

l

j

i





m

ll











3

33

22

11









ll

kk

ll

ll

G











2

3

ll

m







Mean strain

3

kk

m







Mean stress

=3

3

3

3

2

3









m

m

m

G









m

m

K





3





3

2

3



 G

K

Prawo zmiany objętości





m

m

G







3

2 





































m

m

m

m

m

m

K













0

0

0

0

0

0

3

0

0

0

0

0

0

ij

m

ij

m

K















3

Volume change law

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12/16

M.Chrzanowski: Strength of Materials

SM1-10: Continuum Mechanics: Constitutive

equations

t

T



























































m

m

m

m

m

m

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

0

0

0

0

0

0

33

32

31

23

22

22

13

12

11

33

32

31

23

22

22

13

12

11

 

ij

t

T

ij

m

ij

m

ij

ij

t

t

t

t















t

D

t

A

=

+

3

/

kk

m

t

t 

deviator

axiato
r





ij

m

ij

t

t

t

D





 

ij

m

t

t

A



Decomposition of symmetric matrix (tensor) into deviator and

volumetric part (axiator)

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13/16

M.Chrzanowski: Strength of Materials

SM1-10: Continuum Mechanics: Constitutive

equations



































m

m

m

m

m

m

K













0

0

0

0

0

0

3

0

0

0

0

0

0

ij

m

ij

m

K















3





KA

A

3



Volume change law

Distortion
law













ij

m

m

ij

K

G









3

3

2











ij

m

ij

kk

ij

ij

m

ij

K

G

















3

2



















ij

m

m

ij

G

G











3

3

2

2









ij

m

ij

ij

m

ij

G

G

G





















2

2

2





GD

D

2





3

2

3



 G

K

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14/16

M.Chrzanowski: Strength of Materials

SM1-10: Continuum Mechanics: Constitutive

equations





KA

A

3

















2

1

1





E





GD

D

2









GD

KA

T

2

3





K

A

A

3

/







G

D

D

2

/







G

D

K

A

T

2

/

3

/



















E

v

ij

kk

ij

ij













 1

ij

kk

ij

ij

G











2











1

2

E

G



3

2

3



 G

K

-1



ν



1/2





G



E/3

E/9



K





- 















2

1

3 

E

K

0

No volume change:

incompressible

material

Constants cross-
relations:

0

No shape

change:

stiff material

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M.Chrzanowski: Strength of Materials

SM1-10: Continuum Mechanics: Constitutive

equations

x

1

x

2

x

3

Volume change

Composed (full)
deformation

x

1

x

2

x

3

Shape change

x

1

x

2

x

3

G

D

K

A

T

2

/

3

/











+

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16/16

M.Chrzanowski: Strength of Materials

SM1-10: Continuum Mechanics: Constitutive

equations

x

1

x

2

x

3

Volume change

Composed (full)
deformation

x

1

x

2

x

3

Shape change

x

1

x

2

x

3

G

D

K

A

T

2

/

3

/











+

FAST

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M.Chrzanowski: Strength of Materials

SM1-10: Continuum Mechanics: Constitutive

equations



stop