PSI - Issue 2_A

Roberto Brighenti et al. / Procedia Structural Integrity 2 (2016) 2788–2795

2790

Author name / Structural Integrity Procedia 00 (2016) 000–000

b

z

(b)

(a)

ϕ 0 (< r >)

(0)

b

ϕ (< r >, t )

λ

stretch

(1)

r

y

O

Chains microscopic end to end distance density probability - r*

r*

x

Chains end-to-end distance, r = < r >

Fig. 2. Scheme of a polymer chain and definition of its end-to-end vector (a), probability distribution functions of the chains lengths (b)

[

] 3

[

] ) 3 −

( ) λ

2 2 2 + + −

T

½ Ψ = − ⋅ ∆ = + T S

λ λ λ ,

Ψ =

kT

( ) ½ F

F F

tr(

kT

(2)

x

y

z

where the last expression of

( ) F Ψ provides the work in a general 3D framework, while the work for the simple case

( ) λ Ψ . In other words,

( ) λ Ψ represents

of the material stretched only along the three coordinate axes is given by

the strain-free energy per chain contained in an elastomer stretched by . The above energy per single chain (Eq.2) can be used to determine the stored energy per unit volume of material. To this end, it is necessary to define the chain end-to-end distance distribution in the material that is usually assumed to be a Gaussian function ( , ) t r ϕ in elastomers: z x y λ λ λ , ,

   

=    

2

3/ 2

  

  

r

2 3

(

) 2

3

c

⋅

−

=

⋅

−

ϕ

( , ) r t

r

exp

exp

c

(3)

2

2

3/ 2

π

π

2

Nb

Nb

c being the links’ concentration per unit volume. The energy stored in the unit volume of polymer can be expressed by: ∫ +∞ −∞ Ψ = r r r r t U d t V ( , ) ( ) ( , ) ϕ with [ ] 2 0 2 0 2 2 2 ( ) 2 3 2 3 ( ) r Fr r r − = = Nb kT Nb kT U (4) where ( ) r U is the energy per single chain. The application of a given stretch – represented by the deformation gradient F – produces a modification of the distribution function ( , ) t r ϕ since the chains’ length changes and, therefore, the concentration of links for a given end-to-end distance is not equal to that in the stress-free state (Fig. 2b) (Doi (2013)): ( ) [ ] = − ⋅ ⋅ = − Ψ = ∫ ∫ ∫ ∞ −∞ ∞ −∞ ∞   / 3 2 2 0 2 0 2 0 2 ( ) ) 3 tr( 2 3 ( , ) ( ) 2 3 ( ) Nb T V dr dr Nb nkT dr N dN Nb nkT r F F r Fr r F ϕ ϕ (5)

ν

nkT

AkT

[

]

[

] ) 3 −

T

T

=

− =

F F

F F

tr(

) 3

tr(

2

2

N ∑ = i

. In the above expression, the relation n A ν = has been used, where ν is the number

2

2

2

=

=

being

r

b

Nb

0

i

1

of linked moles per unit volume, A is the Avogadro’s number, and n is the number of linked chains per unit volume. In a simple 1D case with a stretch along the x -direction ( x λ ), it is possible to write explicitly the stress strain relationship assuming an incompressible behaviour (an isochoric transformation is mathematically represented by the condition det 1 = = F J ) as typically occurs for such a class of materials. Such an incompressible constraint