PSI - Issue 2_A

Roberto Brighenti et al. / Procedia Structural Integrity 2 (2016) 2788–2795 Author name / Structural Integrity Procedia 00 (2016) 000–000

2791

4

x σ is given

leads to the stretches

along the non-loaded Cartesian directions, while the Cauchy stress

2 = =

λ λ λ 1/ 2

y

z

x

by:

   

   

  

  

  

  

Ψ ( )

Ψ ( )

d

0 0 A L d d V

2

2

F

0 0 A L d d

nkT

nkT

2 λ x

V

=

−

σ

x λ

= =

=

=

+ −

3

2

A L

(6)

0 0

x

2 x λ

x λ

λ

λ

2

2

A

A dL

0

0

x

x

Eq. (6) provides the initial tangent elastic modulus of the stress-strain curve,

, and represents

E d d x x / = ε σ

=

3

nkT

the stress-strain constitutive relationship for the so-called neo-Hookean materials.

3. Failure in elastomers

3.1. Microscopic failure: a model based on the evolution of the chains’ end-to-end distance distribution

Failure in elastomers can be assumed to occur when stretching exceeds the maximum chains length, once the entanglement of the chains disappears because they reach their maximum possible elongation equal to Nb (Fig. 1c). Physically, this corresponds to the detachment of the chains’ cross-links when the maximum elongated length is reached. The chains that loose their reciprocal cross-linking do not contribute anymore to the load bearing capacity of the material, and must not be accounted for in the evaluation of the internal energy of the stretched material (Eq. 5). This microscopic failure occurring at the chains’ joints leads to a stress relaxation in the material. Such a microscopic failure can be quantified by properly updating the probability distribution function ( , ) t r ϕ due to the chains’ failure. The evolution law for ( , ) t r ϕ can be written by considering its variation in the time interval dt : [ ] dt t t t t t F F r r r r r ( , ) ( , ) ( , ) ( , ) ( , ) ϕ ϕ δϕ δϕ δϕ   + = + = xl xl (7)

( , ) t r ϕ 

where the term

is the time derivative of

( , ) t r ϕ (evaluated by assuming no changes in the actual existing

xl

cross-links, subscript xl ), and

is the time derivative of

( , ) t r ϕ for a fixed deformation tensor

r = ∂ ( , ) / ϕ t

∂

δϕ

r ( , ) t

t

F

F

F . The two above variations (

) can be evaluated separately. The first term can be written as follows:

r r ( , ) , ( , ) t t ϕ ϕ   xl

F

 

)  

(8)

ϕ

∂

(

)

(

r r

( , ) t

−

−

1

1

δϕ

+ F F r r ϕ δ

δ

≅ −

( , ) r t

F F

( , )tr t

xl

∂

Eq. (8) provides the variation of the function’s value in the time interval dt , determined by changing the stretch of the chains from F to dt F F F F  + = + δ . The second contribution F r ( , ) t ϕ  in Eq. (8) can be expressed as follows: ( , ) ( , *) ( , ) r r r r F t H t ⋅ = ϕ ϕ  (9)

r ( , ) t ϕ  with time related to the amount of the failed links (represented by the part of

and gives us the variation of

F

Nb = * r , Fig. 2b).

( , *) r r H is the Heaviside

the distribution that exceeds the maximum allowable chain’s length

function (i.e. * r r ≥ ). Eq. (7) is the total variation of the distribution of the chains length, expressed as the sum of their variation with respect to the applied stretch for a fixed cross-links configuration (term xl ( , ) t r ϕ  ) and the variation with respect to time for a fixed stretch, F r ( , ) t ϕ  , taking into account the loose of the broken links in the polymer network. The above relationships give the variation of the distribution function with respect to the current state ( ( , ) t r ϕ ), and can be considered to fall in the Lagrangian formulation framework. It can easily be deduced that the current concentration of the active cross-links is given by ∫ +∞ −∞ = r r t d c t ( , ) ( ) ϕ (graphically it represents the area under the ( , ) t r ϕ curve, Fig. 2b). ( , *) 0 = r r H for * r r < and ( , *) 1 = r r H for