Wagner model (original) (raw)

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For the American labour relations model, see Wagner Act.

Wagner model is a rheological model developed for the prediction of the viscoelastic properties of polymers. It might be considered as a simplified practical form of the Bernstein-Kearsley-Zapas model. The model was developed by German rheologist Manfred Wagner.

For the isothermal conditions the model can be written as:

σ ( t ) = − p I + ∫ − ∞ t M ( t − t ′ ) h ( I 1 , I 2 ) B ( t ′ ) d t ′ {\displaystyle \mathbf {\sigma } (t)=-p\mathbf {I} +\int _{-\infty }^{t}M(t-t')h(I_{1},I_{2})\mathbf {B} (t')\,dt'} {\displaystyle \mathbf {\sigma } (t)=-p\mathbf {I} +\int _{-\infty }^{t}M(t-t')h(I_{1},I_{2})\mathbf {B} (t')\,dt'}

where:

M ( x ) = ∑ k = 1 m g i θ i exp ⁡ ( − x θ i ) {\displaystyle M(x)=\sum _{k=1}^{m}{\frac {g_{i}}{\theta _{i}}}\exp({\frac {-x}{\theta _{i}}})} {\displaystyle M(x)=\sum _{k=1}^{m}{\frac {g_{i}}{\theta _{i}}}\exp({\frac {-x}{\theta _{i}}})}, where for each mode of the relaxation, g i {\displaystyle g_{i}} {\displaystyle g_{i}} is the relaxation modulus and θ i {\displaystyle \theta _{i}} {\displaystyle \theta _{i}} is the relaxation time;

The strain damping function is usually written as:

h ( I 1 , I 2 ) = m ∗ exp ⁡ ( − n 1 I 1 − 3 ) + ( 1 − m ∗ ) exp ⁡ ( − n 2 I 2 − 3 ) {\displaystyle h(I_{1},I_{2})=m^{*}\exp(-n_{1}{\sqrt {I_{1}-3}})+(1-m^{*})\exp(-n_{2}{\sqrt {I_{2}-3}})} {\displaystyle h(I_{1},I_{2})=m^{*}\exp(-n_{1}{\sqrt {I_{1}-3}})+(1-m^{*})\exp(-n_{2}{\sqrt {I_{2}-3}})},

The strain hardening function equal to one, then the deformation is small and approaching zero, then the deformations are large.

The Wagner equation can be used in the non-isothermal cases by applying time-temperature shift factor.