Nonlinear stress relaxation of carbon black-filled rubber vulcanizates under various types of deformation

Nonlinear stress relaxation behavior of carbon black-filled rubber vulcanizates under large strain is investigated for the three types of deformation (equibiaxial stretching, pure shear, and uniaxial stretching) with various sets of two principal ratio (λ₁ and λ₂). The stress relaxation curves in the wide time (t) region of t > 10⁻¹ s at a reference temperature of 296 K are obtained by the time-temperature superposition principle. We demonstrate that stress relaxation component Δσ(λ₁, λ₂, t) ([identical with] σ[λ₁, λ₂, t] - σ[λ₁, λ₂, [infinity]], where σ[λ₁, λ₂, [infinity]] is the equilibrium stress) is separable into the time (t)- and deformation (λ)-dependent functions, that is, Δσ(λ₁, λ₂, t) = Δσtotal(λ₁, λ₂)ψ(t), where Δσtotal represents the total relaxation strength. We further show that the time-dependent term ψ(t) of the filled rubbers is almost similar to that of the unfilled rubbers in the time region examined here, whereas Δσtotal for filled rubbers is considerably larger than that for unfilled ones. In addition, the reduced relaxation strength Δσtotal(λ₁, λ₂)/σ(λ₁, λ₂, [infinity]) is almost independent of the degree and type of deformation, and the value for the filled rubber is larger than that for the unfilled one. These results indicate that (1) the strain-driven collapse of filler networks causes a large degree of stress relaxation, but it does not affect the relaxation dynamics in the time region investigated ([infinity] > t > 10⁻¹ s), and (2) the relaxation dynamics in filled rubbers is governed by that in the rubber matrix. These findings considerably facilitate the full description of the finite deformation with time effect for filled rubbers.