 Fading-memory representation of the Oldroyd-A model

ID: fading-memory-representation-of-the-oldroyd-a-model

Put

S = σ^{d} - 2 μ_{r} D

in the Oldroyd-A model. It obeys

D_{l} S + S / τ = 2 (μ - μ_{r}) D / τ

. Pulling back as

F^{T} SF

converts this to an ordinary linear relaxation equation. With decayed remote-past initial memory, its solution is

σ^{d} (t) = 2 μ_{r} D (t) + \frac{2 ( μ - μ _{r} )}{τ} \int_{- \infty}^{t} e^{- (t - s) / τ} F (t)^{- T} F (s)^{T} D (s) F (s) F (t)^{- 1} d s .

At a finite initial time, include the homogeneous transported term

e^{- (t - t_{0}) / τ} F (t)^{- T} F (t_{0})^{T} S (t_{0}) F (t_{0}) F (t)^{- 1}

. The integral-only law is not equivalent to all arbitrary initial-value solutions without a memory condition.

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