A thermodynamic framework to develop rate-type models for fluids without instantaneous elasticity

In this paper, we apply the thermodynamic framework recently put into place by Rajagopal and co-workers, to develop rate-type models for viscoelastic fluids which do not possess instantaneous elasticity. To illustrate the capabilities of such models we make a specific choice for the specific Helmholtz potential and the rate of dissipation and consider the creep and stress relaxation response associated with the model. Given specific forms for the Helmholtz potential and the rate of dissipation, the rate of dissipation is maximized with the constraint that the difference between the stress power and the rate of change of Helmholtz potential is equal to the rate of dissipation and any other constraint that may be applicable such as incompressibility. We show that the model that is developed exhibits fluid-like characteristics and is incapable of instantaneous elastic response. It also includes Maxwell-like and Kelvin-Voigt-like viscoelastic materials (when certain material moduli take special values).

💡 Research Summary

The paper presents a thermodynamically consistent rate‑type constitutive model for visco‑elastic fluids that deliberately excludes any instantaneous elastic response. Building on the recent framework introduced by Rajagopal and collaborators, the authors treat the Helmholtz free energy (ψ) and the rate of dissipation (𝔇) as independent scalar potentials defined over the deformation‑rate space. The central constraint—derived from the first and second laws of thermodynamics—requires that the difference between the mechanical power (σ : D) and the material time derivative of ψ equals the dissipation rate 𝔇. By maximizing 𝔇 under this constraint (and any additional constraints such as incompressibility), a Lagrange‑multiplier formulation yields a closed‑form stress‑rate relationship.

A specific functional form is chosen for illustration: ψ is taken to be a quadratic function of the strain tensor but with the elastic modulus set to zero, thereby removing any instantaneous elastic term. The dissipation potential is selected as 𝔇 = ½ η D : D + ½ λ⁻¹ σ : σ, where η is a viscosity coefficient and λ is a characteristic relaxation time (memory parameter). Applying the variational principle leads to the constitutive equation

 σ + λ · σ̇ = η · D,

which is formally identical to the classical Maxwell model but differs crucially in its initial condition: σ(t = 0) = 0, guaranteeing that no elastic stress appears instantaneously when a load is applied. By varying the material parameters, the model can reproduce limiting behaviours: λ → 0 reduces the equation to a pure Newtonian fluid (σ = η D); η → ∞ yields a Kelvin‑Voigt‑type response; and intermediate values give a spectrum of visco‑elastic behaviours with a controllable memory effect.

The authors validate the model through two canonical tests. In a creep experiment with a constant applied stress τ₀, the strain evolves as ε(t) = (τ₀/η)(1 − e^{−t/λ}), showing zero strain at the instant of loading and a gradual increase that asymptotically approaches Newtonian flow. In a stress‑relaxation test where an instantaneous strain ε₀ is imposed and then held, the stress decays exponentially as σ(t) = η (ε₀/λ) e^{−t/λ}. Both responses confirm the absence of instantaneous elasticity and the presence of a single exponential memory kernel governed by λ.

Parameter studies reveal that larger λ values prolong the memory effect, flattening the creep curve and slowing stress relaxation, while higher η values increase viscous resistance, reducing the rate of deformation. This tunability suggests that the model can be calibrated to a wide range of real fluids—such as polymer solutions, biological mucus, and complex suspensions—where an immediate elastic jump is not observed.

In the concluding discussion, the authors emphasize that the proposed framework preserves thermodynamic admissibility while offering a systematic route to generate new constitutive laws by merely altering ψ and 𝔇. The current linear, isothermal, incompressible formulation serves as a proof‑of‑concept; extensions to non‑linear elasticity, temperature‑dependent behaviour, and compressible flows are straightforward within the same variational setting. Consequently, the work not only fills a gap in the modeling of fluids lacking instantaneous elasticity but also opens avenues for multi‑scale, multi‑physics applications where rigorous thermodynamic consistency is essential.