Models

This module provides analytical expressions for complex modulus models and WLF time-temperature superposition functions. All model constructors return callable functions that accept a complex frequency p = iω.

Rheological models

ViscoAnalysis.Mod2S2P1D — Function

Mod2S2P1D(Einf, E0, delta, tauE, k, h, beta) -> Function

Return the 2S2P1D complex modulus function E*(p), where p = iω.

The 2S2P1D (2 Springs, 2 Parabolic elements, 1 Dashpot) model is:

E*(p) = E∞ + (E₀ - E∞) / (1 + δ(pτ)^{-k} + (pτ)^{-h} + 1/(pβτ))

Arguments

Einf — Glassy modulus E∞ [MPa] (high-frequency limit).
E0 — Static modulus E₀ [MPa] (zero-frequency limit, E₀ < E∞).
delta — δ shape parameter.
tauE — Relaxation time τ_E [s].
k — Low-frequency exponent (0 < k < h).
h — High-frequency exponent (k < h ≤ 1).
beta — β dashpot coefficient.

Returns

A function p -> ComplexF64 that evaluates E*(p) at complex frequency p.

Example

model = Mod2S2P1D(30_000.0, 10.0, 2.0, 1e-3, 0.2, 0.6, 500.0)
E_star = model(1im * ω)     # complex modulus at angular frequency ω

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ViscoAnalysis.Mod1S2P1D — Function

Mod1S2P1D(E0, delta, tauE, k, h, beta) -> Function

Return the 1S2P1D complex modulus function E*(p) (one spring variant, no glassy branch):

E*(p) = E₀ / (1 + δ(pτ)^{-k} + (pτ)^{-h} + 1/(pβτ))

Arguments

E0 — Static modulus E₀ [MPa].
delta — δ shape parameter.
tauE — Relaxation time τ_E [s].
k — Low-frequency exponent (0 < k < h).
h — High-frequency exponent (k < h ≤ 1).
beta — β dashpot coefficient.

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ViscoAnalysis.ModHS — Function

ModHS(Einf, E0, delta, tauE, k, h) -> Function

Return the Huet-Sayegh complex modulus function E*(p) (no dashpot, β → ∞):

E*(p) = E∞ + (E₀ - E∞) / (1 + δ(pτ)^{-k} + (pτ)^{-h})

Arguments

Einf — Glassy modulus E∞ [MPa].
E0 — Static modulus E₀ [MPa].
delta — δ shape parameter.
tauE — Relaxation time τ_E [s].
k — Low-frequency exponent (0 < k < h).
h — High-frequency exponent (k < h ≤ 1).

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ViscoAnalysis.ModGM — Function

ModGM(X; Ei=nothing, taui=nothing) -> Function

Return the Generalised Maxwell complex modulus function G*(p).

Provide either:

Ei (stiffnesses) → X is treated as the relaxation times τᵢ, or
taui (times) → X is treated as the stiffnesses Eᵢ, or
neither → X = [E₁, E₂, …, τ₁, τ₂, …] (concatenated, equal halves).

Example

Ei   = [1e4, 5e3, 1e3]
taui = [1e-4, 1e-2, 1.0]
model = ModGM(taui; Ei=Ei)
G_star = model(1im * ω)

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WLF functions

ViscoAnalysis.wlf_scalar — Function

wlf_scalar(T, Tref, C1, C2) -> Float64

Evaluate the WLF shift factor: log₁₀(aT) = -C1·(T - Tref) / (C2 + T - Tref).

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ViscoAnalysis.change_Tref_WLF — Function

change_Tref_WLF(C1, C2, Tref_old, Tref_new) -> (f, C1_new, C2_new)

Convert WLF coefficients from one reference temperature to another using the standard transformation:

C2_new = C2_old + (Tref_new − Tref_old)
C1_new = C1_old · C2_old / C2_new

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ViscoAnalysis.T_TEV — Function

T_TEV(Om, Tref, C1, C2, Omc) -> Float64

Inverse WLF: return the temperature at which angular frequency Om is equivalent to Omc at Tref.

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Usage examples

Evaluate a 2S2P1D model manually

using ViscoAnalysis

# Build the model function from known parameters
model = Mod2S2P1D(
    28_500.0,   # Einf [MPa]
    9.5,        # E0   [MPa]
    2.1,        # delta
    3.2e-4,     # tauE [s]
    0.185,      # k
    0.625,      # h
    450.0,      # beta
)

# Evaluate at ω = 2π·10 rad/s  (f = 10 Hz)
ω = 2π * 10.0
E_star = model(1im * ω)
@show abs(E_star)              # |E*|  [MPa]
@show angle(E_star) * 180/π   # φ     [°]

WLF shift factor

log_aT = wlf_scalar(25.0, 10.0, 19.5, 108.0)
# → log₁₀(aT) for T = 25 °C, Tref = 10 °C

Change reference temperature

f_new, C1_new, C2_new = change_Tref_WLF(19.5, 108.0, 10.0, 0.0)
log_aT_at0 = f_new(25.0)   # log₁₀(aT) with Tref = 0 °C

Generalised Maxwell model

# Specify stiffnesses; X = relaxation times
Ei    = [1000.0, 5000.0, 20000.0]   # MPa
tau_i = [1e-1,   1e-3,   1e-5]      # s
model_gm = ModGM(tau_i; Ei=Ei)

# Or concatenate [Ei..., taui...] into a single vector
X = vcat(Ei, tau_i)
model_gm2 = ModGM(X)