8  L06 — Viscoelasticity

Linear Viscoelastic Models and Numerical Implementation

📽 Slides: Open presentation

8.1 Motivation: Rate-Dependent Elastic Behavior

Viscoelastic materials exhibit:

• Creep: increasing strain under constant stress
• Stress relaxation: decreasing stress under constant strain
• Hysteresis: energy dissipation in cyclic loading

Examples: polymers, soft biological tissues, asphalt, filled rubber.

Key distinction from plasticity: fully recoverable in principle (no permanent set), but time-dependent.

8.2 The Maxwell Model

Spring (stiffness \(E\)) in series with dashpot (viscosity \(\eta\)):

\[ \dot{\varepsilon} = \frac{\dot{\sigma}}{E} + \frac{\sigma}{\eta} \]

Relaxation time \(\tau = \eta/E\). Under constant strain \(\varepsilon_0\): \[ \sigma(t) = E\varepsilon_0\,e^{-t/\tau} \]

→ Captures relaxation, but predicts indefinite creep.

8.3 The Kelvin-Voigt Model

Spring and dashpot in parallel: \[ \sigma = E\varepsilon + \eta\dot{\varepsilon} \]

Under constant stress \(\sigma_0\): \[ \varepsilon(t) = \frac{\sigma_0}{E}\left(1 - e^{-t/\tau}\right) \]

→ Captures creep saturation, but cannot relax instantaneously.

8.4 Standard Linear Solid (SLS)

One spring in parallel with a Maxwell element. Captures both creep and relaxation:

\[ \sigma + \tau_\sigma\dot{\sigma} = E_\infty\varepsilon + E_0\tau_\sigma\dot{\varepsilon} \]

Relaxed modulus \(E_\infty < E_0\) (instantaneous).

8.5 Generalized Maxwell Model (Prony Series)

Multiple Maxwell branches in parallel: \[ \sigma(t) = \varepsilon_0\left[E_\infty + \sum_{i=1}^N E_i\,e^{-t/\tau_i}\right] \]

• Excellent fit to broad relaxation spectra
• Each branch adds two parameters (\(E_i\), \(\tau_i\))
• DMA experiments needed for identification

8.6 Hereditary Integral Representation

The stress history in terms of the relaxation kernel \(G(t)\): \[ \sigma(t) = \int_0^t G(t-s)\,\dot{\varepsilon}(s)\,ds \]

For the Prony series: \[ G(t) = E_\infty + \sum_{i=1}^N E_i\,e^{-t/\tau_i} \]

8.7 Numerical Implementation: Recursive Update

Each Maxwell branch maintains an internal variable \(h_i\) (history stress). Efficient recursive update over time step \(\Delta t\):

\[ h_i^{n+1} = e^{-\Delta t/\tau_i}\,h_i^n + E_i(1 - e^{-\Delta t/\tau_i})\,\Delta\varepsilon \]

Total stress: \[ \sigma^{n+1} = E_\infty\varepsilon^{n+1} + \sum_i h_i^{n+1} \]

Algorithmic tangent: \(\partial\sigma/\partial\varepsilon = E_\infty + \sum_i E_i(1-e^{-\Delta t/\tau_i})\Delta t/\tau_i\) (approx).

8.8 3D Finite Viscoelasticity

For finite deformations, use volumetric-isochoric split of the free energy:

\[ \Psi = \Psi_\text{vol}(J) + \bar{\Psi}_\infty(\bar{\mathbf{C}}) + \sum_i \Gamma_i(\bar{\mathbf{C}}, \tilde{\mathbf{C}}_i) \]

where \(\tilde{\mathbf{C}}_i\) are internal (viscous) deformation variables.

Notation — three distinct Cauchy-Green–like tensors in this section. The symbol \(\mathbf{C} = \mathbf{F}^T\mathbf{F}\) is the usual right Cauchy-Green tensor (see L03). The overbar \(\bar{\mathbf{C}} = \bar{\mathbf{F}}^T\bar{\mathbf{F}}\) is the isochoric (unimodular) part of \(\mathbf{C}\) obtained from the volumetric–isochoric split \(\mathbf{F} = J^{1/3}\bar{\mathbf{F}}\) so that \(\det\bar{\mathbf{C}} = 1\). The tilde \(\tilde{\mathbf{C}}_i\) denotes internal viscous deformation variables (one per Maxwell branch \(i\)) — they are state variables evolved by the viscoelastic rate equation, not kinematic quantities derived from \(\mathbf{F}\). In particular, \(\tilde{\mathbf{C}}_i \neq \mathbf{C}\) and \(\tilde{\mathbf{C}}_i \neq \bar{\mathbf{C}}\) at a generic instant.

Evolution equation for each internal variable mirrors the 1D recursive formula but in tensor form.