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theory.html
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<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml" lang="" xml:lang="">
<head>
<meta charset="utf-8" />
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<title>theory</title>
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<script
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<body>
<h2 id="theory-of-linear-viscoelasticity">Theory of linear
viscoelasticity</h2>
<p><a
href="https://en.wikipedia.org/wiki/Viscoelasticity">Viscoelastic</a>
materials exhibit both viscous and elastic characteristics when
undergoing deformation. The theory of linear viscoelasticity describes
ideal materials for which there is a linear relationship between
mechanical stress and mechanical strain. An extensive review can be
found in <a
href="https://link.springer.com/book/10.1007/978-1-4899-7485-3">Brinson
H. F., Brinson L. C., Polymer Engineering Science and Viscoelasticity,
Springer, (2015)</a>. A brief summary of the equations used in this
notebook is provided below and was extracted from <a
href="https://onlinelibrary.wiley.com/doi/full/10.1002/pip.3257">Springer,
M., and Bosco N. Prog Photovolt 28.7 (2020): 659-681</a>.</p>
<h3 id="mathematical-description">Mathematical description</h3>
<p>The rate-dependent material behavior can be described in either the
time or frequency domain. In the time domain, the uniaxial, nonaging,
isothermal stress-strain equation for a linear viscoelastic material can
be represented by a Boltzmann superposition integral,</p>
<p style="text-align: center; "><span class="math display" style="text-align:center;">\[ \sigma(t) = \int_0^t E(t-\tau)
\frac{\mathrm{d \varepsilon(\tau)}}{\mathrm{d} \tau} \mathrm{d} \tau
\quad , \]</span></p>
<p>where <span class="math inline">\(\sigma(t)\)</span> is the stress
response over time, <span class="math inline">\(t\)</span>; <span
class="math inline">\(E(t)\)</span> is the relaxation modulus; <span
class="math inline">\(\varepsilon\)</span> denotes the strain; and <span
class="math inline">\(\tau\)</span> is the integration variable. The
limiting moduli for the viscoelastic material are defined as the
instantaneous modulus, <span class="math inline">\(E(t=0) =
E_0\)</span>, and the equilibrium modulus, i.e., <span
class="math inline">\(E(t) \rightarrow E_{\infty}\)</span> for <span
class="math inline">\(t \rightarrow \infty\)</span>.</p>
<p>The <a
href="https://en.wikipedia.org/wiki/Generalized_Maxwell_model">Generalized
Maxwell Model</a> is commonly used to represent the stress-strain
response of polymers. The relaxation modulus derived from this model is
given by,</p>
<p><span class="math display">\[ E(t) = E_{\infty} +
\sum\limits_{i=1}^{m} E_i \exp\left(-\frac{t}{\tau_i} \right) \quad ,
\]</span></p>
<p>where <span class="math inline">\(\tau_i\)</span> (relaxation times),
<span class="math inline">\(E_i = E_0 \alpha_i\)</span> (relaxation
moduli) are material properties, and <span
class="math inline">\(m\)</span> is the number of terms in the series.
The above equation is often referred to as Prony series, and the
equilibrium modulus can be defined by the Prony series as,</p>
<p><span class="math display">\[ E_{\infty} = E_0 \left[1-\sum_{i=1}^N
\alpha_i \right] \quad . \]</span></p>
<p>The material properties can be directly obtained from relaxation or
frequency-dependent test data. Material properties measured in the time
domain can be converted into the frequency domain, and vice versa, by
making use of a Fourier transformation,</p>
<p><span class="math display">\[ E'(\omega) = E_{\infty} +
\sum\limits_{i=1}^{m}\frac{\omega^2 \tau_i^2 E_i}{\omega^2 \tau_i^2 + 1}
\quad ,\]</span> <span class="math display">\[ E''(\omega) =
\sum\limits_{i=1}^{m} \frac{\omega \tau_i E_i}{\omega^2 \tau_i^2 +1}
\quad . \]</span></p>
<p>Herein, <span class="math inline">\(E'(\omega)\)</span> is the
storage modulus, <span
class="math inline">\(E''(\omega)\)</span> is the loss modulus,
and <span class="math inline">\(\omega = 2 \pi f\)</span> is the angular
frequency, where <span class="math inline">\(f=1/t\)</span> is the
frequency in Hertz and <span class="math inline">\(t\)</span> is the
time period in seconds, respectively. The complex modulus, <span
class="math inline">\(E^*(\omega)\)</span>, and the loss factor, <span
class="math inline">\(\tan(\delta)\)</span>, are given as,</p>
<p><span class="math display">\[ E^*(\omega) = E'(\omega) + i
E''(\omega) \quad \text{and} \]</span> <span
class="math display">\[
\tan(\delta) = \frac{E''(\omega)}{E'(\omega)} \quad ,
\]</span></p>
<p>respectively.</p>
<h3 id="experimental-characterization">Experimental
characterization</h3>
<p>An efficient way to determine the storage modulus, <span
class="math inline">\(E'(\omega)\)</span>, and the loss modulus,
<span class="math inline">\(E''(\omega)\)</span>, is by dynamic
mechanical analysis (DMA) or dynamic mechanical thermal analysis (DMTA).
The material under test is excited to mechanical steady-state
oscillations, either load- or displacement-controlled, and the
corresponding response is measured. Alternatively, relaxation
experiments in the time domain can be conducted to determine the
relaxation modulus, <span class="math inline">\(E(t)\)</span>.</p>
<p>Measurements at very low frequencies (long time periods) can be very
time-consuming and might be unfeasible for practical applications. On
the other side of the spectrum, measurements at very high frequencies
can be limited by the instrumentation or unintended heating of the
sample during cyclic deformation. To avoid such situations, the <a
href="https://en.wikipedia.org/wiki/Time%E2%80%93temperature_superposition">time-temperature
superposition principle (TTSP)</a> is applied for thermo-rheologically
simple materials. For such materials, the viscoelastic response at one
temperature is related to the viscoelastic response at another
temperature by changing the time scale (or frequency). This way, the
time scale for the materials characterization can be extended by
conducting the same frequency measurements at different temperatures.
Afterward, a reference temperature is selected, and the isothermal
measurements are shifted on a logarithmic time (or frequency) scale to
form a so-called master curve.</p>
<figure>
<img
src="https://raw.githubusercontent.com/NREL/pyvisco/main/figures/TTSP_small.png"
alt="TTSP" />
</figure>
<p>Time-temperature shift factors, <span
class="math inline">\(a_{\mathrm{T}}(\theta)\)</span>, are defined as
the horizontal shift that must be applied to individual measurements at
a constant temperature, <span class="math inline">\(\theta_i\)</span>,
to form the master curve at the reference temperature, <span
class="math inline">\(\theta_{\mathrm{ref}}\)</span>.</p>
<p>The determined shift factors, <span
class="math inline">\(a_{\mathrm{T}}\)</span>, are used to define a
shift function that describes the temperature dependence of the
viscoelastic material. Different shift functions for various materials
are available in the literature. A commonly used shift function is the
the <a
href="https://en.wikipedia.org/wiki/Williams%E2%80%93Landel%E2%80%93Ferry_equation">Williams-Landel-Ferry
(WLF)</a>,</p>
<p><span class="math display">\[ \log(a_{\mathrm{T}}) =
-\frac{C_1\left(\theta-\theta_{\mathrm{ref}}\right)}{C_2 +
\left(\theta-\theta_{\mathrm{ref}}\right)} \]</span></p>
<p>where <span class="math inline">\(\theta_{\mathrm{ref}}\)</span> is
the reference temperature of the master curve. <span
class="math inline">\(\theta\)</span> is the temperature of interest,
and <span class="math inline">\(C_1\)</span> and <span
class="math inline">\(C_2\)</span> are calibration constants. The TTSP
is based on the kinetic theory of polymers, which is strictly speaking
only valid above the glass transition temperature, <span
class="math inline">\(\theta_g\)</span>. Although the TTSP is thought to
be valid also for temperatures below <span
class="math inline">\(\theta_g\)</span>, the exact lower limit is not
well defined, and the principle is commonly applied to temperatures
below <span class="math inline">\(\theta_g\)</span> as long as the
measurement data are shiftable to form a smooth master curve.
Alternatively, different shift functions can be fitted. Below we provide
a routine to fit the polynomial shift function up to degree four.</p>
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