## Introduction

In previous parts of this series I covered the theory of linear viscoelasticity based on stress relaxation and constant strain rate tension tests. In this article I will show how a linear viscoelastic material model can be used to predict the response due to a sinusoidal strain history. The derivation will show how the storage and loss moduli are defined, and how they can be determined from the stress response.

## Why Care About the Cyclic Loading Response?

- In many applications the load is cyclic.
- It is easy to measure the steady-state cyclic response using a Dynamic Mechanical Analysis (DMA) machine.
- We can calibrate a linear viscoelastic model to dynamic data (E’ and E’’).

*Figure 1. DMA test machine.*

## Cyclic Response - Full Solution

In this example I will determine the stress response of a linear viscoelastic material loaded with a sinusoidal strain history: \( \varepsilon(t) = \varepsilon_0 \sin(\omega t)\), and I will only consider times \(t \ge 0\). In part 1 of this series I showed that the stress for **any** strain history can be obtained from:

\[\sigma(t) = \displaystyle \int_0^{t} E_R(t – \tau) \dot{\varepsilon}(\tau) d\tau. \]

(1)

Inserting the sinusoidal strain history into this equation gives:

\[\sigma(t) = \displaystyle \int_0^{t} E_R(t-\tau) \varepsilon_0 \omega \cos(\omega \tau) d\tau.\]

(2)

Rewrite the integral based on the variable substitution: \( s = t – \tau\):

\[\sigma(t) = \displaystyle \int_0^t E_R(s) \varepsilon_0 \omega \cos[\omega (t-s)] ds.\]

(3)

## Example - Full Solution

To evaluate the integral and calculate the actual stress response we need to specify the stress relaxation modulus. Let’s assume the following 1-term Prony expression: \( E_R(t) = E_0 e^{-\alpha t}\). Inserting this into Equation (3), and evaluating the integrals gives:

\[\sigma(t) = \displaystyle\frac{E_0 \varepsilon_0 \omega}{\alpha^2 + \omega^2} \left[ \alpha \cos(\omega t) + \omega \sin(\omega t) + e^{-\alpha t} \cdot f(\alpha, \omega, t) \right]\]

(4)

Note 1: I did not write out the whole equation, I left some terms in the function \(f(\alpha, \omega,t)\). Note 2: In steady state, part of the stress is in-phase with the strain, and part of the stress is out-of phase with the strain.

The predicted stress-strain response due to a sinusoidal strain history can also be calculated using MCalibration (see Figure 2). In this case the steady-state response is reached after about one cycle.

*Figure 2. Predicted cyclic response from a one-term Prony series model.*

## Cyclic Response - Steady State

(5)

which can also be written:

\[\sigma(t) = \varepsilon_0 \left[ E'(\omega) \sin(\omega t) + E”(\omega) \cos(\omega t)\right]\]

(6)

## Tan Delta

From Equation (6) we know that the stress is given by a sin and a cos term, which can also be written: \(\sigma(t) = \sigma_0 \sin(\omega t + \delta)\).

This equation can also be written: \( \sigma(t) =\sigma_0 \sin(\omega t) \cos\delta + \sigma_0 \cos(\omega t) \sin\delta \).

Which gives:

\[ \tan\delta = \displaystyle \frac{E”}{E’}\]

## Summary

- The storage (E’) and loss modulus (E’’) are often measured using DMA experiments.
- The Prony series terms can be determined from the dynamic data.
- For more info about any of these topics see my book “Mechanics of Solid Polymers“.