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=Linear Viscoelastic Response=

A brief summary of linear viscoelastic response of amorphous polymers based on Lawrence E. Malvern's Introduction to the Mechanics of a Continuous Medium pages 306-313

1. Introduction to Viscoelastic Polymers
A viscoelastic material displays viscous properties of a Newtonian fluid as well as elastic properties of an elastic solid (Reference 3.1). Such materials can be divided into two broad classifications: amorphous polymers and crystalline. Amorphous polymers, of interest in this article, can be further subdivided into crosslinked and uncrosslinked. Uncrosslinked polymers are intertwined, long-chain polymers that are not chemically bonded to adjacent polymer chains. Untreated, natural rubber is an example of an uncrosslinked polymer. Crosslinked polymers are bonded - either ionically or covalently - between polymer chains forming a three dimensional "network." Vulcanized rubber is an example of a crosslinked polymer because the polymer chains are randomly, covalently bonded by sulfur atoms during the vulcanization process (Reference 3.2). Crosslinked and uncrosslinked materials have different viscoelastic responses. Also, both crosslinked and uncrosslinked polymers behave differently at different temperatures. In fact, two phases have been identified: glassy and rubbery, with a transition phase in between. The temperature at which the phase transition occurs is called the glass-transition temperature (Tg). The viscoelastic behavior of polymers is characterized by three methods: creep tests, stress relaxation tests, and dynamic response to sinusoidal loads. Constitutive equations for this behavior can be developed by analogies to spring and dashpot models using the Kelvin-Voigt or Maxwell elements or by the Boltzmann superposition principle and the continuous spectra.

2.1 Creep Tests
The methodologies for creep testing crosslinked and uncrosslinked polymers are different because of the differences in the behavior.

2.1.1 Creep Testing Crosslinked Polymers
During creep tests of crosslinked polymers, the creep compliance defined in Equation 2-1 is determined as a function of time at constant temperature and constant traction, T12. Though creep tests can be carried out in several ways, the most common creep test is a tensile test. However, in the configuration presented here and in Malvern (Reference 3.1), two identical samples are bonded between three parallel plates. The outer two plates remain stationary during the shear creep test while a constant, longitudinal load is applied to the middle plate. The shear strain ($${\gamma}_{12}$$) is measured between when the load is initially applied, t0, and the time when the load is removed, t1. The viscoelastic response is linear if the compliance is the same for any load. Four typical creep curves of a crosslinked polymer at four different temperatures are shown in Figure 2.1. The independent variable is time and the dependent variable is the creep compliance. At all four temperatures, the creep compliance immediately increases at t0 to the glass compliance, Jg, and the creep compliance immediately decreases at t1 by the same amount. In the glassy phase, well below the glass transition temperature, the creep compliance remains constant at the glass compliance, Jg. At low temperatures within the transition phase, or the glassy transition, the glass compliance steadily increases until t1. After the immediate decrease by Jg the samples eventually, ideally, return to their initial state. For higher temperatures within the transition phase, the rubbery transition, the rubbery elastic plateau is ideally reached at equilibrium compliance, Je. After the load is removed, the sample returns to the initial state in about the same time, ($$t_{1}-t_{0}$$). Within the rubbery phase, the creep compliance almost immediately reaches the equilibrium compliance.



2.1.2 Creep Testing Uncrosslinked Polymers
Creep testing for uncrosslinked polymers is typically conducted in a viscometer with two coaxial, rotating cylinders. The response is qualitatively similar to the crosslinked polymers except at equilibrium compliance above the transition temperature. Uncrosslinked polymers in the glass phase remain at equilibrium compliance until the load is removed and ultimately return to the initial state similar to the crosslinked polymers displayed in Figure 2.1. However, the creep compliance continues to increase at a constant rate, the creep rate, during creep tests of uncrosslinked polymers above the glass transition temperature as shown in Figure 2.3. The creep rate is defined in Equation 2-2 where η is the viscosity. Also, uncrosslinked polymers in the transition and rubbery phases will not return to the initial state after the load has been removed at t1. The permanent deformation depends on the time when the constant load was removed.



2.2. Stress-Relaxation Tests
Stress-relaxations tests of crosslinked polymers are similar to creep tests except the sample is subjected to a constant strain instead of constant traction and the relaxation modulus in Equation 2-3 is measured instead of creep compliance. With constant strain, stress-relaxation can take several decades. Therefore, a typical relaxation curve of a viscoelastic polymer in the transition phase, shown in Figure 2.3, presents both the relaxation modulus and time on a logarithmic scale. A master curve like this would typically be generated from several tests at various temperatures using the time-temperature superposition principle (Reference 3.3), described in Equation 2-4. In Equation 2-4, aT is the horizon translation or shift factor, a constant that depends on the test temperature (T) and the constant temperature (T0) of the master curve. The blue line represents measurements taken in a single test at temperature(T) with blue arrows pointing horizontally back to the green line of the master curve at temperature T0 to illustrate the time-temperature superposition with a shift factor less than 1. Again, the viscoelastic polymers will relax from the glass modulus, Gg, to the equilibrium modulus, Ge, at the rubbery plateau. After the rubbery plateau, uncrosslinked polymers relax to zero stress in a rubbery flow as shown by the dashed green line in Figure 2.3. Crosslinked polymers will also slowly decrease from the equilibrium position as illustrated by the dotted green line.



2.3 Dynamic Response to Sinusoidal Loading
Creep compliance and stress relaxation tests are transient experiments. A third type of test is the dynamic response to sinusoidal loading in which the traction is periodic, sometimes called dynamic mechanical analysis (Reference 3.4). Assuming uniform stress and strain throughout the sample, the traction, Equation 2-5, and the strain, Equation 2-6, are out of phase by an angle called the phase shift, δ, where T0 here is the maximum traction and ω is the angular velocity.

The strain can be expressed as a combination of the in phase component, J'T0, and out of phase component, J"T0, as shown in Figure 2.4. Where J' is the storage compliance, J" is the loss compliance, and the combination defined in Equation 2-7 is the complex compliance, J*, where i is the imaginary number, $$\sqrt{-1}$$. Similarly, the traction can be broken down into its in phase and out of phase components, G'γ0 and G"γ0, respectively, as shown in Figure 2.5. Where G' is the storage modulus, G" is the loss modulus and the complex combination defined in Equation 2-8 is the complex modulus, G*.



The simple harmonic Traction and strain can be expressed as a function of the complex amplitudes of traction and strain and the angular velocity by Equations 2-9 and 2-10.

Therefore, the complex compliance and relaxation modulus can be expressed as a product of the complex amplitudes in Equations 2-11 and 2-12.

Finally, the ratio of the storage compliance to the loss compliance equals the ratio of the storage modulus and the loss modulus and is defined as the loss tangent, tan δ.

2.4 Volumetric Response
Volumetric, or dilatational, response and tensile response can be qualitatively similar to shear tests where the bulk modulus is defined by Equation 2-13.

3. References
3.1 Malvern, Lawrence E., Introduction to the Mechanics of a Continuous Medium, 1969. 3.2 http://en.wikipedia.org/wiki/Vulcanization. 3.3 http://en.wikipedia.org/wiki/Time-temperature_superposition 3.4 http://en.wikipedia.org/wiki/Dynamic_mechanical_analysis