Viscoelasticity

Maxwell model

The Maxwell model can be represented by a purely viscous damper and a purely elastic spring connected in series, as shown in the diagram. The model can be represented by the following equation:

$${\displaystyle \sigma +{\frac {\eta }{E}}{\dot {\sigma }}=\eta {\dot {\varepsilon }}}$$

Under this model, if the material is put under a constant strain, the stresses gradually relax. When a material is put under a constant stress, the strain has two components. First, an elastic component occurs instantaneously, corresponding to the spring, and relaxes immediately upon release of the stress. The second is a viscous component that grows with time as long as the stress is applied. The Maxwell model predicts that stress decays exponentially with time, which is accurate for most polymers. One limitation of this model is that it does not predict creep accurately. The Maxwell model for creep or constant-stress conditions postulates that strain will increase linearly with time. However, polymers for the most part show the strain rate to be decreasing with time.[2]

This model can be applied to soft solids: thermoplastic polymers in the vicinity of their melting temperature, fresh concrete (neglecting its aging), and numerous metals at a temperature close to their melting point.

Schematic representation of Kelvin–Voigt model

The Kelvin–Voigt model, also known as the Voigt model, consists of a Newtonian damper and Hookean elastic spring connected in parallel, as shown in the picture. It is used to explain the creep behaviour of polymers.

The constitutive relation is expressed as a linear first-order differential equation:

$${\displaystyle \sigma =E\varepsilon +\eta {\dot {\varepsilon }}}$$

This model represents a solid undergoing reversible, viscoelastic strain. Upon application of a constant stress, the material deforms at a decreasing rate, asymptotically approaching the steady-state strain. When the stress is released, the material gradually relaxes to its undeformed state. At constant stress (creep), the model is quite realistic as it predicts strain to tend to σ/E as time continues to infinity. Similar to the Maxwell model, the Kelvin–Voigt model also has limitations. The model is extremely good with modelling creep in materials, but with regards to relaxation the model is much less accurate.[7]

This model can be applied to organic polymers, rubber, and wood when the load is not too high.

The standard linear solid model, also known as the Zener model, consists of two springs and a dashpot. It is the simplest model that describes both the creep and stress relaxation behaviors of a viscoelastic material properly. For this model, the governing constitutive relations are:

Maxwell representation	Kelvin representation
${\displaystyle \sigma +{\frac {\eta }{E_{2}}}{\dot {\sigma }}=E_{1}\varepsilon +{\frac {\eta (E_{1}+E_{2})}{E_{2}}}{\dot {\varepsilon }}}$	${\displaystyle \sigma +{\frac {\eta }{E_{1}+E_{2}}}{\dot {\sigma }}={\frac {E_{1}E_{2}}{E_{1}+E_{2}}}\varepsilon +{\frac {E_{1}\eta }{E_{1}+E_{2}}}{\dot {\varepsilon }}}$

Under a constant stress, the modeled material will instantaneously deform to some strain, which is the instantaneous elastic portion of the strain. After that it will continue to deform and asymptotically approach a steady-state strain, which is the retarded elastic portion of the strain. Although the Standard Linear Solid Model is more accurate than the Maxwell and Kelvin–Voigt models in predicting material responses, mathematically it returns inaccurate results for strain under specific loading conditions.

The Burgers model consists of either two Maxwell components in parallel or a Kelvin–Voigt component, a spring and a dashpot in series. For this model, the governing constitutive relations are:

Maxwell representation	Kelvin representation
${\displaystyle \sigma +\left({\frac {\eta _{1}}{E_{1}}}+{\frac {\eta _{2}}{E_{2}}}\right){\dot {\sigma }}+{\frac {\eta _{1}\eta _{2}}{E_{1}E_{2}}}{\ddot {\sigma }}=\left(\eta _{1}+\eta _{2}\right){\dot {\varepsilon }}+{\frac {\eta _{1}\eta _{2}\left(E_{1}+E_{2}\right)}{E_{1}E_{2}}}{\ddot {\varepsilon }}}$	${\displaystyle \sigma +\left({\frac {\eta _{1}}{E_{1}}}+{\frac {\eta _{2}}{E_{1}}}+{\frac {\eta _{2}}{E_{2}}}\right){\dot {\sigma }}+{\frac {\eta _{1}\eta _{2}}{E_{1}E_{2}}}{\ddot {\sigma }}=\eta _{2}{\dot {\varepsilon }}+{\frac {\eta _{1}\eta _{2}}{E_{1}}}{\ddot {\varepsilon }}}$

This model incorporates viscous flow into the standard linear solid model, giving a linearly increasing asymptote for strain under fixed loading conditions.

Schematic of Maxwell-Wiechert Model

The Generalized Maxwell model, also known as the Wiechert model, is the most general form of the linear model for viscoelasticity. It takes into account that the relaxation does not occur at a single time, but at a distribution of times. Due to molecular segments of different lengths with shorter ones contributing less than longer ones, there is a varying time distribution. The Wiechert model shows this by having as many spring–dashpot Maxwell elements as necessary to accurately represent the distribution. The figure on the right shows the generalised Wiechert model.[8] Applications: metals and alloys at temperatures lower than one quarter of their absolute melting temperature (expressed in K).

The second-order fluid is typically considered the simplest nonlinear viscoelastic model, and typically occurs in a narrow region of materials behavior occurring at high strain amplitudes and Deborah number between Newtonian fluids and other more complicated nonlinear viscoelastic fluids.[3] The second-order fluid constitutive equation is given by:

$${\displaystyle \mathbf {T} =-p\mathbf {I} +2\eta _{0}\mathbf {D} -\psi _{1}\mathbf {D} ^{\triangledown }+4\psi _{2}\mathbf {D} \bullet \mathbf {D} }$$

where:

• ${\displaystyle \mathbf {I} }$ is the identity tensor
• ${\displaystyle \mathbf {D} }$is the deformation tensor
• ${\displaystyle \eta _{0},\psi _{1},\psi _{2}}$ denote viscosity, and first and second normal stress coefficients, respectively
• ${\displaystyle \mathbf {D} ^{\triangledown }}$denotes the upper-convected derivative of the deformation tensor where
  $${\displaystyle \mathbf {D} ^{\triangledown }\equiv {\dot {\mathbf {D} }}-(\bigtriangledown \mathbf {v} )^{\mathbf {T} }\bullet \mathbf {D} -\mathbf {D} \bullet \bigtriangledown \mathbf {v} }$$

  
  and ${\displaystyle {\dot {\mathbf {D} }}\equiv {\frac {\partial }{\partial t}}\mathbf {D} +\mathbf {v} \bullet \bigtriangledown \mathbf {D} }$is the material time derivative of the deformation tensor.[3]

The upper-convected Maxwell model incorporates nonlinear time behavior into the viscoelastic Maxwell model, given by:[3]

$${\displaystyle \mathbf {\tau } +\lambda \mathbf {\tau } ^{\triangledown }=2\eta _{0}\mathbf {D} }$$

where ${\displaystyle \mathbf {\tau } }$ denotes the stress tensor.

The Oldroyd-B model is an extension of the Upper Convected Maxwell model and is interpreted as a solvent filled with elastic bead and spring dumbbells. The model is named after its creator James G. Oldroyd.[10][11][12]

The model can be written as:

$${\displaystyle \mathbf {T} +\lambda _{1}{\stackrel {\nabla }{\mathbf {T} }}=2\eta _{0}(\mathbf {D} +\lambda _{2}{\stackrel {\nabla }{\mathbf {D} }})}$$

where:

• ${\displaystyle \mathbf {T} }$ is the stress tensor;
• ${\displaystyle \lambda _{1}}$ is the relaxation time;
• ${\displaystyle \lambda _{2}}$ is the retardation time = ${\displaystyle {\frac {\eta _{s}}{\eta _{0}}}\lambda _{1}}$;
• ${\displaystyle {\stackrel {\nabla }{\mathbf {T} }}}$ is the Upper convected time derivative of stress tensor:
  $${\displaystyle {\stackrel {\nabla }{\mathbf {T} }}={\frac {\partial }{\partial t}}\mathbf {T} +\mathbf {v} \cdot \nabla \mathbf {T} -((\nabla \mathbf {v} )^{T}\cdot \mathbf {T} +\mathbf {T} \cdot (\nabla \mathbf {v} ));}$$

  
• ${\displaystyle \mathbf {v} }$ is the fluid velocity;
• ${\displaystyle \eta _{0}}$ is the total viscosity composed of solvent and polymer components, ${\displaystyle \eta _{0}=\eta _{s}+\eta _{p}}$;
• ${\displaystyle \mathbf {D} }$ is the deformation rate tensor or rate of strain tensor, ${\displaystyle \mathbf {D} ={\frac {1}{2}}\left[{\boldsymbol {\nabla }}\mathbf {v} +({\boldsymbol {\nabla }}\mathbf {v} )^{T}\right]}$.

Whilst the model gives good approximations of viscoelastic fluids in shear flow, it has an unphysical singularity in extensional flow, where the dumbbells are infinitely stretched. This is, however, specific to idealised flow; in the case of a cross-slot geometry the extensional flow is not ideal, so the stress, although singular, remains integrable, although the stress is infinite in a correspondingly infinitely small region.[12]

If the solvent viscosity is zero, the Oldroyd-B becomes the Upper Convected Maxwell model.

Wagner model is might be considered as a simplified practical form of the Bernstein–Kearsley–Zapas model. The model was developed by German rheologist Manfred Wagner.

For the isothermal conditions the model can be written as:

$${\displaystyle \mathbf {\sigma } (t)=-p\mathbf {I} +\int _{-\infty }^{t}M(t-t')h(I_{1},I_{2})\mathbf {B} (t')\,dt'}$$

where:

• ${\displaystyle \mathbf {\sigma } (t)}$ is the Cauchy stress tensor as function of time t,
• p is the pressure
• ${\displaystyle \mathbf {I} }$ is the unity tensor
• M is the memory function showing, usually expressed as a sum of exponential terms for each mode of relaxation:
  $${\displaystyle M(x)=\sum _{k=1}^{m}{\frac {g_{i}}{\theta _{i}}}\exp \left({\frac {-x}{\theta _{i}}}\right),}$$

  
  where for each mode of the relaxation, ${\displaystyle g_{i}}$ is the relaxation modulus and ${\displaystyle \theta _{i}}$ is the relaxation time;
• ${\displaystyle h(I_{1},I_{2})}$ is the strain damping function that depends upon the first and second invariants of Finger tensor ${\displaystyle \mathbf {B} }$.

The strain damping function is usually written as:

$${\displaystyle h(I_{1},I_{2})=m^{*}\exp(-n_{1}{\sqrt {I_{1}-3}})+(1-m^{*})\exp(-n_{2}{\sqrt {I_{2}-3}})}$$

If the value of the strain hardening function is equal to one, then the deformation is small; if it approaches zero, then the deformations are large.[13][14]

Shear rheometers are based on the idea of putting the material to be measured between two plates, one or both of which move in a shear direction to induce stresses and strains in the material. The testing can be done at constant strain rate, stress, or in an oscillatory fashion (a form of dynamic mechanical analysis).[24] Shear rheometers are typically limited by edge effects where the material may leak out from between the two plates and slipping at the material/plate interface.

Extensional rheometers, also known as extensiometers, measure viscoelastic properties by pulling a viscoelastic fluid, typically uniaxially.[25] Because this typically makes use of capillary forces and confines the fluid to a narrow geometry, the technique is often limited to fluids with relatively low viscosity like dilute polymer solutions or some molten polymers.[25] Extensional rheometers are also limited by edge effects at the ends of the extensiometer and pressure differences between inside and outside the capillary.[3]

Despite the apparent limitations mentioned above, extensional rheometry can also be performed on high viscosity fluids. Although this requires the use of different instruments, these techniques and apparatuses allow for the study of the extensional viscoelastic properties of materials such as polymer melts. Three of the most common extensional rheometry instruments developed within the last 50 years are the Meissner-type rheometer, the filament stretching rheometer (FiSER), and the Sentmanat Extensional Rheometer (SER).

The Meissner-type rheometer, developed by Meissner and Hostettler in 1996, uses two sets of counter-rotating rollers to strain a sample uniaxially.[26] This method uses a constant sample length throughout the experiment, and supports the sample in between the rollers via an air cushion to eliminate sample sagging effects. It does suffer from a few issues – for one, the fluid may slip at the belts which leads to lower strain rates than one would expect. Additionally, this equipment is challenging to operate and costly to purchase and maintain.

The FiSER rheometer simply contains fluid in between two plates. During an experiment, the top plate is held steady and a force is applied to the bottom plate, moving it away from the top one.[27] The strain rate is measured by the rate of change of the sample radius at its middle. It is calculated using the following equation:

$${\displaystyle {\dot {\epsilon }}=-{\frac {2}{R}}{dR \over dt}}$$

where ${\displaystyle R}$ is the mid-radius value and ${\displaystyle {\dot {\epsilon }}}$ is the strain rate. The viscosity of the sample is then calculated using the following equation:

$${\displaystyle \eta ={\frac {F}{\pi R^{2}{\dot {\epsilon }}}}}$$

where ${\displaystyle \eta }$ is the sample viscosity, and ${\displaystyle F}$ is the force applied to the sample to pull it apart.

Much like the Meissner-type rheometer, the SER rheometer uses a set of two rollers to strain a sample at a given rate.[28] It then calculates the sample viscosity using the well known equation:

$${\displaystyle \sigma =\eta {\dot {\epsilon }}}$$

where ${\displaystyle \sigma }$ is the stress, ${\displaystyle \eta }$ is the viscosity and ${\displaystyle {\dot {\epsilon }}}$ is the strain rate. The stress in this case is determined via torque transducers present in the instrument. The small size of this instrument makes it easy to use and eliminates sample sagging between the rollers. A schematic detailing the operation of the SER extensional rheometer can be found on the right.

Schematic of the SER extensional rheometer. The sample (brown) is held to two cylinders (grey) which are then counterrotated at varying strain rates. The torque required to strain the sample at these rates is calculated via a set of torque transducers present in the instrument. These torque values are then converted to stress values, and the stresses and strain rates are then used to determine the viscosity of the sample.

Though there are many instruments that test the mechanical and viscoelastic response of materials, broadband viscoelastic spectroscopy (BVS) and resonant ultrasound spectroscopy (RUS) are more commonly used to test viscoelastic behavior because they can be used above and below ambient temperatures and are more specific to testing viscoelasticity. These two instruments employ a damping mechanism at various frequencies and time ranges with no appeal to time–temperature superposition. Using BVS and RUS to study the mechanical properties of materials is important to understanding how a material exhibiting viscoelasticity will perform.[29]