Viscoelasticity is a material property that combines both viscous and elastic characteristics. Many materials have such viscoelastic properties, especially materials that consist of large molecules. Polymers are viscoelastic because their macromolecules can have temporary entanglements with neighbouring molecules, causing elastic properties. After some time these entanglements disappear and the macromolecules flow into other positions where new entanglements occur (viscous properties). A viscoelastic material shows elastic properties on short time scales and viscous properties on long time scales. These materials exhibit behavior that depends on the time and rate of applied forces, allowing them to both store and dissipate energy. Viscoelasticity has been studied since the 19th century by researchers such as James Clerk Maxwell, Ludwig Boltzmann, and Lord Kelvin. Several models are available for the mathematical description of the viscoelastic properties of a substance:
Constitutive models of linear viscoelasticity assume a linear relationship between stress and strain. These models are valid for relatively small deformations only. Constitutive models of non-linear viscoelasticity are based on a more realistic non-linear relationship between stress and strain. These models are valid for relatively large deformations. The viscoelastic properties of polymers are highly temperature dependent. From low to high temperature the material can be in the glass phase, rubber phase or the melt phase. These phases have a very strong effect on the mechanical and viscous properties of the polymers. Typical viscoelastic properties are:
A time-dependent stress in the polymer under constant deformation (strain). A time dependent strain in the polymer under constant stress. A time and temperature dependent stiffness of the polymer. Viscous energy loss during deformation of the polymer in the glass or rubber phase (hysteresis). A strain rate-dependent viscosity of the molten polymer. An ongoing deformation of a polymer in the glass phase at constant load (creep). The viscoelasticity properties are measured with various techniques, such as tensile testing, dynamic mechanical analysis, shear rheometry and extensional rheometry.
Background In the nineteenth century, physicists such as James Clerk Maxwell, Ludwig Boltzmann, and Lord Kelvin researched and experimented with creep and recovery of glasses, metals, and rubbers. Viscoelasticity was further examined in the late twentieth century when synthetic polymers were engineered and used in a variety of applications. Viscoelasticity calculations depend heavily on the viscosity variable, η. The inverse of η is also known as fluidity, φ. The value of either can be derived as a function of temperature or as a given value (i.e. for a dashpot).
Depending on the change of strain rate versus stress inside a material, the viscosity can be categorized as having a linear, non-linear, or plastic response:
When a material exhibits a linear response it is categorized as a Newtonian material. In this case the stress is linearly proportional to the strain rate. If the material exhibits a non-linear response to the strain rate, it is categorized as non-Newtonian fluid. There is also an interesting case where the viscosity decreases as the shear/strain rate remains constant. A material which exhibits this type of behavior is known as thixotropic. In addition, when the stress is independent of this strain rate, the material exhibits plastic deformation. Many viscoelastic materials exhibit rubber like behavior explained by the thermodynamic theory of polymer elasticity. Some examples of viscoelastic materials are amorphous polymers, semicrystalline polymers, biopolymers, metals at very high temperatures, and bitumen materials. Cracking occurs when the strain is applied quickly and outside of the elastic limit. Ligaments and tendons in the human body are viscoelastic, so the extent of the potential damage to them depends on both the rate of the change of their length and the force applied. A viscoelastic material has the following properties:
hysteresis is seen in the stress–strain curve stress relaxation occurs: step constant strain causes decreasing stress creep occurs: step constant stress causes increasing strain its stiffness depends on the strain rate ε ˙ {\displaystyle {\dot {\varepsilon }}} or the stress rate σ ˙ {\displaystyle {\dot {\sigma }}}
Elastic versus viscoelastic behavior
Unlike purely elastic substances, a viscoelastic substance has an elastic component and a viscous component. The viscosity of a viscoelastic substance gives the substance a strain rate dependence on time. Purely elastic materials do not dissipate energy (heat) when a load is applied, then removed. However, a viscoelastic substance dissipates energy when a load is applied, then removed. Hysteresis is observed in the stress–strain curve, with the area of the loop being equal to the energy lost during the loading cycle. Since viscosity is the resistance to thermally activated plastic deformation, a viscous material loses energy through a loading cycle. Plastic deformation results in lost energy, which is uncharacteristic of a purely elastic material's reaction to a loading cycle. Specifically, viscoelasticity is a molecular rearrangement. When a stress is applied to a viscoelastic material such as a polymer, parts of the long polymer chain change positions. This movement or rearrangement is called creep. Polymers remain a solid material even when these parts of their chains are rearranging to accommodate the stress, and as this occurs, it creates a back stress in the material. When the back stress is the same magnitude as the applied stress, the material no longer creeps. When the original stress is taken away, the accumulated back stresses causes the polymer to return to its original form. The material creeps, which gives the prefix visco-, and the material fully recovers, which gives the suffix -elasticity.
Linear viscoelasticity and nonlinear viscoelasticity Linear viscoelasticity is behavior in which the function is separable in both creep response and load. All linear viscoelastic models can be represented by a Volterra equation connecting stress and strain:
ε ( t ) = σ ( t ) E inst,creep + ∫ 0 t K ( t − t ′ ) σ ˙ ( t ′ ) d t ′ {\displaystyle \varepsilon (t)={\frac {\sigma (t)}{E_{\text{inst,creep}}}}+\int _{0}^{t}K(t-t'){\dot {\sigma }}(t')dt'}
or
σ ( t ) = E inst,relax ε ( t ) + ∫ 0 t F ( t − t ′ ) ε ˙ ( t ′ ) d t ′ {\displaystyle \sigma (t)=E_{\text{inst,relax}}\varepsilon (t)+\int _{0}^{t}F(t-t'){\dot {\varepsilon }}(t')dt'}
where
t is time
σ ( t ) {\displaystyle \sigma (t)} is stress
ε ( t ) {\displaystyle \varepsilon (t)} is strain
E inst,creep {\displaystyle E_{\text{inst,creep}}} and E inst,relax {\displaystyle E_{\text{inst,relax}}} are instantaneous elastic moduli for creep and relaxation K(t) is the creep function F(t) is the relaxation function Linear viscoelasticity is usually applicable only for small deformations. In nonlinear viscoelasticity, the function is not separable. It usually happens when the deformations are large or if the material changes its properties under deformations. Nonlinear viscoelasticity also elucidates observed phenomena such as normal stresses, shear thinning, and extensional thickening in viscoelastic fluids. An anelastic material is a special case of a viscoelastic material: an anelastic material fully recovers to its original state on the removal of load. When distinguishing between elastic, viscous, and forms of viscoelastic behavior, it is helpful to reference the time scale of the measurement relative to the relaxation times of the material being observed, known as the Deborah number (De) where:
D e = λ / t {\displaystyle De=\lambda /t}
where
λ {\displaystyle \lambda } is the relaxation time of the material
t {\displaystyle t} is time
Dynamic modulus
Viscoelasticity is studied using dynamic mechanical analysis, applying a small oscillatory stress and measuring the resulting strain.
Purely elastic materials have stress and strain in phase, so that the response of one caused by the other is immediate. In purely viscous materials, strain lags stress by a 90-degree phase. Viscoelastic materials exhibit behavior somewhere in the middle of these two types of material, exhibiting some lag in strain. A complex dynamic modulus G can be used to represent the relations between the oscillating stress and strain:
G = G ′ + i G ″ {\displaystyle G=G'+iG''}
where i 2 = − 1 {\displaystyle i^{2}=-1} ; G ′ {\displaystyle G'} is the storage modulus and G ″ {\displaystyle G''} is the loss modulus:
G ′ = σ 0 ε 0 cos δ {\displaystyle G'={\frac {\sigma _{0}}{\varepsilon _{0}}}\cos \delta }
G ″ = σ 0 ε 0 sin δ {\displaystyle G''={\frac {\sigma _{0}}{\varepsilon _{0}}}\sin \delta }
where σ 0 {\displaystyle \sigma _{0}} and ε 0 {\displaystyle \varepsilon _{0}} are the amplitudes of stress and strain respectively, and δ {\displaystyle \delta } is the phase shift between them.
Constitutive models of linear viscoelasticity
Viscoelastic materials, such as amorphous polymers, semicrystalline polymers, biopolymers and even the living tissue and cells, can be modeled in order to determine their stress and strain or force and displacement interactions as well as their temporal dependencies. These models, which include the Maxwell model, the Kelvin–Voigt model, the standard linear solid model, and the Burgers model, are used to predict a material's response under different loading conditions. Viscoelastic behavior has elastic and viscous components modeled as linear combinations of springs and dashpots, respectively. Each model differs in the arrangement of these elements, and all of these viscoelastic models can be equivalently modeled as electrical circuits. In an equivalent electrical circuit, stress is represented by current, and strain rate by voltage. The elastic modulus of a spring is analogous to the inverse of a circuit's inductance (it stores energy) and the viscosity of a dashpot to a circuit's resistance (it dissipates energy). The elastic components, as previously mentioned, can be modeled as springs of elastic constant E, given the formula:
σ = E ε {\displaystyle \sigma =E\varepsilon }
where σ is the stress, E is the elastic modulus of the material, and ε is the strain that occurs under the given stress, similar to Hooke's law. The viscous components can be modeled as dashpots such that the stress–strain rate relationship can be given as,
σ = η d ε d t {\displaystyle \sigma =\eta {\frac {d\varepsilon }{dt}}}
where σ is the stress, η is the viscosity of the material, and dε/dt is the time derivative of strain. The relationship between stress and strain can be simplified for specific stress or strain rates. For high stress or strain rates/short time periods, the time derivative components of the stress–strain relationship dominate. In these conditions it can be approximated as a rigid rod capable of sustaining high loads without deforming. Hence, the dashpot can be considered to be a "short-circuit". Conversely, for low stress states/longer time periods, the time derivative components are negligible and the dashpot can be effectively removed from the system – an "open" circuit. As a result, only the spring connected in parallel to the dashpot contributes to the total strain in the system.
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:
σ + η E σ ˙ = η ε ˙ {\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 increases linearly with time. However, polymers for the most part show the strain rate to be decreasing with time. 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. The equation introduced here, however, lacks a consistent derivation from more microscopic model and is not observer independent. The Upper-convected Maxwell model is its sound formulation in terms of the Cauchy stress tensor and constitutes the simplest tensorial constitutive model for viscoelasticity (see e.g. or ).
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:
σ = E ε + η ε ˙ {\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. This model can be applied to organic polymers, rubber, and wood when the load is not too high.
Standard linear solid model
The standard linear solid model, also referred as the Kelvin or Zener model, consists of two springs and a dashpot. This model is referred as Kelvin model by some authors, as Lord Kelvin (William Thomson) observed that the rate of dissipation of energy increases less rapidly than the square of the frequency as predicted by a simple Voigt Model. Finally, he concluded that simple Voigt Model does not corresponds to reality . However, the standard linear solid model formally proposed by John Henry Poynting and Joseph John Thomson in 1902 . 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:
Under a constant stress, the modeled material instantaneously deforms to some strain, which is the instantaneous elastic portion of the strain. After that it continues 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.
Jeffreys model The Jeffreys model like the Zener model is a three element model. It consist of two dashpots and a spring.
It was proposed in 1929 by Harold Jeffreys to study Earth's mantle.
Burgers model
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:
This model incorporates viscous flow into the standard linear solid model, giving a linearly increasing asymptote for strain under fixed loading conditions.
Generalized Maxwell 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. Applications: metals and alloys at temperatures lower than one quarter of their absolute melting temperature (expressed in K) and characterizing elastic efficiency.
Constitutive models for nonlinear viscoelasticity Non-linear viscoelastic constitutive equations are needed to quantitatively account for phenomena in fluids like differences in normal stresses, shear thinning, and extensional thickening. Necessarily, the history experienced by the material is needed to account for time-dependent behavior, and is typically included in models as a history kernel K.
Second-order fluid
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. The second-order fluid constitutive equation is given by:
T = − p I + 2 η 0 D − ψ 1 D ▽ + 4 ψ 2 D ⋅ D {\displaystyle \mathbf {T} =-p\mathbf {I} +2\eta _{0}\mathbf {D} -\psi _{1}\mathbf {D} ^{\triangledown }+4\psi _{2}\mathbf {D} \cdot \mathbf {D} }
where:
I {\displaystyle \mathbf {I} } is the identity tensor
D {\displaystyle \mathbf {D} } is the deformation tensor
η 0 , ψ 1 , ψ 2 {\displaystyle \eta _{0},\psi _{1},\psi _{2}} denote viscosity, and first and second normal stress coefficients, respectively
D ▽ {\displaystyle \mathbf {D} ^{\triangledown }} denotes the upper-convected derivative of the deformation tensor where D ▽ ≡ D ˙ − ( ∇ v ) T ⋅ D − D ⋅ ∇ v {\displaystyle \mathbf {D} ^{\triangledown }\equiv {\dot {\mathbf {D} }}-(\nabla \mathbf {v} )^{\mathbf {T} }\cdot \mathbf {D} -\mathbf {D} \cdot \nabla \mathbf {v} } and D ˙ ≡ ∂ ∂ t D + v ⋅ ∇ D {\displaystyle {\dot {\mathbf {D} }}\equiv {\frac {\partial }{\partial t}}\mathbf {D} +\mathbf {v} \cdot \nabla \mathbf {D} } is the material time derivative of the deformation tensor.
Upper-convected Maxwell model
The upper-convected Maxwell model incorporates nonlinear time behavior into the viscoelastic Maxwell model, given by:
τ + λ τ ▽ = 2 η 0 D {\displaystyle \mathbf {\tau } +\lambda \mathbf {\tau } ^{\triangledown }=2\eta _{0}\mathbf {D} }
where τ {\displaystyle \mathbf {\tau } } denotes the stress tensor.
Oldroyd-B model
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. The model can be written as:
T + λ 1 T ∇ = 2 η 0 ( D + λ 2 D ∇ ) {\displaystyle \mathbf {T} +\lambda _{1}{\stackrel {\nabla }{\mathbf {T} }}=2\eta _{0}(\mathbf {D} +\lambda _{2}{\stackrel {\nabla }{\mathbf {D} }})}
where:
T {\displaystyle \mathbf {T} } is the stress tensor;
λ 1 {\displaystyle \lambda _{1}} is the relaxation time;
λ 2 {\displaystyle \lambda _{2}} is the retardation time = η s η 0 λ 1 {\displaystyle {\frac {\eta _{s}}{\eta _{0}}}\lambda _{1}} ;
T ∇ {\displaystyle {\stackrel {\nabla }{\mathbf {T} }}} is the upper convected time derivative of stress tensor: T ∇ = ∂ ∂ t T + v ⋅ ∇ T − ( ( ∇ v ) T ⋅ T + T ⋅ ( ∇ v ) ) ; {\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} ));}
v {\displaystyle \mathbf {v} } is the fluid velocity;
η 0 {\displaystyle \eta _{0}} is the total viscosity composed of solvent and polymer components, η 0 = η s + η p {\displaystyle \eta _{0}=\eta _{s}+\eta _{p}} ;
D {\displaystyle \mathbf {D} } is the deformation rate tensor or rate of strain tensor, D = 1 2 [ ∇ v + ( ∇ v ) T ] {\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. If the solvent viscosity is zero, the Oldroyd-B becomes the Upper Convected Maxwell model.
Wagner 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:
σ ( t ) = − p I + ∫ − ∞ t M ( t − t ′ ) h ( I 1 , I 2 ) B ( t ′ ) d t ′ {\displaystyle \mathbf {\sigma } (t)=-p\mathbf {I} +\int _{-\infty }^{t}M(t-t')h(I_{1},I_{2})\mathbf {B} (t')\,dt'}
where:
σ ( t ) {\displaystyle \mathbf {\sigma } (t)} is the Cauchy stress tensor as function of time t, p is the pressure
I {\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: M ( x ) = ∑ k = 1 m g i θ i exp ( − x θ i ) , {\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, g i {\displaystyle g_{i}} is the relaxation modulus and θ i {\displaystyle \theta _{i}} is the relaxation time;
h ( I 1 , I 2 ) {\displaystyle h(I_{1},I_{2})} is the strain damping function that depends upon the first and second invariants of Finger tensor B {\displaystyle \mathbf {B} } . The strain damping function is usually written as:
h ( I 1 , I 2 ) = m ∗ exp ( − n 1 I 1 − 3 ) + ( 1 − m ∗ ) exp ( − n 2 I 2 − 3 ) {\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.
Prony series
In a one-dimensional relaxation test, the material is subjected to a sudden strain that is kept constant over the duration of the test, and the stress is measured over time. The initial stress is due to the elastic response of the material. Then, the stress relaxes over time due to the viscous effects in the material. Typically, either a tensile, compressive, bulk compression, or shear strain is applied. The resulting stress vs. time data can be fitted with a number of equations, called models. Only the notation changes depending on the type of strain applied: tensile-compressive relaxation is denoted E {\displaystyle E} , shear is denoted G {\displaystyle G} , bulk is denoted K {\displaystyle K} . The Prony series for the shear relaxation is
G ( t ) = G ∞ + ∑ i = 1 N G i exp ( − t / τ i ) {\displaystyle G(t)=G_{\infty }+\sum _{i=1}^{N}G_{i}\exp(-t/\tau _{i})}
where G ∞ {\displaystyle G_{\infty }} is the long term modulus once the material is totally relaxed, τ i {\displaystyle \tau _{i}} are the relaxation times (not to be confused with τ i {\displaystyle \tau _{i}} in the diagram); the higher their values, the longer it takes for the stress to relax. The data is fitted with the equation by using a minimization algorithm that adjust the parameters ( G ∞ , G i , τ i {\displaystyle G_{\infty },G_{i},\tau _{i}} ) to minimize the error between the predicted and data values. An alternative form is obtained noting that the elastic modulus is related to the long term modulus by
G ( t = 0 ) = G 0 = G ∞ + ∑ i = 1 N G i {\displaystyle G(t=0)=G_{0}=G_{\infty }+\sum _{i=1}^{N}G_{i}}
Therefore,
G ( t ) = G 0 − ∑ i = 1 N G i [ 1 − e − t / τ i ] {\displaystyle G(t)=G_{0}-\sum _{i=1}^{N}G_{i}\left[1-e^{-t/\tau _{i}}\right]}
This form is convenient when the elastic shear modulus G 0 {\displaystyle G_{0}} is obtained from data independent from the relaxation data, and/or for computer implementation, when it is desired to specify the elastic properties separately from the viscous properties, as in Simulia (2010). A creep experiment is usually easier to perform than a relaxation one, so most data is available as (creep) compliance vs. time. Unfortunately, there is no known closed form for the (creep) compliance in terms of the coefficient of the Prony series. So, if one has creep data, it is not easy to get the coefficients of the (relaxation) Prony series, which are needed for example in. An expedient way to obtain these coefficients is the following. First, fit the creep data with a model that has closed form solutions in both compliance and relaxation; for example the Maxwell-Kelvin model (eq. 7.18-7.19) in Barbero (2007) or the Standard Solid Model (eq. 7.20-7.21) in Barbero (2007) (section 7.1.3). Once the parameters of the creep model are known, produce relaxation pseudo-data with the conjugate relaxation model for the same times of the o